Felix Klein: Visions for Mathematics, Applications, and Education 9783030757854, 3030757854

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Vita Mathematica 20

Renate Tobies

Felix Klein Visions for Mathematics, Applications, and Education

Vita Mathematica

Volume 20

Edited by Martin MattmRuller

More information about this series at http://www.springer.com/series/4834

Renate Tobies

Felix Klein Visions for Mathematics, Applications, and Education Revised by the Author and Translated by Valentine A. Pakis

Renate Tobies Friedrich-Schiller-Universität Jena Jena, Germany Translated by Valentine A. Pakis Saint Paul, MN, USA

Originally published in German under the title: Felix Klein: Visionen für Mathematik, Anwendungen und Unterricht. Berlin: Springer Spektrum, 2019.

ISSN 1013-0330 ISSN 2504-3706 (electronic) Vita Mathematica ISBN 978-3-030-75785-4 (eBook) ISBN 978-3-030-75784-7 https://doi.org/10.1007/978-3-030-75785-4 © Springer Nature Switzerland AG 2019, 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Book layout: Stefan Tobies Cover illustration: Felix Klein, 1875, Private Estate Hillebrand, Scheeßel This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Figure 1: Felix Klein, 1875 [Hillebrand].

“Whoever shall live on in the memory of the wide world must have had an impact on that world.” (BLUMENTHAL 1928, p. 2)

PREFACE Richard Courant spoke euphorically about Felix Klein (1849-1925): “His life was full of intellectual vigor and the will to act, both spurred by a brilliant imagination that was always contriving more and more new designs. He was entirely the sort of wise man and ruler described in Plato’s Republic.”1 With his Erlangen Program, Klein convincingly redefined geometry: geometric properties as invariants of transformation groups. He systematized mathematical theories by recognizing and explaining the interrelations between different disciplines. His visionary programs concerned mathematics and its applications, but also history, philosophy, and pedagogy from kindergarten through higher education. He was extraordinarily engaged, as his admirers would say, in raising awareness for the “eminent cultural significance of mathematics and its applications.”2 In 1892, the famous Austrian theoretical physicist Ludwig Boltzmann extolled Klein’s all-encompassing activity: […] Klein’s work encompasses almost all areas of mathematics. Especially noteworthy are his contributions to the following areas: 1 Algebra and its application to the theory of algebraic forms, number theory, geometry, the resolution of higher equations. 2 General theory of functions, theory of elliptic, Abelian, θ-functions and of Riemann surfaces; 3 Theory of differential equations; 4 Foundations of geometry, curvature and other shape relations of curves and surfaces, also newer geometry and projectivity, the application of geometry to mechanics.3

The present book deals with Klein’s multifaceted programs and the development of his works. It sheds light on how Klein became a scientist who was able to attract students – male and female alike – to follow his visions. In 1870, Klein became the first German mathematician to seek personal contact with French mathematicians since Plücker, Dirichlet, and Jacobi had done this some decades before. Klein traveled several times to the British Isles, to Italy, to the United States, etc. He was at the center of the first international congresses of mathematicians and was elected the first chairman of the International Commission on Mathematical Instruction (ICMI) in 1908. In Germany, Felix Klein steered the fortunes of the German Mathematical Society three times as its chairman and, as a professor emeritus, he was still considered the “foreign minister” of mathematics. In the 1890s, the French mathematician Charles Hermite gushingly 1 2 3

COURANT 1926, p. 211. [UAG] Math.-Nat. Fak. 25, Valentiner (report from July 19, 1924). Quoted from HÖFLECHNER 1994, pp. 173–74 (Boltzmann to Paul von Groth). – Regarding the context, see Section 6.5.2.

vii

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referred to Klein as “a new Joshua in the promised land.”4 Klein became a citizen of the world, explicitly condemning national chauvinism (see Section 8.4). A precocious student, Klein had completed secondary school at the age of sixteen, earned a doctoral degree at the age of nineteen, and completed his postdoctorate (Habilitation) at the age of twenty-one. He was offered his first full professorship at the age of twenty-three, at the University of Erlangen (1872). This was followed by positions at the Polytechnikum in Munich (1875), the University of Leipzig (1880), and the University of Göttingen (as of 1886). More than just focusing on Klein’s professional achievements, this book will also be concerned with Klein as a person. At the age of twenty-six, he married Anna Hegel, the granddaughter of the great philosopher. Her extant letters to Felix Klein document their good relationship and demonstrate that she was often involved in his academic work. Of their four children (one son, three daughters), their son would go on to pursue a technical career. Their youngest daughter studied mathematics, physics, and English in Göttingen and at Bryn Mawr College in the United States. She achieved a distinguished career as a teacher and school principal until 1932; later, she was demoted during the Nazi regime. Klein cultivated a cooperative working style. At the age of twenty, he found his most important partner in the Norwegian Sophus Lie. Klein wanted to work together, not in competition. Nevertheless, he had to deal with opponents, competitors, different views and interests. David Hilbert, who, on the occasion of Klein’s sixtieth birthday in 1909 also invited Henri Poincaré and Gösta MittagLeffler to Göttingen, referred in his speech then to Klein’s opponents and supporters and expressed his own affinity for Klein.5 Klein was not, from the outset, the “Zeus enthroned above the other Olympians,” as Max Born experienced him during his own years as a student (“He was known among us as ‘the Great Felix’,” Born went on, “and he controlled our destinies”).6 We will instead encounter a mathematician who was often plagued by self-doubt and who worried that he might not be able to live up to his own high standards. Early translations of his work and his efforts as the chief editor of the journal Mathematische Annalen brought him fame and influence. With his finger on the pulse of international trends, Klein left a lasting mark on many areas of mathematics, its applications, and organization in Germany. In an astounding number of areas, he was in fact a pioneer.7 At the University of Göttingen, Klein had laid the foundation for a new golden era and had pointed the way ahead, as Hilbert put it (see Appendix 12). This meant that he appointed the best scientists (among them Hilbert, Hermann Minkowski, Carl Runge, Ludwig Prandtl, Edmund Landau) to work beside him, that he found new ways to retain them in Göttingen, and that he established new institutes by acquiring funds from industry – inspired by the example of American universities. 4 5 6 7

For the context of this quotation, see Section 8.2.2 of this book. See TOBIES 2019b, pp. 513–14, Engl. trans. in ROWE 2018a, pp. 198–99. BORN/BORN 1969, p. 16. For a summary of Klein’s pioneering achievements, see Section 10.2.

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Right up into old age, Klein was open to new mathematical, scientific, and technical theories. Thus he also identified open problems in the fields of fluid dynamics and statics. In partial collaboration with Emmy Noether, he made significant contributions to the theory of relativity, acknowledged by Albert Einstein. Klein recognized the specific talents of his students with great foresight. He promoted gifted persons regardless of their religion, nationality, and gender. He guided more than fifty doctoral students, including two women (an Englishwoman and an American) as well as further students from abroad, to new results. During his lifetime, he received numerous honors, and his versatility is still widely recognized today. Since the year 2000, the European Mathematical Society has awarded a “Felix Klein Prize” to young scientists for outstanding research in applied mathematics (this award was initiated by the Fraunhofer Institute for Industrial Mathematics in Kaiserslautern). Since 2003, moreover, the ICMI has presented a “Felix Klein Award” for lifetime achievements in the field of mathematical pedagogy. In Germany, several institutions have been named after him. There is a Felix Klein Lecture Hall and a Felix Klein Colloquium at the Heinrich Heine University in Düsseldorf (Klein’s birthplace) and at the University of Leipzig as well. There is a Felix Klein building at the University of Erlangen and a Felix Klein program at the Technische University in Munich (including a “Felix Klein Teaching Prize”). In Göttingen, there is a secondary school named after Felix Klein, and the meeting room of the Mathematical Institute of the University is adorned by the original Max Liebermann portrait of Klein. The names of the donors who funded this painting are an expression of Klein’s worldwide network, which extended as far as India and Japan.8 After the late Leipzig historian of mathematics Hans Wußing had encouraged me to study the life and work of Felix Klein, it was the American historian of mathematics David E. Rowe who first enabled me – when Germany was still divided – to study the archival materials pertaining to Klein in Göttingen. The mathematician Helmut Neunzert invited me to give lectures at the University of Kaiserslautern with the following words: “We like to use Klein’s arguments to promote the applications of mathematics even today!” A Felix Klein Center was established there in 2008. Robert Fricke, the mathematician and erstwhile rector of the Technische Hochschule in Braunschweig, aptly compared Felix Klein (an uncle of Fricke’s wife) to a triptych, the central panel of which should be devoted to Klein the researcher, while the two flanking panels should depict him as an academic teacher and an outstanding organizer.9 The goal of this book is to put this triptych into words and enrich it with a human dimension. Jena, March of 2021 8 9

Renate Tobies

See Section 8.5.2, and Appendix 10, Fig. 43. – The portrait of Hilbert in the same room was painted in 1928 by Eugen Spiro, who was forced to emigrate in 1935. On Hilbert, see in particular Sections 6.3.7.3 and 7.9 in this book. FRICKE 1919, p. 275. – See the genealogy in Figure 2.

x

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Figure 2: An excerpt from the Klein-Hegel Family Tree (my own design, from [Hillebrand])

CONTENTS Preface................................................................................................................... vii List of Tables ......................................................................................................xviii List of Figures .....................................................................................................xviii 1 Introduction ......................................................................................................... 1 1.1 The State of Research ................................................................................... 3 1.2 Guiding Questions......................................................................................... 8 1.3 Editorial Remarks ....................................................................................... 13 2 Formative Groups ............................................................................................. 17 2.1 The Klein–Kayser Family ........................................................................... 17 2.1.1 A Royalist and Frugal Westphalian Upbringing .............................. 17 2.1.2 Talent in School and Wide Interests as Gifts from His Mother’s Side ................................................................................................. 20 2.1.3 Felix Klein and His Siblings ............................................................ 21 2.2 School Years in Düsseldorf......................................................................... 22 2.2.1 Earning His Abitur from a Gymnasium at the Age of Sixteen ........ 23 2.2.2 Examination Questions in Mathematics .......................................... 25 2.2.3 Interests in Natural Science During His School Years .................... 26 2.3 Studies and Doctorate in Bonn.................................................................... 28 2.3.1 Coursework and Seminar Awards.................................................... 29 2.3.2 Assistantship and a Reward for Winning a Physics Contest ........... 34 2.3.3 Assisting Julius Plücker’s Research in Geometry............................ 36 2.3.4 Doctoral Procedure .......................................................................... 40 2.4 Joining Alfred Clebsch’s Thought Community .......................................... 45 2.4.1 The Clebsch School ......................................................................... 47 2.4.2 The Journal Mathematische Annalen ............................................... 53 2.4.3 Articles on Line Geometry, 1869 ..................................................... 58 2.5 Broadening His Horizons in Berlin ............................................................. 61 2.5.1 The Professors in Berlin and Felix Klein ......................................... 62 2.5.2 Acquaintances from the Mathematical Union: Kiepert, Lie, Stolz .. 66 2.5.3 Cayley’s Metric and Klein’s Non-Euclidean Interpretation ............ 71 2.6 In Paris with Sophus Lie ............................................................................. 73 2.6.1 Felix Klein and French Mathematicians .......................................... 74 2.6.2 Collaborative Work with Sophus Lie............................................... 78 78 2.6.2.1 Notes on W-Configurations xi

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2.6.2.2 Principal Tangent Curves of the Kummer Surface 80 2.6.3 A Report on Mathematics in Paris ................................................... 82 2.7 The Franco-Prussian War and Klein’s Habilitation ................................... 83 2.7.1 Wartime Service as a Paramedic and Its Effects.............................. 84 2.7.2 Habilitation....................................................................................... 88 2.8 Time as a Privatdozent in Göttingen........................................................... 90 2.8.1 Klein’s Teaching Activity and Its Context ...................................... 91 2.8.2 An Overview of Klein’s Research Results as a Privatdozent .......... 98 2.8.3 Discussion Groups ......................................................................... 110 110 2.8.3.1 A Three-Man Club with Clebsch and Riecke 2.8.3.2 The Mathematical and Natural-Scientific Student 113 Union 2.8.3.3 A Scientific Circle: Eskimo 115 2.8.3.4 The “Social Activity” of Bringing Mathematicians Together 117 3 A Professorship at the University of Erlangen ............................................. 123 3.1 Research Trends and Doctoral Students ................................................... 125 3.1.1 The Vision of the Erlangen Program ............................................ 126 3.1.2 Klein’s Students in Erlangen.......................................................... 132 3.1.3 New Research Trends .................................................................... 138 139 3.1.3.1 On a New Type of Riemann Surface 3.1.3.2 The Theory of Equations 143 3.2 Inaugural Lecture: A Plan For Mathematical Education .......................... 144 3.3 First Trip to Great Britain, 1873 ............................................................... 147 3.4 Trips to Italy .............................................................................................. 153 3.5 Developing the Mathematical Institution .................................................. 158 3.6 Family Matters .......................................................................................... 160 3.6.1 His Friends Marry and Klein Follows Suit .................................... 161 3.6.2 Klein’s Father-in-Law, the Historian Karl Hegel .......................... 164 3.6.3 Anna Hegel, Felix Klein, and Their Family................................... 166 4 A Professorship at the Polytechnikum in Munich ....................................... 171 4.1 A New Institute and New Teaching Activity ............................................ 173 4.1.1 Creating a Mathematical Institute .................................................. 174 4.1.2 Reorganizing the Curriculum ......................................................... 176 4.2 Developing His Mathematical Individuality ............................................. 178 4.2.1 The Icosahedron Equation ............................................................. 179 4.2.2 Number Theory .............................................................................. 183 4.2.3 Elliptic Modular Functions ............................................................ 184 4.2.4 Klein’s Circle of Students in Munich ............................................ 191 191 4.2.4.1 Phase I: 1875–1876 4.2.4.2 Phase II: 1876–1880 193 4.3 Discussion Groups in Munich ................................................................... 201

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4.3.1 A Mathematical Discussion Group with Engineers and Natural Scientists ...................................................................................... 201 4.3.2 The Mathematical Student Union and the Mathematical Society.. 204 4.3.3 The Meeting of Natural Scientists in Munich, 1877 ...................... 205 4.4 “Ready Again for a University in a Small City” ....................................... 208 5 A Professorship for Geometry in Leipzig ..................................................... 213 5.1 Klein’s Start in Leipzig and His Inaugural Address ................................. 215 5.2 Creating a New Mathematical Institution ................................................. 218 5.3 Teaching Program ..................................................................................... 221 5.3.1 Lectures: Organization, Reorientation, and Deviation from the Plan ............................................................................................... 221 5.3.2 The Mathematical Colloquium / Exercises / Seminar ................... 227 5.4 The Kleinian “Flock” ................................................................................ 232 5.4.1 Post-Doctoral Mathematicians ....................................................... 233 5.4.2 Klein’s Foreign Students in Leipzig .............................................. 243 5.4.2.1 The First Frenchman and the First Briton 244 245 5.4.2.2 The First Americans 246 5.4.2.3 The Italians 5.4.2.4 Mathematicians from Switzerland and AustriaHungary 248 5.4.2.5 Russian and Other Eastern European Contacts 250 5.5 Fields of Research ..................................................................................... 252 5.5.1 Mathematical Physics / Physical Mathematics .............................. 253 5.5.1.1 Lamé’s Function, Potential Theory, and Carl Neumann 253 5.5.1.2 On Riemann’s Theory of Algebraic Functions and Their Integrals 255 5.5.2 Looking Toward Berlin .................................................................. 260 260 5.5.2.1 Gathering Sources 5.5.2.2 The Dirichlet Principle 261 5.5.2.3 Klein’s Seminar on the Theory of Abelian Functions (1882) 264 5.5.2.4 Openness vs. Partiality 266 5.5.3 Looking Toward France ................................................................. 267 267 5.5.3.1 French Contributors to Mathematische Annalen 269 5.5.3.2 Klein’s Correspondence with Poincaré 5.5.4 Three Fundamental Theorems ....................................................... 272 5.5.4.1 The Loop-Cut Theorem (Rückkehrschnitttheorem) 273 273 5.5.4.2 Theorem of the Limit-Circle (Grenzkreistheorem) 5.5.4.3 The (General) Fundamental Theorem 277 279 5.5.4.4 Remarks on the Proofs 5.5.5 The Polemic about and with Lazarus Fuchs .................................. 282 5.5.6 The Icosahedron Book ................................................................... 286 5.5.7 A Book on the Theory of Elliptic Modular Functions ................... 291

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291 5.5.7.1 Supplementing the Theory 294 5.5.7.2 Who Should Be the Editor? – Georg Pick 5.5.8 Hyperelliptic and Abelian Functions ............................................. 298 5.6 Felix Klein and Alfred Ackermann-Teubner ............................................ 300 5.7 Felix Klein in Leipzig’s Intellectual Communities ................................... 307 5.7.1 A Mathematicians’ Circle .............................................................. 308 5.7.2 The Societas Jablonoviana ............................................................ 308 5.7.3 The Royal Saxon Society of Sciences in Leipzig .......................... 310 5.8 Turning His Back on Leipzig .................................................................... 314 5.8.1 Weighing Offers from Oxford and Johns Hopkins ........................ 314 5.8.2 The Physicist Eduard Riecke Arranges Klein’s Move to Göttingen ...................................................................................... 316 5.8.3 The Appointment of Sophus Lie as Klein’s Successor – and the Reactions ................................................................................ 320 6 The Start of Klein’s Professorship in Göttingen, 1886–1892 ...................... 325 6.1 Family Considerations .............................................................................. 326 6.2 Dealing with Colleagues, Teaching, and Curriculum Planning ................ 328 6.2.1 The Relationship Between Klein and Schwarz .............................. 328 6.2.2 The Göttingen Privatdozenten Hölder and Schoenflies ................. 329 6.2.3 Klein’s Teaching in Context .......................................................... 332 6.3 Independent and Collaborative Research .................................................. 337 6.3.1 The Theory of Finite Groups of Linear Substitutions: The Theory of Solving Equations of Higher Degree ................... 337 6.3.2 Hyperelliptic and Abelian Functions ............................................. 339 6.3.3 The Theory of Elliptic Modular Functions (Monograph) .............. 341 6.3.4 The Theory of Automorphic Functions (Monograph) ................... 343 6.3.5 The Theory of Lamé Functions and Potential Theory ................... 344 6.3.6 Refreshing His Work on Geometry ............................................... 347 6.3.7 Visions: Internationality, Crystallography, Hilbert’s Invariant Theory .......................................................................................... 352 6.3.7.1 An Eye on Developments Abroad 352 6.3.7.2 Arthur Schoenflies and Crystallography 356 6.3.7.3 Felix Klein and Hilbert’s Invariant Theory 357 6.4 Bringing People and Institutions Together ............................................... 361 6.4.1 The Professorium in Göttingen ...................................................... 361 6.4.2 A Proposal to Relocate the Technische Hochschule in Hanover to Göttingen .................................................................................. 362 6.4.3 The Idea of Reorganizing the Göttingen Society of Sciences ....... 364 6.4.4 Felix Klein and the Founding of the German Mathematical Society .......................................................................................... 367 6.5 The Pivotal Year of 1892 .......................................................................... 373 6.5.1 Refilling Vacant Professorships in Prussia .................................... 373

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6.5.1.1 Berlin, Breslau, and Klein’s System for Classifying Styles of Thought 373 6.5.1.2 Hiring a Successor for H.A. Schwarz in Göttingen 377 6.5.2 A Job Offer from the University of Munich and the Consequences ............................................................................... 379 7 Setting the Course, 1892/93–1895 .................................................................. 383 7.1 Klein’s Assistants and His Principles for Choosing Them ....................... 385 7.2 The Göttingen Mathematical Society........................................................ 392 7.3 Turning to Secondary School Teachers .................................................... 397 7.4 A Trip to the United States ....................................................................... 401 7.4.1 The World’s Fair in Chicago and the Mathematical Congress ...... 401 7.4.2 Twelve Lectures by Klein: The Evanston Colloquium .................. 404 7.4.3 Traveling from University to University ....................................... 406 7.4.4 Repercussions................................................................................. 407 7.5 The Beginnings of Women Studying Mathematics .................................. 411 7.6 Actuarial Mathematics as a Course of Study ............................................ 418 7.7 Contacting Engineers and Industrialists .................................................... 421 7.8 The Encyklopädie Project ......................................................................... 425 7.9 Klein Succeeds in Hiring David Hilbert ................................................... 434 8 The Fruits of Klein’s Efforts, 1895–1913 ...................................................... 437 8.1 A Center for Mathematics, Natural Sciences, and Technology ................ 438 8.1.1 The Göttingen Association............................................................. 439 8.1.2 Applied Mathematics in the New Examination Regulations and the Consequences .................................................................. 445 8.1.3 Aeronautical Research ................................................................... 450 8.2 Maintaining His Scientific Reputation ...................................................... 454 8.2.1 Automorphic Functions (Monograph) ........................................... 455 8.2.2 Geometric Number Theory ............................................................ 457 8.2.3 A Monograph on the Theory of the Spinning Top......................... 461 8.2.4 Inspiring Ideas in the Fields of Mathematical Physics and Technology ................................................................................... 465 466 8.2.4.1 Hydrodynamics / Hydraulics 468 8.2.4.2 Statics 8.2.4.3 The Theory of Friction 471 472 8.2.4.4 The Special Theory of Relativity 8.3 Program: The History, Philosophy, Psychology, and Instruction of Mathematics ............................................................................................ 474 8.3.1 The History of Mathematics .......................................................... 477 8.3.2 Philosophical Aspects .................................................................... 481 8.3.3 Psychological-Epistemological Classifications ............................. 490 8.3.4 The “Kleinian” Educational Reform .............................................. 493 500 8.3.4.1 Suggestions for Reform

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8.3.4.2 A Polemic about the Teaching of Analysis at the University 508 8.4 International Scientific Cooperation ......................................................... 510 8.5 Early Retirement and Honors .................................................................... 514 8.5.1 Recovering and Working in the Hahnenklee Sanatorium .............. 515 8.5.2 Max Liebermann’s Portrait of Felix Klein ..................................... 519 8.5.3 The Successors to Klein’s Professorship ....................................... 522 9 The First World War and the Postwar Period............................................. 525 9.1 Political Activity During the First World War.......................................... 526 9.1.1 The Vows of Allegiance of German Professors to Militarism....... 527 9.1.2 A Plea for Studying Abroad ........................................................... 531 9.2 History of Mathematics, the “Cry for Help of Modern Physics,” and Edition Projects ....................................................................................... 534 9.2.1 Remarks on Klein’s Historical Lectures ........................................ 536 9.2.2 Felix Klein and the General Theory of Relativity.......................... 538 9.2.3 The Golden Anniversary of Klein’s Doctorate, and Edition Projects ......................................................................................... 545 9.3 Mathematical Education – International and National ............................. 548 9.3.1 The International Commission on Mathematical Instruction ........ 549 9.3.2 Countering the Restriction of Mathematics and the Natural Sciences ........................................................................................ 551 9.4 Support for Research ................................................................................. 556 9.4.1 The Emergency Association of German Science ........................... 557 9.4.2 The Gauss-Weber / Helmholtz Society .......................................... 560 9.5 End of Life ................................................................................................ 564 10 Concluding Remarks .................................................................................... 569 10.1 A Summary of Findings .......................................................................... 570 On the Continuity of Klein’s Field of Research ...................................... 570 Creating Favorable Conditions for Good Scientific Work ..................... 572 Focusing on Problems of Mathematical Instruction at Schools ............. 573 Klein’s Handling of His Health Problems .............................................. 573 A Summary of the Aspects that Guided the Research for the Present Biography ..................................................................................... 574 10.2 A Pioneer................................................................................................. 585

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Appendix: A Selection of Documents ............................................................... 593 1) A letter from Felix Klein to Heinrich von Mühler, the Prussian Minister of Religious, Educational, and Medical Affairs (Minister of Culture). .................................................................................................. 593 2) An application submitted by Felix Klein to the Academic Senate of the University of Erlangen for funding to improve the collection of the University Library’s mathematical section (November 15, 1872). ........ 594 3) Nomination of Dr. Felix Klein, full professor of mathematics at the Technische Hochschule in Munich, to be made an extraordinary member of the mathematical-physical class of the Royal Bavarian Academy of Sciences, June 7, 1879. ....................................................... 597 4) A report by the Philosophical Faculty at the University of Göttingen concerning its decision to propose Felix Klein as the successor to Moritz Abraham Stern, along with separate opinions by the professors Ernst Schering and Hermann Amandus Schwarz (January 1885). ...................................................................................................... 598 5) On the scientific polemic between Felix Klein and Lazarus Fuchs. An excerpt of a letter (in draft form) from Felix Klein to Wilhelm Förster (a professor of astronomy at the University of Berlin), January 15, 1892. .................................................................................... 603 6) Letters concerning the potential successor to H.A. Schwarz’s full professorship at the University of Göttingen. ......................................... 605 7) Felix Klein on the draft of Ludwig Bieberbach’s dissertation, which was supervised by the Privatdozent Paul Koebe at the University of Göttingen. ................................................................................................ 607 8) Dr. Klaus, a neurologist at the Sanatorium for Neurology and Internal Medicine in Hahnenklee: two reports on the state of Felix Klein’s health. ...................................................................................................... 608 9) Nomination of Felix Klein to be made a corresponding member of the Royal Prussian Academy of Sciences in Berlin, February 27, 1913. ..... 609 10) Speeches given on May 25, 1913 upon the presentation of Max Liebermann’s portrait to Felix Klein. ..................................................... 611 11) Virgil Snyder from Ithaca (New York) to Felix Klein, a letter, dated July 4, 1924, concerning the International Congress of Mathematicians in Toronto, Canada from 11 August to 16 August 1924. ............ 618 12) David Hilbert’s eulogy for Felix Klein, delivered at the session of the Göttingen Mathematical Society held on June 23, 1925, one day after Klein’s death. .................................................................................. 621 Bibliography ....................................................................................................... 623 Index of Names ................................................................................................... 655

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LIST OF TABLES Table Title Number

Page

1

Evaluations of Felix Klein’s achievements from his Abitur diploma (August 3, 1865).

24

2

Examination questions in mathematics (1865).

26

3

Courses attended by Felix Klein at the University of Bonn (1865–1868).

31

4

A list of course offerings in mathematics, physics, and astronomy at the University of Göttingen for the summer semester of 1871.

93

5

On the Erlangen Program.

127

6

Participants in Klein’s Research Seminars, 1880/81–1885/86.

230

7

Lectures at the Göttingen Mathematical Society, 1892/93.

392

8

Applied Mathematics in the Prussian Examination Regulation for Teaching Candidates at Secondary Schools as of 1898. Members of the Commission for Education in the Upper House (Herrenhaus) of the Prussian Parliament, formed on March 19, 1909.

446

Felix Klein’s Courses and Other Activity, 1914–1922.

535

9 10

500

LIST OF FIGURES Figure Title Number

Page

1

Felix Klein, 1875.

v

2

An excerpt from the Klein – Hegel Family Tree.

x

3

An excerpt of a letter from Felix Klein to Sophus Lie dated April 1, 1872.

16

4

Felix Klein at the age of two (unknown illustrator).

18

5

Felix Klein’s Doctoral Certificate, December 12, 1868.

43

6

Alfred Clebsch.

46

7

The title page of volume 6 of Mathematische Annalen (1873).

55

8 9

A Kummer surface with 16 real nodes. A title page of the Bulletin des Sciences Mathématiques et Astronomiques.

60 76

10

An excerpt of a letter from Klein to Lie dated July 29, 1870, including a sketch of the asymptotic curves between two double points on a Kummer surface.

85

11

Clebsch’s diagonal surface, the first model of a cubic surface on which all of its 27 lines are real.

106

12

An illustration of a cubic surface with four real nodes.

109

13

Eduard Riecke.

112

14

The title page of Klein’s Erlangen Program (October 1872).

122

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15

Klein’s circle in Erlangen, 1873. Felix Klein (on the right) with Ferdinand Lindemann, Wilhelm Bretschneider, Siegmund Günther, Adolf Weiler, and Ludwig Wedekind.

134

16

Charles Xavier Thomas’s arithmometer. Serial No. 759, built in 1868; dimensions (mm): 460 long, 180 wide, 93 high.

146

17

Felix Klein’s certification as a foreign member of the London Mathematical Society, 1875, and the De Morgan Medal, which he became the fourth mathematician (after Cayley, Sylvester, and Rayleigh) to receive in 1893.

152

18

Anna Hegel and Felix Klein’s engagement announcement – January 9, 1875.

162

19

A photograph from Anna and Felix Klein’s silver wedding anniversary – Sunday, August 19, 1900.

167

20

Klein’s modular figure, derived from Dedekind.

185

21

Klein’s “main figure” (Hauptfigur) with 2 × 168 circular arc triangles.

187

22

Adolf Hurwitz.

197

23

Carl Linde, 1872.

202

24

The Klein bottle.

257

25

The title page of Klein’s book On Riemann’s Theory of Algebraic Functions and Their Integrals.

259

26

An excerpt of Klein’s drafted letter to A. Ackermann-Teubner, December 31, 1899.

304

27

Felix Klein’s home in Göttingen, Wilhelm-Weber-Straße 3.

326

28

Rohns Tavern on the Hainberg.

327

29

The founding members of the German Mathematical Society (Deutsche Mathematiker-Vereinigung, DMV), September 18, 1890.

370

30

Felix Klein’s certification as a foreign member of the Società Italiana delle Scienze, 1896.

382

31

The Göttingen Mathematical Society, 1902.

394

32

Grace Chisholm and Luise Klein.

414

33

An ENCYKLOPÄDIE trip to Wales. Felix Klein (seated in the middle) and Arnold Sommerfeld (left) with George Hartley Bryan (standing in the middle) and Bryan’s family.

432

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The Göttingen Association for the Promotion of Applied Physics and Mathematics. An invitation to the celebration of its tenth anniversary, February 22, 1908.

441

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The title page of the French edition of Riemann’s Collected Works (1898), including Felix Klein’s speech (discours) on Riemann, KLEIN 1894.

464

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Klein’s updated plan for the volume Die mathematische Wissenschaften of the project Die Kultur der Gegenwart [The Culture of the Present], August 1912.

476

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The title page of the first issue of the journal L’Enseignement mathématique (1899), which has been the journal of the ICMI since 1908.

495

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Committees (etc.) in which Klein discussed educational issues.

497

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Bockswiese-Hahnenklee in the Harz mountains, a view of the sanatorium.

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Max Liebermann’s portrait of Felix Klein (1912).

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Felix Klein, a drawing by Leonard Nelson.

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Felix Klein’s diploma for his honory doctorate from the Jagiellonian University of Krakow (Uniwersytet Jagiellonski w Krakowie), 1900.

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A list of donors who sponsored Max Liebermann’s painting of Klein’s portrait in 1912.

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The certification of Felix Klein’s election as a foreign associate of the National Academy of Sciences of the United States of America, April 21, 1898.

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Felix and Anna Klein’s gravestone in Göttingen’s old city cemetery.

622

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Felix Klein’s diploma for his honory doctorate (doctoris rerum politicarum dignitatem et ornamenta) from the University of Berlin, April 25, 1924.

654

1 INTRODUCTION Perhaps it would also benefit all mathematicians to a great degree if someone would endeavor to synthesize the drastically divergent branches of mathematics into a comprehensive whole while maintaining what is particular to each.1

In the letter quoted above to the Norwegian mathematician Sophus Lie, Felix Klein reported, among other things, about his attempt to combine Kummer surfaces with hyperelliptic functions. He did not yet have, however, a sufficient understanding of the latter. Klein’s ambition to understand and combine as many research areas as possible led to the further development of mathematical theories and to the establishment of new disciplines. These efforts, along with his “incomparable prophetic vision,”2 provided him with an overview of the disciplines that also enabled him to oversee the edition of the comprehensive Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen [Encyclopedia of the Mathematical Sciences, Including Their Applications] (1898–1935).3 Klein experienced and influenced several transformational processes in the field of mathematics. As a young mathematician, he himself was involved in transforming geometry. He attempted to penetrate into other areas as well. As a middle-aged man, he also promoted the study of set theory, ultimately accepting Hilbert’s axiomatic approach to constructing theories and making significant contributions of his own to the expansion of peripheral areas of mathematics: its applications in branches of theoretical physics and technology, and the history, didactics, and philosophy of mathematics. The period of disciplinary change in the field of geometry had a strong effect on Klein’s mathematical thinking as a young man. Erhard SCHOLZ (1980) has described this process in his history of the concept of manifolds: a retreat from the dominance of Euclidean geometry, the spread of new approaches to the field (nonEuclidean geometries, higher-dimensional geometry), and the synthesis of various research areas. The use of analytic methods had led to differential geometry (a preliminary high point in this regard was Gauss’s work from 1828). On the basis of algebraic methods, Jean-Victor Poncelet’s projective geometry (with synthetic and analytic methods) was further developed into algebraic geometry. Projective algebraic geometry in turn gave rise to the approaches of higher-dimensional geometry. The latter acquired a particular shape with the publication of Hermann 1 2 3

[Oslo] A letter from Felix Klein to Sophus Lie, dated December 25, 1873. CARATHÉODORY 1925, p. 2. Hereafter referred to as the ENCYKLOPÄDIE. This reference work consists of six volumes in German. Some but not all of the volumes also appeared in a revised French edition; see TOBIES 1994; GISPERT 1999; and Section 7.8 below.

© Springer Nature Switzerland AG 2021 R. Tobies, Felix Klein, Vita Mathematica 20, https://doi.org/10.1007/978-3-030-75785-4_1

1

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1 Introduction

Graßmann’s Ausdehnungslehre [Theory of Extension] in 1844. Today, Graßmann’s book is recognized as one of the first attempts to formulate the abstract concept of vector space. Bernhard Riemann initiated a geometric research program whose guiding ideas would have effects beyond the nineteenth century and are still reflected in mathematical disciplines such as topology, geometric function theory, modern differential geometry, and the general theory of relativity, among others. As a young student under Julius Plücker, Felix Klein delved into what was then called “newer geometry” (neuere Geometrie). His orientation was broadened by Alfred Clebsch’s algebraic-geometric school, and he ultimately became an active contributor to these new developments. Hermann Weyl described Klein’s methodology as follows: The most prominent characteristic of his scientific methodology is his passion for mixing and folding things together, thus allowing a wide variety of disciplines to permeate one another. His mathematical brilliance is based on his ability to recognize internal connections and relations between concepts whose foundations are entirely distinct.4

As a student, Klein experienced the disputes between various mathematical schools and sensed the disadvantage of having a one-sided orientation. He saw that every method can have advantages and disadvantages. Regarding the conflict between analytic and synthetic methods in geometry, he held the following opinion: “A healthy development will use both methods and enjoy the fruits of their interaction.”5 Thus, from early on, he attempted to familiarize himself with a broad range of mathematical branches. The physiologist Carl Ludwig once told Klein that the best way not to be absorbed by a single school of thought was to travel about 600 kilometers away from one’s home and then reexamine the situation from that remove. “One will certainly be astonished,” Ludwig said, “to find that so many seemingly obvious views simply fall away.”6 Klein’s efforts to systematize and synthesize methods and disciplines involved reconceptualizing mathematical patterns of order. He reclassified geometry and function theory, for instance, by the group-theoretical approach to geometry and his principle of level separation (Stufentheorie) within the theory of elliptic functions.7 Klein always tried to see the complete picture and, looking back at his career, he considered himself to have been a “romantic, not a classicist” (according to Wilhelm Ostwald’s distinction between the two).8 Unlike a quietly ruminating classicist, who works on a single field in detail “with classical soberness” and sometimes never finishes this task, Klein numbered himself among the “re-

4 5 6 7 8

Weyl, Obituary of Felix Klein, 1928 ([UBG] Cod. MS. F. Klein 117). KLEIN 1979 [1926], p. 104. Ibid., p. 105. See also Section 5.7.3 below. On mathematics as a science of potential patterns of order, see NEUNZERT and ROSENBERGER 1991, p. 130. Regarding the scientific practice of classification, see LÊ and PAUMIER 2016. [UBG] Cod. MS. F. Klein 1.22 (notes dated December 12, 1918). Regarding the distinction between romantics and classicists in science, see OSTWALD 1909.

1.1 The State of Research

3

volutionaries” in science who possess a “romantic spirit of conquest” and who are blessed with an overabundance of ideas, plans, and quick reactions. His creed was as follows: Certainly the keystone of every mathematical structure is to furnish compelling proof for all its assertions. Certainly mathematics condemns itself if it renounces compelling proofs. Yet the secret of the productivity of genius will always lie in posing new questions and divining new theorems that shall disclose valuable results and connections. Without the creation of new points of view, without new goals being set up, mathematics, for all the rigor of its proofs, would soon exhaust itself and begin to stagnate.9

Klein was happy to let others work out the concrete details of his ideas. That said, he was also especially proud when he devised a convincing proof on his own. A clear example of this is his successful proof, first formulated in 1876, of one of Kronecker’s theorems (see Section 4.2.1). Klein’s famous book on the icosahedron would in fact culminate in the proof of this theorem (see 5.5.6).10 This biography will pay special attention to Klein’s manner of posing new questions, his recognition of new connections between fields, and his ability to identify fresh talents and new collaborators to advance many of his ideas. The remainder of this introduction will cast light on the state of research, highlight certain central questions, and describe my methodological approach and use of sources. 1.1 THE STATE OF RESEARCH Klein wrote a short autobiography,11 and his brother Alfred left behind a family chronicle. The twenty-nine volumes of preserved protocols from Klein’s seminars are a unique source that has yet to be analyzed in detail.12 These volumes, which are available online, range from the first seminar that Klein co-taught with Clebsch in the summer of 1872 all the way up to the year 1912. Klein himself was still able and resilient enough to furnish his collected mathematical writings – Gesammelte mathematische Abhandlungen (GMA, 1921–23) – with addenda and commentary; vol. 3 contains an appendix, including (chronological) lists: of his lectures, seminars, supervised doctoral theses, assistants, articles, and books.13 Even when some of his expositions no longer withstand critical scrutiny, they are still valuable sources that reveal his motives, points of departure, and the connections that he made. This is also true of his lecture courses, in which he often classified topics historically. His lectures on the development of mathematics in the nineteenth century (Vorlesungen über die Entwicklung der Mathematik im 19.

9 10 11 12 13

KLEIN 1979 [1926], pp. 254–55. See KLEIN 1884 (the first English edition of this book appeared in 1888). KLEIN 1923a. See CHISLENKO/TSCHINKEL 2007; TOBIES 2014; HELLER 2015; and ECKERT 2019b. See online: https://gdz.sub.uni-goettingen.de/id/PPN237839962.

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1 Introduction

Jahrhundert, 1926–27)14 were edited after his death; however, some of the texts Klein prepared are missing there. The occasion of Klein’s seventieth birthday was celebrated with the publication of a special issue of the journal Die Naturwissenschaften with contributions from R. Fricke, A. Voss [Voß], W. Wirtinger, A. Schoenflies, C. Carathéodory, A. Sommerfeld, H.E. Timerding, and L. Prandtl. Mostly free of hagiography, this issue reflects the spectrum of Klein’s productivity in an abbreviated form. Obituaries complete the picture. In the mentioned ENCYKLOPÄDIE, which Klein directed, a number of his earlier judgements and scholarly disputes (with Camille Jordan or Lazarus Fuchs,15 for instance) are smoothed over, relativized, or not mentioned at all. Typically, international experts present the state of research at the time. Here, as well as in later works of this sort,16 one finds numerous references to concepts that were either named after or coined by Klein. Without a doubt, Klein used the ENCYKLOPÄDIE to disseminate and classify his own findings. However, his correspondence with Henri Poincaré (see Section 5.5.5) and with Wilhelm Killing (see Section 3.3) are good examples of the facts that Klein strove for historical accuracy in his depictions of mathematical developments and that he never wanted his name to be associated with concepts or ideas originally formulated by others. Of the numerous mathematical concepts conceived by Klein and discussed at length in encyclopedia entries, some of the most significant should be listed here: Klein’s line coordinates (see Section 2.3.3); the Cayley-Klein metric (see Section 2.5.3); Klein’s Erlanger Programm, known in English as the Erlangen Program, which grew from his earlier research (see Sections 2.8.2 and 3.1.1); the problem of Clifford-Klein space forms (Section 3.3); Klein-Riemann surfaces (see Sections 4.2.3 and 5.5.1.2); Klein’s (homogeneous) space;17 Klein’s level theory for classifying the theory of (elliptic) modular functions (see Section 4.2.4.2);18 the Klein bottle (Section 5.5.1.2); Klein’s so-called “four-group” and his Formenproblem (Section 5.5.6); Klein’s fundamental theorems, later known as uniformization theorems (see Section 5.5.4); Klein’s (transcendental) prime form;19 and Klein’s oscillation theorem (see Sections 5.5.1.1 and 6.3.5). 14 There is only an English edition of the first volume, 1979. 15 Regarding a conflict between Klein and Jordan, see especially BRECHENMACHER 2011. On the polemic between Klein and Fuchs, see Section 5.5.5 and Appendix 5 in this book. 16 See NAAS/SCHMIDT 1961/1984; WEISSTEIN 2003, etc. 17 Klein space (Kleinscher Raum) is another name for a homogeneous space. Klein’s Erlangen Program interprets geometry as a theory of certain homogeneous spaces. A topological space X is called homogeneous if there is a topological transformation group G that transitively acts on X. This means that for every points x, y from X there is a transformation g from G such that gx = y. Intuitively, a homogeneous space X looks the same everywhere; observers of any of its points will see the same picture. 18 See the article by Robert Fricke in ENCYKLOPÄDIE, vol. 2.2 (1913). 19 See the article by Franz Meyer on “Invariantentheorie” in ENCYKLOPÄDIE, vol. 1, p. 297 (1899); and by Wilhelm Wirtinger on “Algebraische Funktionen und ihre Integrale” in ibid., vol. 2.2, pp. 115–75 (1901). Klein’s ideas on this topic were first published in his article “Zur Theorie der Abel’schen Functionen,” Math. Ann. 36 (1890), pp. 1–83.

1.1 The State of Research

5

Klein’s work is multifaceted, inspiring, and fascinating, and so it comes as no surprise that a great many (male and female) mathematicians and historians of mathematics, science, technology, and education have written about various aspects of his activity. Thomas Hawkins, an American historian of mathematics, has earned accolades for his studies of Klein’s contributions. HAWKINS (1984) deserves special mention because it places Klein’s Erlangen Program in its proper context and analyzes its reception. I.M. YAGLOM (1988) examines Klein’s role in the evolution of the concept of symmetry. In 2015, Lizhen LI and Athanase PAPADOPOULOS published an anthology of articles devoted to investigating the impact of the Erlangen Program in various branches of physics and mathematics. Peter SLODOWY (1993, Engl. 2019) provides a new edition of Klein’s book on the icosahedron (KLEIN 1884), together with a commentary from a modern perspective. In 1986, Gerd FISCHER published a two-volume collection of mathematical models that were relevant to Klein’s research approach (this book was reprinted in 2018). FISCHER (1985) is an edition of Klein’s Abitur examinations in the field of mathematics. Regarding the topic of mathematical models, contributions by David E. ROWE (2013, 2017, 2019b) and Anja SATTELMACHER’s dissertation (2017) deserve special mention. Recently, too, the Spanish mathematician Roberto RODRÍGUEZ DEL RÍO (2017) has written a book about Klein’s new vision for geometry; his book has already been translated into Italian and French.20 SCHOLZ’s aforementioned dissertation from 1980 is not only an excellent history of the manifold concept from Riemann to Poincaré; it also includes a detailed discussion of Klein’s contributions to this area of study. SCHOLZ’s 1989 Habilitation thesis, which focuses on the group concept, and some of his other studies are likewise relevant to our topic. Regarding the history of the group concept, reference should also be made to the work of Hans WUßING (1969, 1984, 2007). ZIEGLER (1985) analyzes Plückerian line geometry and its further development by Klein and his student Ferdinand Lindemann within the framework of the history of projective geometry and geometric mechanics. More recent studies – BIOESMAT-MARTAGNON (2010) and PLUMP (2014) – have refined Ziegler’s work. In the middle of the 1980s, David E. Rowe, one of the foremost authorities on the mathematics of Felix Klein, began to publish articles on Klein’s work, and at the University of Mainz, he encouraged some students to focus on Klein as well. One noteworthy result of this encouragement is the Habilitation thesis by Moritz EPPLE (1999), which is concerned with knot theory. ROWE (2018a) is a convenient collection of the author’s earlier essays, and ROWE (2019a) offers a clear analysis of the early works of Klein and Sophus Lie and their mutual influence. Thirty years earlier, David E. Rowe had organized an international symposium on the history of mathematics,21 the proceedings of which were published in

20 Unfortunately, it is not very reliable in numerous details. 21 The symposium was held between June 20th and 24th in 1988 at Vassar College (Poughkeepsie, New York). In the published proceedings (ROWE/MCCLEARY 1989), it is falsely stated that the conference took place in 1989.

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1 Introduction

three volumes as History of Modern Mathematics. Around that same time, Rowe began to publish his first studies on Klein and Sophus Lie’s early collaborative work in the field of geometry (see ROWE 1989), a topic that Eldar Straume (Norges Teknisk-Naturvitenskapelige Universitet, Trondheim) and Leslie Kay (Virginia Tech) have also been working on for some years. At the symposium in 1988, Jeremy J. Gray gave a talk about algebraic geometry in the nineteenth century. Additional studies by Gray on Lazarus Fuchs and his theory of differential equations (1984), on Poincaré (2013),22 and his recent argument (2019) that Klein helped to create the twentieth-century definition of Galois theory are important to understanding Klein’s mathematical influence. There are also many pertinent works by French historians of mathematics, including Frédéric BRECHENMACHER’s studies of Camille Jordan and related areas, a number of contributions by Hélène Gispert, with whom I co-authored an article in which we compared the mathematical societies in France and Germany (GISPERT/TOBIES 1996). Noteworthy, too, are Catherine GOLDSTEIN’s 2011 study of Charles Hermite, Barnabé CROIZAT’s 2016 doctoral thesis on Gaston Darboux, and several papers by François LÊ – particularly his 2015 study of Alfred Clebsch’s diagonal surface, a topic that held Klein’s interest for many years.23 For her MA thesis at the University of Jena, Tina RICHTER (2015) translated and annotated the correspondence that Darboux addressed to Klein. On the basis of additional sources, it has been possible to study Klein’s relationship with French mathematicians in greater detail (see TOBIES 2016). Konrad JACOBS (1977) published a selection of facsimiles and transcriptions from Klein’s substantial Nachlass, which is carefully archived at the Göttingen State and University Library. Only a small portion of Klein’s vast correspondence has been edited for publication. The letters exchanged by Klein and Hilbert were edited by FREI (1985). TOBIES and ROWE (1990) contains Klein’s correspondence with Adolph Mayer, where the main topic of discussion is the activity of editing the journal Mathematische Annalen. Smaller editions of Klein’s correspondence – between Klein and Otto Stolz (BINDER 1989), Klein and E.S. Fedorov (BURCKHARDT 1972), Klein and A. Gutzmer (TOBIES 1988), Klein and A.A. Markov (TOBIES 2018), Klein and Paul Koebe (TOBIES 2021a) – round out this picture. Umberto Bottazzini has written at length about Riemann’s influence on Italian mathematicians. Together with Jeremy Gray, he analyzed the development of complex function theory (BOTTAZZINI/GRAY 2013). Bottazzini and other Italian historians of mathematics have done much to reveal the close connections that Klein had with the Italian algebraic-geometric school of thought. Of particular interest are two collections of articles, one on mathematicians in Bologna (COEN 2012) and the other on the work of Corrado Segre (CASNATI et al. 2016). Several

22 This biography focuses primarily on Poincaré’s most significant publications; see the review by Scott A. Walter in Historia Mathematica 44 (2017), pp. 425–35. 23 On the occasion of Klein’s 150th birthday, a ceramic model of this surface (1.4 meters wide and 2.5 meters high) was installed at the University of Düsseldorf (see KAENDERS 1999).

1.1 The State of Research

7

editions of correspondence, moreover, document the fact that Klein maintained a close relationship with mathematicians in Italy ever since his first trip there in 1874 (see CREMONA 1992–99, LUCIANO/ROERO 2012, and ISRAEL 2017). I was personally able to examine a collection of Klein’s letters held in Pisa. Since the publication of his dissertation in 1986, Klaus VOLKERT has made several contributions to topics that are relevant to this study (the history of Anschauung, non-Euclidean geometry, etc.). Karen PARSHALL and David E. ROWE’s book from 1994 not only illuminates the prominent role that Klein played in the emergence of the American mathematical research community; it also provides a lucid outline of Klein’s most significant mathematical findings. Reinhard SIEGMUND-SCHULTZE has published a number of books and articles that I have drawn upon, including his work on Richard von Mises, who – recognized by Klein – enhanced the domain of applied mathematics. Ulf HASHAGEN’s thesis on Walther Dyck (2003) and Michael ECKERT’s studies on Arnold Sommerfeld (2013) and Ludwig Prandtl (2017, Engl. 2019a) are also rich and reliable sources for Klein’s biography. There have only been a few assessments of Klein’s political positions. The role of Jewish mathematicians in German-speaking academic culture is the topic of BERGMANN et al. (2012), and the prominence of Jewish mathematicians in Göttingen during Klein’s years there was discussed as early as ROWE (1986). Cordula TOLLMIEN’s detailed analysis (1993) has made it possible to reinterpret the fact that Klein’s signature appears on the nationalistic “Manifesto of the Ninety-Three” from 1914. The older study by Karl-Heinz MANEGOLD (1970) investigated the main aspects of Klein’s activity as an organizer of scientific research. Susann HENSEL et al. (1989) analyzed the anti-mathematical attitude of German engineers in the nineteenth century – an attitude that Klein was ultimately able to overcome. Bernhard vom BROCKE (1991) is an insightful collection of articles on the so-called “Althoff system” that prevailed at the Prussian Ministry of Culture, which would turn increasingly to Klein as a reliable adviser. Several scholars have written about Klein’s contributions to reforming mathematical education. Here I should refer in particular to the pioneering studies by Gert SCHUBRING and the recent translation of Klein’s book series Elementary Mathematics from a Higher Standpoint by Schubring, Martha Menghini, and Anna Baccaglini-Frank in 2016. WEIGAND et al. (2019) brings together a variety of international research approaches; the book arose from a series of panels devoted to the “legacy of Felix Klein” at the 13th International Congress on Mathematical Education, which was held in Hamburg in 2016. Masami ISODA’s contribution in the latter book – along with Harald KÜMMERLE’s excellent doctoral dissertation (2018), published as a book in 2021 – have shed light on the extent to which Klein’s influence reached as far as Japan (see also DAUBEN/SCRIBA 2002). Finally, this book is the culmination of many years of archival research, and it draws upon much of my own earlier work on a broad spectrum of topics (see the Bibliography).

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1.2 GUIDING QUESTIONS In order for this book to remain readable, I have necessarily had to limit the scope of my presentation. Each of the scholars cited above understandably approached the topic of Klein’s biography from a particular perspective. As I learned as a young historian of mathematics and education, however, a proper assessment of a problem or a scientist should ideally take all sides into account. Even though it would of course be impossible in this biography to write a detailed history of all the disciplines to which Klein contributed, my attempt at an omnivorous approach has enabled me to reach new conclusions about Felix Klein as a mathematician and as a person. My thesis is that Klein’s way of working was characterized, above all, by its continuity. This is counter to the prevailing opinion that his approach changed drastically in 1882 (because of health problems) and in 1895 (when Hilbert joined the faculty in Göttingen). It will be shown that during the early stages of his career, when his research primarily concerned geometry, algebra, function theory, and number theory, Klein had already developed a consistent attitude toward applied mathematics, social issues, educational policy, and scientific administration. Conversely, during his later years, when his focus shifted to so-called peripheral areas of mathematics, Klein did not cease from promoting works of pure mathematics. It will also be necessary to reevaluate Klein’s health-related issues and his political attitude. The aspects that have guided my research for this biography are the following: First: I will examine how and by what means Felix Klein became an internationally recognized mathematician who significantly influenced the development of mathematics, its applications, and education during his lifetime. Second: Klein’s manner of conducting research and leading people was based on cooperation. He brought students and young researchers from numerous countries within his fold. I will show who his most important collaborators were for his research programs and projects during specific periods of his life, and I will also discuss when and why competition happened to arise instead of cooperation. Third: Klein’s lifetime coincided with the German Empire and the beginning of the Weimar Republic. He was briefly a paramedic during the Franco-Prussian War, and he experienced the First World War. He secured financial backing from industry and also from the military in order to build up his teaching and research programs in Göttingen. I intend to discuss the political attitude that underlay his pursuits. In my efforts to ascertain the structural features of Felix Klein’s career, Ludwik Fleck’s concept of the “thought collective” has proven to be especially useful. Fleck’s work is recognized today as a precursor to Thomas Kuhn’s The Structure of Scientific Revolutions. In 1935, and thus long before Kuhn, Fleck – a PolishJewish microbiologist, physician, and theorist of science – analyzed the genesis and structure of research communities, which he referred to as thought collectives.

1.2 Guiding Questions

9

To characterize the views that prevail within a thought collective (Denkkollektiv), he coined the term “thought style” or “style of thinking” (Denkstil): If we define a “thought collective” as a community of persons mutually exchanging ideas or maintaining intellectual interaction, we will find by implication that it also provides the special “carrier” for the historical development of any field of thought, as well as for the given stock of knowledge and level of culture. This we have termed a “thought style.”24

Fleck referred both to the group-formational and social effect of commonly shared views and concepts and to the special role that such collectives play in introducing young researchers to a given area of study. Group norms experienced in one’s youth can often be formational throughout one’s life. At the same time, acquiring experiences in one area and trying them out in other communities can be points of departure for becoming a member of a special thought collective or for forming a collective on one’s own. According to Fleck, too, each member of such a collective can simultaneously belong to other communities (scientific, political, cultural) and thus introduce varying points of view. At this point, it can be said that one of Klein’s major characteristics was his willingness and ability to form, manage, and lead associations or groups. His work ethic was based on the values of his family and early education. His connection to the internationally networked Plücker and Clebsch allowed him to become a member of a community associated with “newer geometry,” which still had to be formed on the national level. The early deaths of Plücker and Clebsch contributed to the fact that Klein was able to become the head of a thought collective that aimed to carry out Riemann’s agenda in the field of geometry. At the time, Leo Koenigsberger, who was Karl Weierstrass’s first prominent student, sensed that a shift in the predominant style of thinking was taking place: Even we younger mathematicians all felt at the time as though the Riemannian views and methods did not belong to the strict mathematics of Euler, Lagrange, Gauß, Jacobi, Dirichlet, and others – as always seems to be the case when science is penetrated by a great new idea, which at first needs time to be processed in the minds of the living generation. Thus the achievements of the Göttingen school were not appreciated by many of us (or at least not by most of us) as much as their great significance deserved, and we seldom gave this work the place of distinction that science would soon award it.25

Klein remained rooted in this geometric style of thinking, even though he would go on to integrate the methods of other orientations into his concepts. Thus, nearly at the age of sixty, he ultimately accepted the axiomatic style of thinking that, thanks above all to David Hilbert, came to prevail in Germany.26 Although Klein supported Hilbert’s new approach to invariant theory, he was at first not fully on board with the latter’s “abstract” number theory. Surprisingly, sources reveal that Klein’s much-discussed talk on “arithmeticization” (1895) had originally been 24 FLECK 1979 [1935], p. 39. See also TOBIES 2012, pp. 7–8. 25 KOENIGSBERGER 2004 [1919], p. 29 (http://histmath-heidelberg.de/txt/koenigsberger/leben.pdf). 26 In this regard, see also Klein’s notes on Hilbert’s problems (1900) in Section 10.1.

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1 Introduction

directed against Hilbert’s new “abstract” approach, something that Klein later, in 1908, himself referred to as subjective (see Sections 6.3.7.3, 8.2.2, 8.3.2). Contrary to Herbert MEHRTENS’s (1990) classification of mathematicians as either modern (Hilbert et al.) or anti-modern (Klein et al.),27 it seems to me as though, in Klein’s case, it is possible to speak of a particular sort of modernity on whose basis new domains such as modern numerical analysis, actuarial mathematics, and financial mathematics were able to develop. These would later give rise to fields such as engineering mathematics and business mathematics.28 In order to establish the necessary institutional and personnel resources for the study of mathematical applications in technical fields, Klein sought and acquired new sources of funding by following the example of the Carl Zeiss Foundation in Jena and that of American universities. Mitchell Ash has underscored, as a scientific-historical concept, the extent which the availability of “financial, […] cognitive, instrumental, personnel, institutional, or rhetorical” resources have influenced the development of science.29 This concept of resources provides a fitting way to understand Klein’s efforts to secure and utilize financial support – from all available sources (the state, industry, the military) – for teaching, research, instruments, and institutes. Conversely, the interest of donors can be seen as an interest in scientific results. At the same time, this need for resources does much to explain Klein’s activity as a non-partisan representative of the University of Göttingen in the Prussian Upper House of Lords (Herrenhaus, the first chamber of the Prussian parliament) as well as his rhetoric and approval of declarations during the First World War (see Sections 8.3.4 and 9.1). Ash’s concept also fits well with Klein’s approach to hiring personnel. Klein always sought out the best candidate for a specific job in question, without regard to gender, nationality, or religious affiliation. This is evidenced by his (hitherto unknown) engagement on behalf of Georg Pick (see 5.5.2.4) and his support for the careers and publications of Max and Emmy Noether, Adolf Hurwitz, Arthur Schoenflies, Georges Brunel, Gino Fano, Irving W. Stringham, and many others. In order to present all of this information in a readable fashion, I have organized this book around the following three points (I to III): I. Arranging the book according to the stages of Klein’s life and career has made it possible to best represent the early-established diversity of his activity. Chapter 2 thus examines the formative communities of which he was a part. The latter include his childhood family and school years, which formed the basis of his prodigious work ethic. This period was followed by his time as a university student (in Bonn, Göttingen, Berlin, and Paris), when he met influential teachers (Plücker, Clebsch) who opened up opportunities for him to join international communities of intellectuals and who made it easier for him to find collaborative partners. Klein’s time as a Privatdozent in Göttingen, where he was part of 27 For a critical view of this classification, see also ROWE 1997 and BAIR et al. 2017. 28 See NEUNZERT/PRÄTZEL-WOLTERS 2015 and FRAUNHOFER ITWM 2018. 29 ASH 2002, p. 32. See also ASH 2016.

1.2 Guiding Questions

11

Clebsch’s tight community, ultimately had the effect that, as the youngest member of Clebsch’s circle, he was able to carry on, with the greatest intensity, the latter’s program of bringing together disciplines and people, and that he was even able to promote the careers of older members of the group. Chapters 3, 4, and 5 are devoted to Klein’s activity as a full professor at the University of Erlangen (1872–75), at the Polytechnikum in Munich (1875–80; the latter institution was designated a Technische Hochschule in 1877), and at the University of Leipzig (1880–86). In 1886, Klein transferred to the University of Göttingen, and his range of activity there is treated in four chapters (6 through 9), which warrant summaries of their own. Chapter 6 covers the years 1886 to 1892. These years in Göttingen were made difficult by the fact that Klein had been hired against the wishes of the two professors of mathematics who were already on the faculty there, H.A. Schwarz and E. Schering (see Section 5.8.2 and Appendix 4) – a fact that long went unrecognized. Klein cultivated contacts within the overall framework of the university, within the national framework of mathematicians, and abroad. In Göttingen, he concentrated on his academic work and sent memoranda with wide-reaching aims to the Prussian Ministry of Culture. It was not until 1892, when Schwarz accepted a new position in Berlin and Klein rejected an invitation to move to the University of Munich, that Klein was able to realize many of his goals. Chapter 7 concerns the years 1892 to 1895. Supported by Heinrich Weber and oriented toward foreign – particularly American – examples (Klein’s first trip to the United States was in 1893), Klein was able to form the basis for his success in the coming years. During this brief period, his most significant accomplishments included hiring, for the first time, mathematical assistants at Göttingen; founding the Göttingen Mathematical Society; establishing contact with secondary-school teachers and industrial leaders; supporting the right of women to study at the university level; creating an official course of study for actuarial mathematics; beginning the ENCYKLOPÄDIE project; and, finally, hiring Hilbert in April of 1895. Chapter 8, which treats the period from 1895 to 1913, examines the fruits of Klein’s previously determined course of action and illuminates the interrelated nature of his projects. During this time, he completed his monograph on automorphic functions (FRICKE/KLEIN) and worked on a book about the spinning top theory (KLEIN/SOMMERFELD; Klein’s interest in this topic can be dated back to his second stay in Paris in 1887). Within the framework of his intensively managed program for mathematical physics and mechanics (an idea he formulated as early as 1881; see Section 5.5), Klein became a significant originator of ideas for young researchers (see Section 8.2.4). He was able to forge alliances between Göttingen scientists and industrialists, and he was also able to convince the latter that it was in their interest to provide financial support. Moreover, this support was meant to assist not only research but also educational reform, which Klein coordinated and promoted both nationally and internationally. It should become clear from this chapter that Klein’s practical committee work (regionally, nationally, and internationally) was closely interrelated both to his research-oriented teaching and to his theoretical work on book projects such as

12

1 Introduction

the ENCYKLOPÄDIE, the Abhandlungen über den mathematischen Unterricht (KLEIN 1909–16), and Kultur der Gegenwart (KLEIN 1912–14). Klein’s early idea to prepare an (ultimately unrealized) final volume for the ENCYKLOPÄDIE meant that he devoted considerable energy to intensively studying the history, philosophy, psychology, and didactics of mathematics (see Section 8.3). Chapter 9 concerns Klein’s time as an emeritus professor (1913–25), a period that also includes the years of the First World War. After a respite spent in a sanatorium, he continued to manage committees, teaching, and editorial projects. He made substantial contributions to the general theory of relativity and he remained, in the words of Abraham Fraenkel, the “foreign minister” (Außenminister) of German mathematics.30 It should be stressed that Klein’s term as a representative of the University of Göttingen in the Prussian parliament, 1908–18 (see Sections 8.3.4 and 9.1.2), where he focused on educational policy, and his immediate action after the November Revolution of 1918 can be seen as a remarkable example of continuity across periods of political upheaval. The extent of these activities, which concerned research funding, issues of hiring, and new educational reforms, will be discussed at length here for the first time. Chapter 10 will revisit the guiding questions of this book and summarize the many pioneering aspects of Klein’s work. The Appendix of this book contains a selection of important primary sources. In order to help the reader work through the book, I decided to make the Index of Names more comprehensive than is customarily the case. There, in addition to including biographical dates, I have also indicated whether a given person conducted doctoral research under Klein, contributed to the ENCYKLOPÄDIE, or became member of the German Mathematical Society (DMV). II. At the beginning of each chapter, I provide a brief overview of Klein’s activity during the timeframe under discussion, with respect to how this activity related to research topics, teaching, committees, or to new local, national, and international processes. References in the book to previous or later developments are meant to facilitate the recognition of broader contexts and continuities. III. Topics that remained relevant across extended periods of time are discussed at greater length upon their first mention. Topics of this sort include, for instance, the members of Clebsch’s intellectual community (Section 2.4.1); the role of the journal Mathematische Annalen and Klein’s position on its editorial board (Section 2.4.2); the members of Klein’s family (Section 3.6); Klein’s advisory position for the B.G. Teubner publishing house and his influence over its publications (Section 5.6); Klein’s work for and within the German Mathematical Society (Deutsche Mathematiker-Vereinigung, DMV; Section 6.4.4); the ENCYKLOPÄDIE project, which began as an idea proposed by Franz Meyer (Section 7.8); and other innovations that I discuss in Chapter 7. This approach made it possible for me to concentrate, in Chapters 8 and 9, on the new activities that Klein began to undertake during his later years. 30 FRAENKEL 2016 [1967], p. 138.

1.3 Editorial Remarks

13

1.3 EDITORIAL REMARKS This section provides an overview of my treatment of sources, my citation style, and other editorial issues. On the basis of TOBIES (1981a), the late Emil Fellmann asked me in 1982 to prepare a comprehensive biography devoted to Felix Klein for his series Vita Mathematica. Due to the political and territorial divide in Germany, however, I was unable to accomplish this at the time, mainly because I had yet to study Klein’s comprehensive Nachlass in Göttingen. In 1985, David E. Rowe made it possible for me, for the first time, to spend four weeks examining this archive. Since then, a number of additional works have been published. Our edition of Klein’s correspondence (TOBIES/ROWE 1990), for instance, was the result of fruitful collaboration. In the summer of 2015, Annika Denkert of Springer Spektrum in Heidelberg invited me to write a biography of Felix Klein for her Portfolio series (TOBIES 2019). At that point, I asked David E. Rowe if he might be willing to co-author such a book with me, but his other obligations regrettably prevented him from doing so. In the present book, I have made an effort to allow Felix Klein to speak for himself as often as possible. One of my primary aims has been to examine overlooked or infrequently analyzed sources in order to discover the reasoning behind Klein’s decisions, to expose the origins of his approaches, and to present the arguments and opinions of his contemporaries. Over the years, I have had the opportunity to study and analyze numerous primary sources. These include countless letters by and to Klein, for the latter documents possess the highest degree of authenticity. It was Klein’s custom to write and save drafts of his most important letters. Typically, these drafts differ little from the letters that he ultimately sent, as a quick comparison of his correspondence with the Prussian Ministry of Culture shows [StA Berlin]. The drafts of his letters are archived in Klein’s Nachlass in Göttingen [UBG]. In October of 1878, however, Klein burned all of his previous correspondence in an “impetuous act of foolishness,” as he explained to Max Noether in 1899.31 In order to assess matters before this time, therefore, the letters to his correspondence partners that survive elsewhere (Oslo, Paris, Pisa, St. Petersburg, etc.) are especially important. Beyond this, documents preserved in additional Nachlässe (Ferdinand Lindemann’s, Robert Fricke’s, among others), the private estate of Meinolf Hillebrand’s family (Hillebrand was one of Klein’s great-grandsons), and materials housed at Klein’s Gymnasium in Düsseldorf have been extremely valuable. Another excellent source is the unedited correspondence between Klein and Adolf Hurwitz; this entire collection of letters is archived in Göttingen. Revealing, too, is the correspondence between H.A. Schwarz and Karl Weierstrass [BBAW], for the two had much to say about Klein. An overview of the primary sources that I have consulted can be found in the list of archives at the beginning of this book’s bibliography. 31 [UBG] Cod. MS. F. Klein 12: 651 (a letter from Klein to M. Noether dated March 1, 1899).

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1 Introduction

Abbreviated references in my footnotes are spelled out in full in the bibliography. Citations in [square] brackets refer to archival sources, while the names of authors in SMALL CAPITALS refer to works of scholarship. Certain secondary sources, which are relevant only to a specific context, have been cited in full in the footnotes and not added to the bibliography. The transliteration of Cyrillic names is not always consistent, for the simple reason that they appear in different forms in different sources. Acknowledgements First and foremost, I would like to thank the editors at the Birkhäuser/Springer publishing house in Basel/Cham (Switzerland) – Thomas Hempfling, Sarah Goob, Sabrina Hoecklin, and Martin Mattmüller – for including this book in the series Vita Mathematica. The editor of this series, Martin Mattmüller, read the entire manuscript diligently, and the author is deeply grateful for his helpful advice. I would like to continue by emphasizing that I am deeply indebted to the directors and staff of the numerous archives whose sources I have already been able to use for the German version of this biography. Special thanks go to Mrs. Bärbel Mund, Head of the Manuscript Department at the University Library of Göttingen, for her ongoing support. I am also indebted to numerous colleagues who read the original German edition of this book and have since shared and discussed additional insights with me in a wide variety of settings: at the Max Planck Institute for Mathematics in the Sciences (Leipzig), the Polish Mathematical Society and at the Institute of the History of Sciences of the Polish Academy of Sciences in Warsaw, and the Mathematical Institute of the Jagiellonian University in Kraków, at meetings of the German Mathematical Society and the Austrian Mathematical Society, at the Kepler Symposium hosted by the Johannes Kepler University in Linz, at the Toeplitz Colloquium organized by the University of Bonn, at a lecture held at the Felix Klein Center in Kaiserslautern (organized by the Fraunhofer Institute for Industrial Mathematics) – among other places. This book could not have been written without the help and collaboration of my international colleagues. David E. Rowe deserves special thanks for his early support and for providing numerous valuable references for the English edition. For many years of fruitful cooperation and for his many fundamental contributions to this book, I owe special thanks to Reinhard Siegmund-Schultze (Kristiansand, Norway). As far back as 1992, the mathematician Tito M. Tonietti (Pisa) enabled me to have access to the archives at the Scuola Normale. Christa Binder (Vienna) kindly made her transcriptions of the correspondence between Klein and Otto Stolz available to me. I sincerely thank Frédéric Brechenmacher for sending me his publications as well as letters from Felix Klein to Camille Jordan. For their collaboration on several projects, my gratitude extends to Hélène Gispert, MarieJosé Durand-Richard, and Dominique Tournès. Marie-José Durand-Richard also went above and beyond to ensure that I could acquire copies of Klein’s letters held in the Darboux Nachlass in Paris. An effort to assist Eldar Straume (Trond-

1.3 Editorial Remarks

15

heim, Norway) and Leslie Kay (Virginia Tech) with their project of editing Klein’s letters to Sophus Lie provided me with an occasion to study archival material in Oslo, and I have to thank Eldar Straume for kindly inviting me (after my work in Olso) to deliver a lecture in Trondheim, and also Reinhard SiegmundSchultze, who invited me to Kristiansand. Without the help of Sergei S. Demidov in Moscow, my work at the archive of the Academy of Sciences in St. Petersburg would have been far less successful. Danuta Ciesielska (Warsaw/Kraków) generously made her analyses of the work of Polish mathematicians in Göttingen available to me, and she supported my research by inviting me to give lectures. Martina Bečvářová (Prague), who has published important works on Czech mathematicians, brought many new ideas to my attention at a workshop on “Women and Mathematics” hosted by the Oberwolfach Institute for Mathematics. I had the pleasure of organizing the latter event with Nicola Oswald (Wuppertal) and Tinne Hoff Kjeldsen (Denmark). I would also like to express my sincere thanks to Elisabeth Mühlhausen (Göttingen) and Cordula Tollmien (Hann. Münden) for their valuable collaboration in this field of research. Harald Kümmerle (Halle) familiarized me with his latest results on Japanese mathematicians who studied with Felix Klein. I would like to thank Michael Rahnfeld (Weddingstedt) and Henning Heller (Vienna) for comments on the effects of Klein’s work on aspects of the philosophy of mathematics. In this context, I would like to mention that Henning Heller completed his undergraduate thesis “Beiträge Felix Kleins zur Gruppen- und Invariantentheorie” [Felix Klein’s Contributions to Group and Invariant Theory] with me at the University of Jena in 2015, holds a master’s degree from the University of Bristol, and is currently working on Klein’s mathematics and philosophy at the University of Vienna. Furthermore, the interdisciplinary spirit at the Friedrich Schiller University in Jena (at the Faculty of Mathematics and Computer Science and the Ernst Haeckel House) deserves to be acknowledged here because it formed an important basis for my work. As I mentioned above, several colleagues have read the German version of this book in detail and have provided me with valuable references and feedback. I have made every effort to ensure that their insights have been included in the English translation. In this regard, my gratitude is also due to Reinhard Bölling (Potsdam), Peter Bussemer (Gera), Günter Dörfel (Dresden), Michael Fothe (Jena), Rita Meyer-Spasche (Munich), Rainer Schimming (Potsdam), and Gert Schubring (Rio de Janeiro). Finally, it remains for me to thank the translator himself, Valentine A. Pakis, for his constructive, reliable, and excellent work. The translator would like to acknowledge his great debt to Reinhard Siegmund-Schultze and Martin Mattmüller, who each read the draft in full and made invaluable corrections throughout. He would also like to thank Lizhen Li (University of Michigan) and Li Peng (Higher Education Press, Beijing) for making materials available to him that were out of his reach. Although I have relied on many people while writing this book, it goes without saying that I alone am responsible for any remaining errors or inaccuracies.

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1 Introduction

Figure 3: An excerpt of a letter from Felix Klein to Sophus Lie dated April 1, 1872 [Oslo] (Regarding the context, see Section 2.8.3.4)

2 FORMATIVE GROUPS This chapter will examine the people and communities that shaped Felix Klein’s views, work ethic, manners of behavior, and style of thinking. The following groups will be discussed: – his family; – his classmates and teachers in Düsseldorf; – his intellectual community while studying in Bonn under Julius Plücker; – the algebraic-geometric school led by Alfred Clebsch; – his circle of friends in Berlin and Paris; and – the circle of Privatdozenten in Göttingen (not only mathematicians). 2.1 THE KLEIN–KAYSER FAMILY A strong work ethic and certain other factors were instilled in Klein from an early age. Both his father’s and his mother’s side of the family played a role in this, as both Felix Klein and his brother Alfred agreed. 2.1.1 A Royalist and Frugal Westphalian Upbringing A stubborn will, unrelenting diligence, a sober sense of reality, absolute reliability, and wellconsidered frugality – these are the traditional characteristics of this rugged German clan, which were also purely embodied in my father.1

His family (Kleine, Kleinen) stemmed from the Sauerland area of Westphalia.2 His ancestors included farmers and representatives of the region’s small-scale iron industry. His great-grandfather, the farmer Friedrich Peter Kleine (b. 11/11/1731), married Catharina Margarethe Schürfeld on July 27, 1776. Their first-born son, Johann Peter Friedrich Klein (b. 9/18/1777, d. 11/22/1858), was married to Maria Catharina Hammerschmidt (b. 3/31/1787, d. 10/6/1871), the daughter of a timber merchant. Her father could not write, but he was excellent at making calculations in his head. Felix Klein’s grandfather Peter Klein established a blacksmith shop that produced miner’s lamps and other small devices out of iron. These grandparents led a very simple and frugal life: a bucket in the yard was used for washing themselves. The family acquired a modest degree of prosperity through hard work: they used their small savings to purchase plots of forest, which they 1 2

KLEIN 1923a, p. 12 (Felix Klein’s autobiography). [Hillebrand] A family chronicle by Alfred Klein (begun in 1910 and expanded in 1918).

© Springer Nature Switzerland AG 2021 R. Tobies, Felix Klein, Vita Mathematica 20, https://doi.org/10.1007/978-3-030-75785-4_2

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2 Formative Groups

would then clear themselves and convert into farmland. Of their six children, the eldest was Felix Klein’s father: Peter Caspar Klein, who was born on August 11, 1809 in Voerde and died on January 26, 1889 in Düsseldorf. His old-fashioned Prussian-Protestant disposition was, as Felix Klein described it in 1923, “in stark contrast to the more light-hearted and joyful manner of Rhinelanders.”3 Alfred Klein (1910) characterized his father as a “robust Westphalian, industrious, brusque, an organizational and financial talent, a man of independent mind and character, firm in his convictions, hard on himself and others.”

Figure 4: Felix Klein at the age of two (unknown illustrator) [Hillebrand].

Because Caspar Klein had been frail as a child, he was spared from having to lead the hard life of a farmer or craftsman. At the age of fifteen, he was hired as a clerk at the mayor’s office on Enneper Straße (Enneper Street). The office’s jurisdiction included the districts of Haspe, Voerde (where Felix Klein’s grandparents lived), Vorhalle, Waldbauer, and Westerbauer. Enneper Straße, which extended for two miles along the Ennepe river, had been built as a military road by the Prussian government at the end of the eighteenth century. It was lined with numerous ironrelated production facilities. For the sake of serving his mandatory military service, Caspar Klein went to Düsseldorf, where he was active from 1829 to 1831 as a brigade clerk, sergeant and additionally worked for the “Royal Government.” After the Congress of Vienna in 1815, the city of Düsseldorf had become the seat of government for the Rhineland Province of the Prussian Kingdom. Here, Felix Klein’s father worked as a civil servant and climbed the ranks to become a presidential secretary. In 1845, moreover, he took on an ancillary position as an inspector for the Jägerhof and Benrath Palaces. As a close confidante of two noble government presidents – Adolph Theodor Freiherr von Spiegel-Borlinghausen und zu Peckelsheim (in office from 1837 to November 1849) and Karl Friedrich

3

KLEIN 1923a (autobiography), p. 12.

2.1 The Klein–Kayser Family

19

Leo Freiherr von Massenbach (from 1850 to 1866) – Caspar Klein remained faithful to the royal house. During the German revolutions of 1848–1849, he sat in fear with his family on packed suitcases. The family survived unscathed and, on August 7, 1850, Caspar Klein was awarded for his loyalty by being made a Knight of the Fourth Class of the Order of the Red Eagle.4 At the end of 1853, Caspar Klein was promoted to provincial treasurer (Landrentmeister), which was the highest financial office at the Royal Treasury in Düsseldorf. On Felix Klein’s Abitur diploma, his father’s profession is listed as provincial treasurer and government councilor. In 1889, Caspar Klein left behind a fortune of approximately 700,000 Mark, “which he had amassed through strict diligence, great frugality, and financial savviness.”5 He was able to support his son Felix when the latter had financial difficulties in Munich during the 1870s (see Section 4.4). Already in the 1860s, Caspar Klein made it possible for Felix, who was interested in science and technology, to tour factories, but he adhered to the principle that his sons should orient themselves, from an early age, toward career goals and toward earning their own income. In 1871, while he was a Privatdozent in Göttingen, Felix Klein still had to comply with his family’s strict principle of frugality. In a letter to the Norwegian mathematician Sophus Lie, he complained about this rigid, unbending regime as follows: Dear Lie! Today I have to relate some sad news to you. I will not be able to come to Norway in the fall. My previous calculation has been spoiled by a letter that I received from home yesterday. I had devised a plan whereby I would have been able to make the trip if only the yearly sum that I receive from home would have been made available to me in advance. My father, however, rejected this idea categorically; I should become accustomed, “as though I were a civil servant,” to receiving regular quarterly payments. “I could take great trips as soon as I had greater means at my disposal.” I had to restrain myself with all my might so as to avoid starting a long conflict, so little do my parents understand me and my disposition. These people – and yet I must say that, relatively, my parents do not do this to the greatest extent – appraise the value of life according to the money that one earns and saves! And they have no notion whatsoever of how beautiful it is to live for an idea; they simply don’t believe me when I tell them that science is undertaken for its own sake […].6

As one can see, Felix Klein’s assessment here applies to both of his parents. On September 10, 1844, at the age of thirty-five, his father had married Sophie Kayser, who was ten years his junior, but he only did so after he had managed to save a sum of 2,000 Reichsthaler.7

4 5 6 7

His son Felix would receive this same honor in the year 1889 (see Section 6.3.7.1). [Hillebrand] Alfred Klein (1910), p. 3. (700,000 Mark in the year 1889 is equivalent to about 4,800,000 euros in 2020). [Oslo] Klein to Sophus Lie, May 22, 1871. On the status of Privatdozenten (lecturers), see Section 2.8. [Hillebrand] Alfred Klein (1910), p. 4. From 1821 to 1871, a special Reichsthaler (=30 silver pennies) was valid in Prussia.

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2.1.2 Talent in School and Wide Interests as Gifts from His Mother’s Side My mother likewise came from people who had moved away from the industry of Aachen. She was more cheerful and had more flexible opinions than my father. Her multifaceted interests were therefore the main source of intellectual life in the house. Admittedly, this greater degree of activity was tied to an inclination toward nervous exhaustion, which, as a feature inherited from my maternal background, often affected me later in life.8

Felix Klein remained very close to his mother and was known to have sent her a letter every Sunday.9 His opinion of his mother was shared by his brother Alfred, who wrote, “My mother represented what was good and genial in the house; she had strong pedagogical and speculative-scientific interests and a great memory for historical dates.” In addition, Alfred Klein underscored the intellectual activity of both parents and their “rationalistic way of thinking about religious issues.”10 The Kayser family tree can be traced back to the Reformation. In the Free Imperial City of Aachen, the ancestral home of the Kayser family, there was a long period of “religious unrest” during the decades of the Reformation. Merchants and educated citizen were among the first to convert to Protestantism, and they fled from the city in increasing numbers. The Kaysers, a family of fabric manufacturers, left for Thuringia. In addition to innkeepers and farmers, Sophie Kayser’s ancestors also included church dignitaries, surgeons, pharmacists, an estate manager, a civil servant, a wine merchant, and an instrument manufacturer. One ancestor (Privy Councilor of Commerce C.G. Jaeger) left behind a career as a dye manufacturer and indigo dealer to become a banker; upon his death in 1852, 1/32 of his estate (12,500 Reichsthaler) was inherited by Sophie Kayser. Felix Klein’s maternal grandfather, Christian Gottfried Kayser (b. 10/30/ 1791), followed the family tradition and became a wool and fabric merchant. On March 24, 1817, he married Eleonore Schleicher (b. 3/10/1793, d. 5/22/1875), who hailed from Stolberg. This couple’s four children were Felix Klein’s mother Sophie (b. 4/22/1819 in Aachen), Mathilde (married name: Fischer), and Alfred and Ivan, who both followed their father’s example and became fabric manufacturers and wool merchants.11 By Caspar Klein’s side, Sophie Kayser developed into a very frugal wife. The household budget had to be managed with embarrassing precision. According to Felix Klein’s brother Alfred, “Mother would have to calculate precisely whether she could afford to buy a single bread roll; any spending beyond the budget was unthinkable.”12 8 9

KLEIN 1923a, p. 12. This is according to an obituary for Felix Klein that was written by F. Lindemann and printed in the Münchner Neueste Nachrichten on July 9, 1925. 10 [Hillebrand] Alfred Klein (1910) 11 Alfred Klein’s family chronicle (1910, 1918) does not relate when his mother’s siblings were born and died. 12 [Hillebrand] Alfred Klein (1910), p. 3.

2.1 The Klein–Kayser Family

21

2.1.3 Felix Klein and His Siblings Caspar and Sophie Klein had four children together. Their second child Felix was born on April 25, 1849, in Jägerhofstraße 11 in Düsseldorf. His birthday is easy to remember because it consists of the squares of three prime numbers, 2, 5, and 43, as Klein himself was happy to point out. His parents predestined him to a life of good fortune by naming him Felix, which means “happy” or “lucky” in Latin. He was endowed with a readiness of mind and the “good-natured humor of his Rhineland home,” features that would characterize him throughout his entire life.13 On May 1, 1869, Felix Klein’s older sister, Aline Leonore (b. 8/19/1847), married the businessman August Hermann Flender, whose first wife had died in 1867 after six years of childless marriage. Flender owned ironworks in Düsseldorf and Benrath, and he left behind a considerable estate after his early death on January 3, 1882. His businesses lived on in the form of joint-stock corporations such as Brückenbau Flender and Balcke, Tellering & Company in Benrath.14 The marriage between Aline and August Flender produced eight children, four sons and four daughters. Among these, I should draw attention to their third-born child, Hermine Adolfine Leonore (b. 3/2/1873, ob. 8/28/1912), because she would marry Felix Klein’s doctoral student Robert Fricke after the latter had become a professor at the Technische Hochschule in Braunschweig on April 1, 1894. Felix Klein’s brother Alfred, who was born on October 15, 1854, studied law, earned a doctoral degree, and received the title of Justizrat (judicial counselor). On April 25, 1880, he settled down as a lawyer in Düsseldorf.15 With his first wife, Magda Schulz (b. 9/12/1864, ob. 5/24/1893), he had two children, and with his second wife, Helene Portig (b. 10/30/1873), he had four more. The letters between Alfred and his brother Felix document their close relationship. Among other things, Alfred Klein maintained contact with Felix Klein’s son while the latter was living in the United States. He advised his brother in legal affairs, as in the matter, for instance, of preparing a will: “In my opinion, a joint will is simpler and fully satisfies your wishes.”16 In 1894, Felix Klein called upon his brother to help him establish contact with industrialists in the Rhineland (see Section 7.7). Later, Alfred Klein wrote anxious remarks about the 1918 November Revolution and the Spartacists: “My dear Felix! […] I have been overtaken by pessimism since the 15th of November; the future looks very bleak. Here we have plenty of opportunities to witness the insanity of the Spartacists.”17 Felix Klein’s rational activity at this same time will have to be judged with greater nuance. He remained optimistic and active, and he had no reservations whatsoever about the new government of the Weimar Republic (see Sections 9.1.2 and 9.3.2).

13 14 15 16 17

KIRCHBERGER 1925, p. 2. [Hillebrand Private Estate] Alfred Klein (1918), p. 8. [UBG] Cod. MS. Klein 10, p. 399 (a letter from Alfred to Felix Klein dated April 24, 1900). Ibid., p. 403 (quoted from a letter dated December 17, 1913). Ibid., pp. 400, 419 (quoted from a letter dated April 1, 1919).

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Felix Klein’s younger sister, Eugenie (b. 1/20/1861, d. 2/30/1910) never married. She worked as a nurse, volunteer, and board member at the Evangelical Hospital in Düsseldorf and, in addition, she found time to take her nieces on long trips, as Alfred Klein commented in 1910. The commemorative photograph taken on Felix Klein’s silver wedding anniversary in 1900 contains fifteen people, among them his siblings Eugenie and Alfred (see Section 3.6.3, Fig. 19). 2.2 SCHOOL YEARS IN DÜSSELDORF Having first been taught to read, write, and do basic arithmetic by his mother, Felix Klein was sent at the age of six to attend a private elementary school for two and a half years. As of the fall of 1857, he then attended the Gymnasium in Düsseldorf. Known as the Görres-Gymnasium today,18 it is one of the oldest schools of this sort in the German-speaking area, with a history that goes back to the year 1545. Klein’s future professor in Bonn, Julius Plücker, had been graduated from this same school in the year 1819. During Klein’s years there, the school building was located on Alleestraße, which is Heinrich-Heine-Straße today, for the famous poet, who came from a Jewish family, had attended the (Christian) Gymnasium from 1807 to 1810. Following a decree instigated by Wilhelm von Humboldt on November 12, 1812, Gymnasium became the formal designation in the Kingdom of Prussia for schools that prepared pupils directly for university. Klein called his school, which focused on classical languages, a humanistic Gymnasium.19 Of course, then mathematics belonged to the basic components of Prussian secondary schools20, but in retrospect, Klein emphasized the rather formal character of this mathematical instruction. Throughout the second half of the nineteenth century, so-called Realgymnasien (where Latin was the only classical language taught) and Oberrealschulen (which were more strongly oriented toward the natural sciences and modern languages) were created. In the year 1900, Felix Klein did much to ensure that all three types of secondary schools would be treated equally in Prussia (see Section 8.3.4.1). The school systems differed strongly in the German states. As of 1844, the Gymnasium in Düsseldorf was directed by the historian Karl Kiesel, who was qualified to teach history, geography, classical languages, Hebrew, German, and philosophy at all levels, and mathematics up to the intermediate level.21 Klein described his school principal as a strict and excellent teacher; the latter evaluated his proficiency in Greek. According to the school’s records,22 the evaluations of his teachers, the high quality of his Abitur essay on German

18 19 20 21 22

Since 1947, the school has borne the name of Joseph Görres, a philosopher and publisher. KLEIN 1923a (Autobiography), p. 12. See SCHUBRING 1983; and SCHUBRING 2012. [BBF] Personnel records. [Gymnasium Düsseldorf] Felix Klein’s graduation certificate (1865).

2.2 School Years in Düsseldorf

23

literature, and his later comments on his Gymnasium instruction corroborate the view that his time spent at the school served to strengthen the diligent work ethic that had already been instilled in him at home. 2.2.1 Earning His Abitur from a Gymnasium at the Age of Sixteen It is only through effort that a person can achieve a feeling of happiness (Klein 1865).

Thus wrote Felix Klein in his Abitur essay on the topic “The Toils of Life Alone Teach us to Value the Blessings of Life” (Goethe)23. He concluded this German essay with the words: “The most blessed are not those who, born in the lap of luxury, have possessed and experienced everything desirable from early age but rather those who, by gradually toiling with the hard struggles of life, have climbed from one step to the next.” Referring to Psalm 90:10, he added: “Indeed, if a life has been precious, it has been, as the Psalmist says, one of toil and trouble.” The topic of the essay had been determined by August Uppenkamp, who, like Klein’s family, came from Westphalia. Having earned a doctoral degree in 1847, Uppenkamp became a senior teacher at the Gymnasium in Düsseldorf in 1851 and was later made the principal of the school.24 His evaluation of Klein’s essay was as follows: “The topic has essentially been treated properly. Although the language is without ornamentation, it is sufficiently correct. Klein’s earlier essays were typically somewhat better. Satisfactory.”25 Uppenkamp, who was also responsible for teaching Latin, appraised Klein’s examination in this subject (translation and an essay) as good. For the essay, pupils were asked to write about the following maxim by Cicero: “In omnibus saeculis pauciores viri reperti sunt, qui suas cupiditates, quam qui hostium copias vincerent” (“In all ages, fewer have been found capable of conquering their own passions than of defeating hostile armies”). The examination in theology involved writing an essay in response to the following question: “What does the Holy Scripture teach us about the person of the Savior?” Hugo Deussen, who had been teaching religion at the Gymnasium and the Realschule in Düsseldorf since 1864 and was also an assistant pastor at the local Protestant church,26 gave Klein a grade of satisfactory for this examination, but he also added a few critical remarks: In the present work, however, I would have wished to see a more precise explication of what the Holy Scripture teaches us about the person of our Savior, whereas certain other topics, such as Jesus’s futility27, could have been discussed more briefly. It was likewise out of place to offer a defense of what the Scripture teaches us. A fundamental flaw of this work is that it does not discuss the practical significance of the represented teachings, even though the

23 24 25 26 27

Source: Goethe, Torquato Tasso, 1807 (Act V, Scene 1, Antonio speaking to Alfons). [BBF] Personnel records. [Gymnasium Düsseldorf] Felix Klein’s school records. KÖSSLER 2008, p. 68. Deussen had really written “Sinnlosigkeit” (senselessness/futility).

24

2 Formative Groups words “our Savior” in the essay topic implied this. Moreover, the style of this essay would have benefited from greater clarity and stronger organization.28

Klein’s written examination in Hebrew was evaluated by a teacher named Krahle – about whom I was unable to find any further information – and it was given a grade of good. His examination in French was especially impressive; it contained “few and minor mistakes,” and the essay topic was “Victoire de Sobieski à Lemberg”29 (which suggests that the instructional content of the class was focused on military history). Klein’s knowledge of languages would facilitate his later studies in Paris and his correspondence with colleagues from France, Italy, Russia, and elsewhere. His Abitur diploma contains evaluations of the knowledge that Klein acquired in nine different subjects (Table 1) and attests to his strong sense of ethics and general diligence: “Klein has worked without any problems and good cheer; he participated constructively in his classes; he always behaved in a morally upright manner, and we have high expectations regarding his future success.” Table 1: Evaluations of Felix Klein’s achievements from his Abitur diploma (August 3, 1865) 1. In religion, he is able to provide information about the content of the class. He was also able to place excerpts of entire teachings into their context, and although he did not go into matters in great depth, he demonstrated participation and knowledge. His proficiency in the subject is thus satisfactory. 2. In German, he treats his assignments properly and in appropriate language. He is proficient in logic and also familiar with literary history. His knowledge of the subject is thus satisfactory. 3. In Latin, he is skilled at writing, reading, and speaking the language, so that his knowledge of this subject can be called good. 4. In Greek, he has a detailed understanding of the language and is also familiar with the content of the texts that were read. In this subject, too, his knowledge can be called good. 5. In French, he understands the texts so easily and correctly and his German translations are so skillful that he has to be attested as having a good knowledge of this subject. 6. In Hebrew, he has a precise understanding of grammar and is able to understand select passages of the Old Testament. His knowledge of this subject is therefore good. 7. In mathematics, his understanding is quick and certain, and he is always able to recall and apply what he has learned. His knowledge of this subject is thus good. 8. In history and geography, he has acquired a complete overview of the material, and he is able to call to mind and categorize historical details. His knowledge of these subjects is therefore good. 9. In natural history, he has a quite sound understanding of what was taught, and he is able to discuss this material easily, clearly, and comprehensively. His knowledge of this subject is therefore good.

28 [Gymnasium Düsseldorf], see the German original in TOBIES 2019b, p. 17 – At that time, elements of Protestant theology were turning away from salvation-historical interpretations of Jesus’s blood sacrifice; see BERNSTORFF 2009, p. 79. 29 John III Sobieski (1629–1695) was King of Poland and the Grand Duke of Lithuania. He drove back a Turkish invasion at Lviv in 1675.

2.2 School Years in Düsseldorf

25

Although Klein’s teachers graded his achievements merely as good and satisfactory, these assessments are clearly stronger than they might seem today. This Gymnasium provided a sort of education that trained the memory and stressed logic and grammar. In his autobiography from 1923, Klein recalled the pleasure and satisfaction that he experienced after translating stanzas from Schiller’s “The Cranes of Ibycus” into Greek verse, though he doubted that he had fully captured the content and poetic value of the poem. The school’s curriculum, he thought, covered a massive amount of material but did not provide much vital or creative insight. Topics such as “poetry, cultural history, and folklore” were neglected: The view at the time was that the best possible education could be achieved if the pupil had to struggle, by hard work, through a nearly unmanageable amount of material. Even though this method ignored the imagination and any sort of artistic sensibility and deprived us of a great deal of genuine knowledge, it nevertheless provided us with a valuable skill: We learned to work and to work some more.30

2.2.2 Examination Questions in Mathematics In his autobiography, Felix Klein does not mention his mathematics teacher, Dr. Jakob Schneider. Only in passing does he refer to the “strictly formal character” of the mathematical instruction, which came easy to him but which lacked any references to applications or more recent methods. Schneider taught at the Gymnasium in Düsseldorf from 1858 to 1888. In 1840, he earned a doctoral degree from the University of Bonn – the title of his dissertation was “Ueber electrische Figuren, mit Rücksicht auf verwandte Erscheinungen des electrischen und magnetischen Gewitters” [“On Electric Figures, with Regard to Related Phenomena of Electrical and Magnetic Storms”] – and he later received numerous awards, particularly for his archeological and historical studies. His personnel files reveal that he was qualified to teach mathematics, physics, chemistry, botany, and mineralogy to all levels as well as Latin, history, and geography to the lower levels.31 The questions and answers of Klein’s Abitur examination in mathematics were published by Gerd FISCHER (1985). Of the four questions, that concerned with algebra (1) is especially noteworthy; Klein treated a quintic equation (see below Table 2). Whereas, as is well known, the general quintic equation is not solvable by radicals,32 this particular problem could be solved. Later, equations of the fifth degree and higher became one of Klein’s significant areas of research.

30 KLEIN 1923a, p. 13. 31 [BBF] Personnel records. See also https://de.wikipedia.org/wiki/Jacob_Schneider (accessed November 4, 2019). 32 In 1824, Niels Henrik Abel formulated the first complete proof demonstrating this. On the basis of Galois theory, George Paxton Young and Carl Runge later provided an explicit criterion for determining whether a given quintic equation can be solved with radicals.

26

2 Formative Groups

Table 2: Examination questions in mathematics (1865)33 (1) The difference between two numbers is 2, the difference between their fifth powers is 2882. Identify these numbers. [Schneider’s comment on Klein’s solution was: There is nothing essentially wrong with this work.]34 (2) Place a chord through a given point in a given circle in such a way that its segments have a given relation to one another. [Entirely successful] (3) Calculate the angles of a triangle with the following sides: a = 11, b = 9, d = f. [There is nothing essentially wrong with this calculation.] (4) Calculate the volume of a rectangular parallelepiped whose diagonal plane is a square of area 122 and whose base edges have a ratio of 5 to 7.

Schneider evaluated Klein’s solutions as follows: Algebra: good Planimetry: good Trigonometry: good Stereometry: good This work is fully in accordance with his other achievements, which have been praiseworthy in every respect.

Felix Klein did not have any classmates who were particularly interested in mathematics. Most of his graduating class were pursuing a theological career and became Catholic priests.35 One of his “oldest and best friends,” however, was Albert Wenker, who completed his Abitur two years after Klein and who, in 1869 and 1870, supported his construction of mathematical models.36 When, as a university student, Klein would return home to Düsseldorf during his semester breaks, Wenker “was the only person with whom I could have scientific conversations,” as he later informed Sophus Lie. Klein was deeply saddened when, in early February of 1871, Wenker died of typhus as a result of the Franco-Prussian War.37 2.2.3 Interests in Natural Science During His School Years Klein fondly recalled the interests that were awakened in him at his elementary school in Düsseldorf, which was then located at the corner of Grabenstraße and Kanalstraße. The son of this private school’s founder, who was named Krumbach, was a student teacher (Referendar) who had failed his initial teaching examination. Klein wrote the following words about him: “To him I owe my earliest excitement for and instruction in the natural sciences, and I can still clearly remem-

33 Published with Klein’s solutions in FISCHER 1985. 34 Klein described the two equations I) x – y = 2; II) x5 – y5 = 2882; he raised I to the fifth power, transformed the system in two quadratic equations with one unknown, and received the results x = 5 or –3; y = 3 or –5. 35 See KLEIN 1923a, p. 13. 36 See KLEIN 1922 [GMA, vol. 2, pp. 7–10]; and Sections 2.3.4 and 2.7.2 below. 37 [Oslo] A letter from Klein to Lie dated March 11, 1871.

2.2 School Years in Düsseldorf

27

ber the details of these lessons. He also cultivated, in an excellent way, my ability to do arithmetic in my mind.”38 In contrast, the (humanistic) Gymnasium offered little by way of scientific instruction. In Wilhelm Ruer, however, Klein found a friend at school who was supportive of such interests: The lack of stimulation in the natural sciences, however, was compensated for by my friend and classmate Wilhelm Ruer. At his father’s pharmacy, my relentless questions were always answered with friendly and instructive answers. These were supplemented with numerous excursions, and thus I gained my earliest knowledge of chemistry, botany, and zoology. A corresponding enthusiasm for the abstract side of things was fostered in me by the small astronomical observatory in Düsseldorf. Its director, Robert Luther, was a quiet scholar who devoted himself with unceasing diligence to the then difficult task of discovering small planets. He introduced me to the praxis of astronomy and, to a certain extent, he allowed me to participate in his research. Of course, I experimented and tinkered around to the best of my abilities […].39

Robert Luther had been the director of the Düsseldorf-Bilk Observatory since 1851.40 From 1852 to 1865, the year in which Klein completed his Abitur, Luther discovered fourteen asteroids, and he would go on to discover ten more by the year 1890. He was a pioneering astronomer in many respects, and he was internationally recognized for his work. Later, as a professor in Göttingen, Klein was not only keenly interested in ensuring that the astronomy faculty remained strong there; he also supported the construction of (educational) observatories. A mainbelt asteroid discovered by Paul G. Comba in 1997 was named after Felix Klein.41 Another graduate of the Gymnasium in Düsseldorf, Adolph Kirdorf, went on to have a career in the mining industry. In 1894, Klein was able to convince him to serve on a committee tasked with promoting scientific and technical research (see Section 7.7). Ever since his early factory tours, Klein was interested in “the natural sciences in the broadest sense, from the purely intellectual side all the way to the virtuoso technical side,” and less interested in the commercial side of enterprises.42 This accords with his decision, in 1916, to reject an invitation to join the German Federation of Technical and Scientific Organizations, for the main purpose of this federation was to promote the economic interests of enterprises. On Klein’s Abitur diploma, it is noted that he intended to study mathematics and the natural sciences. The Abitur examination committee released him into the world “under the higher condition that he acquire, with continued devotion to his task, the means of useful and noble activity.” Klein chose to attend the nearby University of Bonn, which was then the only university in what would now be the state of North Rhine-Westphalia. KLEIN 1923a, p. 12. See also Section 9.2.3. Ibid., pp. 13–14. For the German original, see TOBIES 2019b, p. 20. Today, Bilk is a district of Düsseldorf. On R. Luther, see NDB, vol. 15 (1987), pp. 561–62. https://en.wikipedia.org/wiki/Meanings_of_minor_planet_names:_12001–13000#818, (12045 Klein [1997 FH1]), accessed May 15, 2020. 42 KLEIN 1923a, p. 14. 38 39 40 41

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2.3 STUDIES AND DOCTORATE IN BONN Bonn belonged to the Prussian Rhine Province. Located within the government district of Cologne, the city had 22,492 residents according to the census taken on December 3, 1864, and thus it was approximately twice the size of Düsseldorf, Klein’s birthplace. In 1818, King Friedrich Wilhelm III had established the University of Bonn on the bank of the Rhine for his new province. It was the sixth Prussian university, after those in Greifswald, Berlin, Königsberg,43 Halle, and Breslau.44 Oriented toward the Humboldtian educational ideal of the unity of teaching and research, the University of Bonn also invoked the “spirit of free research for students of all confessions.”45 It had a theological faculty with two equal parts (one for the Catholic and one for the Protestant denomination) as well as faculties for law, medicine, and philosophy. In the nineteenth century, the primary task of the philosophical faculty was to educate future secondary-school teachers; by tradition, it also included mathematics and the natural sciences.46 Mathematics was not an independent course of study at the time. Not until 1895 did Klein manage to introduce actuarial mathematics as a new degree program in Prussia (see Section 7.6). Klein himself was able to complete his studies by passing either a teaching examination or a doctoral examination. Felix Klein was a student at the University of Bonn from October 5, 1865 to December 15, 1868.47 Plagued from early on with poor health (asthma), he was less inclined than others to get into mischief. Whereas it was typical for the noble students, such as the seventeen-year-old Otto von Bismarck in Göttingen,48 to run riot and find themselves in detention, Felix Klein felt duty-bound to the work ethic of his socially climbing family. We have to imagine him as an assiduous student who was soon making frequent visits to his professor Julius Plücker’s house. Only after Klein had secured his first professorship did he no longer conduct mathematics (at all times) with such “grim seriousness” but rather became, in his words, “mostly a very jolly person.”49 The themes of the next four sections are as follows: – Klein completed a wide-ranging course of study in mathematics and the natural sciences within six semesters, and he was awarded on account of his seminar activities.

43 Founded in 1544 as a Protestant university by Duke Albrecht von Brandenburg-Ansbach, the University of Königsberg remained Prussian until the end of the Second World War (Königsberg has been the Russian city of Kaliningrad since 1946). 44 Known as the Schlesische Friedrich-Wilhelms-Universität from 1811 to 1945; since 1945 it has been the Polish University of Wrocław. 45 SYBEL 1868, p. 101. 46 On the profession of mathematics teachers in the 19th century, see SCHUBRING 1983. 47 [UA Bonn] F. Klein’s diploma. 48 See KRAUS 2005 and the same author’s essay in FREUDENSTEIN 2006, pp. 102–04. 49 [Oslo] A letter from Klein to Lie dated August 26, 1872.

2.3 Studies and Doctorate in Bonn

– – –

29

The geometric-intuitional approach that characterized Klein’s work derived from his cooperation with Julius Plücker, who was renowned internationally for his contributions to both physics and mathematics. At the fifty-year anniversary celebration of the university, Klein was awarded a prize for winning a problem-solving contest in theoretical physics. Moreover, he had to determine and develop his own ideas for a dissertation in mathematics, because Plücker had died on May 22, 1868.50 2.3.1 Coursework and Seminar Awards

In the University of Bonn’s matriculation record for the winter semester of 1865/66, Felix Klein’s name is listed second, after that of a medical student. Of the just 355 students attending the university, only fourteen were enrolled for mathematics and the natural sciences; still only sixteen years old, Felix Klein was the youngest among them.51 All but one of his thirteen fellow students came from the surrounding area, the exception being A. Gontschareff, who came from Simbirsk in Russia (the city is known as Ulyanovsk today). Like Klein, eight of his fellow students stayed in Bonn for several semesters. Among these, Bernhard Pontani and Ernst Sagorski are especially noteworthy. They both acquired multiple teaching qualifications and enjoyed careers as teachers. Pontani, in addition, earned a doctoral degree in physics under Adolf Wüllner.52 Like Klein, Sagorski was also honored for winning a student prize, and he acted as an “opponent” in Klein’s doctoral procedure. Later, as a teacher at the Landesschule Pforta near Naumburg, which was then part of Prussia, Sagorski was – like Klein – active in Prussian educational reform.53 Klein was an active member in the Bonn Natural Sciences Seminar and, as of 1867, in the newly established Mathematical Seminar as well. A Seminar was a type of institution at German universities designed to prepare students for their later teaching jobs at secondary schools. Under the guidance of a professor, students gave presentations, held discussions, and solved problems. Seminars for mathematics had been founded following the example of seminars for language teacher candidates; one of the first seminars for mathematics and physics had been created at the University of Königsberg in 1834.54 One can surmise from Klein’s later remarks, however, that the Bonn Natural Sciences Seminar made it relatively easy for students to earn their teaching qualifications: 50 GRAY 2013 (p. 88) erroneously states that Plücker died in 1871. 51 [UA Bonn] Matriculation record AB-07, WS 1864–1872. 52 [BBF] Personnel records. On Wüllner’s treatment of infinitesimal calculus in his Lehrbuch der Experimentalphysik (6th ed., 1907), see KLEIN 2016 [1924], p. 235. 53 [BBF] Personnel records. See also KÖSSLER 2008, vol. 18, p. 18; Ernst Sagorski, Analytischgeometrische Untersuchungen (Naumburg: Heinrich Sieling, 1875). 54 See KÖNIG 1982; BIERMANN 1988, p. 97; OLESKO 1991. – On the structural difference between a Seminar and an Institute, see SCHUBRING 2000b.

30

2 Formative Groups In the Natural Sciences Seminar in Bonn, which had been founded in 1819,55 students alternated in giving presentations about sections from the general textbooks of the time. Five days a week, from four to five o’clock in the afternoon, they held discussions on chemistry, botany, physics, zoology, and minerology. By today’s standard, this was an entirely elementary operation. Within this seminar, the advanced students could take an exam [a colloquium, which meant that they gave a lecture, and a discussion followed]. That was the entire teaching examination for scientific subjects! In this way, however, one could comfortably complete one’s studies in three years.56

In this seminar, Klein gave presentations about various topics in physics, crystal systems in mineralogy, vascular cryptogams and the alternation of generations in botany, butterflies in zoology, and sodium in chemistry.57 He made it known that he was particularly interested in the descriptive natural sciences. Then residing at the Poppelsdorf Palace, which had belonged to the university since 1818 and was surrounded by its own botanical garden, Klein had easy access to such fields: Regarding lectures, I mainly occupied myself – at least after Easter of 1866 – with botany and other descriptive natural sciences, to which end my apartment at the time in the Poppelsdorf Palace, which also housed the university’s natural-scientific collections, offered me favorable opportunities. I attended only a few lectures in mathematics, because the highly elementary lectures held by Lipschitz, whose scientific significance I only learned to appreciate much later, and the underdeveloped mathematical-physical course offerings in Bonn did not captivate me in any way.58

In his first semester, Klein studied a broad range of subjects and attended typical introductory courses (see Table 3): experimental physics (taught by Julius Plücker), analytic geometry (taught by Rudolf Lipschitz), and differential calculus (taught by the Privatdozent Franz Gehring, whom he considered “confused”).59 Klein took a logic course taught by the theology professor Joseph Neuhäuser, whose expertise was mainly ancient philosophy (Anaximander, Aristotle). For astronomy, which had interested him since his Gymnasium years, he enrolled in a course taught by Friedrich Wilhelm Argelander, who had been the director of the observatory in Bonn since 1837 and had also participated as an advisor in the construction of the Düsseldorf-Bilk observatory. Later, when he was able to determine his own course content, Klein would integrate the theme of Argelander’s lecture – the “method of least squares” developed by Gauss and Legendre – into his curriculum for applied mathematics (see Section 8.1.2). During this semester, Klein also attended a lecture on Goethe by the art historian Anton Springer.60 55 SCHUBRING 1989b states correctly that this seminar was founded in 1825. 56 [Hecke] A lecture by Klein (1910/11), pp. 246–47. 57 See SCHUBRING 1989a, p. 210. In botany, cryptogams are plants that reproduce with spores and not with seeds (algae, moss, ferns, lichens, fungi, etc.). 58 KLEIN 1923a, p. 14. 59 Quoted from LOREY 1916, p. 166. Gehring, who earned his doctorate in Berlin in 1860, became a Privatdozent in Bonn 1862, and worked in the same capacity in Vienna beginning in 1873 ([UA Bonn] PF-Pa 158). Later, he became a music publicist. 60 This would later help Klein to develop a relationship with the Goethe scholar Friedrich Zarncke at the University of Leipzig (see Section 5.2).

2.3 Studies and Doctorate in Bonn

31

There were two categories of instruction: private lectures had to be paid for, while public courses were free.61 Table 3: Courses attended by Felix Klein at the University of Bonn Private Lectures

Public Courses

Winter Semester 1865–66 1) 2) 3) 4)

Prof. Dr. Joseph Neuhäuser: Logic Prof. Dr. Rudolf Lipschitz: Analytic Geometry Prof. Dr. J. Plücker: Experimental Physics PD Dr. Franz Gehring: Differential Calculus

1) 2) 3)

Lipschitz: Mathematical Exercises Prof. Dr. Anton Springer: Goethe in Bonn Prof. Dr. Friedrich Wilhelm Argelander: Method of Least Squares

Summer Semester 1866 1) 2) 3) 4) 5)

Plücker: Electricity Theory Prof. Dr. Hans Landolt: Inorganic Chemistry Prof. Dr. Johannes von Hanstein: General Botany Lipschitz: Number Theory Plücker: Mechanical Practicum

1)

Plücker: Mathematical Exercises

1)

Landolt: Selected Chapters (Chemistry) Hanstein: On Reproduction Lipschitz: Mathematical Exercises Plücker: Mathematical Exercises Prof. Dr. Franz Hermann Troschel: Natural Sciences Seminar

Winter Semester 1866–67 1) 2)

Lipschitz: Analytical Mechanics Landolt: Organic Experimental Chemistry

2) 3) 4) 5) Summer Semester 1867 1) 2) 3)

Lipschitz: Newton’s Law Troschel: Zoology Prof. Dr. Johann Jakob Nöggerath: Mineralogy

1) 2) 3)

Lipschitz/Plücker: Mathematical Seminar Hanstein: Natural Systems Troschel: Natural Sciences Seminar

Winter Semester 1867–68 1)

Landolt: Chemical Practicum

1) 2)

61 [UA Bonn] Felix Klein’s registration record.

Lipschitz/Plücker: Mathematical Seminar PD Dr. Eduard Ketteler: Interference Phenomena

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Summer Semester 1868 1)

Lipschitz: Differential Equations

1) 2)

Gehring: Calculus of Variations Lipschitz/Plücker: Mathematical Seminar

1)

Prof. Dr. Gustav Radicke:62 Analytical Statics PD Dr. Ernst Pfitzer: On Parasitic Fungi

Winter Semester 1868–69 1)

Lipschitz: Number Theory

2)

In the vita appended to his dissertation, Klein also refers to a lecture by Karl Gustav Bischof,63 a cofounder of geochemistry in Germany. This lecture could have spurred on his interest in the earth sciences, which he would later promote at the University of Göttingen. The breadth of Klein’s course of study led to him acquiring qualifications to teach a range of subjects. That said, even though Klein participated in courses taught by the chemist Hans Landolt, the mineralogist and geologist Johann Jacob Nöggerath, the zoologist Franz Hermann Troschel, and the botanist Johannes von Hanstein, it was Julius Plücker who influenced him most profoundly. Plücker had studied in Bonn, Berlin, Heidelberg, and Paris. In 1825, he completed his Habilitation in Bonn, where he became a professor extraordinarius three years later. In 1835, after holding intermediary positions in Berlin and Halle, he was appointed full professor of mathematics in Bonn, and in 1836 he was additionally made a full professor of physics there. Regarding Klein’s presentations in his physics course in 1865-66, Plücker offered the following assessment: “Through his talent, knowledge, and diligence, Klein stands out above all other participants in the seminar.” He added: “Klein displayed an eminent talent for mathematical physics as well as for experimental physics.”64 The course offerings in mathematics and physics, however, were somewhat less developed than those in the descriptive natural sciences, and this is because of Plücker’s dual appointment. At first there was no Mathematical Seminar, for Julius Plücker had opposed its creation when it was proposed in 1864 by the newly hired Rudolf Lipschitz.65 A former student of Dirichlet in Berlin and an expert in mathematical analysis, Lipschitz was hired as a full professor of “pure” mathematics in Bonn against Plücker’s wishes. Plücker would have preferred for the position to be offered to Alfred Clebsch, whose approach to mathematics was 62 Gustav Radicke was a professor extraordinarius (see ADB 27 [1888], p. 135). 63 This vita is included in the copy of his dissertation kept at the University Library in Bonn. Klein also mentioned Bischof in the vita included in his Habilitation thesis. See TOBIES 1999a, p. 85. 64 Quoted from SCHUBRING 1989a, p. 210. 65 Regarding the early history of the mathematics department in Bonn, see ERNST 1999, pp. 33– 37; and SCHUBRING 1985.

2.3 Studies and Doctorate in Bonn

33

closer to his own. In 1864, Plücker had met Clebsch in Gießen, where the 39th Annual Meeting of the Gesellschaft deutscher Naturforscher und Ärzte [Society of German Naturalist and Physicians] (GDNÄ) took place. Plücker had given a lecture on his current research topic “Über eine neue Auffassung des Raumes vermöge der Geraden als Raumelement” [On a New Conception of Space by Virtue of the Straight Line as a Spatial Element].66 There he discussed his results with Clebsch, who would later relate these methods in his own lectures. Plücker’s animosity toward Lipschitz rubbed off on Klein, so much so that he would only later come to recognize Lipschitz’s importance to mathematics. When, in the fall of 1866, the Mathematical Seminar was finally formed, Plücker and Lipschitz each directed his own exercises, in which students had to give talks and solve problems. Klein attended the seminar exercises of both professors. Studying under Plücker, Klein was able to familiarize himself with his teacher’s latest research. Lipschitz, in contrast, had to lower the level of his exercises on account of the insufficient educational background of his students. The following quotation comes from a report by the university’s Kurator dated December 16, 1867: In the winter semester of 1866/67, Professor [Julius] Plücker chose the principle of reciprocity as the subject of his exercises, which required that the students had previous familiarity with the elements of analytic geometry. Here, the main consideration was the parallelism between geometric and analytic approaches. During the summer semester of 1867, Plücker divided his exercises into two weekly courses. In one course (for the more advanced students), he treated one of the more significant chapters of recent analytic geometry: “The significance of the number of constants in the equations of algebraic curves and surfaces.” In the second course, he discussed the analytic representation of the straight line in space in the two instances where it is determined by points that lie on it or by planes that intersect in it. He considered this discussion a basic introduction to a new geometry of space, whereby a straight line is regarded as a spatial element. Professor [Rudolf] Lipschitz created two courses and devoted one hour per week to each course. In the lower-level course, his intention was to develop an elementary foundation of convergence theorems for power series; the knowledge, however, that the majority of participants had brought with them from Gymnasium was so inadequate for this task that there was nothing left to do but go down a level and discuss the binomial theorem with positive exponents, the basics of making calculations with fractional powers and exponentials in a series. Only in the second semester was it possible for him to execute his original plan. In the upper-level class during the first semester, Lipschitz gradually developed the theory of systems of linear differential equations with constant coefficients. This was followed in the second semester by the application of the theory of functional determinants to systems of differential equations. Throughout, effort was made to bring to life the connection between the proposed analytic problems and corresponding problems in mechanics.67

According to Gert Schubring, the Prussian Ministry of Culture allocated 85 Thaler per semester for seminar awards (Seminarprämien) and 50 Thaler per year for a reference library.68 Felix Klein received a seminar award for three semesters. A

66 See TOBIES/VOLKERT 1998, p. 236. 67 Quoted from ERNST 1933, pp. 33–37. 68 See SCHUBRING 1985, p. 149.

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comparison with similar Seminar prizes at the University of Berlin shows that the Ministry requested to see performance evaluations of the students who earned such awards (see Section 2.5.1). On March 9, 1868, a Mathematical and Naturalist Student Union (known as MNV Marsia) was created in Bonn, and Felix Klein is still listed as one of its most prominent members today.69 Unions of this sort already existed in Greifswald, Breslau, Berlin, and Halle.70 At different universities throughout his career, Klein would participate in similar student unions and even instigate their formation himself. One of Klein’s fellow members in Bonn was Friedrich Neesen, with whom Klein maintained a long-lasting friendship (see 2.8.3.2). 2.3.2 Assistantship and a Reward for Winning a Physics Contest When, for the summer semester of 1866, Julius Plücker needed a new assistant for his course on experimental physics, he chose Felix Klein. In the vita appended to his dissertation, Klein wrote: I had the good fortune of developing a closer relationship with one of the most significant representatives of these sciences [mathematics and physics], Professor Plücker, who appointed me his assistant for two years at the Physics Institute in Bonn and who included me in his mathematical research. This good relationship lasted until his death on May 22, 1868.71

Plücker’s significant achievements in physics include his discovery of crystal magnetism (1847) and the discharge spectra of diluted gases under magnetic influence (1857). Plücker recognized that every gas has its own characteristic spectrum, and he observed the first three lines of the hydrogen spectrum. He thus created a basis for modern spectral analysis and, together with the glass blower and instrument maker Heinrich Geißler, he also created the foundations for modern vacuum engineering.72 It is no surprise that the young Klein considered pursuing a career in physics. This idea was strengthened when he won a physics contest on the topic of “ether vibrations,” which Plücker had formulated in 1867 on the occasion of the upcoming celebration of the university’s fiftieth anniversary.73 Although there is no surviving record or evaluation of what Klein did to win this contest, the vita contained in the record of his Habilitation procedure states that “a historical-critical treatise on the question of the direction of vibrations in polarized light” had been

69 See https://de.wikipedia.org/wiki/Arnstädter_Verband (accessed November 8, 2019). 70 In 1868, the Mathematical and Natural-Scientific Student Unions at various universities came together to form a single cooperative union. From 1909 to 1933, this association was known as the Arnstadt Union (Arnstädter Verband). 71 Quoted from the vita contained in the copy of Klein’s dissertation held by the University Library in Bonn (see TOBIES 2019b, p. 26). 72 See DÖRFEL 2014, and DÖRFEL/MÜLLER 2006. 73 [UA Bonn] PF 6401: Record of contests.

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asked for.74 This is in line with the fact that Klein had attended Eduard Ketteler’s lecture on interference phenomena during the winter semester of 1867/68.75 The award ceremony for student prizes took place on the last day of the anniversary festivities, which ran from August 1 to August 4, 1868. Plücker did not live to see the ceremony, but the list of attendees was impressive. The nineteenyear-old Klein met people there who would turn out to be important to him later in his life. Universities and Academies had sent deputies to be in attendance, including the famous Karl Weierstrass, who gave an official address on behalf of the Berlin Academy of Science (the speech was co-signed by the other two renowned mathematicians in Berlin, Ernst Eduard Kummer and Leopold Kronecker).76 Klein would soon be on his way to Berlin, for the winter semester of 1869/70, to continue his studies under this triumvirate of mathematicians (see 2.5). The delegate from the University of Erlangen was the historian Karl Hegel. Here, Hegel was able to see his future son-in-law in the limelight (see 3.6.2). The event on August 4, 1868 began at eleven o’clock in Bonn’s Protestant Church and opened with a performance of Carl Maria von Weber’s Jubel-Ouvertüre (Op. 59).77 After a speech delivered in Latin by Friedrich Heimsoeth – a classical philologist, musicologist, and art historian – nine students were honored: one from the Catholic Theological Faculty, two from the Protestant Theological Faculty, one from the Medical Faculty, and five from the Philosophical Faculty, the latter bestowing one prize for philosophy, two for chemistry, one for theoretical physics, and one for philology. The aforementioned Ernst Sagorski was honored for his work in chemistry; Felix Klein was honored next for his achievement in theoretical physics. The last student to receive an award was Otto Lüders, who had won the socalled “Welcker Contest.” Since 1819, the classical philologist and archaeologist Friedrich Gottlieb Welcker (ob. 12/17/1868) had done much to promote a liberal atmosphere in Bonn, and the celebrated student Lüders followed in his footsteps. Through the close friendship between Lüders and Ulrich von Wilamowitz-Moellendorff, who since 1867 had likewise been a student in Bonn, a line can be drawn to Felix Klein. Based on his studies of Plato, Wilamowitz-Moellendorff had also taken an interest in classical mathematical texts.78 Later, he would be one of Klein’s close allies in his effort to reorganize the Göttingen Academy of Sciences.

74 See TOBIES 1999a, p. 85. 75 See Table 3. – Ketteler worked as a Privatdozent in Bonn as of 1865, and he became an influential figure in the field of optical research. One of his students, for example, was Carl Pulfrich, who went on to become an important industrial researcher at the famous Zeiss optical company in Jena, which was directed by Ernst Abbe. Later, Klein would hold both Abbe and the Zeiss company in high esteem (see Section 8.1.1). 76 See SYBEL 1868. 77 See ibid., p. 112. 78 For instance, Eva Sachs, whose doctoral dissertation was directed by Wilamowitz (De Theaeteto Atheniensi Mathematico, 1914), later published the article “Die fünf platonischen Körper:  

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Directly following the prize ceremony for students, honorary doctorates were awarded. The selection of honorees is indicative of the liberal atmosphere mentioned above and reflects an unusually emancipated way of thinking on the part of the decision makers. One of the honorary doctors was Charles Darwin, whose magnum opus On the Origin of Species had first been published in 1859 and translated into German as early as 1860. Darwin’s highly controversial theory of evolution was received much more warmly in Germany than in other countries; already in the early 1860s, it was firmly supported by Ernst Haeckel, a professor in Jena.79 From early on, Klein was open to new natural-scientific theories, and thus it is no surprise that biologists would later turn to him for support (see 8.3.4). Honorary doctorates were also awarded to the optician and microscope designer Eduard Hartnack and to the aforementioned instrument maker Heinrich Geißler, who made unique scientific contributions to the early stages of gas discharge research. Klein’s later efforts to establish a technical college for precision mechanics in Göttingen (see Section 8.1.1) may have had its roots here. Other honorees included the chemists August Wilhelm Hofmann and August Kekulé. In 1867, Kekulé had come from Gent to take a position in Bonn, which then had the largest chemistry institute in the world (located next to the Poppelsdorf Palace, where Klein resided as a student). In addition, honorary doctorates were presented to, among others, the liberal British philosopher and economist John Stuart Mill, who was a social reformer and promoter of women’s emancipation; the French chemist and microbiologist Louis Pasteur; the cartographer August Heinrich Petermann; and the botanists Nathanael Pringsheim and Julius Sachs, who came from Jewish families. If we consider these honorees through the eyes of the young Felix Klein, we can imagine the opinions, ideas, and plans that they might have sparked: inspiration to respect academic fields beyond his own and to take an interest in emerging disciplines and theories, and motivation to value talented researchers regardless of their nationality, religion, or gender. 2.3.3 Assisting Julius Plücker’s Research in Geometry Even though Klein served as an assistant for Plücker’s course on experimental physics and won the prize in physics discussed above, the goal of becoming a physicist fell into the background. Julius Plücker had returned his attention to mathematical research, and he included his student in this work. Klein wrote:   Zur Geschichte der Mathematik und der Elementarlehre Platons und der Pythagoreer” (1917) in Wilamowitz and Kiessling’s journal Philologische Untersuchungen. 79 See HOßFELD/OLSSON 2009. Ernst Haeckel spoke about Darwin’s theory at several Annual Meetings of the GDNÄ, first in 1863. On the meeting in 1877, where Klein was involved in its organization, see Section 4.3.3.

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In addition, I was also involved in Plücker’s own scientific work, which, after a period of making discoveries in physics, had taken on a predominantly mathematical orientation. I had to help Plücker develop his investigations, and I enjoyed this sort of “work-study” arrangement, as one would call it today. It was this experience that inspired, more than anything else, my own work in the subject of line geometry. That said, this work also prevented me at the time from acquiring a broader understanding of mathematics and physics.80

In his first geometric phase, Plücker had been influenced in Paris by Jean-Baptiste Biot’s lectures on analytic geometry and, indirectly, by the work of Gaspard Monge.81 Plücker strove to reconceptualize analytic geometry, and he published two volumes under the title Analytisch-geometrische Entwicklungen [AnalyticGeometric Developments] (1828, 1831). Like Hermann Grassmann and August Ferdinand Möbius, he contributed to the rise of analytic methods over the previously dominant synthetic methods in the field. His conflict with Jakob Steiner, a Berlin-based proponent of synthetic geometry, seemed inevitable.82 In Plücker’s second mathematical phase, which took place while Klein was his student, his focus was on line geometry, which he approached with both synthetic and analytic methods. Synthetic methods helped him to gain insights into structures.83 But an accepted proof necessarily had to be analytic. In Plücker’s work of this sort, Mechthild Plump has detected points of departure for Klein’s much-discussed paper on the “arithmetizing of mathematics” (see Section 8.3.2).84 As early as 1846, Plücker had proposed the idea of a four-dimensional projective geometry in which lines and their “line coordinates” function as basic entities of three-dimensional space. In the case of a line that is given by an equation, the coordinates can ultimately be established directly by means of the equation in question.85 Plücker was inspired by James Joseph Sylvester, whom he had met in 1863 at the Annual Meeting of the British Association for the Advancement of Science, to pursue this field of research further. In 1864, as noted above, he discussed his findings at the Annual Meeting of the GDNÄ in Gießen, and in December he sent a paper to the Royal Society of London86. He also presented

80 KLEIN 1923a, p. 14. 81 On Monge see BARBIN et al. 2019, pp. 3–8; On Plücker’s work and how it compared to that of Gergonne, Poncelet, Möbius, Jacobi, Cayley, and others, see the obituary that Clebsch wrote with Klein’s assistance (CLEBSCH 1872); see also ZIEGLER 1985; and PLUMP 2014. 82 On Plücker’s visiting professorship in Berlin and Steiner’s aversion to Plücker, see BIERMANN 1988, pp. 67–68. See also KLEIN 1979 [1926], pp. 87–120. 83 Klein likewise felt about himself that he was able to gain new knowledge by way of geometric intuition (see especially Section 4.2). 84 PLUMP 2014, pp. 155–56. 85 Given a line l in complex projective three-space P3 defined by two points (x1, x2, x3, x4) and (y1, y2, y3, y4), Plücker represented it by six homogeous coordinates pij, 1≤ i < j ≤ 4, defined by pij := xi yj – xj yi. The coordinates (pij) represent a line in this space if and only if they satisfy the Plücker relation p12 p34 + p13 p42 + p14 p23 = 0. The fundamental objects in this line geometry – line complexes – are defined by homogeneous equations in the line coordinates pij. (See ROWE 1992, pp. 48–52; PARSHALL/ROWE 1994, p. 156). 86 Plücker, J. (1865). “On a New Geometry of Space.” Philos. Trans. Royal Society 14: 53-58.

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(wooden) models of his geometric objects in Great Britain and France.87 The results of his research had important implications pertaining to a general theory of algebraic curves and surfaces. Esteemed abroad, Plücker was awarded the Copley Medal in 1866 by the Royal Society of London (its highest honor).88 The Académie des Sciences in Paris made him a corresponding member in 1867. Plücker was also in close contact with Luigi Cremona in Milan, and Felix Klein would benefit from all of these international connections. Klein’s so-called “work-study” with Plücker consisted of learning about how to manufacture mathematical models and helping him with his book on line geometry. As can be read in letters from Plücker to Klein, Plücker wanted his assistant to be back in October of 1867 before the beginning of the semester so that he could support him in the final phase of his book Liniengeometrie (Part I). Klein was asked to read through the proofs of the book, and he also came immediately when the already terminally ill Plücker summoned him on April 25, 1868 (Klein’s birthday).89 Klein’s early concentration on this subject had obvious advantages and disadvantages. The disadvantage was that it limited the scope of his studies and, at least at first, tied him too closely to a single scientific school of thought. The advantage was that it helped him to acquire, as a young man, the ability to conduct his own creative work and that it gave him deep exposure to a relatively new area of mathematics, one with links to scholars in France, Great Britain, and Italy. Klein quickly felt at home in this intellectual community, which was attempting to accomplish something new. He considered it his good fortune that Plücker “introduced him to the methods of the newer geometry and involved him in the geometric work that he was doing at the time.”90 Klein explained the concept of line coordinates as follows: The equation of the straight line u1x1 + u2x2 + u3x3 = 0 is completely symmetric in the coefficients u and the coordinates x. Plücker then interpreted the u as the variable quantities, so that the equation comes to represent the system of lines through the fixed point x. He called the u “line coordinates”; in terms of them the above equation represents the pencil of lines through the point, and thus the point itself. Just as I can interpret the linear relation as the equation of a line in point coordinates, so too I am entitled to see it as the equation of a point in line coordinates. With this idea of the arbitrary “space element” that can be chosen as the starting point for geometry came a full elucidation of the Poncelet-Gergonne principle of duality. Because the equation for the united configuration of point and line (in space, of point and plane) is symmetric in the two elements, it follows that one can interchange these two words in any statement that is based on a simple juxtaposition of the two elements.91

87 88 89 90

See PLUMP 2014, pp. 108–12. In 1912, Felix Klein would become the fourth German mathematician to win this award. Plücker’s letters to Klein are published in PLUMP 2014, pp. 125–26. Quoted from the vita contained in Klein’s Habilitation thesis, which is published in TOBIES 1999a, pp. 84–85. 91 KLEIN 1979 [1926], p. 112. See also LORENAT 2015.

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Klein and others embraced the fundamental concepts developed by Plücker, such as axis and ray coordinates, complexes, congruences, higher ruled surfaces, complex surfaces, etc.92 Klein called the “Plücker formulae” his teacher’s a main achievement. The latter, he explained, “connect the order n of a curve (the degree of its equation in point coordinates) with its class k (the degree of its equation in line coordinates) and its simple (so called ‘necessary’) singularities.”93 Plücker’s projective thinking94 and his approaches to “newer geometry” were highly inspiring to his student Klein, who found an idea for his dissertation in this field. Later, in 1876, Klein would make use of Plücker’s formulas to develop one of his own in order to determine the behavior of geometric forms with respect to their reality.95 Plücker’s work was not only the point of departure for Klein’s intuitive (anschaulich) geometric thinking, which Hilbert would praise so strongly in 1909.96 It is also the origin of why Klein always considered “working with spatial ideas as such, that is, the geometric imagination” to be his strongest tool for discovering new results. He underscored this point as late as 1922.97 He discussed this aspect of being able to imagine something spatial with the help of geometric models or in one’s own mind in Note III of his Erlangen Program from October of 1872. Also, Plücker’s use of both synthetic and analytic methods is reflected in Klein’s early insight about the relation between these methods, which, unlike many geometers at the time, he did not regard as oppositional. This would be the point in Note I of his Erlangen Program (see Section 3.1.1). When Plücker died in May of 1868, only the first part (226 pp.) of his book Neue Geometrie des Raumes, gegründet auf die Betrachtung der geraden Line als Raumelement [A New Geometry of Space, Based on the Consideration of the Straight Line as a Spatial Element] was ready for print. Plücker’s son Albert sent this first part to the Prussian Minister Heinrich von Mühler.98 The enclosed letter mentioned that Julius Plücker had intended to present this part at the University of Bonn’s anniversary celebration and that the second part would be edited by “Mr. Klein, whom my father counted among the most talented of his many young students.”99 In his preface to Plücker’s first part, Alfred Clebsch noted that there was only a little completely elaborated for the promised second part, […] but fortunately, Plücker’s former assistant in his physics lectures, Mr. Klein, who already participated in many ways in the editing of the book and has internalized its spirit and

92 See KLEIN 1976 [1926], pp. 108–14. 93 Ibid., p. 112. 94 See Helmut Karzel, “Wandlungen des Begriffs der projektiven Geometrie (1959),” in KARZEL/SÖRENSEN 1984, pp. 13–19. 95 Felix Klein, “Eine neue Relation zwischen den Singularitäten einer algebraischen Kurve,” Mathematische Annalen 10 (1876), 199–209; reprinted in GMA II. For an explanation of this work, see KLEIN 1979 [1926], pp. 113–14. 96 See TOBIES 2019b, p. 514, translated in ROWE 2018a, p. 198. 97 KLEIN 1922 (GMA II), p. 5. 98 Heinrich von Mühler was the Prussian Minister of Culture from 1862 to 1872. 99 Quoted from ERNST 1933, p. 40.

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2 Formative Groups method, has been put in the position, through an oral agreement with the deceased, to fill in the gaps of the manuscript according to Plücker’s intended meaning. […] These continuations will have as their object the further development of his theory of second-degree complexes in a way that, in accordance with Plücker’s ideas, is analogous to the theory of second-degree surfaces. Plücker’s methods will thus be maintained as faithfully as possible.100

Plücker’s family had commissioned the young Felix Klein to edit these unpublished materials.101 Klein spent a few days off at the home of his teacher’s widow, Antonie Plücker (née Altstätter), whom he also consulted in order to facilitate Clebsch’s task of writing Julius Plücker’s obituary.102 Plücker’s book on line geometry included applications in optics and mechanics. For this reason, Klein attended a course on analytic statics during his last semester in Bonn (see Table 3). Later, Klein would encourage his doctoral student Ferdinand Lindemann to develop a mechanics of rigid bodies based on line geometry. As a professor, Klein would also often teach analytical mechanics, first in Erlangen during the winter semester of 1873/74.103 In May of 1869, he completed the second part of Plücker’s Liniengeometrie (378 pp.). In his preface, Klein referred to his own contributions to the book and to a pertinent study by the Italian mathematician Giuseppe Battaglini, which would require a separate exposition. It was Battaglini’s work in particular that inspired Klein’s dissertation, which he had already defended on December 12, 1868. 2.3.4 Doctoral Procedure The idea for his dissertation – “Über die Transformation der allgemeinen Gleichung zweiten Grades zwischen Linien-Coordinaten auf eine canonische Form” [On the Transformation of the General Second-Degree Equation in Line Coordinates Into a Canonical Form]104 – came to Klein during the summer of 1868 while he was working on Plücker’s book. Through Clebsch, he learned about Battaglini’s work,105 mentioned above, and about Jacob Lüroth’s Habilitation thesis, which applied Plücker’s methods.106 To provide himself with a broader international orientation, Klein studied the textbooks by the Irishman George Salmon on 100 Clebsch’s preface (dated June 8, 1868), in PLÜCKER 1868, pp. III-IV. 101 Klein was also authorized by Plücker’s family to communicate on their behalf with the Leipzig-based publishing house B.G. Teubner; see ACKERMANN/WEIß 2016, p. 31. 102 [Canada] A letter from Felix Klein to Antonie Plücker dated November, 10, 1871. I am indebted to Eisso Atzema for bringing this reference to my attention. 103 For an edition of these lectures, see KLEIN 1991. 104 Originally published in Bonn by the C. Georgi press (1868); revised in Math. Ann. 23 (1884), pp. 539–78, reprinted in KLEIN 1921 (GMA I), pp. 5–48. English trans. by D.H. Delphenich: https://neo-classical-physics.info/uploads/3/0/6/5/3065888/klein_-_canonical_forms.pdf. 105 Giuseppe Battaglini, “Intorno ai sistemi di rette di primo grado” and “Intorno ai sistemi di rette di secondo grado,” Giornale di Matematiche 6 (1868 [2nd ed.]), pp. 24–36, 239–58 (the first edition was published in 1866). 106 Jacob Lüroth, “Zur Theorie der windschiefen Flächen,” Crelle’s Journal 67 (1867), 130–52.

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projective and algebraic geometry, which the mathematician Wilhelm Fiedler had translated into German. In this regard, Klein explained: It was not entirely easy for me to move from the more elementary methods of Plücker’s representation to the strict method of projective coordinates, as it was used by Battaglini. Studying the textbooks by Salmon-Fiedler and several original papers helped me to get over this difficulty. I soon noticed that the canonical form of second-degree complexes at the basis of Battaglini’s work could not be the general form. Thus I had the topic that I hoped I might be able to form into a dissertation, namely the development of a truly general canonical form.107

In 1866, Battaglini had treated the theory of forms defined by one, two, or three algebraic equations of the first or second degree with more modern methods than Plücker’s. Klein recognized that the general linear transformation of line coordinates was still lacking and that it is necessary for solving further problems. This was the point of departure for the first argument in his dissertation (see below). Klein spent the September of 1868 at his parents’ house in Düsseldorf (Bahnstraße 15) and worked out the initial ideas of his dissertation. He showed his preliminary work to Rudolf Lipschitz in Bonn, who was assigned to supervise his doctoral procedure. Lipschitz recommended that he should examine not only the simplest case but possibly all special cases as well. With this in mind, he gave Klein a recently published work by Weierstrass, in which the latter presented his theory of elementary divisors for a given variable n.108 Klein used Weierstrass’s theory as a tool for classifying quadratic line complexes. His dissertation opens with the following definition of line complexes: “A line complex of degree n encompasses a triply-infinite number of straight lines that are distributed in space in such a manner that those straight lines which go through a fixed point form a cone of order n, or – put another way − will envelope a curve of class n.”109 Later, the Italian mathematician Gino Fano would write: “One noteworthy application of this theory [the theory of elementary divisors] is the systematic classification of second-degree line complexes, which was begun by F. Klein in his dissertation and completed by A. Weiler and C. Segre.”110 In addition to the algebraic aspect of his study, Klein was also keenly interested in the geometric problem that had originally inspired Plücker. Thus, a few years later, Klein had a discussion with Sophus Lie about the question of how to find all second-degree complexes and how to visualize the various types of quadratic line complexes within a given scheme of classification. Klein’s doctoral stu107 KLEIN 1921 (GMA I), p. 3. – See SALMON 1862. 108 Karl Weierstrass, “Zur Theorie der quadratischen und bilinearen Formen,” Monatshefte der Akademie der Wissenschaften zu Berlin (May, 1868), pp. 310–38. – On the history of this area, see also HAWKINS 1977, and BRECHENMACHER 2016a. 109 KLEIN 1921 (GMA I), p. 11. On Kleinian line coordinates, see ROWE 1992c; Konrad Zindler, “Algebraische Liniengeometrie” (1921), in ENCYKLOPÄDIE, vol. 3 (C 8), p. 1112; Emil Müller, “Die verschiedenen Koordinatensysteme” (1910), in ENCYKLOPÄDIE, vol. 3 (1.1, B 7), pp. 732–35; and VOSS 1919, p. 281. 110 Gino Fano, “Kontinuierliche geometrische Gruppen: Die Gruppentheorie als geometrisches Einteilungsprinzip” (1907), in ENCYKLOPÄDIE, vol. 3.1.1, pp. 289–388, at 384–85.

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dent Adolf Weiler would later deal with this very issue (1873).111 The Italian mathematician Corrado Segre translated this theory into the language of multidimensional geometry, while Rudolf Sturm applied it within the framework of synthetic geometry. Segre completed his doctorate in 1883 in Turin, for which he adopted Klein’s motto – “Line geometry is like the geometry of an M 4( 2 ) in R5” – and further developed the topic.112 After the appearance of Segre’s study, Klein published the aforementioned revised version of his dissertation in volume 23 of Mathematische Annalen (1884). He referred to Segre’s work, and he also accepted two of Segre’s articles (one coauthored by Gino Loria) for publication in the same volume of Annalen.113 Here it should be noted that some of Klein’s studies contain inaccuracies, insufficient proofs, or even errors. His way of dealing with this was to acknowledge the findings of others, correct and supplement the work in question, and republish it, always with references to the scholars who had achieved new results. In the revised version of his dissertation published in the first volume of his collected works (1921), Klein is sure to acknowledge, once more, Segre’s approach from 1884.114 He also refers there to Konrad Zindler’s overview of line geometry published in the ENCYKLOPÄDIE (1921) – Zindler had previously published a textbook on the subject – as well as to studies by Ernst Steinitz, who had written a thorough analysis of line geometry and the configurations investigated by Klein.115 Klein had dedicated his dissertation “to his unforgettable teacher Julius Plücker, in grateful memory.” Lipschitz evaluated Klein’s performance in the oral doctoral examination as summa cum laude.116 After all the internal examination proceedings had been completed, a public disputation took place on December 12, 1868 (this was still part of the process at the time). It involved formulating theses that did not necessarily have to be related to one’s dissertation. In addition, every doctoral candidate had to hold a debate with three opponents, who had to discuss the candidate’s theses in a public forum. One of the opponents was the physicist Emil Budde, who was seven years older than Klein and already held a doctoral degree. Like Klein, Budde had attended the Gymnasium in Düsseldorf, had studied mathematics and physics in Bonn, and had worked as Plücker’s assistant. He went on to become a significant internationally-oriented and mathematically-oriented (industrial) physicist. 111 For further discussion of Weiler’s work and how it relates to Klein’s and Lie’s approaches to line geometry, see ROWE 2019a, pp. 193–94. 112 See TERRACINI 1926, pp. 211–15; LUCIANO/ROERO 2012; and CASNATI et al. 2016. 113 Corrado Segre and Gino Loria, “Sur les différentes espèces de complexes du 2e degré des droites qui coupent harmoniquement deux surfaces du second ordre,” Math. Ann. 23 (1884), pp. 213–34; and Corrado Segre, “Note sur les complexes quadratiques dont la surface singulière est une surface du 2e degré double,” Math. Ann. 23 (1884), pp. 235–43. 114 KLEIN 1921 (GMA I), p. 4. 115 See ROWE 1989a, pp. 218–24. 116 [UA Bonn] Prom.-Album, p. 84, no. 526. There is no surviving copy of Lipschitz’s report.

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Figure 5: Felix Klein’s Doctoral Certificate, December 12, 1868 ([UBG] Cod. MS. F. Klein 101).

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The second opponent was Klein’s fellow student Ernst Sagorski, and the third, Johannes Seeger, completed his doctoral degree seven days after Klein in the field of physics (likewise summa cum laude).117 The five theses put forward by Klein demonstrate his wide range of interests, but they are also partly indicative of the limited state of knowledge at the time: 1. The canonical equation form that serves as the basis for Battaglini’s work on complexes of the second degree,

∑

2

x

a x p x = 0, is not the general form.

2. The application of the principles that Cauchy developed in his Méthode générale propre à fournir les équations de condition relatives aux limites des corps […] to differential equations of a given order does not seem to be beyond all doubt.118 3. When explaining the phenomena of light one cannot circumvent the assumption of lightether. 4. Positive and negative electricity should not be regarded as equal opposites. 5. It is desirable that, alongside Euclidean methods, newer methods of geometry should be incorporated into the Gymnasium curriculum.119

Regarding the existence of light-ether (Lichtäther) mentioned in his third thesis, Klein still held on to this belief as late as 1904,120 but he abandoned it upon recognizing the mathematical basis of Einstein’s special theory of relativity (see Section 8.2.4.4). This is just one example of Klein’s maxim that one should not remain committed to obsolete ideas, as so many experimental physicists (and even the theoretical physicist Max Abraham) did in the case of ether.121 Klein’s interest in mathematical instruction at the secondary-school level (thesis 5) would remain with him his entire life. He also pointed out necessary curriculum changes in an early letter to the French mathematician Gaston Darboux: At the moment, for instance, a lively struggle is taking place here about whether the methods for teaching geometry at a Gymnasium, which have remained static for God knows how long, should not be changed to reflect the progress that geometry has made since Monge’s time, even though it is impossible to dispute such things in a reasonable way […].122

117 Ibid., no. 527 (December 19, 1898). The title of Seeger’s dissertation was “Ueber die Gleichgewichtsvertheilung der statischen Electricität auf drei und vier leitenden Kugeln” [On the Equilibrium Distribution of Static Electricity on Three and Four Conductive Spheres]. 118 Here, Cauchy had studied the change of the behavior of solutions of a differential equation system if a small perturbation term, which assumes considerable values only in a small domain of the independent variable, is added to the right-hand side. He applied his theory to linear systems of a given order and referred to the behavior of physical quantities in the proximity of boundary surfaces. As Cauchy’s preconditions remained vague, Klein could have seen in this a point of critique. – The author would like to thank Hans Fischer, Eichstätt, for important information. 119 KLEIN 1921 (GMA I), p. 49. 120 See KLEIN 1904. 121 See HENTSCHEL 1990. 122 [Paris] 59–60: Klein to Darboux, March 21, 1872 (the original German quotation is published in TOBIES 2019b, p. 37).

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2.4 JOINING ALFRED CLEBSCH’S THOUGHT COMMUNITY When, in 1869, the young, nineteen-year-old Dr. Felix Klein moved to Göttingen to continue his studies, the small city of 15,000 residents had belonged to Prussia for just three years. Ever since a railroad line had been made operational there on July 31, 1854, the city’s train station has served as its gateway. From there, one of the first things that newly arrived visitors see is the Hotel Gebhard, whose owner, recognizing the signs of the times, built his facility in a modern style six years after the train line had opened. Later, as a Privatdozent in Göttingen, Klein would be a regular face at the hotel’s beer hall, and he would also later stay at the Hotel Gebhard when visiting the city from out of town.123 During the first half of 1869, he lived at the home of a widow named Fobbe (Groner-Tor-Straße 25).124 Göttingen was characterized above all by its educational facilities. Alongside the Royal Society of Sciences, which had been established in 1751 (since 1942, it has been known as the Göttingen Academy of Sciences and Humanities), the University of Göttingen (also known as the Georg-August University), which had been founded in 1737 by the provincial government of Hanover and was now under Prussian control, formed the center of the city. Until the end of the 1860s, most of the businesses in Göttingen had some connection to scientific activity. There were seven facilities in the city, for instance, that produced scientific instruments, and together they employed approximately fifty specialists. By the year 1900, these numbers would increase to twelve facilities with 270 employees and apprentices. Established during the Enlightenment, the University of Göttingen had long afforded its professors the freedom to develop their ideas. Such scholars included the experimental physicist and aphorist Georg Christoph Lichtenberg, the zoologist and anthropologist Johann Friedrich Blumenbach, the pioneering organic chemist Friedrich Wöhler, and the multifaceted thinker Carl Friedrich Gauss. Felix Klein described Gauss, a professor of astronomy, as an outstanding and singular phenomenon in German mathematics and its applications.125 Gauss’s scientific breadth would later serve as an example for Klein when, toward the end of the century, he would finally be in a position to expand and restructure the university’s institutions. Gauss’s successors, Peter Gustav Lejeune Dirichlet and Bernhard Riemann, had been able to build Göttingen into an internationally recognized center for mathematical research, but for various reasons they produced only few students of

123 Regarding the Hotel Gebhard, see E. Böhme’s article in FREUDENSTEIN 2016, pp. 106–12. The hotel’s beer hall was established in 1865. Felix Klein would even compose some of his correspondence there; see, for instance [Oslo] A letter from Klein to Lie dated April 6, 1873. 124 See NISSEN 1962, p. 92. 125 See KLEIN 1979 [1926], pp. 7–57.

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their own. Alfred Clebsch, on the contrary, who had succeeded Riemann in October 1868,126 already had a circle of students by the time Klein arrived. Clebsch’s thought community in the area of algebraic geometry can be regarded as the first significant mathematical school in the nineteenth century (see Section 2.4.1).127

Figure 6: Alfred Clebsch (TOBIES/ROWE 1990, p. 10).

Together with the mathematician Carl Neumann, Clebsch founded the journal Mathematische Annalen in 1868. During its early years, this journal, which still exists today, developed into the internationally oriented mouthpiece of the Clebsch school. Later, thanks in large part to Klein’s leadership, it would broaden its scope (see Section 2.4.2). During his time in Göttingen from January to August of 1869, Klein still considered himself a student. As a young doctor, he joined the Mathematical and Natural-Scientific Student Union, which had been founded through Clebsch’s initiative on December 7, 1868. Here, Klein made contacts with people who would remain part of his network for years to come, as the example of the biologist Karl Kraepelin shows (see Section 8.3.4.1). While in Göttingen, Klein continued to attend lectures, though his main focus was on the aforementioned edition of Plücker’s Liniengeometrie. He also completed five additional studies (see 2.4.3).

126 See Göttinger Nachrichten (1875), p. 282. 127 For a discussion of the term “mathematical school,” see TOBIES 2008c, pp. 149–76.

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2.4.1 The Clebsch School Born in Königsberg, Alfred Clebsch benefited from the education offered by his East Prussian home university, particularly from its Mathematisch-physikalisches Seminar, which had been founded in 1834 by Jacobi and the physicist Franz Neumann. This seminar produced numerous mathematicians whom Klein would encounter throughout his career, among them Adolph Mayer, Karl Von der Mühll, Carl Neumann, and Heinrich Weber.128 Clebsch’s approach to algebraic geometry had been influenced by Jacobi’s student Otto Hesse,129 and his approach to mathematical physics had been shaped by Franz Neumann. Another of his influences was Friedrich Julius Richelot, who spoke enthusiastically about Jacobi and about the “still little-known Riemannian intuitions.”130 After earning his doctoral degree in 1854 under Franz Neumann, Clebsch devoted his early studies to mathematical problems in optics, hydrodynamics, and elasticity theory. He completed his Habilitation at the University of Berlin in 1858 and, in the same year, was made a professor of theoretical mechanics at the Polytechnikum in Karlsruhe. In 1863, Clebsch took a new position as a professor of mathematics at the University of Gießen, and in 1868 he came to Göttingen, where Felix Klein joined his circle. Clebsch’s article “Ueber die Anwendung der Abel’schen Functionen in der Geometrie” [On the Application of Abelian Functions in Geometry] is considered the foundational work of algebraic geometry.131 Here, he combined geometry with elliptic and Abelian functions, which yielded, in a new way, theorems that Hesse had formulated about third- and fourth-order curves. Clebsch also used Plücker’s methods and combined the traditions of Jacobi and Jakob Steiner and the work of the Englishmen Cayley and Sylvester and the Irishman Salmon with Riemannian ideas. Clebsch’s use of Abel’s theorem would significantly influence both Klein and Klein’s later students.132 Clebsch created a new type of scientific thought collective. Later, the famous Russian mathematician Igor R. Shafarevich named the following members of this circle: “Gordan, Brill, Lüroth, Zeuthen, and the most famous of all: [Max] Noether and Klein.”133 The latter scholars would take this research program in a wide variety of directions. It will be necessary below to provide a brief profile of this cast of characters, especially as Klein would remain in touch with them for many years. When he himself was offered a professorship before some of the older members of the Clebsch school, he found himself in the position of being able to promote their careers. They thanked him by collaborating with him in many of his undertakings.

128 129 130 131 132 133

See OLESKO 1991; and TILITZKI 2012. See KLEIN 1975; M. Noether, “Otto Hesse,” Zeitschr. Math. Physik 20 (1875), pp. 77–88. According to Heinrich Weber, this was in the year 1854; see KOENIGSBERGER 2004, p. 109. The article was published in Crelle’s Journal 63 (1864), pp. 189–243. See Section 5.5.2.3 of this book; and also PARSHALL/ROWE 1994, p. 157. SHAFAREVICH 1983, p. 140.

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Klein’s first contact with Clebsch’s circle took place on May 31, 1868 (Whitsunday), shortly after Plücker’s death. Clebsch invited him to go on a hike along with Brill, Gordan, Lüroth, Otto Hesse, Christian Wiener, Carl Neumann, and others.134 Klein referred to the people around Clebsch as “Clebsch’s school,” and wrote: “Like Jacobi, Clebsch was one of those blessed teachers who understood how to draw out young talents and make them independent researchers.”135 According to Ferdinand Lindemann – a proud member of the Clebsch school, who would earn a doctorate under Klein and edit Clebsch’s lecture courses on geometry – the first generally recognized member of this school was Olaus Henrici (b. 1840).136 Henrici began his studies in 1859 at the Polytechnikum in Karlsruhe and ultimately found a permanent position in London thanks to the support of Hesse, Sylvester, Cayley, Hirst, and Clifford. Later, Henrici acted as a contact person for Klein and his students when they traveled to Great Britain. He also translated some of Klein’s papers into English (see Section 4.2.3). Henrici’s models and instruments were used at many German universities. It was because of Henrici’s relationship with Klein that the University of Göttingen received the first version of his “new harmonic analyzer” for finding the coefficients of the Fourier series of a function. Klein presented this instrument at the Royal Society of Sciences in Göttingen and used it in his courses.137 The oldest member of Clebsch’s school by age – and, according to Klein, its most significant one – was Paul Gordan (b. 1837).138 Gordan earned his doctorate in Berlin in 1862 and completed his Habilitation one year later under Clebsch’s supervision in Gießen. Gordan and Clebsch coauthored the book Theorie der Abelschen Functionen [Theory of Abelian Functions] (1866), which was based on a geometric-algebraic interpretation of Riemann’s results. Leo Koenigsberger, one of Weierstrass’s students, praised the book, but he mentioned a dispute about the precedence of some results.139 However, Koenigsberger also wrote: “Among the mathematicians in Berlin, it was Weierstrass alone who quickly recognized that his results concerning hyperelliptic functions and his theorems concerning general Abelian functions were superseded by Riemann’s investigations.”140 Gordan took this area of study in an algebraic direction. He simplified the symbolic calculus for the calculation of algebraic invariants that had been developed by Clebsch and Siegfried Aronhold. For this effort, Gordan came to be called the “king of invariant theory,” and he would not be removed from this throne until David Hilbert entered the scene (with Klein’s support; see Section 134 “Wanderung an der Bergstraße,” LOREY 1916, pp. 213–14; KLEIN 1922 (GMA II), p. 3. 135 KLEIN 1979 [1926], p. 278. 136 See Ferdinand Lindemann, “Olaus Henrici,” Jahresbericht DMV 36 (1927) Abt. 1, pp. 157– 62, at 157; and CLEBSCH 1876/1891. 137 See Olaus Henrici, “Ueber einen neuen harmonischen Analysator (Auszug aus einem Briefe an Herrn F. Klein),” Göttinger Nachrichten (February 3, 1894), pp. 30–32. 138 KLEIN 1979 [1926], p. 278. 139 KOENIGSBERGER 2004 [1919], p. 38. 140 Ibid., pp. 28–29.

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6.3.7.3). The theory of Clebsch-Gordan coefficients would earn him fame later. Such coefficients are numbers that appear in the representation theory of Lie groups, and they were later applied in quantum physics. In 1872, when Gordan was running out of career prospects in Gießen, Felix Klein (twelve years his junior) would come to his aid (see Section 3.5). Alexander Brill (b. 1842) studied at the Polytechnikum in Karlsruhe while Clebsch and Christian Wiener (one of Brill’s uncles) were teaching there. Brill completed his studies in Karlsruhe with exams in architecture and in pedagogy, and he earned a doctoral degree under Clebsch in Gießen on July 13, 1864.141 After a brief period of study in Berlin (on Clebsch’s recommendation), Brill returned to Gießen in 1867 to complete his Habilitation. In 1869, he was offered a professorship at the newly established Polytechnikum in Darmstadt, where Klein visited him on several occasions. When Klein was given a position at the Polytechnikum in Munich, he was able to arrange for Brill to be hired as a professor there at the same time (see Chapter 4). Alexander Brill and Max Noether’s coauthored article “Ueber die algebraischen Funktionen und ihre Anwendung in der Geometrie” [On Algebraic Functions and Their Application in Geometry] can be regarded as another significant development of algebraic geometry.142 Twenty years later, Klein invited them to write a thorough report on the history of algebraic functions for the DMV, a work that represents yet another trace of Clebsch’s wide-reaching influence.143 Max Noether (b. 1844) earned his doctoral degree (without having to submit a thesis) in 1868 at the University of Heidelberg under the supervision of Otto Hesse. In 1868, when Hesse took a new position at the Polytechnikum in Munich, Noether came to study under Clebsch. Noether pursued Clebsch’s program of working out the implications of Riemann’s theory of complex functions for algebraic geometry, and his particular interest was the birational geometry of curves.144 Klein’s cooperation with Noether during the beginning of 1869 in Göttingen led to a finding that Noether cited in a study published that year (see 2.4.3). After Noether had moved to Heidelberg in 1870 to complete his Habilitation, he and Klein maintained an intensive correspondence; they also hiked together during the semester break (e.g. a Rhine tour in summer 1871). Noether, who came from a Jewish family, received, thanks to Klein, his first professorial position in 1875 in Erlangen (see Section 3.5). Noether supported Klein as a reviewer for the journal Mathematische Annalen, and he repeatedly fulfilled Klein’s wish of writing obituaries for important contributors to this journal. These biographical articles provided, in Klein’s judgment, “an excellent resource for the study of the

141 See FINSTERWALDER 1936, p. 654. 142 The article appeared in Math. Ann. 7 (1874), pp. 269–310. 143 The report was published in the Jahresbericht DMV 3 (1894), pp. 107–566. For further discussion of Alexander Brill’s life and work, see F. Severi, “A. von Brill zum 80. Geburtstag,” Jahresbericht DMV 31 (1922) Abt. 1, pp. 89–96. 144 See SCHOLZ 1980, Appendix 2.

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whole [mathematical] period.”145 Later, Klein would come to the assistance of Max Noether’s daughter Emmy (see Sections 7.5 and 9.2.2), and he involved Max Noether’s son Fritz in his project concerned with the theory of the spinning top.146 Jakob Lüroth (b. 1844) had likewise earned his doctoral degree under Hesse (1865), and later he would edit Hesse’s collected works. After a short time in Berlin, where he attended lectures by Weierstrass, he came to Gießen to continue his studies under Clebsch, whose research orientation was closer to his own. After completing his Habilitation in Heidelberg (1868) and holding a deputy professorship there, he became a professor at the Polytechnikum in Karlsruhe in 1869. Klein, who had profited from Lüroth’s Habilitation thesis in his dissertation (see 2.3.4), arranged for Lüroth to be his successor at the Technische Hochschule in Munich (see Section 4.4). In 1883, Lüroth took a position at the University of Freiburg im Breisgau, where he would remain.147 When Klein later recommended Lüroth as a potential member of the first board of the DMV (see Section 6.4.4), he was thus considering appointing an old ally from Clebsch’s intellectual circle. According to Igor Shafarevich, the Danish mathematician Hieronymus Georg Zeuthen (b. 1839) was also a member of Clebsch’s circle, though Max Noether did not regard him as an immediate member of the school.148 Zeuthen built upon the work of the French mathematician Michel Chasles to contribute to enumerative methods in geometry, and he worked closely with Clebsch’s group. Klein and Zeuthen would forge a long-lasting relationship; Zeuthen would go on to publish sixteen articles in Mathematische Annalen.149 He would also prove to be a reliable contributor to the ENCYKLOPÄDIE project (he provided an essay on enumerative methods)150 and to Klein’s project on the Kultur der Gegenwart [Culture of the Present] (see Section 8.3.1). While serving as the general secretary of the Royal Danish Academy of Sciences in Copenhagen, Zeuthen made sure that Klein was appointed a foreign member of this society (on March 8, 1892). Hermann Schubert (b. 1848), who maintained an active research agenda as a secondary-school teacher, was likewise closely associated with Zeuthen’s field of study and with Clebsch’s circle. Klein met Schubert for the first time during his semester in Berlin (see 2.5.5), and he later arranged for Schubert to serve as the representative for Gymnasium instructors on the DMV’s board of directors (see 6.4.4). Schubert’s calculus of enumerative geometry was also based on Chasles’s methods.151 In 1900, David Hilbert included among his famous (then) unsolved 145 146 147 148 149 150

KLEIN 1977 [1926], p. 145. KLEIN/SOMMERFELD 1897–1910, vol. IV. See A. Brill and Max Noether, “Jakob Lüroth,” Jahresbericht DMV 20 (1911), pp. 279–99. See Max Noether, “Hieronymus Georg Zeuthen,” Math. Ann. 83 (1921), pp. 1–23. See TOBIES/ROWE 1990, p. 39. This article gave rise to Zeuthen’s Lehrbuch der abzählenden Methoden der Geometrie (Leipzig: B.G. Teubner, 1914). The latter study, however, failed to consider certain contributions by Eduard Study, who had a public dispute with Zeuthen in which Klein was involved. See Section 5.4.1 and HARTWICH 2003, pp. 73–88. 151 Hermann Schubert, Kalkül der abzählenden Geometrie (Leipzig: B.G. Teubner, 1879).

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problems the matter of establishing “a rigorous foundation for Schubert’s enumerative calculus” (see Section 10.1). Schubert published fifteen articles in Mathematische Annalen,152 and he would also contribute to Klein’s ENCYKLOPÄDIE (his article in vol. I concerns the foundations of arithmetic). He was also quick to recognize and promote the mathematical talents of the Hurwitz brothers while they were still schoolboys,153 and he ultimately recommended that Adolf Hurwitz should study under Felix Klein (see Section 4.2.4.2). Hurwitz would become Klein’s most accomplished doctoral student and one of his notable collaborators. Aurel Voss (b. 1845), whose path to becoming an academic was paved by Clebsch, owed his later career primarily to Felix Klein. On March 17, 1869, Voss was awarded a doctoral degree for a thesis entitled Über die Anzahl reeller und imaginärer Wurzeln höherer Gleichungen [On the Number of Real and Imaginary Roots of Higher Equations], which was supervised by Moritz Abraham Stern in Göttingen,154 and then he became a teacher. Still inspired by one of Clebsch’s courses, which he attended along with Klein in 1869, he decided to return to Göttingen to complete a Habilitation.155 When Clebsch suddenly and unexpectedly died, Voss followed the young Professor Klein to Erlangen. Voss remained thankful to Klein throughout the rest of his life for facilitating his Habilitation process, and the two became close friends. As an established professor, Voss would also contribute to the ENCYKLOPÄDIE and to Klein’s project on the Kultur der Gegenwart. About the young Klein, Voss wrote of a “youthful Dozent” with “unusually multifaceted talents, a divinatory feel for science, and a range of original ideas.”156 With these words, Voss was describing the youngest member of Clebsch’s thought collective; though young, Klein was nevertheless the quickest to acquire a complete overview of Clebsch’s work, and he was best suited to represent it systematically and to perpetuate Clebsch’s wide-ranging research program.157 The fact that Clebsch’s influence extended beyond the circle of scholars discussed above is evident from the correspondence between Richard Dedekind and Heinrich Weber. Both would carry out Clebsch’s initiative of editing Riemann’s work.158 Heinrich Weber, who would work as Klein’s colleague in Göttingen for a while (see Chapter 7), developed Riemann’s ideas further in a purely algebraic manner. Klein, who since the late 1870s had also been intensively engaged with Weierstrass’s lectures, remarked (in his historical lectures) that Weierstrass’s theory of Abelian functions would not become widely known until later, but it was “simpler, more systematic, and much more rigorous” than Riemann’s.159 At first, 152 153 154 155 156 157 158 159

See TOBIES/ROWE 1990, p. 39. See OSWALD/STEUDING 2014. This dissertation was published by the Göttingen-based press of E.A. Huth in 1869. [UB Frankfurt] B.I.1, No. 441: A letter from Klein to Lorey dated Febr. 26, 1919. VOSS 1919, p. 286. See CLEBSCH 1874 (an obituary for Clebsch written by his friends and colleagues). See SCHEEL 2014, pp. 43–51. KLEIN 1979 [1926], p. 289. Klein’s intensive study of Weierstrass’s lectures began with Hurwitz; see Sections 4.2.4.2 and 5.5.2.

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however, he pursued Clebsch’s intended geometric-algebraic (Riemann) program, whose participants would take it in a variety of directions. During his first stint in Göttingen, which began in January of 1869, Klein attended lectures by Moritz Abraham Stern (theory of numerical equations), Bernhard Minnigerode (theory of partial differential equations and their application to mathematical physics),160 and Alfred Clebsch. During the winter semester of 1868/69, the latter taught a course (five hours per week) on the analytic geometry of space and gave a public course devoted to exercises in geometry (one hour per week). In the summer of 1869, from April 15 to August 15, Clebsch offered two four-hour weekly courses: one on determinants, elimination, and algebraic forms, the other on the mathematical theory of light. Aurel Voss, who attended these lectures along with Klein, reminisced about them as follows: The beautiful form of his lectures; the pleasure that this incomparable teacher seemed to feel when he presented to his audience the thoughts with which he and his scientific friends A. Cayley, C. Jordan, and L. Cremona were vividly engaged at that very time; the elegance with which he treated, in the first lectures on the geometry of space ever to be held in Göttingen, all of the new resources at hand, from homogeneous coordinates and the principle of duality to the theory of representing algebraic surfaces in conjunction with the Abelian theorem, and not least the new geometry of space by J. Plücker – all of this introduced his students to a whole new world, connected them to the present in a lively fashion, and instigated them to study such scholarly literature.161

Clebsch taught the latest results in his field. These included works not only by Cayley, Jordan, and Cremona but also Plücker’s more recent contributions to line geometry. As Voss reported, the students were surprised when Clebsch cited the young Felix Klein, who was present among them in the lecture hall, as an expert in this area of research: “We were astounded that this young man, whose lovable personality appeared mature and original beyond his years, was referred to in one of Clebsch’s lectures as an authority of this new geometry of space, with which Plücker had so enriched science during the last years of his life.”162 Clebsch’s work and his ability to discover and systematize connections between individual research areas that were previously thought to be heterogeneous would serve as a model for Klein’s own approach. He remarked: “In my view, the most important aspect of Clebsch’s influence was the moral influence he exerted by instilling in us, in addition to a deep interest in science, a confidence in our own powers.”163 Clebsch’s refined lecture style, the organization of his seminar, his way of dealing with people, “communicating his thoughts bounteously and without reservations,” and in sum, his “great program of unification (of people as

160 See the vita appended to Klein’s Habilitation (published in TOBIES 1999a, p. 85); and the lists of lectures to be held at the University of Göttingen, which are printed in Göttinger Nachrichten (1868), no. 14, p. 310; (1869), no. 6, p. 90. 161 VOSS 1919, p. 280. 162 Ibid. 163 KLEIN 1979 [1926], p. 278.

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well)” would serve as an example to Klein.164 Clebsch’s program involved the synthesis of function theory, algebra, and geometry. According to Klein, his classification principles in geometry, his symbolic representation of invariant theory (based on Aronhold’s work), and his term “connex” (introduced as a basic entity in the geometry of surfaces)165 were important steps toward combining newer geometric studies with the theory of first-order differential equations. When Clebsch died suddenly on November 7, 1872, his students and friends got together to discuss how his obituary should be written. As early as November 15, Klein wrote the following in a letter to Max Noether: We have decided to write a more comprehensive scientific biography, much like the one that Clebsch himself composed about Plücker. Thus we have to turn to you and others for help. We have identified six periods in Clebsch’s scientific activity: Mathematical physics (Neumann) Partial differential equations and second variation (Mayer) Bordered determinants – the beginnings of newer algebra (Lüroth) Abelian functions (Brill) Newer algebra (Gordan) Surface mapping (Noether).166

The Leipzig-based professors Carl Neumann and Adolph Mayer, both educated in Königsberg, were members of the editorial board of the journal Mathematische Annalen, for which the obituary was being prepared. In the end, Carl Neumann handed over the section on mathematical physics to his colleague Karl Von der Mühll, who had likewise studied in Königsberg. Klein edited the text into a coherent whole, a process that required many rounds of discussion. This circle of friends and colleagues also created a Clebsch Foundation in order to support his wife and sons (who were still not of working age).167 Intellectually, this group found itself in heavy competition with mathematicians trained or teaching in Berlin. Without their leader Clebsch, they had to implement their new approach against traditional, “rigorous” mathematics. This involved establishing the legitimacy of Riemann’s way of thinking. Like Leo Koenigsberger (see Section 1.2, p. 9), we could regard this as a new style of thinking in mathematics. 2.4.2 The Journal Mathematische Annalen In May of 1868, while on their aforementioned hike along the Bergstraße, Alfred Clebsch and Carl Neumann had decided to create a new mathematical journal to compete with the existing Journal für die reine und angewandte Mathematik.168 Later that same year, Neumann was offered a professorship at the University of 164 165 166 167 168

[UBG] Cod. Ms. F. Klein (a manuscript “25 Jahre moderner Mathematik”, February 2, 1893). See Section 2.8.3.1. [UBG] Cod. MS. F. Klein 12, No. 554: Klein to Noether, November 15, 1872. [UBG] Cod. MS. F. Klein 7L, pp. 1–72 (Clebsch Foundation). Briefly Crelle’s Journal called, after its founder, see Section 2.5.

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Leipzig. While there, he got in touch with the Leipzig-based publishing house B.G. Teubner. In a letter dated June 10, 1868, he proposed that the press should establish Mathematische Annalen as a new journal and he recommended Clebsch to be the editor. “Through his talent and energy,” Carl Neumann predicted, “the journal will soon surpass all other mathematical journals in Europe with respect to the richness of its content, its elegance, and its dissemination.”169 The first issue came out on December 22, 1868. This journal and the authors associated with it would help B.G. Teubner, which had been founded in 1811, to develop into one of the most important publishers of mathematical scholarship.170 As editors of the journal, Clebsch and Neumann aimed at an international orientation. Articles could be submitted in German, English, French, and Italian. The very first volume contained two articles by Camille Jordan in Paris and one each by Arthur Cayley (Cambridge), Eugenio Beltrami (Bologna), and Zeuthen (Copenhagen) – mathematicians with whom Klein would quickly be associated. Klein’s first contribution to the journal appeared in issue 2 of the second volume (1870). After Clebsch’s death, Carl Neumann sought out new collaborators. Beginning with the sixth volume (1873), he expanded the editorial board to include his Leipzig colleagues Adolph Mayer and Karl Von der Mühll as well as Felix Klein and Paul Gordan as representatives of the Clebsch school (see Fig. 7). As early as 1873, Klein articulated his special sense of responsibility for the journal, noting that “caring for the Annalen, to the extent that it lies within my powers, is my primary concern.”171 Klein contributed his own articles; he encouraged the editorial board to publish dissertations by four of his Erlangen students (see Section 3.1.2), the Habilitation thesis by Aurel Voss, etc. Klein informed Neumann that he had organized an exchange between journals: the Teubner publishing house sent a copy of each new issue of Mathematische Annalen to the editors of the French Bulletin de la Société Mathématique, and in return they received copies from Paris. Neumann was less fond of editorial activity. He knew that he had put the journal in good hands, when, beginning with its tenth volume (1876), he handed over its chief editing responsibilities to Klein and Adolph Mayer and assumed a lesser role on the editorial board. Klein and Mayer expanded the journal’s subscription base. They attracted new authors and proved to be especially perspicacious by publishing Georg Cantor’s studies on set theory in a series of articles that appeared from 1879 to 1884. Leopold Kronecker in Berlin was reluctant to publish Cantor’s paper in Crelle’s Journal, and he ultimately became a decisive opponent of set theory.172 169 170 171 172

For a facsimile of Neumann’s letter, see SCHULZE 1911, pp. 300–01. See ibid.; WEIß 2018; and also Section 5.6 of this book. [UBG] Math. Archiv 165a, p. 3 (a letter from Klein to Neumann dated April 27, 1873). Cantor submitted his first article, “Ein Beitrag zur Mannigfaltigkeitsklehre,” to Kronecker on July 11, 1877; it was published in vol. 84 (1878), pp. 242–58. The series of Cantor’s articles in Math. Ann. began with “Ueber unendliche, lineare Punktmannichfaltigkeiten [sic],” Math. Ann. 15 (1879), pp. 1–7. See also PURKERT/ILGAUDS 1987, and FERREIRÓS 2007.

2.4 Joining Alfred Clebsch’s Thought Community

Figure 7: The title page of volume 6 of Mathematische Annalen (1873).

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When Gottlob Frege submitted his article “Booles rechnende Logik und die Begriffsschrift” [Boole’s Calculating Logic and Conceptual Notation] to the Mathematische Annalen, Klein was less far-sighted than usual. He recommended that the author should submit the text to a philosophical journal.173 When, in addition to Crelle’s Journal, another competing publishing venue appeared on the market – Acta Mathematica, founded in 1882 by the Swedish mathematician Gösta Mittag-Leffler – Klein engaged in a special sort of politics. He offered his cooperation and even submitted some of his own work for publication, which was rejected. Sophus Lie, one of the initiators and board members of the Scandinavian journal, wrote the following from Paris to Mittag-Leffler in October of 1882: “Of course, Klein wishes to be on friendly terms with the journal. […] Klein is far and away a more outstanding mathematician than most German function theorists would admit or even understand.”174 In a letter to Klein sent in December of 1882, Lie referred to Mittag-Leffler as a good person but also as a scheming diplomat.175 Mittag-Leffler had many merits. He had studied under Hermite in Paris and under Weierstrass in Berlin, and he contributed important findings to function theory. He not only arranged for Sofya Kovalevskaya to become a professor in Stockholm in 1884 but also for her to join the editorial board of “his” journal. Mittag-Leffler received support for Acta Mathematica from the Swedish king and, as well, from the governments in Paris and Berlin. Klein was annoyed by such support for the Swedish journal; “his” Annalen existed without requesting funds from a German government. Mittag-Leffler ensured, too, that the most prominent mathematicians from Berlin were decorated with Swedish awards. Despite his support of the journal, Lie would ultimately not publish in Acta Mathematica, preferring instead Klein’s Mathematische Annalen. Yet Klein offered the following advice to his best student, Adolf Hurwitz: From a general perspective, it seems to me desirable for you to publish from time to time in Mittag-Leffler’s journal. If we limit our publications to the Annalen, then we too easily put ourselves in an isolated position (I myself would have been happy to see my work appear in Mittag-Leffler’s and Kronecker’s journals, if it had not been rejected by both editors). That said, I would indeed like you to reserve your main studies for the Annalen.176

Klein and Mayer made continuous efforts to expand the international authorship represented in the Annalen by including more and more contributions from Western and Eastern Europe and also from overseas (see 5.4.2.5, 5.5.3.1, and 6.3.7.1). Later, Klein regularly made changes to update the journal’s editorial board. On the one hand, he did this for personal reasons to reduce his organizational duties (thus his appointment in 1887 of his former doctoral student Walther Dyck), and on the other hand he did so to maintain a high level of content and to allow room for new directions in mathematics. As of volume 42 (1893), for instance, the 173 174 175 176

Klein’s letter to Frege (dated August 14, 1881) is printed in TOBIES/ROWE 1990, p. 37. Quoted from STUBHAUG 2002, p. 291. See ibid., p. 297; and also ROWE 1992b, pp. 610–12. [UBG] Math. Archiv 77, p. 192 (a letter from Klein to Hurwitz dated January 15, 1888).

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editorial board was joined by Heinrich Weber and Max Noether. Sophus Lie declined Klein’s offer to take such a position; he was only willing to serve as one of the journal’s chief editors, but Klein doubted that Lie could fulfill this role. After years of resistance from members of the old Clebsch school, Klein did manage, however, to appoint David Hilbert to the editorial board (as of 1898) and, in 1902, Klein named him one of the main editors of Mathematische Annalen.177 Subsequently, Klein made sure that the journal printed a rich selection of studies devoted to the most modern topics at the time. Thus he insisted, for instance, in a letter to Hilbert on July 31, 1901: “From the latest issues of the Annalen, no one would know how intensively you and your students are working in such a variety of new directions!”178 It was Klein who instigated Hilbert’s first doctoral student Otto Blumenthal, who was by then a professor at the Technische Hochschule in Aachen, to join himself, Hilbert, and Dyck as one of the journal’s main editors as of 1906 (instead of simply serving on the broader editorial board). Klein remarked that Blumenthal “has acquitted himself so well as a member of the editorial board that, in my opinion, we should no longer keep this recognition from him.”179 As of 1909, after the death of A. Mayer, Klein and Hilbert arranged for Hermann Minkowski and Otto Hölder to join the editorial board.180 Following the deaths of Paul Gordan (1912), Karl Von der Mühll (1912), and Heinrich Weber (1913), Klein supported the recommendation of Otto Blumenthal – who, in the meantime, had taken on the lion’s share of editorial duties – to invite Brouwer and Carathéodory to join the editorial board beginning with volume 76 (1915).181 When, during the First World War, a new situation arose at the B.G. Teubner publishing house (see 5.6), it was Klein again who managed the affairs of the journal. Teubner considered its mathematical publications to be a losing venture. In contrast, the Julius Springer Verlag in Berlin was on the upswing. As of 1918, the latter press began to publish a new mathematical journal, which (like Crelle’s Journal and Mathematische Annalen) still exists today: the Mathematische Zeitschrift, then edited by Leon Lichtenstein. The Springer publishing house was building up its mathematics program and was interested in acquiring an additional, more applications-oriented journal. Mathematische Annalen switched from Teubner to Springer. As the journal’s fourth main editor (replacing Dyck), Klein appointed Albert Einstein to encourage and oversee contributions in theoretical physics (see Section 9.2.2). Klein himself remained one of the main editors until volume 92 (1924). The following year, he stepped down to become one of the 177 Klein had wanted to appoint Hilbert to the editorial board as early as 1894. When he failed to do so, Klein arranged for Hilbert to receive complimentary copies of the journal from then on (see FREI 1985, p. 95). See also Section 5.6. 178 Quoted from FREI 1985, p. 129. From 1898 to July of 1901, twenty students completed dissertations on topics suggested by Hilbert (see HILBERT 1935, p. 431). 179 [UBG] Cod. MS. F. Klein 22 L: Klein’s memorandum to the members of the editorial staff of Math. Ann. (Hilbert, Mayer, Gordan, M. Noether, Dyck, and H. Weber) dated April 23, 1906. 180 See FREI 1985, pp. 135–37. Minkowski, however, died on January 12, 1909. 181 [UBG] Cod. MS. F. Klein 8, p. 138 (a letter from Blumenthal to Klein dated June 5, 1914).

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members of the broader editorial board. This happened primarily because of disagreements between Klein and Brouwer about which articles should be accepted for publication.182 When Klein passed away, the editors of Mathematische Annalen emphasized: If, as much this has been humanly possible, the journal Mathematische Annalen has encompassed, in equal measure, all areas of mathematics that are being developed in a lively manner, this is due to Felix Klein. He made sure that different mathematical orientations were represented on the editorial board and that the members of this board could all work with him on an equal footing.183

His first articles to appear in this yournal originated in Göttingen in 1869. 2.4.3 Articles on Line Geometry, 1869 After Klein had cleared his desk of Plücker’s Liniengeometrie on May 25, 1869, he was free to write about his own findings. From June 4 to August 4, 1869, he completed five articles, which Clebsch accepted for publication either in the Göttinger Nachrichten or Mathematische Annalen. Klein referred to the influence of Clebsch’s circle as follows: In comparing my dissertation to the work that I completed soon thereafter, one can detect the stimulating influence that the environment in Göttingen exerted on me. I have chosen this rather vague expression because, in addition to Clebsch, the still small number of specialist students that he had already taken under his wing also influenced me in the most vibrant ways. At the time, Clebsch himself had taught us above all about the rational mapping of the lowest algebraic surfaces to the plane, which he himself discovered, and he had convinced Noether in particular to continue these principal investigations and to expand them to include multidimensional structures.184

Max Noether’s study “Zur Theorie der algebraischen Funktionen mehrerer komplexer Variablen” [On the Theory of Algebraic Functions of Several Complex Variables] is especially instructive regarding our assessment of why Klein chose to enter this particular research area of Clebsch’s circle.185 Noether’s article began by mentioning how Riemann, in his theory of algebraic functions of one complex variable, had established a classification of equations that define these functions. Noether then cited Clebsch’s expansion of this theory to functions of two variables and his introduction of the concept of the genus of a surface.186 Noether himself extended this area of inquiry by developing a technique for mapping the lines

182 [UBG] Klein 8: 143/A, 146 (a letter from Blumenthal to Klein). See also the third chapter of ROWE/FELSCH 2019. 183 Obituary “Felix Klein†,” Math. Ann. 95 (1926), p. 1. See also VAN DALEN 2005, pp. 601–33. 184 KLEIN 1921 (GMA I), p. 51. 185 Max Noether’s article appeared in Göttinger Nachrichten (1869), pp. 298–306. 186 See Alfred Clebsch, “Sur les surfaces algébriques,” Comptes Rendus de l’Académie des Sciences 67 (1868), p. 1238. – Regarding the concept genus, see also François LȆ 2020.

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of a linear complex to points in three-dimensional space. This is where Klein could bring his own special knowledge into play. In the article, Noether wrote: “I am indebted to Dr. Klein for the simplest mapping of a second-degree line complex.”187 Klein later reprinted this single page of Noether’s article in his own collected works,188 and on the same page he noted that Sophus Lie had, from a different point of departure, arrived at this same way of mapping first-degree line complexes. Thus it is clear that there were already points of contact between Klein’s and Lie’s work even before they would soon meet in Berlin (see 2.5.2). In an earlier paper – “Über eine Abbildung der Komplexflächen vierter Ordnung und vierter Klasse” [On a Mapping of Complex Surfaces of the Fourth Order and Fourth Class] (completed June 14, 1869) – Klein had already shown that he would be able to contribute to Clebsch’s program (based on Plücker’s own) for classifying curves and surfaces.189 This was one of several programs that Klein would ultimately keep in mind over many years, promote with contests, and disseminate through his own lectures. Klein considered another work completed in June of 1869 – “Zur Theorie der Linienkomplexe des ersten und zweiten Grades” [On the Theory of First- and Second-Degree Line Complexes] – to be especially important, noting that, after his dissertation, it was really with this study that “I earned my spurs.”190 Here, Klein introduced the concept of a Kummer surface, an idea that Kummer had first described in 1864.191 Clebsch published this article in Göttinger Nachrichten in 1869 (vol. 13, pp. 258–76), and an extended version of it soon appeared in Mathematische Annalen. In it, Klein formulated the following theorem: Those points whose complex cone degenerates into two planes – the so-called singular points – form a Kummer surface of the fourth order and class, with 16 double points and 16 double planes. The same surface is enveloped by the singular planes – i.e., such planes whose complex curve has resolved into a system of two points. In what follows, a surface of this sort will be called a Kummer surface. In relation to the complex, it is called a singularity surface.192

Voss later remarked that Klein, “through a perspicacious combination of fundamental complexes and their positional relations,” recognized that the singularity surface of the general second-degree line complex, which was supposedly discovered by Plücker, had already appeared in Kummer’s work as a self-dual form of the fourth order and class with 16 double planes and 16 double points.193 In this work, Klein also mentioned that determining the singularities of a Kummer surface depends on resolving a sextic equation and several quadratic

187 188 189 190

Max Noether, “Zur Theorie …,” p. 305. KLEIN 1921 (GMA I), p. 89. KLEIN 1921 (GMA I), pp. 87–88 (originally published in Math. Ann. 2 (1870), pp. 371–72). KLEIN 1979 [1926], p. 155. – For the article in question, see KLEIN 1921 (GMA I), pp. 53–80. Trans.: http://neo-classical-physics.info/uploads/3/0/6/5/3065888/klein_-_line_complexes.pdf 191 For a detailed discussion of Kummer’s approach, see ROWE 2019a. 192 Math. Ann. 2 (1870), pp. 198–226, quotation p. 214 and in KLEIN 1921 (GMA I), p. 69. 193 VOSS 1919, p. 281.

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equations. He emphasized Camille Jordan’s discovery from 1868 that the 16thorder equation of the dual elements could be reduced to a sextic equation and several quadratic equations.194 Klein’s achievement was that he confirmed this geometrically. His reference to an algebraically solvable sextic equation, however, formed a significant point of departure for his later studies concerned with the solution of algebraic equations.195 François Lê has rightly underscored that Klein’s use here of so-called “geometric equations” was an important step toward developing his Erlangen Program.196 In 1899, Otto Hölder already devoted a brief section of his ENCYKLOPÄDIE article to these equations.197 In his “spur-earning” work, Klein also determined how a Kummer surface could be constructed (see Fig. 8). Of course, he dutifully referred to Kummer’s own model from 1864.198

Figure 8: A Kummer surface with 16 real nodes (FISCHER 1986, fig. 34).

Among the four models of second-degree line complexes that were produced by Klein’s friend Albert Wenker during the summer and fall of 1869, two are of Kummer surfaces.199 Both Klein and Wenker went about the construction of their models in a way that differed from Plücker’s approach. Plücker, as Klein explained, “constructed his models of complex surfaces only empirically, by assuming appropriate values for the constants present in the equation, from the equations of the horizontal sections or from those of the ‘meridian sections’ crossing through 194 195 196 197

KLEIN 1921 (GMA I), p. 71. See VOSS 1919, p. 208; and Section 4.2.1. LÊ 2015b. Otto Hölder, “Galois’sche Theorie mit Anwendungen,” in ENCYKLOPÄDIE, vol. I.1 (3.c., d.), esp. pp. 518–20. 198 KLEIN 1921 (GMA I), p. 71. See also VOSS 1919, p. 281. 199 See KLEIN 1922 (GMA II), pp. 7–10.

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the z-axis.” Klein’s models, in contrast, were based on a geometric construction, in that he “used the planes that touch the surfaces along entire conic sections.”200 One of the four models designed by Wenker would later play a special role, as Klein explained: “In particular, we spent an extraordinary amount of effort on the model of the general complex surface by looking for a case that did not possess any special symmetry but rather symmetries that could provide us with an overview and also facilitate the construction.”201 Klein presented this model of the general Plückerian complex surface at a conference held by the Berlin Physical Society in March of 1870,202 and he would also use it in his Habilitation lecture in January of 1871 (see Section 2.7.2). Klein was familiar with Kummer’s work on the theory of linear ray systems, of which the surfaces named after him were a part, and he had also made successful use of Weierstrass’s theory of elementary divisors in his dissertation. All of this was reason enough for him to spend a semester in Berlin. Between September 5 and October 15 of 1869, Klein had also prepared a special work for Kummer.203 All in all Klein had achieved some independent results, which was a precondition to be allowed to participate in the Mathematical Seminar of Kummer and Weierstrass. 2.5 BROADENING HIS HORIZONS IN BERLIN Despite the favorable conditions in Göttingen, I felt compelled to expand my horizons, for I wanted to move beyond the confines of scientific “schools.”204

From the fall of 1869 to March 17, 1870, the twenty-year-old Dr. Felix Klein spent a semester studying in Berlin, even though both Plücker and Clebsch had advised him not to do so. On his trip, Klein brought along the prejudices of his teachers, but he wanted to get to know this famous center for mathematical research on his own. He lived at Karlstraße 11 (Reinhardtstraße today), which was near the university and also near the apartments of his closest academic friends there.205 From there it was also convenient to visit professors personally at home. Klein found himself having to participate in numerous “social obligations.”206 It was common at the time, for instance, to introduce oneself personally at the homes of professors. 200 Quoted from KLEIN 1922 (GMA II), p. 3 (see also TOBIES 2019b, p. 50–51). 201 Ibid., p. 3. After Wenker’s death, Klein had the models constructed by a mechanical workshop in Cologne. 202 Klein’s lecture “Über ein Modell einer Plücker’schen Komplexfläche” [On a Model of a Plückerian Complex Surface] is cited in Die Fortschritte der Physik 24 (1868 [1872]), p. vii. 203 On this study, see ROWE 2000, p. 64; ROWE 2013, p. 2; and ROWE 2019a, pp. 182–84. 204 KLEIN 1923a (autobiography), p. 15. 205 AMTLICHES VERZEICHNIS (1869), p. 24. Sophus Lie lived at Kronenstraße 52, Otto Stolz at Schumannstraße 1b (parallel to Karlstraße), and Ludwig Kiepert lived at Dessauer Straße 7. 206 [Oslo] A letter from Klein to Lie dated April 13, 1870.

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The topics covered below will include the events that Klein attended and what the professors in Berlin thought of the young doctor (Section 2.5.1); the new friends and collaborators whom Klein met at the Mathematical Union (Section 2.5.2); and how, for the first time, Klein used his ability – fostered by Clebsch – to discover connections between seemingly distinct areas of research (Section 2.5.3). It should be mentioned in advance that Klein also attended meetings related to physics: the Physical Colloquium, which had been founded by Gustav Magnus in 1843 and given rise to the Berlin Physical Society in 1845.207 The latter was oriented toward interdisciplinary research; it included non-academic members, and it had established the review journal Die Fortschritte der Physik [Advances in Physics],208 which was the prototype for the Jahrbuch über die Fortschritte der Mathematik, the first German review journal for mathematics, founded in 1870 (see Sections 2.6.1 and 2.8.3.4). As early as 1869, Klein joined the Physical Society as a member, and in a meeting on March 11, 1870, he spoke on his model of a general (Plückerian) complex surface [allgemeine Complexfläche], as mentioned above in Section 2.4.3. This presentation may seem less unusual if we consider that Gustav Magnus and Julius Plücker knew each other personally and had exchanged ideas about models and line coordinates as early as the 1840s.209 2.5.1 The Professors in Berlin and Felix Klein When Klein came to Berlin in the fall of 1869, the Friedrich Wilhelm University, which had existed since 1810, was under the leadership of its Rektor Emil du Bois-Reymond (Carl Runge’s father-in-law). This renowned physiologist, who had also studied mathematics (in addition to theology, philosophy, and geology) in Bonn, numbered among the founding members of the Berlin Physical Society and argued, from early on, for the use of graphic methods in medicine. Later, Klein considered him a pioneer of educational reform, a movement that would reach its peak shortly after the year 1900 (see Section 8.3.4). In 1869, the mathematician Ernst Eduard Kummer was the Prorektor (the Rektor’s deputy) of the University of Berlin.210 In 1855, he had been hired as a professor to replace Dirichlet,211 and this has been considered the beginning of the “golden age” of Berlin mathematics.212 Kummer managed to create a position for the forty-one-year-old Karl Weierstrass, who, like him, had spent more than ten years earning a living as a secondary-school teacher. The third member of the tri207 See SCHREIER et al. 1995. 208 The first issue of the Fortschritte der Physik, which appeared in 1847, reviewed the scholarly literature that had appeared in 1845. 209 See PLUMP 2014, pp. 106–07, 114, 302–03. 210 AMTLICHER BERICHT (1869), p. iii. 211 Kummer and Dirichlet were related to each other through their wives (the cousins Ottilie and Rebecca Mendelssohn). – On Dirichlet’s mathematical biography, see MERZBACH 2018. 212 See BIERMANN 1988, pp. 79–152; and KLEIN 1979 [1926], p. 264.

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umvirate was Leopold Kronecker, who had been Kummer’s pupil at the Gymnasium in Liegnitz (today the Polish city of Legnica). After earning his doctorate in Berlin in 1845, Kronecker worked as a businessman. In 1855, he settled in Berlin as an independent scholar; he came from a wealthy Jewish family. The relationship among these three luminaries was not without friction,213 but these professors attracted numerous students to the Prussian capital. Between 1860 and 1870, the number of mathematics students doubled there.214 In the winter semester of 1869/70, seventy-four students were enrolled in the subject at the University of Berlin.215 One of the focal points of mathematics in Berlin was the Journal für die reine und angewandte Mathematik [Journal for Pure and Applied Mathematics], which had been established in 1826 by August Leopold Crelle and is still commonly known as Crelle’s Journal. Under Crelle’s editorship, the journal had a national and international orientation,216 but its subsequent editor, Carl Wilhelm Borchardt, gradually turned it into a mouthpiece for mathematicians in Berlin. In 1869, the editorial board consisted of Karl Heinrich Schellbach,217 Kummer, Kronecker, and Weierstrass. Borchardt had earned his doctoral degree under Jacobi and had completed his Habilitation in 1848; he did not hold a professorship (he had private means). As early as 1861, he retired from teaching, and he wanted to step down from his position as the editor of Crelle’s Journal, too, but then “Clebsch’s” journal appeared on the scene. Borchardt remained in his position largely because of his longstanding feud with Clebsch over the latter’s edition of Jacobi’s Vorlesungen über Dynamik [Lectures on Dynamics] (1866).218 According to Ferdinand Lindemann, it was this same feud that had prompted Clebsch to establish the journal Mathematische Annalen.219 Borchardt continued to hold this grudge even after Clebsch’s death. He wrote about how Mathematische Annalen had been founded as a slight to him and about how the journal had become a showcase for superficiality – a development for which he blamed the deceased Clebsch: “[Clebsch] did not use his talent to conduct profound research but rather to achieve occasional, obvious, and merely superficial accomplishments. Among his students, [Felix] Klein is probably the one who abets this superficiality the most.”220 Borchardt was not as welcoming to modern developments as Clebsch, and he even held strict and meticulous sway over former young scientists from Berlin.

213 On the falling out between Weierstrass and Kronecker, see BIERMANN 1988, pp. 137–39; and Reinhard Bölling’s article in KÖNIG/SPRENKELS 2016, pp. 95–100. 214 BIERMANN 1988, p. 103. 215 These numbers are tallied from the AMTLICHES VERZEICHIS 1869, pp. 1–54. 216 See NEUENSCHWANDER 1984, p. 11; and ECCARIUS 1976. 217 Schellbach’s mathematical-pedagogical seminar, which was associated with the FriedrichWilhelms-Gymnasium in Berlin, was attended by Clebsch, Carl Neumann, and many others. 218 See BIERMANN 1988, p. 81. 219 [Lindemann] Memoirs, p. 48. See also Weierstrass’s preface in the supplement to Jacobi’s Gesammelte Werke (Berlin: Reimer, 1884), pp. 3–4. 220 Quoted from NEUENSCHWANDER 1984, p. 52 (Borchardt to Lipschitz on December 25, 1875).

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Such was the account of Weierstrass’s former doctoral student Leo Koenigsberger.221 Even after Borchardt’s death, when Kronecker and Weierstrass took over the main editing duties of Crelle’s Journal (from 1881 to 1888), the venue suffered from their disagreements, be it about the founding of modern analysis or about Georg Cantor’s work on set theory. Soon, therefore, Weierstrass left Kronecker to do the work alone.222 Klein would publish only one article in Crelle’s Journal; it appeared years after Kronecker’s death, but it concerned Kronecker’s theorem (see Section 5.5.6). Kronecker made significant contributions to number theory, the theory of elliptic functions, and algebra. Since 1861, he had been a “lecturing Academy member” at the University of Berlin, which meant that, as a member of the Berlin Academy of Sciences, he was able to offer lectures. Lipschitz had written Klein a letter of recommendation for Kronecker,223 and Klein attended his lecture course on the theory of quadratic forms and he began to understand something about number theory.224 Later, Klein recalled that Kronecker was able to grasp “many fundamental relations as though by presentiment,”225 but he opposed the latter’s tendency to apply one-sided intellectual norms to all varieties of mathematical work (see Sections 5.4.2.4 and 6.5.1.1). Weierstrass is considered a founder of modern analysis based on logical and arithmetical methods.226 He created a school of function theory,227 but he hardly ever referred to other works. Because Weierstrass seldom published his findings, those interested in his new ideas could only encounter them in his lectures and their transcripts. He designed a notable lecture cycle: introduction to the theory of analytic functions, the theory of elliptic functions, applications of elliptic functions to problems in geometry and mechanics, the theory of Abelian functions, the application of Abelian functions to solving select geometric problems, and – in addition – the calculus of variations.228 In 1869, Weierstrass dominated the scene, as Felix Klein informed: Pretty much the entire interest of the students here has so far concentrated on Weierstrass’s investigations. However, the understanding of these does not keep up with the interest. Weierstrassiana are not judged from an independent point of view, but the students are completely dominated by them. One reason could be that there is something very impressive about him personally, especially in conversation.229

221 See KOENIGSBERGER 2004 [1919], pp. 28–29. 222 See BIERMANN 1988, pp. 137–39. Kronecker acted as the sole editor of Crelle’s beginning with volume 104 (1889), while Weierstrass, Hermann von Helmholtz, Heinrich Eduard Schroeter, and Lazarus Fuchs served on the editorial board (“unter Mitwirkung”). 223 See SCHARLAU 1986, p. 178–79 (Lipschitz to Kronecker on August 7, 1869). 224 [UBG] 12: 527/3 (Klein to Max Noether on December 17, 1869). 225 KLEIN 1979 [1926], p. 264. See also BIERMANN 1988, p. 85. 226 See for example KÖNIG/SPREKELS 2016. 227 See BEHNKE 1966; BÖLLING 1994. 228 See BIERMANN 1988, p. 104; see also BÖLLING 2016. 229 [UBG] 12: 527/3 (Klein to Max Noether on December 17, 1869).

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In 1869, Klein was put off (as Sophus Lie was, too) by the nature of Weierstrass’s lectures on function theory.230 From Plücker and Clebsch, Klein was accustomed to a different type of lecturing style. While in Berlin, Klein made a transcript of Weierstrass’s lecture notes on elliptic functions (see also Section 2.5.2), and later in life he would pay a copyist to transcribe several of Weierstrass’s other lectures (see Section 5.5.2.1). As Klein reported to Max Noether, however, the interests had now become more varied, “because Sophus Lie, a certain Dr. [Otto] Stolz, who is a private lecturer in Vienna and also mainly dealt with geometry, and I came here.”231 Their geometrical interests were favoured by the fact that Ernst Eduard Kummer had chosen “the theory of ray systems, especially those of the third order” as the topic for his winter semester seminar. “Kummer has no special student here, so Lie and I are the only ones who have some knowledge of the ray systems,” Klein continued. Ernst Eduard Kummer had made important contributions to various branches of mathematics, including function theory, number theory, geometry, etc. Kummer was a man of independent judgement; he praised, for instance, Friedrich Prym’s dissertation (1863) because of its use of Riemann’s geometric function theory, despite the fact that Weierstrass was highly critical of it (see 5.5.2.2).232 Before Weierstrass was promoted to full professor in 1864, most mathematicians in Berlin had formally been supervised by Kummer, including Paul du Bois-Reymond in 1859 (a brother of the aforementioned Emil du Bois-Reymond) and Hermann Amandus Schwarz in 1863, who was offered a full professorship at the Polytechnikum in Zurich in 1869 and who, as a scion of the Berlin mathematics, would never develop an especially warm relationship with Felix Klein. During Klein’s semester in Berlin, Kummer was no longer giving lectures about his latest research findings. His well-attended lectures were instead devoted to topics that were already established. By that time, Kummer limited his discussion of more recent research to the mathematical-scientific seminar,233 which he and Weierstrass had started in April of 1861. Understandably, Klein was more interested in participating in this seminar than he was in attending Kummer’s lectures. By rule, participation in the seminar was restricted to twelve people with demonstrable scientific talent.234 Later, Klein explained: “I did not attend any great lectures in Berlin, and therefore I was all the more excited to participate in Kummer and Weierstrass’s mathematical seminar, in which the participants gave presentations on topics of

230 [Oslo] II (a text written by Klein on November 1, 1892), printed in ROWE 1992a, p. 589. 231 [UBG] 12: 527/3 (Klein to Max Noether on December 17, 1869). 232 Ibid., pp. 94, 352. The findings in Prym’s influential dissertation would be used and further developed by Clebsch, Klein, Hilbert, and others. 233 This was in distinction to Schellbach’s mathematical-pedagogical seminar. On this mathematical-scientific seminar, see BIERMANN 1988, pp. 89, 96–100, 279–81. 234 See BIERMANN 1988, p. 280.

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their choice.”235 In the vita submitted with his Habilitation materials, Klein noted that, at the end of this semester, “I was pleasantly surprised by a friendly letter from Professor Kummer, in which I was informed that I would be the recipient of one of the two seminar prizes.”236 Kummer and Weierstrass assessed: Berlin – January 31, 1870 Dr. Felix Klein from Düsseldorf completed his mathematical studies at the Universities of Bonn and Göttingen to the extent that he has already earned the degree of Doctor of Philosophy237 with honors. He has also already published a few good mathematical studies and, upon the request of the late Prof. Plücker in Bonn, he has edited and published the incomplete work that the latter left behind. For the winter semester of last year, he came to Berlin and enrolled in the university here in order to join the Mathematical Seminar as an ordinary member. Since then, he has participated enthusiastically in the seminar’s exercises and has given several presentations which, with respect to their form, are to be deemed entirely excellent. Because he has also conducted his scientific research with lively zeal and tireless diligence – supported by his good talent – it can be expected that he will distinguish himself scientifically as a teacher of mathematics and that he will continue to be highly fruitful and productive. Kummer (signed) Weierstrass (signed)238

This report, which was sent to the Ministry of Culture, was certainly supportive of Klein’s future career. The second prize of the semester went to the Viennese Privatdozent Otto Stolz, with whom Klein would remain in close contact. In other semesters, Ludwig Kiepert, Eugen Netto, Georg Frobenius, and others were distinguished in the same way.239 Klein’s much-discussed seminar presentation on Cayley’s metric took place after the semester had officially ended (see Section 2.5.3). 2.5.2 Acquaintances from the Mathematical Union: Kiepert, Lie, Stolz The number of mathematics students in Berlin exceeded the limit of twelve who, per semester, were allowed to participate in Kummer and Weierstrass’s Mathematical Seminar. For this reason, those who were not accepted to join the seminar had formed, in 1861, a Mathematical Union (Mathematischer Verein) in Berlin240 that was open to everyone and would serve as a model for similar organizations elsewhere. The union collected membership dues, maintained a library, and was intended to deepen the mathematical knowledge of its members via lectures, discussions, and problem-solving.

235 KLEIN 1923a (autobiography), pp. 15–16. 236 Quoted from TOBIES 1999a, p. 85. 237 The title “Doctor of Philosophy” comes from the tradition of mathematics belonging to the Philosophical Faculty. 238 Quoted from LOREY 1926, p. 150. 239 See BIERMANN 1988, pp. 107–10. 240 [UAB] No. 559: Records of the Math. Student Union, May 1862–November 1935.

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It was here that Felix Klein met Ludwig Kiepert, Sophus Lie, Otto Stolz, Heinrich Bruns, Eugen Netto, Hermann Schubert, Max Simon, and others, most of whom (with the exception of Lie, Stolz, and Schubert) completed their doctoral degrees under Weierstrass and Kummer between 1867 and 1871.241 Schubert was the academically active secondary-school teacher, mentioned above, who was closely associated with Clebsch’s algebraic-geometric school (see 2.4.1). He earned his doctorate in 1870 from the University of Halle. Heinrich Bruns and Klein would reunite later as colleagues in Leipzig, where Bruns became a professor of astronomy. In 1869, however, it was Kiepert, Lie, and Stolz who proved to be especially important for Klein’s development as a mathematician, for they inspired him to seek new research findings in a variety of different directions. Ludwig Kiepert helped to familiarize Klein with Weierstrass’s areas of research. While looking for contacts among the young mathematicians in Berlin, Klein had asked Weierstrass for recommendations about whom, in particular, he ought to know. Weierstrass recommended Kiepert, who attended his lectures from October of 1865 to April of 1871. Kiepert had also served as Weierstrass’s “blackboard writer” (since 1861, health issues had prevented Weierstrass from writing on the blackboard himself). Kiepert reported that Weierstrass’s lectures on Abelian functions in the summer semester of 1869 began with 107 attendees but ended with just seven remaining students. “A master researcher […] but had many difficulties as a teacher” was how Kiepert described his Doktorvater,242 who assigned to him a research topic on the theory of elliptic functions that he himself had been unable to resolve.243 Even in his fourth year, Kiepert was still having difficulties with Weierstrass’s lectures, and thus it is no surprise that Klein did not devote too much of his time to them during his stay in Berlin. Kiepert, who soon became one of Klein’s close friends, later wrote: “Most of all, I owe thanks to Weierstrass for creating my friendship with Felix Klein.”244 Klein and Kiepert informed each other about their teachers’ lectures (Plücker’s and Weierstrass’s, respectively). They traveled together, visited one another later in life, and achieved – each in his own way – similar research results on such topics as the transformation of elliptic functions, the solution of quintic equations, and the complex multiplication of elliptic functions.245 In his Leipzig lectures on the theory of elliptic functions, Klein would cite Kiepert’s findings in his explanations of transformation theory: “Kiepert’s determinants,” “The Jacobian summation formula according to Kiepert.”246 In Klein’s estimation, Kiepert was the one student of Weierstrass who contributed the most to the area of elliptic func-

241 242 243 244 245 246

BIERMANN 1988, p. 353. KIEPERT 1926, p. 59. See Weierstrass’s evaluation of Kiepert’s dissertation in BIERMANN 1988, pp. 117–18. KIEPERT 1926 (memories about Weierstrass), p. 62. Ibid., pp. 62–64; and [UBG] Cod. MS. F. Klein 10, pp. 49–120 (Kiepert’s letters to Klein). Felix Klein, Theorie der elliptischen Funktionen, 2 parts (1884), transcribed by Biedermann (held in the library of the Mathematical Institute, University of Leipzig), pp. 411–12.

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tions. Klein would later delve deeper into Weierstrass’s methods (see Section 5.5.2), and he acknowledged the influence that Weierstrass had on his work: [B]ut in fact Weierstrass has strongly influenced all of us who, having grown up on another soil, came to elliptic and related function. […] In my works on hyperelliptic and Abelian functions of 1886–89 […], I have carried over to higher cases the idea of σ, and I mean, that I have given their definitive form to Weierstrass’s thoughts on the decomposition of algebraic functions on a Riemann surface into prime factors and units.247

Klein and Kiepert wanted to do what they could to overcome the historical aversion that existed between the representatives of their respective schools. In 1881, Kiepert, who had been a professor at the Technische Hochschule in Hanover since 1879, expressed this unequivocally in a letter to Klein: “Nothing came from the animosity between [Jakob] Steiner and Plücker or of Kronecker’s behavior towards Clebsch etc. Thus it is all more pleasing to me that you have remained my faithful friend even though we come from entirely different schools.”248 On account of his differences with Kronecker, Kiepert decided in 1884 no longer to publish his results in Crelle’s Journal. Since then, he instead sent his articles to Klein, who published them in Mathematische Annalen or in the Göttinger Nachrichten. While working as a professor in Hanover, Kiepert also served as a consultant for a life-insurance company there. In the middle of the 1890s, he helped Klein to establish the first ever university seminar devoted to actuarial science in all of Germany (see Section 7.6). The Norwegian mathematician Sophus Lie, whom Klein also first met in Berlin, would turn out to be his most important mathematical collaborator.249 Klein was able to cooperate with Lie immediately, because their work shared similar points of departure. In Norway, a lecture by Hieronymus Zeuthen in 1868 had inspired Lie to immerse himself in the works of Poncelet, Chasles, Plücker, and others.250 Lie had sent his first published work to Clebsch, so Klein was already familiar with it before he went to Berlin. Klein reported to his mother: Among the young mathematicians here, I have made an acquaintance that seems very promising to me. The person in question is a Norwegian called Lie, whose name was already familiar to me from an article that he had published in Christiania. We have been working in particular on similar subjects, so that there is no lack of material to discuss with him. We are not only united by this common love but also by a certain repulsion to the way in which mathematics here is made to seem superior to the mathematical achievements of others, especially foreigners.251

Among the seventy-four mathematics students in Berlin during the winter semester of 1869/70, only a few were from abroad: three from Switzerland, one each from Poland and Italy, the Norwegian Sophus Lie, and the Austrian Otto Stolz. 247 248 249 250 251

KLEIN 1979 [1926], p. 274–75. [UBG] Cod. MS. F. Klein 10: A letter from Kiepert to Klein dated October 15, 1881. On Lie’s biography, see STUBHAUG 2002; F. ENGEL 1899; and M. NOETHER 1900. STUBHAUG 2002, p. 103. Klein’s letter dated October 31, 1869; quoted from LIE 1934 (GMA I), p. 636.

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Klein had heard condescending remarks about foreign science from university administrators as well as professors, for instance when he paid an introductory visit to the meteorologist Heinrich Wilhelm Dove.252 Because Weierstrass cited other scholars so infrequently in his lectures, he was able to foster such an attitude in his students, as the example of H.A. Schwarz would show (see Section 5.5.2.4). Lie had his first publication (mentioned above) translated into German, and submitted it to Crelle’s Journal.253 Klein was enthusiastic about their common approach, and he began to edit an article that Lie was working on at the time: “Über die Reciprocitäts-Verhältnisse des Reye’schen Complexes” [On the Reciprocity Relations of the Reye Complex], which Clebsch accepted for publication in Göttinger Nachrichten.254 The article contained references to Max Noether’s technique for mapping the lines of a linear complex to points in three-dimensional space and to Theodor Reye’s tetrahedral configuration (Reye complex).255 Just as Max Noether had thanked Klein in his article from 1869 (see Section 2.4.3), Sophus Lie would write the following at the end of his latest study: “For the last two theorems I am indebted to Dr. Klein, with whom I hope to work together on a more comprehensive study of the pertinent congruences.” This article contained important ideas in an embryonic stage: the idea of contact transformation (with which straight lines could be transformed into spheres; that is, Lie turned Plückerian line geometry into sphere geometry), and the idea that the Reye complex defines a first-order partial differential equation. Klein would ultimately incorporate these findings into his Erlangen Program. Lie found it difficult to express his many important ideas in writing. Klein was happy to assist him with this in order to immerse himself in Lie’s material. He helped Lie to present his ideas systematically and he strengthened Lie’s argumentation with conclusions by analogy. He boosted Lie’s self-confidence, and he presented the latter’s work on the Reye complex in the Berlin Mathematical Seminar because Lie still felt uncertain about his German skills.256 Lie wrote home: “I regard it as an extraordinary stroke of luck that Klein, who is an outstanding (if still young) pupil of Plücker and Clebsch, has remained in Berlin this semester. We are traveling to Paris together, and if I get the stipend in question, also to Milan and Cambridge.”257 Klein was to become the first German mathematician to study in Paris again, after Plücker, Dirichlet and Jacobi, who had gone there some decades before. In Paris, Klein and Lie would indeed successfully carry on with their collaborative work (see Section 2.6). Later, Klein assessed:

252 Ibid. 253 LIE 1934 (GMA I), pp. 1–11 = “Ueber eine Darstellung des Imaginären in der Geometrie,” Crelle’s Journal 71 (1869), pp. 346–53. See also STUBHAUG 2002, p. 109. 254 The article appeared in Göttinger Nachrichten (February 16, 1870), pp. 53–66. 255 See Max NOETHER 1900, p. 5; and, for further detail, ROWE 2019a. 256 STUBHAUG 2002, p. 136. 257 Quoted from ibid., p. 137.

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In 1874, when Lie recommended that his student Elling Holst should study under Klein, he wrote that Klein was “equally eminent as a researcher and teacher” and that he possessed the ability to adapt to the thoughts of others.259 Later still, Klein would support Lie’s career in an entirely extraordinary way (see Section 5.8.3). Klein’s third influential acquaintance from Berlin, the Viennese Privatdozent Otto Stolz, came to Germany in the fall of 1869 with a travel stipend from the Austrian government. From Stolz, Klein learned of the existence of non-Euclidean geometries, and he also gained access to Karl Georg Christian von Staudt’s projective geometry. In Klein’s words: “All the while, Stolz was not only my stern critic but also my guide to scholarly literature. He had studied the works of Lobachevsky, János Bolyai, and von Staudt in detail, which I never could have forced myself to do, and he was happy to answer all of my questions.”260 It should be highlighted that von Staudt had freed projective geometry from a number of methodological shortcomings.261 Poncelet’s projective geometry grew out of three-dimensional (Euclidean) geometry, in that it supplemented space or planes with so-called “infinitely distant points” and replaced the term “parallel” with the idea of lines “meeting in infinity.” This led to statements such as: “Two different lines in a plane always intersect at exactly one point.” At first, however, metric concepts such as distances and angles were still used, particularly for defining cross-ratios. In general, projective geometry sought to synthesize geometric results exclusively by using the concepts of “join” and “intersection.” In contrast to his predecessors, von Staudt, in his book Geometrie der Lage [Geometry of Position], strove for a metric-free conceptualization of geometry and only relied on assumptions concerning the position and arrangement of points, lines, etc. Klein recognized the possible connection between Cayley’s metric, non-Euclidean geometry, and von Staudt’s new projective concept of the cross-ratio (also called double ratio, a number associated with four collinear points). Klein outlined his basic ideas while still in Berlin (see Section 2.5.3), and he worked on the topic in greater detail after his stay in Paris and the interruption caused by the FrancoPrussian war. In doing so, he had the good fortune that Otto Stolz had decided to continue his studies with him in Göttingen (see Section 2.8.2). Stolz remained one of Klein’s collaborators for many years. He published fifteen articles in Mathematische Annalen and he was a reliable reviewer. 258 259 260 261

[Oslo] II (a text written by Klein on November 1, 1892), printed in ROWE 1992a, p. 589. Quoted from STUBHAUG 2002, p. 236. KLEIN 1923 (GMA I), p. 52. – Regarding Otto Stolz and Klein, see also BINDER 1989. See Max Noether, “Zur Erinnerung an Karl Christian von Staudt,” Jahresbericht der DMV 32 (1923) Abt. 1, pp. 97–118, esp. 105, 112–14.

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2.5.3 Cayley’s Metric and Klein’s Non-Euclidean Interpretation This was the first research topic in which Klein intuited a connection that no one had thought of before. He approached this projective metric from the perspective of Plücker’s line geometry. Klein had been familiar for some time with the work of Arthur Cayley, whose projective metric (first formulated in 1859) made it possible “to add the actual measurement itself to the general projective concept of the cross-ratio.”262 Cayley had demonstrated that the usual (Euclidean) geometry could be understood as being a part of projective geometry.263 Klein had first become aware of Cayley’s theory in the summer of 1869, when he was reading Wilhelm Fiedler’s German translation of George Salmon’s A Treatise on Conic Sections. In Berlin, Klein and Sophus Lie studied selections of Cayley’s work together. In a letter to Lie, written after the latter had already left Berlin for Göttingen, Klein explained: “Something new that I can tell you is that Cayley has sent me his two works on cubic surfaces and reciprocal surfaces, selections of which we had studied together from the Proceedings.”264 These studies turned out to be useful for a lecture that Weierstrass had asked Klein to deliver after the end of the semester (Weierstrass often continued his seminars after the official end of the term).265 Klein had already informed Lie on March 8, 1870: “Unfortunately, I will not be able to leave Berlin as early as I had intended, namely on Thursday the 17th. On Friday evening I met with Weierstrass, and he charged me with the task of delivering, on the 16th, my promised seminar lecture about Cayley’s generalization of the concept of distance.”266 In his article from 1859, Cayley had introduced metrics to projective geometry via a fixed conic that he referred to as the “absolute,” which is determined by considering projective transformations acting on a complex projective plane. From this, he deduced the projective metric that is named after him. Klein recognized the connection between Cayley’s study and non-Euclidean geometry after Otto Stolz had informed him about Lobachevsky’s work. In Klein’s letter from the time to Lie about his presentation, there is no sign of frustration about Weierstrass’s attitude (Klein would often express such frustration later): “Last Wednesday evening, I delivered my lecture about Cayley in Weierstrass’s seminar. On the next day, when I visited him to say farewell, Weierstrass discussed things with me at length. Kummer had far less to say; he gave me a copy of his work on ray systems to pass along to you.”267 In fact, it seems as 262 See Max NOETHER 1895, p. 468. 263 Arthur Cayley, “A Sixth Memoir Upon Quantics,” Philos. Trans. Royal Society of London 149 (1859), pp. 61–90. 264 [Oslo] A letter from Klein to Lie dated March 10, 1870. 265 See HILDEBRANDT/STAUDE-HÖLDER 2014, p. 8. 266 [Oslo] A letter from Klein to Lie dated March 8, 1870. 267 [Oslo] A letter from Klein to Lie dated March 29, 1870. Kummer’s article was “Über Strahlensysteme, deren Brennflächen Flächen vierten Grades mit sechzehn singulären Punken sind,” Gesammelte Werke, ed. A. Weil (Berlin: Springer, 1975), vol. 2, pp. 418–32.

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though Klein’s conversation with Weierstrass led him to think more precisely about the definition of the distance between two points: “My lecture about Cayley was an occasion for me to propose the following definition of distance, which agrees with Cayley’s definition in substance but has the advantage of providing the related function entirely […] and thus transparently. Namely, I have replaced the arccosine with the relevant integral.”268 Klein’s first intuitive idea did not persuade Weierstrass in the least, but later he interpreted his approach as simply being a different way of discovering new research results: In February 1870 I gave a lecture on the Cayley metric in Weierstrass’s seminar, closing with the question of whether this work didn’t extend and agree with Lobachevsky’s. As an answer I was told that these were two completely different separate spheres of thought, and that the first thing to be considered in the foundations of geometry is the idea of a line as the shortest distance between two points. I let myself be impressed by this rejection and put aside the idea I had already formed. With respect to the logicians’ criticisms, which lay further from my interests, I was always timid. Only very much later did I come to understand that this was a matter of a difference in natural dispositions, and that the psychology of mathematical research conceals great problems. Weierstrass’s nature was obviously more attuned to careful inquiry, to building a path to the summit step by step. It was less in his nature clearly to discern the outlines of distant mountain peaks; at least in this case he made no use of such a view from the distance.269

In anticipation of his future work in 1871, it should be mentioned in passing how Klein classified Euclidean and non-Euclidean geometries with projective methods. In his historical lectures, he explained his approach as follows: From the beginning there was the task of studying the Cayley metric in the various cases that arise by distinguishing the varieties of the second degree from the projective point of view. Considering only varieties with real equations, these cases are: a) true surfaces of the second degree: 1. real ruled (one-sheeted hyperboloid, hyperbolic paraboloid), 2. real unruled (ellipsoid, elliptic paraboloid, two-sheeted hyperboloid), 3. imaginary. b) true curves of the second degree: 1. real (ellipse, parabola, hyperbola), 2. imaginary. c) point-pairs: 1. real, 2. imaginary. d) double point.

Klein explained further: Case b, 2 gives the usual metric, if one takes the basic conic section – what Cayley calls “the absolute” – to be the spherical circle. Cases a, 2 and a, 3 lead to just the two kinds of nonEuclidean geometry that were distinguished by Gauß, Lobachevsky, Bolyai, and Riemann and that are obtained from the usual geometry by taking the sum of the angles of a triangle to 268 [Oslo] A letter from Klein to Lie dated March 29, 1870. 269 KLEIN 1979 [1926], p. 140. Klein often referred to Weierstrass as a “logician.”

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be less than or greater than π. So these systems too have now been included in projective geometry and lost everything paradoxical. This is the simplest way to arrive at their characteristic and at a conviction of their consistency.270

Gauss, Lobachevsky, Bolyai, and Riemann had already demonstrated that the negation of the Euclidean “parallel postulate” leads to different (non-Euclidean) geometries. It still remained to be shown, however, that these non-Euclidean geometries lack contradictions. Klein’s work was significant because he proved that Cayley’s projective metric could serve as a model for non-Euclidean geometries and because he derived the non-contradictory nature of the latter from that of projective geometry.271 It would cost him much time and effort to convince those who were skeptical about these non-Euclidean theories (see 2.8.2 and 2.8.3.3). 2.6 IN PARIS WITH SOPHUS LIE My plan to receive an explicit order from the Ministry has failed. I have to restrict myself simply to submitting an application in which I request diplomatic recommendations to travel to Paris and London. Such recommendations may be advantageous, for instance, for gaining access to the École Polytechnique or for viewing the larger collections of research materials.272

From April 19, 1870 until the outbreak of the Franco-Prussian War in July, Klein spent his time in Paris. His father had recommended that he should request, for this trip, an official order from the Prussian Ministry of Culture. The Ministry responded to this request as follows: “We have no need for French or English mathematics.”273 His second application, however, was successful. Felix Klein received diplomatic recommendations and was later required to submit a report about his activities abroad (see Appendix 1). On April 17 (Easter Sunday), Klein was still at his parents’ home in Düsseldorf. From there he traveled to Aachen, where he visited relatives and also paid a visit to the Theodor Reye, whose work had been central to Lie and Klein’s research (see Section 2.5.2). In 1870, Reye had been made a professor of geometry and graphical statics at the newly founded Royal Rhenish-Westphalian Polytechnical School (Königlich Rheinisch-Westphälische Polytechnische Schule, as of 1880: Technische Hochschule Aachen). Reye had attended lectures by Riemann in Göttingen and had ultimately been influenced by Karl Culmann in Zurich. Reye thanked Culmann (known from his book Die graphische Statik [Graphical Statics], published in 1866) for his reference to von Staudt’s Geometrie der Lage [Geometry of Position]. Based on von Staudt’s work, Reye wrote a more 270 Ibid., p. 138. – For an interpretation of Klein’s classification with the terminology of mathematical structuralism, see Francesca BIAGIOLI (2020). 271 See Hans REICHARDT 1985, pp. 239–40. – Regarding the concept of “model,” see also SCHUBRING 2017. 272 [Oslo] A letter from Klein to Lie dated March 8, 1870. 273 KLEIN 1923a (autobiography), p. 16.

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intelligible Geometrie der Lage (1866, 1868), which saw several editions. The topic addressed in the second volume of his book – complexes – was a point of reference for Lie and Klein. From Aachen, Klein traveled on the overnight train to Paris, where Lie picked him up at the Gare du Nord in the morning of April 19, 1870.274 The two of them stayed in adjacent rooms at the Hôtel Molinié (Rue de l’École de Médicine 32), and at the beginning of June they moved to the Hôtel Bellevue (Boulevard de Montparnasse 35).275 Due to his knowledge of French, Klein gained access to Parisian institutions without a hitch, while Sophus Lie – who had arrived there a few weeks before – was only able to make any contact with French mathematicians with Klein at his side.276 In addition to their mathematical studies, they also enjoyed the Parisian life. As Klein wrote to his mother on May 6, 1870, they dreamily roamed the streets arm in arm.277 Lie reminded Klein of this great time several years later, while they were attempting to schedule another trip to Paris together: Do you not also think about a trip to Paris? It would be remarkable to meet in Paris again. Then once again we could go to Sceaux and drink coffee among the trees, and even once more admire the hippopotamuses in the zoological gardens, and perhaps even meet once more at Closerie des Lilas. Think of it!278

2.6.1 Felix Klein and French Mathematicians Gaston Darboux proved to be the key person in Klein’s growing network of French mathematicians.279 Camille Jordan, who was twelve years older than Klein, also gave a warm welcome to the mathematicians who came to him from Clebsch.280 Every Monday, Klein and Lie were able to meet the seventy-sevenyear-old Michel Chasles at the Institute, where they also met the sixty-one-yearold Joseph Liouville,281 who had published Evariste Galois’s group-theoretical writings in 1846 in his Journal de Mathématiques Pures et Appliquées. While in Paris, Klein and Lie found a receptive community for their geometrical work, which they had largely lacked in Berlin. Ever since being appointed a professor of advanced geometry at the Sorbonne in 1846, Chasles – Darboux’s 274 [Oslo] A letter from Klein to Lie dated April 13, 1870. 275 [Oslo] XXXIII, p. 8 (a letter from Klein to his mother dated May 6, 1870). 276 In Berlin, they had prepared for their stay in Paris by participating in a French conversation course, but Lie had soon stopped attending. ([Oslo] XXXIII, p. 3, Klein to his mother, January 15, 1870). 277 [Oslo] XXXIII, p. 8. 278 Quoted from STUBHAUG 2002, p. 288. 279 See TOBIES 2016; RICHTER 2015; and CROIZAT 2016. 280 Jordan had visited Clebsch in Göttingen in 1869 (HARTWICH 2005, p. 14). By that time, Lie also considered himself to be a member of Clebsch’s school (see STUBHAUG 2002, p. 138). 281 See STUBHAUG 2002, p. 143; see also LÜTZEN 1990; VERDIER 2009.

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doctoral supervisor – had developed an influential school in the area of projective and metric geometry. These French geometricians made important contributions in the field by introducing, as real structures, the spherical circle (Kugelkreis) in the infinitely distant plane and circular points on the infinitely distant line. Klein later described their influence on his and Sophus Lie’s work.282 After Darboux had introduced Klein and Lie to him, Chasles invited them for dinner on April 27th.283 Moreover, Chasles would usher Klein and Lie’s collaborative work from June of 1870 to publication (see Section 2.6.2.1). Two years later, he became the first president of the Société Mathématique de France. In 1870, Gaston Darboux had taken over the editorial duties of the Bulletin des Sciences Mathématiques et Astronomiques with Jules Hoüel and Jules Tannery, and they steered the journal in an international direction.284 Even before Klein came to Paris, Darboux had surprised him with a letter in which he suggested that they should work together. Klein responded as follows in a letter dated March 25, 1870: “A few weeks ago, I looked at the first issue of your valuable journal with Clebsch in Göttingen, and I look forward to the opportunity of collaborating on such an up-to-date undertaking.”285 Klein contributed to this publication venue, which he described as a review journal, from its first (1870) to its eleventh volume (1876), and he was credited on the title page of the Bulletin (see Fig. 9). Not until 1876, when Klein became the chief editor (with Adolph Mayer) of Mathematische Annalen, did he step down from his position as a reviewer for the Bulletin, but he made sure that the two journals would continue to work closely together. In their report from July 7, 1870 to the Mathematical Student Union in Berlin, Klein and Lie expressed their admiration for the Bulletin: Allow us to go into greater detail regarding the Bulletin des Sciences Mathématiques et Astronomiques. We believe that such a journal is a very useful but also a very difficult undertaking that can only fully achieve its goals if it has a large number of predominantly native collaborators in the disciplines that it covers. The Bulletin is not yet in such a favorable position. And, indeed, it is not difficult to point to a number of imperfect judgements in the volumes that have already appeared. However, the personality of the editor, G. Darboux, whose talents we believe to be extraordinarily suited to this very purpose, strikes us as one that will ensure that the Bulletin will become better and better over time. Most of its reviews are distinguished by their expertise and clarity.286

They were pleased, moreover, by “one of the Bulletin’s main tendencies […], which is to make the hitherto little-known modern branches of geometry and algebra more familiar in France.” The journal published not only reviews but also articles, including (translated) contributions by Klein and Lie. 282 283 284 285 286

KLEIN 1979 [1926] pp. 132–35. [Oslo] XXXIII, p. 8–9. See GISPERT 1987; NEUENSCHWANDER 1984; CROIZAT 2016, and HENRY/NABONNAND 2017. [Paris] 41: A letter from Klein to Darboux dated March 25, 1870. See here, and for the following quotation, [Oslo] A report by Klein and Lie dated July 7, 1870. (The German original report is published in TOBIES 2015).

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Figure 9: A title page of the Bulletin des Sciences Mathématiques et Astronomiques.

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Their collaborative studies in Paris were also influenced by Camille Jordan, who just had published Traité des substitutions et des équations algébriques, (Paris: Gauthier-Villars, 1870). It has often been discussed how the Traité shaped Klein’s thinking about group theory. In their mentioned report, Klein and Lie referred to the work as a “singular phenomenon.” Even though they had yet to study it in detail, the book introduced them to certain fundamental ideas: C. Jordan, ingénieur des mines, absorbed himself in Galois’s theory of equations for about five years, and he has made extraordinarily important advances in this discipline. In particular, he solved the problem of indicating all algebraically solvable equations of a given degree. If we are not mistaken, his success is partly due to the idea of introducing the Galois imaginary into the theory of linear substitutions. Without this instrument, one can reduce integer linear substitutions to a canonical form only in very special cases; with it, one can reduce all of them.

In his Traité, Jordan not only developed Galois’s theory further. His aforementioned visit to Clebsch in 1869 had expanded his perspective to include the findings of Hesse, Clebsch, Kummer, and their (geometrical) equations in conjunction with the related idea of (substitution) groups,287 something which would lead Klein and Sophus Lie to their first application of the group concept (see Section 2.6.2.1). Later, Klein expressed himself inconsistently about the extent to which Jordan’s Traité had influenced him. In 1886, he asked Jordan for a (further) copy for the new reading room at the University of Göttingen.288 In 1892, Klein said it initially had only an“indirect stimulus” on his research. In 1921, he wrote that the work at first seemed to them like “a book with seven seals.”289 On October 25, 1924, Klein wrote to Friedrich Engel in no uncertain terms that Jordan’s work had revealed to them the “general significance of the group concept,” and that Jordan’s article “Sur les groups de mouvement” and Galois’s work demonstrated that “every equation possesses a particular group as soon as one recognizes the area of rationality in which one operates.”290 In Jordan’s article on motion groups (Annali di matematica pura ed applicata 2/3 [1868], pp. 167– 215, 322–45), “groups of transformations” are defined that in modern terminology would be called semi-groups.291 Upon Clebsch’s request, Jordan had published an introduction to Galois’s work in the first volume of Mathematische Annalen,292 a text that Klein and Lie had already read before their trip to Paris.293

287 See, for example, pages 305–08, 427–30 in Jordan’s Traité. This could have been a starting point for the international Galois field network, described by BRECHENMACHER 2016b. 288 [Paris-ÉP] 97: Klein to Camille Jordan, July 3, 1886. 289 [Oslo] II (a text by Klein dated November 1, 1892); KLEIN 1921 (GMA I), p. 51. 290 This letter is appended to Engel’s article “Gruppentheorie und Grundlagen der Geometrie,” Mitteilungen aus dem Mathematischen Seminar der Universität Gießen 35 (1945), pp. 1–22 (the letter appears on pp. 22–24, and the quotation here is from page 23). 291 See HOFMANN 1992; HOLLINGS 2014; and PORUBSKÝ 2018. 292 Camille Jordan, “Commentaire sur Galois,” Math. Ann. 1 (1868), pp. 141–60. 293 See Max NOETHER 1900, p. 8.

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Whereas, with Jordan’s Traité, group theory was regarded as an indispensable instrument for the theory of equations (in which, Klein remarked, “a substitution means a permutation of letters”), Klein and Lie ultimately attempted “to work out the significance of group theory for various domains of mathematics.”294 Of course, their group concept was quite different from its modern abstract formulation.295 Klein later discussed the process of development from their intuitive (anschaulichen) group concept to the abstract formulation – but he could not refrain from remarking that the latter, though “excellent for proofs,” “is not at all directed to the discovery of new ideas and methods.”296 2.6.2 Collaborative Work with Sophus Lie Klein’s letters to Lie show how he attempted to be an equal collaborator. Even before their trip to Paris, he had written the following to him: In Paris, I hope that I can be of more value to you than was possible in Berlin, where, as you know, I was occupied with a wide variety of social obligations. No disruptions of this sort will burden me in Paris, and I have no intention to let such things impose upon me there. Your investigations are profoundly interesting to me. I feel infinitely unworthy to contribute to them, especially when I think of the promise that I made myself to work on complexes as much as possible. Since the last time I wrote to you, I have not yet spent even a quarter hour working on mathematics.297

Yet even before their trip, they had both thought about publishing something together in Paris. Even before the letter quoted above, Klein had written to Lie: “I am sticking to our project with the Paris Academy and, ever since I conceived of it, I have not doubted its feasibility for a second. I share your conviction that our studies will at least be as valuable as any of the multitude of articles published in the Comptes Rendus.”298 They would indeed publish two articles together on socalled W-configurations (see Section 2.6.2.1), and in July of 1870 they achieved new results related to the Kummer surface (see Section 2.6.2.2). 2.6.2.1 Notes on W-Configurations Klein and Lie had come to an understanding that they should publish their results in the Comptes Rendus hebdomadaires des séances de l’Académie des sciences de Paris (Comptes Rendus for short), the publication venue of the Académie des Sciences. The reason for their confidence was Lie’s study on the Reye complex,

294 295 296 297 298

KLEIN 1979 [1926], p. 315. See SCHOLZ 1989, pp. 103–09; and WUßING 2007. KLEIN 1979 [1926], p. 316. [Oslo] A letter from Klein to Lie dated April 13, 1870. [Oslo] A letter from Klein to Lie dated March 9, 1870.

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which Klein had edited for the Göttinger Nachrichten. In the proceedings of the Parisian Académie for June 6th and 13th of 1870, Chasles indeed published their “Deux notes sur une certaine famille de courbes et de surfaces” [Two Notes on a Certain Family of Curves and Surfaces], which were five and four pages long, respectively.299 In Paris, the publication process was simpler than it was in Berlin: The articles in the Comptes Rendus typically serve the purpose of providing a preliminary overview. The editors are very liberal in their acceptance of articles, perhaps too liberal. In any case, the possibility of having one’s work published within eight days is extremely pleasant. A facility of this sort is lacking in Germany, with the exception of the academy proceedings that are published so quickly in the Göttinger Nachrichten.300

Their notes in the Comptes Rendus concerned previously overlooked curves or surfaces that, “through a continuous closed family of ∞1 or ∞2 linear tetrahedral transformations are mapped to themselves: W-curves and W-surfaces.”301 The terms “W-curve” and “W-surface” derive from the fact that Sophus Lie, in his conversations with Klein, used von Staudt’s projective term Wurf (throw) for the cross-ratio. Later, Halphen would speak of courbes anharmoniques.302 Even though their notes on W-configurations were not without inaccuracies – Klein corrected and supplemented these texts in his collected works (GMA I) – they nevertheless brought to light, “for the first time, the significance of the concept of a group of linear space transformations.”303 The expanded version of this text, which was prepared by Klein for Mathematische Annalen and dated March 1871, contains a reference to Jordan’s Traité des substitutions et des équations algébriques and the following footnote: “The expression ‘a closed system of transformations’ fully corresponds to what one would call, in the theory of substitutions, ‘a group of substitutions’.”304 Max Noether wrote about Klein and Lie’s “equal familiarity with the operations of linear substitutions,” and he stressed: “Their introduction of the concept of a closed system of transformations (Abgeschlossenheit einer Transformationsschar), like the concept of commutativity (Vertauschbarkeit), derives from the influence of Galois’s ideas, which, in the domain of discontinuous substitution groups, were disseminated at the time through C. Jordan’s recently published book and through his commentary on Galois that had been published a year earlier in the first volume of Mathematische Annalen.”305 299 300 301 302 303

Comptes Rendus (June 1870), pp. 1222–26, 1275–79. (KLEIN 1921 (GMA I), pp. 415–23). [Oslo] A report by Klein and Lie dated July 7, 1870 (published in TOBIES 2015). Max NOETHER 1900, p. 6. See KLEIN 1921 (GMA I), pp. 424–549, esp. 436; and NABONNAND 2008. [Oslo] II: Klein’s notes dated November 1, 1892 (printed in ROWE 1992a, p. 591). According to HAWKINS (1989, p. 284), “when Klein arrived in Berlin, he may have been more disposed than Lie to perceive groups arising in a geometrical context.” 304 Felix Klein and Sophus Lie, “Ueber diejenigen ebenen Curven, welche durch ein geschlossenes System von einfach unendlich vielen vertauschbaren Transformationen in sich übergehen,” Math. Ann. 4 (1871), pp. 50–84, at p. 56 (KLEIN 1921 (GMA I), pp. 424–59). 305 Max NOETHER 1900, p. 8.

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In 1892, Klein described how he and Sophus Lie divided their labor in their work on W-configurations: Everything new that we stated there about differential equations undoubtedly belongs to Lie; on the other hand, I took it upon myself to work out the relationships to invariant theory and to deal with a great number of details. Thus I was the first to observe that the famous theory of the logarithmic spiral is subsumed here and that there is an analogous theory of the loxodrome. From today’s understanding of things, this seems self-evident. At the time, however, we were surprised that our projective ideas could be transferred to such transcendent configurations of metric geometry.306

In his collected works (KLEIN GMA I), Klein classified these studies on W-configurations as belonging to the period leading up to his Erlangen Program. 2.6.2.2 Principal Tangent Curves of the Kummer Surface Klein and Lie achieved their main collaborative result at the beginning of July in Paris. Because of the outbreak of the Franco-Prussian war, they were not able to complete this study until later in the year. At the end of July, however, while Klein was back at his parents’ house in Düsseldorf, he sent a letter to Lie in Paris with additional findings. This letter contains an explanatory sketch (see Fig. 10) that would be included in their joint article, which was published by Ernst Eduard Kummer in Berlin at the end of 1870. Because Klein’s letter with the sketch was written during the war, it will be discussed below in Section 2.7.1. However, as the insights in this article largely derive from their time together in Paris, they should briefly be explained here.307 Even while in Berlin, Klein and Lie had sought to discover new results about the properties of objects in three-dimensional space by using so-called “transfer principles.” In Paris, French works on sphere geometry – in particular Darboux’s works on confocal cyclides – inspired them to reach new conclusions via analogy. Ultimately, Lie was able to discover analogies between the sphere geometry being practiced in France and Plücker’s line geometry, and Klein discovered analogies between line geometry and metric geometry. Incidentally, it should be mentioned that, for Klein, so-called Dupin cyclides would later play a special role (see Section 2.8.2).308 Their transfer principles allowed them to recognize the fundamental similarities between the lines of curvature on cyclides and the principal tangent curves (also known as “asymptotic curves”) on a Kummer surface. Toward the 306 [Oslo] II (printed in ROWE 1992a, p. 591). Lie commented: “The actual existence of W-curves and their equations were your achievements” ([Oslo] LXI, No. 4, p. 23). 307 For a brief outline of Klein and Lie’s ideas in this article, see also PARSHALL/ROWE 1994, pp. 164–65. 308 See KLEIN 1921 (GMA I), p. 51; and ROWE 2019b. Dupin cyclides are surfaces enveloped by a two-parameter family of spheres. Klein would use Dupin’s Theorem, which states that the surfaces in orthogonal families intersect along lines of curvature, to formulate a proof of a theorem by analogy.

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end of their article, they explained that it had been Sophus Lie who first observed that the asymptotic curves of the Kummer surface (see Fig. 8 in 2.4.3) are algebraic curves of the sixteenth order.309 On the basis of this observation, Klein found the correlation between those curves and the second-degree complexes that belong to the Kummer surface, and he determined their singularities.310 When Klein prepared this co-authored article to be reprinted in his collected works, he added the following comment about the day of their insight in Paris: In the beginning of July, 1870, I got up early one morning and wanted to start directly, when Lie, who was still in bed, called me from his room and described to me the connection between the principal tangent curves of a surface and the curvature curves of another surface that he had found in the night in such a way that I did not understand a word. (It was concerned with the line-sphere transformation, but instead of spheres, he operated, semi-intuitively, with rectilinear hyperboloids that went through a fixed real conic section.) In any case, he convinced me that the principal tangent curves to the Kummer surface must be algebraic curves of order 16. Later that morning, while I was visiting the Conservatoire des Arts et Métiers, the thought occurred to me that we must be dealing with those very curves of order 16 that appeared already […] in my “Theorie der Linienkomplexe ersten und zweiten Grades,” and I quickly succeeded in deducing the geometric considerations […] independently of the Lie transformation. When I returned to our hotel at four o’clock in the afternoon, Lie had gone out, and I left him a summary of my results in a letter.311

Klein did, after all, understand Lie’s line-to-sphere mapping, and he realized that Lie’s claim regarding the principal tangent curves (asymptotic curves) of Kummer surfaces was correct. As David Rowe has described in detail, Klein was now able to deduce other properties of these curves via his theory of one-parameter families of quadratic line complexes with a fixed singularity surface.312 Through these collaborations with Sophus Lie, Klein achieved a new general understanding of the methods and objectives of geometry. That is, he recognized that, in addition to the projective interpretation of algebraic surfaces, there is an equally valid way of looking at things, according to which spheres play the same invariant role that lines do in the projective approach. Here he also found an im309 During the same year, Lie would publish a separate article on this topic in Christiania as well as in Paris, where it appeared as “Sur une transformation géometrique,” Comptes Rendus 71 (October 31, 1870), pp. 579–83. 310 See Felix Klein and Sophus Lie, “Über die Haupttangentencurven der Kummerschen Fläche vierten Grades mit 16 Knotenpunkten,” Sitzungsberichte der Berliner Akademie (December 15, 1870), pp. 891–99. This article was reprinted in Math. Ann. 23 (1884), pp. 579–86; and in KLEIN 1921 (GMA I), pp. 90–97. For an English translation, see Felix Klein and Sophus Lie, “On the Principal Tangent Curves of the Fourth-Degree Kummer Surface with 16 Nodes,” trans. D.H. Delphenich (http://www.neo-classical-physics.info/uploads/3/4/3/6/34363841/ klein_lie_-_principal_tangent_curves_of_kummer_surfaces.pdf; accessed Dec. 10, 2019). 311 KLEIN 1921 (GMA I), p. 97. Klein, however, overemphasized his independence in this quotation from 1921. In 1916, in his report to Friedrich Engel (see ENGEL/HEEGAARD 1922, p. 60), Klein did not mention a letter but only that he understood the idea over the course of the day and that he knew how to determine the characteristics of such curves of order 16. In the evening, Klein had told this to Lie. See [Oslo] XXXIII, p. 11. 312 See ROWE 2019a, pp. 188–91.

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petus for the ideas that he would present in his Erlangen Program. Published in 1872, the latter text begins by referring to the relation between projective methods, metrical properties, and the imaginary circle at infinity common to all spheres, and it also refers to the idea of transfer principles.313 2.6.3 A Report on Mathematics in Paris Sophus Lie’s Nachlass contains a multi-page report on the state of mathematics in Paris. Dated July 7, 1870, it is written in Klein’s hand and signed by both Klein and Lie.314 This report about their experiences there, which was composed for the Mathematical (Student) Union at the University of Berlin, was also sent to the Prussian Minister of Culture Heinrich von Mühler (see Appendix 1): “I have written to the Minister, to whom, as you know, I was obliged to send reports about French and English mathematics. As evidence that I have worked according to my proposed plan, I have also submitted to him our article in the Comptes Rendus and a copy of the report that we sent to the Student Union in Berlin.”315 With their report, Klein and Lie provided an impressive overview of the status of French mathematics at the time. They discussed the way in which mathematics was taught there at various institutions, the students who were earning degrees, the enrollment numbers in the lectures, people (mathematicians and also engineers) who published new mathematical results, and the mathematical work that had been produced there in recent years. Unlike the case of German universities, all the university lectures in Paris were free and open to the public. Klein and Lie were not very enthusiastic about this because it increased the student-teacher ratio. At the same time, they missed the seminar activity that was common in Germany as well as a mathematical library (Lese-Institut). While in Paris, it was only through their private contacts that they were able to access new scholarly literature. Moreover, they regretted that the French mathematicians did not, in their opinion, have a very close relationship with one another – a situation that would soon improve with the aforementioned establishment of the Société Mathématique de France. Their comparison of German and French mathematical journals and the respective ways in which articles were written in the two countries is especially noteworthy. They preferred the French approach: Compared to the German way, the French manner of editing mathematical studies has the advantage of incomparably greater clarity or – we should rather say – simplicity. In Germany, one often adopts the method of condensing mathematical analyses as much as possible and editing them in such a way that they can only be understood by those working in the same

313 KLEIN 2020 [1872], p. 1. See also sections 4 (on “Transfer of Properties by Mapping”) and 5 (on “Hesse’s Transfer Principles”) in this work. 314 As mentioned above, this report is published with commentary in TOBIES 2015. 315 [Oslo] A letter from Klein to Lie dated December 30, 1870.

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discipline. By the manner of presentation that is customary here, this drawback is remedied, even though the space that each communication takes is larger. It only takes a moment to recognize that the French manner of presentation has the absolute advantage. For the only reasonable purpose of a mathematical study is to be understood; its purpose should not be to arouse admiration for its author.316

All in all, Klein’s stay in Paris had a lasting influence on him. Throughout his career, he would maintain a regular exchange of ideas with French mathematicians; he would always discuss the latest works by French authors in his seminars; he would send his students to study in Paris; and he would encourage his French colleagues to contribute to Mathematische Annalen and to his later book projects. Moreover, he supported students from France who came to study under him (on Darboux’s recommendation above all). During the German reform of mathematical education, the French example repeatedly served as an argument for making certain changes.317 Klein’s experience in Paris, and especially his tour of the collections at the Conservatoire des Arts et Métiers, also increased his enthusiasm for mathematical models318 and instruments. Among the latter, he was particularly pleased to learn about new and less expensive reproduction methods by means of lithography – a technique was used by mathematicians in France (and Italy) earlier than it was in Germany.319 2.7 THE FRANCO-PRUSSIAN WAR AND KLEIN’S HABILITATION We met two older gentlemen – that is, assessors – from Bonn, who led us over the battlefield on August 19th. During the foot march, I spoke with them at length about my Habilitation plans and other things. A wonderful coincidence: one of them was the same man who would later be of the greatest importance to me: Althoff […].320 Nevertheless, now that the opportunity has arisen to write you a letter, I should not refrain from offering you a sign of life […].321

Klein and Sophus Lie had to end their stay in Paris earlier than planned. On the Saturday after the outbreak of the war (July 16, 1870),322 Klein traveled to his parents’ home in Düsseldorf323 and, still in July, sent a letter to Lie in which he discussed the spherical circle, complexes, and similar concepts (see below). This 316 317 318 319 320 321 322

Quoted from TOBIES 2015. See, in particular, Sections 5.5.3; 7.4; 8.3.4; and 9.3.2 below. See also BRECHENMACHER 2017. See KLEIN 1923a (autobiography), pp. 22–23. [UBG] Cod. MS. F. Klein 22L, p. 3 (Klein’s “wartime reminiscences” for his children). [Paris] 42: Klein to Darboux, February 14, 1871. In one letter, Klein stated that the war had been declared on July 16th (see Appendix 1). It was known that there had been a turbulent session in the French Senate on July 15th, in response to which the German states of Bavaria and Prussia began to mobilize their troops on July 16th. 323 Both STUBHAUG (2002, p. 13) and PATTERSON (2016, p. 129) falsely claim that Klein traveled to Berlin.

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gave rise to the well-known story of Lie coming under suspicion of being a German spy. Lie had Klein’s letter in his bag when he was detained in Fontainebleau (approximately 55 kilometers southeast of Paris) in August of 1870. He was held in custody until the 10th of September because he did not have a valid passport. Lie turned to Chasles, Bertrand, and others for help, and Darboux ultimately traveled to Fontainebleau and successfully arranged for his release. Immediately thereafter, Lie set off on his intended course through Switzerland toward Italy.324 In the first of the following two sections, I will discuss Klein’s participation as a paramedic in the Franco-Prussian War from August 16 to October 2, 1870,325 and I will consider the influence that the war had on his career and on his contacts in France (Section 2.7.1). In the second (Section 2.7.2), I will focus on how the war barely interrupted Klein’s mathematical train of thought and how, even on the battlefield, Klein kept his intended Habilitation plan in mind. 2.7.1 Wartime Service as a Paramedic and Its Effects This is no place to provide a detailed account of the war. It should only be stressed that the French officially declared war on July 19, 1870, and that the German forces were victorious over those of Napoleon III. On January 18, 1871, the Prussian king Wilhelm I was crowned the German Emperor at the Palace of Versailles. Likewise at Versailles, a preliminary peace treaty was signed on February 26, 1871. The German Empire was unified, and Otto von Bismarck was named its first chancellor. Regarding the politics of higher education, it was relevant that, as stipulated by the Treaty of Frankfurt am Main (ratified on May 10, 1871), AlsaceLorraine and thus the University of Strasbourg became part of Germany – a situation that would last until 1918. On July 29, 1870, Klein wrote from Düsseldorf to Sophus Lie in Paris about his new ideas concerning their research on principal tangent curves (asymptotic curves), and he included in this letter the sketch that I mentioned above (see Fig. 10).326 This same illustration would be printed in their joint article, which E.E. Kummer published in the Sitzungsberichte of the Berlin Academy.327 Klein would later even incorporate this image into the design of a gown for his fiancee (see Section 3.6.1). In this letter, Klein also reported about his unsuitability for military service: “Yesterday I was examined to determine my fitness for military service and, for the time being, I was found to be unsuitable. Yesterday, too, I received a 324 See Gaston Darboux, “Sophus Lie,” Bull. Amer. Soc. 5 (1899), pp. 367–70; M. NOETHER 1900, p. 14; and STUBHAUG 2002, pp. 145–46. 325 See Klein’s vita from December 5, 1870 in TOBIES 1999a, p. 85. GRAY’s remark (2013, p. 489) that Klein “served in the Prussian army” misleadingly suggests that he was active in the armed service. 326 In his letter to Lie, Klein added that the dashed line in this sketch (Fig. 10) was improperly (that is, asymmetrically) drawn. The published article contains the correct version. 327 Reprinted in KLEIN 1921 (GMA I), p. 94.

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preliminary response to the application in which I made myself available to the Ministry of War.” He did not, however, receive the position of an intendance official (an administrative official in the army), which he had expected.

[…] Yesterday, too, I received a preliminary response to the application in which I made myself available to the Ministry of War. It seems probable that I will be deployed as an intendance official in Muenster. Aside from that, things are going extremely poorly for me. Even less than usual, I have no idea of what I should be doing, and I am feeling crazier than ever before. Although I have plenty of delightful time to work, I dawdle away all of this time by doing nothing at all (etc.) in the manner with which you are familiar. – A few days ago, Wenker was deployed to serve in the infantry. Farewell. Greet everyone on my behalf (Darboux, Jordan, Liouville, et al.), and please let me hear something from you soon. Yours, Felix Klein Figure 10: An excerpt of a letter from Klein to Lie dated July 29, 1870, including a sketch of the asymptotic curves between two double points on a Kummer surface [Oslo].

After Klein had devoted eight days to second-degree complexes, pondering all the while that he might “quietly work on mathematics by myself here throughout the duration of the war,” he traveled to Bonn and joined an emergency volunteer organization (Nothelferverein). This organization equipped people with backpacks, caps, and Red Cross armbands in order to send them onto the battlefield “with the purpose of seeking out the wounded and providing them with refreshment, etc., hearing their final wishes, writing the necessary letters for them, etc.”328

328 [Oslo] A letter from Klein to Sophus Lie dated August 8, 1870.

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Klein was serving as a paramedic in the Walloon region of Belgium when, on August 19, 1870, he met Friedrich Althoff, who after the war would work as a legal advisor and consultant for ecclesiastical and educational affairs at the (now) Prussian university in Strasbourg and who, in 1882, would move to Berlin to take a new position at the Prussian Ministry of Culture. Ten years older than Klein, Althoff was already married and was likewise serving as a paramedic on the battlefield. Their conversations during their march to Couvin, a municipality in the province of Namur, concerned Klein’s plan for his Habilitation and created a foundation for their future encounters. Klein’s service as a paramedic was brief. When, approximately four weeks later, he next wrote to Sophus Lie, he was convalescing in the Belgian city of Bouillon. Having tended to wounded soldiers in Metz and Sedan, and having evacuated field hospitals, his experiences were certainly not for the faint of heart, though he acknowledged that he himself did not have to be “in the true fire.”329 His recovery, however, was soon followed by another illness: “I fell ill […] and had to leave Château Thierry, which we had reached on foot. For more than fourteen days, I have been at home receiving medical attention. My illness, which consists of a gastric fever, is not at all dangerous, but it has lasted a long time.”330 Whereas Klein’s friend from school Albert Wenker died of typhus, Klein himself was able to recover with plenty of bedrest at home. He kept to a strict diet, and by the middle of November in 1870 he was healthy enough to receive a visit from Sophus Lie, to finish their joint article on the principal tangent curves of the Kummer surface, to submit his Habilitation materials, and to regain the attention of the Prussian Ministry of Culture (see Appendix 1). The peace treaty between Germany and France had not yet been signed, but all that mattered to Klein and Lie was mathematics. From Düsseldorf, they sent a letter in mid-November to Ernst Eduard Kummer in Berlin about their “research on the fourth-degree surface with sixteen nodes,” to which Kummer responded on the 26th of November with the suggestion that they should send him “what [they] had discovered about this fourth-degree surface in a fully substantiated, if short, form as a brief self-standing article.” Kummer hoped to present this article to the board of the Academy and have it published, but he also expressed a tinge of skepticism, noting that their work “might be seen by the mathematicians in Berlin as a corruption of the development of geometry if the mere results are published without the necessary justification.”331 This remark was probably meant as a slight to many of the articles published in the Comptes Rendus and the Göttinger Nachrichten as well. However, Klein and Lie were able to fulfill Kummer’s wishes remarkably quickly. On December 14, 1870, Klein sent the manuscript of the coauthored article to Kummer, who presented it to the Academy a day after he had

329 [Oslo] A letter from Klein to Lie dated September 13, 1870. 330 [Oslo] A letter from Klein to Lie dated October 18, 1870. 331 [Oslo] A letter from Kummer to Klein and Lie dated November 26, 1870.

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received it.332 The range of mathematicians to whom Klein thought to send offprints of this article is an indication of how widely his network already extended: “Cayley, Sylvester, Salmon, Cremona, Battaglini, Beltrami, the mathematicians from Berlin, Geiser, Sturm, Schroeter, Reye, Brill, Lüroth, and so on.”333 Klein’s later remark that the Franco-Prussian War, “though it affected all of our experiences so profoundly, did not disrupt our scientific relationships as much as one might now suppose,”334 is confirmed by his correspondence at the time. Even before the peace treaty was signed, Klein had already reestablished contact with Paris. A letter that he sent to Darboux on February 14, 1871 began with the following words: “Dear Mr. Darboux! I don’t know where or how this letter will find you or even if you will welcome it, coming as it does from my country.” The twenty-one-year-old Klein described the course of events that his life had taken and ended his letter with a hopeful wish for lasting peace: After I so suddenly had to leave Paris, I joined a relief corps as a voluntary paramedic. As such, I spent some time – until the end of September – on the front lines. My discharge from this activity was not voluntary, for I had become significantly sick. I spent the entire time until the new year as a convalescing patient, during which period I could not bring myself to engage in any scientific work. A pleasant diversion for me was a visit from Lie, who came to me on his way back to Christiania and who had completed his Habilitation in the meantime. After the new year, I resumed my usual scientific activity; I went to Göttingen and settled at the university. Of course, the enrollment of students there is extremely low at the moment. People are not yet ready to resume the quiet activities of peacetime. I hope that things will soon be better and remain so for a long time!335

For his part, Darboux reacted positively, and their exchange of letters resumed. Klein would not return to Paris until 1887, but in the meantime he sent a number of young mathematicians there (Ferdinand Lindemann, Walther Dyck, Eduard Study, David Hilbert); he established contact with additional French mathematicians; and he guided several young French students, who had been sent to study under him by Darboux, toward producing publishable research.336 Whereas the First World War would later cause enormous differences between French and German scientists, the relationship between Klein and Darboux in 1870 and 1871 was unaffected by such severe nationalistic excesses. This is not to say that such passions were not heated in the scientific community at the time. Among mathematicians, for instance, Ernst Eduard Kummer made patently antiFrench remarks,337 while Camille Jordan was decidedly anti-German. Later, how-

332 333 334 335 336 337

[Oslo] Klein to Lie, Dec. 13, 1870. For the article, see KLEIN 1921 (GMA I), pp. 90–97. [Oslo] An undated letter from Klein to Lie (probably around the end of February in 1871). KLEIN 1921 (GMA I), p. 51. [Paris] 42: Klein to Darboux, February 14, 1871 (published in TOBIES 2016, p. 106). See Section 5.4.2.1. [Oslo] Kummer to Klein, Nov. 26, 1871: “In this war, France has shown itself to be a morally depraved nation.” According to STUBHAUG (2002, p. 136), “Kummer’s ill-will toward the French no doubt stemmed from his traumatic childhood experience, from the period when Napoleon’s army invaded his hometown of Sorau and infected the populace with typhus.”

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ever, Jordan would overcome his overly nationalist tendencies when Klein sent young mathematicians to study under him in Paris.338 In 1870, the anti-French attitude of the meteorologist Heinrich Wilhelm Dove, which Klein had encountered in Berlin, was reflected in the stance of his eldest son, Richard Wilhelm Dove, in a drastic manner. The latter, who was then the Prorektor of the University of Göttingen, officially endorsed the idea that Paris should be bombarded with cannon fire, and he was celebrated throughout Germany for doing so.339 Richard Wilhelm Dove, a professor of canon law at the University of Göttingen and member of the National Liberal Party, is also worth mentioning here because the non-partisan Felix Klein would later succeed him as the University of Göttingen’s representative in the Prussian House of Lords (see Section 8.3.4.1). 2.7.2 Habilitation Then I spent three days in Göttingen. It seems fairly certain that I will complete my Habilitation there, which, as you also expressed, would be the most reasonable thing for me to do.340

Klein had discussed the next steps of his career with Sophus Lie and, with Clebsch’s advice, he had already determined what he ought to do even before his trip to Paris. On December 5, 1870, Klein sent the following application to the Philosophical Faculty of the University of Göttingen: Allow me to present to the esteemed Faculty of Philosophy of the University of Göttingen my humble request to be appointed a Privatdozent of mathematics there on the basis of the following enclosed documents: 1. A doctoral diploma, 2. A curriculum vitae, 3. Copies of the publications listed in my curriculum vitae. I have submitted this application to the esteemed Faculty from my hometown because I am presently recovering there from the consequences of a long illness that will detain me until the new year. While convalescing, however, I do not wish to delay this application any longer, for I had hoped to submit it before the beginning of the semester. So as not to waste any further time, please allow me to suggest that I deliver a probationary lecture on one of the following three topics: 1. A demonstration of a model of Plücker’s general complex surface; 2. On those curves that are satisfied by a linear differential equation of the first order; 3. On the fourth-degree Kummer surface with sixteen nodes.

338 Commenting on a letter that Jordan had written to Clebsch, Klein remarked that Jordan’s response was “not something to be expected from a thoughtful man; every line was full of patriotic (that is, French) passion. He terminated his membership with the Academy here and thereby broke off all relations.” [Oslo] A letter from Klein to Lie dated July 12, 1871. – [Lindemann] Memoires, p. 68. On April 2, 1886, Hilbert wrote to Klein that Jordan had gone out of his way to ensure that they (Hilbert and Study) would warmly greet Klein on his behalf and inform him that all of his eight children could speak German (see FREI 1985, p. 4). 339 [Lindemann] Memoires, p. 39. 340 [Oslo] A letter from Klein to Lie dated March 29, 1870.

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If possible, I would like to request that this lecture should be scheduled to take place during the first days of the new year. With the utmost respect, Dr. Felix Klein341

The copies of publications listed in Klein’s curriculum vitae included his doctoral dissertation, his edition of the second volume of Plücker’s Liniengeometrie, his articles from 1869, and his co-authored articles with Sophus Lie that appeared in the Comptes Rendus. Klein did not have to submit an independent Habilitation thesis, but this was not unusual at the time. Clebsch himself had earned his Habilitation without submitting such a thesis.342 Two days after receiving Klein’s application, Alfred Clebsch wrote: Dr. F. Klein, whom I have known personally and through his writing for a long time, has shown through his talent, knowledge, and relatively early achievements that he can meet the highest expectations, and I believe that the Faculty can only rejoice that he has decided to pursue his first professional activity here in Göttingen. Among the proposed topics of his probationary lecture, I would vote for the first.343

It already seemed somewhat certain in advance that Clebsch would choose the first topic, since Klein had informed Lie early on: “Incidentally, I have already applied to Göttingen and, as the topic of my probationary lecture, I have chosen to give a demonstration of Wenker’s model.”344 Klein had already given a lecture in Berlin on Albert Wenker’s model of Plücker’s general complex surface (see Section 2.4.3). After Klein had arrived in Göttingen on January 2, 1871,345 the dean of the Philosophical Faculty, Karl Hoeck,346 issued an invitation to attend “The probationary lecture and colloquium of Dr. Felix Klein on next Sunday, the 7th of January, at six o’clock,” and he assigned Clebsch the task of leading the colloquium. Later, Klein reported about this event to Wilhelm Lorey: It was much simpler then than it is now. My previously published studies were kindly accepted as my Habilitation thesis. At the dean’s house, where wine and cake were served, I gave a lecture to the assembled honorary faculty (ca. eight members), who were all sitting around the table, about a model that I had made of Plücker’s general complex surface, and then I answered a few questions from Clebsch on that topic.347

In addition to the dean and Clebsch, the following other participants were in attendance: the historian Georg Waitz, the botanist Friedrich Gottlieb Bartling, the physicist Wilhelm Weber, the philosopher Hermann Lotze, the geologist Wolf-

341 [UAG] Phil. Dek. 156 (1870/1871), pp. 510–12 (quoted from TOBIES 1999a, p. 84). 342 See BIERMANN 1988, pp. 363–68. In Berlin, the Habilitation thesis did not become a requirement until 1883, and the first to be submitted there were by Johannes Knoblauch (on March 15, 1883) and Carl Runge (on June 6, 1883). 343 Quoted from TOBIES 1999a, p. 86 (a handwritten document by Clebsch dated Dec. 7, 1870). 344 [Oslo] A letter from Klein to Sophus Lie dated December 12, 1870. 345 [Oslo] A letter from Klein to Lie dated January 15, 1871. 346 A classical historian, Hoeck was also the director of the Göttingen University Library. 347 Quoted from LOREY 1916, p. 191.

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gang Sartorius von Waltershausen,348 the theologian and orientalist Ernst Bertheau, the Germanist Wilhelm Müller, and the Protestant theologian Friedrich Ehrenfeuchter.349 In 1885, Wilhelm Müller would be the dean of the Faculty when Klein’s appointment to Göttingen as a full professor was discussed (see Section 5.8.2). Georg Waitz, Wilhelm Weber, and Hermann Lotze would likewise serve as important contacts for Klein later on. On January 7, 1871, Dean Karl Hoeck made the following remarks in Klein’s Habilitation file: “He has met all academic requirements in an outstanding way; even the external form of his lecture met with general approval.” The Faculty submitted an application to the Royal University Curatorium in which it was recommended that “[…] Dr. Klein be provisionally granted the venia legendi in the subject of mathematics.” This request was approved – “provisionally for a period of two years” – on January 13, 1871.350 2.8 TIME AS A PRIVATDOZENT IN GÖTTINGEN Dr. Klein, who is fully here now, has told me much about the pleasant time that he experienced in Paris. As always, he is very industrious […]; I am happy to have gained such an active and amiable colleague here.351

In the letter quoted above to Camille Jordan, Clebsch demonstrated his appreciation for Klein, who produced new research results, took up his ideas, and even took the time to read through the proofs of Clebsch’s articles. Klein gave his lecture courses and exercises, supervised advanced students, and participated in the many social engagements to which the Privatdozenten felt obliged: attending balls, carousing in pubs, and going to academic sessions.352 In a letter to Lie from February 9, 1872, Klein remarked: “Fathers who have daughters invite me, even if I have yet to pay a formal visit. What is one supposed to do in such a situation?” On November 4, 1871, with Clebsch’s endorsement, Klein was made an Assessor (committee member, assistant) of the Mathematical Class of Göttingen’s Royal Society of Sciences (known as the Academy of Sciences since 1942). At the same meeting, Arthur Cayley (Cambridge) had been named an external member of this society, and Ludwig Schlaefli (Bern) and Hermann Grassmann (Stettin) had been appointed corresponding members.353 Klein had already made successful 348 Wolfgang Sartorius von Waltershausen had contributed to Gauss’s research on the earth’s magnetic field. A close friend of Gauss, he wrote the book Gauß zum Gedächnis [In Memory of Gauss] (Leipzig: Hirzel, 1856). 349 [UAG] Phil. Dek. 156 (1870/1871), p. 509. 350 Ibid., pp. 517–18. 351 A letter from Clebsch to Jordan dated March 5, 1871 (quoted from LÊ 2015, p. 171). 352 [Oslo] Two letters from Klein to Sophus Lie (one undated from the beginning of 1871 and the other dated January 1, 1871). 353 Klein’s and also Grassmann’s appointment were approved by sixteen of the eighteen voting members; Cayley and Schlaefli were elected unanimously. [AdW Göttingen] Pers. 12, p. 288.

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use of studies by Cayley, Schlaefli, and Grassmann, and now, as an Assessor, he could submit his own results for publication in the Academy’s proceedings. Clebsch repeatedly drew attention to Klein’s work in his publications.354 He helped his favorite student receive a professorship in Erlangen early on, and he still ensured on October 12, 1872, before his untimely death from diphtheria on November 7, that Klein was named a corresponding member of the Academy in Göttingen (along with Sophus Lie and Adolph Mayer).355 As a Privatdozent, Klein was still financially dependent on his parents, and in 1871 he had to cancel a trip that he had planned to visit Lie in Norway (see Section 2.1.1). His only source of income was the fees that students had to pay to attend his “private” courses. Klein’s courses are considered in the context of the members of the Seminar of mathematics and physics (see Section 2.8.1). The many works that Klein completed during this time are evidence of a higher level of productivity in his research life: further studies on the relationship between line geometry and metric geometry; essential new ideas for the theory of equations; the development of his ideas on non-Euclidean geometry; fundamental thoughts on how to systematize various branches of geometry; ideas about how to model and classify cubic surfaces. All of this was accomplished nearly simultaneously, and his studies from this time involved closely related approaches to those that he would later develop more deeply in his Erlangen Program. I will investigate the trends that defined Klein’s research there and the conditions that enabled this creative phase, during which he also supervised his first doctoral student (see Section 2.8.2). It will also be shown how Klein was engaged in mathematical and nonmathematical circles in Göttingen and elsewhere. He followed Clebsch’s idea of unification, but also his own social drive (Section 2.8.3). 2.8.1 Klein’s Teaching Activity and Its Context Every semester, the Göttinger Nachrichten announced the courses that would be taught at the university.356 After being granted the venia legendi on January 13, 1871, Klein offered exercises during the ongoing semester on “select chapters of geometry” (two hours per week),357 and he explained to Sophus Lie: “I am living quite happily and, as a so-called privatissimum, I have already begun to teach mathematical exercises, for which I have six students.”358 In the subsequent sum354 See, for instance, Göttinger Nachrichten (Dec. 6, 1872), pp. 621–23; CLEBSCH 1872, p. 26. 355 [AdW Göttingen] Pers. 20 (Clebsch’s proposals). Clebsch fell ill while travelling to the Ministry of Culture in Berlin to discuss a job offer that he had received from the University of Vienna. See Göttinger Nachrichten (1875), p. 265. 356 See https://www.sub.uni-goettingen.de/sammlungen-historische-bestaende/alte-drucke-15011900/historische-vorlesungsverzeichnisse/#c9191. 357 See KLEIN 1923 (GMA III), Appendix, p. 4. 358 [Oslo] A letter from Klein to Lie dated January 22, 1871.

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mer semester in 1871, Klein formulated his course offerings without indicating their exact times or hours per week (see Table 4). His lectures on Plücker’s complexes were attended by five students, and his lectures on theoretical optics were attended by nine.359 In order to provide a better idea of Klein’s teaching activity as a Privatdozent, it will be necessary to introduce the Royal Mathematical and Physical Seminar (Kgl. mathematisch-physikalisches Seminar) as well as his colleagues there. Both are of interest, because the Seminar and most of these colleagues would still be there when, fifteen years later, Klein would return as a professor to Göttingen and serve as one of the directors of this Seminar. In 1850, Moritz Abraham Stern (b. 1807)360 had founded the Royal Mathematical and Physical Seminar, which had been based on a similar institution at the University of Halle (founded in 1839). The stated aim was to offer a “coherent and systematic curriculum” that would encourage students, future secondary school teachers of mathematics and physics, “to stay longer in Göttingen.” The founding statute of the Seminar obliged its members to participate, on a weekly basis, in two hours of mathematical exercises and up to four hours of exercises in physics.361 Within the framework of the Seminar, M.A. Stern and Georg Ulrich (b. 1798) directed the mathematics division, while Wilhelm Weber (b. 1804) and J.B. Listing (b. 1808) led the physics division.362 Ernst Schering (b. 1833) offered exercises on magnetism. In addition, the following other course listings were announced for the Seminar: “Prof. Ulrich, mathematical exercises, Wed., 8 o’clock,” “Prof. Stern on certain properties of continued fractions, Wed., 8 o’clock,” “Prof. Klinkerfues, one meeting per week: instruction in astronomical observations,” “Prof. Listing, exercises in physics.”363 Whereas Gauss, Dirichlet, and Riemann had never participated in this Seminar, Clebsch followed the example of the Königsberg school364 and intentionally used it as a venue for teaching the latest research in the field. This is also evidenced by the seminar that he and Felix Klein co-taught in the summer of 1872.365 For Klein, this manner of teaching would serve as a model throughout his career.

359 Klein kept a list of all the students who participated in his lectures and seminars from 1872 to 1920, a document that is preserved in [UBG] Cod. MS. F. Klein 7 E. The figures cited here are from pages 1–2 of this list. 360 Regarding M.A. Stern see also below, and SCHMITZ 2006. 361 A brief history of this Seminar is contained in [UBG] Cod. MS. F. Klein 2 E, pp. 13–14. 362 [UBG] Cod. MS. F. Klein 2 E, pp. 13–14. 363 See Göttinger Nachrichten (1871), pp. 64, 66; and [UAG] Math. Nat. 0012 (current affairs). 364 See also OLESKO 1991. 365 See below Section 2.8.2, and [Protocols] vol. 1.

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Table 4: A list of course offerings in mathematics, physics, and astronomy at the University of Göttingen for the summer semester of 1871.366 Instructor

Course Title

Days

Time

Prof. Ulrich

Stereometry with Spherical Trigonometry

M, T, Th, F

10 AM

Prof. Ulrich

Practical Geometry with Exercises in the Field

4 hours/week

5–7 PM

Prof. Clebsch

Analytical Geometry of Surfaces

M, T, Th, F

12 PM

Prof. Clebsch

Select Chapters of Higher Geometry

M, T

11 AM

Prof. Stern

Theory of Numerical Equations

4 hours/week

8 AM

Prof. Stern

Differential and Integral Calculus

5 hours/week

7 PM

Prof. Enneper

Theory of Definite Integrals

M, T, W, Th, F

10 AM

Prof. Schering

Functions of Complex Variables, esp. Elliptic, Abelian, and Riemannian Functions

4 hours/week.

9 AM

Dr. Klein

On Plücker’s Complexes

1 or 2 hours/week (free of charge)

Dr. Minnigerode

Theory of Linear Partial Differential Equations and Their Applications to Mathematical Physics

4 hours/week

Dr. Klein

On Theoretical Optics

4 hours/week

Prof. Clebsch

Exercises on Topics in Newer Algebra

W (public)

12 PM

Prof. Schering

Magnetic Exercises (for members of the Math. and Physical Seminar)

Friday

6 PM

Dr. Klein

Mathematical Exercises on Some Aspects of Geometry

Prof. Klinkerfues

Spherical Astronomy

M, T, Th, F

12 PM

Prof. Weber

Physics, First Part

M, T, Th, F

5–6 PM

Prof. Listing

Optics, Including Crystal Optics

4 hours/week

12 PM

Prof. Listing

On the Eye and the Microscope

Private instruction during available hours

Prof. Listing

Exercises in Practical Physics

Saturday

366 Göttinger Nachrichten (1871), pp. 63–65.

10–12

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Along with Gauss as professor of astronomy, Georg Ulrich had been a full professor of mathematics since 1831. In addition, he was a government councilor and was responsible for examining teaching candidates in mathematics and physics. His lectures concerned practical geometry, mechanics, analysis, and other areas of geometry, but he did not incorporate the latest research methods in his teaching.367 Moritz Abraham Stern was the first unbaptized mathematician from a Jewish family to become a full professor at a German university, though he was first offered this position thirty years after his doctoral examinations, which Gauss had evaluated as summa cum laude in 1829. Already a Privatdozent in 1829, Stern had to wait nineteen years before becoming an associate professor (Professor extraordinarius). His lucid lecturing style was generally praised.368 According to Aurel Voss, however, the content of Stern’s lectures seldom extended beyond the mathematics of Fourier’s time (d. 1830).369 Ferdinand Lindemann, who attended Stern’s lectures on algebraic analysis in 1870/71, reported that Stern based his lectures on his own book, “in a fairly modern style, because all operations were at first only supposed to have a symbolic meaning, and only afterwards did he mention their application to the numerical system.”370 Klein had already attended some of Stern’s lectures in 1869 (see Section 2.4.1), and he maintained a good relationship with both Stern and his son Alfred for many years. When Klein returned to Göttingen as a full professor in 1886, it was Stern’s position that he was appointed to fill (see Section 5.8.2). Around the year 1871, the physicist Wilhelm Weber, who had once collaborated with Gauss (famous for the Gauss-Weber telegraph, 1833), was said to have conducted mostly unsuccessful experiments. He also maintained a heated polemic with Hermann von Helmholtz (b. 1821).371 Klein, on the contrary, integrated Helmholtz’s law on the conservation of force into his teaching and did not take much interest in Weber’s field of research at the time.372 Johann Benedict Listing, who had earned his doctoral degree under Gauss in 1834, was made an associate professor of physics in 1839 and a full professor of mathematics in 1849. He coined such terms as “geoid” and “topology,” though mathematicians still – for a long time – used the older notion analysis situs for the latter area of research. Aurel Voss, who had attended Listing’s lectures, remarked

367 368 369 370

See VOSS 1919, p. 280. It was appreciated, for instance, by Richard Dedekind. See LOREY 1916, pp. 81–82. VOSS 1919, p. 280. [Lindemann] Memoirs, p. 40. The book in question is M.A. Stern, Lehrbuch der algebraischen Analysis (Heidelberg: C.F. Winter, 1860). 371 [Lindemann] Memoirs, pp. 40–41. In particular, Weber took issue with Helmholtz’s book On the Conservation of Force, which was originally published in German in 1847. 372 See KLEIN 1923a (autobiography), p. 15. On the polemic between W. Weber and Helmholtz and the latter’s plea to orient physics around empirically tested natural laws, see Helmholz’s preface to Thomson and Tait’s Handbuch der theoretischen Physik (Braunschweig: Vieweg, 1871, pp. VIII–IX); and KLEIN 1979 [1926], pp. 22–23.

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that “his over-abundant use of ingenious terminology was not always well-suited for promoting genuine insight.”373 This was a time in which surface topology was developing into an independent subdiscipline and topological methods were first being incorporated into the study of projective geometry. Later, Felix Klein would play a role in these developments: By establishing the idea of a group and the notion of space as a number manifold, Klein managed to produce a concise summary of Listing’s definitions, which one could formulate as follows: the task of analysis situs consists in determining all of those properties of spatial configurations that behave invariantly toward the group of all continuous transformations of space.374

Alfred Enneper (b. 1830) had likewise studied under Gauss. Although he had completed his Habilitation in 1859, he would not become an associate professor until 1870. Klein made use of his findings in differential geometry and also sent them to Sophus Lie.375 According to Aurel Voss’s judgement, Enneper had a good knowledge of recent French and Italian scholarship, and he prepared his lectures meticulously. Nevertheless, his lectures were sparsely attended (Lindemann reported the presence of only two or three students). The whole time, Enneper only looked at the blackboard and wrote down his lecture systematically and flawlessly, without any written notes.376 Ernst Schering, who had been a full professor of mathematics since 1868 and was one of the directors of the observatory for theoretical astronomy and geodesy,377 was primarily engaged in editing Gauss’s collected works for the Göttingen Royal Society of Sciences – a project that Klein would carry on after Schering’s death (see Section 8.3.1). In 1885, Schering would oppose hiring Klein as a full professor in Göttingen on the grounds that Klein belonged to a different scientific school (see Appendix 4.2). For the same reason, Clebsch, too, after he had moved to Göttingen to take up his professorship in 1868, had to endure Schering’s utter lack of friendliness “to an unbelievable extent.”378 An associate professor as of 1867, the astronomer Ernst Friedrich Wilhelm Klinkerfues (b. 1827) was for various reasons unable to teach his scheduled courses at the time.379 Bernhard Minnigerode (b. 1837), who was one of Klein’s fellow Privatdozenten in Göttingen, had studied under Riemann and had completed his Habilitation in 1866. In 1874, he became an associate professor in Greifswald, where he was promoted to full professor in 1885. While in Greifswald, Minnigerode focused on classifying crystal groups by means of a geometric group concept. Later,

373 374 375 376 377 378 379

VOSS 1919, p. 280. DEHN/HEEGAARD 1907, p. 154. See also SCHOLZ 1980, pp. 142–79. [Oslo] A letter from Klein to Lie dated February 1, 1872. See VOSS 1919, p. 280; and [Lindemann] Memoirs, pp. 40, 42. See Göttinger Nachrichten (1875), p. 282. [Deutsches Museum] No. 1968-2/2 (Clebsch to M.A. Stern on August 8, 1868). [Lindemann] Memoirs, p. 44.

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this approach would be superseded by Arthur Schoenflies’s studies of elementary spatial groups – studies encouraged by Klein (see Section 6.3.6.1). Alfred Clebsch impressed his students not only with the content of his lectures but also with his rhetorically sophisticated lecturing style. In the summer of 1871, between twenty and thirty students attended his lectures.380 In the summer of 1872, he delivered lectures on the theory of elliptic functions to more than seventy students, including Klein and Lindemann.381 Whereas, in the future, Klein would use his semester breaks to prepare new lectures, he at first had little time for that: “I prepared from one lecture to the next as best as I could, for which my friends in physics, Riecke and Neesen, were especially helpful.”382 That he much rather would have been teaching and conducting research in mathematics instead of physics is evident from his letters at the time. In order to earn more money during his first two semesters, he had to concentrate on theoretical physics. In his later autobiography, Klein interpreted this as “his plan to become a physicist.”383 In September of 1871, however, he wrote the following to Gaston Darboux: “My own work has been slowed down considerably by my lecture on theoretical optics.”384 To Sophus Lie, Klein wrote: Yet perhaps one also sees things with less prejudice if one possesses an overview of neighboring disciplines (I count theoretical physics as one of these). This is approximately how I motivate myself when I now have to deal with physics, etc. The next and only cogent reason that I do this, however, is due to the conditions at the university here, which barely allow me to teach anything but physics. That said, I am quite attracted right now to the notion of mathematical physics as it is practiced by W.[illiam] Thomson. Just as he has regenerated mathematical physics by placing its physical content in the foreground, I think it would be possible to regenerate geometry in a similar way.385

Klein’s lectures in physics were thus a temporary solution. Nevertheless, he immersed himself in the material and even conducted experiments, as Eduard Riecke reported (see Appendix 10.1). In the winter semester of 1871/72, eleven students attended Klein’s lectures “On the Interaction of Natural Forces and the Law of the Conservation of Force.” In these lectures, Klein discussed the theory of heat and electricity, for which he relied especially on William Thomson and Peter Guthrie Tait’s Treatise on Natural Philosophy (vol. 1, 1867),386 which had been recommended to him by his Scottish friend William Robertson Smith.387 In a letter to Plücker’s widow, Klein still expressed certain reservations about having to teach physics: “I returned here from Düsseldorf in September. Fourteen 380 381 382 383 384 385 386

[Lindemann] Memoirs, p. 41 See LOREY 1916, p. 161. [UBG] Cod. MS. F. Klein 22 L: 4, p. 5. KLEIN 1923a, p. 17. [Paris] 45: A letter from Klein to Darboux dated September 5, 1871. [Oslo] A letter from Klein to Lie dated October 1, 1871. Trans. into German by G. Wertheim, edited by H. v. Helmholtz (Handbuch der theoretischen Physik). Braunschweig: Vieweg, 1871. 387 For further discussion of Klein’s relationship with W.R. Smith, see Section 3.3.

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days ago, the semester began with its dreaded lectures. I have ten to twelve regular students and can thus be quite content. In general, the mathematical conditions are as lovely as ever; there have never been so many students.”388 It should be mentioned that both of Klein’s physics lectures, in the summer of 1871 and the winter of 1871/72, were attended by Richard Börnstein, who later became a prominent physicist and meteorologist. At the time, the latter was close to completing his doctoral degree under Wilhelm Weber’s supervision in 1872.389 In the case of theoretical physics, Klein filled a gap at the university; his own field, geometry, was already represented by Clebsch at the time. During the winter semester of 1871/72, Klein’s hour-long public lecture, “On the Application of Transformations in Geometry,” was attended by only four students. Among these were his friend Friedrich Neesen and Carl Rodenberg, who went on to create – and assemble a collection of – mathematical models. In the summer semester of 1872, Clebsch handed over his introductory geometry lectures to Felix Klein, and conducted a joint research seminar with him. This allowed Klein to concentrate on geometry in teaching and research. From April 1 to July 27, 1872, he lectured four times per week at 8 AM on the analytic geometry of the plane. These lectures were attended by thirty-eight students, including his later doctoral student Adolf Weiler and, once again, Neesen and Rodenberg.390 Ferdinand Lindemann, who enrolled in these lectures later, reported that Klein had based his course on Karl Georg Christian von Staudt’s Theorie des Imaginären [Theory of the Imaginary],391 which he had first gotten to know while studying non-Euclidean geometry. It is worth emphasizing that Klein would soon be offered a professorship in Erlangen, where von Staudt had taught as a professor himself. From the course listings for the winter semester of 1872/73, we know that, if Klein had remained in Göttingen, he would have offered the following lectures: “Analytical Geometry of Space” (three lectures per week), and “On Higher Elements of Plane Geometry” (four lectures per week).392 In October of 1872, when Klein became professor in Erlangen, the Privatdozent Friedrich Neesen took over his teaching duties (i.e., the geometry program) in Göttingen for the winter semester of 1872/73. Neesen was followed in this capacity by Aurel Voss, with Klein’s endorsement (see 3.1.2). Clebsch’s professorship was not filled until April 1, 1874, namely by Lazarus Fuchs, a Berlin-trained mathematician with expertise in the theory of linear differential equations.393 Even though Fuchs’s time in Göttingen was brief, he and Weierstrass ensured that the Russian student Sofya Kovalevskaya was able to earn a doctoral degree 388 [Canada] A letter from Klein to Antonie Plücker (née Altstätter) dated November 10, 1871. 389 [UBG] Cod. MS. F. Klein 7 E, pp. 1–2. Börnstein is perhaps best known for the book Physikalisch-chemische Tabellen [Physical-Chemical Tables], which he co-edited with Hans Heinrich Landolt. First published in 1883, a version of the book is still in use today. 390 [UBG] Cod. MS. F. Klein 7 E, pp. 5–8. 391 [Lindemann] Memoirs, p. 45. 392 See Göttinger Nachrichten (July 24, 1872), p. 360. 393 See GRAY 1984.

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there in absentia, thus becoming the first female mathematician in Europe to be awarded such a degree in the 19th century.394 Klein admired Kovalevskaya’s work and would himself become a staunch supporter of the right of women to study at the university level (see Section 7.5). In Göttingen, Fuchs was succeeded in 1875 by H.A. Schwarz, who had likewise completed his doctorate in Berlin with Kummer and Weierstraß. In 1868, moreover, Schwarz married Marie Luise Kummer, a daughter of Ernst Eduard Kummer. In 1885, Schwarz and Schering would (unsuccessfully) oppose hiring Klein as a full professor in Göttingen (see Section 5.8.2). 2.8.2 An Overview of Klein’s Research Results as a Privatdozent During his three semesters as a Privatdozent, the industrious and amiable Klein – as Clebsch called him – completed sixteen manuscripts. Four of these studies were placed by Clebsch in the 1871 volume of the Göttinger Nachrichten, and, in 1872, three others were placed by Klein himself in that journal, for which he was then active as an Assessor. Nine articles (two of which were unmodified reprints from the Göttinger Nachrichten) were published by Clebsch in Mathematische Annalen. In addition, Klein worked on four models and was preparing his aforementioned Erlangen Program (1872), which would ultimately appear as the summation of his previous work. Some of these studies and their approaches have already been discussed above in their proper context, so that here I can limit myself to providing an overview meant to demonstrate the wide range of methods that Klein employed during this brief time period. First. In communication with Lie and in cooperation with Clebsch, Klein refined the results that he had formulated in Paris concerning W-configurations (see Section 2.6.2.1). In this case, Klein restricted himself to making a systematic study of various types of W-curves that occur on a plane. He had set aside his intended study of spatial W-configurations, for at first he found it too difficult and he was already immersed in too many other ideas. Klein ultimately submitted the article for publication in March of 1871.395 Beforehand, he reported enthusiastically to Lie that Clebsch had recognized certain connections to Abelian functions and that the differential equation of the complex in question could be integrated: Clebsch has now made me aware that, by the nature of φ, the integrals are precisely those Abelian integrals for which the inversion problem can be solved with the sums of three integrals. The theory of second-degree complexes is thus an illustration of the theory of Abelian functions for p = 3. Similarly, line geometry is in general an illustration of the theory of Abelian functions for p = 4, and the Kummer surface is an illustration of this theory for p = 2.396

Here, p is the topological invariant later called the genus (the maximum number of non-separating loop-cuts [Rückkehrschnitte]), which Riemann introduced in 394 For a detailed discussion of Kovalevskaya’s doctoral procedure, see TOLLMIEN 1997. 395 See KLEIN 1921 (GMA I), pp. 424–59. 396 [Oslo] A letter from Klein to Lie dated January 15, 1871 (emphasis original).

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order to analyze surfaces and their (conceivable) mappings (see Section 3.1.3.1). In the same letter to Lie, Klein added: “At the moment, I am entirely absorbed by these ideas; I believe that it would be very fruitful to pursue them, but first I have to learn more about Abelian functions.”397 Here we can see the origin of Klein’s intensive study of Abelian functions. Lie’s desire to further pursue the unprocessed spatial W-configurations was his main motivation for visiting Klein in the summer of 1872 (he had announced his plan to do so as early as January of that year). By the time he came, however, both of them had become occupied with other topics. The topic of W-configurations would be further advanced by others.398 Second. Through his correspondence with Lie and Darboux, Klein further investigated the relations between line geometry and metric geometry. He wrote his own studies, edited texts for Lie, and oriented himself toward French scholarship.399 One of Klein’s results, which he developed by analogy, is especially noteworthy. In Paris, he had already familiarized himself with Dupin cyclides (surfaces enveloped by a two-parameter family of spheres). Klein would use Dupin’s Theorem, which states that the surfaces in orthogonal families intersect along lines of curvature, to prove a theorem concerning the relationships between line complexes and principal tangent curves of the Kummer surface (completed on March 4, 1871): “Über einen Satz aus der Theorie der Linienkomplexe, welcher dem Dupinschen Theorem analog ist” [On a Theorem from the Theory of Line Complexes that is Analogous to Dupin’s Theorem].400 As late as 1913, mathematicians in Berlin would cite this finding in their proposal to appoint Klein a corresponding member of the Academy there (see Appendix 9). In a letter to Darboux dated September 27, 1871, Klein explained the connections that he saw between their respective work: As it seems to me, the problem that you have mentioned – “déterminer une surface, connaissant une propriété de ses sphères principales” – is identical with the problem treated by Lie in his article published in Christiania […]. Regarding my treatment of the integration of the general second-degree complex, I have thus used the elliptic line coordinates that I analyzed in my first note published in the Göttinger Nachrichten. […] Moreover, I have hardly any doubt that, for your part, you have taken almost the exact same path; it is very remarkable how your studies and those by Lie tend to agree so closely with my own. The reason for this, however, is not so coincidental; your studies of metric problems had captured our attention to a great degree; there Lie discovered the connection between line-geometric problems and the problems of metric geometry, and he dragged me along toward this line of investigation.401

397 [Oslo] A letter from Klein to Sophus Lie dated January 15, 1871. 398 See, for instance, Anders Wiman, “Über die W-Kurven im dreidimensionalen Raume,” Acta Mathematica 64 (1935), pp. 243–52. 399 For a discussion of Klein’s collaboration with French mathematicians at that time, see also ROWE 2019a. 400 See KLEIN 1921 (GMA I), pp. 98–105; and KLEIN 1979 [1926], p. 72; [Oslo] Letters from Klein to Lie dated January 21 and February 25, 1871. 401 [Paris] 49–50: Klein to Darboux, Sept. 27, 1871 (German original in TOBIES 2019b, p. 87).

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Further studies were soon to follow. In October of 1871, Klein completed his article “Über Liniengeometrie und metrische Geometrie” [On Line Geometry and Metric Geometry], an important early paper in which he offered the insight that “line geometry is equivalent to metric geometry in four variables.”402 In November of 1871, Klein submitted his article “Über gewisse in der Liniengeometrie auftretende Differentialgleichungen” [On Certain Differential Equations that Appear in Line Geometry] to Mathematische Annalen. Even though Aurel Voss would later find a few inaccuracies in this work, it was a classic example of how Klein classified ideas and drew connections: between Sophus Lie’s spherical geometry and Darboux’s work, to Kummer, to Hermann Schubert’s theory of characteristics, to Moritz Pasch’s Habilitation thesis, to Lüroth’s theory of skew surfaces, and to the possible definition of the focal surface of a congruence as a special complex. Later, as Eisso Atzema has shown, the latter idea was taken in yet another direction by Julius Weingarten.403 All of this led to what Klein would integrate and systematize in his Erlangen Program. Klein remained in touch with Darboux; he encouraged the Leipzig-based mathematician Adolph Mayer to write to Darboux in order to make his own research results known in Paris;404 and he introduced his later doctoral students (Staude, Domsch, Bôcher) to Darboux’s work. Whereas Klein dutifully cited the work of other authors, it upset him somewhat that DARBOUX (1873) made no mention at all of the relevant studies by himself and Lie. That said, they were both pleased that Darboux printed positive reviews of their work in his Bulletin.405 Third. Klein’s regular interaction with Clebsch led to him making contributions to the field of algebraic equations. Klein’s first result in this area of research was completed in May of 1871: “Über eine geometrische Repräsentation der Resolventen algebraischer Gleichungen” [On a Geometric Representation of the Resolvents of Algebraic Equations].406 Drawing on Galois theory and the work of Clebsch and Jordan, Klein discovered the basic principle that, “in the theory of equations, the invariant theory of those forms is relevant that pass into one another by means of particular discontinuous groups of linear substitutions.” This principle would serve as an indirect impetus for Klein’s Erlangen Program as well as the basis of his later work on transcendent automorphic functions.407 Fourth. Along with Otto Stolz, Klein delved deeper into the field of non-Euclidean geometry (see Section 2.5.3). Beginning in May of 1871, the two of them 402 Math. Ann. 5 (1872), pp. 257–77; reprinted in KLEIN 1921 (GMA I), pp. 106–26. – Engl. trans.: http://neo-classical-physics.info/uploads/3/0/6/5/3065888/klein_-_line_geometry.pdf 403 Math. Ann. 5 (1872), pp. 278–303 (KLEIN 1921 [GMA I], pp. 138–52). – Moritz Pasch, Zur Theorie der Komplexe und Kongruenzen von Geraden (Gießen, 1870); ATZEMA 1993. 404 See TOBIES/ROWE 1990, pp. 62–63 (a letter from Klein to A. Mayer dated Dec. 8, 1871). 405 [Oslo] A letter from Klein to Lie dated June 28, 1873. In the 2nd edition of Darboux’s book (published in 1896), there is a single reference (on p. 227) to Klein’s first article on nonEuclidean geometry: “[…] un Mémoire important de M. Klein (Math. Annalen, t. IV).” 406 This article appeared in Math. Ann. 4 (1871), pp. 346–58. 407 [Oslo] II: Klein’s notes dated November 1, 1892 (printed in ROWE 1992a, p. 599).

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lived in the same house in Göttingen. Not only did they work together but they also drank “a glass of beer every evening.” It is clear that they often had more than just one glass; as Klein wrote to Lie, “[…] I have a mild hangover, for yesterday evening I once again spent too much time at the pub.”408 Klein had reason to celebrate, for, as was seldom the case in his life, he happened to be “truly content” with his work and proud of what he had achieved: I am now able to prove what, at the time [i.e., in Berlin], I only had a vague notion of, namely that Cayley’s general metric leads to the exact same ideas as those of so-called non-Euclidean geometry, which Gauss, Bolyai, and Lobachevsky developed by disregarding Euclid’s eleventh axiom. This connection between two heterogeneous things seems all the more interesting to me because it sheds an entirely new light both on the true meaning of non-Euclidean geometry and on the significance of Cayley’s investigations – if, for once, I may express myself with such pride.409

Regarding the awkwardly formulated “eleventh axiom” (= fifth postulate) from Book I of Euclid’s Elements, it should be noted that there are statements that are equivalent to it, for example: “Given a line and a point not on it, one and only one line parallel to the given line can be drawn through the point” (thus the parallel postulate), or “The sum of the angles in every triangle is 180°.” After centuries of attempts to prove this postulate, mathematicians finally realized that a new, nonEuclidean type of geometry would arise if it were accepted that this does not hold. Around the year 1870, however, there was still no evidence of the consistency of non-Euclidean geometries. Klein worked on his new findings in non-Euclidean geometry until August of 1871. Clebsch presented a short version of these results in the Göttinger Nachrichten and a detailed version in the Mathematische Annalen.410 Klein called the two types of non-Euclidean geometry “hyperbolic geometry” (infinitely many parallels; the sum of the angles in a triangle less than 180°; developed independently by Gauss, Lobachevsky, and Bolyai) and “elliptic geometry” (no parallels; sum of angles greater than 180°; Riemann). And he referred to the usual Euclidean geometry as “parabolic geometry.” As mentioned above, Klein used a model borrowed from Cayley as a proof of consistency. With the help of projective geometry based on Cayley’s approach, he showed how it is possible to conceptualize both types of non-Euclidean geometry: by means of a second-degree surface as a so-called fundamental surface. That is, Klein detected that the inner points of a (real) conic surface ‒ a hyperboloid, paraboloid or ellipsoid (the sphere being a special case of the ellipsoid) ‒ can be interpreted as a model for non-Euclidean plane geometry. He did not make an illustration, and he did not consider the special case of the circle. When Klein published these articles in his collected works,

408 [Oslo] Klein’s letters to Sophus Lie dated April 15, April 31, and October 1, 1871. 409 [Oslo] A letter from Klein to Lie dated July 2, 1871. 410 Felix Klein, “Über die sogenannte Nicht-Euklidische Geometrie,” [On So-Called Non-Euclidean Geometry] Göttinger Nachrichten 1871, Nr. 17, pp. 419–33; and Math. Ann. 4 (1871), pp. 574–625 (KLEIN 1921 GMA I, pp. 244-305).

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he commented that one usually regards the circle as a Fundamentalkegelschnitt411; today, the circular disk model (Kreisscheibenmodell) is generally used.412 Darboux immediately commissioned the translation of Klein’s article (the short version) into French for publication in his Bulletin: “Sur la géométrie dite non euclidienne.”413 The translation was done by Jules Hoüel, one of the Bulletin’s coeditors. Hoüel may have introduced a few minor inaccuracies via his translation,414 but it was still Klein’s first publication to be translated by someone other than himself. Hoüel had also made the work of the Italian mathematician Eugenio Beltrami – whom Klein cited – available in France, where Liouville also dealt with this area.415 Wilhelm Killing referred to Klein’s Annalen article when he stressed “the important and beautiful theorem that non-Euclidean spatial forms can be derived from projective geometry by replacing the distance between two points with the logarithm of a certain cross-ratio.”416 Some established mathematicians reacted positively to Klein’s work on nonEuclidean geometry. In September of 1871, Klein was pleased to receive a letter from Beltrami in which the latter approved of his findings. Wilhelm Fiedler informed Clebsch that he intended to incorporate Klein’s results into his German translation of Salmon’s Analytic Geometry of Three Dimension. Klein wrote about this to Lie: “It is not difficult to see why such a thing is understood while others are not, and this is because it is met by a ready audience and does not offer much that is new but rather provides an overview of a series of things with which people are already familiar.”417 That said, Klein’s early work in this field did incite critique from certain mathematicians and philosophers.418 As Klein wrote to Lie, “On account of my work on non-Euclidean geometry, [Richard] Baltzer in Giessen has depicted me as the most objectionable and depraved human being because, in his estimation, I have brought things together that have nothing at all to do with one another.”419 In Hungary, where Bolyai’s work was well known, Julius König expressed doubts 411 KLEIN 1921 GMA I, p. 242. 412 Later, Henri Poincaré also developed a model of 2-dimensional hyperbolic geometry, a conformal disk model. See also GRAY 1985; 2006; 2013, pp. 38–55. 413 Bulletin 2 (1871), pp. 341–51. 414 [Paris] 59: Klein to Darboux, March 21, 1872. 415 Regarding Hoüel, see HENRY/NABONNAND 2017. – Eugenio Beltrami had developed a differential geometric metric that operated along the same lines as Cayley’s metric on a submanifold of the projective plane: “Teoria fondamentale degli spazii di curvatura costante,” Annali di Mat. Pura App., Ser. II 2 (1868–69), pp. 232–55. See Nicola Arcozzi’s article in COEN 2012, pp. 1–30; and SCHOLZ 1980, pp. 101–13, 125–41. 416 Wilhelm Killing, Die nicht-euklidischen Raumformen in analytischer Behandlung (Leipzig: B.G. Teubner, 1885), p. 262. 417 [Oslo] A letter from Klein to Lie dated September 29, 1871. In a letter on January 24, 1872, to Lie, Klein compared the warm reception of his work on non-Euclidean geometry with the scholarly community’s lack of interest in their collaborative work on W-configurations. 418 On the reception of Klein’s work by philosophers, see Section 2.8.2.3. 419 [Oslo] A letter from Klein to Sophus Lie dated April 16, 1872 (emphasis original).

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about Klein’s theory, but he was open to the idea.420 In 1873, König would come to Göttingen to attend a mathematics conference (see 2.8.3.4). Later, he arranged for Gusztáv Rados to study under Klein; together, König and Rados saw that Klein would be elected as a member of the committee that awarded the Bolyai Prize (see Section 5.4.2.4). Arthur Cayley remained generally skeptical of Klein’s contributions to non-Euclidean geometry.421 In Max Noether’s opinion, however: By means of his projective metric, [Cayley] had even done a service to philosophy: for since, as F. Klein stressed, the correlation of Staudt’s Würfe to cross-ratios is independent of our metric, the result is the ultimate subordination of the metrical to the projective, the identity of this metric with that of general hyper-Euclidean geometry in the space of constant curvature (as Beltrami implicitly suggested), and thus a new intuitive understanding of concepts of space that are independent of the parallel postulate.422

British mathematicians such as Andrew R. Forsyth and William Kingdon Clifford were quick to appreciate Klein’s results. Clifford expanded the definition of parallels to suit elliptic space geometry (thereby making non-Euclidean geometry realizable on a sphere: here there are no parallels, since the lines are simply great circles and these always intersect). Clifford’s expansion made it possible to construe two “parallels” (or “skew lines”) to a given line through a given point.423 This gave rise to further results in the field, which were soon outlined by Federigo Enriques.424 One more persuasive study was needed to establish the general idea that Euclid’s “parallel postulate” is independent of the other postulates and that a different, non-Euclidean geometry would necessarily arise if this postulate is no longer considered to be valid.425 The ongoing skepticism of many scholars prompted Klein to refine his ideas about the topic and to write “a long philosophical study,” which culminated in his (third) essay “Über die sogenannte Nicht-Euklidische Geometry” [On So-Called Non-Euclidean Geometry], dated June 8, 1872. In this article, Klein emphasized the utility of such investigations: In the case of mathematics, they introduced the new concept of “an arbitrarily extended manifold of constant curvature,” and, in the case of physics, he prophetically cited Riemann’s idea “that the reconceptualization of traditional spatio-mechanical ideas should not be hindered by the narrowness of concepts, and progress in our knowledge of the connection between things not inhibited by traditional prejudices.” 426 420 [Oslo] Klein to Lie on May 18, 1872. See also Julius König, “Ueber eine reale Abbildung der s.g. Nicht-Euclidischen Geometrie,” Göttinger Nachrichten 9 (1872), pp. 157–64. 421 See the discussion in JI/PAPADOPOULOS 2015, pp. 91–136. 422 Max NOETHER 1895, p. 479. 423 In three-dimensional geometry, skew lines are lines that do not intersect but are not parallel. 424 Federigo Enriques, “Prinzipien der Geometrie,” in ENCYLOPÄDIE, vol. 3 (1.1), pp. 112–17. 425 SCHOENFLIES 1919, p. 289. 426 Klein had sent the manuscript of this article to Lie in two installments, the first on June 5, 1872 and the second on June 8, 1872. – Because of a strike by the typesetters at the press, the publication of this article was delayed. It first appeared in Math. Ann. 6 (1873), pp. 112–45 (quotation, p. 114); (reprint in KLEIN 1921 (GMA I), pp. 311–43).

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According to Schoenflies, one of Klein’s main contributions was that he stripped non-Euclidean geometry of its “metaphysical accessories” and elevated it to “one of the most attractive and applicable realms of knowledge.”427 Because of these findings, Schoenflies regarded Klein as a “thoroughly cognizant pioneer of the general axiomatic-geometric approach to research […], an approach that would fully mature approximately ten years later, first in the work of [Moritz] Pasch and afterwards in that of Hilbert.”428 Recent scholars of the philosophy of mathematics see in Klein’s works the origin of structuralism.429 Klein’s third article on non-Euclidean geometry anticipated a number of significant aspects of his Erlangen Program: his discussion of groups of transformation; his reference to the fact that his definition of this concept derived from an analogous conceptual formulation in substitution theory; the example of groups of motions, first proposed by Jordan; the concept of the principal group (Hauptgruppe), with unchanging/invariant geometric properties;430 and his insight that, with more extensive groups, the number of invariant properties is smaller. In this article from June of 1872, as Erhard Scholz has underscored, Klein generalized the way that metric geometry could be integrated into projective geometry in such a way that other subdisciplines could also be included by specifying a manifold and assigning a transformation group to it: “In accordance with the importance that projective geometry possessed for Klein, he introduced the notion of a manifold (Mannigfaltigkeit) as a reformulation of n-dimensional projective space […].”431 Fifth. Klein’s Erlangen Program, the actual title of which was Vergleichende Betrachtungen über neuere geometrische Forschungen [Comparative Considerations on Recent Research in Geometry] (see Fig. 14), synthesized a number of the approaches discussed above. The starting point for writing this work may have been Klein’s vision, from November 20, 1871, “to write an essay of a very general sort on recent methods in geometry, in which I would like to show how each method (or nearly each) can be subsumed under the following general claim: to develop the properties of geometric things that are preserved in a given cycle of transformation.”432 This agrees with what Klein would write much later – on October 25, 1924 – to Friedrich Engel: The basic idea behind my Erlangen Program came to me in November of 1871, while I was attempting to synthesize the work of Hamilton and Grassmann under a single point of view. In the third volume of my collected writings, however, I stressed that Möbius’s complete works are borne by the same thought, though he never formulated it explicitly.433

427 428 429 430 431 432 433

SCHOENFLIES 1919, p. 289. Ibid. See BIAGIOLI 2016, 2020; and SCHIEMANN 2020. In modern mathematics, the term automorphism group is used for this. SCHOLZ 1980, p. 131. [Oslo] A letter from Klein to Lie dated November 20, 1871 (emphasis original). Quoted from Mitteilungen aus dem Mathematischen Seminar der Universität Gießen 35 (1945), p. 22. – Regarding A.F. Möbius, see also FAUVEL/FLOOD/WILSON 1993.

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In their work, Klein and Lie at first spoke of “systems,” “families,” or “cycles” of transformations. The concept of a group first appeared in their correspondence in December of 1871. On December 25, 1871, Klein wrote again about his idea to write an essay with “an overview of existing geometric methods,” which he intended to provide by “placing them into groups, each according to the cycle of transformations that they present.”434 This is confirmed by his later explanation that, in December of 1871, he came to the opinion that, for the study of a manifold, there are as many different manners of treating it as one can construe, within the manifold, continuous groups of any sort of transformations, and that the Euclidean and non-Euclidean metrics are just as surely included in the projective approach as their “groups” – given the appropriate choice of coordinates – are contained in the entire group of projective transformations.435

As Erhard Scholz has discussed, Klein at first used a limited concept of the manifold, which consisted of “the combination of a projective space (real or complex) with a group of transformations.” Not until 1874 did Klein begin to employ a broader concept of the manifold in his work.436 His concept of groups was intuitive, geometric: “The combination of an arbitrary number of transformations of space always results in a single transformation. If now a given system of transformations has the property that any transformation obtained by combining transformations of the system belongs to that system, it shall be called a group of transformations.”437 As he informed Lie, Klein had become enthusiastic about Grassmann’s approach,438 he had begun to read Hamilton, and he had begun to study Hankel’s Theorie der komplexen Zahlen [Theory of Complex Numbers] and Chasles’s Rapport sur les progrès de la géométrie en France (1870), the latter of which he also reviewed. Klein’s Scottish friend W.R. Smith proved to be a helpful catalyst yet again by providing Klein with reports about the discussions that were being held at the time among Peter Guthrie Tait and his students in Edinburgh. Regarding the idea that a general principle for classifying as many geometric approaches as possible lies in the concept of the group, Klein derived this primarily from (1) his own non-Euclidean considerations; (2) the inspiration that came to him from studying Sophus Lie’s work; and, most importantly, (3) Hamilton’s quaternion theory from 1843 (which comprised a theory of the invariants of motion in Euclidean space) and Grassmann’s extension theory (1844, 1865), which 434 435 436 437

[Oslo] A letter from Klein to Lie dated December 25, 1871. [Oslo] II: Klein’s notes dated November 1, 1892 (printed in ROWE 1992a, p. 601). SCHOLZ 1980, pp. 132–36, 170–74. KLEIN 2020 [1872], p. 3. For a discussion, with references to Klein and Lie, of the modern understanding of groups and semi-groups, see HOFMANN 1992. 438 Grassmann’s work is frequently cited in Klein’s Erlangen Program. On May 3, 1870, one of Grassmann’s sons had attended the Royal Seminar in Göttingen (see [UAG] Math. Nat. 0012, n.p.), where he presented his father’s book Ausdehnunglehre [Extension Theory] as a gift. See also TOBIES 1996a; SCHUBRING 1996; and PETSCHE 2006. Klein provided an overview of Grassmann’s ideas in his book Elementary Mathematics from a Higher Standpoint (vol. 2, pp. 37–48 in the English translation).

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provided the foundation for a geometry of multi-dimensional space. On January 1 and 5, 1872, Klein sent a draft of his long essay to Sophus Lie. After receiving Lie’s reply (which is lost), Klein remarked that, “in a certain sense, the difficulty of the work and its merit lie in its presentation.” On this basis, he was able to complete his Erlangen Program (see Section 3.1.1). Sixth. In the meantime, Klein also devoted effort to an additional area of research: the classification of cubic surfaces. This reflected his interest in tangible models, which inspired him to think of new ideas (see Section 2.4.3). In this regard, Klein oriented himself toward Clebsch, who combined algebraic representations of cubic surfaces with their physical representation, a prime example being Clebsch’s diagonal surface with twenty-seven real lines (Fig. 11).439

Figure 11: Clebsch’s diagonal surface, the first model of a cubic surface on which all of its 27 lines are real (FISCHER 1986, fig. 10). 439 For a clear description of Clebsch’s surface, see https://www.maths.ox.ac.uk/aboutus/departmental-art/cubic-surfaces/clebsch-diagonal-surface (accessed December 30, 2019); see also LÊ 2013; and TOBIES 2017a.

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In 1999, as I mentioned in my introduction, a large ceramic model of this diagonal surface was installed at the University of Düsseldorf on the occasion of Klein’s 150th birthday. Klein used this surface to understand higher-order spatial configurations in an intuitive manner. He would return to it, too, in future stages of his research life. He would later, for instance, employ group theory to analyze the problem of twenty-seven real lines on a cubic surface (see Section 6.3.1). Even while editing his collected works at the end of career, Klein corrected certain proofs and added a supplement – “Über die durch die 27 reellen Geraden vermittelte Zerlegung der Clebschschen Diagonalfläche” [On the Division of Clebsch’s Diagonal Surface Effected by Its 27 Real Lines]440 – to the reprint of his article “Ueber Flächen dritter Ordnung” [On Cubic Surfaces], which was first published in 1873. It was in the latter paper, as its title suggests, that Klein attempted to classify such surfaces.441 The idea to undertake the classification of cubic surfaces first came to Klein in Göttingen in 1871. It was prompted by Klein’s supervision of the doctoral student Joseph Diekmann. The latter submitted his dissertation on June 9, 1871: “Ueber die Modificationen, welche die ebene Abbildung einer Fläche 3ter Ordnung durch Auftreten von Singularitäten erhält” [On the Modifications that Are Obtained in the Planar Mapping of a Cubic Surface by the Appearance of Singularities].442 Steered in this direction by Klein, Diekmann analyzed the types of nodes on cubic surfaces that appeared in the work of Ludwig Schlaefli (1863),443 and he used “a geometric interpretation of the mapping associated with Grassmann’s manner of creating such representations.” As a full professor, Clebsch was the official reviewer of the dissertation. In his vita, Diekmann also thanked, in addition to Clebsch and M.A. Stern, the Privatdozent Klein.444 From Klein’s correspondence with Lie, however, we learn that Klein was in fact Diekmann’s primary supervisor. Klein send a copy of Diekmann’s dissertation to Lie and remarked: “I helped the author a great deal. I was unfortunately unable to get very far with him, however, because he is not the most clear-headed person.”445 Diekmann would go on to become an accomplished teacher and author of textbooks.446 Klein delved deeper into the subject both while Diekmann was writing his dissertation and afterward. His letters to Darboux, Lie, Max Noether, and Otto Stolz provide a picture of his ongoing research and his enthusiasm for the topic. In a letter to Darboux dated February 3, 1872, for instance, we read that Klein had been “thinking about a model of a cubic surface that shows its 27 lines in a clear 440 KLEIN 1922 (GMA II), pp. 56–62. Klein’s research assistant Vermeil gave a lecture on this supplement at the annual meeting of the German Mathematical Society in Leipzig in 1922. See Jahresbericht DMV 31 (1922) Abt. 2, pp. 103–04. 441 Math. Ann. 6 (1873), pp. 551–81 (Reprint KLEIN 1922, GMA II, pp. 11–44). 442 This dissertation was published in Math. Ann. 4 (1871), pp. 442–75. 443 On Schlaefli’s work in this area of research, see KELLERHALS 2010, pp. 169–70. 444 [UAG] Prom. Phil. Fak. 156, pp. 395–99. 445 [Oslo] A letter from Klein to Sophus Lie dated July 29, 1871. 446 See LOREY 1916, p. 84.

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grouping.” Six days later, Klein thus explained to Lie that he hopes to have “built a model of a general F3” [a cubic surface (Fläche)] by the time of the next Academy meeting. On March 13th of the same year, he informed both Lie and Max Noether about “the basic forms of cubic surfaces.” “I can now,” he went on, “determine all such surfaces with respect to their shape.”447 Relying on Schlaefli’s work, Klein identified five types of cubic surfaces. He explained these types in a letter to Darboux (March 21, 1872) and in one to Otto Stolz (March 30, 1872), where he was happy to remark that he had once again been able to demonstrate a connection between different areas of research: I have recently been engaged in an entirely new study, which I think I will be able to complete. Its aim is to develop the forms of F3. Like Schlaefli, I have arrived at five types, each according to the reality of its lines. I will also show, however, that these five types can be defined by their interrelation in the Riemannian sense, for the surfaces are connected in a 4-fold, 3-, 2-, 1-, or 0-fold manner. This would thus be a link between algebra and analysis situs, and it brings me great pleasure to have recognized this. I am especially pleased because I am able to prove that the five types in question exhaust all of the formal possibilities of F3. I intend to write this up.448

This topic was also the subject of the first (and only) research seminar that Clebsch and Klein led together. Announced to be about “various, mainly geometric topics,” it took place on Tuesdays from May 7th to the end of June in 1872. Thirteen students gave presentations on the work of French, Italian, Norwegian, Swiss, and German mathematicians: Abel, Chasles, Clebsch, Cremona, Dedekind, Frobenius, Hesse, Carl Neumann, Puiseux, Schlaefli, and others. The primary focus of the seminar was on cubic surfaces and recent scholarship on algebra.449 Noteworthy attendees include Adolf Weiler and Wilhelm Bretschneider, who would later be Klein’s doctoral students (see Section 3.1.2); Carl Rodenberg, who also attended Klein’s lectures in Göttingen, earned a doctoral degree in 1874, still inspired by Clebsch (who died in 1872),450 and became known for creating a series of twenty-six mathematical models made of plaster451; and Friedrich Neesen, Klein’s friend from Bonn, who, on June 25, 1872, presented to the seminar a “model of a cubic surface with four nodes, completed according to Dr. Klein’s specifications.”452

447 [Paris] 59: Klein to Darboux, March 21, 1872; [Oslo] A letter Klein to Sophus Lie dated March 13, 1872. See also VOSS 1919, p. 285. 448 The German original is quoted in BINDER 1989, p. 5 (see also TOBIES 2019b, p. 95). 449 [Protocols] vol. 1, pp. 1–28. 450 Rodenberg, meanwhile a secondary school teacher in Plauen, submitted his doctoral thesis at the University of Göttingen: Das Pentaeder der Fächen dritter Ordnung beim Auftreten von Singularitäten (Göttingen: E.A. Huth, 1874, 31pp.). 451 Rodenberg published this series in Flächen dritter Ordnung (Darmstadt: Ludwig Brill, 1881). There are several online presentations. For an explanation of the context, see Franz Meyer (1928), “Flächen dritter Ordnung,” in ENCYKLOPÄDIE vol. III.2.2b, p. 1505. 452 [Protocols] vol. 1, pp. 15–16.

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Figure 12: An illustration of a cubic surface with four real nodes (Klein 1922 [GMA II], p. 15).

On August 3, 1872, Clebsch and Klein presented models created for their aforementioned joint seminar at the meeting of the Göttingen Royal Society. Clebsch showcased two models constructed by Weiler (among them a model of the diagonal surface with twenty-seven real lines), while Klein presented his model built by Neesen. Here Klein described the significance of how he was able to derive, from this model, additional forms of cubic surfaces by means of continuous variation: Because a surface with four nodes does not have any absolute invariants, all other surfaces with four real nodes can be derived from the present example by means of real collineation. With respect to their behavior at infinity, five main types have to be distinguished. If one deforms such surfaces in the finite by means of continuous processes, whereby the sections touching one another at a single node can either come together or fully separate, one thus schematically obtains the shapes of other cubic surfaces. It can be proved that all cubic surfaces can be produced in this way, so that it is thereby possible to obtain a complete overview of all possible forms of cubic surfaces in general.453

453 Alfred Clebsch and Felix Klein, “Über Modelle von Flächen dritter Ordnung,” Göttinger Nachrichten 20 (1872), pp. 402–04, here at p. 404.

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Because Klein, on account of being offered a professorship in Erlangen and on account of Clebsch’s early death, had to focus on different priorities, he would not be able to flesh out the details of this contribution to the classification of cubic surfaces until somewhat later. On May 5, 1873, he submitted a short version of his revision (six pages) to the Sitzungsberichte [Proceedings] of the Physical and Medical Society in Erlangen, and on June 6, 1873 he submitted the aforementioned longer version to Mathematische Annalen. For the latter article, Klein asked Adolf Weiler to create a plaster model and an illustration of a cubic surface with four real nodes (see Fig. 12). Even though Klein accomplished a great deal while working as a Privatdozent, he repeatedly expressed how discontent he was with himself for not achieving even more: “At the moment, I am idle, for I have more or less forgotten how to work on detailed questions.” He hoped that Lie’s presence would provide him with fresh inspiration: “For me, your visit here is a matter of scientific life and death.”454 Lie arrived in Göttingen on September 8, 1872. During that time, Klein not only completed his Erlangen Program but also helped to edit the article that Lie was working on during his visit (see Section 3.1.1). 2.8.3 Discussion Groups Klein, who had experienced a rather distant relationship with the professors in Berlin, was far more integrated into the scholarly community as a Privatdozent in Göttingen, where he was taken under Clebsch’s wing. In addition to leading a seminar together, Klein and Clebsch met in a small group to discuss their research. Klein also participated in other groups and associations, including the student union, gatherings of his fellow Privatdozenten in Göttingen, and his circle of contacts from out of town. The following sections will outline the role that Klein played in these various communities. 2.8.3.1 A Three-Man Club with Clebsch and Riecke As a Privatdozent, Klein was a frequent visitor at the home of Professor Alfred Clebsch, who since 1867 had been married to his second wife Minna Rays, the daughter of a district judge. Among other things, they discussed the articles by Klein and Lie that Clebsch would soon accept for publication.455 In early April in 1871, this gathering took on the character of a regular club with three participants, as Klein informed Lie: “Clebsch, the physicist Riecke, and I have recently formed a club consisting of the three of us. We get together once a week to discuss things that happen to be on our minds. During our next meeting, I plan to speak about 454 [Oslo] A letter from Klein to Sophus Lie dated July 18, 1872. 455 [Oslo] Klein’s letters to Lie dated February 4, 1871; March 11, 1871; May 27, 1872, etc.

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your theory of the imaginary and about our other collaborative studies.”456 Within this circle, Clebsch would have to be considered the leader, but he encouraged collaboration on an equal footing. In November of 1871, Klein wrote enthusiastically to Plücker’s widow about his regular interaction with Clebsch: “Things are going very well for me in general. My constant contact with Clebsch has been extremely valuable and a vital element of my life.”457 During Klein’s time as a Privatdozent, Clebsch developed his “fundamental study of plane geometry; he investigated the general connections (he calls them connexes) that are represented by an equation containing a series of point coordinates and a series of line coordinates, each of them in a homogeneous way.”458 With his concept of the connex, Clebsch made advances in the theory of differential equations.459 Klein would use this concept in his Erlangen Program to characterize groups of contact transformations in a consistent manner.460 Hardly anything has been written about the fact that Eduard Riecke (Fig. 13) was just as closely associated with Clebsch as Klein was, and that Klein benefited from this relationship. Klein and Riecke first met in 1869, when they attended Clebsch’s lectures together, and they soon realized that they were on the same wavelength (see Appendix 10.1). Because Riecke would later be a driving force behind bringing Klein back to Göttingen – against the wishes of the mathematicians there (see Section 5.8.2) – he should be introduced here in greater detail. Born in Stuttgart as the son of a physician, Riecke studied mathematics and physics at the Polytechnikum there and at the University of Tübingen. He passed his teaching examinations in 1869 and moved to Göttingen to continue his studies under the guidance of Friedrich Kohlrausch, Wilhelm Weber, and Alfred Clebsch. On April 30, 1871, Riecke submitted his dissertation – “Ueber die magnetische Natur des weichen Eisens” [On the Magnetic Nature of Soft Iron]461 – and shortly thereafter he completed his Habilitation with a thesis entitled “Über eine Art allgemeiner Kugelfunctionen” [On a Type of General Spherical Functions]. Clebsch served as the chair of Riecke’s Habilitation procedure because Weber was sick, and on June 18, 1871 he wrote in his evaluation: “Already in his doctoral dissertation, Riecke had demonstrated a level of mathematics that is far beyond what is typical.” Riecke’s Habilitation thesis concerned, in Clebsch’s

456 [Oslo] A letter from Klein to Lie dated April 7, 1871. 457 [Canada] A letter from Klein to Antonie Plücker dated November 10, 1871. 458 [Oslo] A letter from Klein to Lie dated June 29, 1872. The work in question is Alfred Clebsch, “Ueber ein neues Grundgebilde der analytischen Geometrie der Ebene,” Göttinger Nachrichten 22 (1872), pp. 429–49. 459 Clebsch’s connex is a geometric concept that includes, as special cases, the curve considered as a point locus and as a line envelope. For a detailed description of this concept, see Emil Müller, “Die verschiedenen Koordinatensysteme,” in ENCYKLOPÄDIE, vol. 3 (1.1), pp. 755–56. See also CLEBSCH 1874, p. 50. 460 KLEIN 2020 [1872], p. 20; KLEIN 1921 (GMA I), p. 486. 461 See Göttinger Nachrichten (1871), p. 620.

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summary, “a class of functions which one often encounters when studying the theory of electric currents and magnetism.”462

Figure 13: Eduard Riecke ([UBG] Math. Arch. 52).

Riecke obtained the venia legendi for mathematics and physics; worked as an assistant in the physics lab; and continued his work on the topic of magnetism, which, in 1872, resulted in the publication of three articles in the Göttinger Nachrichten. Among the latter was a critical examination of Helmholtz’s law of “electrodynamic interactions,” a topic that had been brought to Riecke’s attention by W. Weber.463 On March 14, 1873, Riecke was offered an associate professorship, and in 1881 he succeeded Weber as a full professor of experimental physics.464 Here he remained, until his death in 1915, one of Felix Klein’s closest allies. Riecke’s two-volume Lehrbuch der Experimental-Physik [Textbook of Experimental Physics], which was first published in 1896, has gone through numerous editions, the most recent of which appeared in 2015.

462 [UAG] Phil. Dek. 156 (1870/1871), pp. 307–09, 529–35. 463 Göttinger Nachrichten (August 14, 1872), pp. 394–402. 464 See Göttinger Nachrichten (1875), pp. 279, 285; and NDB, vol. 21 (2003), pp. 562–63.

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Riecke’s mathematical theory of magnetism and electrodynamics was one of the topics discussed at the gatherings of this three-person club. Klein included the topic in his lectures on theoretical physics. Riecke himself first lectured on it in the summer of 1872,465 when Klein was finally able to concentrate on geometry. Klein’s letters to Max Noether reflect the close personal relationships between Klein and Riecke. On August 9, 1871, Klein remarked: “Riecke was just in my room and picked me up to go to the Stegemühle [a restaurant near Göttingen], where we often spend the evening.” He ended his letter of January 12, 1871 with the words: “Enough for today; there’s a Friday club at [Wilhelm] Weber’s, where I have to go with Riecke.”466 There were other circles with whom they willingly socialized. 2.8.3.2 The Mathematical and Natural-Scientific Student Union Felix Klein had joined the Student Union in Göttingen in 1869, and he did much to promote the solidarity of its members. As a Privatdozent, too, he attended the union’s weekly meetings almost regularly, and often together with Riecke.467 He had an especially close relationship with the students who had served as president of the union (Diekmann, Neesen, Lindemann) at various times. On June 16, 1871, under the leadership of Joseph Diekmann – who had just completed his dissertation under Klein’s supervision (see Section 2.8.2) – the student union bestowed the title of honorary member upon Klein and emphasized that the organization was “currently flourishing largely thanks to you.”468 Friedrich Neesen, who has already been mentioned several times, took over the presidency after Diekmann. He had studied mathematics and physics under Plücker in Bonn, where he and Klein had started a student union at that university. Like Klein’s, Neesen’s doctoral procedure was chaired by Rudolf Lipschitz. After completing his doctoral degree, Neesen moved to Göttingen, where he participated in Klein and Clebsch’s research seminar and constructed the aforementioned model for Klein. Already before his trip to Paris, Klein spent many days at Neesen’s parents’ house in Cleve,469 where Neesen’s father owned a gas plant. One sign of their lifelong friendship, among others, is the fact that Neesen and his daughter were the only non-family members in attendance at Klein silver wedding anniversary (see Section 3.6.3, Fig. 19). Neesen’s dissertation – “Ueber die Abbildung von leuchtenden Objekten in einem nicht centrirten Linsensystem” [On the Mapping of Luminous Objects in a

465 466 467 468 469

See Göttinger Nachrichten (1872), p. 118. [UBG] Cod. MS. F. Klein 12: 537; 542 (Klein’s letters to Max Noether). [Lindemann] Memoirs, p. 45. [UBG] Cod. MS. F. Klein 114, No. 1. [Oslo] Letters from Klein to Lie dated March 29, 1870, April 13, 1870, and Sept. 29, 1871. Cleve belonged to the administrative district of Düsseldorf, where Klein’s parents lived.

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Non-Centered Lens System] (December 1871)470 – concerned the optic calculation of rays. Whereas his predecessors in this field (Euler, Lagrange, Gauss, Möbius) had used centered lens systems in their examinations, Neesen investigated what influence a non-centered lens system would have on the position of the images. At the same time, he built upon the works of Gauss and Möbius by expanding the application of the collinear relationship between image and object. Over the years, Klein’s work had several points of contact with the field of optics. This can be said of his research on Plücker’s line geometry and, most obviously, of his own lectures on theoretical optics in the summer of 1871 (see Table 3), which he discussed with Neesen.471 It should also be mentioned here that Klein, in the inaugural lecture that he delivered in Erlangen on December 7, 1872 (see Section 3.2), would underscore “so-called geometric optics” as an area of application for mathematics.472 Later, Klein recognized the implications of Hamilton’s results for practical applications in optical instruments.473 He expanded upon these findings and published his results in the Zeitschrift für Mathematik und Physik at the time when the journal was being reconfigured as a publication venue for work on applied mathematics (see Section 5.6).474 In 1901, Klein sent his two articles to the optical company Carl Zeiss in Jena, where his results were appreciated.475 Neesen and Klein also shared a common opinion about the use of more recent methods of geometry in secondary-school education, an opinion which they expressed in similar Thesen as part of their doctoral procedures.476 After completing his Habilitation in physics in the fall of 1872, Friedrich Neesen became an assistant at the physics department in Göttingen, where he directed, together with Eduard Riecke, the “practical exercises” in the physics laboratory. In the fall of 1873, Neesen moved to Berlin, where he taught at the military academy and also as a Privatdozent and associate professor at the University of Berlin.477 Like Klein, Neesen became a member of the Physical Society in Berlin. 470 This dissertation was published in Bonn in 1871 by the Carl Georgi press. It contains a vita on page 32. 471 See LOREY 1916, p. 191. 472 Quoted from JACOBS 1977, p. 9. 473 Felix Klein, “Ueber neuere englische Arbeiten zur Mechanik,” Jahresbericht DMV 1 (1891), pp. 35-36. 474 See KLEIN 1922 (GMA II), pp. 603–12. The titles of the articles are “Über das Brunsche Eikonal” [On Bruns’s Eikonal] and “Räumliche Kollineationen bei optischen Instrumenten” [Spatial Collineations in Optical Instruments]. 475 [UBG] Cod. MS. F. Klein 8, p. 492 (a letter from Czapski to Klein dated Sept. 20, 1901). – Georg Prange continued Klein’s initiative of making Hamilton’s optical papers known, and he published a German edition supported by the Carl Zeiss foundation; see TOBIES 2020b. 476 Appendix to Neesen, “Über die Abbildung” (doctoral thesis), p. 33. 477 See Göttinger Nachrichten (1873), p. 141; and (1875), p. 276. Neesen’s lectures on geometric optics at the University of Berlin in the summer semester of 1889 were attended by, among others, Moritz von Rohr, who would become an important “calculating optician” at the Carl Zeiss company. See TOBIES 2017c, p. 122.

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Ferdinand Lindemann assumed the presidency of Göttingen’s Mathematical and Natural-Scientific Student Union after Neesen had completed his Habilitation. Lindemann, who is best known today for formulating the first proof of the transcendence of π (1882), had been studying in Göttingen since 1870, and he joined the student union at the end of the 1871/72 winter semester. He wrote in his memoirs that one of the customs of the union was to discuss difficult or unsolved problems at the end of each meeting. Klein remarked that Lindemann was able to solve these (often geometric) problems with ease.478 Klein, in Lindemann’s recollection, “happened to be present” when Lindemann gave a presentation about Klein’s recently published articles on non-Euclidean geometry: “A few days later, I received a letter from Klein in which he asked me to visit him. There he immediately suggested a topic for my doctoral dissertation concerning mechanics in non-Euclidean geometry.”479 Following Klein’s suggestion, Lindemann immersed himself in the scholarship on this topic during the following semester break. The result was his doctoral thesis: “Ueber unendlich kleine Bewegungen und über Kraftsysteme bei allgemeiner projectivischer Massbestimmung” [On Infinitely Small Motions and On Systems of Forces in the General Projective Metric].480 2.8.3.3 A Scientific Circle: Eskimo From the memoirs of the philosopher Carl Stumpf, we learn that Klein was also a driving force behind an informal group known as “Eskimo”: Klein, who was already then an active organizer, founded with me a group called “Eskimo,” a loose association in which young natural scientists could give lectures and engage in friendly interaction. My role in this group had been to represent the philosophical side of things. Professors were excluded. To my knowledge, the club still exists today, though it now has looser rules.481

The members of the club were primarily Privatdozenten. They gathered in one or another’s apartment for lectures on a specific theme and then continued the discussion in the pub (Gebhard’s beer hall). This small circle included, in addition to Klein and Stumpf, the physicist Eduard Riecke, the anatomist Friedrich Siegmund Merkel, the chemist Bernhard Tollens, and Max Bauer, who had completed his Habilitation in the fields of mineralogy and geology. Regarding Klein’s intellectual contribution to the group, this mostly had to do with his studies at the time on non-Euclidean geometry. He was able to convince Carl Stumpf, a close friend, of the validity of his ideas, even though Immanuel

478 479 480 481

[Lindemann] Memoirs, p. 46. Ibid., p. 45. The dissertation was published in Math. Ann. 7 (1874), pp. 56–143. See also Section 3.3. STUMPF 1924, p. 212. Later, professors also participated, such as the chemist Otto Wallach, who first joined the group in 1899. As a professor, Klein himself did not participate. See BEER/REMANE 2000, pp. 163–64.

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Kant’s pronouncement that three-dimensional Euclidean geometry was necessarily true was a maxim to Stumpf’s doctoral supervisor in Göttingen, Hermann Lotze. To Lotze, non-Euclidean geometry and n-dimensional space seemed unimaginable. Klein reported: “None less than Lotze had just then proclaimed all non-Euclidean geometry to be nonsense,” and he recalled the “endless […] discussions that I had with my friends in Gebhard’s Tunnel482 every evening of the winter of 1871/72.”483 Philosophers such as Hermann Lotze and Eugen Dühring dismissed non-Euclidean geometry “as a pointless intellectual game or as something mystical and bizarre.”484 Carl Stumpf earned his doctoral degree under Lotze’s supervision in 1868, and in 1870 he obtained the venia legendi for philosophy with a thesis, composed in Latin, on mathematical axioms. About the latter work, he remarked: “I never published the text because the non-Euclidean manner of thinking, to which Felix Klein had introduced me, was ultimately over my head.”485 Later, when both Carl Stumpf and Felix Klein were full professors, they remained in close contact (see Section 8.3.2). Klein’s circle of fellow Privatdozenten also included members of the historical seminar led by Georg Waitz. A legal historian and medievalist, Waitz had been a professor in Göttingen since 1848. He practiced Leopold von Ranke’s approach to historiography, and he was closely acquainted with Klein’s future father-in-law Karl Hegel (see Section 3.6.2). Ranke’s so-called historicism was based on systematic and source-critical methods, as compared to the philosophical treatment of historical events that had been favored before. Already at this point, Klein appreciated the significance of source-critical, systematic historical research and the “possibilities and advantages of an organized course of specialized study.”486 A letter from Klein to Otto Stolz refers to his close contact with members of Waitz’s seminar such as Ernst Steindorff and David Peipers: “Everything is going very well for all of us – that is, for Kiepert and the group from Göttingen including Riecke, Stumpf, Peipers, Steindorff, etc. etc. We were extremely happy together throughout our entire time in Göttingen.”487 This group of like-minded scholars also included the church historian Richard Zoepffel, whose wedding Klein attended,488 and Alfred Stern, a son of the mathematician M.A. Stern who completed his Habilitation in history at the University of Göttingen in 1872.489 In 1873, Alfred Stern became an associate professor at

482 Regarding the beer hall at the Hotel Gebhard, see the beginning of Section 2.4. 483 KLEIN 1979 [1926], pp. 140–41. In this work, Klein also laments Lotze’s misunderstanding of the term “curvature” (Krümmungsmaß). 484 CLEBSCH 1891, p. 554. On Dühring, see VOLKERT 2013, p. 204. Regarding the philosophical context of non-Euclidean geometry, see TORRETTI 1978 and BIAGIOLI 2016. 485 STUMPF 1924, p. 211. [UAG] Phil. Dek 156 (1870/1871), pp. 498–507 (Habilitation Stumpf); Stumpf’s habilitation thesis was later published by Wolfgang Ewen, see STUMPF 2008. 486 KLEIN 1923a, p. 15. 487 [Innsbruck] A letter from Klein to Otto Stolz dated March 30, 1872. 488 Ibid. A letter from Klein to Otto Stolz dated July 28, 1872. 489 See Göttinger Nachrichten (1872), p. 345.

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the University of Bern (Switzerland), and he was promoted to full professor there in 1878. One of Klein’s close friends, Alfred Stern took a firm stance against the anti-liberal, anti-Semitic, and nationalistic historiography practiced by the influential (and also misogynistic) Berlin-based historian Heinrich von Treitschke. Later, Stern would repeatedly host Klein in Switzerland, and he would also develop a close relationship there with Klein’s doctoral student Adolf Hurwitz.490 Klein’s return to Göttingen in the year 1886 was facilitated not only by Riecke’s support. The Philosophical Faculty there, which still comprised naturalists, humanists and mathematicians, had, in the meantime, been joined by the following professors: the historian Ernst Steindorff, who carried on the work of his teacher Waitz and had married the latter’s daughter, Clara; the philosopher and classical philologist David Peipers;491 and Bernhard Tollens, who served as director of the Agricultural-Chemical Laboratory. In the meantime, too, Friedrich Merkel had become a full professor of anatomy. Even if, for health reasons, Klein would later not be able to socialize as much as he might have liked, his involvement in this circle is evidence of his efforts to reach beyond the strict boundaries of a single academic discipline; interdisciplinary efforts of this sort, according to Richard Courant, could indeed be interpreted as a “key element of Klein’s life.”492 2.8.3.4 The “Social Activity” of Bringing Mathematicians Together The mathematics students in Göttingen and Berlin felt the differences between their respective teachers, as when Ferdinand Lindemann reported: “We in Göttingen could not understand it at all when a student coming from Berlin would have no idea about Hesse’s theorems concerning curves of the third order, for instance.”493 Klein, who regularly used his semester breaks to discuss his work and cultivate contacts, similarly complained from Berlin in a letter to Sophus Lie: “It is unfortunately impossible to relate our ideas to the people here, because there are no points of connection whatsoever.” He went on: Thus I have been engaged all the more energetically in creating a sort of social arrangement that might, first, bring together the mathematicians here and then all German mathematicians (by means of a periodically recurring conference); if such a meeting existed, it would be easier to promote the general scientific standpoint in the sense that seems desirable to us. Thereby I would also be able to satisfy my ever-present need for social activity. I am discontent when only working abstractly on pure science.494

490 [UBG] Cod. MS. F. Klein 10, p. 1160B. See also [Deutsches Museum] Sondersammlung 1968–4/2 (a letter from Klein to Alfred Stern, 1919). 491 David Peipers is probably best known for his book Untersuchungen über das System Plato’s (Leipzig: B.G. Teubner, 1874). 492 COURANT 1926, p. 197. 493 [Lindemann] Memoirs, p. 48. 494 [Oslo] A letter from Klein to Lie dated April 1, 1872. A page of this letter is reproduced in Figure 3 (p. 16).

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Though Klein would later write about himself that he considered “social activity to be a substitute for lost genius,”495 the sources indicate not only that he had this social streak from early on but also that he regarded it as one of his most important qualities. In a specific sense, Klein followed Clebsch’s idea of bringing people together in order to promote a mutual understanding for different approaches to mathematical research. Attendance was usually sporadic at the annual meetings of the Society of German Natural Scientists and Physicians (Gesellschaft deutscher Naturforscher und Ärzte, GDNÄ), which had been founded in 1822. Participation depended greatly on where the event was held. Just as other disciplinary factions had begun to split from the GDNÄ,496 Clebsch proposed, at the organization’s annual meeting held in 1867 in Frankfurt am Main, that a separate mathematical association should be formed. Not long thereafter, on Whitsunday of 1868, Clebsch invited twenty mathematicians – the student Klein among them – to join him on the aforementioned hike along the Bergstraße. Here, Klein not only got to know Clebsch and the other mathematicians. He was also introduced to Christian Wiener’s model of a cubic surface with twenty-seven real lines (asymmetrical and based on an empirical construction) and he witnessed the discussion that led to the creation of the journal Mathematische Annalen. These early stages of Clebsch’s efforts to bring scholars together were interrupted, however, by the Franco-Prussian War. After the war, Clebsch put the Privatdozent Klein on the task. The latter recruited the Leipzig-based mathematician Adolph Mayer and Max Noether to serve as members of a planning committee: Perhaps you recall that we spoke last Easter about how desirable it would be to organize a gathering of mathematicians in the not-so-distant future. Since then, I have asked around and received the impression that a conference held early next year – in late May, for instance – would be agreeable to many sides. Clebsch, who will incidentally be in Leipzig at that time, is also entirely in favor of the plan. I would like now with this letter to ask whether you might perhaps be willing to play an organizational role in the matter. I have posed this same question to Noether in Heidelberg, whom I know personally very well. The three of us: you, Noether, and I could perhaps form a committee “for the purpose of organizing a mathematics conference.”497

When Klein heard that another meeting was being planned for Berlin, he formulated his general principle of “not allowing any fragmentation.”498 The twentytwo-year-old coordinated with the organizers of the other meeting and followed Max Noether’s suggestion to appoint a Berlin-based mathematician to the planning committee. They recruited Carl Ohrtmann, a senior teacher at a Realgymasium in Berlin who – along with Felix Müller, a teacher at the Königliches Luisengymnasium – had founded Germany’s first review journal devoted to mathe-

495 496 497 498

Quoted from JACOBS 1977, p. 5. On these developments, see the detailed discussion in TOBIES/VOLKERT 1998. Quoted from TOBIES/ROWE 1990, p. 59 (Klein to Mayer on October Oct. 10, 1871). [UBG] Cod. MS. F. Klein 12: 542 (Klein to Max Noether on November 19, 1871).

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matics: Jahrbuch über die Fortschritte der Mathematik [Yearbook on Advances in Mathematics].499 Recommended by Hermann Schubert and Otto Stolz, Klein assumed the duty of writing reviews about “line geometry and some algebra” as of the second volume of the Jahrbuch.500 Because he was discontent about how works were being distributed for review, Klein chose the topic of reviewing as one of the discussion points for the conference that he was planning. A letter from Klein to Otto Stolz indicates how he prepared to implement his ideas: I would like to take the occasion of the conference to restructure the manner in which works are reviewed for the Jahrbuch, in the sense that each reviewer should review the work in question completely and not simply focus on his special area of expertise, so the journal will be more coherent than it has been. I would also like to recruit new reviewers, such as [Alexander] Brill. I am not yet able to write anything more specific about this, for in the next few weeks I intend to learn more about the relationships between the people whose names might come up in this regard. In general, however, if you are able to attend the meeting around Easter time or if potential reviewers are voted on otherwise, may I be secure in knowing that you approve of this idea?501

Before the planned national conference took place in Göttingen from the 16th to the 18th of April in 1873, another meeting of approximately fifty people had been held in Berlin during the Easter break of 1872. Here, Klein met the Austrian physicist Ludwig Boltzmann for the first time.502 In addition, the planning committee for the national meeting acquired two further members: Ludwig Kiepert, a friend of Klein and student of Weierstrass, and Emil Lampe,503 who was then a Corrector for Crelle’s Journal. When Clebsch died during the conference’s planning stages, Klein had meanwhile begun his first professorship (at the University of Erlangen), and he stuck to the plan: “My program is to unify German mathematics, even its isolated members (insofar as they are animated by genuine interest).”504 The professors whom Klein especially wished to be in attendance received an extra invitation: Regarding the conference, it seems necessary to write personally again to those men who seem especially important to us. On the basis of suggestions from Kiepert and [Max] Noether, I would like to ask you to send a special invitation to Richelot, [Eduard] Heine, and Aronhold, while Kiepert will write to the mathematicians in Berlin; Noether to [Lazarus]

499 See MÜLLER 1904 and also SIEGMUND-SCHULTZE 1993. 500 [Oslo] Letters from Klein to Lie dated March 29, 1871 and May 31, 1871. Klein wrote reviews for the Jahrbuch until 1879, at which point he passed on this task to Aurel Voss ([UGB] Cod. MS. F. Klein 12, p. 70: a letter from Voss to Klein dated February 18, 1879). 501 [Innsbruck] A letter from Klein to Stolz dated February 3, 1873. 502 Ibid. Klein’s letter to Otto Stolz dated March 30, 1872. – Boltzmann spent 1871/72 with Helmholtz, and in 1872 he published one of his famous papers on statistical mechanics. 503 At that time, Lampe was a senior teacher at the Friedrich-Werdersche trade school in Berlin. Later he became a professor at the Technische Hochschule in Berlin, and as of 1885 he served as chief editor of the Jahrbuch über die Fortschritte der Mathematik. 504 Quoted from TOBIES 1991, p. 35 (a letter from Klein to Max Noether dated December 1872).

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2 Formative Groups Fuchs, Schwarz, and [Otto] Hesse; [Alexander] Brill to Christoffel, Reye, and [Heinrich] Weber; and I will write to Lipschitz, [Wilhelm] Fiedler, Grassmann, and Schlaefli.505

In the end, however, none of these specially invited professors attended the prepared conference, which was held in Göttingen in April of 1873.506 The participants included former members of Clebsch’s circle (A. Brill, Gordan, Klein, Lindemann, Lüroth, M. Noether, H. Schubert, Voss, Weiler); scholars associated with the University of Göttingen (Enneper, Klinkerfues, Listing, Meyerstein, Minnigerode, Neesen, Riecke, Schering, M.A. Stern, Georg Ulrich, Wilhelm Weber); representatives of the journal Mathematische Annalen (Adolph Mayer and Karl Von der Mühll, in addition to Klein and Gordan); representatives of the journal Jahrbuch über die Fortschritte der Mathematik (Emil Lampe, Felix Müller, Carl Ohrtmann); and representatives of the journal Archiv der Mathematik und Physik (Reinhold Hoppe). Participants from abroad included Zeuthen from Denmark, Julius König and Mór Réthy from Hungary, Ernst Pasquier from Belgium, and Otto Stolz from Austria. In addition, the event was also attended by the professors Moritz Pasch (from Giessen) and Rudolf Sturm (from Darmstadt).507 Also present were the algebraists Eugen Netto and Ernst Schröder, who did not yet hold professorships at the time, and Max Simon, a mathematics teacher in Strasbourg who was to become a well-known historian of mathematics. The organizers had done their best, but the meeting ultimately fell short of their goal, which was to make connections between a broad variety of mathematical schools of thought. Nevertheless, Klein wrote optimistically to Sophus Lie: “I have to tell you briefly about our conference. As far as I’ve heard, the latter took place to the satisfaction of everyone involved. The collaborators working on Clebsch’s biography held long discussions, which I will take into account over the course of the summer as I finish the matter.”508 The conference had also been enriched by an exhibit of models. Klein was able to realize his ideas for the Jahrbuch as well, which were based on his experiences as a collaborateur for the French review journal – Darboux’s Bulletin (see Section 2.6.1). On March 11, 1871, Klein received the first volume of the Jahrbuch, which contained reviews of works that had appeared in 1868. Klein thought that it left “a good, objective impression,” and he had compared: “Darboux’s Bulletin is somewhat analogous but far more subjective.”509 In addition, a main result of the conference in Göttingen was the decision to plan a follow-up conference. Klein was elected, along with Carl Ohrtmann and Alfred Enneper, to be part of a preparatory committee for a meeting to be held in

505 Quoted from TOBIES/ROWE 1990, p. 70 (a letter from Klein to Mayer dated Jan. 25, 1873). 506 For a full list of participants, see GUTZMER 1904, p. 23. 507 Rudolf Sturm would later write the three-volume book Die Gebilde ersten und zweiten Grades der Liniengeometrie in synthetischer Behandlung (Leipzig: B. G. Teubner, 1893–96), the third volume of which repeatedly refers to Klein’s analytic results. 508 [Oslo] A letter from Klein to Lie dated May 4, 1873. 509 Ibid. Klein to Lie, March 11, 1871.

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Würzburg in 1875. For several reasons, however, this plan for Würzburg was never realized. Already on October 24, 1874, Klein wrote to Moritz Abraham Stern in Göttingen about a planning meeting that he had scheduled with Adolph Mayer, Ohrtmann, and Enneper in Leipzig: “I am actually going about it with very mixed feelings, for we have no reason to expect great success this second time around, and it is very well possible that, in light of this circumstance, we might seek a way to postpone the conference for a while.”510 Other challenges arose that Klein also had to manage. In November of 1874, he accepted a new position in Munich (see Chapter 4). At the same time, while still in Erlangen, he began to make his private life a priority, so that his free days were planned differently. Moreover, the interest among German professors of mathematics to form a separate association of mathematicians remained minimal at the time, even though the French Société Mathématique had already been founded in 1872. The French society was the first national association devoted exclusively to mathematics. Before its establishment, there had only been local mathematical societies (such organizations were founded, for instance, in Hamburg in 1690, in Prague in 1862, in London in 1865, in Moscow in 1867, in Tokyo in 1877, in Palermo in 1884, and in New York in 1888). In Germany, the first national organization of mathematicians – the German Mathematical Society (Deutsche Mathematiker-Vereinigung) – was founded in 1890 (see Section 6.4.4), after which similar societies would be formed in other countries.511

510 [UBG] Cod. MS. F. Klein 10, p. 1160B (a letter from Klein to Stern dated October 24, 1874). 511 For details concerning the establishment of these organizations, see also TOBIES 1986a; and TOBIES 1989b.

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Figure 14: The title page of Klein’s Erlangen Program (October 1872).

3 A PROFESSORSHIP AT THE UNIVERSITY OF ERLANGEN I have to say that the hour of fulfillment has struck for me, too. Yesterday I received an offer to become a full professor in Erlangen, where [Hans] Pfaff (von Staudt’s successor) died two or three months ago. This morning I accepted the offer, and I will probably move to Erlangen as early as next semester.1

Thus wrote Klein to Sophus Lie, who had just become a professor in Christiania a few months before. To Darboux, Klein wrote: “It is von Staudt’s old chair, and it brings me not a little pleasure to be his successor, because I have been studying his work repeatedly over the past few years.”2 To the Leipzig-based mathematician Adolph Mayer, he sent the following news: “The matter came so unexpectedly” that he had little time to think about it. Klein ended his lectures before the close of the term in order to embark upon a three-week tour of the Tyrolian Alps with his Scottish friend William Robertson Smith. Otto Stolz had recommended the route that they should take: “We undertook,” as Klein wrote to him, “the tour that you suggested with the utmost diligence but also with the greatest pleasure, except that, instead of the Kreuzspitze, we hiked up the Similaun, which is not much more difficult but is higher and more interesting on account of all the snow.”3 Klein was athletic at the time, and he had become a full professor at such an early age that he was still too young to vote in the German Empire’s first parliamentary election – a fact that he was amused to relate.4 Klein arrived in Erlangen on September 29, 1872. The city, after changing hands between several political territories, was now part of Bavaria. The university had been founded in 1743 in what was then the principality of BrandenburgBayreuth. Under Bavarian rule, it was renamed the Royal Friedrich Alexander University, and the only reason why it was not closed down was because it possessed the only Protestant theological faculty in the otherwise Catholic state. Erlangen’s population then was approximately 12,500, and thus it was an even smaller town than Göttingen. There were fewer than four hundred students enrolled at the university. The palace, the palace garden, and the orangery had belonged to the university since 1818. The palace contained the university library, lecture halls, seminar rooms, and scientific collections. Here, after much effort, Klein acquired a room for conducting mathematical exercises (see Section 3.2). 1 2 3 4

[Oslo] A letter from Klein to Lie dated August 3, 1872. [Paris] 62: Klein to Darboux, August 28, 1872. TOBIES/ROWE 1990, pp. 64–66 (a letter from Klein to A. Mayer dated August 28, 1872); and [Innsbruck] A letter from Klein to Stolz dated August 29, 1872. The Similaun is a mountain in the Ötztal Alps with an elevation of 3,599 meters. See CARATHÉODORY 1925, p. 2. Only men older than 25 were eligible to vote. The first election of the German parliament took place on March 3, 1871.

© Springer Nature Switzerland AG 2021 R. Tobies, Felix Klein, Vita Mathematica 20, https://doi.org/10.1007/978-3-030-75785-4_3

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After the death of Karl Georg Christian von Staudt (Erlangen’s first mathematician with an international reputation), his professorship was briefly held by Hermann Hankel (1868–69) and Hans Pfaff (1869–72). After Pfaff’s death on May 20, 1872, the Philosophical Faculty of the University of Erlangen submitted, as early as June of that year, a list of just two potential candidates to fill the position to the Bavarian Ministry of Culture in Munich. Klein’s name stood at the top of the list, and Johannes Thomae’s was in second place. From a letter written by the dean of the Philosophical Faculty, Eugen Lommel (Klein’s future brotherin-law), we learn that, for financial reasons, the university was looking for a young man who could teach a broad range of subjects. Clebsch had sent a glowing recommendation for Klein, and this was reflected in the letter dated June 26, 1872, to the Bavarian Ministry of Culture in support of Klein’s candidacy: Just twenty-three years old, Klein has managed, through the number and quality of his publications, to earn the unreserved respect and admiration of his colleagues. […] Recently, his article “On Non-Euclidean Geometry” stirred lively discussion and became a sensation in the widest circles – here and abroad. […] The majority of his work is in the area of analytic geometry, a discipline which, by its nature, occupies a mediating position between the various approaches of contemporary mathematics mentioned above; moreover, I am also aware that Clebsch has introduced him in the most thorough manner to modern algebra, and thus, to a high degree, he possesses the multifaceted education that is so desirable to us. Klein, however, is not only impressive as an author but is also, according to the judgement of his experienced colleagues, a highly effective teacher. […] If we may add to this that, in addition to his abilities as a scholar and teacher, his sterling private character, his vigor, his vitality, and his amiability are also unanimously praised, then we must recognize in Klein, in every respect, the man above all to whom we would like to entrust the vacant professorship.5

The Bavarian king, Ludwig II, signed Klein’s letter of appointment on August 21, 1872. Klein began his new position on October 1, 1872 with a yearly salary of 2,000 Gulden. Upon request, he was allotted 400 Gulden to cover his moving expenses.6 He rented an apartment on the first floor of Gabelsberger Straße 16, which was just a short walk away from the palace and the botanical garden. On December 9, 1872, Klein was made a member of the Societas Physicomedica Erlangensis, which was founded in 1808 and still exists today.7 This academic society pursued the goal of “exchanging thoughts, observations, and experiences from all areas of the natural sciences, technology, and medicine.” Klein was the first, as Max Noether reported, to integrate mathematics into its program.8 That is, Klein made use of the society’s Sitzungsberichte [Proceedings] – as he had done with the Göttinger Nachrichten before – to publish his findings and

5 6

7 8

For the original German letter, see TOBIES 1992a, p. 767 (TOBIES 2019b, p. 108). [UA Erlangen] R. Th. II. Pos. 1, No. 15 (Hans Pfaff’s personnel file, which contains records of Klein’s hiring process). By comparison, we know that in 1869 the mathematician Gustav Bauer was hired as a professor by the University of Munich with an annual salary of 1,200 Gulden. See VOSS 1907, p. 61. [UB Erlangen] MS 2565 [8] List of members; and [10] Minutes from the meeting. M. NOETHER 1908, p. 81. I am indebted to Cordula Tollmien for this reference.

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those of his students quickly. When, after five semesters in Erlangen, Klein moved to Munich to take a new position, the Societas Physico-medica Erlangensis made him an honorary member on May 10, 1875, so that this publication venue remained open to him. In all, he published sixteen articles in the Sitzungsberichte. Klein’s Erlangen Program, which arose from discussions with Sophus Lie, was published as an independent booklet (see Fig. 14). Klein regretted that his teacher Clebsch had not lived to see it. After its publication, Klein spent a good deal of time writing Clebsch’s academic biography and continuing Clebsch’s program of bringing disciplines and people together. Although Klein’s higher lectures in Erlangen were never attended by more than eight students, he supervised, within the framework of his own research program, the doctoral studies of a surprisingly high percentage of young mathematicians at the time (Section 3.1). Klein was aware that most mathematics students would not become researchers but rather teachers at secondary school. For this reason, he made it a point in his obligatory inaugural lecture as a professor to formulate his far-sighted goals for education (Section 3.2).9 Klein continued his efforts to familiarize himself with different scientific schools of thought. In 1873, he undertook a long-planned trip to Great Britain (see Section 3.3), and in 1874 he traveled to Italy for the first time (Section 3.4). Even though Klein left Erlangen after just five semesters there, he did much to build up the university’s mathematical institution, which, before his arrival, had existed in name (“Mathematisches Institut”) only (Section 3.5). In Erlangen, too, he set the course for his familiy life (Section 3.6). 3.1 RESEARCH TRENDS AND DOCTORAL STUDENTS Of course the enjoyment of independent productivity will always remain accessible to only a few.10

Klein felt from early on that he numbered among the few people who could make creative contributions to mathematics. He knew that such activity “requires a special disposition not given to everyone,” and he compared it to musical creativity: “[O]nly very few persons are musically creative, but still most people have a more or less cultivated understanding for a finished piece of music. The class of those with no musical sense whatsoever is decidedly limited. Similarly, there are also those otherwise normally gifted persons, again not many, who have absolutely no head for mathematics, and are utterly unable to follow even the simplest mathematical argument.”11 Klein was quickly able to recognize the mathematical talents of others, and he took pleasure in guiding young mathematicians. 9

Klein’s inaugural lecture (Antrittsrede) in Erlangen is often confused with his Erlangen Program. For a somewhat recent example of this confusion, see STUBHAUG 2002, p. 165. 10 Quoted from Klein’s inaugural lecture in Erlangen (see Section 3.2), which he delivered on December 7, 1872. The English translation given here is from ROWE 1985, p. 136. 11 Ibid., p. 137.

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The development of the Erlangen Program – a text that, as a newly appointed professor in Erlangen, Klein was required to publish – has already been discussed above (see Section 2.8.2). My focus here will be on the vision of the work and its effects. In addition, I will also outline the research interests of Klein’s students in Erlangen and discuss how they took Klein’s own research in new directions. 3.1.1 The Vision of the Erlangen Program Klein wrote the following to Darboux on November 29, 1872: “Lie visited me for nearly two months, September and October. What we discussed will be evident to you from the publications that arose from our conversations: Lie’s note in the Göttinger Nachrichten and my inaugural program.”12 By “inaugural program,” Klein meant his Erlangen Program (KLEIN 1872, 48pp.), whose basic idea and motivation he later explained in the first volume of his collected writings: Even as early as my time in Bonn, my interest was directed toward understanding, in the conflict between feuding mathematical schools, the mutual aspects of their concurrent approaches, which were outwardly dissimilar from each other and yet related by their nature. I hoped to resolve their contradictions by means of a unifying and all-encompassing concept. Within the field of geometry, there was still plenty for me to do in this respect.13

The broad spectrum of geometric approaches in which Klein had immersed himself and for which he sought an ordering principle included Julius Plücker’s line geometry, projective geometry, the birational geometry of Riemann and Clebsch, sphere geometry (as developed in particular by French mathematicians), Hermann Grassmann’s extension theory, Karl Georg Christian von Staudt’s geometry of position, two types of non-Euclidean geometry (one developed by Gauss, Lobachevsky, and Bolyai; the other by Riemann), and Sophus Lie’s new insights into contact transformations and their applications to systems of partial differential equations. Klein recognized that these different approaches to geometry could be classified with correlated transformation groups, with each of such groups determining a specific set of invariants. Klein drew a connection between these individual approaches to geometry by replacing his aforementioned idea of a “principal group” (see Section 2.8.2) with a more comprehensive group (which would preserve only part of the geometric properties). The transition to a larger group thus corresponded to a transition toward a less specific geometry. Table 5 arranges traditional Euclidean geometries into a schema in order to illustrate their interrelations. The schema itself does not appear in Klein’s booklet but was formulated later on to provide an overview.14 The Erlangen Program, for 12 [Paris] 64v; Sophus Lie, “Zur Theorie partieller Differentialgleichungen erster Ordnung, insbesondere über eine Classification derselben,” Göttinger Nachrichten (1872), pp. 473–89. 13 KLEIN 1921 (GMA I), p. 52. See also HAWKINS 1984; and ROWE 1989a. 14 See also WUßING 1968.

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instance, does not address the affine geometry cited here. The motion of a geometric figure (its displacement or rotation, for instance) alters its position. The length of line segments and the property of lines to be orthogonal thereby remain invariant. Parallel lines cross over again into parallel lines. The partition ratio of three points and the cross-ratio of four points maintain the same value. In the case of a projective mapping (a central projection, for instance) the lengths in a figure do not accord with the corresponding lengths in its image. Parallel lines can be transformed into intersecting lines. Only the cross-ratio of four points and incidence relations remain invariant. Klein only mentioned non-Euclidean geometries in an appendix to his Program (see below). Table 5: On the Erlangen Program

Position Extent Orthogonality Parallelity Partition ratio Cross-ratio Incidence

Motion group not invariant invariant invariant invariant invariant invariant invariant Metric geometry

Equiform group not invariant not invariant invariant invariant invariant invariant invariant Equiform geometry

Affine group not invariant not invariant not invariant invariant invariant invariant invariant Affine geomety

Projective group not invariant not invariant not invariant not invariant not invariant invariant invariant Projective geometry

The content of the Erlangen Program has been widely discussed in the Englishspeaking world. A recent study by Gray bases its analysis on the early English translation by Klein’s doctoral student Haskell (1893).15 David Rowe, who considers Haskell’s translation outdated, has recently translated the text anew. The Erlangen Program followed a lofty vision, as Klein remarked twenty years after its publication: “My Program, as we believed, was meant to be an outward sign of the redevelopment of geometry, comparable to the redevelopment that Poncelet had set in motion fifty years before.”16 By “we,” Klein meant himself and Sophus Lie, with whom he had discussed his basic ideas. By the summer of 1873, however, the work had hardly caused the splash that Klein had hoped it would. To a question from Lie, Klein responded in June of 1873: You ask what people have thought about my Program. Well, I have hardly heard any opinions about it at all. A Frenchman, Pasquier, who was in Göttingen over Easter, told me that Darboux has criticized it. Mansion, on the contrary, is delighted with it and hopes for it to be translated. [Max] Noether, who also understood our work on W-curves, wrote to me in approval. Gordan, whom I asked directly about it, told me that it does not appeal to him in the least; he seemed to regard the whole thing as an exercise in style. In contrast, the old [Moritz Abraham] Stern, as someone wrote to me from Göttingen, is very pleased with it. That is just

15 See Jeremy Gray’s article in JI/PAPADOPOULOS 2015, pp. 59–73. 16 [Oslo] II: Klein’s notes dated November 1, 1892 (printed in ROWE 1992a, p. 202).

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3 A Professorship at the University of Erlangen about everything that I have heard. My colleagues here have unanimously told me that they did not understand it.17

The (Belgian) mathematician Ernest Pasquier had participated in the seminar led by Clebsch and Klein in 1872.18 Paul Mansion, another Belgian mathematician, translated works by Riemann, Plücker, Clebsch, and others into French. The Austrian Otto Stolz numbered among the few who were quick to recognize the potential of Klein’s Erlangen Program. He sent a draft of his review of the booklet to Klein before it appeared in the Jahrbuch über die Fortschritte der Mathematik,19 and Klein was very pleased about Stolz’s sympathetic assessment. Klein requested, however, that Stolz should “emphasize Lie’s name somewhat more: It is essentially by exchanging ideas with him that I formulated these general ideas.” Friedrich Engel later reported, however, that, for Sophus Lie, “Klein’s thought that many areas of mathematics to that point could be conceptualized as an invariant theory of certain known groups […] was new and surprising.”20 Klein stressed Lie’s inspiration at several points in the Program, but he himself recognized that there were still unsolved problems. In a letter to Stolz, he elaborated: In your review of my work, you considered my unification of various branches of geometry to be complete; I think that is an exaggeration. Only one step has been made, and although I think that this step is an important one, the synthesis still has to become much tighter. I believe that I am now in a position to take things a step further and to set aside the limitations that I mention in § 9 (in the middle). Yet what I have in mind is still so undefined that I hardly know how to capture it in words.21

Section 9 of his text concerned the group of all contact transformations, and it owed much to the work of Lie and Clebsch. Later, in his collected works, Klein would cite two main imperfections in his Program from 1872: the unsatisfactory identification of projective geometry with linear invariant theory, and the restricted notion of a function that he used.22 In comparison with Lie, as Rowe has underscored, Klein had transformation groups in mind that could be applied globally (not locally) to a manifold.23 The distinction between the global and local properties of a manifold, which already appears in Riemann’s work, is not fully expressed in Klein’s Erlangen Program. Scholz has pointed out that Klein used a mixed concept between projective and topological mapping, and that he would overcome this problem in later works.24 Klein regarded his Program as a suggestion to systematically study the invariants and covariants of known groups. This was an ambitious goal, but it is only according to this measure that the Program

17 18 19 20 21 22 23 24

[Oslo] A letter from Klein to Lie dated June 28, 1873 (see TOBIES 2019b, p. 111). [Protocols] vol. 1, pp. 12–13, 21–22. The review was published in the 1875 issue of the Jahrbuch (pp. 234–35). F. ENGEL 1899, p. xxxix. [Innsbruck] A letter from Klein to Otto Stolz dated January 8, 1874. KLEIN 1921 (GMA I), p. 414. ROWE 2003, p. 671. SCHOLZ 1980, p. 136.

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should be judged. In this respect, one can consider the researchers working today who continue to rely on Klein’s basic ideas.25 Klein appended seven notes to his program that should be understood as an immanent component of his visionary ideas:26 I. In the first note, he stressed that, for him, the distinction between synthetic and analytic approaches was no longer essential, as I have already mentioned above in the context of Plücker. II. Klein referred to the state of geometry at the time – that is, its division into separate disciplines – as something that is “hopefully only provisional.”27 III. Here he stressed the value of spatial perception or intuition (Anschauung) both with respect to teaching and as a fundamental method of geometric research. This is something that would occupy him throughout his life: “A model – whether constructed and observed or only vividly imagined – is for this geometry not a means to an end but rather the object of inquiry itself.” IV. In this note, Klein engaged with the theory of manifolds of any number of dimensions, and he discussed the development of this theory by Plücker, Grassmann, Gauss, and Riemann. V. Klein discussed non-Euclidean geometry as a special case because, in his words, it was still associated with “a multitude of non-mathematical speculations.” VI. In this note, he explained in greater detail how line geometry could be understood as the investigation of a manifold of constant curvature. He cited in particular his article “Ueber Liniengeometrie und metrische Geometrie” [On Line Geometry and Metric Geometry]. VII. Finally, Klein interpreted “binary forms” with reference to the corresponding theory by Clebsch,28 and in doing so he drew a connection between sphere geometry (the interpretation of x + iy on the surface of the sphere), the vertices of a symmetrical tetrahedron, and the theory of biquadratic equations.29 We have to keep these notes in mind, too, when we say that Klein continued to follow the program, for then it is possible to agree with his later assessment of the work: “This Erlangen Program has always remained the major guideline for my later investigations, and its ordering principle can still be applied to a number of other areas, including function theory, mechanics, and physics.”30 In 1884, Lie wrote as follows to Klein: “If you are collecting your older works for republication in Mathematische Annalen, do you not also want to reprint your Erlangen Program there? It is certainly your most significant work from around 1872, and it would be better understood now than it was then.”31 As early as 1883, 25 26 27 28 29 30 31

See, for instance, KISIL 2011; JI/PAPADOPOULOS 2015; RATAJ/ZÄHLE 2019. KLEIN 2020 [1872], pp. 23–27; KLEIN 1921 (GMA I), pp. 490–97. Here and in the following, see KLEIN 2020 [1872], p. 24–25. A “binary form” is a polynomial with two variables. This aspect would be taken up by Klein’s doctoral student Ludwig Wedekind (see below). KLEIN 1923a (autobiography), p. 18. See also CARATHÉODORY 1919, and Section 9.2.2. [UBG] Cod. MS. F. Klein 10: 695/3.

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the Greek mathematician Cyparissos Stéphanos (then in Paris) had suggested that he could translate the Erlangen Program into French and publish it in MittagLeffler’s Acta Mathematica, which sometimes incorporated translations (especially of the works of Georg Cantor). Klein agreed, Poincaré and Sophus Lie supported the idea. However, Mittag-Leffler rejected Stéphanos’s proposal. Klein remained diplomatic and eventually abandoned the plan.32 When, in November of 1889, Corrado Segre, a professor in Turin, requested permission to translate the work into Italian,33 Klein gave his assent. Segre’s student Gino Fano, who would subsequently study under Klein, did the translation, and Klein supplemented the text with commentary.34 In Turin, the Privatdozent Mario Pieri also made use of Klein’s Program, and he even referred to Klein as “a personal hero.”35 The work was finally translated into French by Henri Padé, who studied under Klein in 1890 and 1891. Klein had hoped that Padé would use the Italian version as the base text, because Klein himself had updated the latter to include more recent scholarly literature. Padé, however, used the German original from 1872 because he had a better command of German than Italian. Klein therefore furnished the French edition with a preface.36 In France, it now became more apparent that, as early as 1872, Klein had stressed that his main points of departure were Camille Jordan’s work on groups of motions (“Sur les groupes de mouvements”) and Michel Chasles’s intuitive approach of regarding metric properties as projective relationships to a fundamental configuration (the infinite spherical circle). The French translation was followed in 1893 by the aforementioned English translation by M.W. Haskell, who, in 1890, had completed his doctoral degree under Klein’s supervision in Göttingen. Additional translations would subsequently appear in Polish (by S. Dickstein in 1895), in Russian (by D.M. Sintsov in 1896), and in Hungarian (by Lajos Kopp in 1897).37 Klein also agreed with Lie’s idea to reprint the Erlangen Program in Mathematische Annalen, but he wanted to publish it along with reprints of his and Lie’s early collaborative studies (with explanatory notes). In the early 1890s, however, the two had begun to develop diverging opinions (see Section 6.3.6), so that Klein gave up on this plan and decided to publish only his Program.38 He added a few corrections and notes, referring to developments over the previous twenty years

32 On Stéphanos, see PHILI 2009, and [UBG] Cod. MS. F. Klein 11 (Stéphanos to Klein, September 29, 1883). Regarding the context, see also DÉCAILLOT 2011, p. 35, and ROWE 1992b, p. 612 (a letter from Klein to Mittag-Leffler dated June 21, 1885). 33 [UBG] Cod. MS. F. Klein 9: 991 (a letter from Segre to Klein dated November 19, 1889). 34 Felix Klein, “Considerazioni comparative intorno a ricerche geometriche recenti,” Annali di matematica pura ed applicata 17 (1890), pp. 307–43. 35 See MARCHISOTTO/SMITH 2007, p. 56. 36 [UBG] Cod. MS. F. Klein 11: 157, 158 (letters from Padé to Klein dated Oct. 27 and Nov. 3, 1890). Felix Klein, “Considérations comparatives sur les recherches géométriques modernes,” Annales de l’école normale supérieure 3/8 (1891), pp. 87–102, 173–99. 37 See KLEIN 1923 (GMA III), Appendix, p. 17; also GRAY 2005. 38 This reprint appeared in Math. Ann. 43 (1893), pp. 63–100.

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and especially to Lie’s findings. This reprint, the many translations, Klein’s lectures on advanced geometry, and his students all served to make the ideas of his Erlangen Program known to a broader audience. For example, Klein’s American scientific heir Edward Kasner based his doctoral thesis, “The Invariant Theory of the Inversion Group: Geometry Upon a Quadric Surface” (1899), on it.39 Of course, Sophus Lie’s contribution and that of other mathematicians to the influence of the Erlangen Program are not a matter of dispute. Thomas Hawkins is certainly right to remark that it was Lie who did the most to promote the Erlangen Program and to ensure its widespread influence.40 It should be kept in mind, however, that Lithographic copies of Klein’s hand-written lectures circulated more widely than Hawkins supposes. They were available in Italy, for instance, where they can still be found (among other places) at the Scuola Normale Superiore in Pisa, and they were eagerly awaited in France by Émile Picard and his father-in-law Hermite.41 Furthermore, even though Gino Fano underscored the works of Italian mathematicians and Eduard Study in his article on group theory in the ENCYKLOPÄDIE and paid little attention there to the Erlangen Program, as Hawkins points out,42 this was done in full agreement with Klein, who guided the ENCYKLOPÄDIE project. Élie Cartan wrote the extended French version of this encyclopedia article (“La théorie des groupes continus et la géométrie”), and the question of how Klein’s view of transformation groups could be imported into a differential geometric setting played a crucial role in his further research (as well as in the research of Hermann Weyl).43 In 1914, the Dutchman J.A. Schouten published a book with Teubner in which he used Klein’s classification principle for the investigation of geometric quantities in vector analysis, and Klein had written a preface to it. The geometrician Wilhelm Blaschke, who completed his doctorate under Wilhelm Wirtinger in Vienna and his Habilitation under Eduard Study in Bonn, emphasized the following point in the preface to his edition of Klein’s lectures on geometry (which was published after Klein’s death): Klein’s group-theoretical development of geometry, as first formulated in his Erlangen Program from 1872 and further refined in his Introduction to Higher Geometry, is just as important and vital today as ever before for the further development of geometry – and of physics as well.44

Klein’s Erlangen program influenced famous philosophical systems as well. Its impact on Ernst Cassirer’s “system of invariants of experiences” has been well investigated by IHMIG (1997). SCHIEMER (2020) called Klein’s Program an important origin of structuralism in the philosophy of mathematics. 39 40 41 42 43 44

published in Transactions of the American Mathematical Society 1 (1900) (4), pp. 430–98. See HAWKINS 1984, p. 452; and HAWKINS 2000. See TOBIES 2016. HAWKINS 1984, p. 454. See SHARPE 1997; and in particular SCHOLZ 2012. KLEIN/BLASCHKE 1926, preface. See also Sections 9.2.2 and 9.2.3.

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3.1.2 Klein’s Students in Erlangen Like Clebsch, Klein dreamed of founding his own scientific school. But he recognized the difficulties of achieving such a lofty aim, because in Bavaria there had been no math teachers at secondary schools for some decades.45 Klein feared the worst: “Until now, there has been no mathematics in Bavaria, and the southern German student seems to be a lazy thinker. That said, my rather strong social drive – if this is understood as the desire to affect other people – will presumably be satisfied.”46 About a month after the beginning of the semester, Klein communicated the following to Darboux: Here in Erlangen I was at first very isolated, especially scientifically; my only interactions were with beginners. Since then, two older people have come here from Göttingen who work independently on geometry. I hope very much that this number increases next semester and that, in Erlangen, I will gradually succeed in establishing a school of geometric productivity such as that which had gotten off to such a great start in Göttingen while Clebsch was still there.47

After receiving his certificate of employment, Klein had informed the senate of the University of Erlangen on September 3, 1872 that he planned to offer a lecture, Monday through Friday from 11 to 12 o’clock, “on elementary aspects of algebra in connection with the analytic geometry of the plane,” and to teach “mathematical exercises” one hour per week.48 However, when he showed up to deliver his first lecture, on November 5th, only two students were in attendance. Aurel Voss and Adolf Weiler transferred to Erlangen to continue their studies under Klein, after Alfred Clebsch had died on November 7th. In general, the university facilities in Erlangen were rather underdeveloped at the time. During the winter semester of 1869/70, for instance, there were only 374 students enrolled at the entire university, and only two in mathematics. Klein’s course for beginners during his first semester in Erlangen (1872/73) was ultimately attended by five students, who remained there for several semesters and would later become teachers. At first, only one student, Adolf Weiler, registered for his advanced lecture “Select Chapters of Newer Geometry, with Practical Exercises.”49 In response, Klein changed the lecture topic to projective metrics, thereby attracting the attendance of Aurel Voss and Siegmund Günther in addition to Weiler. Weiler, in fact, would write his doctoral thesis on this topic, and it also informed Voss’s Habilitation thesis, which, with Klein’s help, he was able to submit to the University of Göttingen.50

45 46 47 48 49 50

See SCHUBRING 2007, p. 2.; SCHUBRING 2012. [Oslo] A letter from Klein to Lie dated August 3, 1872. [Paris] 64: Klein to Darboux, November 29, 1872, emphasis original. [UA Erlangen] I–II, Pos. 1, No. 27: Felix Klein. [UBG] Cod. MS. F. Klein, 7E. Klein thanked M.A. Stern for fulfilling Klein’s request and allowing Aurel Voss to do this ([UBG] Cod. MS. F. Klein 11: 1160A, a letter from Klein to M.A. Stern dated July 26, 1874).

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Siegmund Günther, who was already a Privatdozent when Klein arrived in Erlangen, gave lectures on the history of mathematics there.51 Although he attended Klein’s courses, he never incorporated their material into his own research. When Klein, in 1874, arranged for Paul Gordan to receive an associate professorship (see Section 3.5), Günther felt passed over. On July 17, 1874, he transferred as a Privatdozent to the Polytechnikum (as of 1877: Technische Hochschule) in Munich. When Klein revealed to him that he himself has accepted a position there, Günther decided to teach at a Bavarian secondary school.52 Finally, in 1886, he became professor of geography at this Technische Hochschule. Aurel Voss, in contrast, considered himself especially fortunate to have interacted with Klein for four months on a nearly daily basis and to have learned from his example. In awe, he wrote about Klein’s “remarkable ability to identify, in the works of others, the point that related specifically to his own ideas” and about his “talent of directing each of his students to the topic that best suited the latter’s particular gifts and stage of development.”53 Despite his small number of students, Klein’s efforts in Erlangen to create a productive geometric school were successful. Over the course of just five semesters there, he supervised not only Voss’s Habilitation research but also the doctoral dissertations of six mathematicians, four of whom would go on to have university careers. During Klein’s second semester in Erlangen, after Voss had left for Göttingen to defend his Habilitation, five students attended his advanced lecture on invariant theory: Wilhelm Braun, Wilhelm Bretschneider, Ferdinand Lindemann, Ludwig Wedekind, and the aforementioned Adolf Weiler. With these potential doctoral students, Klein initiated, on April 22, 1873, and together with the Privatdozent Siegmund Günther, his mathematical research seminar in Erlangen.54 Just two months later, Klein informed Sophus Lie: “Two dissertations will soon be completed here: the first concerns my classification of second-degree complexes, and the second is about the problem of projective dynamics (kinematics and statics).”55 Weiler was the first to defend, on July 19, 1873, and Lindemann soon followed on August 2, 1873. Klein had already made the following suggestion to Lindemann at a conference in Göttingen shortly after Clebsch’s death: As you know, Clebsch’s lectures ought to be edited, if possible, and especially his lectures on geometry. Do you have any interest in editing the latter under my oversight? The plan for

51 See Uebersicht des Personal-Standes bei der Kgl. Bayerischen Friedrich-Alexander-Universität Erlangen 1869/70 (Erlangen: Kunstmann), pp. 13–25; and LOREY 1916, p. 193. 52 [UBG] Cod. MS. F. Klein 9: 505 (a letter from Günther to Klein dated February 7, 1875); JACOBS 1977, p. 2; [Lindemann] Memoirs, p. 54. 53 VOSS 1919, p. 286 (emphasis original). 54 Until January 21, 1874, Günther delivered several lectures in this seminar on continued fractions, which was the topic of his Habilitation thesis ([Protocols] vol. 1, pp. 29–30, 44–46, 64– 66, 70–71, 80). Günther’s thesis was published as Darstellung der Näherungswerthe von Kettenbrüchen in independenter Form (Erlangen: E. Besold, 1873). 55 [Oslo] A letter from Klein to Lie dated June 28, 1873.

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3 A Professorship at the University of Erlangen this, which Gordan and I have agreed upon, is as follows: You take your doctoral exams, which can happen by the end of the summer. […] You complete the edition while remaining in daily contact with me, that is, probably in Erlangen. […] I will take it upon myself to write a preface for the publication. […] I can think of no better opportunity for you to use your abilities and also make yourself known in an advantageous way!56

Klein had switched roles. Just as he himself had edited Plücker’s work on line geometry under Clebsch’s aegis, the twenty-year-old Lindemann was now working under Klein’s supervision to edit Clebsch’s lectures. To facilitate Lindemann’s relocation to Erlangen, Klein rented an apartment for him; he induced Lindemann to give presentations on topics related to Clebsch in his seminar, and he met with him regularly.57

Figure 15: Klein’s circle in Erlangen, 1873. Felix Klein (on the right) with Ferdinand Lindemann, Wilhelm Bretschneider, Siegmund Günther, Adolf Weiler, and Ludwig Wedekind (TOBIES/VOLKERT 1998, p. 132).

56 [Lindemann] Memoirs, pp. 49–50 (a letter from Klein to Lindemann dated Dec. 27, 1872). 57 Ibid., pp. 52–54. Regarding Lindemann’s presentations, see [Protocols] vol. 1, pp. 69, 84.

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Lindemann described how Klein assembled and maintained his circle in Erlangen, which was joined by Axel Harnack in the fall of 1873. Once a week, Klein invited his students to his home, where one of them would give a lecture. Afterwards, they would all go out to dinner at eight o’clock. They would often eat lunch together at the Gasthof Walfisch,58 where professors in other disciplines were also regular patrons, including the zoologist Ernst Ehlers, who was the president of the Societas Physico-medica Erlangensis and whom Klein encountered again as a professor in Göttingen. After lunch, the whole group enjoyed the famous cheesecake at Café Mengin.59 Weather permitting, they would take a postprandial walk along the canal or toward the Rathsberg range of hills; in bad weather, they would go instead to the botanical garden and the greenhouses. During these excursions, they discussed mathematical problems, wrote “formulas and figures in the sand in the surroundings of Erlangen,” and heard several lectures by the botanist Maximilian Reeß, who was an expert in mycology. The professor of botany Reeß had been elected as a member of the Societas Physico-medica Erlangensis on the same day as Klein. Klein had been interested in biology since his student years. In 1874 and 1875, he even participated in the practical zoological exercises led by Emil Selenka, Ehlers’s successor. Klein was never fully certain of his own mathematical creativity, and he was still somewhat undecided about whether, at some later point, he should shift his research agenda to the natural sciences.60 But Klein had not yet run out of mathematical ideas. In this intensive work atmosphere, further dissertations were completed. Wilhelm Bretschneider, who had given presentations in Klein’s seminar on December 17, 1873, and February 25, 1874, submitted his dissertation – “Über Kurven 4. Ordnung mit 3 Doppelpunkten” [On Curves of the Fourth Order with Three Double Points] – on March 10, 1874. While working as a secondary school teacher in Württemberg, Bretschneider had acquired funding to continue his studies. At Klein’s instigation, he developed an analytic approach to a topic that Heinrich Schröter had treated synthetically.61 Ludwig Wedekind passed his doctoral examination on March 11, 1874. In his dissertation – “Beiträge zur geometrischen Interpretation binärer Formen” [Contributions to the Geometric Interpretation of Binary Forms] – he drew upon Klein’s Erlangen Program and his new principle of transference, which involved applying the geometric interpretation of x + iy to the surface of a sphere (Riemann’s number sphere) in order to develop a theory of binary forms. Wedekind’s ideas about the complex cross-ratio helped Klein to formulate his theory of the

58 The Walfisch inn, where Goethe had once spent the night, was located on Calvinstraße 5. In 1912, the building was demolished to make way for a bank. [Lindemann] Memoirs, p. 52. 59 Café Mengin (Schloßplatz 5) still exists in Erlangen today. 60 See JACOBS 1977, p. 2; and [Oslo] A letter from Klein to Lie dated February 22, 1875. 61 [Protocols] vol. 1, p. 90. See also G. Kohn and G. Loria, “Spezielle ebene algebraische Kurven,” in ENCYKLOPÄDIE, Vol. 3 (C 5), p. 560.

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icosahedron (see Section 3.1.3.2). Klein published Wedekind’s findings directly after a related article of his own in Mathematische Annalen.62 Axel Harnack gave two presentations in Klein’s seminar (on December 10, 1873 and January 14, 1874) about the work of Ferdinand Minding, from whom he had learned differential geometry in his birthplace of Dorpat (now Tartu, Estonia). Klein recognized Harnack’s analytic background and encouraged him to draw a connection between the theory of elliptic functions and the geometry of cubic curves in order to discover new sets of questions. To do so, Harnack used another of Klein’s important results: a new type of Riemann surface (see Section 3.1.3.1). Harnack quickly developed his own theorems, which he presented in Klein’s seminar on February 11, 1874, and he submitted his dissertation on July 18th of that year: “Ueber die Verwerthung der elliptischen Functionen für die Geometrie der Curven dritten Grades” [On the Utilization of Elliptic Functions for the Geometry of Third-Degree Curves].63 Wilhelm Braun’s dissertation, which he submitted on July 22, 1874, grew from the seminar lectures that he delivered on January 28 and April 21, 1874, about the geometric peculiarities of Lissajous’s tuning-fork curves: “At the instigation of my esteemed teacher, Professor Klein, to whom I owe the greatest thanks for his kind support, I attempted to study the oscillating curves, which result from two pendulum motions on the plane, as an object in itself in the sense of projective geometry and to ascertain their singularities.”64 Klein promoted the further advancement of his students. Just as Clebsch had published Klein’s early work in Mathematische Annalen or in the Göttinger Nachrichten, Klein likewise used the Annalen and the Sitzungsberichte of the Societas Physico-medica Erlangensis to publish the works of his doctoral students.65 It was in Erlangen, too, that the first foreign students came to study under Klein. These were Scandinavians recommended by Sophus Lie. In 1874, the Swede Victor Bäcklund, who already held a doctoral degree, came to Erlangen on a six-month travel stipend. He had achieved results in the field of algebraic geometry, for which he made use of Lie’s invariant theory of contact transformations. Klein took Bäcklund along as a guest to the meetings of the Erlangen Societas, he submitted the latter’s work for publication in the society’s Sitzungsberichte, and he also persuaded him to publish articles in Mathematische Annalen.66 62 Felix Klein, “Über binäre Formen mit linearen Transformationen in sich selbst,” Math. Ann. 9 (1875), pp. 183–208; Ludwig Wedekind, “Beiträge zur geometrischen Interpretation binärer Formen,” Math. Ann. 9 (1875), pp. 209–17. See also VOSS 1919, p. 286. 63 [Protocols] vol. 1, pp. 85–87; and Math. Ann. 9 (1875), pp. 1–54. 64 Wilhelm Braun, Die Singularitäten der Lissajous’schen Stimmgabelcurven (Erlangen: E. Th. Jacob, 1875), p. 4. 65 [UB Erlangen] MS 2565 [10]: Minutes from the society’s meetings held on July 28, 1873 and July 13, 1874. 66 [UB Erlangen] MS 2565 [10]: Minutes from the society’s meetings held on April 11, 1874 and March 6, 1876; and TOBIES/ROWE 1990.

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At the end of November in 1874, the Norwegian Elling Holst arrived as well. Holst had only just passed his master’s examination under Sophus Lie in July of that year. Lie had advised Holst not to continue his studies in Berlin but rather to go to Klein: “Believe me, there is nothing for you in Berlin unless you were to be as lucky as I and meet another Klein there.”67 Holst attended Klein’s lecture on “Select Chapters of Newer Geometry” (1874/75), and he gave three presentations in Klein’s seminar,68 which, in addition to a talk by Paul Gordan, also featured lectures by the freshly minted doctors Lindemann, Harnack, and Wedekind. Klein sent positive reports about Holst to Lie and recommended him for a Norwegian stipend so that he could continue his studies.69 When Klein moved to Munich in April of 1875 to begin a new position as a professor at the Polytechnikum there, Holst, Lindemann, Harnack, and Wedekind went with him and played an active role in the Mathematical Colloquium during Klein’s first year in Munich. The careers of Klein’s students advanced quickly. In December of 1876, Holst submitted his first article to Mathematische Annalen, where the dissertations by Lindemann, Weiler, Harnack, and Wedekind also appeared. Shortly after completing his Habilitation (Leipzig, 1876), Harnack became an associate professor at the Polytechnikum in Darmstadt, and he was appointed an associated professor in Dresden the following year. Wedekind completed his Habilitation in Karlsruhe, where he rose to become an associate professor in 1880 and a full professor in 1883. Adolf Weiler, who was Swiss, completed his Habilitation in 1875 at the Polytechnikum in Zurich, and in 1899 he was appointed associate professor at the University of Zurich. Lindemann, who at Klein’s instigation studied abroad in 1876, finished his Habilitation in Würzburg in 1877. In 1878, he became an associate professor in Freiburg im Breisgau, and he was promoted to full professor there the following year. In 1883, he ultimately became the first mathematician with an academic genealogy linked to Clebsch to be offered a full professorship at a Prussian University (Königsberg). The main reason for this appointment was Lindemann’s successful proof of the transcendence of π. But Klein considered this professional development especially important for other reasons as well: Your job offer from Königsberg is a benefit to us all, for it is a victory for our principle (namely our struggle against the cliquish nature of academia). Do you recall that, just ten years ago, a certain influential person stated that no student of Clebsch would ever come to Prussia with his consent?70

67 Quoted from STUBHAUG 2002, p. 236 (a letter from Sophus Lie to Holst). 68 [Protocols] vol. 1, pp. 143, 149, 151 (Holst’s lectures, Dec. 15, 1874; Jan. 26, 1875; and March 5, 1875). On Holst and especially his contribution to the reform of mathematical education in Norway around 1900, see JONASSEN 2004; and SIEGMUND-SCHULTZE/SØRENSEN 2006, pp. 167–74. 69 [Oslo] Letters from Klein to Lie dated December 2, 1874 and February 22, 1875. 70 [Lindemann] Memoirs (a letter from Klein to Lindemann dated March 30, 1883). – Further simplified proofs of the transcendence of π would follow, and Klein played a role in their publication. For a detailed discussion of these developments, see ROWE 2015.

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In Aurel Voss’s opinion, Klein was successful because, when interacting with his students, “he scattered the gold flakes [Goldkörner] of his rich talent without any concern about how they might later be put to use, for he never held the petty view of those who were inclined to see their own students as future competitors.”71 3.1.3 New Research Trends Whenever I think of you, it so often feels as though I have long and perhaps hopelessly been separated from my better self. And yet I repeatedly tell myself that, given my young age and my highly different aptitude, things would have to come to this. Who knows where my scientific activity will turn? Like you, I would like to formulate great new theories. Then again – and I have been following this plan with some consistency – I would also like to gain, to the extent that this is possible, a complete overview of all of mathematics in order to put an end once and for all to the fragmentation that has decidedly been such a great misfortune for all of us. Often, too, I think that my education in mathematics must at some point enable me to accomplish something reasonable in physics.72

Here, in 1875, Klein expressed his vision to integrate as many areas of mathematics as possible. Later, when he was no longer in a position to do so by himself and with his own students, he initiated the ENCYKLOPÄDIE project (see Section 7.8), which would serve to bring together various branches of mathematics, its applications, and its practitioners. In doing so, he would continue to focus on unsolved problems (see, in particular, Section 8.2.4). During this early phase of Klein’s research, which was predominantly oriented toward geometry, Klein missed cooperating with Sophus Lie and he increasingly sought to forge his own paths ahead. His publications during his time in Erlangen concerned the following areas of research: First, old topics such as cubic surfaces, Plücker’s complex surfaces, and nonEuclidean geometry. In this regard, it came to light that some of Klein’s articles had been published prematurely and needed to be corrected. After Klein had published an addendum on non-Euclidean geometry,73 Darboux later wrote to him on December 8, 1879 that one of his additions to a definition by von Staudt was superfluous. In response, Klein published a correction in Mathematische Annalen74 along with Darboux’s extensive letter: “Sur le théorème fondamental de la géométrie projective.”75 At the same time, he commented in a letter to Otto Stolz, with whom he had worked on the topic: “The matter is of course very frustrating to me, but nothing helps except admitting openly that one has erred.”76

71 72 73 74 75 76

VOSS 1919, p. 288. [Oslo] A letter from Klein to Lie dated February 22, 1875 (emphasis original). Math. Ann.7 (1874), pp. 531–37. Math. Ann. 17 (1880), pp. 52–54. Math. Ann. 17 (1880), pp. 55–61. [Innsbruck] A letter from Klein to Stolz dated April 28, 1880.

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Second, some new individual problems. Klein extended Pascal’s well-known theorem about a hexagon inscribed in a conic to n-space,77 a topic to which later mathematicians would also devote themselves.78 Here, with reference to Wedekind’s dissertation, Klein formulated his aforementioned new principle of transference, “by virtue of which the sphere, which serves to represent x + iy, simultaneously obtains the significance of the fundamental surface of a projective metric.”79 Klein and his students used this principle as a tool for generalization; it would also later prove to be fruitful for Klein’s theory of the icosahedron. In this area of research, analyzing things according to principles of transference or analogies was already an important methodological approach in Hesse’s work.80 In 1873, moreover, Klein also engaged in a philosophical discussion concerning the concept of a function in order to understand why continuous functions could exist without differential quotients.81 Weierstrass had been the first to provide an example of a non-differentiable continuous function. Klein took up this subject immediately with the help of his friend Otto Stolz, who would also later bring Bernhard Bolzano’s early results into the picture.82 (For further discussion of Klein’s forays into philosophy, see Section 8.3.2). Third, Klein devoted himself to new approaches and to questions concerning the connection of surfaces as related to a new type of Riemann surface and to aspects of the theory of equations, between which he drew a close connection. 3.1.3.1 On a New Type of Riemann Surface In accordance with his research plan, Klein chose to immerse himself deeply in the topics that had been worked on by Clebsch. Such topics included Riemann’s ideas, and in this case Klein relied more heavily on Riemann’s original ideas than on Clebsch’s results, with which he was not entirely satisfied. Clebsch had preferred an algebraic approach, whereas Klein tried to think about Riemann’s problems in geometric terms. Following Plücker, he preferred an intuitive approach. Klein sought a connection between the analytic results and the geometric form described by them. Ultimately, he found a “new type” of (projective) Riemann surface, which he later described as follows:

77 This article appeared in the Erlanger Sitzungsberichte on November 10, 1873. 78 See, for example, Sahib Ram Mandan, “Pascal’s Theorem in n-Space,” Journal of the Australian Mathematical Society 5/4 (1965), pp. 401–08. 79 Ludwig Wedekind, “Beiträge zur geometrischen Interpretation binärer Formen,” Math. Ann. 9 (1875), p. 213. 80 See Otto Hesse, “Ein Uebertragungsprincip,” Crelle’s Journal 66 (1866), pp. 15–21. 81 KLEIN 1922 (GMA II), pp. 214–24 (Erlanger Sitzungsberichte on December 8, 1873). 82 [Innsbruck] A letter from Klein to Stolz dated November 23, 1873; and Otto Stolz, “B. Bolzanos Bedeutung in der Geschichte der Infinitesimalrechnung,” Math. Ann. 18 (1881), pp. 255–79. Klein explained Weierstrass’s function at some length in the third volume of his Elementary Mathematics from a Higher Standpoint. See KLEIN 2016 [1928], pp. 41–46.

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3 A Professorship at the University of Erlangen In Riemann’s work, the number p is a characteristic for the “connection” of a closed surface. For me, it was an especially tormenting problem to figure out what this idea might have to do with the form of the related algebraic curves, and I was fortunate when, by constructing a “new” Riemann surface, I managed to find an extremely simple answer to this question.83

In order to determine whether two orientable surfaces can be mapped continuously and on a one-to-one basis, Riemann used characteristic invariants: p, the maximum number of possible loop-cuts [Rückkehrschnitte] that neither fragment the surface nor intersect one another; and μ, the number of boundary curves. In the case of closed surfaces, which were Klein’s main consideration, μ = 0 and the surfaces are characterized by p alone. As Klein had already learned in Göttingen in 1869 (see Section 2.4.3), Clebsch referred to the number p as the “genus” (Geschlecht) of the surface or of the equation F (ζ, z).84 Klein later explained how this was relevant to his further research: Equations F (ζ, z) = 0 can be biuniquely and continuously related to each other if and only if they have the same p. […] Thus Riemann has given a first characteristic to distinguish all algebraic equations that can be gotten from one another by biunique – or, as one says from the standpoint of formulas, by birational – transformations: they necessarily have a numerical invariant, the number p. Varieties with the same p are then further distinguished by their essential constants, the so-called “moduli.” Riemann finds their number to be zero for p = 0, one for p = 1, and 3p – 3 for p > 1.85

The theory of Riemann surfaces came about because the analytic continuation of holomorphic functions is not univalent. That is, one can obtain different function values along different paths. By means of a multiple-sheeted surface (a covering surface) as a domain of definition, the univalence of the analytic continuation can be achieved (as an example, think of the Riemann surface of the complex logarithm). Many researchers were working on this topic at the time. Scholz has described Riemann’s ideas about the “connection of surfaces” (their topology) and has compared the various approaches of Riemann, Möbius, Carl Neumann, Jordan, Schläfli, and Klein.86 In Scholz’s opinion, this comparison revealed that Klein, in his definition of the connection of surfaces, relied more heavily on Camille Jordan’s work than on Riemann’s. Klein began to publish on this topic in February of 1874, both in the Erlanger Sitzungsberichte and in Mathematische Annalen. In his seminar, he gave a lecture entitled “Ueber den Zusammenhang der Flächen” [On the Connection of Surfaces] on February 18, 1874, and on May 12th of that year he presented the lecture “Ein neuer Beweis über das p algebraischer Curven” [A New Proof Concerning the p of Algebraic Curves].87 As mentioned above, Klein attempted to “ascertain 83 KLEIN 1922 (GMA II), p. 5. – See also PARSHALL/ROWE 1994, pp. 168–69. 84 See also KLEIN 1979 [1926], p. 295. 85 Ibid., p. 242. Klein explained this repeatedly with examples: a sphere has no closed cuts, so p = 0; a torus has one such cut, that is, p = 1; and for p = 2, Klein used the example of the double torus (the shape of a pretzel). See also KLEIN 2016 [1925], pp. 124–25. 86 SCHOLZ 1980, pp. 163–79. 87 [Protocols] vol. 1, pp. 87–89.

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the number for the genus of a curve immediately from the form of the curve in question.”88 In his seminar lecture, he stressed: “The means to do this is provided by a new way of conceptualizing the course of an algebraic function by means of a Riemann surface that is closely connected to the curve represented by the function.”89 His article “Ueber eine neue Art der Riemann’schen Flächen” [On a New Type of Riemann Surfaces] begins as follows: When investigating the algebraic functions y of a variable x, one can make use of two different intuitive tools. One can either represent y and x equally as coordinates of a point on the plane, where only the real values of the same become evident and where the image of the algebraic function becomes the algebraic curve – or one can extend the complex values of one variable x to a plane and designate the functional relationship between y and x by means of the Riemann surface constructed over the plane. In many respects, it must be desirable to have a way to cross over between these two intuitive images.90

Klein explained that he understood the contact point (for real tangents) and the single real point (for imaginary tangents) as an image of the pair of curves or as an image of the place in the algebraic configuration. As Wirtinger later assessed, Klein combined von Staudt’s theory of the imaginary and the concept of the Riemann surface with the general concept of the Riemannian manifold.91 In dialogue with Ludwig Schläfli, Klein developed his work further, clarified his fundamental concepts, and implicitly expanded his concept of the manifold. Whereas, in his Erlangen Program, Klein had only considered relative properties, he now began to distinguish between relative and absolute properties in the sense of analysis situs (topology): I call absolute those properties that belong to the manifold in question independent of the surrounding space, in which they can be taken to lie. Relative properties depend on the surrounding space; they are invariant in the case of distortions of the manifold that take place within the space in question, but they are not invariant under arbitrary distortions.92

As an example of an absolute property, Klein cited the (non-)orientability of a surface. Among the novel approaches that Klein developed in the area of surface topology was the idea of understanding the projective plane as a “double surface,” which, in modern terms, means replacing it by its orientation covering.93 Klein’s

88 On the different terminology used to denote the topological invariant “genus” (Geschlecht) in the nineteenth century, see SCHOLZ 1980, p. 168. 89 [Protocols] vol. 1, p. 105. – For a lucid discussion of Riemann surfaces from a modern perspective, see LAMOTKE 2009, which also takes Klein’s work into account. 90 KLEIN 1922 (GMA II), p. 89. Originally published in Math. Ann. 7 (1874), p. 558. 91 See WIRTINGER 1919, p. 287. 92 Felix Klein, “Ueber den Zusammenhang der Flächen,” Math. Ann. 9 (1875), pp. 476–482, at p. 478 (= KLEIN 1922 [GMA II], p. 67). It should be noted that, in this context, EPPLE 1999 (pp. 164–66) regards Klein’s relativization of the knot problem as a new epistemic approach. In topology today, a distinction is drawn between the local and global properties of spaces. 93 See Felix Klein, “Bemerkungen über den Zusammenhang der Flächen,” Math. Ann. 7 (1874), p. 550 (= KLEIN 1922 [GMA II], p. 64).

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interest in non-orientable surfaces also led him to design his eponymous “Klein bottle,” a model of which he described in 1881.94 During his time in Munich (see Chapter 4), Klein, in his own words, got to the genuine Riemann, which means that he no longer used the algebraic equation to define an algebraic curve but rather derived Riemann’s existence theorem directly from the surface.95 He further developed his understanding of the Riemann surface in conjunction with works on geometric function theory: From a differential-geometric basis, he managed to characterize the complex structure on a real two-dimensional manifold, which made it possible to separate the concept of the Riemann surface from its elementary, immediate specifications by means of the sheet and branch structure above the complex plane.96

On the whole, Klein’s geometric manner of problem-solving led him to understand intuitively that Riemann’s approach to function theory would lead him farther than Weierstrass’s immediately could. Reinhard Bölling, today’s foremost expert in Weierstrass’ ideas, has convincingly shown how Weierstrass’s critique of Riemann initially caused mathematicians to look down on those who drew upon his work, but also how Riemann’s concept ultimately proved to be farther-reaching than Weierstrass’s.97 Klein’s comparison of Riemann and Weierstrass was as follows: Riemann was the man of brilliant intuition. Through his comprehensive genius he surpassed all his contemporaries. Where his interest had been awakened, he began anew, without letting himself be led astray by tradition and without submitting to the constraints of systematization. Weierstrass was primarily a logician; he proceeded slowly, systematically, step by step. Where he worked, he strove for definitive form.98

True to his motto of testing out all possible approaches, however, Klein would also incorporate Weierstrass’s approaches to function theory into his own arsenal of methods (see especially Section 5.5.2). Klein repeatedly returned to the topic of Riemann surfaces with new approaches; he held a series of lectures on them, which has recently been newly edited.99 These lectures provide a picture of how Klein integrated different mathematical subdisciplines and heuristic approaches while also taking into account the latest research results in the field. This is a fitting place to mention that Hermann Weyl, while lecturing during the winter semester of 1911/12 on Riemann surfaces in light of recent findings in set theory and topology, was able to rely on the willing support and helpful advice of Felix Klein, to whom he dedicated the first two editions of his book Die Idee

94 95 96 97 98 99

KLEIN 1893 [1882], p. 74 (repr. KLEIN 1923 [GMA III], p. 571). – See Section 5.5.1.2. See KLEIN 1922 (GMA II), p. 5. SCHOLZ 1980, p. 181. BÖLLING 2016, pp. 83–85. See also BOTTAZZINI 1986. KLEIN 1979 [1926], p. 231. See KLEIN 1986.

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der Riemannschen Fläche [The Idea of the Riemann Surface] “in gratitude and admiration” (see also Section 5.5.1.2).100 3.1.3.2 The Theory of Equations On December 4, 1873, Klein wrote to Lie about where his research might take him next: “Perhaps I will turn my focus to equations that, in the representation of a complex variable on the surface of a sphere, are formulated by means of the regular solids.”101 Further following his principle that, “in the theory of equations, such forms are a matter of invariant theory that pass into one another by means of particular discontinuous groups of linear substitutions” (see Section 2.8.2), Klein spoke in his seminar on July 7, 1874 about how “equations with one variable that possess in themselves linear transformations are peculiar with respect to their solvability.” In this regard, he particularly studied the equation of the twelfth degree, which, “if considered in light of the vertices of the regular icosahedron, can be reduced in such a way that the group of this equation, after the adjunction of the irrationality 5 , consists of 60 substitutions, and that one can derive its solution to an equation of the fifth degree with an adjoint product of differences.”102 Klein wrote three articles about the theory of binary forms and the equation of the twelfth degree, which he submitted to the Erlanger Sitzungsberichte (on May 11, 1874; December 14, 1874; and July 12, 1875), expanded for publication in Mathematische Annalen, and combined into a single work in the second volume of his collected writings. In these studies, he demonstrated above all that finite groups of linear substitutions can be derived from the regular polyhedra (the tetrahedron, cube, octahedron, dodecahedron, and icosahedron; later, he added the dihedron).103 He explained their transformations (that is, their rotations and reflections) and the dual reciprocation of the polyhedra, and he showed that the transformations which align a regular solid with itself form a group. By employing his principle of transference and referring to the Riemann sphere (which con100 The first edition of Weyl’s book was published in 1913, the second in 1923. 101 [Oslo] A letter from Klein to Sophus Lie dated December 4, 1873. Since regular polyhedra (also known as Platonic solids) will come up again in subsequent chapters, I should perhaps note that Klein was of course familiar with Euclid’s Elements, where, at the end of Book 13, it is shown that there are only five polyhedra whose faces are all identical regular polygons, identically arranged at all vertices. Euclid derives in previous theorems about how these polyhedra can be inscribed within a sphere. Because Plato mentioned these shapes in his Timaeus dialogue, they are also referred to as Platonic solids. 102 [Protocols] vol. 1, p. 123. 103 See KLEIN 1979 [1926], p. 320: “In these investigations I discovered an additional regular solid, the ‘dihedron’ […]: if we imagine the portion of the plane bounded by the sides of a regular n-gon to be doubled, we can consider this configuration as a regular solid, which maps to itself under n rotations about its principal axis and n reflections [Umklappungen] about lines in its equatorial plane. (This agrees exactly with the usual definition of a regular solid, only here the enclosed space-content is zero).”

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tains the vertices of the polyhedra), Klein was able to prove that no additional finite groups of linear transformations exist beyond those that he had identified by means of polyhedra. In cooperation with Paul Gordan, Klein continued to work on this topic in Munich, and he would often revisit it in later years.104 3.2 INAUGURAL LECTURE: A PLAN FOR MATHEMATICAL EDUCATION Still, the lack of widespread knowledge of mathematics is only a symptom of a deeper, much more serious problem. It is a symptom of the fateful division that has taken hold of our education all too strongly, and which from some persons has even found approval as a matter of principle: I am referring to the division between humanistic and scientific education. Mathematics and those fields connected with it are hereby relegated to the natural sciences, and indeed the indispensability of mathematics for these warrants this placement. On the other hand, its conceptual content belongs to neither of these two categories.105

Klein wrote these words down in November of 1872, when he was preparing his inaugural lecture at the University of Erlangen. He delivered this address on December 7, 1872, before a largely non-mathematical audience, which included the university’s Rektor, the legal scholar August Bechmann. The twenty-three-yearold Klein possessed a firm opinion about the essence and the role of mathematics: One should not believe that the essence of mathematics lies in the formula; the formula is only a precise designation for the thought connections involved. […] But the time is gone when the formula played the sole sovereign role at the expense of the thoughts behind it, and in which one regarded a mathematical work as finished so long as the computations were accessible. Today it is different: we require an inner understanding of the ongoing development, and consider a mathematical result complete only when it can be regarded from beginning to end as self-evident.106

At the same time, Klein philosophized about the place of mathematics within the system of the sciences and within society at large. He emphasized both the formal educational value of mathematics as well as the value of its applications, in which regard he especially underscored “the theoretical services performed by mathematics in the development of other sciences.”107 He cited examples of this from theoretical physics: the theory of light, molecular theory, geometric optics, the theory of heat conduction, and potential theory – topics in which he had immersed himself as a Privatdozent in Göttingen. He was aware that mathematicians were not appreciated at the time for the practical applications of their field that were “somewhat removed from the academic outlook,” among which he mentioned the predictive calculations of astronomers, the precision of geometric measurements, 104 See KLEIN 1922 (GMA II), pp. 255–61. 105 Klein’s inaugural address in Erlangen, quoted here from ROWE 1985, p. 135. For the German text, see JACOBS 1977, p. 19 (and TOBIES 2019b, p. 124). 106 Quoted from ROWE 1985, pp. 137–38. 107 Ibid., p. 137.

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and the accomplishments of engineers.108 This academic outlook was somewhat dismissive of technical applications – an attitude that Klein would do much to change in later years. In order to overcome the widespread opinion at secondary schools “that mathematics is just not important,”109 Klein argued that teachers should implement an intuitive teaching method and that the standards should be raised for the mathematical education of future teaching candidates. In this regard, he made the following pronouncement: “If we educate better teachers, then mathematics instruction will improve by itself, as the old consigned form will be filled with a new, revitalized content!”110 Klein intended to promote both logical exposition – the art of separating the essential from the inessential – as well as the use of geometric drawings and models. He argued that attention should be given both to mathematical exercises and to seminars with student participation. In this respect, he compared mathematical exercises to the “practica” used in teaching the natural sciences and technical subjects. Already here, he recommended that students should spend time at one of Germany’s Polytechnika (technical colleges), two of which he himself had visited and closely inspected (in Berlin and Darmstadt).111 Twenty-six years later, Klein did indeed make sure that spending one or more semesters at a Polytechnikum (Technische Hochschule) would become acceptable and would be recommended to teaching candidates in Prussia (see Section 8.1.2). In Erlangen, it took some time for Klein to acquire a classroom in which to conduct mathematical exercises. At first, he was given just a closet in the lecture hall for storing his models. Not until April of 1874 did the mineralogist Friedrich Pfaff, a brother of the late mathematician Hans Pfaff, offer Klein a spare room “for practical mathematical exercises in drawing, modeling, etc.” The room was located within the mineralogical collection in the palace. Klein saw this as a ray of hope, even though the facility had considerable technical shortcomings, such as bad flooring and problems with its heating and sanitation.112 At 50 Gulden, the annual budget of Erlangen’s Mathematical Institute was extremely meager. Klein therefore submitted special applications for funding to acquire instruments such as a polar planimeter and one of Charles Xavier Thomas’s arithmometers (see Fig. 16), which was the first mass-produced mechanical calculator (1,500 of these devices were produced from 1820 to 1878).113 To acquire the latter, Klein applied for additional funding of 100 Gulden on March 2, 108 109 110 111 112

Ibid. Ibid., p. 139. Ibid. Ibid. See also LOREY 1916, p. 150. [UA Erlangen] Ph. Th .I. Pos. 20 V, No. 8 (Klein’s requests for a classroom, dated December 19, 1872; April 16, 1874; July 23, 1874; and November 9, 1874). 113 See Rita Meyer-Spasche, “On the Impact of Mechanical Desktop Calculators on the Development of Numerical Mathematics” (2017), http://www2.ipp.mpg.de/~rim/e_art_wittenberg174.pdf (accessed January 12, 2021).

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1874; as he explained in his application, “such a machine would not only be extremely valuable for teaching but also for all of the institutes at the university whose professors occasionally have to make large numerical calculations.”114 He demonstrated how to use this calculator not only in his research seminar but also in a meeting of the Societas Physico-medica Erlangensis.115 Klein was the first person in Germany to acquire such a calculator for a university institute. Later, he continued to use this apparatus in his teaching, and he would commission Rudolf Mehmke and Aurel Voss to write about the development of mechanical calculators for the ENCYKLOPÄDIE.116

Figure 16: Charles Xavier Thomas’s arithmometer. Serial No. 759, built in 1868; dimensions (mm): 460 long, 180 wide, 93 high. Photograph by Georg Pöhlein and courtesy of the Informatik-Sammlung, University of Erlangen

Klein purchased mathematical models from the Delagrave publishing house in Paris. During his trip to Great Britain in 1873, he learned about more models and instruments that he soon incorporated into his teaching: mathematical models by Olaus Henrici, a tide-predicting machine by William Thomson (based on the methods of harmonic analysis and Fourier analysis), and a simple mechanical apparatus that “generated linear motion by means of mere circular motion.”117

114 [UA Erlangen] Ph. Th. I. Pos. 20 V, No. 8 (request for a math. apparatus, March 2, 1874). 115 [Protocols] vol. 1, pp. 96, 149; and [UB Erlangen] MS 2565 [10] Minutes of the meeting of the Societas Physico-medica Erlangensis held on May 11, 1874. 116 See Rudolf Mehmke, “Numerisches Rechnen,” in ENCYKLOPÄDIE, vol. 1.2, pp. 938–1079; and Aurel Voss, “Differential- und Integralrechnung,” in ENCYKLOPÄDIE, vol. 1.1, pp. 128–34 (an appendix to Voss’s article). – Klein also explained the theory of Amsler’s polar planimeter in the second volume of his Elementary Mathematics from a Higher Standpoint; see KLEIN 2016 [1925], pp. 15–20. – See also DURAND-RICHARD 2010. 117 [Protocols] vol. 1, pp. 67–68 (a lecture delivered by Klein on November 5, 1873).

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3.3 FIRST TRIP TO GREAT BRITAIN, 1873 I think that I am only at the beginning of my real education. Let me go to England and maybe also to Italy to meet the mathematicians there. Then I will come to you with some more original thoughts and be fully ready to participate, once again, in your manner of thinking.118

Klein wrote this to Sophus Lie on April 23, 1873. He postponed his trip to Norway because he did not want to visit Lie, who devised his theories very quickly, without any new creative ideas of his own. As his letters to his friend repeatedly suggest, Klein wanted to meet Lie on the same scientific level. In order to prepare for his trip to the British Isles, Klein took courses in English along with his doctoral student Ludwig Wedekind: “At the moment, I am taking four hours of English instruction per week. I am not learning much, however, because the amount of work that I am putting into it at home = 0 and because I am not well disposed to learning languages. Nevertheless, I am looking forward to the trip.”119 His remark about learning languages was certainly an understatement, given that he already had a good command of French and the classical languages at the age of sixteen (see Section 2.2.1). On August 8, 1873, Klein described his travel plans: I am set to leave the day after tomorrow. I will meet my friend [William Robertson] Smith in Leipzig and travel with him via Hamburg to Edinburgh. I intend to stay in Scotland for a few weeks and attend the meeting of the British Association in Bradford in the middle of September. Cayley, whom I contacted, wrote me a very obliging letter in response, and I am very much looking forward to meeting the old man. I am very curious about what my scientific gains will be from my trip (although that is not the only reason why I’m going), and especially whether the direction of my research will become more clearly defined than it has recently been. I am genuinely attracted to all aspects of mathematics. Usually, the way that mathematics is presented in books does not satisfy me in the least, or, on the other hand, things seem so self-explanatory that I have no desire to work on them (over the past weeks, incidentally, I have had no time to produce any results).120

Years later, in his autobiography, Klein had fond and grateful memories of his Scottish friend William Robertson Smith: “A mathematician and physicist by nature, Smith nevertheless shifted his attention to theological and oriental studies. Later, he invited me to England and made it much easier for me to access the academic circles there.”121 Smith, who had studied in Bonn in 1868, transferred to Göttingen the following year, where he was influenced by the theologian and orientalist Julius Wellhausen. Klein and Smith had spent much time together discussing works by Thomson and Tait, by R. Clausius (second law of thermodynamics), and others.122 They had undertaken an Alpine tour in August of 1872, and began their journey on August 10, 1873; Klein would return on October 16. 118 119 120 121 122

[Oslo] A letter from Klein to Lie dated April 23, 1873. [Oslo] A letter from Klein to Lie dated June 28, 1873. [Oslo] A letter from Klein to Lie dated August 8, 1873. KLEIN 1923a (autobiography), p. 15. [UBG] Cod. MS. F. Klein 12: 550 (Klein to Max Noether on July 5, 1872)

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After five weeks in Great Britain, Klein sent a report to Lie from Kirkcaldy, a city on the Scottish coast, about the attitude of British scientists toward geometric research. In essence, he felt that his own work did not quite overlap with what was being done there. In Scotland, that is, he encountered a different approach to mathematics, according to which “mathematical research is only undertaken and appreciated if it has immediate applications.”123 To Klein, the main proponent of this approach was Peter Guthrie Tait, who was especially interested in the applications of mathematics to physics. The researchers working in this way had little respect for Cayley, whom Klein and other German mathematicians had regarded as a shining star. Before Klein’s trip, seven of Cayley’s articles had been published in Mathematische Annalen, and others would follow.124 Surprised by this contrast in Great Britain, Klein compared it (in the aforementioned letter to Lie from September 14, 1873) to the familiar contrast in Germany between geometry and function theory. Klein’s conclusion was to bring the two sides together. In the future, that is, he hoped both to take into account the applications of geometric research and to gain a firmer understanding of function theory. Shortly thereafter, Klein participated in the forty-third annual meeting of the British Association for the Advancement of Science, which took place from September 17 to 24, 1873 in Bradford. Here he met Arthur Cayley, who occupied a respectable position in the Association. Cayley was one of the vice presidents of “Section A: Mathematics and Physics,” and received financial support from the Association for the publication of his book Mathematical Tables – a grant that was twice the amount of what Tait received for his Thermo-Electricity.125 Thus it is conceivable that Klein’s first impression of mathematics in Great Britain (formed by Tait in Edinburgh) was not representative. After Klein had met several mathematicians in Bradford – including Cayley, James Joseph Sylvester,126 Henry Smith, and William Kingdon Clifford – he wrote the following to Lie: I have to tell you about England! Indeed, this is terribly difficult to do in few words. Cayley is an extraordinarily friendly man who takes an interest in everything that is presented to him. Sylvester is entirely different. When he has something on his mind, he tells everyone about it and is briefly but entirely absorbed by the topic. I wished that he worked more steadily. There is no doubt that he is more brilliant than Cayley, and everyone in London is generally of the same opinion. One of the finest, incidentally, is [Henry] Stephen Smith, who visited you in Christiania. I wished that I could have spent more time with him. Then, finally, there is Clifford – the “divine one,” as they call him. Among the younger mathematicians, he is decidedly the best, and to me he is a highly interesting man to the extent that his interests encompass not only nearly all branches of mathematics but also, and to the same degree, the natural sciences and philosophy from a mathematical perspective.127

123 [Oslo] A letter from Klein to Sophus Lie dated September 14, 1873 (see also Section 10.1). 124 See Max Noether, “Arthur Cayley,” Math. Ann. 46 (1895), pp. 462–80. 125 See REPORT 1874, pp. xlvi, and lx. – On Cayley and his contributions to mathematics in Cambridge, see BARROW-GREEN/GRAY 2006. 126 On Sylvester’s life and work, see PARSHALL 2006. See also Section 5.8.1. 127 [Oslo] A letter from Klein to Lie dated November 4, 1873.

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In the same letter, Klein recommended that Sophus Lie should always send his results to the London Mathematical Society, and especially to Clifford, who had been a professor at University College London since 1871. Clifford died in 1879 at the young age of thirty-three. Nevertheless, he left behind an impressive body of mathematical work, about which I will only mention here that he had been influenced by Riemann’s work on differential geometry and that some of his findings were closely related to Klein’s own. Thus, Clifford’s lecture in Bradford – “On a Surface of Zero Curvature and Finite Extent” – motivated Klein to expand his concept of the manifold and his understanding of topology.128 Clifford had discovered a closed two-dimensional submanifold of elliptic space that is locally isometric to the Euclidean plane. This inspired Klein, in his future lectures on the theory of automorphic functions, to pursue the task of determining all manifolds of constant Riemannian curvature. Wilhelm Killing later called this the problem of Clifford-Klein space forms. Although Klein had proposed the term “space form with multiple connectivity,” Killing insisted: “The essence of these new space forms lies in a thought that was first raised entirely by you.” He concluded: “You see, however, that important reasons occasioned me to introduce the term ‘Clifford-Klein,’ and I hope that the term establishes itself.”129 It has certainly stuck. At the conference in Bradford, Henry John Stephen Smith, a professor at Oxford University, served as the president of “Section A: Mathematics and Physics.” Klein had begun to study Henry Smith’s Report on the Theory of Numbers in 1871, and this book would form an important basis in Klein’s later work on number theory (see Section 4.2.2). In Bradford, Smith gave a lecture “On Modular Equations,” a subject that would become one of Klein’s central research topics. In his address as president of the section, Henry Smith remarked that a committee had been formed some years before “to aid the improvement of geometrical teaching in this country.” In England, as in Germany, Euclidean geometry was dominant in the curricula at secondary schools, and Smith believed that it should be supplemented or replaced by more recent geometric methods. With respect to the “triumphs of modern geometry” over Euclidean geometry and the aforementioned “parallel postulate” (likewise referred to here as the “eleventh axiom”), he mentioned Arthur Cayley and Felix Klein by name: Two of those whose labours have thrown much light on this difficult theory are present at this Meeting – Prof. Cayley, and a distinguished German mathematician, Dr. Felix Klein; and I am sure of their adherence when I say that the sagacity and insight of the old geometer are only put in a clearer light by the success which has attended the attempt to construct a system of geometry, consistent within itself, and not contradicted by experience, upon the assumption of the falsehood of Euclid’s eleventh axiom.130

128 See SCHOLZ 1980, pp. 170–74; SCHOLZ 1999, p. 31; and KLEIN 1921 (GMA I), pp. 241, 253– 82. 129 [UBG] Cod. MS. F. Klein 10, p. 191 (Killing to Klein on June 9, 1891). See Wilhelm Killing, “Über die Clifford-Kleinschen Raumformen,” Math. Ann. 39 (1891), pp. 257–78. 130 REPORT 1874, p. 5 (“Mathematics and Physics.” Address by Prof. H.J.S. Smith).

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In this same address, Henry Smith also highlighted James Clerk Maxwell’s recently published Treatise on Electricity and its underlying mathematical theory. This did not escape Klein’s attention. He studied this theory and later commissioned H.A. Lorentz to write the article on it for the ENCYKLOPÄDIE (see Section 7.8). In 1881, Henry Smith sent his student Arthur Buchheim to study under Klein (see Section 5.4.2.1); and when Smith died in 1883, Cayley would recommend Klein to apply for the vacant chair at Oxford University (see Section 5.8.1). In his lecture at the annual meeting in Bradford in 1873, the Irishman Robert Stawell Ball also referred to Klein. He spoke in the section on mechanics and physics about “Contributions to the Theory of Screws.” Here he made use of Klein’s theory of first-degree and second-degree line complexes in his discussion of six fundamental complexes.131 In the mechanics of solid bodies, the theory of the screw is useful for describing static and kinematic systems. When he returned to Erlangen, Klein instructed his doctoral student Lindemann to take into account Ball’s approach and to write an addendum to his dissertation: “Ueber unendlich kleine Bewegungen und über Kraftsysteme bei allgemeiner projectivischer Massbestimmung” [On Infinitely Small Motions and Systems of Force in a General Projective Metric].132 Later, Klein would return to the topic himself in order to systematize this area of research with the methods of invariant theory and group theory (in this article, he also drew upon his Erlangen Program).133 After his stay in Great Britain, Klein immersed himself in the “study of functions.” In December of 1873, he delved more deeply into hyperelliptic functions, and he planned to read more about Abelian functions during the summer of 1874. His aim was to master geometric topics such as “the Kummer surface and inscribed ∞ 5 tetrahedra with hyperelliptic functions.” With the hope of cooperating further with Sophus Lie, he wrote to him: “When I come to visit you in fall, I will be more familiar with certain areas of research than I was before, and thus I can perhaps be useful.”134 As mentioned above, new mathematical models and instruments were presented at the conference in Bradford, and Klein immediately incorporated these into his teaching (see Section 3.2). Just as he considered such instruments useful for his university, he also thought that it would be valuable to initiate an exchange between British and German academic journals. At a meeting held on November 10, 1873, the zoologist Ernst Ehlers, president of the Societas Physico-medica Erlangensis, informed the society’s members that “Prof. Klein has taken steps to initiate an exchange of publications with the Royal Society of Edinburgh, the 131 See REPORT 1874, p. 27: “A group of six coreciprocals is intimately connected with the group of six fundamental complexes already introduced into geometry by Dr. Felix Klein (see ‘Math. Ann.’ Band ii, 203).” 132 This addendum was published in Math. Ann. 7 (1874), p. 144. 133 Felix Klein, “Zur Schraubentheorie von Sir Robert Ball,” Zeitschrift für Mathematik und Physik 47 (1902), pp. 237–65; reprinted in Math. Ann. 62 (1906), pp. 419–48 (= KLEIN 1921 [GMA I], pp. 503–32). See also KLEIN 1991. 134 [Oslo] A letter from Klein to Lie dated December 12, 1873.

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Mathematical Society of London, and with the editors of the journal Nature, and it has been decided to send these societies copies of our published proceedings.”135 This exchange of journals was an important means of keeping Klein and his students up to date with the latest international research; at the time, in Klein’s opinion, the holdings of the university library in Erlangen were drastically antiquated. As early as November 15, 1872, he therefore applied for additional funding so that the library might acquire more works of scholarship pertinent to mathematics, stressing “that mathematics is a thoroughly international science and that the progress of the productive mathematician is inhibited in an essential way unless he is fully aware of the contemporary ideas of others.” His application is a testament to his overview of scholarly literature and to his extremely skillful approach to achieving his aims (see Appendix 2). Because the library’s overall budget was limited, this free exchange of journals served to provide a helpful supplement to its collection. It should be noted that, in his later positions, Klein would likewise be a stubborn advocate for university libraries to increase their holdings in the field of mathematics.136 In Bradford in 1873, Felix Klein was made a corresponding member of the British Association for the Advancement of Science.137 In 1875, Klein was appointed as a foreign member of the London Mathematical Society (see Fig. 17), whose president at the time was Henry John Stephen Smith.138 As of 1884, the latter society has awarded, every three years, the De Morgan Medal, which is named after its cofounder and first president Augustus de Morgan. In 1893, Klein became the fourth mathematician (after Cayley, Sylvester, and Lord Rayleigh) to receive this honor, which to this day has been given to just a few non-British mathematicians.139 On June 6, 1878, the Cambridge Philosophical Society named Klein an honorary member.140 On December 10, 1885, the Royal Society of London, which has existed since 1660, elected Klein as a foreign member, and it made him an honorary member on November 7, 1902. In 1912, Klein became the fourth German mathematician to be awarded the Royal Society’s Copley Medal, after Gauss (1838), Plücker (1866), and Weierstrass (1895). Since 1731, this prize has been awarded to a researcher working in any branch of science, and it is the highest distinction bestowed by the Royal Society.

135 [UB Erlangen] MS 2565 [10], p. 21. 136 While working in Munich, Klein and his assistant Walther Dyck assessed the holdings of the Bavarian State Library and compiled a list of (mostly non-German) mathematical books and journals that the library ought to acquire. ([BStBibl] Halmania VI: letters from Klein to the director of the library dated January 20, 1879 and February 17, 1879). Regarding Klein’s similar activity in Leipzig see [UB Leipzig] MS. 0800: Nachlass of the librian Koehl (Klein’s letter to Koehl dated July 24, 1881); on Göttingen, see FREWER-SAUVIGNY 1985. 137 See REPORT 1874, p. 81. 138 See BERICHT 1876, p. 8. Regarding the presidents of this society, see OAKES et al. 2005. 139 See https://www.lms.ac.uk/prizes/list-lms-prize-winners#DeMorgan_medal (accessed Jan. 26, 2020). 140 See BERICHT 1878, p. 28.

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Figure 17: Felix Klein’s certification as a Foreign Member of the London Mathematical Society, 1875 ([UBG] Cod. MS. F. Klein 114: 51), and the De Morgan Medal, which he became the fourth mathematician (after Cayley, Sylvester, and Rayleigh) to receive in 1893 ([UBG] Cod. MS. F. Klein 115: 10).

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3.4 TRIPS TO ITALY In the meantime, I am now preparing for another longer trip. Now that you are getting married and I therefore can’t come to Norway, I would like to travel to Italy and perhaps spend the entire fall there.141

Although a trip to Italy had long been on Klein’s agenda, it was not part of his plans in the summer of 1874. At that time, Klein wanted to work with Sophus Lie and finally to embark on a trip to Norway – a trip that had already been postponed several times. On February 10, 1874, he wrote to Lie: “I consider it a complete certainty that I am coming to Norway in the fall.”142 On April 4th, he informed Lie that he would like to visit from the middle of August to the middle of September and that he intended to take Norwegian classes in advance from a Danish student in Erlangen. Klein was looking forward to this trip to Norway “with all the eagerness of which I am capable.” Yet Lie’s alternative proposal, which was for Klein to meet him in Paris during Lie’s honeymoon, was not an option for Klein. Klein had oriented his recent work in such a way as to “culminate” in collaboration with Lie. This plan fell through, and Klein drew the following conclusion: I will now have to find my way more independently. This path ahead is already laid out for me by the lectures that I have begun to prepare on Abelian functions. I will attempt – and my article on Riemann surfaces is a first step in this direction – to provide a geometrically intuitive account of this entire area of study.143

Klein wanted to seek out Italian mathematicians and expand his knowledge, and he was met in Italy with open arms. The political unification of the country in 1861 had also inspired its mathematicians, who in part had participated in patriotic struggles, had held political offices over the course of the nation gaining independence, and had fostered contacts with foreign mathematicians.144 Alfred Clebsch, for instance, was a member of academic societies in Milan and Bologna. Felix Klein’s was integrated into this network from early on. In his dissertation (see Section 2.3.4), Klein had generalized a result by Giuseppe Battaglini, who held professorships in Naples (as of 1860) and Rome (as of 1874) and who had cofounded the Giornale di matematiche. As of 1867, the latter journal published Battaglini’s works on non-Euclidean geometry as well as Jordan’s articles on group theory. Battaglini served as the journal’s editor until his death in 1894, and as late as 1893 he published a lecture that Klein had given in Chicago. Klein organized an exchange between the Giornale and Mathematische Annalen. At the same time, he also recruited Italian authors for “his” journal – scholars such as Bertini, Brioschi, Ascoli, and D’Ovidio.

141 [Oslo] A letter from Klein to Lie dated July 23, 1874. 142 [Oslo] A letter from Klein to Lie dated February 10, 1874. 143 [Oslo] A letter from Klein to Lie dated April 26, 1874 (see the original quotation in German TOBIES 2019b, p. 132). 144 On these details, see NEUENSCHWANDER 1983.

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Enrico D’Ovidio had studied under Battaglini while the latter was still in Naples, and his work drew upon Klein’s articles on non-Euclidean geometry.145 As of 1872, D’Ovidio held the professorship for algebra and analytic geometry at the University of Turin, where he taught line geometry (as developed by Plücker and Klein) and where he became known for formulating the law of sines in ndimensional curved spaces. His students included Corrado Segre and Giuseppe Veronese, who contributed to the geometry of higher-dimensional spaces.146 Both Veronese and Francesco Gerbaldi, who was D’Ovidio’s assistant for several years, would conduct postdoctoral research under Klein during the 1880s (see Section 5.4.2.3). Since 1888, Corrado Segre held a professorship alongside D’Ovidio in Turin. He collaborated with Klein and some of Klein’s students for many years.147 During a trip to Italy in March and April of 1899, which Klein undertook in the interest of the ENYCLOPÄDIE, he stayed in Turin for some time. Afterwards he delivered a lecture entitled “Mathematik in Italien” [Mathematics in Italy] at the Göttingen Mathematical Society.148 As early as February of 1899, Klein had tried to hire Segre’s student Gino Fano as a professor in Göttingen (see also 8.1.2).149 Fano, who had translated Klein’s Erlangen Program into Italian (see 3.1.1) and studied under Klein in 1893 and 1894, contributed to the ENCYKLOPÄDIE. Fano’s translation of the Erlangen Program was published in 1890 in Annali di matematica pura ed applicata. This journal was characterized by the work of significant Italian mathematicians of the older generation. Its chief editor at the time was Francesco Brioschi (Milan), and its editorial board included Luigi Cremona (Rome), Enrico Betti (Pisa), Eugenio Beltrami (Pavia), and Felice Casorati (Pavia). Brioschi and Betti are associated with the beginnings of algebraic geometry in Italy. In 1858, both of them (and Casorati) studied abroad together in France and Germany; they met Riemann in Göttingen and later invited him to Italy. They translated his papers and lectured about his work. In November of 1872, Klein had ordered Casorati’s book Teorica [sic!] delle funzioni di variabili complesse (Pavia 1868) for the university library in Erlangen (see Appendix 2). While serving as the general secretary of the Italian Ministry of Education, Brioschi founded the Politecnico di Milano and the Academy associated with it. He was named a corresponding member of the Göttingen Royal Society of Sciences in 1869 and a foreign member in 1870. His research on the theory of equa145 See SCHOENFLIES 1919, p. 294. 146 Klein recruited Segre to write the article on multi-dimensional geometry for the ENCYKLOPÄDIE (“Mehrdimensionale Geometrie,” in ENCYKLOPÄDIE, vol. 3 [2.2.A], pp. 772–972). 147 See Section 2.3.4 above and CASNATI et al. 2016. For the correspondence between Segre and Klein from 1883 to 1923, see [UBG] Cod. MS. F. Klein 9: 952–998B. 148 [UBG] Cod. MS. F. Klein 22F (a draft of Klein’s lecture). 149 See Klein’s letter to Fano dated Febr. 5, 1899 (printed in TERRACINI 1952, p. 486). Fano expressed his thanks but said that this would only be a “fantasy.” He preferred to find a professorship in Italy, which he ultimately did (in Messina in 1899 and then in Turin in 1901). ([UBG] Cod. MS. F. Klein 9: 4A (a letter from Fano to Klein dated Febr. 10, 1899).

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tions, which was published in the Göttinger Nachrichten and in Mathematische Annalen, influenced Klein’s thinking about the theory of the icosahedron (see Section 4.2.1, and 4.2.3). Klein remained in close contact with Brioschi’s doctoral students Luigi Cremona and Eugenio Beltrami. As early as December 30, 1868, Klein had sent a copy of his dissertation to Luigi Cremona: The study is concerned with an aspect of the theory of second-degree complexes. After his trip to Northern Italy this last fall, the now-deceased Plücker expressed more than once that you are the only person who fully understood him. I thus find it of the utmost importance to subject my first academic work to your esteemed judgement, all the more so because I achieved a result that deviates from that which lies at the basis of Battaglini’s study of second-degree complexes.150

Cremona became a corresponding member of the Göttingen Academy of Sciences (again, then called the “Royal Society”) in 1869 and a foreign member in 1880, and he was a contributor to Mathematische Annalen as early as 1871, when the journal was still edited by Clebsch. Cremona’s research was concerned above all with third-order space curves, descriptive geometry, and graphical statics.151 Before he went to Italy, Klein had already informed Cremona about his edition of Plücker’s Liniengeometrie and had asked him what he thought of Clebsch.152 Cremona thought highly of him, as Klein mentioned in Clebsch’s obituary, which appeared in Mathematische Annalen in 1873. Later, Max Noether also characterized Clebsch and Cremona as having the same essential feature: “Neither was much concerned with solving particular individual questions or with abstract questions of principle; rather, they were interested in the creative methodology of their science.”153 This could similarly be said of Klein. When Klein was planning his first trip to Italy for August and September of 1874, he wrote to Cremona on July 23 that he intended to come to Rome via Switzerland with a friend (whose command of Italian was presumably better than his own). Klein stressed that he would especially like to meet Cremona, Beltrami, and Battaglini,154 and he later provided further details of his travel plans: Genoa – August 25, 1874 My esteemed colleague and friend! As you can see from the heading of this letter, I have by now entered lovely Italy, if only its northern region, and I will now take this opportunity, in response to your kind letter from August 4th, to share with you the details of my travel plans. On Thursday, August 27th, I will take a steamship from here to Naples, and I think that I will spend the next ten to twelve days there and in the surrounding area. Then I will travel to Rome, for which I have set aside approximately three weeks. If, as your letter suggests, you will still be in Sorrento until the end of August […] then I will come together with my travel companion (Dr. Carl Schmidt, a Privat-

150 151 152 153 154

[Rome] 2589 (a letter from Klein to Cremona dated Dec. 30, 1868). See also MENGHINI 1993. See also Marta Menghini’s article in BARBIN et al. 2019, pp. 57–68. [Rome] 2590–2593 (letters from Klein to Cremona). Max Noether, “Luigi Cremona,” Math. Ann. 59 (1904), pp. 1–19, at p. 19. [Rome] 2596 (a letter from Klein to Cremona dated July 23, 1874).

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3 A Professorship at the University of Erlangen dozent of theology at the University of Erlangen) to visit you there for a day, which should be easy to do from Naples, and discuss with you where and how I might be able to speak with you again in Rome. I regret terribly, of course, that my knowledge of Italian is almost nonexistent and that I thus must fear that our conversation, which might be easiest to conduct in French, might cause you considerable inconvenience. Luckily, it is somewhat easy to communicate about mathematics – though this may also be the reason why I understand so little Italian, as I have simply been content to understand mathematical articles written in it. I also have to thank you for the information regarding Prof. Battaglini and Beltrami. I have not yet received a letter from the latter, since I have already been away from Erlangen for a rather long time. We will probably still be able to make arrangements, however. In the hope of meeting you soon and being able to listen to you and learn from you, I remain sincerely yours, Felix Klein155

Klein met with Beltrami in Venice.156 Their work on non-Euclidean geometry had common points of reference.157 Beltrami’s article “Zur Theorie des Krümmungsmaasses” [On the Theory of Curvature Measure] appeared in the first volume of Mathematische Annalen. Since January of 1873, Klein had been studying “Beltrami’s treatise on differential parameters” and had been discussing it in his lectures, as he reported to Sophus Lie: “The work is very beautiful, but in the case of our concepts of invariants or the connex, as established by Clebsch, it can be expanded in an essential way.”158 Klein was looking for a more general method, and he wanted to discuss the matter with Beltrami. Beltrami had been influenced by Gauss’s work; he had met Riemann in Pisa and had kept in close contact with Clebsch.159 Later, Beltrami sent some of his own students to study under Klein, and thus G. Morera came to Leipzig (see 5.4.2.3) and E. Pascal to Göttingen. Klein’s plan to meet Enrico Betti in 1874 fell through. Betti held professorships for advanced geometry and analysis (1859), theoretical physics (1864), and celestial mechanics (1870) in Pisa, where he also had supported Riemann (who died in Italy in 1866). On September 8, 1874, Klein wrote to Betti from his hotel in Naples (Hôtel Minerva): “Yesterday I learned from Prof. Cremona that you are back in Pisa, where, recently, I unfortunately looked for you in vain while spending a day in Livorno as part of my steamship tour from Genoa to Naples.” Klein informed Betti further that he intended to be in Florence from the 24th to the 28th of September, and that, “during those three days, I would be willing to come to Pisa, even though it would be more pleasant for me to meet you in Florence, because I also really want to get to know the city of Florence […].” Klein wanted to have a discussion with Betti “about questions concerning the connection of higher

155 [Rome] 2597 (Klein to Cremona, August 25, 1874). – For Carl Smith see UEBERSICHT 1874, p. 5. 156 See JACOBS 1977, p. 2; and [Rome] A letter from Klein to Cremona dated Nov. 21, 1874. 157 On Beltrami’s work, see Nicola Arcozzi’s article in COEN 2012, pp. 1–30. 158 [Oslo] A letter from Klein to Lie dated January 22, 1873. 159 See Ernesto Pascal, “Eugenio Beltrami,” Math. Ann. 57 (1903), pp. 65–107.

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spaces, about which I hope to learn a great deal from you.”160 Betti had announced that he would arrive in Florence on Saturday, but then changed his plans without notice to come on Sunday. Klein spent the Saturday waiting for him in vain, and ultimately decided to spend that Sunday sightseeing.161 Klein’s move from Erlangen to Munich in 1875 meant that he was even closer to Italy than he had been. Brioschi thus visited him while traveling to and from Gauss’s centenary celebration in Göttingen.162 Cremona and Giuseppe Jung, a professor of graphical statics in Milan, came to Munich in 1876 and 1877 (see Section 4.3.3). Moreover, Klein undertook a second trip to Italy during the Easter vacation in 1878 (April 2nd to 25th). This time he traveled with his wife (see Section 3.6), and they visited Pisa, Florence, Bologna, and Venice.163 Klein had informed Cremona in advance that he had “undetermined travel plans to come to Italy with my wife, but probably only to northern Italy, because the present times and the circumstances might make it difficult to travel any farther.”164 His son Otto, who was not yet three years old (see Section 3.6.3), would be taken care of by his relatives in Erlangen during this time. On this trip in 1878, Klein finally managed to meet Enrico Betti, whom he had failed to meet in 1874. Klein wrote to him: “My work has taken a turn that has made me increasingly reliant on your algebraic investigations.”165 While in Pisa, Klein also met Ulisse Dini, who succeeded Betti in 1871 as the professor of advanced geometry and analysis there; Eugenio Bertini, who was a professor of geometry there from 1875 to 1880; and Ernesto Padova, who, with Betti’s support, was made a professor of theoretical mechanics in Pisa in 1875. Betti would remain later, too, a point of reference for Klein. As late as 1882, and with respect to his work on Riemann’s function theory, Klein sent an inquiry to Betti “about the use of closed Riemann surfaces in space and about the extent to which Riemann himself had already developed a theory of stationary currents of incompressible fluids in three-dimensional spaces that are somehow curved but closed.” In particular, he asked: “Have you personally worked to develop this theory further and, if so, have you ascertained how the related (or at least similarly classifiable) statements by Helmholtz and [William] Thomson should be understood?”166 In February of 1877, the Reale Istituto Lombardo di Scienze e Lettere in Milan made Klein a corresponding member of its “Classe di scienze matematiche e naturali.” Brioschi had presented Klein’s results before this body. Furthermore, the first two of Klein’s honorary doctoral degrees – he was awarded ten in total – were bestowed by Italian universities, the first by the University of Turin (1880) and the second by the University of Bologna (1888). 160 161 162 163 164 165 166

[Pisa] 826: Klein to Betti dated September 8, 1874 (emphasis original). [Pisa] 828: A letter from Klein to Betti sent from Munich and dated October 2, 1874. See TOBIES/ROWE 1990, p. 88 (a letter from Klein to A. Mayer dated May 11, 1877). [Pisa] 830: A letter from Klein to Betti dated April 30, 1878. [Rome] 2602: A letter from Klein to Cremona dated November 4, 1877. [Pisa] 829: A letter from Klein to Betti sent from Munich and dated March 30, 1878. [Pisa] 832: A letter from Klein to Betti dated March 13, 1882 (emphasis original).

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In 1896, when Klein was elected a foreign member of the Società Italiana delle Scienze (the Italian Academy of Science) in Rome (see Fig. 30), Klein thanked the president Luigi Cremona for this surprising honor167 and emphasized, in a personal letter to him, that he had frequently been discontent with his scientific achievements and still often thought he was searching for his own way. Klein referred here to “his old love of physics” and enclosed a lecture on technical physics that he had delivered in Hanover (see Section 7.7). He concluded this letter to Cremona by thanking him for the many inspirations that he had received from Italian mathematicians and, above all, for sending to him “capable students, who have all been especially valuable to me.”168 3.5 DEVELOPING THE MATHEMATICAL INSTITUTION Gordan and I complement one another so well that the mathematics program in Erlangen can now be seen as being equal to that of every other university.169

Klein, who was especially creative when cooperating with other mathematicians, saw that his collaborative work with Sophus Lie was coming to an end, and thus he reoriented his approach. He had not only strengthened his contacts with British and Italian mathematicians, but he was also able to establish an associate professorship for mathematics in Erlangen. This was a rather amazing feat, given that, during the summer semester of 1874, there were still only eleven students of mathematics and physics combined at the university there, whose total enrollment was 442 students (166 of whom were students of theology).170 Before his first trip to Italy, Klein informed Lie that his “highest hope” was for Paul Gordan to be hired in Erlangen. Klein worked with Gordan on the editorial board of Mathematische Annalen, and he wanted to collaborate further with him on “geometry and algebra in all of the advanced ways.”171 Klein’s application for establishing a new associate professorship for mathematics, which the Philosophical Faculty had submitted to the Academic Senate on July 20, 1874, contained the names of three of Clebsch’s students as possible candidates for the position: 1) Paul Gordan, 2) Max Noether, 3) Aurel Voss.172 The application emphasized Gordan’s pioneering achievements, particularly his 1868 proof of the so-called theorem of finitude, which states that every binary form possesses a “finite system of forms,” i.e., that every covariant and invariant of a given binary

167 There were only twelve foreign members, in addition to fourty Italian members. 168 [Rome] 2834: A letter from Klein to Cremona, 1896. – On Italian students in Klein’s courses, see Sections 4.2.4.2, 5.4.2.3, and 6.2.3. 169 [Oslo] A letter from Klein to Lie dated July 23, 1874. 170 See UEBERSICHT 1874, p. 28. 171 [Oslo] Klein’s letters Klein to Lie dated July 23, and October 5, 1874. 172 For the complete text of this application, see TOBIES 1992a, pp. 768–70.

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form is a rational function of a finite number of such covariants and invariants.173 At the same time, it was stressed that Gordan was being barred from advancing his career in Gießen, where he had been an associate professor since 1864. The application also highly praised Noether’s and Voss’s research and teaching achievements to bring them to the Bavarian government’s notice in view of future applications for positions in Bavaria. Paul Gordan came to Erlangen on October 1, 1874, and his yearly salary was 1,700 Gulden. As early as November 8, 1874, Klein recommended that Gordan should be made a member of the Societas Physico-medica Erlangensis, and he was elected to join the society on December 14th. Like every newly hired professor in Erlangen, Gordan had to present a written “program” to join the Philosophical Faculty. Gordan’s program, which was published in 1875 as Über das Formensystem binärer Formen [On the Form System of Binary Forms], was composed in a peculiar way: Gordan dictated and Klein wrote down what he said.174 In his own seminar, which Gordan attended, Klein gave three lectures about Gordan’s program before it was published (these lectures took place on February 10, 17, and 26, 1874).175 Klein’s research at the time concerned “all finite groups of real motions of space […] that transfer a sphere to itself and have a fixed point within this same sphere.” He was therefore busy studying the regular solids (particularly the icosahedron), and he believed that Gordan’s “general investigations of systems” could help him in important ways, especially with the algebraic aspects of his work.176 Klein and Gordan were both mathematicians who benefited from, and even needed, collaboration. Gordan, who in Noether’s words was “clumsy with the pen,” often depended on his close friends (Clebsch, then Klein and Noether) to prepare the final versions of his work.177 Klein valued Gordan’s brilliant ideas, as he did Lie’s. Thus he informed Lie: “Moreover, I will always have to edit Gordan’s texts, which I am pleased to do in order to do him a favor and familiarize myself with his work.”178 After Klein’s new position in Munich had been finalized, he immediately turned to the issue of his successor in Erlangen. In a letter dated December 12, 1874, to the Bavarian State Ministry concerning the “replacement of the full and possibly also the associate professorship for mathematics at the University of Erlangen,” Paul Gordan was the only person recommended as a candidate for the 173 Paul Gordan, “Beweis, dass jede Covariante und Invariante einer binären Form eine ganze Function mit numerischen Coefficienten einer endlichen Anzahl solchen Formen ist,” Crelle’s Journal 69 (1868), pp. 323–54. For further discussion of this proof and additional related proofs by Gordan himself and by Hilbert, see Max NOETHER 1914, pp. 11–18. 174 [Lindemann] Memoirs, p. 54; and for Gordans paper: CATALOGUE 1908, p. 150. 175 [Protocols] vol. 1, p. 150. – Gordan lectured in Klein’s seminar (vol. 1, p. 148) on his article “Über den größten gemeinsamen Factor,” Math. Ann. 7 (1874), pp. 433–48. 176 See M. NOETHER 1914, p. 21. 177 Ibid., p. 21. 178 [Oslo] A letter from Klein to Sophus Lie dated February 22, 1875.

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full professorship. He was given the position on December 28, 1874, and took it up on April 1, 1875, with a yearly salary of 2,000 gulden.179 The proposed candidates for the associate professorship were Max Noether and Aurel Voss. Noether, the preferred candidate, assumed this position on April 1, 1875 with an annual salary of 1,500 gulden.180 As early as January 1875, Klein had explained the Erlangen institution to Noether: “The Seminar has two directors, one for the mathematical department, the other for the physics department. Lommel and I currently hold these positions.” Eugen Lommel and Klein had written the seminar’s statutes, which also allowed the associate professor of mathematics (first Gordan, then Noether) to announce exercises for teaching candidates.181 Klein was interested in finding positions for the most accomplished representatives of Clebsch’s algebraic-geometric school. The religious affiliations of the candidates did not play any role in Erlangen’s hiring process. Both Noether and Gordan came from Jewish families, though Gordan had been baptized at the age of eighteen.182 Klein’s and his colleagues’ later campaigns to endorse Noether’s candidacy for a full professorship failed repeatedly on account of explicit antiSemitism.183 Not until 1888 did Max Noether become a full professor alongside Gordan in Erlangen, even though Gordan, Klein, and Eugen Lommel had been submitting formal proposals in favor of his promotion since 1882. By the time Klein left Erlangen in 1875, he had developed a Mathematical Institution there with improved personnel and facilities, a wider selection of models and instruments, and a more substantial collection of mathematical scholarship in the university library. Before his arrival, the library there did not even have copies of Euler’s Introductio and Lagrange’s Mécanique analytique (see Appendix 2). 3.6 FAMILY MATTERS By securing a full professorship at such a young age, Felix Klein was able to seek a life partner around the same time that his older friends were doing the same. On December 2, 1874, he wrote to Sophus Lie that he was not only “occupied with Gordan” but also with “dancing and socializing.”184

179 180 181 182

[UA Erlangen] T. II, Pos. 1, No. 23 (Paul Gordan). [UA Erlangen] T. II, Pos. 1, No. 6 (Max Noether). [UBG] Cod. MS. F. Klein 12: 569 (Klein’s letter to Max Noether on January 19, 1875). Gordan was baptized on July 21, 1855 in Berlin. This information was brought to my attention by Cordula Tollmien, who came upon it while conducting research for her biography of Emmy Noether (on her first two volumes, see TOLLMIEN 2021). 183 [UBG] Cod. MS. F. Klein 9: 429, 430 (Gordan to Klein, October 18 and 23, 1882). 184 [Oslo] A letter from Klein to Lie dated December 2, 1874.

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3.6.1 His Friends Marry and Klein Follows Suit In the summer of 1872, Gaston Darboux, who was seven years Klein’s senior, had sent Klein an announcement of his marriage. Klein congratulated him and wrote: When I was in Paris, we spoke about little else but scientific matters; this made sense, given the nature of our original relationship and the stage of development that Lie and I were in at the time. Since then I have returned my attention, to a greater degree, to general human interests; at the expense of my academic activity, I have become a society man. So far, I cannot report such a joyful event as you have announced to me. I hope, however, to be in a position to do so at some point in the years to come.185

Sophus Lie then got engaged, and Klein responded to this news as follows on February 1, 1873: I never really thought that you would get engaged before me. Do you remember our conversation while we were traveling from Nuremberg to Fürth, and we were both in agreement that it would be a good thing to land in the safe haven of domesticity? That you have now begun to do so amuses me all the more because I always used to tell myself, whenever a similar thought stirred within me: No, you can’t do it, because then you will not be able to work, and Lie would never allow for that.186

Lie’s marriage, of course, did nothing to dampen his friendship with Klein, as is clear from the fact that, in October of 1874, he and his wife traveled on their honeymoon through Cologne, where they not only visited the cathedral but met up with Felix Klein and Adolph Mayer, and then they went together to Klein’s parents in Düsseldorf, where Klein edited Lie’s (first) article on transformation groups.187 Klein’s interest in the personal lives of his other friends is a further indication of his close connections. “It will interest you to know, as it did me, that Riecke is now engaged,” reported Klein to Otto Stolz on May 4, 1874.188 Later, after the still unmarried Stolz had congratulated Klein upon hearing the news of his engagement, Klein thanked him and noted that Ludwig Kiepert and Alexander Brill were likewise still courting potential brides.189 Felix Klein and Anna Hegel (b. May 24, 1851), who was the eldest daughter of Karl Hegel, a professor of history at the University of Erlangen (and a son of the great philosopher), became engaged on Saturday, January 9, 1875. Ten days later, Klein wrote to Max Noether, for whom he had promised to look for an

185 [Paris] 62: Klein to Darboux, August 28, 1872. 186 [Oslo] A letter from Klein to Lie dated February 1, 1873. 187 [Oslo] Letters dated October 5, 1875 and October 10, 1874. Lie’s article – “Ueber Gruppen von Transformationen,” Göttinger Nachrichten (1874), pp. 529–42 – explicitly mentions Klein’s group-theoretical classifications and its “importance for other mathematical disciplines” (p. 540). – See also STUBHAUG 2002, p. 234. 188 [Innsbruck] A letter from Klein to Stolz dated May 4, 1874. 189 Ibid. A letter from Klein to Stolz dated March 22, 1875.

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apartment (Klein found one for him in March): “At the moment I am of less use for other things than visiting Anna, or going for walks with her etc.”190

Figure 18: Anna Hegel and Felix Klein’s engagement announcement – January 9, 1875 [Oslo]

Lindemann recorded an account of how Felix Klein came to meet his future wife: As stipulated, Klein’s supervision of my edition of Clebsch’s lectures meant that I would bring my manuscript to him and he would go over it with me. After some time, however, he suggested that it would be easier if he came to my place, since we often passed by my apart-

190 [UBG] Cod.MS. F.Klein 12: 569, 572 (Klein to M. Noether on January 19, and March 21, 1875).

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ment during our walks. There I would order a coffee from Mrs. Brater,191 which her daughter Agnes192 would bring up to my room with her friend Miss Hegel, and this would be a pleasant interruption and distraction from our dry work. Only later, when I learned that Klein had become engaged to Miss Hegel, did it become clear to me why Klein preferred to look over the manuscript in my apartment.193

Lindemann also reported that, beginning in the fall of 1874, Klein suddenly and gladly devoted his energies to arranging a costume ball and staging a performance of the play Die Jobsiade (1784).194 This satirical work by Carl Arnold Kortum attracted popular attention in 1872 because the artist Wilhelm Busch had based a series of illustrations on it.195 Klein’s colleagues and doctoral students participated in the performance. Professors at the University of Erlangen appeared on the stage wearing long wigs to examine the idle student of theology Hieronymus Jobs. The role of Jobs was played by Klein’s doctoral student Axel Harnack.196 Klein was fairly certain that his marriage would in fact improve his academic productivity: “In the meantime, I am confidently hopeful that my engagement and marriage will only benefit my work. You will not believe how much more I would have accomplished in the past years if I had enjoyed the tranquility and regularity that I expect to have from having a household of my own.”197 The fact that Klein had a ball gown for his bride decorated with mathematical ornamentation can be seen as a symbolic intertwinement of marriage and mathematics. As Aurel Voss remarked: “The decorative arabesques, which these curves form within the systems of parabolic surface curves, would later be used by Klein as ornamentation on a ball gown for his bride Anna Hegel.”198 In the photograph commemorating Felix Klein’s silver wedding anniversary in 1900, his father-in-law Karl Hegel, who was eighty-six years old at the time, can be seen seated in the middle (see Fig. 19). Between November of 1899 and July of 1900 – not long before this photograph was taken – Hegel had completed his memoirs, and here it seems fitting to summarize his life and relate how it influenced Klein’s own affairs.

191 Pauline Brater, widow of the publicist Karl Brater, was a sister of the late mathematician Hans Pfaff. The widow rented out a room for extra money. 192 Agnes Brater (later married name: Sapper) became a successful author of children’s books. 193 [Lindemann] Memoirs, p. 54. 194 Ibid., p. 55. 195 See Wilhelm Busch, Bilder zur Jobsiade (Heidelberg: Fr. Bassermann, 1872). 196 Axel Harnack and Adolf Harnack (a theologian, later professor of church history and 191130 president of the Kaiser-Wilhelm-Society) were twin brothers. Their father was a professor of theology. Regarding Klein’s interactions with Adolf Harnack, see Section 8.3.4.2. 197 [Oslo] A letter from Klein to Lie sent from Erlangen and dated February 22, 1875. 198 VOSS 1919, p. 283. For an image of the design in question, see Figure 10 in Section 2.7.1.

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3.6.2 Klein’s Father-in-Law, the Historian Karl Hegel Karl Hegel described how he was “always driven ahead by his work and willpower.”199 This is comparable to the work ethic of his son-in-law Felix Klein. Born in Nuremberg as the first of two sons to the great philosopher Georg Wilhelm Friedrich Hegel (see also Section 8.3.2) and his wife Maria von Tucher, who was twenty-one years younger than her husband, Karl Hegel had lived in several places as a child on account of his father’s various university positions. He was educated at the Collège Français in Berlin, where he was the top student of his class and where he developed a love of mathematics from his teacher Johann Philipp Gruson.200 After completing his Abitur at the age of seventeen, he at first studied at the University of Berlin, where his father was the Rektor in 1829 and 1830 (G.W.F. Hegel died in 1831). After further studies in Heidelberg, Karl Hegel earned a doctoral degree in Berlin in 1837 and, the following year, passed his teaching examinations (in classical philology, ancient and modern history, philosophy, and German). While on a trip to Italy, Karl Hegel learned to appreciate the art and culture there, and he discovered his main field of historical research. From his initial goal of writing a history of the governing constitution of Florence emerged a large book on urban constitutions throughout Italy (1847), an accomplishment that brought him “the satisfaction of no longer being regarded merely as the son of my father.”201 Before publishing this book, he had already been able to leave behind his work as a school teacher for a university career; in the fall of 1841, he was hired as an associate professor of history by the University of Rostock. In preparation for this new position, he attended a few lectures by Leopold von Ranke, whose ideas would later be espoused by Felix Klein (see Sections 7.4 and 8.3.1). Ranke’s student Georg Waitz was a friend and colleague of Karl Hegel. As a Privatdozent in Göttingen, Klein was in close contact with Waitz and his circle of students (see Section 2.8.2.3). During the German Revolution of 1848, Felix Klein’s father had remained loyal to the monarchy. Karl Hegel had also witnessed this event. For three years, he acted as a political intermediary between aristocrats and democrats; while in Rostock, he stood up for the freedom of the press and for constitutional reform, but his support did not extend to any deeper democratic causes. Hegel founded a largely royalist journal called the Mecklenburgische Zeitschrift, and this, he believed, played a role in his promotion to full professor.202 Karl Hegel served as the Rektor of the University of Rostock from 1854 to 1856, and in the fall of 1856 he accepted a professorship at the University of Erlangen (Bavaria). In addition, the Bavarian Ministry of Education appointed him

199 200 201 202

HEGEL 1900, p. iii. Ibid., pp. 5, 7. Ibid., p. 115. On the significance of Karl Hegel as a historian, see KREIS 2012. HEGEL 1900, pp. 140–62.

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the “commissioner of examinations for Gymnasien in Erlangen, Schweinfurt, and Hof,” and later also in Munich. This was a special feature of the Bavarian educational system that would also be encountered by Felix Klein, who regularly had to travel from Erlangen to Munich in order to administer and evaluate Abitur examinations (or Absolutorialprüfungen, as they were called in Bavaria) in mathematics.203 In Bavaria, university professors – and not school teachers – were responsible for administering these examinations and reporting their results to the Ministry of Education.204 Karl Hegel’s involvement with a large-scale academy project also served as a model for Klein. Hegel was a member of a historical commission that had been initiated by Ranke at the Royal Academy of Sciences in Munich. The aim of this commission was to edit German parliamentary records, city chronicles, and other historical documents, and it received abundant financial support from the Bavarian government.205 Karl Hegel traveled to numerous places in the name of this project, including Strasbourg and Paris (1867). The trips that Klein would later take for the ENCYKLOPÄDIE project, which was funded by a consortium of Germanspeaking academies, are therefore comparable (see Section 7.8). In 1867, Karl Hegel was named a corresponding member of the Philosophical-Historical Class of the Royal Society of Sciences in Göttingen. He was made an external member there in 1871 on the same day that Klein became an Assessor for the society’s Mathematical and Natural-Scientific Class. While in Erlangen, Karl Hegel practiced and taught Ranke’s approach to historiography (see Section 2.8.3.3). In 1872, Karl Hegel founded the university’s first historical seminar. For Klein, he was a reliable source of information not only about historical matters but also about Italian art and culture. Since May 28, 1850, Karl Hegel had been married to his younger cousin Susanne Tucher von Simmelsdorf, who was thirteen years his junior. During their years together in Rostock, his wife gave birth to three daughters and a son.206 When his wife, whom he outlived by twenty-one years, died after a long illness on New Year’s Eve in 1877, two sons, four daughters, and two sons-in-law were standing by his side.207 Two years before his death in 1899, he preliminarily divided his estate equally among his children.208

203 As a full professor in Erlangen, for instance, Klein spent October 5–17, 1874 in Munich for the sake of these examinations ([Oslo] A letter from Klein to Lie dated October 5, 1874). 204 See HEGEL 1900, p. 176. 205 Ibid., pp. 177–79. 206 Ibid., p. 164. 207 Ibid., p. 207. 208 Felix Klein administered this bequest on the authority of his father-in-law. In this capacity, he resolved a disparity that arose from an advanced payment that Anna (Hegel) Klein had already received. Each of Karl Hegel’s children was due to receive 14,420 Marks, but because his wife had already been advanced 10,100 Marks, her remaining inheritance was only 3,820 Marks ([UBG] Cod. MS. F. Klein 9, pp. 666–71).

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3.6.3 Anna Hegel, Felix Klein, and Their Family Felix Klein and Anna Maria Caroline Hegel’s wedding took place on August 17, 1875 in Munich, after Klein had moved there to begin his new professorship and after he had been granted, on June 2, “official approval to marry Anna Hegel.”209 In Erlangen, on July 6, 1872, Anna’s younger sister Louise Friederike Caroline (b. April 3, 1853) had married the physicist Eugen Lommel, who, as dean of the Philosophical Faculty, had played a role in Klein’s hiring process. Another of Anna’s sisters, Maria (b. 1855), remained at home to take care of her father after her mother’s early death. Her brother Georg (b. 1856) forged a career in the Bavarian military, and her brother Wilhelm Sigmund (b. 1863) became a lawyer and civil servant at a patent office. The youngest sister, Sophie Louise (b. 1861), would become an especially important person in Felix and Anna Klein’s life. Sophie Hegel had not been allowed to marry her great love – a military officer of little means – and devoted herself instead to supporting her sisters. She accompanied Anna and Felix Klein to Leipzig, for instance, in order to help care for their children. On the side, she received training as a singer. Beginning in 1890, Sophie Hegel taught German at a private school in Malvern, England. In 1910, she returned to Germany and lived in Anna and Felix Klein’s home in Göttingen. There she took care of her brother-in-law, who was suffering increasingly from paralysis in his lower limbs, and her sister, who had become hard of hearing.210 When Anna and Felix Klein celebrated their silver wedding anniversary in August of 1900, the following people were in attendance: Anna’s father, two of her sisters and one of her brothers, their four children – Otto (b. August 6, 1876), Luise (b. November 24, 1879), Sophie (b. July 11, 1885), and Elisabeth (b. May 21, 1888) – Luise Klein’s fiancé Fritz Süchting, two of Felix Klein’s siblings, and Klein’s old friend Friedrich Neesen and his daughter (see Fig. 19). In his family chronicle from 1918, Alfred Klein made the following notes about Anna and Felix Klein’s four children:211 Otto, an engineer, spent several years in America after completing his studies and married there his wife Mrs. Myrthel Cram. He works as a factory director in Hanover.212 b) Luise married the engineer Fritz Süchting, who is now a professor in Clausthal.213 They have four children: Otto, Hildegard, Carla, and Peter. c) Sophie married the attorney Eberhard Hagemann in Verden. Their children are Elisabeth, Gabriele, Eveline, Rudolf, and Rose-Marie. d) Elisabeth studied mathematics and music. Her husband, Robert Staiger, whom she married in August of 1914, died in the battle on the river Sambre near Charleroi.214 a)

[Archiv TU München] Personnel files of F. Klein II5. [Hillebrand] An obituary for Sophie Hegel written by Dr. Sigmund Hegel. – See Section 9.5. Ibid., Alfred Klein’s family chronicle (1918), pp. 8–9. Otto Klein, who had earned an engineering degree and later received an honorary doctorate in engineering, died in Göttingen on May 12, 1963 (his death certificate is no. 699/1963 at Göttingen’s registry office). 213 In 1910, Fritz Süchting was employed as the director of electricity in Bremen. 209 210 211 212

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Figure 19: A photograph from Anna and Felix Klein’s silver wedding anniversary – Sunday, August 19, 1900 [Hillebrand]. Standing (from the left): Otto Klein (24 years old), Prof. Friedrich Neesen, Sophie Hegel, Dr. Sigmund Hegel, Luise Klein (20 years old), Dr. Fritz Süchting, Maria Hegel, Dr. Alfred Klein, Elisabeth Klein (12 years old). Seated (from the left): Hanni Neesen, Eugenie Klein, Prof. Karl Hegel, Anna Klein, Felix Klein, Sophie Klein (15 years old).

Among Anna and Felix Klein’s four children, their youngest daughter Elisabeth was the most mathematically gifted, and she was also a talented musician. She studied at the University of Göttingen and at Bryn Mawr College in the United States, and she became qualified to teach mathematics, physics, and English in Germany. She passed her examination with distinction on February 14, 1913.215 After she had become engaged to Robert Staiger, who since 1911 had been the director of the Academic Orchestra Association in Göttingen, she earned an additional degree in music from the Royal Conservatory in Leipzig, where Max Reger was teaching at the time. The musicologist Robert Staiger was on his way toward   214 Robert Staiger, a staff sergeant (Vizefeldwebel), fell in battle in Gozée, a village fifteen kilometers southwest of the Belgian city of Charleroi. The battle that took place there between August 21–23, 1914 on the French-Belgian border is notorious for its gruesome war crimes (German troops perpetrating massacres against the civilian population); see THE MARTYRDOM OF BELGIUM 1915. 215 [BBF] Personnel file; and TOBIES 2008a.

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completing his Habilitation when the First World War broke out.216 They married on August 2, 1914. After her husband had been killed in battle only three weeks later, Elisabeth Staiger had a remarkable career, which led to her becoming the headmistress of an upper school for girls in Hildesheim. In addition, she directed a choir and, like many intellectuals after the First World War, was politically active. She became a member of the German Democratic Party, which had been founded with Einstein’s support and whose members also included the mathematicians David Hilbert, Carl Runge, and Felix Bernstein. In 1933, Elisabeth Staiger expressed her opposition to the firing of her Jewish colleagues and to the conservative image of women promoted by the National Socialists. Because she failed to conform to Nazi politics, she was transferred to Hamburg-Harburg and demoted to a teaching position. In 1945, she was appointed again to the role of headmistress, now at the Kaiserin-Auguste-Viktoria-Schule in Celle (near Hanover).217 Anna und Felix Klein’s other children seem to have been less academically inclined.218 Otto Klein failed his Abitur in the spring of 1894, began working as an apprentice (“Eleve”) for the railway, and ultimately studied mechanical engineering at the Technische Hochschule in Hanover. In 1895, Felix Klein had sought advice from Wilhelm Kohlrausch, an electrical engineer and the Rektor of the Technische Hochschule in Hanover, about selecting an appropriate apprenticeship for his son.219 Otto found his own way, however, by moving to the United States. He reported back to home that he was working as an engineer at a factory in Hamilton, Ohio (December 12, 1903), as a chief engineer in Detroit, Michigan (June 29, 1906), and that he spent the Christmas of 1906 with the German mathematician Oskar Bolza and his family in Chicago (Bolza, who had studied with Klein in Göttingen, held a professorship there). In 1908, Otto Klein became engaged to an American woman. Afterwards, he accepted his father’s help in order find a job in the German machine tool industry.220 To this end, Klein used his connections to the industrialist members of the Göttingen Association (see 8.1.1). Felix Klein’s attempts to find his son a professorship at a Technische Hochschule were unsuccessful. Alwin Nachtweh, a professor of mechanical technology at the Technische Hochschule in Hanover, informed Klein on March 31, 1910, that there was an open professorship there in the field of machine tools and industrial organization and that his son was being considered as a possible candidate for the position. Klein responded immediately with his son’s address and rejoiced: “That is indeed a wonderful combination. Since the beginning of 1909, my son 216 Robert Staiger’s doctoral thesis was published as Benedict von Watt: Ein Beitrag zur Kenntnis des bürgerlichen Meistergesangs um die Wende des 16. Jahrhunderts (Leipzig: Breitkopf & Härtel, 1914). 217 For further discussion of Elisabeth Staiger’s career, see TOBIES 1993a and 2008a. On the establishment of additional political parties after the First World War and the intellectuals who joined them, see TOBIES 2012, pp. 113–17. 218 [UBG] Cod. MS. F. Klein 10: 201–392 (Anna Klein’s letters to her husband). 219 Ibid. 10: 528 (a letter from Kohlrausch to Klein dated February 11, 1895). 220 Ibid. 10: 439, 442, 445 (letters from Otto Klein to his father).

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has been working at the Görlitzer Mechanical Engineering Company, and his address is Dipl.ing. Otto Klein, Görlitz, Reicherstr. 30/III. It would be best if he could discuss his curriculum vitae with you himself.” Otto Klein was the top candidate for this position,221 but he turned down the job offer because he did not think that it would suit him to become an academic teacher. Later, he would also turn down an offer to become a professor at the Technische Hochschule in Danzig.222 Instead of pursuing an academic career, he accepted a position in 1913 at the Hanover-Wülfel Iron Casting Company, which produced military equipment during the First World War and also exploited the labor of war prisoners.223 Fritz Süchting, who married Luise Klein in 1900, sought his father-in-law’s advice about professional decisions. Following Klein’s suggestion, he and Robert Fricke prepared a German edition of John Perry’s book The Calculus for Engineers.224 During this process, Klein introduced Süchting to Maxwell’s equations, with which he had been unfamiliar.225 Süchting directed electricity plants; he enabled students to take practical courses in his Bremen-based business; and in 1912 he became a full professor of mechanical and electrical engineering at the Royal Mining Academy in Clausthal. Today, there is a Fritz Süchting Institute for Mechanical Engineering at the Clausthal University of Technology.226 Süchting maintained a very warm relationship with his father-in-law; he addressed Klein as Lieber Papa and shared his enthusiasm for hiking: “I went on an expedition with […] Luise and the three children to Riefensbeek, deep within the Söse Valley, with a backpack full of cakes and with a stop for coffee at the village inn, just as the Klein family used to hike up the Hainberg hills to Rohns’s café.”227 Sophie Klein often got into trouble at school, and later she was wooed by a number of suitors. On August 15, 1903, her mother Anna Klein wrote to her husband about the “miraculous power of love” that she recognized in her daughter Sophie: “Indeed, a mother feels things doubly, first for her child and then from her own experience. Thus it is quite sad when one’s husband is absent and when he has recently expressed that he is no longer able or willing to be affectionate.”228 This is difficult to interpret, but it is probably an expression of Anna Klein’s [UBG] Cod. MS. F. Klein 11: 1–2 (Nachtweh to Klein on June 17, 1910). Ibid. 10: 403 (Otto Klein to his father on April 23, 1914). Ibid.: 472 (a letter from Otto Klein to his father dated April 4, 1917). See PERRY 1902 [1897]. [UA Braunschweig] A letter from F. Klein to Fricke dated July 14, 1901: “How are you getting along with Süchting? He is not yet familiar with Maxwell’s equations, and I have recommended that he should read, as an introduction, Ebert’s recently published theory of electromagnetism.” The work in question is Hermann Ebert, Magnetische Kraftfelder (Leipzig: J.A. Barth, 1897), a second edition of which would appear in 1904. 226 See https://www.imw.tu-clausthal.de/institut/wissenswertes/ (accessed February 2, 2020). 227 [UBG] Cod. MS. F. Klein 11 (a letter from Süchting to Klein dated October, 1913). Rohns’s café and inn on the Hainberg, which had been built in 1830 in the classical style by the architect Christian Rohns, was a favorite hiking destination for many professors and students in Göttingen (see Figure 28 in Section 6.1). 228 [UBG] Cod. MS. F. Klein 10, p. 274 (Anna Klein to her husband on August 15, 1903). 221 222 223 224 225

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frustrations concerning her husband’s excessive workload and his health problems. Sophie’s suitors at the time were not an issue. Klein evaluated all potential marriage candidates precisely according to their financial conditions. In 1908, Sophie Klein ultimately married Eberhard Hagemann, who was five years her senior. From 1908 to 1931, Hagemann worked as an attorney in Verden (a city on the Aller river in the Prussian province of Hanover), and, beginning in 1924, he additionally served as the chairman of Hanover’s provincial parliament. From 1931 to 1933, he was the president (Landeshauptmann) of Hanover’s provincial government. During the Nazi era, Hagemann left his political office and restricted his activity to practicing law. In 1936 and 1937, he represented the Göttingen-based Protestant minister Bruno Benfey, who came from a Jewish family and had been forced out of his parish.229 Benfey was ultimately able to emigrate to the Netherlands and survive. After 1945, Hagemann received numerous honors, among them an honorary doctoral degree from the University of Göttingen, which was awarded on February 15, 1950.230 Anna Klein’s loving relationship with her husband, her trust in his plans, her willingness to help him think through problems, and her management of his correspondence in his absence are documented in her numerous letters to him, which, as of 1907, she composed with a mechanical typewriter: “This manner of writing is so much fun for me.”231 On the occasion of her husband’s sixty-second birthday – and thus we are jumping quite ahead in Felix Klein’s biography – she wrote the following words to him: Dearest husband! It is almost becoming a rule with us that we will have to be apart on our birthdays and our wedding anniversaries. In this respect, I take comfort in the thought that your multifaceted life and its vigorous demands please you and keep you youthful in a way that is uncommon among other men your age. My wish for your birthday tomorrow is thus that you will be able to remain happy with your work and confident in its success for many years to come. Even if you can no longer go climbing as you once did, your strength has never failed you when you have set out to accomplish something that seems valuable to you. During your years of illness and impediment, moreover, I rightly and repeatedly consoled you with the words: “My grace is sufficient for thee, for my strength is made perfect in weakness.” Among the many things that you have undertaken and striven to achieve, the success of the Göttingen Association has always been the most astonishing and a gratifying to me. Therefore I hope even now that the meeting which you are currently attending might result in a fruitful collaboration with the gentlemen from Berlin.232

229 230 231 232

I am indebted to Oswald Glaser (University of Stuttgart) for this information. See https://de.wikipedia.org/wiki/Eberhard_Hagemann (accessed February 2, 2020). [UBG] Cod. MS. F. Klein 10: 319 (Anna Klein to her husband, January 4, 1907). Ibid. 10: 361 (a letter from Anna Klein to her husband dated April 24, 1911). Invited by the Friedrich Krupp Corporation, the Göttingen Association met on April 24–25, 1911 in Essen. The “gentlemen from Berlin” who attended were representatives of Ludwig Loewe & Company, a German manufacturer of arms and munitions. On the Göttingen Association, see Section 8.1.1. Anna Klein’s quotation is of 2 Corinthians 12:9.

4 A PROFESSORSHIP AT THE POLYTECHNIKUM IN MUNICH Over the course of Bauernfeind’s reorganization of the Technische Hochschule in Munich, the institution received the right, like the university there, to educate teaching candidates in the fields of mathematics and physics. It was only because of this that Klein decided to accept the position, but he was clear about the fact the education of mathematicians there could only really be achieved through the creation of an additional full professorship for mathematics.1

This is Ferdinand Lindemann’s account of things, and it is largely accurate. Upon closer inspection, however, it turns out that Klein did not create an additional professorship but rather merely influenced the hiring process in Munich. Munich, the capital of Bavaria, was first mentioned as a city in the year 1158. By the year 1850, it had a hundred thousand residents, and it grew so quickly over the next thirty years that its population more than doubled. During the reign of King Ludwig II (1865–1886), the music and theater scenes expanded there, and numerous large-scale buildings were erected for art, culture, and education. Since 1871, there have been multiple train stations and railway connections there. A horse-drawn streetcar began operating in the city center in 1876. The Ludwig Maximilians University was relocated from Landshut to Munich in 1826, and already at that time it had more than a thousand students. A polytechnical institution, with a rather turbulent history, has existed in Munich since 1827. Beginning with the École Polytechnique in Paris, which was founded in 1794, a number of polytechnical schools were created throughout Europe.2 Differing in quality, most of the German polytechnical schools improved their status in the second half of the 19th century. At the end of the 1870s, these institutions were renamed Technische Hochschulen, and since the 1960s they have been known as Technische Universitäten. Their primary aim was to produce a qualified workforce for businesses – that is, to educate engineers and architects, for which professions there were no courses of study then at German universities. Ever since he spent time in Berlin and Paris, Klein appreciated such institutions and their strong focus on mathematical teaching and research. In 1868, the Royal Bavarian Polytechnical School (renamed as a Technische Hochschule in 1877), had been restructured. Karl Max von Bauernfeind, a professor of geodesy and the man in charge of the reorganization, modeled his institutional changes on the Polytechnikum in Zurich (Switzerland) and the Polytechnikum in Dresden (Saxony), where both engineers and teaching candidates could complete a full course of study (this was not yet possible at the Polytechnika in Prussia). The Polytechnikum in Munich was divided into five departments: a gen1 2

[Lindemann] Memoirs, p. 56. See BELHOSTE et al. 1994, BARBIN et al. 2019, and KLEIN 1979 [1926], pp. 59–84.

© Springer Nature Switzerland AG 2021 R. Tobies, Felix Klein, Vita Mathematica 20, https://doi.org/10.1007/978-3-030-75785-4_4

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eral department, which included mathematics, and departments of engineering, construction, mechanical engineering, and chemical engineering. Mathematics lectures, on the one hand, became service lectures for the engineering departments and, on the other hand, were meant to train future teachers for the Bavarian Realschulen (secondary schools with a focus on mathematics and sciences).3 In 1868, the Polytechnikum hired numerous new professors who would later be important to Klein, including Carl Linde (a professor of theoretical mechanical engineering), Johann Bauschinger (a professor of technical mechanics and graphical statics), and the mathematician Otto Hesse.4 Since the reorganization of the Polytechnikum, its student enrollment had more than tripled; that is, it soon had, like the University of Munich, more than a thousand students.5 Therefore, on May 31, 1873, Otto Hesse applied for a second professorship at the Polytechnikum with the same teaching duties as his own in the field of analytic geometry, differential and integral calculus, and analytical mechanics. Besides Hesse, Johann N. Bischof taught trigonometry, algebraic analysis, and geometry there. Bischof was also the director of the Polytechnikum’s library. In the department of mechanical engineering, there was also a professorship for descriptive geometry and mechanical technology, and this position was held by Friedrich August Klingenfeld.6 The Bavarian Ministry of Culture approved Hesse’s application not long before the latter died on August 4, 1874. Thus there were now two mathematics professorships that the Polytechnikum had to fill: Hesse’s vacant position and the position that Hesse had encouraged the Polytechnikum to create. Until these two appointments were made, elementary lectures were taught by others: the university professor Gustav Bauer took over the lectures on analytic geometry, and the Privatdozent Siegmund Günther lectured on differential calculus.7 A second Privatdozent, Wilhelm Schüler, held lectures on analytical mechanics; he was also paid to lead students through exercises in mathematics and engineering. The list of candidates to replace Otto Hesse contained just one name: Felix Klein, with reference to the “extraordinary reputation that this young scholar has attained from his academic work as well as from his achievements as a teacher.”8 Klein, who envisaged having a broader sphere of influence and activity than was possible in Erlangen, accepted the position in Munich unconditionally and without delay on November 19, 1874.9 King Ludwig II signed the letter of appointment on 3 4 5 6 7 8 9

The author wishes to thank Gert Schubring for this reference; see also SCHUBRING 2019. According to Klein, Hesse had demonstrated “that the problems of newer geometry could be understood as algebraic problems and be solved with algebraic means” (KLEIN 1875, p. 46). See HASHAGEN 2003, p. 41. BERICHT 1875, pp. 19–20. Ibid., pp. 6, 15. When Klein arrived in April of 1875, Günther took a new position as a secondary school teacher (see BERICHT 1876, p. 6). Later, as a teacher at the Ansbach Gymnasium, he recognized the unusual talent of Heinrich Burkhardt (see Section 6.3.2). TOBIES 1992a, pp. 757–58 (a letter to the Bavarian Ministry of Culture dated Nov. 1, 1874). See ibid., pp. 770–71 (a letter from Klein to Wilhelm Beetz dated November 19, 1874).

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December 2, 1874, which meant that Klein, now a member of the Polytechnikum’s faculty (Lehrerrath), could play a part in deciding who should be hired to fill the second vacant mathematics professorship (even though Klein himself did not begin his new position until April 1, 1875). Klein persuaded the majority of the hiring committee to pursue the candidates of his choice: Alexander Brill ranked first, and Jacob Lüroth second – both of them had studied under Clebsch and were already teaching at polytechnical schools. The committee was chaired by the physicist Wilhelm Beetz, who was the newly appointed director of the Polytechnikum, and it also included the mathematics professors from the University of Munich, Gustav Bauer and Ludwig Seidel.10 Other members of this committee – J.N. Bischof, Johann Bauschinger, and the now acting director of the Polytechnikum, Bauernfeind – had voted for another candidate.11 Klein’s opinion was ultimately able to prevail, but this led to him having a strained relationship with Bauernfeind. For this reason, Klein’s access to the field of geodesy, which Bauernfeind oversaw, was somewhat limited. On April 1, 1875, Felix Klein and Alexander Brill started their new jobs as professors of analytic geometry, differential and integral calculus, and analytical mechanics. Together, they were able to develop their own mathematical institute and to reorganize the teaching curriculum, as discussed in Section 4.1. Section 4.2 provides an overview of how Klein developed, as he put it, his “true mathematical individuality” in Munich, and it also profiles his circle of collaborators and students there. Regarding his research interests at the time, Klein mentioned the following: the intuitive geometry of algebraic entities, the theory of quintic equations based on the icosahedron group, problems of number theory, and geometric function theory, especially modular functions.12 While at the Polytechnikum, Klein befriended numerous scholars working in technical areas of research. From them, he gained the insight that engineering is an important field of application for mathematics and that the two subjects should be combined into an overarching curriculum (this is the topic of Section 4.3). Section 4.4 discusses why Klein ultimately wanted to leave his position in Munich and how he managed to do so. 4.1 A NEW INSTITUTE AND NEW TEACHING ACTIVITY From the outside, it appeared as though Felix Klein and Alexander Brill began their new positions in Munich as equals. However, whereas Klein received an annual salary of 2,500 Gulden (as of January 1, 1876: 5,100 Mark), Brill, who was seven years older than Klein, received only 2,000 Gulden (as of January 1,

10 The university professor Seidel also lectured regularly on astronomical methods (including probability theory) at the Polytechnikum. See BERICHT 1875, p. 15; and 1876, p. 17. 11 [BHSt] MK 19555. 12 KLEIN 1923a, p. 20. See also TOBIES 2019b, p. 151.

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1876: 4,200 Mark). In 1875, both moreover received a cost-of-living pay increase of 350 Gulden annually and compensation for their moving expenses. On October 1, 1877, Klein’s salary was raised to 5,460 Mark.13 In addition, both Klein and Brill could rely on earning more money on account of the high number of students attending their courses. For Klein, this extra income ranged from 870 Mark in the winter semester of 1876/77 to 1,500 Mark in 1879/80. According to Klein’s contract, he was required to teach ten to fourteen hours per week.14 Even before Brill was hired, Klein had applied to create a new Mathematical Institute. He also took the initiative to redesign the mathematics curriculum at the Polytechnikum. This reorganization concerned the mathematics lectures and exercises attended by the large number of engineering students. Within the framework of special courses for teaching candidates and future researchers, Klein and Brill also tried out three new formats: a workshop for constructing models, a mathematical colloquium, and a so-called presentation seminar (Vortragsseminar). 4.1.1 Creating a Mathematical Institute Bearing in mind the difficulties with classroom space that he had experienced in Erlangen, Klein submitted, on December 9, 1874, an application concerning “facilities for a new mathematical institute” to the board of directors at the Polytechnikum in Munich. Klein hoped to create an institute “in which students of mathematics (I only have special mathematicians in mind) can experience a broader education in geometry than what is currently available – a broader education from which, I have no doubt, the science itself will considerably profit.” He argued that the construction of models helps students “to conceptualize abstract geometric relations and thus makes it possible for them to conduct immediately insightful research” (this is reminiscent of Note III in his Erlangen Program). Confidently, he summarized in four points what an institute of this sort would require: I. II. III. IV.

It will have to have its own facilities, that is, in addition to an office for its director, a sufficiently bright and spacious work room and a room for exhibiting its collection of models. It will be necessary to hire an assistant who has an extensive background in geometry and is also a skilled craftsman. It will need to have at its disposal a far higher annual budget than that which is currently allotted for the teaching of mathematics (50 Gulden). Finally, for the purposes of establishing the institute, an additional subsidy will be needed; the amount necessary for this I cannot accurately predict at the moment.15

13 [BHSt] MK19556. FINSTERWALDER 1936 (his obituary of Alexander Brill) contains incorrect information concerning Brill’s and Klein’s hiring process and their equal status. 14 [BHSt] MK 19557. On his teaching load at the time, see KLEIN 1923 (GMA III), Appendix, pp. 4–5. After the Bavarian currency was converted from the Gulden to the Mark in 1875, students had to pay 2.50 Mark for one weekly hour of instruction through the semester. 15 For the full text of this application, see TOBIES 1992a, pp. 771–72.

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Already adept at submitting such applications, Felix Klein went on to emphasize that his first point was the most urgent matter and that he would be willing to discuss his other requests during the next round of budget discussions. After securing a one-time grant of 1,500 Gulden for the Mathematical Institute as well as ongoing funding of 50 Gulden per year to cover operating expenses, Klein and Alexander Brill immediately submitted new applications in May of 1875. They requested 300 Gulden from the administration with the purpose of developing the Institute’s collection of models and, together with J.N. Bischof, they applied for a yearly budget of 150 Gulden to pay for seminar prizes.16 The Mathematical Institute was first mentioned in the Polytechnikum’s annual report for the 1875/76 academic year, where its two divisions are identified: Division 1: Felix Klein; Division 2: Alexander Brill. In order to justify these two divisions (and, later, to hire two assistants), Klein and Brill divided the model collection, which was under development, into “pure geometry” and “differential calculus, mechanics, and mathematical physics,” whereby Klein would be held responsible for models of second- and third-order surfaces, complex surfaces, polyhedra, algebraic space curves (etc.), and Brill would be held responsible for minimal surfaces, representations of shortest distances, the curvature and asymptotic curves of a surface, models of deformed rods (etc.).17 In 1876, after the Mathematical Institute had acquired new rooms in an expanded building of the Polytechnikum,18 Klein and Brill commissioned the construction of wooden and plaster models in their modeling workshop. Klein provided numerous instructions for these projects,19 but over time he left it to Brill to manage them. Brill was ultimately in charge of building the collection, and as of 1878 he directed the entire workshop. The number of models at their disposal – which were constructed by students, purchased, or received as donations from Germany and abroad – grew to around three hundred items in time for the annual conference of the Society of German Natural Scientists and Physicians (GDNÄ) in 1877, which took place in Munich.20 Klein and Brill treated this event as a special occasion to promote the models (see Section 4.3.3), for Ludwig Brill (Alexander Brill’s brother) happened to own a distribution company in Darmstadt, which reproduced and sold them with some success; by the end of the nineteenth century, nearly every mathematical institute in Germany (and many in other countries) had copies of them.21 16 [Archiv TU München] X2d. 17 See MÜNCHEN 1877, pp. 19–20. 18 BERICHT 1877, p. 16. The expansion was built in the courtyard behind the Polytechnikum’s main building, which occupies the block on Arcisstraße across from the Alte Pinakothek. 19 See ROWE 2013; and ROWE 2017. 20 [BHStA] MK 19557 (an overview of the models that had been constructed or acquired). 21 In 1899, Ludwig Brill’s firm was taken over by Martin Schilling in Halle (see SCHILLING 1903). See also POLO BLANCO 2007. – Today, there are some online presentations; see for example http://formpig.com. In 1999, Jonathan Chertok, University of Texas at Austin, begann to create digital reproductions of models from the Brill/Schilling collection.

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4.1.2 Reorganizing the Curriculum In accordance with their job descriptions, Klein and Brill taught analytical geometry, differential calculus, and analytical mechanics. Whereas, during the summer of 1875, only eighteen students attended Klein’s two-hour lectures on analytical mechanics, 207 students attended his lectures on analytical geometry, which were combined with two weekly hours of exercises. His course on differential calculus during the winter semester of 1875/76 had just as many students, and he maintained these high enrollment numbers in the following semesters.22 Soon, therefore, Klein sought to devise a more efficient method of educating engineering students in basic mathematics, for what he really wanted was to devote more time to training future teachers and creative researchers in specialized courses. It should be stressed in advance that Klein succeeded in creating a new curriculum and that this novel approach to instruction and exercises would be perpetuated after his departure from Munich and would also serve as a model for other Technische Hochschulen.23 Klein combined the previous mathematics lectures for engineering students into one lecture course: “Introduction to Higher Mathematics,” which ran for four hours per week for four semesters (architecture students were only required to attend for two semesters), and which involved two weekly hours of exercises. The lecture course on analytical mechanics and other subjects that Otto Hesse had preferred to teach, such as determinants and homogeneous coordinates, were dropped from the curriculum. Klein and Brill began this new teaching cycle in the winter semester of 1877/88. At first, several professors of engineering seemed less than enthusiastic about it, for Klein noted: “There has been some passive resistance, which will be overcome but will leave behind some resentment.”24 Brill’s doctoral student Sebastian Finsterwalder reported that Klein’s “fiery spirit, geniality, and initiative” had been needed to achieve this goal.25 During this process, Alexander Brill had not always shared Klein’s level of “engagement, humor, and lust for life.”26 Much later, Brill humorously reported on Klein’s leading role when he wrote that they had composed verses about this at the house of Klein’s bride (Anna Hegel): “So that your groom [Klein] won’t dictate Brill, rein in your groom with a gentle hand, my Fräulein Hegel!” Brill commented that he lived then with the feeling that he had to maintain his independence, and “Well, I didn’t quite succeed.”27 In order to reconfigure the mathematical exercises according to their vision, Klein and Brill had to part ways with the Privatdozent Wilhelm Schüler, who, as mentioned above, had been in charge of this aspect of the curriculum. Klein 22 23 24 25 26 27

[UBG] Cod. MS. F. Klein 7 E, pp. 32–125. See LOREY 1916, p. 152. See JACOBS 1977 (“Vorläufiges aus München”), p. 2. FINSTERWALDER 1936, p. 657. See HASHAGEN 2003, p. 52. [UBG] Cod. MS. F. Klein 8 (Brill to Klein, Sept. 29, 1912). “Da Dein Bräutigam Brill nicht diktieren will: Mit zarter Hand leg’ an die Zügel dem Bräutigam, mein Fräulein Hegel!”.

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wanted to hire young assistants, whose academic work he could supervise and further develop (see Point II in his application from December of 1874). In addition, he wanted to produce copies of exercise sheets by means of lithography, a copying method that he had learned about in Italy and France. In Germany, this process had first been used to reproduce musical scores and in art.28 The Polytechnikum’s General Department already employed two assistants – one for physics and one for descriptive geometry29 – and so it was somewhat of a struggle to persuade the administration to hire two more. For the winter semester of 1876/77, Klein was able to acquire funding of 550 Mark for one assistant, whose salary he supplemented with his own money. About this, he later noted: “My own assistant = Gottlob Fischer. My wishes were fulfilled, but my income was reduced.”30 Fischer had given two talks in Klein and Brill’s presentation seminar in the summer of 1876 (on Eulerian integrals of the second kind, and on the vortex motion in hydrodynamics). As of the winter semester of 1877/78 (and after repeated applications), Klein and Brill finally each managed to acquire an assistant of his own, whereby Fischer became Brill’s assistant and Klein hired the academically talented Josef Gierster. As of April 1, 1878, when Wilhelm Schüler left the Polytechnikum, these two assistants were given an annual salary of 1,000 Mark each. In 1879, when Gierster left to become a teacher, Walther Dyck, who had meanwhile earned his doctoral degree, was made Klein’s new assistant.31 In April of 1875, Klein and Brill had introduced a Mathematisches Kolloquium, in which they themselves and above all Klein’s students from Erlangen (now post-doctoral students in Munich) gave presentations. This ran until December of 1875; it was cancelled during the first half of 1876 (because Klein’s students went their separate ways); and afterwards it was developed into a mathematical seminar, in which Klein’s research interests were the primary focus.32 From May of 1876 until the summer of 1878, Klein and Brill led the aforementioned presentation seminar, which was later called a proseminar. Among its participants, two in particular would go on to make a name for themselves in science: Max Planck and Adolf Hurwitz. Planck, a future Nobel Prize winner in physics, spoke on three occasions (July 1, 8, and 22, 1877) about the “theory of the rotation of bodies according to Poinsot,”33 and Hurwitz spoke on two occasions (July 6 and 13, 1877) about the “frequency of prime numbers according to Chebyshev.”34 Along with Carl Runge and others, Hurwitz attended Klein’s lectures on number theory during the summer semester of 1877.35 In the fall of 1877, Hurwitz, Runge, and Planck all moved to Berlin to continue their studies. 28 29 30 31 32 33 34 35

See WEIß 1989; WEIß 2017. BERICHT 1875, p. 6. Quoted from JACOBS 1977 (“Vorläufiges aus München”), p. 2. HASHAGEN 2003, p. 672; LOREY 1916, p. 152; KLEIN 1923 (GMA III), Appendix, pp. 4–5. [Protocols] vol. 1. Ibid., vol. 1, pp. 311–14, 320 (Max Planck’s lectures). Ibid., vol. 1, pp. 314–17. [UBG] Cod. MS. F. Klein 7 E (Number Theory, 4 hours per week, 10 Mark, 14 students).

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Carl Runge remained in Berlin, where he earned a doctoral degree in mathematics and completed his Habilitation. He accepted a professorship at the TH Hanover, and from there Klein would ultimately lure him to Göttingen (see 8.1.2). Hurwitz returned to Klein more quickly (see 4.2.4.2), and he would follow Klein’s lead in important fields of mathematical research. Klein also remained in touch with Max Planck. Still two months before his own death, Klein sent Planck a remarkable letter about the theory of relativity and Emmy Noether’s contributions to it (see 9.2.2). 4.2 DEVELOPING HIS MATHEMATICAL INDIVIDUALITY I have always regarded those years, during which I made such decisive progress, as the happiest period of my mathematical production. They were characterized by my frequent meetings with [Paul] Gordan outside of Munich. As a meeting place we chose Eichstätt because it lies in between Erlangen and Munich, and there we often spent Sundays together. Even in later years, Gordan spoke fondly of this era of mathesis quercupolitana […].36

There are already several good overviews of the mathematical results that Klein achieved during his years in Munich.37 Here I will focus on the development of these results and Klein’s methods. In his obituary for Otto Hesse, Klein expressed what his own goals were at the time: “Recently, mathematics is again striving to unify various research areas that have long been treated as separate disciplines.”38 Klein continued his work on surface connections and a new type of Riemann surface, which he had begun in Erlangen; he studied Abelian integrals and contributed, via a geometric approach, to the theory of higher-order algebraic equations. Insights from group theory and invariant theory helped him to develop a general method for solving such equations. By drawing upon number theory and the Riemannian existence theorem, Klein succeeded in classifying elliptic modular functions. Enthusiastic about the value of his geometric approach as a heuristic tool, Klein wrote in an article: “Geometry not only makes things intuitive and easier to understand; it also provided the fundamental basis for discovery in this work.”39 Klein’s approach was characterized by geometric intuition, the combination of approaches from different research areas, his search for general connections, and his use of concrete principles. At the beginning of an article, Klein typically described his goals and methods, contextualized them, and made sure to mention the work of others (especially his students). This can be seen, for instance, in an article published in 1876 on Abelian integrals. Here Klein stressed that his goal 36 KLEIN 1922 (GMA II), p. 259 (mathesis quercupolitana = ‘Eichstättian mathematics’). On the collaboration between Klein and Gordan, see also Max NOETHER 1914, pp. 21–30. 37 See KLEIN 1979 [1926], pp. 315–61; TOBIES/ROWE 1990, pp. 46–52; PARSHALL/ROWE 1994, pp. 147–88; GRAY 2000, pp. 81–87; and ROWE 2018a. 38 KLEIN 1875, p. 50. 39 Felix Klein, “Ueber die Auflösung gewisser Gleichungen vom siebenten und achten Grade,” Math. Ann. 15 (1879), p. 251–82, at p. 252.

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was “to make, in the case of the general algebraic curves of the fourth degree, the course of Abelian integrals immediately intuitive on the curves themselves.” He explained that he had already attempted this for the elliptic integral in the case of third-degree curves (in vol. 7 of Math. Ann.) and that his student Axel Harnack had further developed these ideas (in vol. 9 of Math. Ann.). After this, he referred to the geometric basis of his work and outlined its underlying principle: Here as well as in my earlier article, my main resource has been the new type of Riemann surface […]. I rely to a great extent on the principle of deriving, via a limiting process, complex relationships from simpler relationships, thereby making the discussion accessible. When beginning with a pair of ellipses as a special curve of the fourth degree, the material obtains a constellation and a limitation that many might regard as arbitrary. One will also find that in many places my presentation is only sketchy. What seems valuable to me is the general direction of the considerations and the nature of results that they produce; there is room for hope that, later, I will yet again be able to discuss these same ideas more systematically and completely, perhaps by extending them to curves of the nth degree.40

Klein did not wait until he had figured out every last detail but rather published his preliminary results immediately. As I mentioned elsewhere, this occasionally meant that his articles contained errors, but this approach to publication also allowed Klein’s collaborators and students to think along with him, participate in his process, and refine his results. 4.2.1 The Icosahedron Equation The icosahedron equation had already been part of Klein’s research agenda in Erlangen, where he made the following comment: “Perhaps I will shift my attention to the equations that, in the case of representing a complex variable on the spherical surface, are formed by the regular solids” (see Section 3.2.3.2). In the meantime, Klein had worked out his preliminary results on the connection between group theory, binary forms, and regular solids. In the mathematical colloquium held on April 20, 1875, he explained his solution: Such an equation has simply been known as an equation of degree 12 whose symbolic bilinear covariant vanishes in the fourth iteration. The algebraic group of such an equation must consist of 120 substitutions, corresponding to the 60 motions that bring an icosahedron into congruence with itself, and to the perspective turns of its vertices by means of projection from the barycenter. Considering that this group contains subgroups of 20 and 24 substitutions, there must be resolvents41 of the sixth and fifth degree. Examples of the latter can be elegantly constructed as follows […].42

40 Felix Klein, “Ueber den Verlauf der Abel’schen Integrale bei den Curven vierten Grades,” Math. Ann. 10 (1876), pp. 365–97 (Quotations p. 365); KLEIN 1922 [GMA II], p. 99. 41 In the theory of algebraic equations, a resolvent (or Lagrange resolvent) is an auxiliary quantity that is formed from the roots of a polynomial and the primitive roots of unity. 42 [Protocols] vol. 1, p. 154. – SLODOWY (1993, p. viii) described Klein’s result and the icosahedron equation in the following modern terms: “Let G be the icosahedron group, i.e., the  

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By means of the regular dodecahedron, Klein derived the resolvent of degree six, and he referred to it as a special case of the multiplier equation that arises from the fifth-order transformation of elliptic functions.43 He then explained how a quintic equation can be obtained by means of the octahedron; he demonstrated the latter’s connection to the equation of the sixth degree; and he mentioned that he had compared his approach to Brioschi’s method. On May 5, 1875, Klein delivered a report in his research colloquium about quintic equations, beginning with references to the results of Ruffini and Abel and to Jerrard’s transformation to the simplified form x5 – x – k = 0. The overview of scholarly literature that Klein provided, which included results by Hermite, Jacobi, Kronecker, Brioschi, and Jordan, was mostly borrowed from Brioschi’s older studies. Hermite, Klein noted, had shown “that the dependence of the five roots x on k can immediately be represented by means of elliptic functions.” He also outlined the ideas and proofs that Jacobi, Kronecker, and Brioschi had developed concerning the solution to sextic equations, and he emphasized especially Brioschi’s proof that sixth-degree equations always have a resolvent of the fifth degree. On this basis, Klein developed a different special equation of the fifth degree that can be solved with elliptic functions, and he made the following parenthetical remark: “This is the same equation that appeared in the case of the icosahedron. In Brioschi’s work, the coefficients are not entirely correct. Hermite and C. Jordan likewise have incorrect coefficients; this was first corrected by Joubert in 1867.” Klein further reported that Hermite had asked “whether every equation of the fifth degree can be transformed into this form by means of rational substitution.” Finally, Klein referred to the rather different fundamental idea of Kronecker, who used a cyclic function,44 something that would later be important to Klein’s own approach and to his proof of one of Kronecker’s theorems in 1876. Klein later noted that Kronecker, in comparison with the other mathematicians, “[had] penetrated more deeply into the heart of the theory, though without quite reaching the icosahedron […]. The essential thing is to connect it with the icosahedron equation; bringing in elliptic functions is on the same level as bringing in logarithms to extract roots.”45   group of rotational symmetries of a regular icosahedron. This group operates on the sphere circumscribing the icosahedron, which we identify with the Riemann sphere, that is, with the complex projective line P1. The quotient from P1 to G is in turn identified with P1 and the quotient mapping P1 → P1/G is a ramified covering of degree 60, the order of G. The problem of calculating an original point beneath this mapping can be regarded as the problem of solving an equation of degree 60. Klein called such an equation an icosahedron equation.” 43 In Klein’s work, the concept of a multiplier (Multiplikator) developed various meanings. See KLEIN 1923 (GMA III), p. 137. 44 [Protocols] vol. 1, pp. 161–64, at pp. 163–64. See P. Joubert, “Sur l’équation du sixième degré,” Comptes Rendus 64 (1867), pp. 1025–29; and Leopold Kronecker, “Ueber die Gleichungen fünften Grades,” Crelle’s Journal 59 (1861), pp. 306–10. 45 KLEIN 1979 [1926], p. 338 (emphasis original).

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On July 27 and August 3, 1875, Klein had spoken in his colloquium about Sophus Lie’s new geometric understanding of the theory of partial differential equations.46 Klein had further developed his new type of Riemann surface and had devoted his attention to Abelian integrals. Discontent at first with his creativity, he was finally able to rejoice at the beginning of July in 1876: “The muses are back again!”47 From a letter to Lie dated September 25, 1876, we can gain some insight into Klein’s research program at the time: Of course, my focus first of all is on the analogy with algebraic equations (Galois). In this respect, one can distinguish two research directions: 1) A general direction, which asks: If I have knowledge of certain (non-symmetric) functions of the roots, what can I do then? 2) A special direction, which is more number-theoretical. It states: If an equation is given, which functions of the roots are then known? Either directly as rational numbers or as rational functions of given irrationalities. The case is similar with differential equations. You concern yourself with this problem: If I know certain integrals (say, of a functiontheoretical character), what follows then? I, on the contrary, am working on the following problem: If a differential equation is given, when does it admit 1) rational functions as integrals, 2) algebraic functions, 3) integrals of algebraic functions, etc., where the transition from 1) to 2) to 3) is an iterated adjunction. The three-body problem is dealt with in the sense of 1), whereas it would seem quite demanding to approach it in the sense of 2).48

The two research directions described here ran parallel to one another but also, in certain respects, flowed into another. On June 26, 1876, Klein submitted the article “Ueber lineare Differentialgleichungen” [On Linear Differential Equations] to the Erlanger Sitzungsberichte. This study employed a method “for making it possible to determine whether the integrals of a given second-order linear differential equation with rational coefficients are all algebraic.” Here Klein referred to the points of departure in studies by H.A. Schwarz, Lazarus Fuchs, and Camille Jordan. Francesco Brioschi was also active in this area of research; in August of 1876, he sent Klein a letter that contained a similar result. In response, Klein immediately published portions of Brioschi’s letter as “Extrait d’une lettre de M.F. Brioschi à M. F. Klein” together with his own work from the Erlanger Sitzungsberichte in his journal Mathematische Annalen,49 whose editorship he had recently taken over with Adolph Mayer. Darboux also recognized the importance of Klein’s short article and promptly published it in his Bulletin (1877).

46 47 48 49

[Protocols] vol. 1, p. 177. Quoted from JACOBS 1977 (“Vorläufiges aus München”), p. 2. [Oslo] A letter from Klein to Lie dated September 25, 1876 (emphasis original). Math. Ann. 11 (1877), pp. 115–18 (the quotation earlier in the paragraph is from p. 118 of this publication). Brioschi’s letter was published ahead of Klein’s article, on pp. 111–14.

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Klein pursued this topic further and explained to Adolph Mayer, on October 1, 1876, that he “was on the right track toward finding, in ordinary linear differential equations of the third order (and perhaps also of higher orders), those cases with algebraic integrals.”50 Regarding his search for ways to solve higher-order equations, Klein had, as already mentioned, recognized the connection between the theory of quintic equations and the icosahedron,51 and he used this insight to solve an equation of the twelfth degree. Inspired by Gordan, he inverted the problem and derived the theory of quintic equations from the icosahedron.52 Klein was able to prove one of Kronecker’s hitherto unproven statements: “In the case of arbitrarily given y0 … y4, it is impossible to find a rational function φ (y) that depends on an equation in which there is only one parameter (as in the icosahedron equation).”53 Klein demonstrated, moreover, that this was not only true of quintic equations but also of higher-degree equations. Around November 13, 1876, he rejoiced enthusiastically about this finding with the words O quae mutatio rerum (“Oh, how things have changed!”). On November 18, 1876, he noted: “An area of research that I have long wanted to enter is suddenly open to me. (sleeplessness).”54 On November 23, 1876, Klein sent Adolph Mayer the news: “In certain points, I have gone beyond Kronecker.”55 As usual, Klein published his initial results in the Erlanger Sitzungsberichte (three articles). He mailed a letter to Brioschi, who published it in the organ of the Istituto Lombardo Accademia di Scienze e Lettere (classe di scienze matematiche e naturali),56 an academy that had been founded in Milan by Napoleon in 1797. For Mathematische Annalen, Klein refined his argument even further in a study completed on August 20, 1877.57 All of this work would ultimately be incorporated into his book on the icosahedron (see Section 5.5.6). Klein had already recognized the area’s breadth when he wrote, on December 6, 1876, “I have climbed one mountain, but now an entire mountain range lies ahead of me,”58 and when he informed Adolph Mayer: “The icosahedron is certainly a wonderful object; all the possible theories that I would gradually like to learn about converge in it: invariant theory, the theory of equations, differential equations, elliptic functions, minimal surfaces, number theory.” This led him to a logical conclusion: “Next summer I will try to lecture on number theory.”59

50 51 52 53 54 55 56

Quoted from TOBIES/ROWE 1990, pp. 76–77. – See also GRAY 2008. See Math. Ann. 9 (1875), pp. 183–208. See also Max NOETHER 1914, p. 22. Felix Klein, “Weitere Untersuchungen über das Ikosaeder,” Math. Ann. 12 (1877), p. 559. Klein in JACOBS 1977 (“Vorläufiges aus München”), p. 2. Quoted from TOBIES/ROWE 1990, p. 80. Felix Klein, “Sull’equazione dell’Icosaedro nella risoluzione delle equazioni del quinto grado,” Rendiconti del Reale Istituto Lombardo 2/10 (1877), pp. 253–55. 57 F. Klein, “Weitere Untersuchungen über das Ikosaeder,” Math. Ann. 12 (1877), pp. 503–60. 58 Quoted from JACOBS 1977 (“Vorläufiges aus München”), p. 2. 59 TOBIES/ROWE 1990, pp. 82–83, 84 (Klein to A. Mayer, on January 7 and February 25, 1877).

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4.2.2 Number Theory Number theory is a very old mathematical discipline; important theorems on natural numbers, prime numbers, and the relations between numbers can already be found in Books VII to IX of Euclid’s Elements. In the eighteenth to twentieth centuries, branches of the discipline were developed that worked with different methods: analytical, geometric, and algebraic number theory. Klein used resources from all of these areas before these branches were fully defined. In 1877, when Klein was preparing to give his first lectures on number theory, his knowledge of the field was limited. In any case, that is what he communicated to Otto Stolz: “This is how number theory is going for me: Even though I don’t know anything at all about it, I have announced that I will be offering a four-hour lecture course on the topic during the summer semester, and now I’m studying some of the scholarly literature.”60 Klein prepared intensively for his lectures; he traveled to Erlangen to meet with Gordan for eight days, and he decided not to attend Gauss’s centenary celebration in Göttingen, where the international elite would be gathering (Hermite, Brioschi, the Berlin mathematicians).61 Klein relied above all on Henry Smith’s “Report on the Theory of Numbers,” which his Scottish friend William Robertson Smith had brought to his attention as early as 1871. Back then, after first reading this series of articles, Klein remarked to Sophus Lie: “Even the likes of us can somewhat understand it.”62 In addition, Klein made use of Henry Smith’s approach (from 1874) toward translating one of Hermite’s ideas into geometric terms.63 Moreover, Klein studied works by Lie and Adolph Mayer on Jacobi’s multiplier of a linear partial differential equation.64 Number theory proved to be a useful tool for Klein in his additional work in the area of function theory. In this regard, his approach was oriented toward that of Weierstrass, who in 1876 had published a method for factoring analytic functions into “prime factors.”65 Hurwitz sent Klein the latest results by mathematicians from Berlin. About the winter semester of 1877/78, Klein noted: “A very calm winter. My mathematical orientation is influenced by Berlin. I am working independently and at a higher level of consistency.”66

60 [Innsbruck] A letter from Klein to Otto Stolz dated April 10, 1877. 61 See TOBIES/ROWE 1990, p. 85 (a letter from Klein to A. Mayer dated April 5, 1877). 62 [Oslo] Klein to Lie, on January 28, 1871. Smith’s “Report on the Theory of Numbers” was published in six parts in successive annual reports of the British Association for the Advancement of Science (1859–1865). Reprint in J.W.L. Glaisher, ed., The Collected Mathematical Papers of Henry John Stephen Smith, vol. 1 (Oxford: Clarendon Press, 1894), pp. 38–364. 63 See KLEIN 1923 (GMA III), pp. 7–8. 64 See Adolph Mayer, “Ueber den Multiplicator eines Jacobi’schen Systems,” Math. Ann. 14 (1877), pp. 132–43. 65 Karl Weierstrass, “Zur Theorie der eindeutigen analytischen Funktionen,” Math. Abhandlungen der Kgl. Akademie der Wiss. Berlin (1876), pp. 11–16 (Reprint in Weierstrass’s Mathematische Werke, vol. 2, pp. 77–124). See also KLEIN 1979 [1926], pp. 267–70. 66 Quoted from JACOBS 1977 (“Vorläufiges aus München”), p. 3.

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This work ultimately led to Klein’s classification of elliptic modular functions, whereby he created his so-called “level theory” (Stufentheorie), which involved grouping number moduli into the first, second, third to the nth level (see Section 4.2.3). One aspect of this approach led to Hilbert’s twelfth problem (see Section 10.1), which remains unsolved today. Partial solutions have been offered, for instance in Erich Hecke’s dissertation “Höhere Modulfunctionen und ihre Anwendung in der Zahlentheorie” [Higher Modular Functions and Their Application in Number Theory].67 4.2.3 Elliptic Modular Functions Moreover, I am pleased to stress how the following studies combine group theory, number theory, geometry, and function theory into an inseparable whole, all supported by the fundamental ideas of invariant theory (that is, of projective thinking). That which is known and can be found effortlessly in any given one of these fields is used to solve problems in the others. The method employed here, which of course requires previous study of the approaches that are unique to each individual area of research, can actually be regarded as a basic feature of the studies collected in the present volume.68

The quotation above is from Klein’s introduction to the third volume of his collected works, which contains, among other things, his studies devoted to elliptic modular functions. Regarding this latter topic, Jeremy Gray summarized Klein’s contributions as follows: “Klein connected this study with that of the quintic equation, and so with the theory of transformations of elliptic functions and modular equations as considered by Hermite, Brioschi, and Kronecker around 1858. Klein’s approach to the modular equations was first to obtain a better understanding of the moduli, and this led him to the study of the upper half plane under the action of the group of two-by-two matrices with integer entries and determinant one; his great achievement was the production of a unified theory of modular functions.”69 The course of events that led to the development of elliptic modular functions – a trajectory that ultimately also led to the development of automorphic functions – has already been described at length in previous studies.70 Here I would like to stress that Gauß and Riemann had already worked on elliptic modular functions, and their work was subsequently taken up by numerous researchers. Klein’s earliest work on the theory of elliptic modular functions was inspired by his student years in Berlin, when he had worked together with Ludwig Kiepert. Klein built upon previous investigations; a study by Dedekind (published in Crelle’s Journal in 1877) proved to be an especially important component. Klein used the modular 67 68 69 70

Published in Math. Ann. 71 (1912), pp. 1–37. KLEIN 1923 (GMA III), p. 4. GRAY 2000, p. 101. See KLEIN 1979 [1926], pp. 269–72, 339–353; ENCYKLOPÄDIE, vol. II.2, pp. 277–79 (contributions by Harkness, Wirtinger, and Fricke); and GRAY 2000.

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figure described by Dedekind (see Fig. 20) to serve his own group-theoretical approach. From this he derived so-called fundamental polygons for the corresponding Riemann surfaces, and he succeeded in classifying equations that can be solved with elliptic modular functions. He posed the question: “How must s as a function of J be ramified if the equation φ (s, J) is to be solved by means of elliptic modular functions?”71

Figure 20: Klein’s modular figure, derived from Dedekind (KLEIN 1923 [GMA III], p. 23).

Klein had first encountered this figure in a study by H.A. Schwarz on the hypergeometric series, and he worked on the topic in cooperation with his students. At the annual meeting of natural scientists (GDNÄ) in September of 1877, Klein announced that he would give a lecture on elliptic functions (see Section 4.3.3). In March of 1878, he managed to publish his initial findings in the Erlanger Sitzungsberichte, Mathematische Annalen, and the Proceedings of the London Mathematical Society. In London, Klein’s contribution was translated into English by Olaus Henrici (one of Clebsch’s former students); it was presented to the Society on May 9, 1878 and published shortly thereafter in its Proceedings.72 Toward the end of the century, after Klein had studied Gauß’s oeuvre more closely, he was able to recognize approaches in Gauß’s work that anticipated his own findings during the 1870s. Klein acknowledged this in his historical overview

71 J is the absolute invariant of an elliptic integral. See Felix Klein, “Ueber die Transformation der elliptischen Functionen und die Auflösung der Gleichungen fünften Grades,” Math. Ann. 14 (1879), pp. 111–72, at p. 121: “In the Riemann surface that represents s as a function of J, ramification points can only lie at places where J = 0, 1, ∞. For J = 0, three sheets can connect any number of times; for J = 1, two sheets any number of times; for J = ∞, the ramification can be of any sort.” This article is reprinted in KLEIN 1923 (GMA III), pp. 13–75. 72 Felix Klein, “On the Transformation of Elliptic Functions,” Proceedings of the London Mathematical Society 9 (1879), pp. 123–26.

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of nineteenth-century mathematics, where he again explained the so-called modular figure (this time pointing out that it was first discussed by Gauß).73 On the basis of his work on the icosahedron, Klein further pursued his goal of gaining a better understanding (and simplifying the treatment) of quintic and higher-degree equations in light of the theory of elliptic functions. In 1878, Klein gave several lectures on equations of degree seven and on modular equations. He was encouraged by his success in this area to attempt a group-theoretical approach toward elliptic modular functions; that is, he felt as though he might now be able to answer the question, mentioned above, of which equations could be solved with the theory of elliptic modular functions: The function-theoretical method, which I recently used to investigate the modular equations for the lowest degree of transformation for n = 2, 3, 4, 5, 7, 13, will be applied in what follows to define the resolvents of the fifth, seventh, and eleventh degree, which, according to a famous theorem by Galois, can be established for n = 5, 7, 11.74

As already mentioned, Klein supported his argument with a broad range of scholarly literature; to repeat, he cited works by Brioschi, Betti, Hermite, and Ludwig Kiepert, who had likewise devoted efforts to the topic. Klein also used a resource of his own: the n-layered Riemann surface. In his Annalen article “Ueber die Transformation siebenter Ordnung der elliptischen Functionen” [On the OrderSeven Transformation of Elliptic Functions]75 – dated November, 1878 – Klein formed the corresponding Galoisian resolvent of degree 168, and from this he derived the lower equations. He arrived at a closed, 168-layered Riemann surface of genus p = 3 (see also Section 3.1.3.1). By means of a function, he was able to map this surface onto a polygon that was composed of 168 double triangles of the modular figure (see Fig. 21).76 Klein proved that there must be an algebraic function of degree 168 that is the resolvent of the corresponding modular equation. He showed that there is no other grouping of these 2 × 168 triangles. He used his so-called “main figure” to prove additional theorems. Jeremy Gray explained Klein’s “main figure” as follows: Klein wanted to display the figure in as regular a way as possible, but he knew that there is no solid in three-dimensional space whose symmetry group is G168. […] The aspects of the figure which cannot be realized in three-space have this interpretation: one imagines the octahedron as composed of three hyperboloids of one sheet with axes crossing at right angles, and with opposite edges identified at infinity, thus representing a surface of genus 3. The axes of the hyperboloids may be said to pass through the vertices of the octahedron.77

73 See KLEIN 1979 [1926], pp. 43–44. 74 Felix Klein, “Ueber die Erniedrigung der Modulargleichungen,” Math. Ann. 14 (1879), pp. 417–27, at p. 417. 75 The article was published in Math. Ann. 14 (1879), pp. 428–71. For an English translation of this work, see Levy 1999, pp. 287–331. (http://library.msri.org/books/Book35/files/klein.pdf) 76 For a modern interpretation of this as a 14-edge tessellation, see LAMOTKE 2009, pp. 231–36. 77 GRAY 2000, p. 159.

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Figure 21: Klein’s “main figure” (Hauptfigur) with 2 × 168 circular arc triangles (Source: Math. Ann. 14 (1879), p. 470; KLEIN 1923 GMA III, p.126; KLEIN 1979 [1926], p. 351).

Paul Gordan, who had seen Klein’s article before its publication, wrote to him: “Your study is very good. I would like once again to work on pure invariants” (July 31, 1878); “I have not done any work for a long time; you’re probably doing enough work for the two of us” (September 7, 1878); “I’m busy establishing the system of forms of your order-four curves, and I’m content with my results so far. For instance, I now know the conditions under which a given order-four curve will pass into yours” (December 3, 1878).78 Here Gordan was referring to Klein’s order-four curve λ3μ + μ3ν + ν3λ = 0, which defines an algebraic entity with 168 transformations. Klein published this in his aforementioned article on the order-seven transformation of elliptic functi-

78 [UBG] Cod. MS. F. Klein 9: 400, 401A (Gordan to Klein, July 31, Sept. 7, and Dec. 3, 1878).

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ons. There was, however, more to say about the topic. In a subsequent study – “Ueber die Auflösung gewisser Gleichungen von siebenten und achten Grade” [On the Solution of Certain Equations of the Seventh and Eighth Degree], completed toward the end of March in 1879 – Klein wrote: The modular equation, which corresponds to the order-seven transformation of elliptic functions, has a Galois group of 168 substitutions. Will it be possible, by means of feasible processes, to derive such equations of the seventh or eighth or even 168th degree, which have the same group, from the modular equation? And what are the simplest means that one could use to achieve this end?79

In this article, Klein presented a general method for treating higher equations; that is, he explained not only how to manage the problem with 168 substitutions but also “how one can treat similar problems related to any given higher equation and, what is more important, how one should set them up.”80 Klein developed this approach further with his students, and later authors would contribute to it as well. On September 24, 1879, Gordan wrote the following to Klein: Dear Klein! I just received your package containing the work by Gierster; I’ll read it through and bring it with me to Munich. Now that I have some more free time, I hope to work with you again. I have not made any progress, however, on the seventh-degree equations. It is too convoluted to express, by means of simple functions, the complicated functions of 7 variables with 168 permutations; you will have to provide me with a few suggestions regarding how I ought to proceed.81

Writing in 1914, Max Noether eloquently described how Klein and Gordan collaborated further and successfully in this area of research. In 1878, and on the basis of a function-theoretical approach to the Galois resolvent, Klein had constructed an isomorphic group Γ168 of ternary linear transformations for the corresponding group G168 of 168 substitutions. This was an isomorphic linear group with the fewest possible variables, something he had essentially been pursuing for a long time. Regarding the corresponding curve of the fourth degree, Klein developed “the entire system of forms of its covariants, particularly the cluster of curves of degree 42, Ψ3 – J – Δ7 = 0, which from f = 0 excises the groups of 168 points each. […] Furthermore, Klein succeeded in deriving, as Kronecker had already conjectured in 1858, all equations with the group G168 to the […] modular equations.”82 Gordan developed the algebraic side further, and Klein would later, in his collected works, provide a detailed account of Gordan’s contributions.83

79 Felix Klein, “Ueber die Auflösung gewisser Gleichungen von siebenten und achten Grade,” Math. Ann. 15 (1879), pp. 251–82, at p. 251. 80 Ibid. 81 [UBG] Cod. MS. F. Klein 9: 405, p. 9 (a letter from Gordan to Klein dated September 24, 1879). See Paul Gordan, “Ueber das volle Formenstystem der ternären biquadratischen Form,” Math. Ann. 17 (1880), pp. 217–33, an article in which he thanks Klein multiple times. 82 Max NOETHER 1914, pp. 27–28. 83 See KLEIN 1922 (GMA II), pp. 426–38; and KLEIN 1923 (GMA III), p. 135.

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In 1882, on the basis of this order-four curve, Klein formulated (in close dialogue with Poincaré) three “uniformization theorems” (see Section 5.5.4): The principal importance of our curve probably lay, however, in the fact that the main figure, when inscribed within an orthogonal circle, was the first concrete example of the uniformization of higher-genus algebraic curves, and thus, for me, it provided the firmest support when I was formulating my uniformization theorems.84

In the context of Klein’s work from 1878, a dispute arose with Camille Jordan about the problem of determining all possible finite groups of linear substitutions. There can be hardly any debate over whose ideas came first, however, for Jordan wrote to Klein on October 11, 1878: “Vous avez parfaitement raison.”85 Later, the Swedish mathematician Anders Wiman established that the task of solving this problem for the binary case had first been accomplished by Klein’s geometric considerations and then purely algebraically by Gordan. Jordan, in 1878 and 1880, had dealt with the corresponding task for the ternary case. As Wiman noted: “The difficulty of the task of dealing with more than two homogeneous variables is evident from the fact that Jordan, in his first study, had overlooked a group of 168 collineations of the plane, which in the meantime was derived by Klein by considering order-seven transformations of elliptic functions.”86 In a subsequent article, Wiman stressed that Jordan’s results, though important, did not represent the final word, for in 1889 the Dane Herman Valentiner discovered yet another group of order 360. Wiman was able to show that this group “is holohedrally isomorphic to the group of even permutations of six things.”87 Wiman would also write the chapter on finite groups of linear substitutions for the ENCYKLOPÄDIE.88 Klein’s results from the end of the 1870s quickly made an impact, especially in Italy and England. Klein was a great admirer of Brioschi’s work, especially his contributions to quintic equations, the transformation theory of elliptic functions, and the theory of linear differential equations and hyperelliptic functions.89 Already in 1877, Klein had requested a summary of Brioschi’s results, and he translated it into German himself for publication in Mathematische Annalen.90 Conversely, Brioschi did much to promulgate Klein’s results. He submitted them extremely quickly to the Rendiconti, the publication venue of the Reale Istituto

84 Ibid., p. 136. 85 [UBG] Cod. MS. F. Klein 10, pp. 21 (a letter from Jordan to Klein dated October 11, 1878). See also BRECHENMACHER 2011. 86 Anders Wiman, “Ueber eine einfache Gruppe von 360 ebenen Collineationen,” Math. Ann. 47 (1896), pp. 531–47, at p. 531. 87 Anders Wiman, “Endliche Gruppen birationaler Transformationen in der Ebene,” Math. Ann. 48 (1897), p. 199. See also FRICKE/KLEIN 2017 [1912], pp. 476-81 (Appendix 6). 88 Anders Wiman, “Endliche Gruppen linearer Substitutionen,” in ENCYKLOPÄDIE, vol. I.1 (1899), pp. 522–54. 89 See Max Noether, “Francesco Brioschi,” Math. Ann. 50 (1898), pp. 477–91. 90 Francesco Brioschi, “Ueber die Auflösung der Gleichungen vom fünften Grade,” Math. Ann. 13 (1878), pp. 109–160. – See also Section 4.2.1.

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Lombardo di Scienze e Lettere (Milan).91 As early as 1877, it had been Brioschi who arranged for Klein to become a member of the Reale Istituto Lombardo di Scienze e Lettere (see 3.4). Brioschi also sent Klein’s results to his former student Luigi Cremona, who presented them to the Accademia dei Lincei.92 Founded in Rome in 1603, this latter academy, which was the first private institution for promoting the natural sciences in Europe, accepted Klein as a member in 1883. In Great Britain, where Klein had been a member of the London Mathematical Society since 1875 (see Fig. 17), his latest results were published in the Society’s Proceedings. Arthur Cayley translated into English a summary of Klein’s articles from volumes 14 and 15 of Mathematische Annalen.93 The Bavarian Academy of Sciences made Klein an (associate) member on June 25, 1879 (see Appendix 3). Klein used the occasion of the Academy’s meeting on December 6, 1879 to present his results on the theory of elliptic modular functions. Here he described how, over the course of a series of studies, he “was gradually led to a general and essentially new understanding of elliptic modular functions,” and he explained that the various forms of modular equations, whose relationship to one another had hitherto been confusing, “can be classified according to a simple general principle as very special cases.”94 Klein explained this general principle on the basis of three principles of classification: an algebraic one (subgroups), an arithmetic one (congruence groups),95 and a function-theoretical one (here he introduced the concept of a subgroup’s genus). He demonstrated the application of transformation theory and formulated the following theorem: “Thus we ultimately have, for every degree of transformation n, infinitely many equation systems, all of which can be designated modular equations.”96 When editing his collected works, Klein gave this article the title “Zur [Systematik der] Theorie der elliptischen Modulfunktionen” [On (the Systematics of) the Theory of Elliptic Modular Functions], and he described once again his basic method of using group theory as an ordering principle and how he was able to accomplish the self-imposed task of identifying those algebraic equations which can be solved with elliptic functions.97

91 See Felix Klein, “Sulle equazioni modulari,” Rendiconti del Reale Istituto Lombardo 12 (1879), pp. 21–24; and Felix Klein, “Sulla transformazione dell’ 11° ordine dele funzioni ellittiche,” ibid., pp. 629–32 (The second article had been translated by Giuseppe Jung). 92 Felix Klein, “Sulla risolvente di 11° grado del’ equazione modulare di 12° grado,” Atti della Reale Accademia dei Lincei: Transunti 3 (1879), pp. 177–79. 93 Felix Klein, “On the Transformation of Elliptic Functions,” Proceedings of the London Mathematical Society 11 (1879/1880), pp. 151–55. 94 Klein’s article appeared in the Sitzungsberichte der Münchener Akademie (December 6, 1879) and was reprinted in Math. Ann. 17 (1880), pp. 62–70 (quoted here from p. 62). 95 The constants were subjected to congruence requirements in relation to a number module, from which arose the aforementioned division into levels. KLEIN 1923 (GMA III), pp. 3–4. 96 Math. Ann. 17 (1880), p. 68. 97 KLEIN 1923 (GMA III), p. 3 and pp. 169–78.

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4.2.4 Klein’s Circle of Students in Munich When looking at Klein’s circle of students in Munich, we can distinguish two separate phases. The first phase concerns his first two semesters there, when he still had his doctoral students from Erlangen around him as discussion partners and mentees. The second phase began around the fall of 1876, when he gradually managed to attract new students into his sphere. 4.2.4.1 Phase I: 1875–1876 About this first phase, Lindemann reported that Klein at first continued his traditions from Erlangen: “In the summer, Klein, Harnack, Wedekind, and I met every day at the café near the Hofgarten and took a walk to the English Garden.”98 After Klein had married – on August 17, 1875 – and after his honeymoon was behind him, he cut back on his daily rituals with students. Nevertheless, his discussions with Lindemann about the latter’s edition of Clebsch’s lectures and the meetings of the mathematical colloquium remained constant features in Klein’s life. In 1875, the colloquium took place on Tuesdays from April 13th to August th 10 ; from November 18th to December 21st, five additional meetings were held on various weekends. The speakers at the colloquium included Felix Klein (seven times), A. Brill (three times), F. Lindemann (seven times), A. Harnack (four times), E. Holst (four times), L. Wedekind (once), and Wilhelm Frahm (twice). The only person in this group who was new to Klein was Frahm, and Klein was more than willing to foster his talent. Frahm had earned a doctoral degree in 1873 from the University of Tübingen with a dissertation titled “Ueber die Erzeugung der Curven dritter Classe und vierter Ordnung” [On the Generation of Curves of the Third Class and the Fourth Order], and he was able to complete his Habilitation there in the same year.99 His dissertation was dedicated to Sigmund Gundelfinger (a former student of Clebsch and Gordan). Frahm’s work was also related to Klein’s own, and four of his articles had already been published in Mathematische Annalen (in 1874 and 1875). Moreover, Klein arranged for one of Frahm’s studies – “Über die typische Darstellung bilinearer Formen” [On the Typical Representation of Bilinear Forms] – to appear in the Sitzungsberichte of the Societas Physico-medica Erlangensis.100 Tragically, however, Frahm died in the summer of 1875.101

98 99 100 101

[Lindemann] Memoirs, p. 56. I am indebted to Dr. Gerhard Betsch (Tübingen) for providing me with this information. [UB Erlangen] MS 2565 [10], minutes of a meeting of the Societas held on Dec. 12, 1874. Frahm contracted typhus in August of 1875, and he ended his own life by jumping out of a hospital window. See Lindemann’s introduction to CLEBSCH 1891, p. v.

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Lindemann reported how Klein helped him with his edition of Clebsch’s lectures and also with more mundane things. Klein lent him money so that he could take a trip with Axel Harnack in the spring of 1875. Klein encouraged Lindemann to apply for a stipend from the Bavarian government in order to study abroad; he also sent him to London in his place: “1876. Lindemann with a stipend to London instead of me, and from there to Paris.”102 Klein saw Lindemann as an ambassador who could refresh his own contacts with British and French colleagues, something that would also serve the interests of Mathematische Annalen.103 While in England, Lindemann was able to rely on support from Henrici, Cayley, Clifford, and Henry Smith. While in Paris in 1879, Lindemann received a great deal of attention for his edition of Clebsch’s lectures, the first volume of which had just appeared (Leipzig: B.G. Teubner, 1876; more than 1,000 pages). Felix Klein had written the preface to the book, and there he stressed that it would function well as a textbook. At the same time, it was important for Klein to stress the following sentiment as well: “From him [Clebsch] we also learned to take foreign research fully into account and to incorporate it into our own work.”104 As early as 1880 and 1883, this first volume was translated into French and published in two parts as Leçons sur la géométrie (Paris: Gauthier-Villars).105 During his time in Paris, Lindemann received numerous visits and invitations, and not only from Darboux and Jordan. Even the ninety-year-old Chasles climbed the stairs to Lindemann’s apartment to invite him to a formal dinner at his house. Another visitor was Hermite, who “gave me a copy of his work – still largely overlooked at the time – on the transcendence of the number e, about which he commented that he considered it one of his most significant research results.”106 This study by Hermite formed the basis of Lindemann’s 1882 proof of the transcendence of π. Klein would also play a special role in this as Lindemann’s doctoral supervisor. Before accepting Lindemann’s article for publication in Mathematische Annalen, Klein sent it for review to Georg Cantor, who made important improvements to Lindemann’s argument. In the years that followed, better proofs were formulated by Weierstrass, Adolf Hurwitz, David Hilbert, and Paul Gordan.107 As the editor of Mathematische Annalen, Klein was the intermediary for all these new developments. Later, he promulgated these results in presentations, lectures, and teacher training courses (see Sections 7.3 and 7.4.2).

102 103 104 105

Quoted from JACOBS 1977 (“Vorläufiges aus München”), p. 1. See TOBIES/ROWE 1990, pp. 78–79. Klein’s preface to CLEBSCH 1876, p. iv. The second volume of Lindemann’s edition (CLEBSCH 1891) was not translated. On June 21, 1896, Darboux wrote the following to Klein: “Je vous avouerai que j’ai été très peu satisfait de l’exposition que donne M Lindemann dans le tome II des Leçons de Clebsch. Il m’a même paru qu’il y avait dans cette exposition des erreurs graves en des points essentiels p.e. en ce qui concerne la définition de la distance de deux points sur une ligne droite.” Quoted from [UBG] Cod. MS. F. Klein 8: 505. 106 [Lindemann] Memoirs, p. 70. 107 See ROWE 2015; and ROWE 2018a, pp. 141–42.

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4.2.4.2 Phase II: 1876–1880 As a thinker, Klein was at his best when collaborating with others. On October 1, 1876, he wrote to his Leipzig colleague Adolph Mayer: “It is truly unfortunate: when I have to rely on myself alone, as has been the case during this semester break, then I never complete anything substantial at all. […] I am someone who needs scientific interaction, and I have already been longing for the semester to begin for some time.”108 In the meantime, a few talented students had enrolled in the mathematical colloquium and in the presentation seminar, some of whom were more strongly influenced by Brill while others worked more closely with Klein. Neither Brill’s nor Klein’s students, however, could earn a doctoral degree directly under their supervision, because Polytechnika were still not authorized to grant this degree. In Prussia, Technische Hochschulen had to wait until 1899 for this right, while those in Bavaria had to wait until 1901. Klein’s own experiences in Munich would later motivate him to support the right of Technische Hochschulen to bestow doctorates (see Sections 6.4.2 and 8.1.1). Klein’s and Brill’s doctoral students had to complete their degree requirements at a university. The closest was the University of Munich, where, between 1875 and 1879, the professors Gustav Bauer and Ludwig Seidel did not supervise any doctoral candidates of their own.109 Among Klein’s students, Karl Rohn and Walther Dyck completed their doctoral procedures at the University of Munich; an exception in this regard was Franz Meyer. Other students of Klein during his time in Munich either followed him when he took a new position in Leipzig or submitted their dissertations elsewhere. Regarding the exceptional case of Franz Meyer, Klein mentioned him as one of his doctoral students in several lists, but in other such lists Meyer’s name does not appear.110 Meyer attended courses both at the University of Munich and at the Polytechnikum, but his academic interests were shaped above all by Klein. From the winter semester of 1875/76 until he submitted his dissertation, Meyer gave five presentations either in Klein’s colloquium or in his seminar. His primary area of research was fourth-order curves, and on January 28, 1878 he gave a presentation with the title “Anwendung der Topologie auf algebraische Curven” [The Application of Topology to Algebraic Curves].111 This corresponded to the title of his dissertation – “Anwendungen der Topologie auf die Gestalten der algebraischen Kurven” [Applications of Topology to the Shapes of Algebraic Curves] – which he submitted shortly thereafter to the university (his degree was awarded 108 Quoted from TOBIES/ROWE 1990, p. 76 (a letter from Klein to Mayer dated October 1, 1876). 109 For a list of mathematicians who earned a doctoral degree in Munich during these years, see HASHAGEN 2003, pp. 671–72. 110 [UBG] Cod. MS. F. Klein 22 L: 5, p. 1 (Klein’s list of doctoral students dated March 11, 1913, which includes Franz Meyer); and KLEIN 1923 (GMA III), Appendix, pp. 11–13, where Meyer is not mentioned. 111 [Protocols] vol. 1, p. 243.

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on March 15, 1878). Moritz Epple regards this work as an attempt, instigated by Klein, to apply the theory of knot projections to the theory of curves.112 In the Polytechnikum’s annual report from 1879, Franz Meyer’s dissertation is mentioned as a theoretical study that derived from his seminar work there.113 Meyer was obviously aware that he was indebted to Klein when he wrote to him from Berlin on November 17, 1879: Dear Professor Klein, It has been a long struggle for me to overcome my timidity and bring myself again to your attention. I have left several letters unfinished because I have been repeatedly fraught with misgivings. In the meantime, my need to reestablish contact with you – my esteemed teacher, to whom I owe the most for my education in mathematics – has become so insurmountable that I would rather accept all the reproaches in the world than maintain my depressing silence. […] If you could assign me some work that might spare you some of your own effort, I would be thrilled.114

Klein responded warmly and recommended a Habilitation project to Franz Meyer, which he completed in 1880 in Tübingen. Meyer became an associate professor there, and was later appointed a full professor at the Mining Academy in Clausthal (in 1888) and at the University of Königsberg (in 1897). He remained in regular contact with Klein and he developed, during his time in Clausthal, a plan that would lead to the ENCYKLOPÄDIE project (see Section 7.8). Karl Rohn, who began to study with Felix Klein in the winter semester of 1875/76, earned his doctoral degree with distinction under Gustav Bauer and Ludwig Seidel on August 3, 1878. His dissertation made contributions to the topic of Kummer surfaces (see also Section 4.3.3), and he openly admitted that his actual doctoral supervisor had been Klein. Rohn completed his Habilitation in Leipzig in 1879, and then was able to establish himself as a professor in Saxony, at the University of Leipzig and, for some years, at the Polytechnikum in Dresden. Klein’s next doctoral student, Walther Dyck, also submitted his dissertation – “Über regulär verzweigte Riemannsche Flächen und die durch sie definierten Irrationalitäten” [On Regularly Ramified Riemann Surfaces and the Irrationalities Defined by Them] – to the University of Munich. Although he was ultimately awarded a doctoral degree on July 30, 1879, he faced some resistance along the way. Ulf Hashagen has pointed out that Seidel was “uncomfortable with Klein’s youth and pushiness at the time” and that he had little appreciation for Klein’s orientation toward Riemann’s geometric methods. Those same features of Klein’s were responsible for the fact that none of his students would be able to complete a Habilitation at the University of Munich.115 Seidel was also critical of what he perceived to be Klein’s sloppy use of mathematical terms. About this, Klein would later write to Adolph Mayer that “Seidel, for his part, objects to my work

112 113 114 115

EPPLE 1999, p. 176. BERICHT 1879, p. 8. [UBG] Cod. MS. F. Klein 10: 1151 (a letter from Meyer to Klein dated November 17, 1879). See HASHAGEN 2003, pp. 82–86.

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because I once wrote ‘complex plane’, for instance, instead of ‘plane of the complex variable’.”116 All this aside, Klein would nevertheless be invited, in 1892, to become Ludwig Seidel’s successor at the University of Munich (see 6.5.2). In 1878 and 1879, Alexander Brill likewise sent two doctoral candidates to the University of Munich. By that time, however, Brill was no longer making any contributions of his own to the mathematical colloquium. If Paul Gordan is to be believed, Brill’s knowledge of contemporary mathematics had begun to fall behind. Regarding an article that Wilhelm Heß had submitted to Mathematische Annalen, Gordan, unable to contain his sarcasm, wrote the following to Klein: The studies [that you sent me] are not bad, but there is one genius among them: the name of this honest lad is Heß, and he has submitted a piece on the rational curves of the fourth order. This work was completed under Brill’s supervision, and it is clear that Heß has understood my invariant theory better than Brill ever has. I have also learned a few things from the work, and I am in favor of accepting it as it is for publication in the Annalen. Brill wants to edit it first, which will probably just make it worse, for the errors that stem from the young author alone will do nothing to tarnish the reputation of the journal.117

From 1877 to 1880, Klein’s special lectures were devoted to number theory, elliptic functions, and algebraic equations. At the same time, he also gave presentations on his own new research results in the colloquium and he supervised additional talented students. Among the latter, he stressed four students in particular: I had the good fortune of finding, among those attending my courses, a few outstanding collaborators, who not only supported my own research in essential ways but also – each in his own way – took it a step further. In this respect, I should mention, in chronological order of their significant publications, Gierster, Dyck, Bianchi, and Hurwitz.118

Here, regarding Walther Dyck, Klein mentioned only that he produced “suitably drawn figures” for him and that he ultimately took an interest in work on group theory. The other three students supported Klein’s research at the time more strongly, but they did not complete their doctoral degrees in Munich. Josef Gierster studied in Munich from 1873 to 1877; he participated in Klein’s presentation seminar in 1876/77, in which he gave two talks (on Fourier series and the gamma function); he passed his teaching examinations and he worked as Klein’s assistant, in which capacity he helped Klein make the numerical calculations that were necessary for formulating modular equations.119 Gierster followed Klein’s turn toward number theory and he made important contributions to the theory of class number relations, which formed a building block of Klein’s level theory. Enthusiastic about Gierster’s first results in this field, Klein submitted Gierster’s article to the Göttinger Nachrichten in 1879 and wrote to Darboux about it: 116 117 118 119

Quoted from TOBIES/ROWE 1990, p. 124 (a letter from Klein to Mayer dated Jan. 4, 1881). [UBG] Cod. MS. F. Klein 9: 415, p. 19 (a letter from Gordan to Klein dated Oct. 23, 1881). Klein 1923 (GMA III), p. 5. See Josef Gierster, “Notiz über Modulargleichungen bei zusammengesetztem Transformationsgrad,” Math. Ann. 14 (1878), pp. 537–44.

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4 A Professorship at the Polytechnikum in Munich One subject that is now close to my heart is the relations between the class numbers of quadratic forms of negative determinants, as first established by Kronecker in 1857. (Last summer, Mr. Gierster, who was my assistant then but is now in Bamberg, presumably sent you a copy of his article on this topic from the Göttinger Nachrichten; in the meantime, and in connection to my speculations about the various forms of modular equations, he has pursued this subject further and obtained a great many new results.)120

Basing his work on the findings of Klein, Kronecker, and Henry Smith, Gierster wrote: “In entirely the same way, one can derive, from the modular equations of regular solids introduced by F. Klein, analogous relations, which, in regard to their simple arithmetic construction, exist on the same level as Kronecker’s formulae. Here, in particular, I will convey findings that result from the icosahedral modular equations.”121 Gierster worked as a teacher in Bamberg, and he earned his doctoral degree on February 10, 1881 in Leipzig with a dissertation titled “Die Untergruppen der Galois’schen Gruppe der Modulargleichungen für den Fall eines primzahligen Transformationsgrades” [The Subgroups of the Galois Group of Modular Equations in the Case of a Prime Degree of Transformation].122 Gierster suffered from poor health and died young. Klein arranged for Robert Fricke to write his obituary.123 In the letter quoted above to Darboux, Klein added prophetically: These relations, as you know, are a preliminary step toward the theory of equations for the singular moduli of complex multiplication. I have no doubt that my young collaborators and I will succeed in this, but I will also have to encourage the latter not only to study Kronecker’s published results but also to go far beyond them. First, however, the class number relations will have to be worked out more thoroughly.124

Adolf Hurwitz carried on Gierster’s number-theoretical approach “with resounding success.” Though his background was in algebraic geometry, Hurwitz increasingly concentrated on function theory and number theory.125 As mentioned above, he attended Klein’s lectures on number theory in the summer semester of 1877, and he had already collaborated with Klein even before he made his first contribution to Klein’s presentation seminar (see Section 4.1.2).126 Klein made Hurwitz a part of his family; he worried about Hurwitz’s health, and he would have preferred for Hurwitz to remain in Munich a little longer before continuing his studies in Berlin.127 While in Berlin, a bout of typhus caused

120 [Paris] 69: Klein to Darboux, December 26, 1879. 121 Josef Gierster, “Neue Relationen zwischen den Klassenzahlen der quadratischen Formen von negative Determinante,” Göttinger Nachrichten (June 4, 1879), pp. 277–81; at pp. 277–78. 122 In his dissertation, Gierster thanked Klein for his encouragement and support; see Math. Ann. 18 (1881), pp. 319–65, at p. 321. 123 See Robert Fricke, “Josef Gierster,” Jahresbericht DMV 2 (1893), pp. 44–45. 124 [Paris] 69: Klein to Darboux, December 26, 1879. 125 See KLEIN 1923 (GMA III), pp. 5–6. 126 [UBG] Math. Arch. 77: 7 (a letter from Klein to Hurwitz dated June 27, 1877); and [Protocols] vol. 1, pp. 314–17. 127 [UBG] Math. Arch. 77: 8 (a letter from Klein to Hurwitz dated October 3, 1877).

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him to miss the winter semester of 1877/78. After his recovery, he attended Weierstrass’s lectures on analytic functions and developed a new geometric idea analogous to Chasles’s principle of correspondence, “which is a geometric translation of the theorem that an equation of the nth degree has n roots.”128 Hurwitz worked on this study further in discussion with Klein, and it was accepted for publication in Mathematische Annalen on December 18, 1878.129

Figure 22: Adolf Hurwitz ([UBG] Cod. MS. F. Klein).

Klein advised Hurwitz to keep an open mind: “Besides the pleasure of producing new results, don’t forget to adopt other people’s points of view!”130 In a later letter, Klein remarked: “In the coming winter, you will presumably attend Kronecker’s great lectures on equations; I would be grateful to you if, later on, you could send me a copy of your notes. […] I should add that I would also be happy to receive copies of Weierstrass’s recent lectures, especially those on function theory.”131 Klein wanted to study Kronecker’s lectures mainly in order to see how far Kronecker had advanced with his ideas about quintic equations: “I would like to give his work its due respect, but no more than that.”132 In his lectures on equations, however, Kronecker said nothing about quintic equations; supposedly, he addressed the topic instead in a lecture course on number theory.133 Not until

128 [UBG] Cod. MS. F. Klein 9: 872/1, fol. 1v (Hurwitz to Klein, September 24, 1878). 129 Adolf Hurwitz, “Ueber unendlich-vieldeutige geometrische Aufgaben, insbesondere ueber die Schliessungsprobleme,” Math. Ann. 15 (1879), pp. 8–15. See also HILBERT 1921. 130 [UBG] Math. Arch. 77: 11, p. 17 (a letter from Klein to Hurwitz dated October 6, 1878). 131 Ibid., 13, p. 20 (a letter from Klein to Hurwitz dated October 22, 1878). 132 Ibid., 18, p. 30 (a letter from Klein to Hurwitz dated January 19, 1879). 133 [UBG] Cod. MS. F. Klein 9: 884, fol. 36 (Hurwitz’s letter to Klein dated March 24, 1879).

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March of 1881 did Klein learn that Kronecker had never found a proof for his theorem (see Section 5.5.2.1). During the winter semester of 1878/79, Hurwitz attended Weierstrass’s lectures on elliptic functions,134 and after that term he returned to Klein. He expanded his knowledge under Klein’s tutelage. Klein bombarded him with tasks and mathematical questions, met with him even on Sundays, and openly shared his opinions and advice: “One of the main flaws of mathematicians today is that they do not keep up with the scholarly literature,” or, “Don’t be afraid of Camille Jordan; rather, read at least enough of his book so that you know what he has set out to accomplish in it.”135 Even before the beginning of the winter semester of 1879/80, Hurwitz had studied the work of Klein, Gordan, and Clebsch (in Lindemann’s edition). In Klein’s colloquium, Hurwitz spoke about Hamilton’s quaternion calculus and spherical functions,136 and he formulated proofs for some of Klein’s theorems.137 Klein was sure to acknowledge the contributions of his students; for instance, in his aforementioned study “Zur [Systematik der] Theorie der elliptischen Modulfunktionen” [On (the Systematics of) the Theory of Elliptic Modular Functions], which he first presented at the Munich Academy of Sciences on December 6, 1879, Klein included the following footnotes: “Mr. Hurwitz, who supported my work on these investigations, was led to formulate such elegant equations for the 23rd and 47th degree of transformation […],” and “At first I had operated only with x0 : x1 : x2 : x3; the result as it appears in the text is due to Mr. Hurwitz.”138 It is of some interest how Klein imagined guiding Adolf Hurwitz to the completion of his doctoral degree. Klein had in mind that Hurwitz should submit his dissertation in Leipzig, and he offered, while still in Munich, the following proposal to him: With your permission, I would like to provide you with somewhat more personal guidance than I have offered you in the past – something along the lines of how I worked with Bianchi this summer. So far, I have only had you work on general sets of questions; I have identified the fields in which you ought to immerse yourself; and, just in our conversations, I have helped you to circumvent certain mathematical difficulties that you have encountered. I left it to you to decide how to formulate and approach your research topics. Now I would like to do things differently. For some time, I would like to present you with one special question after another, each of which I will ask you to work out to their conclusion. Of course, these questions will be related to your research topic and, where possible, will grow naturally from our conversations and correspondence. By working in such a way, you will renounce a degree of your independence in order later to be able to deal with the material all the more independently. With this same end in mind, I will perhaps ask you in Leipzig to attend as few other

134 Ibid. 872/2, fol. 3v (Hurwitz’s letter to Klein dated September 24, 1878). Hurwitz signed this letter: “Your admiring former and hopefully future student, Adolf Hurwitz” (fol. 4). 135 [UBG] Math. Arch. 77: 16 and 27/1 (Klein’s letters to Hurwitz dated November 17, 1878 and October 2, 1879). 136 [Protocols] vol. 1, pp. 89–108. 137 See, for example, [UBG] Cod. MS. F. Klein 9: 887 (Hurwitz to Klein dated Sept. 27, 1879). 138 Quoted from the reprint of Klein’s article in Math. Ann. 17 (1880), pp. 62–70, at pp. 69–70.

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courses as possible beside my lecture on function theory (for which I am counting on your support, much as I did last winter for my lectures on analytical mechanics). If I were now to present you with my first question, it would involve working through the theory of new multiplier equations, as I formulated them in Annalen XV, p. 86; Kiepert's recent work also deals with these. Why do these equations exist for prime numbers > 3? How are things in the case of 2 and 3? And in the case of composite numbers? And how, in particular, in case of powers of 2 or 3? Wherever possible, do everything without σ-functions but merely on the basis of your function-theoretical intuitions about modular functions. I hardly need to add how much I myself am interested in this set of questions. For I already told you once that I still don’t have any acceptable proofs for the various statements that I made in the note in question. I was completely convinced of the correctness of these things; otherwise I would not have written the note. However, at least during the last Easter break, when for a long time I thought of taking up these multiplier questions myself, I could not readily or explicitly formulate the relevant proofs.139

Hurwitz gratefully adopted Klein’s program, and their regular interactions culminated in Hurwitz’s dissertation, “Grundlagen einer independenten Theorie der elliptischen Modulfunctionen und Theorie der Multiplicator-Gleichungen erster Stufe” [Foundations of an Independent Theory of Elliptic Modular Functions and a Theory of First-Level Multiplier Equations], which was submitted on March 23, 1881. On December 6, 1880, Hurwitz developed important results in his very first seminar presentation in Leipzig – “Über die Bildung der Modul-Functionen” [On the Formation of Modular Functions], in which he drew upon the work of Gottlob Eisenstein, Cayley, and Weierstrass. Hilbert later emphasized how Hurwitz had adopted and refined Klein’s ideas: Inspired by Klein, Hurwitz used Eisenstein’s approaches to create a theory of elliptic modular functions that was independent of the theory of elliptic functions. [...] One main section of his dissertation concerns so-called multiplier equations, which Hurwitz, drawing upon works by Klein and Kiepert, investigates with characteristic thoroughness and diligence.140

After Klein had visited Enrico Betti, Ulisse Dini and others in Pisa in April of 1878 (see Section 3.4), Gregorio Ricci-Curbastro came from Italy in the fall to study under Klein in Munich (he was Klein’s first Italian student). Ricci attended Klein’s lectures on number theory (in the winter semester of 1878/79) and algebraic equations (in the summer semester of 1879), and he also participated in Klein’s seminar. There he gave a lecture on “Certaines équations du degré 2n,” in which he also addressed the problem of solving quintic equations (this lecture was held on March 3, 1879). In a second presentation, he discussed Henry Stephen Smith’s work on modular functions, which had been published in French in an Italian journal: “Les courbes modulaires: Rapport sur un mémoire de M. Stephen Smith communiqué à la Académie Royale des Lyncées, 1877” (publication date: July 28, 1879).141 During his time in Munich, Ricci did not produce any

139 [UBG] Math. Arch. 77: 38, fols. 60–60v (Klein’s letter to Hurwitz dated August 21, 1880). 140 HILBERT 1921, pp. 161–62. 141 [Protocols] vol. 1.2, pp. 44–51, 75–78. Smith’s article “Sur les équations modulairs” had already been submitted to the Academy in Paris in 1874. See KLEIN 1923 (GMA III), pp. 7–9.

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publishable work. However, when Ricci later developed tensor analysis with his student Tullio Levi-Civita, Klein requested his results immediately for publication in Mathematische Annalen. Klein later explained that Ricci’s work served as the basis for the general theory of relativity.142 Luigi Bianchi, whose first academic term in Munich was the summer semester of 1879, possessed, unlike Ricci-Curbastro, an excellent command of German. He spoke in seven sessions of the colloquium, and he worked intensively with Klein, for the number of participants in the colloquium was low at the time. During the winter semester of 1879/80, five students were enrolled; in the summer of 1880, only Bianchi and Dyck participated. They met on Saturdays or Sundays, and during the summer they visited Klein at his country home, where he had retreated for health reasons. Bianchi’s first presentations in the colloquium were related to his dissertation, which concerned surface curvature and a theorem by Julius Weingarten (the presentations took place on January 17 and 24, 1880). Klein suggested that he should write an article on this topic for Mathematische Annalen.143 In subsequent meetings, Bianchi made contributions to Klein’s level theory and spoke about tetrahedron irrationality (on May 29 and June 5, 1880) and about icosahedron irrationality (on June 13, July 4, and July 11, 1880).144 In his work “Über unendlich viele Normalformen des elliptischen Integrals erster Gattung” [On Infinitely Many Normal Forms of the Elliptic Integral of the First Kind], which was first presented on July 3, 1880 at the Academy of Sciences in Munich and was an effort to expand his level theory with the help of doubly periodic functions, Klein acknowledged Bianchi’s results as follows: “At my request, Mr. Bianchi recently investigated the fifth level, and what I will communicate below are essentially results discovered by him.”145 Bianchi expanded the proofs from this earlier work in an additional Annalen article (published in August of 1880).146 In Klein’s article that appeared in the Proceedings of London Mathematical Society, which I mentioned above, there is also a reference to Bianchi’s findings (on p. 152). About the relationship between Bianchi’s work and his own, Klein later wrote: “By treating, in the summer of 1880 (Math. Ann. 17), the elliptic curves that I would later call elliptic normal curves of the 3rd and 5th order by means of the σ function, he overcame my reservations about using this resource and that of theta series. He [Bianchi] thus did his best to build a bridge from my work to the developments of Weierstrass’s school, particularly to my friend Kiepert’s studies, which were written around the same time.”147 As early as July 29, 1879, Julius 142 143 144 145

See KLEIN 1927, pp. 189–95, 205. See also Section 9.2.2 below. The article appeared in Math. Ann. 16 (1880), pp. 577–82. [Protocols] vol. 1.2, pp. 94–97; and vol. 2, pp. 6–27. KLEIN 1923 (GMA III), pp. 179–85, at p. 183. On Klein’s “principle of dividing levels [Stufenteilung],” see also FRICKE 1913a, pp. 277–79. 146 Luigi Bianchi, “Ueber die Normalformen dritter und fünfter Stufe des elliptischen Integrals erster Gattung,” Math. Ann. 17 (1880), pp. 234–62. 147 KLEIN 1923 (GMA III), p. 6.

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Amelung had discussed Ludwig Kiepert’s work in Klein’s colloquium.148 Klein first needed to collaborate with Bianchi, however, before he was able to integrate Kiepert’s approaches into his own methodological arsenal. Conversely, Bianchi wrote to Klein: “I will never forget what you have done for me. And if I can be useful to you, remember that I am entirely at your disposal.”149 Many years later, in February of 1924, Klein successfully arranged for Bianchi to be made a corresponding member of the Göttingen Academy of Sciences.150 Klein was aware of how important collaboration was for his research. On Saturday, February 14, 1880, he had written in his protocol book: “The undersigned spoke about the current state of his research on elliptic modular functions, with the particular aim of underscoring the significance that the studies by individual colloquium members possess for these investigations. – F. Klein.”151 4.3 DISCUSSION GROUPS IN MUNICH Klein worked together with his students, of course, but as before he was also involved in other circles. These included a mathematical coterie with engineers and natural scientists, a mathematical student union, and a mathematical society (with mathematicians from the University of Munich). In addition, he also played a considerable role in organizing the annual meeting of the Society of German Natural Scientists and Physicians (GDNÄ), which was held in Munich in 1877. 4.3.1 A Mathematical Discussion Group with Engineers and Natural Scientists Klein’s early interests in engineering and technology were reawakened in Munich. In 1870, the aforementioned Johann Bauschinger had founded the first mechanical-technical laboratory at a Polytechnikum, and it soon became a hotbed of government-supported institutions for materials research (Materialprüfanstalten).152 An even more important person in Klein’s life was Carl Linde. In his lectures on theoretical mechanics, Linde taught the mathematical foundations of this research area; he made use of graphical methods and he created the second governmentfunded mechanical-technical laboratory (a research institute for thermodynamics) at the Polytechnikum. About his time in Munich, Klein wrote:

148 149 150 151 152

[Protocols] vol. 1.2, pp. 78–79. [UBG] Cod. MS. F. Klein 8: 91( Bianchi’s letter to Klein dated August 14, 1880). [AdW Göttingen] Pers. 20 (1105). [Protocols] vol. 1.2, p. 108. Johann Bauschinger’s son Julius studied under Klein in 1879 and 1880. Later, from 1920 to 1922, Julius Bauschinger (then a professor of astronomy) would collaborate with Klein within the Emergency Association of German Science; see Section 9.4.1 below.

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4 A Professorship at the Polytechnikum in Munich Through my interaction with outstanding engineers such as Linde, my technical interests were supported on many fronts in the mathematical discussion group that we had created. It was through this group that I became more closely acquainted with the geometric disciplines of mechanical engineering, such as descriptive geometry, graphical statics, and kinematics.153

Klein would later integrate descriptive geometry, graphical statics, and kinematics into his university teaching. In his memoirs, the refrigeration engineer Carl Linde mentioned both this mathematical discussion group and his close relationship with Felix Klein: Whereas my activity in professional associations kept me in ongoing general contact with Munich’s technical circles, a tighter circle of colleagues, which formed during my last years teaching there, proved to be especially valuable for exchanging ideas. After the death of the mathematician [Otto] Hesse, his position was replaced in 1875 by the two young professors Klein and Brill. Together with them and my colleagues von Bezold and Loewe, we created a “mathematical discussion group,” which still exists today in an expanded form and whose rules stipulated that we should meet every two weeks in the office of one of its members, who would then be responsible for lecturing about his current research. It was during these gatherings that my long friendship with Felix Klein began, and it was because of my early impressions of him that I later contributed to the establishment of the “Göttingen Association.”154

Figure 23: Carl Linde, 1872 ([Deutsches Museum] Portrait collection, No. 41926).

153 KLEIN 1923a (autobiography), p. 20. 154 LINDE 1984, p. 34. On Linde’s involvement with the Göttingen institutions, see Section 7.7.

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In 1866, after studying mechanical engineering at ETH Zurich (where he never completed his degree) and after working as a draftsman in Berlin, Carl Linde joined the technical office of the Munich-based locomotive firm Krauß & Co. In 1868, he became an associate professor of theoretical mechanics at the Polytechnikum in Munich. He designed his first refrigeration machine in 1871, and he became a full professor the following year. His refrigerators soon became a hot commodity (so to speak), especially in breweries. Klein noted that, in December of 1875, he met with Linde at the Spaten brewery (Spatenbräu), which was then the largest brewery in Munich (its history dates back to the fourteenth century).155 In 1879, with the backing of two brewing companies, Linde founded the “Society for Linde’s Ice Machines,” a joint-stock company that had immediate success on the market. He gave up his university position, but later he would resume his teaching activities (from 1892 to 1910). Klein recognized that Linde’s research institute was equally engaged in theory and praxis, and later he would strive to establish similar large-scale laboratories in Göttingen. In Klein’s judgement: An excellent example of the capabilities of technical physics from both the practical and theoretical sides is the outstanding air liquefaction apparatus that Linde introduced in 1895. Until that point, the variance that exists between the actual thermodynamic behavior of atmospheric air and the ideal schema of Mariotte's and Gay-Lussac’s law was regarded as incidental; here it is used, with the greatest practical success, as the very principle of the device’s design.156

Klein and Linde both believed in the close relationship between mathematics, physics, and engineering. The initial meetings between the two of them gave rise, at the beginning of 1876, to their mathematical discussion group. In addition to Klein, Linde, and Alexander Brill, the other participants were the physicist and meteorologist Wilhelm von Bezold and the engineer Ferdinand Loewe. Von Bezold had attended Riemann’s courses and, later, he donated his notes from one of Riemann’s lectures (from 1858/59) to the University Library in Göttingen. The lecture in question concerned the problem of the (x + iy) sphere, which had been important to the principle of transference (Übertragungsprinzip) that Klein employed in his work on the icosahedron.157 Von Bezold’s Habilitation thesis – Ueber die physikalische Bedeutung der Potential-Funktion in der Elektricitätslehre [On the Physical Significance of the Potential Function in the Theory of Electricity], which was published in 1861 – likewise fell within Klein’s spectrum of interests. Loewe’s research was devoted to railway and road construction. This work brought him into contact with Georg Krauß, the founder of the aforementioned locomotive company Krauß & Co., whose first locomotive had earned a gold medal at the World’s Fair in Paris in 1867. Krauß was an investor in Carl Linde’s refrigerators. It is thus no surprise that Munich-based businesses would later be

155 See JACOBS 1977 (“Vorläufiges aus München”), p. 1. 156 KLEIN 1904a, pp. 150–51. 157 See KLEIN 1922 (GMA II), p. 256.

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quick to support Klein’s efforts to form a closer bond between science and engineering (see Section 7.7). On December 2, 1898, Klein was awarded Bavaria’s highest honor, the Maximilian Order for Science and Art; in 1905, he received an honorary doctorate from the TH Munich; and, in 1909, he was appointed to the board of the Deutsches Museum in Munich to help enhance its collection of masterpieces from the natural sciences and engineering. None of these honors would have been possible without the influence of his doctoral student Walther Dyck, but they also were due to his close contacts to representatives of technical disciplines. 4.3.2 The Mathematical Student Union and the Mathematical Society The Mathematical Student Union is worth mentioning because it was Felix Klein who initiated its creation.158 In 1877, as the number of students interested in mathematics continued to increase, Klein encouraged them to form an official, registered students’ association like those with which he was familiar from Bonn, Göttingen, and Berlin. Isaak Bacharach,159 Walther Dyck, Joseph Gierster, Franz Meyer, Max Planck, Karl Rohn, and other students of Klein and Brill drafted the founding statutes of this association, which was officially acknowledged in May of 1877 by both the University of Munich and the Technische Hochschule. Klein maintained contact with this union and accepted its invitations to special events. When it became known that Klein would be taking a new position in Leipzig in the fall of 1880, a delegation from the student union officially expressed, on July 31, 1880, how unhappy they were to lose him.160 Professors and Privatdozenten of mathematics and physics at the University of Munich and the Polytechnikum (Technische Hochschule, as of 1877) regularly met in a Mathematical Society. Little is known, however, about the specific content of these meetings.161 The gatherings may have been useful for contributing to the professional development of Klein’s and Brill’s academically talented students. Habilitation topics may have been discussed. Alfred Pringsheim’s Habilitation was completed during this time (in November of 1877), as Klein mentioned in his notes from the period.162 Pringsheim’s research was oriented toward Weierstrass’s analysis and thus corresponded closely to the ideas held by the professors at the University of Munich, who refused to accept the Habilitation theses from 158 See HASHAGEN 2003, p. 64. 159 Bacharach completed his doctorate under M. Noether in Erlangen (1881). His main contribution was in the field of algebraic geometry. His name lives on today in the Cayley-Bacharach theorem. He also wrote an important book on the history of potential theory (Geschichte der Potentialtheorie, Göttingen; Vandenhoeck & Ruprecht, 1883), mentioned in POCKELS 1891, p. 1. Bacharach became a secondary school teacher, and he died as a victim of the Holocaust. 160 See JACOBS 1977 (“Vorläufiges aus München”), p. 4. 161 See HASHAGEN 2003, p. 53. 162 See JACOBS 1977 (“Vorläufiges aus München”), p. 3.

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Klein’s students whose work was oriented toward Riemannian geometry. Later, Klein engaged in a heated public dispute with Alfred Pringsheim over the nature of introductory lecture courses (see Section 8.3.4.1). Despite this disagreement, Pringsheim nevertheless contributed as an author to the ENCYKLOPÄDIE. 4.3.3 The Meeting of Natural Scientists in Munich, 1877 The fiftieth annual meeting of the Society of German Natural Scientists and Physicians (GDNÄ) took place in Munich from the 17th to the 22nd of September in 1877.163 Even as a Privatdozent and young professor, Felix Klein was actively engaged in bringing mathematicians together (see Section 2.8.3.4). Now he used the conference in Munich to recruit the largest possible number of people to participate in its Mathematics Section. At the same time, he served as the chairman of the event’s media and editorial committee.164 In the latter capacity, he edited a booklet that was presented as a gift to each participant in the meeting: München in naturwissenschaftlicher und medicinischer Beziehung: Führer für die Theilnehmer der 50. Versammlung deutscher Naturforscher und Ärzte [The Natural Sciences and Medicine in Munich: A Guide for the Participants in the 50th Meeting of German Natural Scientists and Physicians].165 Daily newsletters were printed and distributed throughout the course of the conference, and the papers given at the event were later published in an official report (Amtlicher Bericht). Eminent researchers such as Charles Darwin and William Thomson (Lord Kelvin) accepted invitations as honorary guests. For the second time, Felix Klein experienced at close range the extent to which many scientists valued Darwin’s theory of evolution,166 but he also encountered its opponents. In his plenary lecture – “Ueber die heutige Entwicklungslehre im Verhältnisse zur Gesammtwissenschaft” [On Today’s Evolutionary Theory in Relation to Science in General] – Ernst Haeckel described Darwin’s theory “as the most significant advancement of our pure and applied sciences,” and he concluded: “From now on, practical philosophy and pedagogy, like theoretical science as a whole, will no longer derive their most important principles from alleged epiphanies (aus angeblichen Offenbarungen) but rather from the natural knowledge of evolutionary theory.”167 Ernst Haeckel’s former teacher in Würzburg, Rudolf Virchow, rallied the opposition in a subsequent talk and elicited shouts of bravo with his imploration: “Teachers, do not teach this!”168 In part because of Virchow’s influence, evolutionary theory and 163 For further discussion of this conference, see TOBIES/VOLKERT 1998. 164 The editorial committee consisted of Prof. F. Klein (chair), Dr. A. Engler, Dr. J. Forster, Dr. E. Hermann, Prof. F. Ratzel, and Dr. E. Schweninger. See AMTLICHER BERICHT 1877, p. iii. 165 This work was published in Leipzig and Munich by the G. Hirth press (1877). 166 Klein’s first encounter with Darwin’s theory had been while he was a student at the University of Bonn (see Section 2.3.2). 167 AMTLICHER BERICHT 1877, pp. 14–22, at p. 20. See also RICHARDS 2009, pp. 312–29. 168 AMTLICHER BERICHT 1877, pp. 65–77, at p. 70. See also ZIGMAN 2000.

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the subjects of botany and zoology were banned, in 1882, from school curricula in Prussia.169 In 1900, biologists fought to remove this ban, and they were able to count on Felix Klein’s support (see Section 8.3.4). Sixteen papers were announced for the meeting’s section on mathematics, astronomy, and geodesy. The chair of this section was Ludwig Seidel, the professor of mathematics at the University of Munich. He led the first panel but did not give a paper himself. The chair of the second panel was Wilhelm Scheibner from Leipzig. He, too, did not give a talk, but later he would play an important role in luring Klein to Leipzig in 1880. The third panel, which was more strongly focused on applied mathematics, was chaired by the geodesist Bauernfeind. The first panel began with Luigi Cremona and Sophus Lie, whom Klein had personally invited. Cremona’s talk was titled “Ueber Polsechsflache bei Flächen dritter Ordnung” [On Polar Hexahedra in the Case of Order-Three Surfaces], and Lie’s paper was “Ueber Minimalflächen, insonderheit über reelle algebraische” [On Minimal Surfaces, Especially Real Algebraic Surfaces]. Lie was followed by Alexander Brill, who gave a presentation on the modelling workshop at the Technische Hochschule and on several of the models that had been constructed there. Finally, Klein lectured “Ueber die Gestalten der Kummer’schen Fläche” [On the Forms of the Kummer Surface]. Brill and Klein both advertised for Ludwig Brill’s firm in Darmstadt, which had produced models designed by the mathematical institute at the TH Munich. Klein presented four types of Kummer surfaces, and he classified them on the basis of his older and more recent results: One can distinguish four types of the general Kummer surface according to its 6 linear fundamental complexes, all or only 4, 2, 0 of which are real. In the first case, the surface has 16 real double points and double planes, in the second only 8, in the third and fourth 4; the latter two cases differ in that, in one, the 4 double planes pass through the 4 nodal points in pairs, whereas in the other this is not so (as in the case of Fresnel’s surface). These four types correspond to the four types of real hyperelliptic integrals (p = 2), which can be distinguished according to the reality of their ramification points and whose course is realized through the surfaces in question. Between these 4 types of the general surface there is a large number of transitional and degenerate cases. These include, above all, Plücker’s complex surfaces. By means of continuous transition, one can derive all of these forms from one another and thus gain a complete overview of the large series of existing possibilities.170

In 1878, Klein’s student Karl Rohn submitted his dissertation – “Betrachtungen über die Kummersche Fläche und ihren Zusammenhang mit den hyperelliptischen Funktionen p = 2” [Considerations Concerning the Kummer Surface and its Connection to the Hyperelliptic Functions p = 2]171 – and he ultimately developed a

169 [StA Berlin] Rep. 76 Vb Sekt. 1, Tit. 5, Abt. V, No. 12, Vol. 1, Fol. 33. 170 AMTLICHER BERICHT 1877, pp. 93–95, at p. 95. 171 This dissertation was published as a book by the Straub publishing house in Munich (1878).

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general method for ascertaining the forms of Kummer surfaces that do not contain multiple straight lines.172 Klein was especially eager to reunite with Sophus Lie at the conference, and he did all that he could to ensure that Lie could come to Munich for an extended period of time. Lie had accepted Klein’s offer to stay at his home in Munich. From Germany, Lie wrote back to Christiania (Oslo) about Klein’s “remarkably amiable” wife and about his “fine strong boy,”173 who had just turned one year old on August 6, 1877. Unfortunately, however, they did not have enough time to accomplish any substantive work together. Lie had already published an article on his paper topic in the Norwegian journal Archiv for Mathematik og Naturvidenskab, of which he was one of the editors. During the conference, it was brought to his attention that he had made a mistake. Lebrecht Henneberg, who in 1875 had completed his doctoral studies in Zurich under the supervision of Hermann Amandus Schwarz (his dissertation concerned minimal surfaces), was the person who identified Lie’s error. After the conference, Lie therefore devoted his full attention to correcting the mistake. Beyond that, he was upset with H.A. Schwarz, who told one and all about Lie’s erroneous results, and he was irritated with Alexander Brill’s reaction to the affair. On top of that, Lie also received some unfortunate news from home, so it is understandable why he and Klein were unable to find time to collaborate.174 The conference report contains an addendum after Lie’s paper: “The author wishes to communicate after the fact that he did not sufficiently consider the case in which the minimal surface becomes a double surface. When this is the case, the formulas of order and class provided by the author should be divided by 2.”175 Klein kept this topic in mind and had his colloquium students give lectures on minimal surfaces (on works by Lie, Schwarz, Weierstrass, and on Henneberg’s theorems).176 Later, he successfully recruited Lebrecht Henneberg, who became an associate professor at the TH Darmstadt in 1878 and a full professor there in 1879, to contribute as an author and consultant to the ENCYKLOPÄDIE project (particularly in the field of technical mechanics).177 The second paper that Klein was scheduled to deliver at the meeting of natural scientists was related to his recent research on the theory of elliptic functions. Because of time constraints, however, he was unable to give this talk. It was announced with the following abstract: One obtains the following theorem: When, in the representation of its values in the complex plane, the absolute invariant g23/∆ of a biquadratic function R (x) [where Δ = g23 – 27 g23 in 172 See Karl Rohn, “Die verschiedenen Gestalten der Kummer’schen Fläche,” Math. Ann. 18 (1881), 99–159. 173 Quoted from STUBHAUG 2002, p. 267. Sophus Lie’s first daughter was born on May 21, 1877. 174 See ibid., pp. 269–70; and [UBG] Cod. MS. F. Klein 10: 655/1 (a letter from Lie to Klein). 175 Amtlicher Bericht 1877, p. 94. 176 [Protocols] vol. 1.2, pp. 22–23, 74–75. 177 [UBG] Cod. MS. F. Klein 9: 680–82 (letters from Henneberg to Klein written between January 13, 1892 and November 9, 1899).

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4 A Professorship at the Polytechnikum in Munich the usual notation] moves across the positive half-space, the value of the period ratio K/K’ of the elliptic integral

∫ dx /

R( x ) passes through a circular arc triangle with the angles 0°,

60°, 90°: Six of these triangles, when arrayed in a particular way according to the laws of symmetry, form a new circular arc triangle with the angles 0, 0, 0, and this, as is well known, is the triangle within which K/K’ moves when the module k2 of the elliptic integral passes through its positive half-space.178

Sophus Lie responded positively to this result and remarked that he himself might want to pursue this direction further. He added: “It is actually remarkable how few people there are who have acquired a really audacious geometric way of thinking. We have probably learned this from Plücker. In any case, we have grounds for being eternally grateful to Plücker.”179 Other presenters in the section included Giuseppe Jung from Italy, Oscar Simony and Simon Spitzer from Austria,180 Paul Gordan, Ferdinand Lindemann, and a few lesser-known researchers.181 Other participants in the conference (who did not present papers) included the historian of mathematics Moritz Cantor (Heidelberg), Sigmund Gundelfinger (Tübingen), Reinhold Hoppe (Berlin), Klein’s friend Ludwig Kiepert (who was then an associate professor in Freiburg), Jacob Lüroth (Karlsruhe; later Klein’s successor in Munich), Klein’s friend Friedrich Neesen (who was then a Privatdozent in Berlin), Max Noether (Erlangen), Theodor Reye (Strasbourg), Rudolf Sturm (Darmstadt), Klein’s old friend Otto Stolz (Innsbruck), and Klein’s student Walther Dyck.182 Klein had thus succeed in enticing a good number of his friends and acquaintances to come to Munich. These attendees did not, however, represent the full spectrum of German mathematics. 4.4 “READY AGAIN FOR A UNIVERSITY IN A SMALL CITY” As early as November of 1876, Felix Klein noted that he was “ready again for a university in a small city.”183 He did not have the rather large city of Leipzig in mind when, in 1877, the university there was searching for someone to do geometry, which had not been represented since August Ferdinand Möbius. For this position, Klein recommended Max Noether, who was still looking for a full professorship, and his doctoral student Ferdinand Lindemann: “If Leipzig really wants to hire a geometrician, my first recommendation would be Noether. My second would be Lindemann, who has just completed his Habilitation in Würzburg. Perhaps, however, it is too early to write any more about such things.”184

178 179 180 181 182 183 184

Amtlicher Bericht 1877, p. 104. Quoted from STUBHAUG 2002, pp. 270–71. Regarding Simony’s research, see EPPLE 1999, p. 179. On the talks given on these panels, see TOBIES/VOLKERT 1998, pp. 241–42. AMTLICHER BERICHT 1877, pp. xvii–xxxiv. Quoted from JACOBS 1977 (“Vorläufiges aus München”), p. 2. Quoted from TOBIES/ROWE 1990, p. 86 (a letter from Klein to A. Mayer dated April 5, 1877).

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There were various reasons for Klein’s desire to leave Munich, some of which have already been suggested: conflicts with engineers over the reorganized curriculum; the lengthy process of hiring assistants; problems with the mathematicians at the University of Munich, who obstructed the doctoral and Habilitation procedures of his students; and his late election as an associate (instead of full) member to the Royal Bavarian Academy of Sciences. The letter that G. Bauer and Seidel wrote in support of Klein’s election to this academy in fact contained several barbs and did not do justice to Klein’s already considerable international reputation (see Appendix 3). In addition, Klein’s relationship with Alexander Brill had soured, even though Klein had attempted to teach with him and take trips with him (in 1876 to Regensburg and to visit Gordan; during the spring of 1877 to hike in the mountains). Brill envied Klein’s position at Mathematische Annalen and his academy memberships. Though seven years Klein’s senior, Brill was not made an associate member of the Bavarian Academy until 1882, and he did not become a full member until 1885. Adolph Mayer wrote that Klein “had become irritated and nervous on account of Brill’s envy, jealousy, and begrudging disposition.”185 Klein informed Darboux that he felt “worn out” and “overworked” to the extent that his health was suffering.186 Klein was constantly working in high gear, and he repeatedly doubted himself. After just his first semester in Munich, he noted: “Tense from the semester. Doubts about my academic abilities.” A little later, he remarked: “I am only content when producing new results.”187 In September of 1877, when his work began to overwhelm him, Klein had written: “I am in a mild state of despair. The work that I have to do after the end of the conference – not to mention dealing with Lie, Lindemann, Gordan, the Annalen, and the icosahedron – is killing me.”188 In addition to editing Mathematische Annalen, Klein had agreed to edit the official conference report for the Society of German Natural Scientists and Physicians (see 4.3.3). At the same time, he was helping Lindemann to prepare his edition of Clebsch’s lectures (vol. 1) for its translation into French. Any hope of collaborating with Lie had vanished, and this caused him strain. He was preparing, too, one of Gordan’s manuscripts for publication. Finally, Klein felt that it was important to translate Brioschi’s results himself for Mathematische Annalen because, “together with my icosahedron study, they conquer a field of research that has yet to be discussed in its pages.”189 As of April 2, 1875, Klein lived on the second floor of Theresienstraße 14, which was close to his office. With his growing family, he moved to Gabelsberger Straße 16 (second floor too).190 From August 8 to October 7, 1879, he retreated to Ebenhausen (20 km outside of Munich) for the sake of his health; there he lived in

185 186 187 188 189 190

Ibid., p. 116 (a letter from Adolph Mayer to Klein dated February 28, 1880). [Paris] 70: A letter from Klein to Darboux dated May 29, 1880. Quoted from JACOBS 1977 (“Vorläufiges aus München”), pp. 1, 4. Quoted from TOBIES/ROWE 1990, p. 93 (a letter from Klein to Mayer dated Sept. 27, 1877). Ibid. (a letter from Klein to Mayer dated October 16, 1877). [Paris] 66-67: A letter from Klein to Darboux dated December 7, 1879.

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the Fischerschlössl, a hunting lodge that is perhaps best known from a painting by Karl Roux (1877). Here, however, Klein hardly had a chance to rest and recover, for he received numerous visits from colleagues and students: In addition to my administrative and academic work, my efforts were devoted to a number of excellent students, and therefore I was already greatly overworked in Munich. It was there that I laid the foundation for my nervous illness, which would later flare up in Leipzig.191

Early in 1880, after his doctor had prescribed gardening as a form of therapy, Klein purchased “a country house with a large garden” (Forstenrieder Straße 8).192 On March 1, 1880, he had appealed to the administration of the Technische Hochschule for “reduced duties during the current semester and the coming summer semester.” His requests were honored, and he was permitted to conclude his lectures one week early during the winter semester of 1879/80. Brill took over his examinations of engineers. For the summer semester, Klein was allowed to cancel his “announced (but not obligatory) special lecture on analytical mechanics (part II) and the mathematical seminar associated with it.”193 Gordan was aware of Klein’s restlessness, and he wrote the following to him on April 24, 1880: “Best of luck with your gardening, and happy birthday tomorrow; if only you could stop working so hard on mathematics!”194 Klein limited his teaching to his seminar with Dyck and Bianchi, who came to his house for the course. Hurwitz, who had likewise fallen sick, remained at his parents’ home in Hildesheim. Klein recommended that he should rest: “Consider me a warning sign.” Admittedly, Klein requested Hurwitz’s presence again as early as April (at his house on Forstenrieder Straße), but on May 10, 1880 he wrote to Hurwitz’s parents that their son, “who combines specific mathematical talents with endearing personal characteristics,” should take a semester off on account of his health.195 His parents agreed to this plan, and Hurwitz, playfully rephrasing a poem by Heinrich Heine, wrote to Klein: “Zieh hinaus bis an das Haus, wo die Moduln spriessen, wenn Du einen Hauptmodul schaust, sag’ ich lass ihn grüßen” [Go out until you reach the house where moduli are blooming; if you happen to look upon a main modular function, extend to it my greetings].196 191 KLEIN 1923a (autobiography), p. 20. 192 [Innsbruck] A letter from Klein to Otto Stolz dated April 28, 1880; and JACOBS 1977 (“Vorläufiges aus München”), p. 4. 193 [Archiv TU München] A letter from Klein to the administration dated March 1, 1880. The same file contains the administration’s agreement to honor Klein’s request (dated March 2, 1880) and the final permission granted by the Ministry of Culture (dated March 8, 1880). 194 [UBG] Cod. MS. F. Klein 9: 407, fol. 11 (a letter from Gordan to Klein dated April 24, 1880). 195 [UBG] Math. Arch. 77: 30, 32, 34 (letters from Klein to Hurwitz dated March 26, April 22, and May 10, 1880). 196 [UBG] Cod. MS. F. Klein 9: 890, fol. 46v (a letter from Hurwitz to Klein from May 1880). The original lines, which are from Heine’s cycle of poems Neuer Frühling [New Spring] are: “Zieh hinaus, bis an das Haus / Wo die Veilchen sprießen. / Wenn du eine Rose schaust, / Sag, ich lass’ sie grüßen” [Go out until you reach the house / where violets are blooming. / If you happen to see a rose, / Extend to her my greetings].

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Although gardening is still recommended today as a therapy to counter nervous exhaustion, Klein’s recovery seemed to have been brought about, above all, by the possibility of being hired by another university. On March 3, 1880, Adolph Mayer wrote to him confidentially: The faculty has finally mustered the courage to ask the Ministry to establish a full professorship for geometry, and your name is on the top of the list for the position. […] As I already said, the prospect is still highly in doubt, but everyone here is working with all our powers on your behalf and we do not want to have anyone else but you.197

The hiring process turned out to be more difficult than expected. On several occasions, Mayer made his way to Dresden to pay personal visits to the appropriate minister, Carl von Gerber, who had once been a professor of law at the University of Leipzig.198 Supposedly, it was Mayer’s offer to allot part of his salary to Klein that ultimately persuaded the hesitant minister to go ahead with the hire.199 Mayer came from a banking family and was not dependent on his salary. For Klein, in contrast, the new position in Leipzig would alleviate his financial situation. Klein had expressed concerns about his finances as early as February of 1876, at which time he had needed to take out a loan. A life-insurance policy purchased in 1879 further restricted his disposable income, and this situation was only made worse by his poor health. Klein had to ask for his father’s assistance to straighten out his affairs.200 Later, he would often seek financial advice from Gordan, who, like Mayer, came from a wealthy banking family. The official job offer came from the Saxon Ministry of Culture on May 21, 1880, and Klein accepted it within a week.201 He was active again as early as June, when he informed Hurwitz that he would of course need him in Leipzig. In addition, Klein was involved in finding a replacement for his Polytechnikum colleague Klingenfeld, who had recently died. Klein made sure that Klingenfeld’s courses on descriptive geometry would be taught by Walther Dyck, so that the latter would be well prepared to carry on such work when he later followed Klein to Leipzig. Arthur Cayley visited Klein and his family several times in Munich: for a month in the summer of 1879 and in June 1880. They discussed their research results. Klein included Cayley’s results in Mathematische Annalen; Cayley prompted Klein to summarise the most recent results of his research, and he submited them to the Proceedings of the London Mathematical Society202 (see also Section 4.2.3).

197 198 199 200 201 202

Quoted from TOBIES/ROWE 1990 (a letter from A. Mayer to Klein dated March 3, 1880). See ibid. (letters from Mayer to Klein dated March 3, April 17, April 19, and April 24, 1880). See WITTIG 1910, p. 41. See JACOBS 1977 (“Vorläufiges aus München”), pp. 1, 4. [StA Dresden] 10282/17, fols. 3–6v. [UBG] Cod. Ms. F. Klein 8: 366; 369-71 (letters from Cayley to Klein dated September 10, 1879; June 6, July 7, and October 13, 1880).

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In early July of that year, moreover, Klein and Brill were visited in Munich by the Swedish mathematician Gösta Mittag-Leffler, who had been traveling through Italy and Switzerland.203 Thus Klein became personally acquainted with MittagLeffler even before the latter founded the journal Acta Mathematica (see Section 2.4.2). Paul Gordan was overjoyed about Klein’s new position in Leipzig, and he provided recommendations concerning who should be chosen as his successor at the Technische Hochschule in Munich: Dear Klein! The news about your new position pleased me to such an extent that it felt as though I was offered the job myself. Leipzig will be your way out of the sluggish conditions in Munich, where you had to wait three years before being made an associate member of the Academy. Do not, however, take the issue of your replacement lightly. My first choice would be Lüroth and my second choice would be Kiepert, but this is between us. Yours truly, Gordan204

In accordance with the wishes of Klein, Brill, and J.N. Bischoff, Klein’s professorship in Munich was offered to Jacob Lüroth, who was a member of Clebsch’s school. Prior to this appointment, Lüroth had been a professor at the Polytechnikum in Karlsruhe (which became a Technische Hochschule in 1885). The secondplace candidate on the list was not Felix Klein’s friend – and Weierstrass student – Ludwig Kiepert but rather Klein’s doctoral student Axel Harnack.205 This sort of hiring politics was a result of the dominant influence of Berlin mathematicians, who hoped to prevent Clebsch’s students (and Clebsch’s students’ students) from rising through the ranks. The hostility was keenly felt by both sides. When the hiring process in Munich became known to mathematicians who had earned their doctoral degrees in Berlin, Ludwig Kiepert expressed to Hermann Amandus Schwarz that “all representatives of the Berlin school” would have to stick closely together and should learn from the hiring politics practiced by the opposition.206 Since 1879, Kiepert had been a professor at the Technische Hochschule in Hanover, where his academic influence was limited (the Prussian Technische Hochschulen did not train teaching candidates in mathematics and they did not offer doctoral degrees until 1899). Kiepert was never, in fact, able to find a new position. Ever hopeful of moving to a new professorship, however, he did not refuse to cooperate with Klein, whom he addressed in letters as “Dear Felix.”207 H.A. Schwarz, on the contrary, actively attempted to hinder Felix Klein’s career (see Section 5.8.2).

203 See STUBHAUG 2010, p. 252. 204 [UBG] Cod. MS. F. Klein 9: 408, fol. 12 (a letter from Gordan to Klein dated May 27, 1880). 205 [BHSt] MK 19557 (a letter from August Kluckhohn to the Bavarian Ministry of Culture dated June 3, 1880). 206 See HASHAGEN 2003, pp. 110–11. 207 [UBG] Cod. MS. F. Klein 10: 49–180 (Ludwig Kiepert’s letters to Klein).

5 A PROFESSORSHIP FOR GEOMETRY IN LEIPZIG Meanwhile, a highly important and long-awaited event in my life has finally been realized. In the fall I will begin a new position in Leipzig as a professor of geometry. I hope that this move, which will provide me with a far larger sphere of influence and more responsibilities, will revitalize and increase my motivation.1 I read in the Literarisches Centralblatt that you will be moving to Leipzig in the fall. I am happy that you will acquire such a great sphere of influence there.2

In October of 1880, Felix Klein arrived in Germany’s leading trade-fair city, whose municipal charter dates to 1165 and whose university had been founded in 1409. The only university in the Kingdom of Saxony, the University of Leipzig had an enrollment of 3,326 students during the winter semester of 1880/81, which was higher than that of any other German university.3 According to the census taken on December 1, 1880, Leipzig had 148,081 residents and was thus the sixthlargest German city after Berlin, Hamburg, Breslau, Munich, and Dresden. With a population of 220,818, Dresden (the capital of Saxony) was home to the Royal Saxon Polytechnikum,4 which focused on training engineers and teachers and where some of Klein’s former doctoral students received professorships. Leipzig was the center of the German publishing industry, and Klein maintained close and longstanding contact with the B.G. Teubner press. Leipzig was an important travel hub. In 1880, there were seven train stations in the city, and its city center was further equipped with horse-drawn streetcars (this was upgraded to a network of tramways in 1896).5 As early as 1701, Leipzig was the third German city, after Berlin and Hamburg, to become equipped with street lights (originally using oil lamps). By the time Klein lived there, gas and electric streetlights had been installed, something which the small town of Göttingen would not acquire for several years to come. Leipzig had also made a name for itself as a center for theater and music, with its St. Thomas Boys Choir (established in 1212) and the Gewandhaus Orchestra, whose roots go back to 1479. Although Klein himself was not very musically talented, his wife, his sisterin-law Sophie, and his youngest daughter Elisabeth were gifted in this regard. Adolph Mayer had procured an apartment for Klein’s family on the third floor of a six-story building on Sophienstraße 10 (today Shakespearestraße), and he also found a nanny to help care for their four-year-old son Otto and their eleven1 2 3 4 5

[Paris] 70: Klein to Darboux, May 29, 1880. [Innsbruck] A letter from Otto Stolz to Klein dated June 18, 1880. See HASHAGEN 2003, p. 94. In 1890, the Polytechnikum was renamed the Technische Hochschule (TH) Dresden. Today, Leipzig’s Central Station is the largest railway terminus in Europe. It was completed in 1915.

© Springer Nature Switzerland AG 2021 R. Tobies, Felix Klein, Vita Mathematica 20, https://doi.org/10.1007/978-3-030-75785-4_5

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month-old daughter Luise.6 This small street was conveniently located near the Bavarian Railway Station and it was within walking distance to the city center, where the university’s main buildings were situated. In May of 1885, the Klein family moved to Sophienstraße 31/II because they were expecting another child.7 Born on July 11, 1885, their daughter was given the name Sophie. In the autumn of 1880, Klein began anew in Leipzig with full intensity. His ambition was “to carry out [his] research, administrative work, and teaching activity with equal energy.”8 However, the metaphor that he used to characterize this time – “I have bitten off more than I can chew”9 – suggests that he was unable to maintain the same high level of energy for all of these areas of activity. In order to understand how big this bite in fact was, we will need to examine the number of parallel activities that Klein was engaged in. Toward the end of September in 1880, while Anna Klein was furnishing their new apartment in Leipzig, Felix Klein traveled to Erlangen. There he edited an article by Gordan on seventh-degree equations, and the two of them discussed various ideas related to Bianchi’s work.10 While in Erlangen, Klein also completed the article that Cayley had requested for the London Mathematical Society (dated Oct. 5, 1880), and wrote the inaugural address required of his new professorship in Leipzig (Oct. 7, 1880). This lecture will be the topic of Section 5.1. As was the case in Munich, Klein’s first initiative in Saxony involved the institutional framework for the university’s mathematicians, whose teaching hitherto took place in the “Augusteum” – the university’s main building on Augustusplatz. Even though Klein, during his hiring negotiations, had followed Adolph Mayer’s advice to avoid the matters of establishing an institute and hiring an assistant,11 he would nevertheless go on to pursue these goals shortly after accepting the position (Section 5.2). In Leipzig, Klein wanted to come closer to realizing his idea of a systematic teaching program. After his first three semesters, however, he was again overtaxed with work, and so he resorted to the same remedy that had worked for him in Munich: he cut back on his program of lecture courses (Section 5.3). In an effort to encourage young researchers to achieve independent results, Klein extended a helping hand to a growing circle of people. The fruits of his efforts need be seen in the context of his numerous students and collaborators from Germany and abroad (Section 5.4). Some of Klein’s mathematical studies during his time in Leipzig involved continuing, refining, and summarizing his previous findings. In doing so, he relied more heavily on the methods of Berlin mathematicians. He figured that his old

6 7 8 9 10 11

See TOBIES/ROWE 1990, p. 123 (a letter from Mayer to Klein dated September 20, 1880). [UBG] Math. Archiv 77: 142 (a letter from Klein to Hurwitz dated June 20, 1885). Quoted from JACOBS 1977 (“Persönliches betr. Leipzig”), p. 2. In German: “Ein Mantel, der mir zu weit ist.” [UBG] Math. Arch. 77: 40 (a letter from Klein to Hurwitz dated September 27, 1880). See TOBIES/ROWE 1990, p. 118 (a letter from Mayer to Klein dated March 7, 1880).

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research program from Munich could be carried out by his students, and he forged a new application-oriented research program of his own. That said, a closer examination of Poincaré’s work inspired, within his old research framework, some of his greatest achievements (the three fundamental theorems related to his theory of uniformization), which he later interpreted to be the zenith of his mathematical creativity (Section 5.5). Klein’s close collaboration with the B.G. Teubner publishing house entered a new phase when Alfred Ackermann-Teubner became a member of the management in 1882. Klein’s work with him also led to the establishment of a prize foundation for achievements associated with the ENCYKLOPÄDIE (Section 5.6). As elsewhere in his career, Klein founded, participated in, and reorganized academic groups. He was involved in a mathematical discussion group, the Societas Jablonoviana, and the Royal Saxon Academy of Sciences (see Section 5.7). Section 5.8 will show why Klein once again sought a change of scenery and how he influenced the choice of his successor in Leipzig. 5.1 KLEIN’S START IN LEIPZIG AND HIS INAUGURAL ADDRESS In Leipzig, Wilhelm Scheibner (54 years old) and Carl Neumann (48 years old) were teaching as full professors, while Adolph Mayer (41 years old) and Karl Von der Mühll (39 years old) held associate professorships. Felix Klein, just thirty-one years old, arrived as the third full professor, with an appointment to teach geometry. Before then, no German university had a full professorship that was devoted only to geometry. Through his editorial work for Mathematische Annalen, Klein had already been familiar with Mayer and Von der Mühll since 1873 (see Section 2.4.2). Scheibner first learned to appreciate Klein at the 1877 meeting of natural scientists in Munich (see Section 4.3.3). In the application for the new professorship, Scheibner made a shrewd argument by citing the large number of professorships in Berlin and Göttingen.12 With the application, he also enclosed a long list of branches of geometry in order to demonstrate that many of them could not be taught in Leipzig without an additional professor on hand. Here is an excerpt of Scheibner’s application to the Saxon Ministry of Culture in Dresden: V. Geometry. a) Analytic geometry of the plane / N.[eumann] and M.[ayer] b) Analytic geometry of space / M.[ayer] c) Theory of curved surfaces / Sch.[eibner] and N.[eumann] d) Descriptive geometry / vacant e) Higher synthetic geometry (Möbius and [Jakob] Steiner) / vacant f) Geometry of position (Staudt and Reye) / vacant g) Theory of invariants and covariants / vacant h) Theory of binary forms / vacant 12 See KÖNIG 1982, p. 92; and SCHLOTE 2004, p. 30.

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Theory of higher algebraic curves and surfaces / vacant Theory of mappings of algebraic surfaces (Cremona and Clebsch) / vacant Theory of complexes (Plücker) / vacant Connection between algebraic curves and the theory of elliptic and Abelian functions / vacant m) Theory of non-Euclidean geometry / vacant n) Theory of higher manifolds / vacant.13

Regarding Felix Klein, who was named as a potential candidate along with two of his students (Axel Harnack and Ferdinand Lindemann), Scheibner wrote: We name as our first choice one of the most prominent students of the late Clebsch, Dr. Felix Klein, a full professor at the Polytechnikum in Munich who, both through his numerous academic publications and through his editorial work for Mathematische Annalen, has done extraordinarily much to further the developments of newer geometry; who has recently achieved, by way of geometric speculations, important new results concerning the theory of algebraic equations and modular functions; and who has distinguished himself by training talented students.14

The Saxon Ministry of Culture offered Klein the position with a starting dated of October 1, 1880. In calculating his pay, the ministry took into account his eight years of civil service in Bavaria. Klein thus received an annual salary of 7,500 Mark and a moving allowance of 1,800 Mark.15 In addition to this, he would also receive fees for his lectures. Per semester, students in Leipzig had to pay a 15Mark fee to attend a four-hour lecture course (this increased to 16 Mark as of the winter semester of 1884/85) and an additional 1.5 Mark of administrative fees. Seminars and exercises cost nothing.16 Also in October 1880, the Saxon Minister of Culture Carl von Gerber appointed Klein to the Examination Committee for Secondary-School Teaching Candidates. This was an important role, because then the primary objective of studying mathematics was – as mentioned before – to prepare oneself to become a secondary school teacher. On October 25, 1880, Klein delivered his inaugural address, which he had prepared in Erlangen a few weeks before and which bore the title “Über die Beziehungen der neueren Mathematik zu den Anwendungen” [On the Relations Between Recent Mathematics and Applications]. Drawing from his experiences in Munich, Klein began: Among all the sciences, hardly any could claim to be more generally applicable than mathematics. Not only do the neighboring natural sciences and the more finely developed aspects of epistemology require a mathematical basis; practical life, too, with its multifaceted undertakings (above all, modern technology) cannot proceed without preliminary education in ma-

13 [StA Dresden] 10210/17, fols. 245–46. 14 Ibid., fol. 250. 15 [StA Dresden] 10281/184 (Klein’s personnel file), fols. 7–9v. Klein returned 300 Mark of the moving allowance because he was soon able to find a new tenant for his apartment in Munich (see ibid., fol. 14). 16 [UBG] Cod. MS. F. Klein 7 E, fols. 128–45. At the TH Munich, the fee for enrolling in lectures was 10 Mark.

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thematics. This is acknowledged and not disputed by any side. And yet, despite this fact, we observe a strange contradiction. No one will deny that, since the beginning of the century, pure mathematics has undergone a powerful and profound development in a variety of directions. With respect to applications, however, all of this development seems to have been nearly useless. The practitioner ignores our progress and is at most inclined to single out a few seemingly paradoxical conclusions and subject them to a rather harsh critique.17

Despite this trend, Klein expressed his “optimistic conviction” that “whatever is of interest to us theoreticians now will later become applicable in a more general sense.”18 However, he also believed that mathematicians themselves would have to change their ways, and this was the basic premise behind his suggestions: First. The “excessive specialization of university instruction and the associated creation of one-sided mathematical schools of thought” must be abandoned. Clebsch’s approach, which fused geometry and algebra as well as geometry and function theory, had “left behind a lasting legacy” only among his students and friends.19 Klein argued in favor of integrating different areas of research, offering general as well as specialized lectures, and designing a systematic teaching program. He also argued that every teacher of mathematics should possess a broad overview of the subject, and that this broad knowledge should be reflected in their teaching and research. Second. With reference to the long-overlooked results of the Leipzig geometrician August Ferdinand Möbius, Klein stressed that geometry could no longer be neglected and that the methods of different approaches to geometry should be incorporated into the curriculum. Klein’s initiatives to reorganize the curriculum and to arrange for editions of the work of Möbius and Grassmann should be seen within this context. Third. Whereas French mathematicians felt compelled early on to present their knowledge in textbooks, this was long an uncommon practice in Germany. During the first half of the nineteenth century, the textbooks published in Germany were often translated works. Klein mentioned this deficiency particularly in the case of analysis, and he pointed out that Cauchy’s Cours d’analyse still formed the foundation of the better books on the subject. Too much time had passed, he thought, before the results of Riemann and Weierstrass – the founders of the theory of complex-valued functions – were more widely available. This changed with the publication of Rudolf Lipschitz’s two-volume Lehrbuch der Analysis [Textbook on Analysis] (Bonn: Fr. Cohen, 1877 and 1880). This was the first German textbook in the field. It was not because he lacked new ideas that Klein began to systematize his own ideas into books; rather, he was aware that it was necessary to do so and to follow the model of French textbooks in order to disseminate his ideas in the first place (see Sections 5.5.1.2, 5.5.6, and 5.5.7).

17 KLEIN 1895a, pp. 535–36. This lecture is reprinted in BECKERT/PURKERT 1987, pp. 40–45. 18 KLEIN 1895a, p. 536. 19 Ibid., p. 537. Klein did not divide his inaugural lecture into numbered points, but such a logical structure is clear.

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Fourth. In order to introduce mathematics to a broader circle of people, Klein offered the following solution: “It is its great abstractness that we have to fight against.”20 He explained the role of illustrations and geometric models, and he showed that geodesists, astronomers, and physicists could also benefit from them. He expressed his belief that the university curriculum should be expanded to include subjects that he had learned to appreciate at the Technische Hochschule: “descriptive geometry, graphic constructions, machine kinematics.” It was this conviction that would motivate his efforts to form, at the University of Leipzig, a mathematical institution with a collection of models and a reorganized curriculum. For the first time at a German university, descriptive geometry was included in the curriculum; of course, it was taught in conjunction with other fields such as projective geometry (see Section 5.3.1).21 Klein concluded his inaugural address by noting that he would only be able to implement this program gradually. The substance of this speech would continue to define his orientation towards applications well beyond his years in Leipzig. In 1895, while he was continuing to advance his program in Göttingen, he sent this address to the Prussian Ministry of Culture, and only at this point did he submit it for publication (KLEIN 1895a). Although he faced many obstacles in Leipzig, he was indeed able to realize several of his goals there. 5.2 CREATING A NEW MATHEMATICAL INSTITUTION Things are going well for me in general, and I am truly content with the conditions here regarding the possible developments that can be made.22

Klein’s letter to Otto Stolz is indicative of his characteristic urge to shape his professional surroundings. Shortly before, the Saxon Minister of Culture Carl von Gerber had approved Klein’s funding request for geometric models and for an appropriate glass case in which to store them. As expressed in his inaugural address, Klein had greater ambitions than merely acquiring models. His experiences from Erlangen and Munich taught him that he would be able to achieve greater status for mathematics by establishing a separate institution devoted to it. On December 5, 1880, Klein, having felt out the conditions at the university, applied not for an Institute (as he had done at the Polytechnikum in Munich) but rather for a Mathematical Seminar. To make his case, he was able to point out that such an arrangement for training teaching candidates already existed at numerous universities. Klein was familiar with such seminars from his own experiences in Bonn, Berlin, and Göttingen. 20 KLEIN 1895a, p. 538. 21 On the development of descriptive geometry in Germany in comparison with France, see Nadine Benstein’s article in BARBIN et al. 2019, pp. 139–66. On the synthesis of projective and descriptive geometry, see Klaus Volkert’s article in the same book (pp. 167–80). 22 [Innsbruck] A letter from Klein to Otto Stolz dated November 10, 1880.

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Klein was granted permission to renovate a little-used building known as the “Czermakeion.”23 On December 7, 1880, he sent a list of desiderata to the university administration with details about renovating and expanding the lecture hall, creating side rooms for seminars, and a library; here Klein also provided detailed information about furniture, heating, blackboards, lighting, and washing facilities.24 Evidently, he intended to create more than a seminar; he desired an institute comparable to that in Munich. Regarding matters of personnel, Klein also made novel decisions. He hired a custodian to take care of facilities; he revived the previously existing position of a Famulus (a student assistant); and, on October 15, 1881, he could hire a regularly salaried assistant (Klein became, in fact, the first professor of mathematics at a German university to have an assistant of this sort).25 Klein’s main justification for creating this budgetary position was to have someone to prepare and oversee the model collection. Unlike his assistants at the Technische Hochschule in Munich and later in Göttingen, Klein’s assistants in Leipzig, Walther Dyck and Friedrich Schur, had already completed their Habilitation; they became Privatdozenten. The Famulus managed the “vouchers” (Belegbogen) for Klein’s private lectures (which students had to pay to attend) and edited his lectures. Each of the five students in Leipzig who went on to hold the position of Famulus also completed his doctoral degree under Klein’s supervision (see Table 6). Klein took over the renovated rooms for the Mathematical Seminar on April 8, 1881. He was also made the director of this seminar as well as the director of the model collection and a co-director of the Czermakeion. The Czermakeion was a classroom building used by all the faculties at the university; its other co-director was the psychologist Wilhelm Wundt. At the time, Wundt established the world’s first institute for experimental psychology, and it would serve as a model for similar institutes elsewhere (including the University of Göttingen). Wundt and Klein first worked together – as members of a doctoral committee – as early as December of 1880.26 At Klein’s request, two colleagues were named co-directors of the Mathematical Seminar: the associate professors Adolph Mayer and Karl Von der Mühll. The younger Klein thereby increased the status of these older colleagues and thus secured two allies. Klein’s effort to have both co-directors promoted to honorary full professors was only successful in the case of Mayer (the promotion was granted in 1881).27 23 The Czermakeion (also “Czermak’s Spektatorium”) was named after the physiologist J.N. Czermak, who left the building (Brüderstraße 34) as a legacy to the university. 24 See König’s article in BECKERT/SCHUMANN 1981, p. 63; and KÖNIG 1982, pp. 127–31. 25 See LOREY 1916, p. 167. For a list of Klein’s Famuli and assistants, see KLEIN 1923 (GMA III), Appendix, p. 14 (here, however, it is wrongly noted that Klein’s assistant in Leipzig began in the summer semester of 1881 instead of in the winter semester 1881/82). 26 Klein and Wundt wrote evaluations for Alfred Donadt’s dissertation, Das mathematische Raumproblem und die geometrischen Axiome (Leipzig: J. Ambrosius Barth, 1881). 27 See TOBIES/ROWE 1990, pp. 25–26.

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In order to build up the Mathematical Seminar, Klein submitted additional applications for funding: for seminar prizes, models, books, furniture, etc. Moreover, he dreamt of having work rooms for students, for which the Czermakeion did not have adequate space. It was far more difficult for Klein to acquire these rooms than some have made it seem.28 Klein’s letters to the Rektor of the university, Friedrich Zarncke, mention several opponents and obstacles. Although Zarncke – a Germanist, Goethe scholar, and the founder of the Literarisches Centralblatt für Deutschland – personally prepared the decisive meeting of the academic senate with Klein, the senate voted against his construction plans and budget projections for new work rooms for mathematics. Klein reacted by submitting an application to the Ministry of Culture, and he sent Zarncke a draft of his application on August 2, 1882: It is in my interest to assure you that I have taken the proper course of action in response to the vote of the academic senate – but it is also in my interest to bring my further plans to your attention. I hope that you view them as sympathetically as you have regarded me hitherto, and I remain hopeful that the time will come when I find full acceptance among the extended circle of colleagues who, at the moment, are confronting me with one obstacle after another.29

Although the Saxon Ministry of Culture did not immediately agree to Klein’s request, a new application, which he completed on October 12, 1882, resulted in mathematics acquiring the desired work rooms on the third floor of Ritterstraße 14, a building known as the “Kleines Fürsten-Collegium.” At the time, no other German university possessed work rooms of this sort for students and younger researchers.30 As of the winter semester of 1883/84, the institution as a whole was called the Mathematical Institute, and it had facilities in two different buildings: a) the Mathematical Seminar on Ritterstraße and b) the model collection (with a modeling room) and a lecture hall for elementary and geometric lectures in the Czermakeion. On March 2, 1886, Klein persuaded the Ministry of Culture to rename these two parts as “Division I” and “Division II” of the institute.31 Otto Hölder first came to Leipzig as a young doctor in April of 1884, and he sent his parents informative letters about Klein’s institution. After completing his doctoral degree under Paul du Bois-Reymond in Tübingen with a dissertation on potential theory (1882), Hölder had continued his studies in Berlin, where he “could not develop any relationships with his peers.” In Leipzig, on the contrary, he felt as though he was integrated immediately: I met with Klein during my first day here, exactly 3 hours after I had arrived in Leipzig. He immediately gave me a key to the rooms of the Mathematical Seminar, and yesterday I already spent 6½ hours there. There is a library room and a work room, and the mathematicians here actually work there all day long. […] Through Klein, one immediately becomes acquainted with all the young mathematicians. He simply sends you into the Seminar, where

28 29 30 31

See KÖNIG 1982; and THIELE 2018. [UB Leipzig] Nachlass Fr. Zarncke (a letter from Klein to Zarncke dated August 2, 1882). [UA Leipzig] Phil. Fak. B1/1423, vol.1, fol.3 (O. Hölder/K. Rohn, “Das math. Institut, 1909”). See Fritz König’s article in BECKERT/SCHUMANN 1981, pp. 67–68.

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everyone introduces himself to everyone else, and then you soon see how everyone gets along with one another […] When working on a project that requires references to scholarly literature, it is very handy to be in the Seminar, which has everything, and no books can be taken out. […] The Seminar facility also makes it immediately possible for the people there to form social relationships; just yesterday I went out with the mathematicians for a beer. […] The work rooms are open every day – even on Sunday – from 7 in the morning to 10 at night.32

5.3 TEACHING PROGRAM Just as in Berlin, where it was “the coordinating hand of Ernst Kummer that designed the curriculum,” and just as Clebsch had functioned in Göttingen, Klein acted in the interest of his students during every semester.33 In what follows, it will be shown how Klein sought to organize the mathematics curriculum in Leipzig and how his own teaching fit into his overall program. Certain aspects should be emphasized in advance: Klein’s competition with Carl Neumann in the area of function theory, the new subject of descriptive geometry, his attempt to reorganize the curriculum, and the temporary reduction of Klein’s teaching load on account of his health and other reasons. In addition, it will be necessary to correct some wrong information that appears in the appendix to the third volume of Klein’s collected writings (see Section 5.3.1). Section 5.3.2 will focus on the specific role of Klein’s research seminar. 5.3.1 Lectures: Organization, Reorientation, and Deviation from the Plan In Munich, Klein had organized the mathematics curriculum with Alexander Brill. In Leipzig, however, he had to convince several older colleagues to go along with his plan. At first, he limited his efforts to producing a “coordinated geometric education for teaching candidates,”34 and he coordinated his plans in advance with his former student Karl Rohn, who had been working as a Privatdozent there since May 15, 1879. Because Rohn would be teaching the lower-level courses during the winter semester of 1880/81 (“Theory of Plane Curves of the Third and Fourth Order,” “Differential and Integral Calculus, with Exercises”), Klein would be able to concentrate on his research-oriented teaching. Already in April of 1880, Klein had written the following to Otto Stolz: “Then in the winter I will focus on function theory, into which, as you know, I have been meaning to immerse myself for a long time but which is still, for me, uncharted territory.”35 While in Munich, Klein had sought to draw a close connec32 Quoted from HILDEBRANDT/STAUDE-HÖLDER, pp. 142, 144 (letters from Hölder to his parents dated April 23 and May 6, 1884). See the German original in TOBIES 2019b, p. 196. 33 On Kummer’s organization of course offerings in Berlin, see BIERMANN 1988, p. 101. 34 See JACOBS 1977 (“Entwicklung meiner Vorlesungen und Arbeiten”), p. 2. 35 [Innsbruck] A letter from Klein to Otto Stolz dated April 28, 1880.

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tion between the fields of geometry, algebra, number theory, and special functions (especially elliptic modular functions). He integrated each of these subjects in his first special lecture course in Leipzig: “Function Theory from a Geometric Approach for Intermediate Students,” which was held four hours per week, Tuesday to Friday from 12 to 1 o’clock.36 Eighty-nine students enrolled in the course, more than attended any other course taught by a mathematician in Leipzig at that time.37 Klein’s Famulus, Ernst Lange, who had already attended Klein’s courses in Munich,38 took lecture notes. Later, in 1892 in Göttingen, Klein requested his student Paul Epstein to produce an autograph copy of this lecture, an edition of which was published by Fritz König in 1987. In a preface (Geleitwort) to this edition, Friedrich Hirzebruch outlined Klein’s basic ideas – the idea of the Riemann surface, algebraic geometry, and the relationships between function theory and algebraic curves – and remarked that such topics, though seldom discussed in contemporary university courses, can still be of great interest to students and teachers alike.39 Under the rubric “mathematics and astronomy,” the courses offered at Leipzig during the winter semester of 1880/81 still, however, indicated a lack of coordination. Both Karl Von der Mühll and Karl Rohn, for instance, were scheduled to teach differential and integral calculus. Von der Mühll, whose expertise was mathematical physics, also offered a course on the mathematical theory of light and conducted mathematical-physical exercises. Adolph Mayer, whose main field was the calculus of variations, offered the introductory lecture course on analytical mechanics, as he usually did in winter semesters. The list was completed by Wilhelm Scheibner (“On the Three-Body Problem,” “On Multiple Integrals”), Carl Neumann (“Mathematical Theory of Electrostatics”), and the associate professor of physics Eilhard Wiedemann (“On Quaternions”). The astronomers, with whom Klein developed good relationships, also offered courses on mathematical subjects: Karl Christian Bruhns, who was then the director of the observatory in Leipzig-Johannisthal (“On the Relations between Differences and Differentials, Sums and Integrals”; “Theoretical Astronomy: On Determining the Orbits of Planets and Comets”; “Colloquium on Topics from Astronomy, Geodesy, and Meteorology”); and Hugo von Seeliger (“Mathematical Geography,” “The Theory of Eclipses and Related Phenomena”). When Bruhns died on July 25, 1881, his professorship and the directorship of the observatory were taken over by Heinrich Bruns. A student of Weierstrass, Bruns knew Klein from the latter’s period of study in Berlin (1869/70). Bruns developed into a top-rate scholar in Leipzig40 and cooperated well with Klein. Together, they co-supervised August Föppl’s doctoral research, which culminated 36 https://histvv.uni-leipzig.de/vv/1880w.html.“Functionentheorie in geometrischer Behandlungsweise für Studierende mittlerer Semester”. 37 See KÖNIG 1982. 38 In Munich, Ernst Lange attended Klein’s lectures from 1878/79 to 1879/80 (higher mathematics, analytical mechanics). See [UBG] Cod. MS. F. Klein 7E. 39 See KLEIN 1987, p. 3. 40 See BIERMANN 1988, pp. 132–34.

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in a thesis on the mathematical theory of structures (1886).41 Later, Klein recruited August Föppl to contribute to the ENCYKLOPÄDIE, and he discussed Föppl’s work in seminars at a time when he himself was trying to systematize problems in the area of statics (see Section 8.2.4). During his second semester in Leipzig – the summer semester of 1881 – Klein took over the beginners’ course “Introduction to the Analytic Geometry of the Plane and of Space” (87 students); K. Rohn taught “Introduction to Analysis”; and Friedrich Schur, who had meanwhile completed his Habilitation, offered a course titled “Elements of Newer Synthetic Geometry” in addition to “Geometric Exercises.” Klein also continued his special lecture course, “Function Theory from a Geometric Approach II,” which had forty-five students and met for four hours per week. The content of his course corresponded to that of his forthcoming book Über Riemanns Theorie der algebraischen Funktionen und ihrer Integrale: Eine Ergänzung der gewöhnlichen Darstellungen [On Riemann’s Theory of Algebraic Functions and Their Integrals: A Supplement to the Usual Treatises] (see 5.5.1.2). By teaching such a course, however, Klein caused a dispute with Carl Neumann, who felt robbed of one of his domains because he regularly gave lectures on function theory.42 In 1865, Neumann had published his book Vorlesungen über Riemann’s Theorie der Abel’schen Integrale [Lectures on Riemann’s Theory of Abelian Functions], and he prepared a second edition, which was published in 1884. Later, Klein wrote: “It is an excellent introduction to the Riemannian circle of ideas.” Comparing Neumann’s book to the aforementioned book by Clebsch and Gordan on the same subject (see Section 2.4.1), Klein remarked that the latter “may be difficult, and it demands the close attention of readers, but it lets them penetrate far more deeply into the problems at hand and it stimulates them to study Riemann’s thought in a serious way.”43 Neumann was reluctant to harmonize the teaching program. Like his Königsberg teacher Richelot, he wanted to teach whatever he pleased – geometry included. Later, Sophus Lie would also feel that Neumann treated him poorly (see Sections 5.5.1.1 and 5.8.3). In any case, Klein avoided further problems with Neumann by designing his own cycle of geometry courses. After teaching “Analytic Geometry” in the summer of 1881, he planned to offer “Projective Geometry and Differential Geometry” during the following semester. Beginning in October of 1881, he introduced a course on descriptive geometry (with exercises), the instruction of which was supported by his assistant Dyck.44 For decades, this discipline, which had been 41 See KÖNIG 1982, A6-4. – From 1912 to 1914, August Föppl’s son Ludwig became Klein’s assistant after he completed his doctoral degree under Hilbert. Like his father, Ludwig Föppl became a professor of technical mechanics. 42 For a list of Neumann’s courses, see http://histvv.uni-leipzig.de/dozenten/neumann_c.html. He taught, for example, “The General Theory of Functions of Complex Variables (According to Cauchy)” (summer, 1876), “Select Chapters from the Theory of Spherical Functions and Potential” (winter, 1877/88), and “The Theory of Functions” (summer, 1882 and 1884). 43 KLEIN 1979 [1926], p. 256. 44 [UBG] Math. Arch. 77: 52 (a letter from Klein to Hurwitz, October 1881).

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established by Gaspard Monge in Paris, had been part of the engineering curriculum at polytechnical schools also in Germany, though here it was usually at a lower level (see BARBIN 2019). Klein, as mentioned above, was the first person to teach it at a German university. His lecture course “Projective Geometry, Part I (in Conjunction with Descriptive Geometry)” attracted more students (102) than any of his other courses during his time in Leipzig. Sixty-six “trainees” (Praktikanten), as he put it, participated in the exercises related to the course.45 Encouraged by the high number of students, Klein felt that it was time to design his first degree program. He discussed the matter with his colleagues, and together they published an article “Bemerkungen über die mathematischen Vorlesungen an der Universität Leipzig” [Remarks on the Mathematical Lectures at the University of Leipzig] in the Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht.46 This journal, which had been published by B.G. Teubner since 1870, circulated widely among teachers. The article listed teaching areas in two categories: “A. Introductory or Elementary Lectures” and “B. More Advanced Mathematical Lectures,” and it described which subjects built best upon one another. Students were advised “to take careful notes – not according to the exact words of the lecture but according to their own understanding – and, in the case of doubtful points, to seek out additional information from the lecturer himself.” It was not expected that every student could become deeply familiar with all areas of mathematics: “Indeed, as regards the education of future Gymnasium teachers, less importance should be placed on ensuring that they develop a particular mathematical specialty than on providing them with a general orientation, ensuring that they can demonstrate – in one direction or another – solid and wellordered knowledge, some degree of familiarity with the appropriate scholarly literature, and a solid level of study.” Students who have chosen mathematics as their minor subject should be allowed to limit their attendance to basic courses. Such care for the interests of students was rather rare at the time. This is evident from a letter by Kiepert, to whom Klein had sent the degree program: Almost every professor, when not teaching certain introductory lectures that bring in the most money, simply teaches his favourite subject, and the students have to see where they can get further knowledge from.47

In the summer semester of 1882, Klein taught the second part of his lecture course on projective and descriptive geometry (70 students), and he returned to the subject of function theory by offering an upper-level course on it. In addition, Klein also gave fifteen lectures on “Single-Valued Functions with Linear Transformations in Themselves.”48 He revised the content of these lectures to form sections III 45 Quoted from JACOBS 1977 (“Persönliches betr. Leipzig”), p. 3. 46 Authorship of this article was attributed to “the professors of mathematics at the University of Leipzig.” See ZmnU 13 (1882), pp. 247–50. 47 [UBG] Cod. MS. F. Klein 10: 87 (a letter from Kiepert to Klein dated May 18, 1882). 48 These lectures were delivered from June 6 to August 4, 1882, and they were transcribed by Eduard Study. See KLEIN 1923 (GMA III), pp. 585, 632, and Appendix, p. 6.

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to V of his article “Neue Beiträge zur Riemannschen Funktionentheorie” [New Contributions to Riemannian Function Theory].49 For the winter semester of 1882/83 (October 15 to March 15), Klein reduced his teaching load. As in Munich, he had to deviate from his already announced lecture program on account of health problems. He cancelled his special lecture course “Select Chapters from Function Theory (Only for Advanced Students).” He handed over his elementary lecture course “The Application of Differential and Integral Calculus to Geometry” (70 students) to Walther Dyck from mid-November to Christmas 1882. Klein explained the situation as follows: I had meanwhile been a patient of sorts. Throughout the entire fall vacation, as you know, I suffered from asthma, and at the beginning of the semester it flared up to such an extent that, by the middle of November, it seemed necessary to hand over my lecture to Dyck and keep only my seminar, to which I have devoted an exceptional amount of energy. The peace and quiet that these and other rational measures have provided me have meanwhile lifted me back up, and thus I intend to take over my lecture again after the New Year. The fact of the matter is that, despite everything, I have taken on too many obligations here in Leipzig. Ever since I made the clear decision to work less, I at least feel more at ease than otherwise.50

For a long time, Klein had regularly suffered from hay fever and asthma, and he also repeatedly suffered from stomach problems and insomnia. Thus, as of the summer of 1881, he usually spent several weeks a year on an East Frisian island in order to benefit from a better climate. This did not always help. On March 19, 1882, for instance, he had written the following to A. Mayer from Norderney: In general, it’s very beautiful here. At the beginning of my trip, however, I did not take the best care of myself, so that I have now been unwell for the past 2 to 3 days: asthma and stomach aches, as so often. By the way, these things are taking their normal course.51

Just a few days later, Klein would formulate one of his most significant theorems – the limit circle theorem (Grenzkreistheorem; see Section 5.5.4). After his conflict with the administration over securing new classrooms in August of 1882, Klein’s sickness returned with particular severity, and it was also improperly treated (cold baths, gymnastics), as he himself diagnosed.52 Nevertheless, on October 2, 1882, he completed the aforementioned article “Neue Beiträge zur Riemannschen Funktiontheorie” [New Contributions to Riemannian Function Theory]. This was more important to him than his health. At this time, he also could not find a way to accompany Sophus Lie to Paris. Even though Klein would later describe his renewed asthmatic illness as a major break in his life – according to Klein, “the center of his productive thinking had been destroyed”53 – his correspondence from this time does not provide any 49 50 51 52 53

Math. Ann. 21 (1883), pp. 141–218 (dated October 2, 1882). [UBG] Math. Archiv 77: 86 (a letter from Klein to Hurwitz dated December 28, 1882). Quoted from TOBIES/ROWE 1990, pp. 129–30 (Klein to Mayer, March 19, 1882). See Klein’s remarks in JACOBS 1977 (“Vorläufiges über Leipzig”), p. 2. See Hellmuth Kneser, “Felix Klein als Mathematiker,” Mitteilungen des Universitätsbundes Göttingen e.V. 26/1 (1949), pp. 1–6, at p. 3.

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evidence that he suffered a personal “fundamental crisis” or even that he experienced any depression. Rather, he resorted to tried-and-true means that had helped him before: He cut back on his lectures and concentrated all the more on his research seminar. As in his last semester in Munich, he invited the four participants in the seminar to his apartment during the fall of 1882, reclined “in a chair for invalids,”54 and even gave six presentations himself (see Section 5.5.2.3). When Klein claimed that he felt as though he had lost his capability for “productive thinking,” we can only understand this in reference to his own standards. His notes from earlier years repeatedly document that he experienced bouts of self-doubt (see Section 4.4). As we shall see, he never completely ran out of ideas, even when he would partially change his working habits and research topics. As of January 14, 1883, Klein resumed teaching the lecture course that he had handed over to Dyck, and he made plans to shorten his summer semester. With a report from his physician, he applied to go on leave from June 15 to August 15, 1883, in order to recover on the North Sea.55 He decided in advance to teach his lecture course on the theory of equations (Thursday to Friday, 12–1 o’clock, privatim) for only half the semester. Accordingly, the sixteen enrolled students only paid half price (8 Mark).56 Dyck agreed to teach Klein’s seminar during the second half of the term. While on leave, Klein wrote his book on the icosahedron (see Section 5.5.6). In a later letter to the Saxon Ministry of Culture, Klein expressed his thanks yet again for having been granted a two-month period of leave: In general, I have good things to report regarding my health. My leave of absence and the following semester break have done much to put me at ease. In large part, my ailment was caused by the excessive amount of work that I had undertaken earlier. Going forward, I will have to take on less work personally and delegate more to others in a way that seems appropriate to me.57

In offering a special lecture course on elliptic functions,58 it is clear that Klein had already begun to think about his next book in the fall of 1883 (see Section 5.5.7). His eagerness to teach the second part of this course in the summer semester of 1884 comes through in one of his letters to Adolph Mayer. Here he mentioned that he wanted to hand over his introductory lecture on analytic geometry to Karl Rohn, and he explained: “Otherwise, I will have to neglect my own academic work overmuch and split my productivity in two.”59 The noticeable decrease in enrollment numbers in Klein’s lectures (39 in 1883/84; 29 in 1884) was then a nationwide trend in Germany, because by that

54 [UA Braunschweig] 04.2.7 (a letter from Klein to Fricke dated December 13, 1911, in which he explains his feelings about the years 1882 and 1883). 55 [StA Dresden] 10281/184 (Klein’s letter dated March 9, 1883). 56 [UBG] Cod. MS. F. Klein 7 E. 57 [StA Dresden] 10281/184, fols.24–24v (Klein’s letter dated December 9, 1883). 58 Klein’s lectures from 1883/84 (transcribed by Otto Fischer) and summer 1884 (transcribed by Paul Biedermann) are kept in the Mathematical Institute of the University of Leipzig. 59 TOBIES/ROWE 1990, pp. 146–47 (Klein to Mayer, November 19, 1883).

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time secondary schools had become overstaffed with teachers – the main job opportunity for graduates in mathematics. In response, Klein cancelled some of his special lecture courses. In the winter semester of 1884/85, his only lecture course was “Higher Algebraic Curves and Surfaces” (33 students), and, during the first three months of the term (November through January), instead he gave presentations of his own within his seminar on the topic of hyperelliptic functions and Kummer surfaces (16 students participated in the seminar).60 In the summer semester of 1885, he offered the lecture course “Introduction to the Analytic Geometry of the Plane and Space” (41 students). He cancelled his advanced lecture on Abelian functions, choosing instead to focus on this subject in his seminar. According to the official records, Klein offered only one lecture course during his last semester in Leipzig: “Introduction to Differential and Integral Calculus” (25 students). In the third volume of his collected works (GMA), Klein wrote that he “gave his own special lecture on the simplest hyperelliptic functions (p = 2),” and that he turned the content of this lecture into an article for Mathematische Annalen (published in April of 1886).61 In letters to Hurwitz, Klein mentioned that he gave “special presentations” (see Section 5.5.8). Before moving on to Göttingen, Klein continued to supervise doctoral students and support the work of other young and talented mathematicians. 5.3.2 The Mathematical Colloquium / Exercises / Seminar Klein’s research seminars are called various things in the published course listings. The announcement for the first semester of the seminar best captured what it was about: “Discussion of recent scholarly literature alongside guidance for the research activities of more advanced students” (Mondays, 6–8 PM, privatissime but gratis). In the summer semester of 1881, the same course was called a “Mathematical Society” and then, 1881/82, “Exercises in the Mathematical Seminar (Colloquium)”; in the summer semester of 1882, it was referred to as the “Mathematical Seminar,” and after that it was consistently called “Exercises of the Royal Mathematical Seminar.”62 Only for the winter semester of 1883/84 was an additional note added to this: “Exercises of the Royal Mathematical Seminar (Select Chapters of Function Theory), in Collaboration with Dr. Dyck.” For the sake of clarity and brevity, I will use the term seminar in my discussion below. While in Munich, Klein had been able to run this sort of research center in large part together with Alexander Brill, but in Leipzig he had no fitting partner in this regard beyond his assistant Walther Dyck. His fellow professors were either not readily willing to cooperate (Carl Neumann), or their approach to mathematics did not align closely enough with Klein’s own (Mayer, Von der Mühll). 60 [Protocols] vol. 6, pp. 155 and 253. 61 KLEIN 1923 (GMA III), p. 321, Appendix, p. 6. 62 “Royal” (königlich), as related to the Kingdom (Königreich) of Saxony.

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At the time when Klein arrived in Leipzig, the University of Berlin still had higher enrollment numbers in mathematics than Leipzig did.63 Thanks to Klein, however, the number of advanced mathematical theses (doctoral dissertations, Habilitationen) completed in Leipzig would soon surpass the numbers in Berlin. During the first semester of the seminar (1880/81), ten people gave a total of eighteen presentations. In the summer semester of 1881, there were fourteen presentations plus four “Riemann evenings” organized by Klein (June 13, June 20, July 4, and July 11, 1881), about which he noted: “Discussion of § 1–12 of Riemann’s theory of Abelian functions.”64 At the time, Riemann’s ideas were still not widely known. Understanding them, according to Hilbert, “elevated one into a higher class of mathematicians.”65 Klein regarded the subsequent two semesters (1881/82 and 1882) as his best in Leipzig.66 They were a period of extreme activity for him, quantitatively and qualitatively. In the winter 1881/82: presentations at twenty-nine seminar meetings (12 people); an additional cycle of ten presentations by Klein (January 18 to February 8, 1882) plus four more presentations (February 15, February 20, February 27, March 6). In the summer of 1882: seventeen presentations in the seminar (nine participants; he gave three presentations himself). On top of this, he delivered the fifteen special lectures mentioned above (from June to August of 1882). All of this work was done alongside research of the highest quality. This was the time of his three “automorphic fundamental theorems,” which were published from January to October of 1882 (see Section 5.5.4). The situation underwent a drastic personnel change in the fall of 1882. In addition to Gierster, Hurwitz, and Staude, Klein’s assistants (Famuli) Ernst Lange and Oskar Herrmann completed their dissertations67 and became secondary school teachers. Thus there were only four advanced students left to participate in the seminar on Abelian functions (from November 6 to December 19, 1882). Yet, this seminar provides further evidence of Klein’s continued productivity and of the excellent research results achieved by its participants (see Section 5.5.2.3). In the summer of 1883, Klein noted about his new circle of people: “Perpetual change of students.”68 A year later, on June 3, 1884, considering the (new) participants in his seminar (on modular functions), he was skeptical about whether they would produce any valuable results:

63 In the fall of 1879, there were 293 mathematics students at the University of Berlin and 186 in Leipzig. See BECKERT/SCHUMANN 1981, p. 57. 64 [Protocols] vol. 3, pp. 51 and 145. 65 HILBERT 1921, p. 162. 66 See his remarks in JACOBS 1977 (“Vorläufiges über Leipzig”), p. 2; and [Protocols] vol. 2. 67 Both dissertations were published in the Zeitschrift für Mathematik und Physik 28 (1883). Oskar Herrmann’s thesis “Geometrische Untersuchungen über den Verlauf der elliptischen Transcendenten im complexen Gebiete” (ibid. pp. 193–210, 257–73) is noteworthy because it treated a prize problem formulated by Klein in October 1881; this work is missing (was forgotten?) in the list of doctoral theses in KLEIN 1923 (GMA III), Appendix, pp. 11–13. 68 Klein in JACOBS 1977 (“Vorläufiges über Leipzig”), p. 3.

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For the time being, I have to keep in mind how unpleasant the conditions are that, even in the best case, define the intermediate stage of a mathematician’s development: much enthusiasm, little success, and few good ideas. Among the most talented is Dr. [Friedrich] Engel, but he will go to Christiania in the fall to study under [Sophus] Lie.69

Klein’s skepticism proved to be unfounded. This and the following seminar included further participants who would go on to earn doctoral degrees, complete Habilitationen, and become important collaborators. During Klein’s eleven semesters in Leipzig, fifty-six people in all attended his seminars (see Table 6). In the last semester, this group included David Hilbert, whom we could call an intellectual grandchild of Klein, as he had done his graduate research with Klein’s former students Lindemann and Hurwitz (see also Section 6.3.7.3). The students who attended only Klein’s lecture courses are not listed in Table 6, though many of them also became outstanding researchers. Theodor Des Coudres, for instance, attended Klein’s lectures for three semesters (1881–83),70 and later, in 1895, Klein would invite him to come to Göttingen to develop the subject of applied electricity (see Section 7.8). From the winter semester of 1880/81 to that of 1885/86, thirty-six students completed their doctoral studies in Leipzig with a dissertation on mathematics. In the case of twenty-two of these, Klein served as the main advisor. Of these, he considered only the sixteen whose dissertation topics arose from his research seminars to be his doctoral students. (Some of the other doctoral candidates were already employed as teachers and submitted their theses externally.) Furthermore, Klein served as the second reviewer on five other dissertation committees. What these numbers mean can only be measured by comparing them to those of other faculty members. During this same period, the two full professors Wilhelm Scheibner and Carl Neumann supervised only five and three doctoral students, respectively. The astronomer Heinrich Bruns supervised four mathematically oriented dissertations, and the psychologist Wilhelm Wundt was the main advisor for two. Even more informative is a comparison with Berlin, where from 1880 to 1886 only twelve students earned a doctoral degree.71 The era of Kummer (b. 1810), Weierstrass (b. 1815), and Kronecker (b. 1832) was coming to an end. Two Habilitationen were completed in Berlin during this time (Johannes Knoblauch and Carl Runge), while five were completed in Leipzig, one of which in the field of astronomy.

69 [UBG] Math. Archiv 77: 155 (a letter from Klein to Hurwitz dated June 3, 1884). 70 See the list of students in [UBG] Cod. MS. F. Klein 7 E. 71 The supervisors were Weierstrass (six times), Kronecker (four), and Kummer (two). Fuchs, who replaced Kummer in 1884, had his first doctoral student (Lothar Heffter) on August 10, 1886. See BIERMANN 1988, pp. 355–56, 365.

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Table 6: Participants in Klein’s Research Seminars, 1880/81–1885/8672 Nationalities: CH Swiss, Fr French, B British, I Italien, A Austro-Hungarian, R Russian (including the Baltic states and Ukraine), US USA V Attendance in Klein’s lecture course, S/s Participation with/without a presentation in the seminar; F Famulus; A Assistant; E Extraordinarius; O Ordinarius; P Article in Math. Ann.; *ENCYKLOPÄDIE author Name

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Baumgart, O.73 *Brunel, G. Fr Büttner, F. Domsch, P. *Dyck, W. Herrmann, O. Hoppe, H. Hurwitz, A. Kollert, J. Lange, E. Nimsch, P. Olbricht, R. *Staude, O. Stöhr, F. Stringham, I. US Veronese, G. I Weichold, G. Biedermann, P. Böttger, A. Buchheim, A. B Dressler, H. Bobek, K. A *Dingeldey, F. *Fischer, O. Herrmann, T. Höckner, G.74 Kantor, S. A

S

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83

V

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84

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85

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P

PP

PPP

PP E

VS

VS

VS

P

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V

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s P

s P

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s

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72 The table is based on KÖNIG 1982, A8–10; KLEIN 1987, p. 239; [UBG] Cod. MS. F. Klein 7 E; [Protocols] vols. 2–8; the journal Mathematische Annalen; and the ENCYKLOPÄDIE. 73 Oswald Baumgart’s dissertation was titled “Über das quadratische Reciprocitätsgesetz: Eine vergleichende Darstellung der Beweise des Fundamentaltheorems” (1885). An English translation has recently been published: The Quadratic Reciprocity Law: A Collection of Classical Proofs, trans. Franz Lemmermeyer (Basel: Birkhäuser, 2015). For more information on Baumgart, see HASHAGEN 2003, p. 126. 74 Höckner completed his doctorate in 1891 under the supervision of Bruns and Scheibner.

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5.3 Teaching Program Name *Papperitz, E. Wirtz, K.75 Wiener, H. Engel, F. *Krazer, A. *Study, E. Friedrich, G. Gerbaldi, F. I *Fricke, R. Krieg v. Hochfelden, F. Molien, T. R Morera, G. I Pick, G. A Reichardt, W. Cole, F. US Fiedler, E. CH Fine, H. US *Hölder, O. Tikhomandritskiy, M. A. R Cornelius, H. Raussnitz,76G. A Richter, O. Struve, L.77 R Waelsch, E. A Weiß, W. A Ameseder, A. A Hildebrand, R. Witting, A.78 *Hilbert, D.

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Vs P

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S P S PP

Most of the participants in Klein’s seminar came from Saxony and went on to become secondary school teachers; some taught at state institutions (Staatsanstalten) in Dresden (Alexander Witting) or Chemnitz (Paul Domsch, Heinrich Hoppe). Erwin Papperitz held professorships at the Technische Hochschule in

75 Karl Wirtz followed Klein to Göttingen. In 1894, he became a professor of electrical engineering at the TH Darmstadt, where he established the field of communications engineering. 76 Also known as Gusztáv Rados. 77 From St. Petersburg, Ludwig von Struve belonged to a famous family of astronomers. 78 In 1886, Alexander Witting submitted his dissertation to Klein in Göttingen.

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Dresden and at the Mining Academy in Freiberg. Friedrich Dingeldey, who came from Darmstadt, became a professor at the Technische Hochschule there, and Robert Fricke, who was born near Braunschweig, became a professor at the Technische Hochschule in that city (he was Dedekind’s successor). Ernst Lange, Klein’s first Famulus, worked as a school director before becoming an official in the Saxon Ministry of Culture.79 Some of the seminar’s participants (A. Böttger, H. Dressler, A. Witting) would make contributions to Klein’s five-volume book project Abhandlungen über den mathematischen Unterricht in Deutschland [Treatises on Mathematical Education in Germany],80 and eleven of them would contribute as authors to the ENCYKLOPÄDIE. 5.4 THE KLEINIAN “FLOCK” As he had done elsewhere, Klein gathered his students closely around him and welcomed them into his family. In May 1884, Otto Hölder spoke of the “Kleinian flock,” which he himself was at first reluctant to join: There are 17 of us altogether in the seminar and everyone knows one another. Among these are 5 doctors, three of them older than I. Aside from the fact that one meets in the work rooms all the time, everyone also gets together on Monday evenings. […] Some of the mathematicians also eat together, but I have avoided this from the first because I want to stay in contact with my friends. This is probably all right: the opportunity to communicate with my peers from mathematics is very nice, but I still don’t want to be entirely swallowed by the Kleinian flock [Heerde].81

Hölder had come to Leipzig with many prejudices from Berlin. Ultimately, however, he was unable to resist Klein’s manner of both encouraging and promoting people. On May 20th, he informed his parents that he had grown closer to Klein, that he had visited him at home, and that he “had been introduced to Mrs. Klein and invited on a boat tour the next day,” which ended in a café in the Rosental park. Klein advised him about how to proceed with his Habilitation and told him that he would be the right man to write the sorely needed textbook on differential and integral calculus.82 Admiringly, Hölder wrote to his parents: “With Klein, there are always interesting people who come to ‘pay him respect’ […]: Mr. Tikhomandritskiy from Kharkiv and Mr. Eneström from Uppsala, both Dozenten.”83 In what follows, I will take a closer look at those members of Klein’s “flock”, who would complete a Habilitation with Klein’s encouragement, and I will also provide profiles of the students who came to study under him from abroad. 79 80 81 82

See LOREY 1916, p. 168. Prompted by the IMUK (ICMI) and edited by Felix KLEIN 1909–1916. Quoted from HILDEBRANDT et al. 2014, p. 144 (Hölder to his parents, May 6, 1884). See ibid., p. 147. Hölder would never write a book of this sort. Klein’s former student Axel Harnack would publish a revised edition of Joseph Serret’s book; see Appendix 2. 83 Ibid., pp. 152–53 (Hölder to his parents, June 24, 1884). Regarding Tikhomandritskiy, see Section 5.4.2.5; on Eneström, see Section 5.6.

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5.4.1 Post-Doctoral Mathematicians In Leipzig, Felix Klein was able for the first time to supervise Habilitation procedures himself. However, new problems arose, so that he once again felt as though he had to recommend the best of his students to other universities. Klein served as the main reviewer in four Habilitation procedures at the University of Leipzig, and he paved the way for five additional successful procedures at other universities. In a letter to Friedrich Althoff, an influential official at the Prussian Ministry of Culture, Klein listed the mathematicians who had attended his seminars and then gone on to have academic careers. The list included the post-doctoral students who completed their Habilitation between 1882 and 1885: Walther Dyck, Friedrich Engel, and Eduard Study in Leipzig, as well as Adolf Hurwitz (Göttingen), Otto Hölder (Göttingen), Adolf Krazer (Würzburg), Otto Staude (Breslau), and Hermann Wiener (Halle).84 In 1881, Friedrich Schur completed his Habilitation in Leipzig; although he was not Klein’s student, Klein served as a reviewer. Friedrich Schur had earned his doctoral degree in 1879 under Kummer’s supervision in Berlin with a thesis that drew upon the methods of Plücker, Klein, and others. Schur had served as Weierstrass’s “blackboard writer”85 and had originally planned to complete his Habilitation under Weierstrass’s student H.A. Schwarz in Göttingen. After a short time, however, he left Schwarz on account of the latter’s rather difficult personality,86 with which Klein and Hurwitz also would have first-hand experience. The Leipzig professor Wilhelm Scheibner agreed that Schur could submit his Habilitation thesis – “Über die durch kollineare Grundgebilde erzeugten Kurven und Flächen” [On the Curves and Surfaces Created by Collinear Basic Figures] – in Leipzig on October 10, 1880. The recently hired Klein acted as a reviewer of this thesis, and he described Schur’s synthetic-geometric approach as being of “systematic importance.” Klein remarked further: “The author was not content simply to formulate general principles or to derive already known results in a different way (as is so often the case in works on synthetic geometry); rather, he succeeded in discovering truly new theorems.”87 Klein was able to coordinate the curriculum with Schur, who was open to his suggestions. Thus Klein, after Dyck’s departure, made Schur his assistant, before Eduard Study would boldly ask for this position.88 Klein arranged for Friedrich Schur to become an associate professor in Leipzig on May 19, 1885.89 As of 1888, Schur would hold full professorships in Tartu, Aachen, Karlsruhe, Straßburg (until 1918), and Breslau.

84 85 86 87 88 89

[UBG] Cod. MS. F. Klein, I B, fol. 23 (Klein to Althoff, October 1, 1885). Ludwig Kiepert had held this same position (see Section 2.5.2). See Friedrich Engel, “Friedrich Schur,” Jahresbericht DMV 45 (1935), pp. 1–31, esp. p. 6. [UA Leipzig] PA 967 (Klein’s evaluation dated November 22, 1880). [StA Dresden] 10281/184, fol. 23b. [StA Dresden] 10281/276.

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In a letter of recommendation for a job opportunity (Marburg, 1892), Klein considered Schur to be a stronger candidate than Walther Dyck. Dyck’s works, he wrote, were “neither numerous, and nor did they meet with more than modest success.” As a teacher and organizer, however, he was “entirely incomparable.” He ranked Schur between Friedrich Schottky and Dyck, and he regarded him as “a respected theoretician and a talented teacher with a level-headed personality”.90 Walther Dyck was Klein’s first doctoral student whose Habilitation research he could supervise himself. Dyck submitted the work on November 11, 1881, but he immediately encountered a problem that did not exist at Bavarian or Prussian universities. According to §1a of the University of Leipzig’s statutes governing Habilitation procedures, it was necessary for candidates to have a secondary school diploma from a humanistic Gymnasium,91 whereas Dyck held his from a Realgymnasium. Many members of the Philosophical Faculty, which included the humanities, natural sciences, and mathematics, did not want to deviate from this rule. Klein had to employ his full power of persuasion to convince eleven faculty members to change their votes. Eight others maintained their dissenting opinion.92 Subsequent candidates who likewise held diplomas from a Realgymnasium (Adolf Hurwitz, Otto Hölder, Adolf Krazer, Hermann Wiener) had to submit their Habilitation theses at other universities. An exception was made for Dyck, who was allowed to submit his thesis – “Gruppentheoretische Studien” [Group-Theoretical Studies] – in Leipzig. In the colloquium held on January 25, 1882, Dyck answered Klein’s questions on group theory in a satisfactory manner but also revealed certain gaps in his understanding of other areas.93 On February 8, 1882, Dyck received the venia legendi for mathematics. By the end of 1883, he was offered professorships by the Technische Hochschule in Hanover and the Technische Hochschule in Munich. On April 1, 1884, he accepted the position in Munich, which was his hometown. In 1887, when Klein sought to reduce his editorial role at Mathematische Annalen, Gordan wrote to him: “Dyck is the last person on my list; I have no concerns about his drive, but I have some reservations concerning his lack of knowledge and achievements.”94 Klein nevertheless appointed Dyck to serve as his “organizational” assistant, while he himself continued to oversee the content of the journal and to find appropriate peer reviewers for submissions. Dyck remained for the rest of his life at the Technische Hochschule in Munich, and he supported Klein in further academic and organizational undertakings. After earning his doctoral degree, Adolf Hurwitz spent one more semester with Klein in Leipzig, one further semester in Berlin, and he was able to complete his Habilitation in Göttingen in May of 1882. Although Klein’s name does not

90 91 92 93 94

[UBG] Cod. MS. F. Klein 1C: 2, fols. 44–44v (Klein to Althoff, May 25, 1892). [UA Leipzig] PA 425. See HASHAGEN 2003, p. 119; and SCHLOTE 2004, p. 69. See HASHAGEN 2003, p. 221. [UBG] Cod. MS. F. Klein 9: 546, fol. 64 (Gordan to Klein, November 22, 1887).

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appear anywhere in Hurwitz’s Habilitation records, it was Klein who made it possible for him to earn the support of the Berlin-based mathematician Leopold Kronecker (see also Section 5.5.2.1). Kronecker knew how to win over the full professors in Göttingen, Ernst Schering and H.A. Schwarz. That is, Kronecker dissuaded Hurwitz from choosing a number-theoretical topic for his Habilitation thesis – noting that “Schwarz does not have a number-theoretical bone in his body” – and made the following recommendation instead: “Create a description of transcendental functions on hyperelliptic figures as products of Weierstrass’s prime functions.” Hurwitz had already worked out this problem, though he was somewhat less interested in this topic.95 He consulted with Klein and agreed to follow Kronecker’s advice.96 During a visit to Göttingen, moreover, Klein succeeded in talking Schwarz out of the idea that Hurwitz should first have to pass a teaching examination before he could earn his Habilitation.97 The rather peculiar behavior of the “fat, unctuous Hermann Amandus Schwarz” – as Albert Einstein later called him in a letter to Hurwitz – was widely known among his contemporaries.98 In his Habilitation application from April 24, 1882, Hurwitz also included his Leipzig dissertation (and five articles), but – apparently for tactical reasons – he did not mention Klein explicitly. He referred only to his studies with Weierstrass and Kronecker in Berlin.99 The title of his Habilitation thesis was “Über die Perioden solcher eindeutiger 2n-fach periodischer Functionen, welche im Endlichen überall den Character rationaler Functionen besitzen und reell sind für reelle Werthe ihrer n Argumente” [On the Periods of Such Single-Valued 2n-Fold Periodic Functions that, at Finite Places, Possess Throughout the Character of Rational Functions and Are Real for Real Values of Their n Arguments]. H.A. Schwarz, who was tasked with reviewing Hurwitz’s work, examined not only the Habilitation thesis, which was closely related to Weierstrass’s ideas, but also the doctoral dissertation. Unlike Hilbert (see Section 4.2.4.2), Schwarz did not acknowledge Klein’s contribution to the field. Rather, he stated that the text deals “with matters to which a large number of mathematicians have devoted themselves in recent years – an area of research that essentially has as its object an appropriate generalization of Jacobi’s concept of the module and Jacobi’s modular and multiplier equations.” But Schwarz rightly judged Hurwitz to be “a young man with an unusual gift for scientific mathematical research.”100 In addition to Schwarz, the committee also included the physicist Eduard Riecke (a friend of Klein’s) as well as Wilhelm Weber, J.B. Listing, and M.A. Stern, so it was otherwise stacked in Hurwitz’s favor.

95 96 97 98 99 100

[UBG] Cod. MS. F. Klein 9: 908 (a letter from Hurwitz to Klein dated January 29, 1882). [UBG] Math. Arch. 77: 59, 60 (Klein to Hurwitz, February 18, 1882; and February 22, 1882). Ibid. 77: 61 (Klein to Hurwitz, March 19, 1882). EINSTEIN 1998, p. 13 (a letter to Hurwitz dated May 4, 1914). [UAG] Phil. Fak. 167a, VIII 4e. [UAG] Phil. Fak. 167a, VIII 4b.

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Hurwitz’s colloquium took place on May 11, 1882, and his public lecture in defense of his Habilitation – “Über die Methoden der neueren Geometrie” [On the Methods of Newer Geometry] – was held on Saturday, May 13th, at noon.101 Ernst Schering, then the Dekan of the Philosophical Faculty, sent a positive report about it to the Prussian minister of culture Gustav von Goßler in Berlin. Hurwitz taught successfully for four semesters, so that, in December of 1883, Stern, Schering, and Schwarz applied for him to receive a Privatdozent stipendium of 1,500 Mark.102 This application helped Hurwitz’s career in Prussia, though only to the extent that he was soon offered a low-paying associate professorship instead. Thus the stipendium did not have to be paid. On April 1, 1884, Hurwitz received a preliminary (nicht-etatmäßig) lecturership at the University of Königsberg with the prospect of soon replacing Johann Georg Rosenhain, who was in poor health, as an associate professor.103 Klein’s former doctoral student Lindemann, who was now full professor in Königsberg, had proposed this at Klein’s instigation.104 On May 25, 1885, Hurwitz informed Klein without any further commentary: “I have recently become salaried; I will receive a 2,000 Mark salary, including an extra 600 Mark for living expenses, although the real budget of this position is 3,660 Mark.”105 Hurwitz would have to be content with such a low annual salary. In Königsberg, Hurwitz impressed the rising stars David Hilbert and Hermann Minkowski with the breadth of his knowledge, which surpassed that of Ferdinand Lindemann. In Hilbert’s judgement, Hurwitz combined the knowledge of “two schools that complement each other excellently: Klein’s geometric school and the algebraic-analytic school in Berlin.”106 For a long time, Hurwitz remained Klein’s most important mathematical contact person. Despite writing glowing recommendations for him, however, Klein failed to secure a full professorship for him in Germany. The main reason for this was the prevalence of anti-Semitism among the members of hiring committees (see also Appendix 6.1 and 6.2).107 Otto Staude had been studying in Leipzig since 1876,108 but it was not until Klein arrived and supervised him that he was able to produce results suitable for a dissertation: “Über lineare Gleichungen zwischen elliptischen Coordinaten” [On 101 [UAG] Kur. 6216, fols. 7–8. 102 Ibid., fols. 9–10. In 1883/84, Hurwitz had fourteen students in his course on number theory and eighteen students in his course on surfaces of the second order. 103 Ibid., fols. 1–6 (a letter dated January 26, 1884). 104 [UBG] Cod. MS. F. Klein 9: 954, 956, 961 (Hurwitz’s letters to Klein). 105 Ibid. 9: 999 (a letter from Hurwitz to Klein dated May 25, 1885). 106 David Hilbert, “Adolf Hurwitz,” Math. Ann. 83 (1921), pp. 161–72, at p. 163. 107 For example, Klein recommended both Max Noether and Adolf Hurwitz for a professorship in Hanover, but Ludwig Kiepert (a professor there) wrote to him on January 25, 1884: “I would not have succeeded here in hiring a Jew” ([UBG] Cod. MS. F. Klein 10: 103). Klein also recommended Hurwitz for a professorship at the Technische Hochschule in Dresden, and wrote to Karl Rohn there: “Hurwitz’s lecture is like the style of his academic work: well thought-through, clear, and insightful” ([StA Dresden] 1020/17, Klein to Rohn, May 3, 1888). 108 See Friedrich Schur, “Nachruf auf Otto Staude,” Jahresbericht DMV 40 (1930), pp. 219–22.

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Linear Equations Between Elliptic Coordinates], which he submitted on March 13, 1881. Just a year later, Staude developed a thread construction of an ellipsoid that Klein thought was one of the most elegant results ever to come out of his seminar.109 Staude’s construction made use of a fixed framework consisting of the ellipse and a hyperbola.110 Klein considered the result so important that he had Scheibner present it as early as March 6, 1882, at a meeting of the Royal Society of Sciences in Leipzig (Klein himself was not yet a member; see Section 5.7.3). A longer version of Staude’s work appeared in Mathematische Annalen, where the author thanked Klein in particular for pointing out to him the geometric significance of the differential equations that he used.111 Staude participated in Klein’s four-person seminar at the end of 1882 (see Section 5.5.2.3). It was in this context that he conceived of his Habilitation thesis, which he was able to submit to the University of Breslau in 1883: “Geometrische Deutung der Additionstheoreme der hyperelliptischen Integrale und Functionen 1. Ordnung im System der confocalen Flächen 2. Grades” [The Geometric Interpretation of the Addition Theorems of the Hyperelliptic Integrals and Functions of the First Order in the System of Confocal Surfaces of the Second Degree].112 Staude trusted that Klein would continue to support his future career.113 In a letter to Axel Harnack, Klein praised Otto Staude’s extraordinary knowledge of scholarly literature, his excellent work on hyperelliptic functions, and his valuable contributions to the edition of Möbius’s work (see Section 5.7.3).114 In 1886, Staude became an associate professor in Tartu, and he was hired as a full professor in Rostock in 1888. For the ENCYKLOPÄDIE (vol. III, Geometry), he wrote the article on surfaces of the second order (1904). The next Habilitation candidate, Adolf Krazer, had earned his doctoral degree under Friedrich Prym in Würzburg in 1881; the title of his thesis was “Theorie der zweifach unendlichen Thetareihen auf Grund der Riemannschen Thetaformel” [The Theory of Twofold Infinite Theta Series on the Basis of Riemann’s Theta Formula]. He had done some postdoctoral research under Weierstrass and Kronecker in Berlin. Yet the topic of his Habilitation thesis did not come to him until he moved to Leipzig.115 In Klein’s seminar in the summer of 1882, Krazer analyzed Paul du Bois-Reymond’s results concerning Fourier integrals.116 He participated in Klein’s four-person seminar in the winter of 1882, and he submitted his Habilitation thesis – “Über Thetafunctionen, deren Charakteristiken aus Dritteln ganzer Zahlen gebildet sind” [On Theta Functions Whose Characteristics Are Formed from Thirds of Whole Numbers] – to the University of Würzburg in 1883. 109 110 111 112 113 114 115 116

See KLEIN 1923 (GMA III), p. 321; and [Protocols] vol. 3, pp. 141–42. Staude’s thread construction is explained in HILBERT/COHN-VOSSEN 1952 [1932], pp. 19–24. Otto Staude, “Ueber Fadenconstruktionen des Ellipsoids,” Math. Ann. 20 (1882), pp. 147–84. The thesis was published in Math. Ann. 22 (1883), pp. 1–69, 145–76. [UBG] Cod MS. F. Klein 11: 1124 (a letter from Staude to Klein dated December 28, 1884). [StA Dresden] 10210/17 (a letter from Klein to Axel Harnack dated January 15, 1885). See Karl Boehm, “Adolf Krazer,” Jahresbericht DMV 37 (1928), pp. 1–33, esp. p. 14. [Protocols] vol. 4, 22–30, 69–82, 133–46, 207–18 (Krazer’s seminar presentations).

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In this thesis, he remarks: “In the present work, which owes its genesis to inspirations gained from my personal interactions with Professor Klein […].”117 Krazer became a professor in Strasbourg and Karlsruhe. He remained in close contact with Klein. They cooperated as members of the German Mathematical Society, and in 1920-22 still worked together in the mathematics division of the Emergency Association of German Science (see 9.4.1). For the ENCYKLOPÄDIE, Krazer wrote the article on Abelian functions and general theta functions.118 On June 23, 1884, Otto Hölder applied for permission to submit his Habilitation thesis in Göttingen, after Klein had steered him toward an appropriate topic. In the vita appended to this text, Hölder referred to the experience that he gained from participating in Klein’s mathematical seminar.119 This Habilitation thesis consisted of Hölder’s doctoral dissertation – “Beiträge zur Potentialtheorie” [Contributions to Potential Theory] – and four additional studies (two from Mathematische Annalen and two unpublished articles). H.A. Schwarz based his official review primarily on Weierstrass’s and Kronecker’s positive opinion of Hölder’s work. After Hölder’s colloquium on July 17, 1884, and after his defense lecture – “Ueber eine Methode, gewisse Grenzübergänge nach einer allgemeinen Regel elementar-geometrisch zu behandeln” [On a Method to Treat Certain Limiting Processes According to a General Rule in an Elementary Geometric Way] – Hölder received the venia legendi for mathematics on July 23rd.120 In comparison to Hurwitz’s, Hölder’s lectures as a Privatdozent in Göttingen were not very well attended: 1884/85 algebraic equations (2 students); 1885 differential equations (5 students), theory of series (Reihentheorie, 1 student), and determinants (11 students); 1885/86 algebraic analysis (12 students); and in 1886 the theory of definite integrals (4 students). Nevertheless, on August 12, 1886, the Philosophical Faculty applied to promote Hölder to the position of associate professor to replace the late Alfred Enneper.121 On April 1, 1887, however, the Ministry of Culture approved no more than a one-year Privatdozent stipendium of 1,500 Mark, with a chance for an extension. Hölder was not promoted to associate professor until April 1, 1889, with a yearly salary of 1,800 Mark. In October of 1889, he accepted a professorship at the University of Tübingen.122 Hölder was a strong individualist. In letters to his parents, he wrote critical remarks about both Schwarz and Klein. When Klein left for Göttingen in 1886, Hölder continued teaching as Privatdozent there. He refused to participate with Klein in a joint colloquium, for he feared that he might lose his “independent character as a teacher”123 (see Section 6.2.3). With the ENCYKLOPÄDIE project, however, Hölder did agree to play along. He not only wrote the article on Galois theory 117 118 119 120 121 122 123

Math. Ann. 22 (1883), pp. 416–49, at p. 417. It is published in vol. II.2 (Analysis), pp. 604–873. The entry is dated December 5, 1920. [UAG] Philos. Fak. 170a (1.7.1884–1885), p. 34n. [UAG] Kur. 5970, pp. 1–3. [UAG] Philos. Fak. 172a, Nos. 75a, 75c–75e. [UAG] Kur. 5970, pp. 4–31. HILDEBRANDT et al. 2014, p. 216 (a letter from Hölder to his parents dated May 6, 1886).

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and its applications for vol. 1 (Arithmetic and Algebra, 1899); he also became a member of the ENCYKLOPÄDIE’s editorial committee (see section 7.4). Hermann Wiener was a son of the mathematician Christian Wiener, whose model of cubic surfaces had inspired Klein when he was a student (see Section 2.8.3.4). Hermann Wiener began his doctoral studies in Munich in 1879. After finishing his doctoral degree there in 1881 – the title of his dissertation was “Über Involutionen auf ebenen Curven” [On Involutions on Plane Curves] – he moved to Leipzig to continue his studies with Klein. He gave presentations in Klein’s seminar on conformal mapping (December 12, 1881) and on the works of Georg Cantor (July 10, 1882), who would ultimately, in 1885, serve as the reviewer of Wiener’s Habilitation thesis in Halle. The title of this thesis was “Rein geometrische Theorie der Darstellung binärer Formen durch Punktgruppen auf der Geraden” [A Purely Geometric Theory of the Representation of Binary Forms by Means of Groups of Points on a Straight Line]. In the meantime, Wiener was also active in Karlsruhe, where he served as his father’s assistant. In 1894, after many years of working as a Privatdozent in Halle, he received a professorship at the Technische Hochschule in Darmstadt. He made especially important contributions to the foundations of geometry. In 1890, Wiener was, like Cantor and Klein, one of the founding members of the German Mathematical Society (see Section 6.4.4). Friedrich Engel’s career path was shaped by the fact that Felix Klein and Adolph Mayer sent him, with a Kregel-von-Sternbach scholarship,124 to study under Sophus Lie in Christiania (Oslo). Engel had been studying in Leipzig since 1879, and Adolph Mayer had already introduced him to Lie’s work. Engel’s dissertation – “Zur Theorie der Berührungstransformationen” [On the Theory of Contact Transformations]125 – was submitted on May 7, 1883. It was officially evaluated by Felix Klein and Scheibner because Mayer (Engel’s true supervisor) was not a full professor and was therefore not permitted to do so. Klein sent Engel’s dissertation to Sophus Lie and notified him at the end of 1883 that “help” was on the way.126 Shortly before, Lie had written the following to Klein: For several years, I abandoned transformation groups and differential equations and concerned myself with geometry. Now I am turning back to them, and I regret, in a way, that I put my most important studies aside, and they therefore remained unnoticed. For each year that passes I become more certain that the theory of transformation groups will shed completely new light on the theory of differential equations. If I could only collect and edit all my results!127

In the summer of 1884, after Engel had completed his doctoral degree, he attended Klein’s lectures on elliptic functions, and he gave presentations in the related seminar. Because his trip to Norway was already planned at this point, Klein

124 Karl Friedrich Kregel von Sternbach, the last male descendent of a noble family, donated a large part of his fortune to the University of Leipzig. 125 This thesis was published in Math. Ann. 23 (1884), pp. 1–44. 126 See also STUBHAUG 2002, p. 310. 127 Quoted from STRØM 1992, p. 8.

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made sure that Freidrich Engel was familiar with all the scholarly literature that he should know (works by Jordan, Sylow, Kronecker, and so on). At the same time, it is clear from Engel’s entries in Klein’s protocol book that he had studied Klein’s lectures closely. In an entry dated July 25, 1884, for instance, Engel noted: “In his course on elliptic functions on July 24, 1884, Prof. Klein presented a very elegant proof of the Abelian relations without the use of series and without second-order functions.”128 In Klein’s eyes, Friedrich Engel was a “ray of hope” (Hoffnungsstrahl), as he put it in a letter to Hurwitz. It was expected that Engel would assist Sophus Lie, who hoped at the time “to compile all my studies on transformation groups into a single book.”129 While in Norway, where Engel arrived in September of 1884, he also produced important results for his Habilitation thesis, “Über die Definitionsgleichungen der continuierlichen Transformationsgruppen” [On the Definition Equations of Continuous Transformation Groups], which Klein evaluated on May 15, 1885. Lie was impressed with Engel’s originality, and he stressed that Engel’s thesis offered “a new general method for determining all continuous groups.”130 Drawing from the judgments of Lie and Mayer, Klein heartily endorsed “the acceptance of the candidate’s Habilitation achievements, and all the more so because the candidate’s earlier publications have already made it easy to recognize that he is a talented and diligent mathematician.”131 Engel completed his Habilitation process on October 26, 1885, with a lecture entitled “Anwendung der Gruppentheorie auf Differentialgleichungen” [The Application of Group Theory to Differential Equations]. Even after Lie had moved to Leipzig in 1886 (see Section 5.8.3), Engel remained his right-hand man throughout the production of Lie’s book Theorie der Transformationsgruppen (3 vols., Leipzig: B.G. Teubner, 1888/1890/1893), where Engel’s assistance is acknowledged on the title page. In 1889, Engel became an associate professor in Leipzig. In 1892, when Klein asked him whether he might want to be the main editor of the works of Hermann Graßmann, Engel did not hesitate to take charge of the project (see Section 5.7.3). After Sophus Lie had returned to Norway, Engel was made an honorary professor in Leipzig. Later, he held full professorships in Greifswald (1904) and Gießen (1913), and he collaborated in the edition of Leonhard Euler’s Opera omnia. Eduard Study completed his Habilitation procedure in Leipzig one day after Engel had done the same. Unlike the other postdoctoral researchers profiled here, Study was an autodidact and an egocentric character who was demanding but gave little in return. This necessarily led to controversies, and Study was involved in many of them, with Klein and others.132 He came to Leipzig after studying zoology under Ernst Haeckel in Jena and mathematics under Theodor Reye in

128 129 130 131 132

[Protocols] vol. 6, pp. 63–75, 90–100, here at p. 100. [UBG] Cod. MS. F. Klein 10: 704, 706/1 (Lie to Klein, in the end of 1884). Ibid. 10: 707 (an undated letter, sent in 1885, from Lie to Klein). [UA Leipzig] PA 436, fols. 4, 4R, here at fol. 4R. See the mathematician Wolfgang KRULL 1970, and the thesis of Yvonne HARTWICH 2005.

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Strasbourg. In the summer semester of 1882, he attended Klein’s lecture course and gave a presentation in Klein’s seminar (on the graphical illustration of Fourier series, May 1, 1882).133 Impressed by this, Klein considered him qualified to edit his aforementioned fifteen special lectures (see KLEIN 1883). Because Klein canceled his advanced lecture course in 1882/83 and only taught his seminar, Study left in order to continue his work in Munich in the fall of 1882. While there, he remained in contact with Klein. Against Klein’s wishes, however, Study submitted his dissertation, which drew upon Graßmann’s Ausdehnungslehre (theory of extension), to Gustav Bauer and Ludwig Seidel at the University of Munich. Admittedly, he acknowledged Klein’s influence in the vita appended to his thesis: “I am obligated to express special thanks to Prof. Klein for much personal inspiration and support.”134 However, he overestimated himself and believed that Klein would publish his entire dissertation in Mathematische Annalen and would choose him to be his assistant in Leipzig. In both cases, he was wrong.135 He nevertheless counted on Klein’s support during the next stages of his career.136 Klein recommended that Study should continue working on a topic that he had already begun to examine while at the Technische Hochschule Munich: the application of the calculus of enumerative geometry (abzählende Geometrie, as established by Chasles, Halphen, Schubert, and others) to the problem of fourth-order space curves. This research culminated in Study’s Habilitation thesis. From today’s perspective, the topic was ambitious but not generally solvable with the methods available at the time. As already mentioned in connection to Hermann Schubert (see Section 2.4.1), Hilbert based Problem 15 on this very issue in his Paris lecture (HILBERT 1900). It would not be solved until the late 1920s, when Bartel Leendert van der Waerden found a general solution with new topological methods. The title of Study’s Habilitation thesis was “Ueber die Geometrie der Kegelschnitte, insbesondere deren Charakteristikenproblem” [On the Geometry of Conic Sections, Particularly the Problem of Their Characteristics].137 Klein recognized Study’s academic achievement in his long official review of the work, which is dated July 6, 1885. Here he described Study’s point of departure, which was the general question, first posed by Chasles some twenty years before, “concerning the number of conic sections that satisfy five given conditions,” and he noted how Chasles had reduced the question “to simple but unproven principles.” Making use of the work of Schubert, Halphen, and others, Study “redirected the question and asked for those enumerative rules (Abzählregeln) that must be taken as a basis if Chasles’s formulae are to be maintained in all cases.” Klein went on:

133 134 135 136 137

[Protocols] vol. 1, pp. 1–5. Quoted from HARTWICH 2005, p. 49. [StA Dresden] 10281/184, p. 22. On this topic, see HARTWICH 2005, pp. 53–60 (an analysis of Study’s letters to Klein). The thesis was published in Math. Ann. 27 (1886), pp. 58–101.

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5 A Professorship for Geometry in Leipzig In this formulation, one sees a characteristic tendency – choosing mathematical definitions in such a way that general theorems remain valid; this is also the case with respect to his approach. The candidate follows the concepts and definitions of invariant theory (Gordan), which he combines with Grassmann’s ideas in a way that leads to a special and very concise representation. I attach less importance to the form of this representation than to the fact that it permits the studies on conic sections to be transferred to other simple geometric entities, such as triples or quadruples of points on a straight line, whereby a broader perspective is gained that the author intends to pursue in later publications. At the same time, his representation of well-known theories yields a number of interesting details, which I will only point out parenthetically […].138

After this positive evaluation, Klein critiqued some formal aspects of the work, which would have to be corrected before its publication, and then he endorsed “permitting the candidate to advance to the final stages of the Habilitation procedure.” In the end, however, he could not refrain from making a personal remark: Dr. Study has an independent and sensitive nature. If he could learn to subordinate, more than hitherto, his subjective impulses to the requirements of a given situation and if, furthermore, his health will allow him to master the efforts needed to penetrate deeper into the essential problems of our science, then I hope that his Habilitation will lead to a fundamental advancement of the mathematical studies conducted at our university.139

Klein helped Study to prepare his Habilitation thesis for publication, and he recommended that he should take a study trip to Paris. Hilbert, who traveled with him, wrote to Hurwitz about Study’s headstrong behavior.140 Study repeatedly rubbed people the wrong way, and he never shied away from conflict. As the editor of Mathematische Annalen, Klein found himself involved in Study’s polemics with various colleagues (Halphen, Zeuthen, Cayley). Klein attempted to mediate and smooth matters over. He ultimately attested that Study was “unwaveringly opinionated” and “intolerant” of other people’s points of view, and he was tired of Study holding him responsible for his own precarious position.141 As a Privatdozent, Eduard Study relocated from Leipzig (Saxony) to Marburg (Prussia) in 1888, and afterwards he tried his luck in the United States. He reunited with Klein at the Chicago World’s Fair in 1893 (see Section 7.4.1), and – unknown until now – it was Klein who, by writing the following favorable evaluation to the Prussian Ministry of Culture, facilitated Study’s return to Germany: Regarding Study, whom I have seen every day for the past three weeks, I have very good things to report. The congress and the subsequent colloquium here have given him the opportunity to forge many personal relationships with other scholars and have allowed his own academic brilliance to shine. For my part, I can add that, in terms of creative power, Study is one of our best young people and that he ranks just beneath Hilbert and Minkowski.142

138 139 140 141 142  

[UA Leipzig] PA 993, pp. 6–7. Ibid., p. 7. See REID 1970, p. 20. For more details about their relationship, see HARTWICH 2005, pp. 60–63, 73–86. [UBG] Cod. MS. F. Klein 1 C 2, fol. 70 (a draft of a letter from Klein to Althoff, dated September 12, 1893). In light of this document, it is clear that the following widely used com-

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In Prussia, Eduard Study would go on to hold an associate professorship (Bonn, 1894) and two full professorships (Greifswald, 1897; Bonn, 1904). He never acknowledged Klein’s advocacy on his behalf, and he continued to stir up conflict. He even caused trouble with his ENCYKLOPÄDIE article on the theory of general and higher complex quantities (vol. 1, pp. 147–83). It took him two years to complete this relatively short article, and he was unwilling to accept any of the changes suggested by Franz Meyer, Hilbert, and Klein. Ever overestimating his own abilities, Study was outraged by the fact that Klein had recommended that he cite one of his own articles: “It’s unbelievable how shamelessly Klein is using this encyclopedia for purposes of self-promotion.”143 Klein’s critique was justified, however, as Study’s article failed to mention certain researchers important in the field, such as Hamilton, who had explained complex numbers as pairs of real numbers and introduced the quaternions.144 Later, Study admitted that he had occasionally treated Klein unfairly, for Klein had possessed knowledge that he himself had only recently come to understand.145 In the end, Study numbered among the many donors who made it possible for Max Liebermann to paint Klein’s portrait (see Fig. 40, and Appendix 10, Fig. 43). Study achieved results that have been described as “the most original and significant contributions to geometry in the spirit of the Erlangen Program.”146 Nevertheless, he continued to act polemically toward his colleagues, including Klein and (later) Hermann Weyl. In an article from 1949 (long after Study’s death), this motivated Weyl’s blunt and rather exaggerated description of Study as “Felix Klein’s contemporary and life-long enemy.”147 5.4.2 Klein’s Foreign Students in Leipzig Because of the central location and the status of the University of Leipzig, and as a result of Klein’s international contacts, a growing number of students came from abroad to study under the young researcher. Just as Sophus Lie, knowing that they would be well supported, had already encouraged Scandinavian students not to go to Berlin but to work with Klein instead, colleagues from France, Great Britain, Italy, Austria-Hungary, Switzerland, and the United States sent some of their students Klein’s way as well.

 

143 144 145 146 147

ment made by the mathematician Gerhard Kowalewski in his autobiography is off the mark: “Of course, Klein never recommended [Eduard] Study for any position” (KOWALEWSKI 1950, p. 140). Quoted from HARTWICH 2005, pp. 96–97. On this topic, see HASHAGEN 2003, p. 453. See HARTWICH 2005, pp. 106–07. Ibid., pp. 106–12. See also ZIEGLER 1985, p. 102. WEYL 1949, p. 535. On the feud between Study and Weyl, see HARTWICH 2005, pp. 129–34.

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5.4.2.1 The First Frenchman and the First Briton Georges Brunel, Klein’s first French student, came to Leipzig for two semesters (beginning with the winter semester of 1880/81), after he had passed the Agrégation (a teaching examination for those seeking careers in secondary schools) at the École Supérieure. Klein’s first British student, Arthur Buchheim, came half a year later (see Table 6). He had been trained by Henry Smith in Oxford.148 Buchheim gave three presentations in Klein’s seminar (on May 5, November 14 and 16, 1881), each of which analyzed the scholarly literature devoted to Abelian integrals. Klein further supported his career by writing a (requested) recommendation on his behalf.149 Although Buchheim died young in 1888, he published twenty-four articles over the course of his brief career, some of which are still cited.150 Eduard Study was influenced by Buchheim’s work on line geometry.151 Research on Alfred North Whitehead has revealed that this famous British philosopher and mathematician used a free semester (in 1885) to go to Germany and take part in Klein’s lectures, although there are no references to this visit in Klein’s or Whitehead’s papers.152 Later, however, Whitehead would cite Arthur Buchheim, whose writings helped him to develop his calculus of extension, and he would also cite Klein’s articles, especially those on non-Euclidean geometry.153 Georges Brunel came with a letter of recommendation from Gaston Darboux.154 Klein felt an obligation to Darboux to guide Brunel toward achieving his own results. After Brunel had given three presentations in his seminar – on determining the genus p (November 20, 1880), on analysis situs (January 31, 1881), and on the theory of manifolds (May 2, 1881),155 Klein wrote to Darboux: I do have to write a few words to you about Brunel. He is remarkably receptive, and there is hardly anything that he has not read and really understood too. That said, the progress with his productivity has been slow; every attempt (and there have been many) to encourage him to work more boldly has so far failed. Only recently has he begun to study curvature radii, and I have of course supported his efforts, from which something publishable will hopefully emerge. This is not to say that he isn’t very valuable to me personally. My interactions with him are friendly, but somewhat awkward. Despite his efforts, the German language still causes difficulties for him.156

148 149 150 151 152

153 154 155 156

Regarding mathematicians in Oxford (UK), see FAUVEL/FLOOD/WILSON 2013. [UBG] Cod. MS. F. Klein 8:324/Anl.: Klein’s Report on A. Buchheim, April 30, 1882). See https://nhigham.com/2013/01/31/arthur-buchheim/ (accessed May 11, 2020). See HARTWICH 2005, p. 36. See LOWE 1990, pp. 151–52. – Michael Rahnfeld (Weddingstedt) is currently working on Whitehead’s metaphysics and the Erlangen Program, and I am grateful to him for informing me about how philosophers made use of Klein’s Erlangen Program to classify philosophical systems. Regarding Ernst Cassirer, see IHMIG 1997. See, for example, Alfred N. Whitehead, A Treatise on Universal Algebra with Applications, vol. 1 (Cambridge: Cambridge University Press, 1898), p. x. See RICHTER 2015; and TOBIES 2016. [Protocols] vol. 2, 76–79, 114–20; vol. 3, 14–19. [Paris] 72–73: Klein to Darboux, May 3, 1881.

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Shortly thereafter, Klein reported that progress had been made: “Brunel has recently brought me a manuscript in which he compiled formulae for the curvature situation in curves in n dimensions; I asked him to prepare it for publication in the Annalen.”157 Klein had found the best way to foster Brunel’s talents by pointing him to Camille Jordan’s “Essai sur la géométrie à n dimensions.” On June 3, 1881, Brunel was thus able to complete his first article: “Sur les propriétés métriques des courbes dans un espace linéaire à n dimensions.”158 At the same time, Klein asked Brunel to reach out to Henri Poincaré; however, some nationalistic undertones in Brunel’s letters to Poincaré were to have an adverse effect on the latter’s relations with Klein (see Section 5.5.3.2). In 1884, Brunel became a professor in Bordeaux, and his later contribution to the German edition of the ENCYKLOPÄDIE (he wrote the article on definite integrals in vol. 2, 1899) can be interpreted as an expression of thanks to Klein. 5.4.2.2 The First Americans Irving W. Stringham, who was the first American to come to Leipzig to work with Klein (and who was two years Klein’s senior), had completed a doctoral degree in 1880 under James Joseph Sylvester at Johns Hopkins University in Baltimore with a dissertation titled “Regular Figures in n-Dimensional Space.” He was enthusiastic about the international, stimulating atmosphere that Klein fostered; he had a strong command of German; and he gave three presentations in Klein’s seminar: on four-dimensional regular bodies (November 29, 1880), on groups of motions of four-dimensional bodies (February 28, 1881), and on self-transformations of groups (April 23, 1881).159 Stringham referred to his dissertation research and, in his third lecture, he presented new results that were based on suggestions by Klein. He analyzed what groups of linear transformations exist in four variables and which of them provide regular bodies, stressing: “The principle on which my consideration of the problem is based comes from a lecture by Prof. Klein from last year: ‘On Motions in Non-Euclidean Space’ (see Protocol Book for the winter semester of 1880/81, p. 97).”160 While still in Saxony, Stringham completed an article based on this third presentation, the results of which originated through his discussions with Klein. This article was published in the oldest American journal devoted to mathematics: the American Journal of Mathematics, which had been founded in 1878 by Sylvester.161 As early as 1882, Stringham was hired as a professor at the University of California in Berkeley.

157 158 159 160 161

[Paris] 74: Klein to Darboux, May 28, 1881. The article appeared in Math. Ann. 19 (1882), pp. 37–55. [Protocols] vol. 2, 46–66, 147–64; vol. 3, 31–50. [Protocols] vol. 3, 32. Irving Stringham, “Determination of the Finite Quaternion Groups,” American Journal of Mathematics 4 (1881), pp. 345–57. – See also DESPEAUX 2008.

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The next two Americans came to Leipzig in the summer semester of 1884, after Felix Klein had declined an offer to replace Sylvester at Johns Hopkins (see Section 5.8.1). Henry Burchard Fine and Frank Nelson Cole arrived with insufficient language skills. In this regard, Otto Hölder made the following remark on April 23, 1884: “Here is an Italian, there an American, and neither can speak German or French.”162 Fine and Cole attended Klein’s lecture courses for two semesters and also participated in his seminar in the summer of 1885, though neither gave a presentation. Nevertheless, they completed their doctoral research relatively quickly. Supported by Eduard Study, Fine submitted his dissertation in Leipzig on May 27, 1885: “On the Singularities of Curves of Double Curvature.”163 Fine was Klein’s only doctoral student to earn his degree (at a German university) with a dissertation not written in German. Later, in 1893 and 1896, Fine invited Klein to Princeton University (see Sections 7.5.3 and 8.1.3). Frank Nelson Cole finished his doctoral degree in 1886 at Harvard University with a dissertation inspired by Klein: “A Contribution to the Theory of the General Equation of the Sixth Degree.” This work was published in the American Journal of Mathematics (vol. 8 [1886], pp. 265–86), in which Fine’s dissertation had also appeared. In addition to this journal, which was now edited by Simon Newcomb,164 there existed at the time just one other mathematics journal in the United States, a fact that sheds some light on the state of research there. Cole, who at first taught at Harvard, received his first professorship at the University of Michigan, where he brought along Klein’s geometric approach to function theory and further disseminated Klein’s enthusiasm for the subject. For example, Cole’s student Edward Kasner based his aforementioned doctoral thesis on Klein’s Erlangen Program, and spent 1899/1900 in Göttingen with Klein and Hilbert. 5.4.2.3 The Italians Giuseppe Veronese had studied engineering and mathematics at the Polytechnikum in Zurich and had then held an assistantship in Rome under Cremona, who recommended that he should continue his studies with Klein. Veronese attended Klein’s lecture courses in the winter semester of 1880/81 and the summer semester of 1881; he also participated in Klein’s seminar, in which he gave presentations of some of his own results that he had already published. Thus, on January 3, 1881, he spoke on the theorie of projective groups and entered this talk into Klein’s protocol book (vol. 2) under the title “Ueber einige merkwürdige Configu-

162 HILDEBRANDT et al. 2014, p. 142 (a letter from Hölder to his parents, April 23, 1884). 163 In: American Journal of Mathematics 8 (1886), pp. 156–77 (Study’s suggestions are acknowledged on p. 158). – “Curves of double curvature are usually just called space curves; the terminology derived from the fact that such curves have non-zero curvature and torsion, whereas the torsion vanishes for plane curves.” Quoted from PARSHALL/ROWE 1994, p. 194. 164 Newcomb was Sylvester’s successor at Johns Hopkins University in Baltimore.

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rationen” [On a Few Noteworthy Configurations]. On April 25, 1881 (Klein’s 32nd birthday), Veronese presented new research in a lecture titled “Ueber die darstellende Geometrie im Raume von 4 Dimensionen” [On Descriptive Geometry in 4-Dimensional Space].165 With Klein’s encouragement, he expanded the ideas presented here for publication in Mathematische Annalen. In volume 18 (1881), p. 448 of the latter, he published a brief excerpt of his work – “Die Anzahl der unabhängigen Gleichungen, die zwischen den allgemeinen Charakteren einer Curve im Raume von n Dimensionen stattfinden” [The Number of Independent Equations that Exist Between the General Characters of a Curve in the Space of n Dimensions] (dated June 1881), and a larger article in the next volume.166 Francesco Gerbaldi, who had already published successfully and had worked as Enrico D’Ovidio’s assistant in Turin, came to Leipzig in the summer of 1883. His presentation in Klein’s seminar – “Ueber die linearen Differentialgleichungen zweiter Ordnung, welche algebraische Integrale besitzen” [On the Linear Differential Equations of the Second Order that Have Algebraic Integrals] – offered a comparison between the works of Klein and Lazarus Fuchs and is especially noteworthy in light of the polemic that had arisen between the two scholars (see Section 5.5.5). In Gerbaldi’s words: The method that we will discuss now, which is the simplest, we owe to Prof. Klein (Math. Ann., vols. IX and XII). Prof. Fuchs (in Borchardt’s Journal, vols. 81 and 85) gave a different method. He reduced the question of the conditions under which a linear differential equation of the second order has algebraic integrals to the following question: In which cases are certain linear differential equations (which can be uniquely derived from the given one, and whose order does not exceed twelve) satisfied by the roots of rational functions? To this end, he introduced the concept of the prime form and demonstrated that the degree of prime forms of the lowest degree never exceeds the twelfth.167

Gerbaldi explained Fuchs’s method and proof, and concluded that Klein’s was simpler. Gerbaldi remained in Leipzig for only one semester, and he did not publish anything in Mathematische Annalen until 1898 (an article closely related to Klein’s, Herman Valentiner’s, and A. Wiman’s results on symmetry groups).168 Giacinto Morera, who like Veronese held a degree in engineering and mathematics, came in the fall of 1883 with a letter of recommendation from Eugenio Beltrami.169 Morera’s first seminar presentation, which took place on July 14, 1884, concerned Klein’s level theory (Stufentheorie) and the “Kleinian fundamental theorem” (from Math. Ann., vol. 15).170 Morera expanded Klein’s ideas, 165 [Protocols] vol. 3, 1–14. 166 The article in question is Giuseppe Veronese, “Behandlung der projectivischen Verhältnisse der Räume von verschiedenen Dimensionen durch das Princip des Projicirens und Schneidens,” Math. Ann. 19 (1882), pp. 161–234. 167 [Protocols] vol. 5, 25–31 (Gerbaldi’s lecture, delivered on May 28, 1883), at pp. 28–29. 168 Francesco Gerbaldi, “Sul gruppo semplice di 360 cillineazione piane,” Math. Ann. 50 (1898), pp. 473–76. 169 [UBG] Cod. MS. F. Klein 8: 77 (Beltrami to Klein, Oct. 7, 1883); see COEN 2012, p. 488. 170 [Protocols] vol. 6, 101–19

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and Klein presented the results in a session of the Royal Saxon Academy of Sciences; moreover, he also accepted them for publication in Mathematische Annalen.171 Hurwitz compared Morera’s findings to his own investigations: The essential difference between Morera’s considerations and my own seems to be that he limits himself to a number module n (which, moreover, is a prime number), while I take into account several modules simultaneously. What we have in common is that both of us implement your general formation process in certain cases. Morera’s execution of this process is to be preferred over mine because it is more direct.172

5.4.2.4 Mathematicians from Switzerland and Austria-Hungary The German-born Wilhelm Fiedler, who had completed his doctorate under August Ferdinand Möbius at the University of Leipzig in 1858, became a Swiss citizen in 1875. Fiedler had integrated Klein’s results into his textbook series (his aforementioned translation of Salmon’s books), and in 1884 he sent his son Ernst to study under Klein. Klein involved Ernst Fiedler in his research (see Section 5.5.7.1) and supervised his doctoral thesis, “Über eine besondere Klasse irrationaler Modulargleichungen der elliptischen Funktionen” [On a Special Class of Irrational Modular Equations of Elliptic Functions] (1885). Wilhelm Fiedler asked Klein several times for his opinion of his son’s scientific abilities. Klein hesitated to answer, but finally he wrote bluntly that Ernst Fiedler lacked mathematical creativity, and that it would be not appropriate to force him to pursue a Habilitation. Father and son accepted Klein’s verdict; nevertheless, Ernst Fiedler achieved a dual-track career. He became a Privatdozent at the Polytechnikum in Zurich in 1886, and a headmaster of a secondary school there.173 The mathematicians from former Austria-Hungary listed in Table 6 came predominantly from Czech regions: Karl Bobek, Seligmann Kantor,174 Emil Waelsch, Wilhelm Weiß, and Adolf Ameseder (a student of the Prague-born Emil Weyr). In addition, this group also included the Hungarian Gustav Raussnitz (Gusztáv Rados) and Georg Pick from Vienna, whose relationship with Klein will be discussed in greater detail in Section 5.5.7.2. Although these students did not complete their doctoral degrees under Klein, he paved the way for them. Three among them earned their doctorates in Erlangen with Gordan and Max Noether. Gordan wrote to Klein: “Another of your students, Mr. Bobek from Prague, recently passed his doctoral examinations with me.”175 Karl Bobek earned his degree on June 23, 1885 with a dissertation titled “Über 171 Giacinto Morera, “Ueber einige Bildungsgesetze in der Theorie der Theilung und der Transformation der elliptischen Functionen,” Math. Ann. 25 (1885), pp. 302–11. 172 [UBG] Cod. MS. F. Klein 9: 991 (a letter from Hurwitz to Klein dated February 11, 1885). 173 Ibid. 9:19 (W. Fiedler to Klein, in December 1885); see also VOLKERT 2018b; 2019. 174 Kantor is perhaps best-known today for his role in developing the Möbius-Kantor graph, which is an important configuration in the field of graph theory. 175 [UBG] Cod. MS. F. Klein 9: 442, p. 47 (Gordan to Klein, September 8, 1885).

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gewisse eindeutige involutorische Transformationen der Ebene” [On Certain Single-Valued Involutory Plane Transformations],176 after he studied under Klein. Wilhelm Weiß and Emil Waelsch had likewise been prepared to complete their studies in Erlangen; Klein induced appropriate presentations from them in his seminar of summer 1885. He described his approach to the field, the theory of algebraic functions of one variable, as follows: It is assumed that Riemann’s results are essentially known. We intend to use and compare both the geometric treatment of function theory and the arithmetic approach to it. On the one hand, we will base the discussions on the older works of Clebsch, Brill and [Max] Noether, and as regards space curves, on the recently published major works by Noether, Halphen, and Valentiner. On the other hand, we will use the lectures by Weierstrass, the works of his followers, and also the papers of Kronecker, Dedekind, and [Heinrich] Weber. In the geometric approach, the concept of the complete system of intersection points plays the same role that the “entire” algebraic function plays in the other approach.177

Waelsch gave a presentation on Alexander Brill and Max Noether’s article “Über die algebraischen Functionen und ihre Anwendung in der Geometrie” [On Algebraic Functions and Their Application to Geometry] (Math. Ann., vol. 7). Weiß reported on Max Noether’s work “Zur Grundlegung der Theorie der algebraischen Raumkurven” [On the Foundation of the Theory of Algebraic Space Curves].178 On December 13, 1885, Gordan wrote to Klein: “The two Austrians whom you sent us are doing well; Weiß, however, works too much, and so I always fear that he won’t be able to endure for long.”179 Weiß completed his doctoral degree in Erlangen in 1887, and Waelsch the following year. Later, too, Klein would send other doctoral students to his friends in Erlangen, such as the Americans H.W. Tyler and W.F. Osgood, who studied under him in Göttingen since the fall of 1887. They continued to work on a thesis problem from Klein while receiving individual instruction from Gordan and Noether.180 The Hungarian mathematician Gustav Raussnitz (Gusztáv Rados) came from a Jewish family from Pest. With an already strong background in number theory from his studies with Julius König in Budapest, he gave presentations in Klein’s seminar (Summer 1885) on Richard Dedekind and Heinrich Weber’s work “Theorie der algebraischen Functionen einer Veränderlichen” [Theory of Algebraic Functions of One Variable] and on the content and methods of Kronecker’s Festschrift, which was titled Über den Zahlbegriff [On the Concept of Number]. Rados analyzed the latter book especially thoroughly, thereby demonstrating that his attitude toward research was closely aligned to Klein’s own. He criticized Kronecker for limiting his focus only to “the development of the features of rational whole numbers and functions without using methods from other areas of 176 [UA Erlangen] Phil. Fak. 923 (F), Akte 923. The dissertation was published in the Sitzungsberichte of the Academy of Sciences in Vienna (vol. 91, 1885). 177 [Protocols] vol. 7, 1–2 (Klein’s introductory lecture in his seminar, April 27, 1885). 178 Abhandl. Kgl. Preuß. Akad. Wiss., Phys.-Math. Kl. (1882), pp. 1–120. 179 [UBG] Cod. MS. F. Klein 9: 443, p. 48 (Gordan to Klein, December 13, 1885). 180 [UBG] Cod. MS. F. Klein 7E; see also BATTERSON 2009, pp. 920–23.

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mathematics.” Very much in the spirit of Klein, Rados promised that, in his discussion, he would use “all existing tools available to us.” In his presentation, that is, he outlined the fundamental concepts of Kronecker’s work and praised its “algebraic materialization of Kummer’s ideal numbers or Dedekind’s ideals,” but also discussed other approaches to the topic (Julius König’s, in particular).181 Klein and Rados remained in touch. Rados received a professorship at the Technische Hochschule in Budapest. When, in 1905, the Hungarian Academy of Sciences established its János Bolyai Prize (10,000 Kronen), Felix Klein and Gaston Darboux were the only two external members who were appointed to the prize committee, which also included Julius König and Gusztáv Rados.182 5.4.2.5 Russian and Other Eastern European Contacts From early on, Mathematische Annalen published works by Russian authors, who submitted their articles in French. These included Aleksandr N. Korkin, a student of Chebyshev who, after earning his doctoral degree, continued his studies in Paris and Berlin and produced significant results in the field of geometric number theory. In 1879, Korkin submitted an early study by his student Andrey A. Markov to the Annalen’s editorial board. Although the article was written in poor French, Adolph Mayer remarked to Klein in a letter that, “for political reasons, we should accept the work for publication.”183 What Mayer had in mind here was their journal’s competition with Crelle’s Journal. When Mittag-Leffler’s journal Acta Mathematica also entered the scene (see Section 2.4.2), Klein expanded, with the help of his seminar participants, the range of contributors to Mathematische Annalen by reaching out to more mathematicians in Eastern Europe. Five years Klein’s senior, the mathematician Matvey A. Tikhomandritsky had already sent Klein a note to be published in Mathematische Annalen – “Ueber das Umkehrproblem der elliptischen Integrale” [On the Converse Problem of Elliptic Integrals] (dated June 20, 1883) – before he arrived in Leipzig. After he had given a presentation in Klein’s seminar (July 21, 1884), a second note was published that is still cited in recent scholarship.184 As of 1883, Tikhomandritskiy was employed as a Dozent at the University of Kharkiv, and he had a broad knowledge of Eastern European mathematical institutions. At Klein’s request, Tikhomandritskiy sent him a multi-page list of researchers, institutions, main research areas, 181 [Protocols] vol. 7, 27–50, 125–50 (lectures, on June 1, July 20, July 27, and August 3, 1885). Kronecker’s Festschrift was also published in Crelle’s Journal 101 (1887), pp. 337–55. 182 In their correspondence, Klein and Darboux agreed that the first Bolyai Prize (in 1905) should go to Poincaré and that the second (in 1910) should be awarded to Hilbert. See TOBIES 2016. 183 Quoted from TOBIES/ROWE 1990, p. 106 (a letter from Mayer to Klein dated May 5, 1879). 184 [Protocols] vol. 6, 121–26; Math. Ann. 22 (1883), pp. 450–54; Math. Ann. 25 (1885), pp. 197–202; and Yurii V. Brezhnev, “What Does Integrability of Finite-Gap or Soliton Potentials Mean?” Philos. Trans. of the Royal Society: Math., Physical and Engineering Sciences 366 (2008), pp. 923–45.

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Eastern European (Russian, Ukrainian, Polish, Hungarian, Czech) scholarly journals, and the researchers involved with them.185 The list included three publications by the Royal Academy of Sciences in St. Petersburg that only published in German and French – Memoires, Bulletins, and Mélanges mathématiques et astronomiques, tirés des Bulletins – along with Russian-language journals printed in Moscow, Kharkiv, Kazan, Odessa, Kiev, Warsaw, and elsewhere. On the basis of this list, Klein decided to pursue exchanges between Mathematische Annalen and some of these journals, thereby attracting additional contributors and students. Theodor Molien was the first student to come to Klein from the Baltics. He studied for three semesters (see Table 6), and his seminar presentations would serve as the basis for his master’s thesis, “Über lineare Transformationen elliptischer Funktionen” [On Linear Transformations of Elliptic Functions].186 In December of 1885, Molien wrote to Klein to inform him “that my promotion to Magister took place on the basis of the research that I conducted under your guidance.”187 After Klein’s former assistant Friedrich Schur had accepted a professorship at the University of Tartu, Molien completed his doctoral studies under him, and Klein published his dissertation in Mathematische Annalen.188 Born in Kazan, A.V. Vasilev, who had studied in St. Petersburg and Berlin (1879/80), then also visited Klein in Leipzig. This we know from a letter that Vasilev wrote to Klein on June 9, 1895. Here, he thanked Klein for his “contribution to the Lobachevsky committee” (see Section 6.3.6) and he informed him that “in today’s meeting of the university senate, you have been elected an honorary member of the Imperial University of Kazan.”189 During that same year, a Russian translation of Klein’s Erlangen Program was published in the Bulletin de la Société de Kasan, a work that was instigated by Vasilev and executed by the mathematician D.M. Sintsov. Both Vasilev and Sintsov became members of the German Mathematical Society (see also Fig. 43). The Eastern European contacts that Klein forged in Leipzig would expand in later years (see Section 6.3.7.1). They also led to Klein being made a member of the Moscow Mathematical Society (1891) and the Academy of Sciences in St. Petersburg (1895), among other honors. The proposal to elect him as a corresponding member of the St. Petersburg academy was written by A.A. Markov and Nikolay Y. Sonin,190 who were longtime contributors to Mathematische Annalen. Klein was quick to recognize new trends, and later he also organized the translation of Russian textbooks, including works by A.A. Markov. This is just one of the many projects that Klein started with the B.G. Teubner publishing house during his Leipzig years (see Section 5.6).

185 186 187 188 189 190

[UBG] Cod. MS. F. Klein 12: 24 (Tikhomandritskiy to Klein, November 14, 1884). [Protocols] vol. 5, 254–56; vol. 6, 25–33 (presentations, on February 4 and May 12, 1884) [UBG] Cod. MS. F. Klein 10: 1283 (Molien to Klein, December 14/26, 1885). Th. Molien, “Ueber Systeme höherer komplexer Zahlen,” Math. Ann. 41 (1893), 83–156. [UBG] Cod. MS. F. Klein 12: 200 (a letter from Vasilev to Klein dated June 9, 1895). [Archiv St. Petersburg] Fond 2, 1–1895 73, pp. 5, 16–17; and JUSCHKEWITSCH 1981.

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5.5 FIELDS OF RESEARCH None of the research that Klein conducted during his years in Leipzig has given rise to more discussion, analysis, and interpretation than his rivalry with Poincaré.191 This relative short phase, during which Klein worked intensively and produced results of the highest level, needs to be integrated with the ensemble of his work. The area of research that fueled the rivalry had occupied Klein’s attention while he was still in Munich, and since then he had passed it along to his students and largely left it aside in his own work (this latter fact is seldom mentioned). On October 24, 1881, Klein expressed as much to Adolf Hurwitz in clear language: The studies that you have written to me about fill me with a degree of envy. You know that number theory has always seemed like a promised land to me, at which I may be able to cast my gaze but in which I might perhaps never set foot. When I think back to my work on elliptic modular functions, I become wistful: so many beautiful perspectives that I will have to leave unexploited! It is some consolation for me to know that you are continuing to work in this direction, and thus I want to encourage you all the more to forge ahead. See to it that you draw an explicit connection between the Legendre symbol and the w-configuration. I think of the matter as follows: […]

Here Klein provided Hurwitz with an array of new ideas that might bear fruit. Regarding his own work, however, he wrote in the same letter: Mechanics or mathematical physics (whatever one might call it) beckons. Only after I have mastered this to the extent that I have original ideas about it can there be talk of flourishing mathematical production on my part. Will I succeed in executing this research program?192

Two aspects of this letter should be emphasized here. First, Klein recognized Hurwitz’s mathematical creativity and repeatedly doubted his own mathematical abilities. By no means did he consider himself to be “the leading German mathematician of his generation and perhaps in the world.”193 Second, Klein’s main intention at the time was to produce his own new results in an applications-oriented discipline, even though in June of 1881 he began his correspondence with Poincaré. Although Klein would deviate from this plan from time to time, his research agenda in Leipzig in fact began with his work on mathematical physics (Section 5.5.1). A corresponding feature of Klein’s work at the time lay in his continued effort to integrate the results developed in Berlin into his own arsenal of methods, even though the mathematicians of the Berlin school continued to retain a skeptical attitude toward him (5.5.2).

191 See ROWE 1992b; ROWE 2018a, pp. 120–27; GRAY 2000; and GRAY 2013, pp. 207–46. 192 [UBG] Math. Arch. 77: 52, pp. 77–78 (Klein to Hurwitz, on October 24, 1881); the original German letter excerpt is published in TOBIES 2019b, p. 224. 193 GRAY 2013, p. 226. Gray’s complete sentence is: “He was ambitious, he had every reason to believe he was the leading German mathematician of his generation and perhaps in the world, and he was beginning to shape himself as the heir of Riemann.”

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Furthermore, Klein was enthusiastic about new research results emerging from France, and not only about Poincaré’s work, though he held it in especially high esteem (5.5.3). Building on this work, Klein produced significant results in a field of research that he had already left behind (5.5.4). The controversy with and about Lazarus Fuchs, which arose in this context, will be given special attention because of its long-lasting repercussions (5.5.5). Klein decided to create systematic expositions of the findings that he had made in Munich. The results of these efforts include his book Ueber Riemann’s Theorie der algebraischen Functionen und ihre Integrale [On Riemann’s Theory of Algebraic Functions and Their Integrals] (5.5.1.2), his book on the icosahedron (5.5.6), and additional ideas and approaches for monographs (5.5.7, 5.5.8). Looking back from the perspective of a modern research mathematician, Dieter GAIER (1990) identified three sources from the nineteenth century that considerably influenced the development of function theory: (1) Riemann’s mapping theorem (1851)194; (2) a theorem by Picard (1879), which states that every entire non-constant function f assumes every value a in the complex plane, with at most one exception; and (3) classical potential theory.195 These points of departure are clearly reflected in Klein’s work. 5.5.1 Mathematical Physics / Physical Mathematics The term “physical mathematics” was first used by Arnold Sommerfeld to describe Klein’s approach.196 Klein, who in his letters to Sophus Lie had repeatedly expressed his interest in physics, was motivated to pursue these interests in Leipzig by Carl Neumann. In doing do, Klein not only achieved novel results in Neumann’s own area of expertise, potential theory. He also used this physicsbased approach as a heuristic principle for describing Riemann’s function theory. 5.5.1.1 Lamé’s Function, Potential Theory, and Carl Neumann As I pointed out in Section 5.3.1, Klein and Carl Neumann had some conflicts concerning their respective teaching assignments. At the heart of this dispute was function theory. In his very first lecture in Leipzig, which was given on October 26, 1880, Klein had stressed that Neumann’s book Vorlesungen über Riemann’s Theorie der Abel’schen Integrale [Lectures on Riemann’s Theory of Abelian Integrals] (Leipzig: B.G. Teubner, 1865) was a fundamental work of scholarship.197 Neumann

194 195 196 197

See for example Reinhold REMMERT 1998, or KRANTZ 2006, pp. 83–107. GAIER 1990, pp. 363–64. SOMMERFELD 1919, p. 301. See KLEIN 1987, p. 12. On C. Neumann’s life and work, see SCHLOTE 2001, 2004, and 2017.

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had been one of the first researchers to engage with Riemann’s work. In 1865, he had even published a second book on the topic with Teubner: Das Dirichlet’sche Princip in seiner Anwendung auf die Riemann’schen Flächen [Dirichlet’s Principle in Its Application to Riemann Surfaces]. In Neumann’s work, Klein thus saw good points of connection to his own physical-mathematical interests. Klein was inspired by his interaction with Carl Neumann to write two articles on “Lamé’sche” functions.198 Klein noted: “In my dealings with Neumann: Lamé’s functions, confocal bodies.”199 However, in his articles “Ueber Lamé’sche Functionen” [On Lamé’s Functions] (submitted in January of 1881) and “Ueber Körper, welche von confocalen Flächen zweiten Grades begränzt sind” [On Bodies that Are Limited by Confocal Surfaces of the Second Degree] (submitted March 14, 1881), there is no mention whatsoever of the impulse that Klein had received from Neumann. Klein merely remarked that Neumann used different notations.200 At the time, Neumann’s most recent contribution to the subject was published in the same volume of Mathematische Annalen as Klein’s article on “Lamé’sche” functions; it appeared immediately ahead of Klein’s study, a gesture that was probably meant to indicate that the two authors had taken a similar approach. Whereas Neumann, at the end of his article, referred explicitly to Klein’s interesting developments (he had read Klein’s studies before their publication),201 Klein made forays into Neumann’s field of research without even citing him extensively. This contradicted Klein’s typical approach, and it potentially could have led to conflict. At any rate, he stated shortly thereafter: “I am beginning to feel antagonism with Neumann.”202 Klein sent his articles to Darboux with the following comment: “In these studies I have proved two theorems that seem to have been unknown until now and yet are of primary importance if one clearly wants to grasp Lamé’s functions.”203 One of these theorems was the oscillation theorem, which Klein first named as such in his later article “Zur Theorie der Laméschen Functionen” [On the Theory of Lamé’s Functions] (1890).204 In the latter work, he expanded his approaches from 1881, he emphasized the role of this theorem in potential theory and the theory of linear differential equations of the second order, and he also introduced the concept of “automorphic functions” in this article for the first time. Later Klein explained this concept as follows:

198 See KLEIN 1922 (GMA II), p. 507. 199 Quoted from JACOBS 1977 (“Persönliches betr. Leipzig”), p. 1. 200 Felix Klein, “Ueber Körper, welche von confocalen Flächen zweiten Grades begränzt sind,” Math. Ann. 18 (1881), 410–427 (Klein makes this remark in the footnote on p. 421). 201 Carl Neumann, “Ueber die Mehler’schen Kegelfunctionen und deren Anwendung auf elektrostatische Probleme,” Math. Ann. 18 (1881), pp. 196–236. 202 Quoted from JACOBS 1977 (“Vorläufiges über Leipzig”), p. 1. 203 [Paris] 75: Klein to Darboux, June 10, 1881. See also KLEIN 1922 (GMA II), p. 507; and Leon Lichtenstein, “Neuere Entwicklungen der Potentialtheorie: Konforme Abbildung,” in ENCYKLOPÄDIE, vol. II.3.1 (1918), pp. 177–377, esp. pp. 189–90. 204 Göttinger Nachrichten 4 (1890), pp. 85–95; reprinted in KLEIN 1922 (GMA II), p. 540–49.

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To give a brief explanation here, automorphic functions are those which satisfy the functional equation ⎛ α z + βi f ⎜⎜ i ⎝ γ i z + δi

⎞ ⎟⎟ = f ( z ) ⎠

for a series of indices i, possibly for infinitely many; thus they are in the broadest sense generalizations of periodic functions, addition of periods being replaced by linear substitutions.205

By 1922, when Klein was preparing volume II of his collected works, the field had evolved. Thus he made an addition to the title of his (second) article from 1881: “Über [die Randwertaufgabe des Potentials für] Körper, welche von konfokalen Flächen zweiten Grades begrenzt sind” [On (the Boundary Value Problem of Potential for) Bodies that Are Limited by Confocal Surfaces of the Second Degree].206 In his words from 1881, Klein treated the “fundamental potential problem of determining, from the values of the potential at the points of a surface, the course of the latter inside the body limited by the surface.”207 A few years later in Göttingen, he succeeded in guiding some of his students toward producing fruitful results in this area of research (see Section 6.3.5). 5.5.1.2 On Riemann’s Theory of Algebraic Functions and Their Integrals In Friedrich Schulze’s history of the first hundred years of the Teubner publishing house, Klein’s work Ueber Riemann’s Theorie … from 1882 is celebrated as his first monograph. Here we read that it provides an exposition of the theory on an intuitive geometric-physical basis in that it attempts to demonstrate that Riemann’s theory of algebraic functions and their integrals is nothing else than a mathematical formulation of those intuitions and facts which the physical study of the theory of stationary currents of heat or electricity brings to light, when one transfers it to the case of connected, closed surfaces.208

The book summarizes many of the results that Klein had achieved since his time in Munich, with the aim of preserving them for posterity: In the meantime, I have been working on a small text on Riemann’s theory, which I would like to see published by Teubner. You are familiar with the issue itself from the summer semester here; yet it is really taking a great deal of effort to present this same material in a clever form. […] You can imagine how much I have to force myself to work it all out, which is a dull task. Yet it is necessary to proceed in such a way; otherwise, all the ideas that I have been engaged with on and off would be lost in the sand.209

205 KLEIN 1979 [1926], p. 258. Lester R. FORD (1915; 1929), who was one of Klein’s intellectual heirs, wrote the first introduction to the theory of automorphic functions in English. – The word automorphic derives from Greek αὐτός ‘self’ and μορφή ‘shape, form’. 206 KLEIN 1922, p. 521 (the brackets characterizing the insertion in the title are original). 207 Ibid. 208 SCHULZE 1911, pp. 306–07. 209 [UBG] Math. Arch. 77: 55 (a letter from Klein to Hurwitz dated September 20, 1881).

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Klein based his physical approach not only on Riemann but also on Thomson and Tait’s Natural Philosophy and on Carl Neumann’s works.210 He presupposed that previous work on Riemann’s theory was already familiar to his readers, but he made sure to draw attention to the results produced by H.A. Schwarz, Georg Hettner, and Friedrich Schottky. From Schottky, Klein learned that the latter’s similar studies had likewise proceeded from “examining the currents of an incompressible fluid” and that only after following Weierstrass’s advice did Schottky replace his physical approach by “taking into account Schwarz’s studies of conformal mapping.”211 Conscious of this preparatory work, Klein claimed for himself the idea “of investigating closed surfaces in space on the basis of function theory and thereby to understand the actual fundamental ideas of Riemann’s theory.” In doing so, he adopted a (supposed) suggestion from Friedrich Prym – who had yet to attend Riemann’s lecture courses himself – and used arbitrary curved surfaces in space as the domains of complex space functions.212 Although “curved surfaces” are not Riemann surfaces, the approach is sensible if one moves from the metric to the conformal structure of the surface.213 Arnold Sommerfeld described the basic ideas as follows: The idea of the Riemann surface, which Riemann introduced in his dissertation and expanded with a suggestion proposed in the conclusion of the same work, was further developed by Klein into the idea of the closed “Klein-Riemann surface.” Just as, in function-theoretical terms, the complex plane is best replaced by the sphere, a ramified Riemann plane of higher genus can be replaced by a closed, singularity-free spatial surface with manifold connectivity. This surface is conceived as having an evenly distributed conductive mass, and it forms a conductor for electric current. The single-valued potentials on the surface form the building blocks for the theory of algebraic functions of surfaces and their integrals. The discontinuity points of the potentials are the sources and sinks of the current; at the same time, they are the points where one should consider the electrodes to be attached, which lead the current toward or away from the conductor. In that one can arrange infinitely many electrodes transversally along a loop-cut (Rückkehrschnitt) of the surface, one obtains as potentials the finite integrals of the surface (integrals of the first kind). Integrals of the second and third kind arise in the case of point-like (connected or separate) electrodes; the single-valued functions on the surface – the algebraic functions of the entity – are constructed as special cases from the potential functions. What is being practiced here is not really mathematical physics; rather, it is physical mathematics.214

210 See KLEIN 1882; KLEIN 1923 (GMA III), pp. 478, 499–573; and KLEIN 1987. 211 Friedrich Schottky, “Ueber die conforme Abbildung mehrfach zusammenhängender ebener Flächen,” Crelle’s Journal 83 (1877), pp. 300–51. Klein classified Schottky’s theorems as special cases among his own results (see KLEIN 1923 [GMA III], pp. 572–73. 212 Beltrami had already dealt with surface segments (Flächenstücke) and had posed the boundary value problem for them. Prym accepted Klein’s approach but denied that the inspiration had come from him. See KLEIN 1923 (GMA III), pp. 479, 501–02 (quoted here from p. 479); and LAUGWITZ 1999, pp. 112–14. 213 I am indebted to Erhard Scholz for this insight. 214 SOMMERFELD 1919, p. 301 (on the original German quotation, see TOBIES 2019b, p. 229).

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In the book, Klein explained that he “endeavoured to obtain a general view of the scope and efficiency of the methods,”215 and he divided the work into three main parts: 1) Introductory Remarks, 2) Riemann’s Theory, and 3) Conclusions. Among his conclusions, Klein considered “surfaces with boundaries and unifacial surfaces,”216 based on his classification of closed surfaces according to the genus p. Klein had already worked on such surface classifications as early as the 1870s, when, in dialog with Ludwig Schläfli, he succeeded in producing results in the area of surface topology. While studying non-orientable surfaces, Klein introduced the term Doppelfläche ‘double (or unifacial) surface’ (see Section 3.1.3.1). These include “certain unifacial surfaces with no boundaries,” among them a special surface with p = 1, a model of which he described in the book (this would later be known as the “Klein bottle”): An idea of these may be formed by turning one end of a piece of india-rubber tubing inside out and then making it pass through itself so that the outer surface of one end meets the inner surface of the other. With reference to all these surfaces it has been established by former propositions that the representation of one surface upon another of the same kind is possible if one, but only one, equation exists among the real constants of the surface; and that the representation, if possible at all, is possible in an infinite number of ways, since a double sign and a real constant remain at our disposal.217

The first illustration of this idea appeared in KLEIN/ROSEMANN (1928); see Fig. 24 below. HILBERT/COHN-VOSSEN (1932, pp. 271–76) contains another illustration. Here, the terms Kleinscher Schlauch ‘Klein tube’ and Kleinsche Fläche ‘Klein surface’ are used synonymously. In the English translation of this book (Geometry and the Imagination, 2nd ed., 1952), the term Klein bottle is used throughout.

Figure 24: The Klein bottle (KLEIN/ROSEMANN 1928, p. 262).

215 Quoted from the English translation of the book, KLEIN 1893 [1882], p. ix (preface). 216 Hardcastle’s “unifacial surfaces (which may or may not be bounded)” in KLEIN 1893 [1882], p. 72, are called “Doppelflächen” in Klein’s original. – For example, the Möbius strip is a non-orientable (one-sided) surface (it is impossible to distinguish its bottom from its top or its inside from its outside). The cylinder strip is a two-sided surface with a boundary. The sphere and the torus are two-sided surfaces without boundary curves. 217 KLEIN 1893 [1882], p. 74 (emphasis original).

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In addition, Klein supplemented further results with proofs that he would incorporate into his lectures, into his article “Neue Beiträge zur Riemannschen Funktionentheorie” [New Contributions to the Riemannian Theory of Functions] (Math. Ann. 21), and into the monographs KLEIN/FRICKE (1890/92) and FRICKE/ KLEIN (1897/1912). Later, in his book Die Idee der Riemannschen Fläche [The Idea of the Riemann Surface] (1913), Weyl explained Riemann surfaces (anew) as real two-dimensional manifolds that have a complex structure and are characterized by local coordinates. Weyl used Hilbert’s proof of the Dirichlet principle for harmonic functions with prescribed boundary conditions in order to introduce complex analytical functions on the surface. In his opinion: This formulation of the concept of a Riemann surface, first developed in intuitive form in F. Klein’s monograph Über Riemann’s Theorie der algebraischen Funktionen und ihrer Integrale, is more general than the formulation that Riemann himself used in his fundamental work on the theory of analytic functions. There can be no doubt but that the full simplicity and power of Riemann’s ideas become apparent only with this general formulation.218

Klein’s book was translated into English (see Fig. 25) by Frances Hardcastle, who came from a famous family. Her maternal grandfather was the astronomer, mathematician, and chemist John Herschel. She wrote to Andrew R. Forsyth, her former professor at Girton College in Cambridge (U.K.), to see whether he could ask Klein for permission to translate the book.219 Hardcastle needed several months to complete the project, which she did while continuing her studies at Bryn Mawr College in Pennsylvania under the supervision of Charlotte Angas Scott and James Harkness. As she informed Klein, she published the translation at her own expense with Macmillan & Co. in London, and she expressed how much she enjoyed the work, writing: “I shall feel more than sufficiently rewarded.”220 Klein had completed this 82-page work (KLEIN 1882) before taking a closer look at Poincaré’s notes, namely on October 7, 1881, on the East Frisian island of Borkum. He had been sent there for health reasons by Ernst Leberecht Wagner, the director of the Medical Polyclinic at the University of Leipzig. In Leipzig, Klein had suffered on hot days from “asthma in conjunction with hay fever.” He used this vacation with his wife and son Otto not only to swim and meet with colleagues (Aurel Voss, Georg Elias Müller),221 but also to write. In the following years, for Klein, different East Frisian Islands (Borkum, Norderney, Spiekeroog, Wangerooge) were on several occasions a destination where he could recover as well as work productively.

218 WEYL 1955 [1913], p. 33 (3rd ed., 1955, pp. 29–30). I am indebted to Erhard Scholz for helpful information about Weyl. On Weyl’s life and work, see also SIGURDSSON 1991. 219 [UBG] Cod. MS F. Klein 9: 65 (a letter from Forsyth to Klein dated February 17, 1893). 220 Ibid. 9: 547 (a letter from Hardcastle to Klein dated July 20, 1893). 221 See Klein’s notes in JACOBS 1977 (“Vorläufiges aus Leipzig”), p. 1. G.E. Müller was on the hiring committee in Göttingen and helped to ensure that Klein was offered the professorship there in 1886 (see Section 5.8.2).

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Figure 25: The title page of Klein’s book “On Riemann’s Theory of Algebraic Functions and Their Integrals” https://archive.org/details/RiemannsTheory

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5.5.2 Looking Toward Berlin Earlier, when I was still fully immersed in geometry, I had little interest in Berlin mathematics. Now that I have a somewhat broader perspective, that has changed […].222

Klein wrote these words to Hurwitz already on March 3, 1879. In the same letter, he even asked Hurwitz to acquire some photographs of Berlin mathematicians for him. When discussing Klein’s relation to the mathematicians in Berlin, we would be wrong to perpetuate the common refrain that he felt some sort of long-lasting enmity toward them. Klein was open to their methods in the sense that he was ever eager to test and combine new approaches in order to produce novel results. Even before Leo Koenigsberger identified the two leading styles of mathematical thinking of his time – that is, the geometric-physical methods stemming from Riemann and the analytic-arithmetic approach in Berlin (see Section 1.2) – Klein had followed Clebsch’s model of combining the best aspects of different approaches. In this vein, he sought to adopt and analyze the latest results from the Berlin school, about which presentations were often given in his seminars. 5.5.2.1 Gathering Sources By the end of the 1870s, Adolf Hurwitz had imbibed three semesters of Berlin mathematics, and he passed on to Klein the contents of the lectures that he attended there. Hurwitz’s own combination of methods can be detected in his work, as already evidenced by two presentations that he gave in Klein’s Leipzig seminar during the winter semester of 1880/81. In these, Hurwitz used both Weierstrass’s function-theoretical approach – in order to derive “numerous number-theoretical theorems about the sums of the powers of a number’s divisor” – as well as the geometric methods of Riemann and Klein.223 In March of 1881, Klein took a trip with his wife to Berlin, where he had delegated his student Walther Dyck shortly before. Klein helped Dyck to establish contact with the Berlin mathematician Leopold Kronecker. In addition, Klein learned something that had long interested him: Kronecker had found no valid proof for his own theorem concerning equations of the fifth degree.224 Thus, Klein 222 [UBG] Math. Arch. 77: 20, fol. 33 (a letter from Klein to Hurwitz dated March 3, 1879). 223 [Protocols] vol. 2, pp. 67–70 (quoted here from p. 70). The titles of Hurwitz’s presentations were “Über die Bildung der Modul-Functionen” [On the Formation of Modular Functions] (delivered on December 6, 1880), and “Über eine Reihe neuer Functionen, welche die absoluten Invarianten gewisser Gruppen ganzzahliger linearer Transformationen bilden” [On a Series of New Functions that Form the Absolute Invariants of Certain Groups of Integer Linear Transformations] (delivered on February 21, 1881). 224 Klein wrote in a letter dated February 3, 1885: “His [Kronecker’s] original proof, which I excerpted from his manuscript in 1881, is wrong.” [UBG] Cod. MS F. Klein 11: 565B, fol. 6v, on the German original see: https://gdz.sub.uni-goettingen.de/id/DE-611-HS-3384691.

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would present his own proof that he had found in Munich (see Section 4.2.1) once more as a crowning argument in his icosahedron book (see Section 5.5.6). During his stay in Berlin, Klein also personally met H.A. Schwarz for the first time. This contact would turn out to be beneficial for Hurwitz, whose career Klein supported. In the case of Kronecker, too, Klein paved the way for Hurwitz. He wrote to Hurwitz that he had “sent a thorough report about you to Kronecker, while I was writing lately about the point in the theory of equations of the fifth degree with which you are familiar. I think that he will receive you with the utmost goodwill.”225 In Berlin, Dyck and Hurwitz heeded Klein’s request to “acquire, for our Leipzig seminar, really useful notes from the lectures by both Weierstrass and Kronecker.”226 Hurwitz hired someone to copy these lectures, and Klein reimbursed him.227 Carl Runge and Klein’s doctoral student Guido Weichold were engaged in procuring additional lectures by Weierstrass.228 Klein’s lectures, publications, and seminar protocols all document the extent to which he valued the results produced in Berlin. On February 27 and March 6, 1882, Klein himself spoke in his seminar about “Weierstrass’s lecture notes on analytic and elliptic functions.”229 The Berlin material was also important for Klein’s aforementioned special series of lectures, which began on June 6, 1882. As he reported to Hurwitz: “The day before yesterday, I began a short special lecture series about single-valued functions with linear transformations into themselves. I hope that Weichold will acquire for me a copy of Weierstrass’s notebook on Abelian functions.”230 5.5.2.2 The Dirichlet Principle Weierstrass’s critique of Dirichlet’s principle concerning potential theory was a focal point in Klein’s analysis of the Berlin material.231 On October 24, 1881, he wrote to Hurwitz that, in his seminar during the winter semester of 1881/82, “the Dirichlet principle will be the focus of our discussions,” and he went on: “I will have to master this manner of investigation, no matter the cost of doing so.”232 In 225 [UBG] Math. Arch. 77: 52 (a letter from Klein to Hurwitz dated October 24, 1881). 226 Ibid. 77: 51 (a letter from Klein to Hurwitz dated September 20, 1881). For further information, see ULLRICH 1988. Regarding Walther Dyck, see HASHAGEN 2003. 227 [UBG] Cod. MS. F. Klein 9: 926 (a letter from Hurwitz to Klein dated May 22, 1882). 228 [Deutsches Museum] No. 1950–6 (Klein’s letters to Runge, dated May 23, 1882; July 22, 1872; and March 16, 1883). 229 [Protocols] vol. 3, p. 142. 230 [UBG] Math. Arch. 77: 80 (a letter from Klein to Hurwitz dated June 8, 1882). 231 See WEIERSTRASS (1895), vol. 2, pp. 49–54. The article in question – “Über das sogenannte Dirichlet’sche Princip” [On the So-Called Dirichlet Principle] – was originally published in 1870. For a recent article on further developments in this field, see Ivan Netuka, “The Transformation of Mathematics Between the World Wars: The Case of Potential Theory,” in BEČVÁŘOVÁ 2021. 232 [UBG] Math. Arch. 77: 52 (a letter from Klein to Hurwitz dated October 24, 1881).

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November of 1881, Klein remarked: “I am delving deep into Dirichlet’s principle, and this morning I had a three-hour conversation with [Carl] Neumann about it. Hopefully, I’ll get to the bottom of the matter. At least now I can really prove all of the theorems that I formulated last summer and are included in my book.”233 In his seminar in 1881/82, Klein had his students analyze, above all, works that sought to replace Dirichlet’s principle with other methods, especially Carl Neumann’s “method of the arithmetic mean” and H.A. Schwarz’s so-called “alternating method,” which was based on conformal mapping.234 Toward the end of the semester, Klein himself delivered several lectures in the seminar: From January 18th to February 8th, the undersigned [Klein] explained, in a total of 10 sequential lectures, a number of passages of his recently published book On Riemann’s Theory of Algebraic Functions and Their Integrals. In conjunction with this, he also elucidated the significance of symmetrical Riemann surfaces to the theory of real algebraic curves and to the related Abelian integrals. Finally, he discussed the so-called Dirichlet principle.235

On February 15, 1882 Klein noted: “Critique of Dirichlet’s principle by the undersigned, F. Klein.”236 He began his lecture as follows: In order to demonstrate the existence of functions on Riemann surfaces, we have used nonrigorous physical methods. After that, some of the presentations (see the protocol book) offered strict mathematical methods [Neumann’s, Schwarz’s] for the existence proofs. Riemann used the Dirichlet principle, which very quickly led him to his goal. In what follows, we will examine this and test its reliability.237

After providing a historical introduction, Klein explained the fundamental objective of potential theory, demonstrated the “geometric features of the potential problem,” and presented a “critical engagement with the Dirichlet principle”: Let u be a surface which passes through the space curve bounding the cylinder that was constructed above the curve of our planar segment by applying the given boundary values. We will now form the integral ⎡⎛ ∂u ⎞ 2 ⎛ ∂u ⎞ 2 ⎤ ∫∫ ⎢⎢⎜⎝ ∂x ⎟⎠ + ⎜⎜⎝ ∂y ⎟⎟⎠ ⎥⎥ dx dy ⎣ ⎦ over this planar segment. Expressed in intuitive, indefinite terms, this integral measures the total slope of the surface u against the xy-plane; it becomes greater, the more the surface rises and falls. We want to set ourselves the task of constructing the surface u = z for which the integral reaches its minimum – a surface that thus lies as flatly as possible against the xy-plane. This is a task for the calculus of variations, and the latter discipline provides the solution by means of its usual methods:

233 Ibid. 77: 54 (Klein to Hurwitz, Nov. 13, 1881). The book is KLEIN 1882 (see Section 5.5.1.2). 234 [Protocols] vol. 3, pp. 91–126. Six of the participants in Klein’s seminar gave presentations on this topic: H. Hoppe, J. Kollert, H. Dressler, F. Dingeldey, H. Wiener, and G. Weichold. 235 [Protocols] vol. 3, p. 136. 236 Ibid., p. 138 (Klein’s lecture, February 15, 1882) 237 These lectures are not contained in the [Protocols]. They survive as an appendix to vol. 2 of Klein’s unpublished lectures on function theory, which are kept in the library of the Mathematical Institute at the University of Leipzig. The quotation here is from p. 364.

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∂ 2u ∂ 2u + = 0. ∂x 2 ∂y 2

This means, however: The aforementioned variational problem, when treated according to the customary methods of this discipline, leads precisely to the same differential equation which we recognized from the outset as that imposed on u, if u is to be a potential function. Now, the Dirichlet principle consists in the following conclusion: Because, among all surfaces passing through the space curve, one must make this integral a minimum or, expressed differently, it must lie most flatly against the xy-plane, there must also be a potential function with the required property. The Dirichlet principle thus assumes as a certainty that every variational problem has a solution, and from this it deduces the existence of this potential function.238

Klein explained this in detail, and he transferred Dirichlet’s ideas to curved surfaces, from which he concluded: “The Dirichlet principle is therefore also sufficient for this.”239 In a section of his lecture concerned with “objections to the Dirichlet principle,” Klein cited examples for the principle’s invalidity and, referring to the related seminar presentations in the protocol book, discussed the extant attempts to deal with these objections. In conclusion, he formulated the following task for his students: “The existence of the potential function is to be confirmed in the Neumannian/Schwarzian manner. After this, it has to be shown (in the spirit of Weierstrass) that it makes the Dirichlet integral to a minimum.”240 All of this information would be incorporated into Klein’s article “Neue Beiträge zur Riemannschen Funktionentheorie” [New Contributions to Riemann’s Function Theory], which he completed on October 2, 1882, and which contains a section on the scholarly research devoted to the Dirichlet principle.241 In the early 1880s, Klein had a high opinion of Weierstrass’s critique of the Dirichlet principle, and he inspired his students to work on the topic further. Tom ARCHIBALD’s (2016) discussion of the counterexamples presented in Weierstrass’s work (including his critique of the Dirichlet principle) demonstrates their importance to further developments in this line of research. Klein had made sure that this topic was integrated into two contributions to the ENCYKLOPÄDIE,242 and in his historical lectures, he himself summarized the history of Dirichlet’s theory, from Gauß (1840) to Hilbert and his students, concluding with a general remark about mathematical knowledge: Here we see how mathematical knowledge, no matter how objective it seems to be, is subject to change. I would like to emphasize this, without offending mathematics or wanting to cast doubt on its foundations and laws.243

238 239 240 241 242

Ibid., pp. 368–69 (emphasis original); see the German quotation in TOBIES 2019, p. 234. Ibid., p. 379. Ibid., p. 386. Math. Ann. 21 (1883), pp. 141–218, reprinted in KLEIN 1923 (GMA III), pp. 630–710. See the articles by H. Burkhardt and F. Meyer on potential theory (1900), and by L. Lichtenstein on recent developments in potential theory (1918) in volume II. 243 KLEIN 1979 [1926], pp. 87–89, 245–50, quotation at p. 250.

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5.5.2.3 Klein’s Seminar on the Theory of Abelian Functions (1882) One expression of Klein’s unbridled productivity was his aforementioned seminar with Krazer, Staude, Domsch, and Weichold, which met at his home on seventeen occasions from November 6 to December 19, 1882. During the first meeting, Klein presented a detailed agenda for the seminar.244 He assumed that Riemann’s work and the contents of his own lecture course from 1880/81 were known, and he planned (I) to supplement this knowledge, (II) to study the concept of Abelian functions, the means to represent them, and (III) their properties and applications. Klein’s entry in the protocol book demonstrates how he organized the material, used the latest results, and how he did not limit himself to geometric methods but rather incorporated various approaches, including those developed in Berlin and Paris. The entry is worth quoting at length: I. Supplements to earlier developments A. The existence proofs. The Riemannian way of inferring the existence of certain related functions by beginning with the “surface” is just one among other possibilities. Other mathematicians have sought to arrive at these existence proofs on different grounds. What is common among them is that they begin with the algebraic equation. Incidentally, I distinguish the following approaches: 1. The modern geometrical thinking. Homogeneous coordinates, determinants. Clebsch and Gordan (1866), Brill & Noether (Annalen 7), more recent works by [Max] Noether. Compare also Clebsch-Lindemann, Lectures. 2. The number-theoretical approach. Method of the largest common divisor. Kronecker in the 91st, Dedekind-[Heinrich] Weber in the 92nd volume of Crelle’s Journal. 3. The method of series developments. Weierstrass’s lectures. – Articles by Christoffel in the Annali. B. Extension of the class of functions. We have so far only considered those functions that 1) are many-valued only due to periodicity, and 2) do not have a substantially singular point. In the first respect, an extension is possible using the functions w which reproduce themselves as αw+β; Prym considered these in volumes 69 and 71 of Crelle’s Journal. If β = 0, and α is a root of unity, then w is algebraic, a so-called root function. In the second respect, the prime functions of Weierstrass will be investigated. C. Relationhips between our functions. In my lecture course, I discussed the following relationships: 1. Abel’s theorem, 2. The Riemann-Roch theorem. The following, which are still missing, also belong here: 3. Certain bilinear relations between the periodicity moduli of the integrals of the first and second kind. 4. The theorems on the exchange of parameter and argument in the case of the third kind. II. Abelian functions. A. The concept and definiteness of Jacobi’s inversion problem. We will proceed like this: 1. Differentiation of the groups of p points ξ1…ξp into ordinary and singular.

244 [Protocols] vol. 4, pp. 111–16.

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2. Verification that each ordinary group is fully defined by its integral sums. 3. Various ways of defining a group of points algebraically. α) by an algebraic function vanishing at the ξ β) by specifying the equation: (w – w1) (w – w2) …. (w – wp), where w is an algebraic function, possibly a root function. 4. Accordingly, definition of the term “Abelian” function. 5. Special formulations and extensions: The “Riemannian” inversion problem, The extended inversion problem of Clebsch-Gordan. B. Treating the inversion problem using ϴ-functions. 1. The empirical introduction of the ϴ-function by Riemann. 2. The transcendental function Tξη of Clebsch-Gordan. 3. The construction of the ϴ-function by Weierstrass. C. Starting from the general ϴ-functions. 1. The doctrine of general ϴ-relations (in modern development); 2. The transition back to the integrals in the case of p = 2, Göpel;245 3. The same in the case of p = 3, Schottky. III. Properties and application of Abelian functions. A. Division and transformation. From Hermite up to the most recent investigations. B. Geometrical applications. 1. Clebsch’s theorems in volume 63 of Crelle’s Journal. 2. The theorems of Roch and Lindemann. 3. Special interpretation of hyperelliptic functions by elliptic coordinates of a higher space.

Klein wanted this seminar to serve as the basis for a later lecture course. He concluded the seminar agenda quoted above with the words: “In the following, the topics mentioned here will not be discussed in a systematic sequence but rather when the right time and opportunity arise.”246 Adolf Krazer gave five presentations on the topic of theta functions, and he would incorporate much from this seminar into his Habilitation thesis (see Section 5.4.1). Otto Staude, who gave two presentations, would likewise write his Habilitation thesis in this area of research. On four occasions, Guido Weichold presented on the theory of hyperelliptic functions according to Weierstrass, and his talks were informed by Weierstrass’s lectures.247 Weichold submitted his dissertation (on symmetrical Riemann surfaces and the periodicity moduli of the related Abelian normal integrals of the first kind) on March 3, 1883.248 Paul Domsch’s dissertation – “Ueber die Darstellung der Flächen vierter Ordnung mit Doppelkegelschnitt durch hyperelliptische Funktionen” [On the Representation of Surfaces of the Fourth Order with a Double Conic Section by Means of Hyperelliptic Functions], which he submitted on October 15, 1884 – concerned a geometric application of this research area.

245 246 247 248

See Crelle’s Journal 35 (1847), pp. 277–312; and pp. 313–17 (Jacobi on Adolph Göpel). [Protocols] vol. 4, pp. 112–16. Ibid., pp. 167–99. Weichold’s dissertation is still cited on occasion; see, for instance, BÉLANGER 2010 (though the latter work is based above all on PONT 1974).

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5.5.2.4 Openness vs. Partiality Klein’s openness in research methods is reflected in his approach to teaching, in his and also in his students’ work. In contrast, the Berlin mathematicians stuck firmly to their specific approach. While on a research trip to Paris in March of 1883, for instance, Walther Dyck, who had applied Kronecker’s methods in his Habilitation thesis, spent time with the Berlin-trained H.A. Schwarz and claimed that Schwarz exclusively sang the praises of his teacher Weierstrass. As Dyck reported to Klein, Schwarz seemed like a conspicuously “Teutonic travel companion.” Schwarz was “anti-French” and his behavior was “crude and almost blustering.” In response, Klein commented (in exaggerated terms) that Schwarz’s behavior was typical of Berlin-trained mathematicians in general: You are now at the point where I would like you to be: You are able to see the state of affairs in Germany from an outsider’s point of view and, above all, you are able to recognize the true character of the Berlin school. Achievement in one’s own narrow area of research and, beyond that, an unbelievably narrow perspective – that is its signature. As to how this common character is expressed in one person or another, this is relatively unimportant. My position is that one should embrace what is good from everyone (as long as the bad doesn’t get too bad), and one should steadily pursue the goals that one regards to be right, regardless of whether they might differ from the commonly held opinion at the time.249

A uniform “Berlin school” did not exist. One could speak, as mentioned above, of a Weierstrass school; Kronecker, in contrast, was not interested in forming a school of mathematical thought. The Berlin mathematicians achieved excellent results in their fields of research, but did not sufficiently recognize other mathematical approaches. Klein rejected this attitude as being biased. Richard von Mises also criticized the one-sided orientation of Weierstrass’ student Schwarz: Thus, despite the intuitive geometric approach of many of his works, he [H.A. Schwarz] nevertheless contributed to the one-sidedness that so strongly characterized the Berlin mathematical school. This attitude was consciously combated and overcome by Felix Klein, and his achievement in this regard accounts for much of the success that the Göttingen school has enjoyed over the last few decades.250

That the mathematicians in Berlin fomented reservations against other styles of thinking is also evident from Otto Hölder’s letters to his parents: Life is very lively here with Klein. Obviously, some of this is also mere posturing [Macherei]. In particular, they have no preference for rigorous mathematical proofs here, so that a student of Weierstrass and [Paul] du Bois-Reymond is often negatively affected by this. I have more than enough to do with reworking that stuff at home, and now Klein wants to give me a problem to present on in the seminar. One must also read a great deal, and I have already gotten into the habit of the superficiality of the men here, which I despise by my very nature.251

249 Quoted from HASHAGEN 2003, p. 158 (a letter from Klein to Dyck dated March 30, 1883). 250 V. MISES 1921, p. 495. On R. von Mises, see SIEGMUND-SCHULTZE 2004, 2018, and 2020. 251 Quoted from HILDEBRANDT et al. 2014, pp. 143–44 (Hölder to his parents, May 6, 1884).

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Hölder’s opinion, which he thus expressed after being in Leipzig for two weeks, was clearly prejudiced by his time in Berlin, and it would soon change (see 5.4.1). In sum, Klein and his students appreciated and used the results from Berlin, whereas mathematicians from there often disregarded results produced elsewhere. Sophus Lie expressed this in clear terms: For your part, you are fair to the mathematicians in Berlin, whose work you understand and appreciate. The Berlin school, on the contrary, has long made efforts to downplay your work, if not ignore it entirely. These men have seldom paid attention to any of your brilliant geometric studies. It was only your relationships – with Poincaré, on the one hand, and with Fuchs on the other (to whom you spoke such harsh truths) – that made your great power clear to these gentlemen, even though they still do not understand you. Although I consider you the winner in this struggle with the Berliners, I believe that it would be good for you to ignore these stressful matters for a few years, and all the more so because you are not always in the best of health.252

Lie wrote these words to Klein at a time when Klein had to decide whether to accept a job offer in the United States (see Section 5.8.1). Klein’s relationship with Henri Poincaré and Lazarus Fuchs, to which Lie referred, will be discussed below. 5.5.3 Looking Toward France Klein followed the latest scholarly literature closely. The correspondence that he began with Henri Poincaré and the rivalry that developed between them in the same area of research can only be explained from a broader perspective. 5.5.3.1 French Contributors to Mathematische Annalen Klein kept an eye on foreign scholarship, but he did so not only in the interest of his own field of research. He was also intent on attracting French authors for Mathematische Annalen. After Camille Jordan’s article was published in volume one of the Annalen, it would not be until volume ten, when Klein and Adolph Mayer took over the editorship, that the next contribution from a French mathematician appeared in the journal: “Lettre de M. Ch. Hermite à M. P. Gordan” (Math. Ann. 10 [1876], pp. 287–88). This is noteworthy for two reasons. First, when Hermite published work in German journals, his venue of choice was at first Crelle’s Journal (which was then known as Borchardt’s Journal).253 Second, Hermite was influential and had numerous students. His opinion of Klein’s work would still be important as late as the 1890s (see Section 7.4.4). Invited by Klein in 1879, Georges Henri Halphen, who had completed his doctoral degree under Hermite in 1878, was the next French mathematician to

252 [UBG] Cod. MS. F. Klein 102: 690/1 (Lie to Klein, in December of 1883, an undated letter). 253 See TOBIES/ROWE 1990, p. 44. Regarding Hermite and Klein, see GOLDSTEIN 2011a.

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contribute to Mathematische Annalen. Already in his Munich seminar, Klein had arranged for his students to analyse Halphen’s publications.254 While still in Munich, too, Klein had sent a letter to Darboux (dated May 29, 1880) in which he inquired about additional French authors: Recently, I have become highly interested (with respect to the many questions that I have faced while studying elliptic functions) in general function theory, about which I was unfortunately only too ignorant until now. Aside from your brilliant studies, I am now captivated, for instance, by the recent notes by Picard. Who is this Picard? Who, moreover, is Appell, whose name one now so often reads in investigations of this sort? When I come to Paris, I hope to get to know all of these gentlemen personally and to speak with them at length. Of course, however, my new position in Leipzig has made it more than doubtful that I will be able to make a trip to Paris this fall.255

Klein would not be able to return to Paris until 1887 (see Section 6.3.6.1). In 1880, Darboux immediately put him in touch with Émile Picard and Paul Appell, both of whom, incidentally, were related to Hermite by marriage. Klein finally got around to writing to Picard, Appell, and Henri Poincaré in June of 1881. Not long thereafter, four French authors published in Mathematische Annalen (vol. 19, 1882) – the three named above and Georges Brunel, who studied under Klein (see Table 6 and Section 5.4.2.1).256 Works by Picard were the topic of presentations during Klein’s first seminars in Leipzig: F. Büttner presented on Picard’s function-theoretical theorems; and Hurwitz gave a talk on some of Picard’s publications in Comptes Rendus (1879, 1880, 1881).257 Hurwitz stressed that Picard’s conception of function theory was closely related to Weierstrass’s, and he was able to associate Picard’s theorems with certain aspects of Riemann’s function theory. As early as this seminar presentation, Hurwitz was already able to generalize one of Picard’s theorems. Going forward during the semester break, Hurwitz sent his results not only to Klein but also to Picard. Because Hurwitz waited in vain for a reply from Picard, Klein wrote the following to him on October 24, 1881: “It seems as though Picard has not published any further. Do not be upset about this; the French are never as honest as we are, and one has to consider this in advance when interacting with them.”258 This remark was based, first, on the experiences of Sophus Lie and Klein, who thought that French mathematicians had not (or had insufficiently) cited their work. That they were upset about this is documented in their letters. Second, Klein had realized that Poincaré – with whom he had since begun to correspond – was not as willing to disclose his methods as Klein himself was accustomed to doing with his students, with Lie, and with his Italian and British colleagues (among others).

254 255 256 257 258

[Protocols] vol. 1, pp. 197, 202. [Paris] 70: Klein to Darboux, May 29, 1880. On the contributions of other French authors in Math. Ann., see TOBIES 2016, pp. 111–16. [Protocols] vol. 2, pp. 165–68 (March 7, 1881); vol. 3, pp. 56–60 (July 25, 1881). [UBG] Math. Arch. 77: 52, fol. 78v (a letter from Klein to Hurwitz dated October 24, 1881).

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At that time, Poincaré’s work was highly relevant to Klein’s own research, and it influenced his thinking considerably. As I already mentioned in Section 4.2.3, the issue at hand was the theory of uniformization (as it would be called later). The importance of this question to the development of mathematics can be measured by the fact that Hilbert’s 22nd problem (1900) involves the following question: “How can analytic relations be uniformized by means of automorphic functions?” Put simply, the goal of uniformization is to parameterize algebraic curves in two variables, that is, to replace the variables with functions that only depend on one variable. For instance, a unit circle, which is given by x2 + y2 = 1, can be parameterized by replacing x and y with cos α and sin α, respectively.259 It was the work done by Klein and Poincaré that largely served, twenty years later, as the starting point for Hilbert’s famous problem. 5.5.3.2 Klein’s Correspondence with Poincaré Klein began his correspondence with Poincaré on June 12, 1881 with the following words: Dear Sir! Your 3 notes in the Comptes Rendus “Sur les fonctions fuchsiennes,” which I read only yesterday and with which I am still only fleetingly acquainted, are so closely related to the considerations and efforts with which I have busied myself in recent years that I had to write to you.260

Klein explained some of his earlier work, cited Halphen regarding a special class of functions, and informed Poincaré that, before Halphen, H.A. Schwarz had already developed the same results (a fact that Poincaré was still unaware of). Toward the end of the letter, Klein formulated his views on this research field in four points. His summary reads like a research agenda that Klein hoped to pursue via his correspondence with the French mathematician: 1. Periodic and doubly periodic functions are merely examples of single-valued functions with linear transformations into themselves. It is the task of modern analysis to determine all of these functions. 2. The number of these transformations can be finite; such is the case with the equations of the icosahedron, octahedron […], which I analyzed in my earlier work (Math. Annalen, vol. 9 [1875/76], vol. 12 [1877]) and which formed, for me, the basis for this set of ideas.

259 For recent discussion of uniformization, see SAINT-GERVAIS 2016 and David Rowe’s review of the original French version of this book in Historia Mathematica 41 (2014), pp. 98–102. – Henri Paul de Saint-Gervais is a fictitious person invented at a meeting at Saint-Gervais; “his” first names refer to Henri Poincaré and Paul Koebe. 260 The correspondence is published in KLEIN 1923 (GMA III), pp. 587–621 (quoted here from p. 587); Acta Math. 39 (1923), pp. 94–132; and in Henri Poincaré’s Œuvres (vol. 1, pp. 26–65). The three notes by Poincaré referred to here by Klein were published on February 14, February 21, and April 4, 1881. For further discussion of the correspondence, see GRAY 2008; GRAY 2013, pp. 224–40; ROWE 1992b; and VERHULST 2012, pp. 34–43. – On Poincaré’s articles from 1880, which have been unknown for a long time, see GRAY/WALTER 2016.

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5 A Professorship for Geometry in Leipzig 3. Groups of infinitely many linear transformations which give rise to usable functions (groupe discontinu, in your terms) can be obtained, for example, by proceeding from a circular arc polygon that intersects a fixed circle at a right angle and whose angles are exact fractions of π. 4. One should deal with all such functions (as you are now in fact beginning to do); but in order to achieve concrete goals, we should limit ourselves to circular arc triangles and especially to elliptic modular functions.261

Klein concludes his first letter to Poincaré by asking him to pass along greetings to Hermite and by expressing his wish to exchange letters with him about their common ideas. At the beginning, this correspondence did in fact develop quite intensively. From June 12 to July 9, 1881, Klein sent five letters to Poincaré, four of which were answered. It was Poincaré who broke off their exchange, at the time when he was moving from Caen to Paris, where he was offered a permanent position as a Maître de conférences at the Faculté des Sciences.262 In these first letters, they communicated about their earlier work. Poincaré recognized and admitted that Klein had produced results in the field before he himself did, and he requested further information. That is, he asked questions, but he did not explain his own methods. Because Poincaré had difficulty accessing German journals in Caen, Klein helped him. He sent Poincaré offprints of his articles, and he asked his students (Dyck, Gierster, Hurwitz) to do the same. Klein mentioned other work (by Riemann, Brill/Noether, Schottky), upon which he had based his research, and he explained his own approach. At the same time, Klein was preparing Georges Brunel, who was then studying under him in Leipzig, to explain his (Klein’s) work to Poincaré. At this point, Brunel did not know Poincaré personally. David Rowe’s analysis of the letters from Brunel to Poincaré makes it clear that Brunel did accomplish the task set by Klein, though he did so with a heavy dose of nationalism. Brunel first wrote to Poincaré on June 22, 1881, and here one reads: As Frenchmen it is our duty to fight the Germans by all possible means, but fairly. By which I mean that we must forthrightly acknowledge what they have accomplished, but we must also not attribute everything to them. If in his theory of modular functions Mr Klein has already published certain special results in the theory of Fuchsian functions, I find it only fair that you should render him justice, as you say. If he has not gone further, so much the worse for him!263

Klein knew nothing of this, and he unselfishly continued to share his latest knowledge in his letters to Poincaré. Moreover, he also fulfilled his promise to Darboux that he would guide Brunel toward producing his own publishable work (see Section 5.4.2.1). Klein’s generous responses to Poincaré demonstrate the sort of unselfishness that he practiced with his students in order to realize his envisioned research pro-

261 KLEIN 1923 (GMA III), pp. 588–89. 262 A Maître de conférences is one rank below a full professorship in France. 263 Quoted from ROWE 2018a, p. 125 (a letter from Brunel to Poincaré dated June 22, 1881).

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gram. Klein expressed how far he had come, but he also identified the places where he had reached an impasse. A good example of this can be seen in his letter dated July 2, 1881, in which he provided detailed answers to questions that Poincaré had asked him (in a letter from June 27, 1881). Klein explained his approach, offered references to related scholarly literature, and wrote about his own progress. However, he also remarked: “I cannot prove that this fundamental space, together with its replications, covers only a part of the complex plane […].264 Here, one can clearly see Klein’s desire to cooperate. In his reply, Poincaré limited himself to asking where he could find the literature that Klein mentioned and whether he could cite one of Klein’s theorems, but he revealed nothing about his own approach. In a letter dated July 9, 1881, Klein responded that Poincaré could of course cite his theorem and that he was puzzled by Poincaré’s insistence on designating classes of functions as fuchsiennes or kleinéennes, recommending instead that they should be called “functions with linear transformations into themselves.” Klein explained his assessment of Dirichlet’s principle – which he had spent so much time studying in preparation for the winter semester of 1881/82 – and the circumstances in which the aforementioned “general existence proof” was valid. Up to this point, Klein only had a general idea of Poincaré’s work, for the latter had, as mentioned, explained very little to him, and Klein had not yet taken the time to look into the matter more deeply. Klein was busy then with his book on Riemann’s theory of algebraic functions and their integrals (see Section 5.5.1.2). Yet even before Klein completed this work (on October 7, 1881), he had written the following to Hurwitz on September 20: In the winter semester, I would now like to make my investigations into the Dirichlet principle the focus of the presentations. Also, in order for my time not to be too divided, I would like to cancel my special lecture course. However, I am excited by these matters related to Poincaré, and I hope to formulate certain beautiful theorems in this regard. That said, it is impossible to do everything at once.265

By October 24, 1881, Klein indeed seemed to have been influenced by Poincaré’s note published in August of that year. Hurwitz thus heard from him: “In a study from the beginning of August, Poincaré asserts that every algebraic irrationality can in fact be solved by fonctions fuchsiennes.”266 After Klein had finally taken the time to read Poincaré’s articles with greater attention, he resumed their correspondence on December 4, 1881. Klein congratulated Poincaré on his proof (from August 8, 1881) of the theorem “that every linear differential equation with algebraic coefficients can be integrated by means of zeta-Fuchsian functions […] and that the coordinates of the points of any algebraic curve can be expressed by means of Fuchsian functions of an auxiliary vari264 KLEIN 1923 (GMA III), pp. 597–98 (emphasis original). 265 [UBG] Math. Arch. 77: 51, fol. 76 (a letter from Klein to Hurwitz dated September 20, 1881). 266 Ibid., 77: 52, fol. 78v (a letter from Klein to Hurwitz dated October 24, 1881).

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able.”267 In addition, Klein suggested that Poincaré should submit an article (of any length) to Mathematische Annalen. Poincaré agreed to do so immediately, but he did not fulfill Klein’s request that he might, in his contribution to the journal, “[…] provide the necessary details concerning the methods of your proofs, that is, about the way that you actually construct the functions in question.”268 Klein did not insist on this, and instead felt incited to continue with his own work. He wrote to Hurwitz: “Poincaré will send me a letter that I will print in the Annalen and furnish with my own commentary. By Easter at the latest, my ‘strict’ existence proofs will have to be in order.”269 Klein succeeded in formulating three new theorems in this research area, all of which were published in 1882. The correspondence between Klein and Poincaré continued until September 22, 1882. In the end, Klein no longer replied to him. For his part, Poincaré was persuaded by Mittag-Leffler to contribute to his journal Acta Mathematica.270 5.5.4 Three Fundamental Theorems Robert Fricke, who later summarized, systematized, and expanded on Klein’s results, made the following classifying remarks in the year 1919: With good reason, what is regarded as Klein’s greatest achievement during his function-theoretical period is his discovery of the theorems that he himself called “fundamental theorems,” and which are known today as “uniformization theorems.” One is familiar with the surprising way in which Abel and Jacobi had contributed to the development of the theory of elliptic functions. Legendre had considered elliptic integrals in their dependence on the integration variables. By investigating all functions of this system of interrelated variables in their dependence on the integral of the first kind, Abel and Jacobi achieved “single-valued” functions; they had recognized the “uniformizing” variable for the system of these functions. In the sense of Riemannian theory, this discovery is related to the algebraic entities of genus 1. It was Klein who discovered the various kinds of uniformizing variables for algebraic entities of any given genus; this is one of the greatest achievements that will remain associated with his name.271

In what follows, I will classify Klein’s three fundamental theorems in the order in which they were created, and this will also provide an opportunity to take a closer look at his correspondence with Poincaré. The first two theorems were later referred to by Fricke as the “loop-cut theorem” (Rückkehrschnitttheorem) and the “limit-circle theorem” (Grenzkreistheorem), respectively, and Klein thought that

267 KLEIN 1923 (GMA III), p. 602. – Klein wrote this in French: « que toute équation différentielle linéaire à coefficients algébriques s’intègre par les fonctions zétafuchsiennes » and « que les coordonnées des points d’une courbe algébrique quelconque s’expriment par des fonctions fuchsiennes d’une variable auxiliaire » 268 Ibid, p. 604 (a letter from Klein to Poincaré dated December 10, 1881), emphasis original. 269 [UBG] Math. Arch. 77: 55, fol. 82 (a letter from Klein to Hurwitz dated December 17, 1881). 270 See also NABONNAND 1999. 271 FRICKE 1919, p. 276. For Klein’s explanation in English, see KLEIN 1979 [1926], pp. 359–61.

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these names were fitting.272 The third, general fundamental theorem includes the other two as the most important special cases. 5.5.4.1 The Loop-Cut Theorem (Rückkehrschnitttheorem) The concept of the loop-cut on Riemann surfaces goes back to Riemann himself (see Section 3.1.3.1). In his letter to Poincaré dated July 2, 1881, Klein had mentioned that he had a possible theorem in mind, which he realized was provable in September of that year while taking a walk on the island of Borkum. After he had worked this out and submitted it for publication in the Annalen (the article is dated January 12, 1882), Klein informed Poincaré: It is possible to solve every algebraic equation f(w, z) = 0 as soon as one has drawn p independent loop-cuts on the related Riemann surface in one and only one way, by means of w = φ (η), z = ψ (η), where η admits a discontinuous group of the type that you mentioned then in connection to my letter. This theorem is so elegant because this group has exactly 3p–3 essential parameters, that is, just as many as there are moduli for equations of the given p.273

The sentence in italics is the loop-cut theorem. Here, η is a function whose complex values Klein interpreted on a sphere. He demonstrated what he had still been unable to do in July of 1881: “Then, on the η-sphere, the image of our cut Riemann surface covers a 2p-fold connected surface part which is simply spread out [i.e., single-layered] everywhere.” Moreover: “The infinitely many analytical continuations of our map do not cover the η-sphere several times.” Klein used an intuitive proof in his article, which bore the title “Über eindeutige Funktionen mit linearen Transformationen in sich” [On Single-Valued Functions with Linear Transformations into Themselves] and which was printed directly after Poincaré’s article in the Annalen.274 In his later judgement: My proof, which clearly forced itself upon me, was such that I could always follow – in my imagination – any modifications of the Riemann surface or its cross-section system; it was therefore quite intuitive in character. […] However, this was the basis for the proof of conti275 nuity.

5.5.4.2 Theorem of the Limit-Circle (Grenzkreistheorem) According to his own account, Klein devised this theorem, too, on an East Frisian island, namely on Norderney, where he spent the Easter vacation in 1882. Because of a storm, he was only able to stay there for eight days. The idea for the 272 KLEIN/FRICKE 2017 [1912], pp. 228, 359 (Here the translation are principal-circle theorem and boundary-circle theorem; we use the traditional terms; see KLEIN 1979 [1926], p. 354). 273 KLEIN 1923 (GMA III), p. 616 (Klein to Poincaré dated January 13, 1882), emphasis original. 274 Math. Ann. 19 (1882), pp. 565–68; reprinted in KLEIN 1923 (GMA III), pp. 622–26. 275 KLEIN 1923 (GMA III), p. 584 (Here, Klein alluded to Koebe’s proofs of continuity in 1912).

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theorem came to him during a sleepless night (from March 22nd to the 23rd), and he worked it out further and wrote it down in the subsequent days. Hurwitz was one of the first to hear about it. On a postcard dated March 22, 1882, Klein informed him that he was traveling to Düsseldorf because he was not feeling well; four days later, Hurwitz received the following news: “I am doing much better, especially as regards mathematics. I have discovered new beautiful theorems about single-valued functions with linear substitutions into themselves.” On March 26, 1882, Klein wrote to him at greater length: I have to write to you about a very simple theorem that I discovered over the past few days and have already sent to Teubner. You are familiar with the table “On Transformations of the Seventh Order” in vol. 14 of the Annalen, where the regular Riemann surface p = 3 is mapped onto a circular-arc tetradecagon [a fourteen-sided polygon] whose edges intersect a principal circle at a right angle. I want to place the latter in the plane of a variable η. Then η is completely unbranched on the surface p = 3, and every unbranched function on the surface is a single-valued function of η, which reproduces itself linearly. The point now is that, precisely in this sense, there exists one and only one η-function on every given Riemann surface! (That is, independent of any special cutting of the surface.) As you see, things are moving ahead. Thus I would like all the more to engage you to work on this as well. For myself alone, the problems are too difficult. Moreover, Rausenberger has also submitted two small notes on the formation law of the single-valued functions in question. If we all work together, soon real progress will be made in all matters related to algebraic functions.276

Klein’s result was the limit-circle theorem: “that one can, in one and essentially only one way, uniformize any Riemann surface with p ≥ 2 without branch points by means of automorphic functions with a limit-circle.”277 His article appeared under the title “Über eindeutige Funktionen mit linearen Transformationen in sich (Zweite Mitteilung)” [On Single-Valued Functions with Linear Transformations into Themselves (Part Two)].278 Klein explained later: “Summarizing the essentials, we can see that the modular functions, or more generally the triangle functions, have led to general automorphic functions, which are single-valued within a circle and have this circle as limit-circle!”279 From the standpoint of uniformization and the theory of automorphic functions, his limit-circle theorem corresponded to Poincaré’s result, which he had achieved by proceeding from differential equations based on the work of Lazarus Fuchs.280 Klein’s claim in his letter to Hurwitz – “For myself alone, the problems are too difficult” – and his call for cooperation were indicative of the way he worked. 276 [UBG] Math. Arch. 77: 64 (a letter from Klein to Hurwitz dated March 28, 1882), emphasis original. The article with the table that Klein mentions here is his own: “Ueber die Transformation siebenter Ordnung der elliptischen Funktionen,” Math. Ann. 14 (1878), pp. 428–70. See also Fig. 21 in Section 4.2.3. 277 KLEIN 1979 [1926], p. 359. – It should be mentioned here once more that Klein introduced the term automorphic functions in 1890 (see Section 5.5.1.1). 278 Math. Ann. 20 (1882), pp. 49–51; reprinted in KLEIN 1923 (GMA III), pp. 627–29. 279 KLEIN 1979 [1926], p. 355. 280 On this topic, see already SCHOLZ 1980, pp. 198–200, and GRAY 1984.

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He recognized that, by formulating the two theorems, he had not closed off this research area but had rather opened it up. Adolf Hurwitz and Otto Rausenberger (a teacher in Frankfurt am Main and a former student of Leo Koenigsberger) were willing to cooperate. Klein had persuaded Hurwitz to turn his seminar presentation from February 21, 1881 into an article. Urged on by Klein, Hurwitz finished this piece even before he would submit his Habilitation application (see Section 5.4.1). Klein’s letter to Hurwitz dated April 16, 1882 indicates how important it was to him for the world to know, precisely, when he had begun working with his students on this new class of functions: In your introduction, could you include a sentence in which you mention, without any especially sharp language, that you began to work on these things at my instigation and independent of Poincaré? The first article by Poincaré is from February 14, 1881. My concern is to underscore the continuity that exists between our current efforts and those of 1879 (in the summer). With the same goal in mind, I referred in my latest note to my table [Klein’s “main figure”; see Fig. 21] on the transformation of the seventh order.281

Hurwitz’s article in Mathematische Annalen had the same title as his seminar presentation: “Ueber eine Reihe neuer Functionen, welche die absoluten Invarianten gewisser Gruppen ganzzahliger linearer Transformationen bilden” [On a Series of New Functions that Form the Absolute Invariants of Certain Groups of Whole-Number Linear Transformations]. He draws upon Klein’s theorems and his geometric methods, and Klein added a footnote on the first page of Hurwitz’s article, explaining that the work stems directly from Hurwitz’s presentation in his seminar and that there is evidence of this in his protocol book.282 The first manuscript that Rausenberger had submitted to the Annalen still contained some errors, but Klein discovered in it a “useful basic idea” for this research area. In response, Klein sent him, on January 4, 1881, a detailed list of references to literature and invited him to Leipzig.283 The results were Rausenberger’s three articles in volume 20 of Mathematische Annalen. In the third of these – “Ueber eindeutige Functionen mit mehreren, nicht vertauschbaren Perioden I” [On Single-Valued Functions with Multiple Non-Interchangeable Periods I] (April 1882) – Rausenberger corroborated Klein’s view of the correct order of historical events: “I would like to take this opportunity to mention that, in letters sent to me 1½ years ago, Mr. Klein defined the essential task of his function-theoretical investigations as follows: To establish all functions with any given linear transformations into themselves.”284

281 [UBG] Math. Arch. 77: 69 (a postcard from Klein to Hurwitz dated April 16, 1882). – See Section 4.2.3. 282 Math. Ann. 20 (1882), pp. 125–34 (with Klein’s note on p. 125). See also [Protocols] vol. 2, pp. 138–46. 283 [UB Frankfurt] Lorey Nachlass, entry no. 424 (Klein to Rausenberger, January 4, 1881). 284 Math. Ann. 20 (1882), pp. 187–212, at pp. 187–88 (emphasis original).

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Whereas Klein had cooperated well with many German and foreign mathematicians, his relationship with Poincaré became increasingly competitive. Yet Klein had offered to cooperate, and later he stressed to him once again: “For me, a lively connection with like-minded mathematicians has always been the precondition for my own mathematical production.”285 In a letter dated May 7, 1882, Klein also explained how he had proven his theorems by means of continuity and he provided a thorough description of his approach. Yet Poincaré merely responded that Klein’s “second lemme […] il est probable que nous l’établissions de la même manière,” and that he had likewise used the continuity method, though their methods differed in certain details.286 When Klein asked him how he had demonstrated the convergence of a corresponding series, Poincaré avoided the question by remarking that it would take too long to explain the two examples in a letter and that his results would be published soon. Nevertheless, Klein maintained his openness. He explained to Poincaré yet again “how I use the continuity method,” and in the same letter (May 14, 1882) he also mentioned additional ideas for proofs: “When I recently visited him in Göttingen (on April 11th), Mr. Schwarz mentioned to me another, quite different proof, which is also based on considerations of continuity.” Klein went on to explain Schwarz’s “beautiful train of thought”287 in the interest of pursuing a common research agenda with Poincaré. Poincaré’s letter from May 18, 1882 makes it clear that he had in fact realized that Klein’s work on the limit-circle theorem and his own recent work yielded nearly identical results. At this point, he opened up to Klein by explaining his approach, he adopted Schwarz’s line of thinking, and he sent Klein some of his older studies (which Klein had requested many months earlier). Erhard Scholz already analyzed how Poincaré used Schwarz’s idea to prove a uniformization theorem concerning analytic curves.288 Poincaré began the article in question by referring to a “beau théorème de M. Schwarz” and to Dirichlet’s principle, which Klein had explained to him.289 Here he makes no mention of “fonctions fuchsiennes,” but he does refer to “la surface de Riemann.”

285 KLEIN 1923 (GMA III), p. 610 (a letter from Klein to Poincaré dated April 3, 1882). 286 Ibid., p. 614 (a letter from Poincaré to Klein dated May 12, 1882): “[…] j’emploie comme vous la continuité, mais il y a bien de manières de l’employer et il est possible que nous différions dans quelques détails.” 287 KLEIN 1923 (GMA III), pp. 615–16 (a letter from Klein to Poincaré dated May 14, 1882). 288 See SCHOLZ 1980, p. 201. 289 Henri Poincaré, “Sur un théorème de la théorie générale des fonctions,” Bulletin de la Société Mathématique de France 11 (1883), pp. 112–25. Regarding this article, see also GRAY 2013, pp. 247–51; and SCHOLZ 1980, pp. 200–03.

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5.5.4.3 The (General) Fundamental Theorem Klein explained his new general theorem to Poincaré in a letter from May 7, 1882 – that is, before his article was even published. In his last letter to Poincaré (September 19, 1882), he discussed the theory in greater detail and stated that his two other theorems (the loop-cut theorem and the limit-circle theorem) could be classified as special cases of it. After pointing out an inaccuracy in Poincaré’s work, Klein wrote: Regarding my own work, I am limiting myself to demonstrating the geometric approach by virtue of which I understand the new functions to be defined in the Riemannian sense. Owing to the nature of the matter itself, there are many points of contact with your geometric approach to the subject. The most general group that I am taking into consideration is created from an arbitrary number of “isolated” substitutions and from a number of groups with a “principal circle” (which can be real or imaginary, or even degenerate to a point) by means of “sliding into one another” (Ineinanderschiebung). The theorems in my two other notes in the Annalen are then subsumed as special cases under the general theorem, which goes something like this: To every Riemann surface with arbitrarily given branching and cutting one and only one η-function of the pertinent type always belongs.290

Klein also told Poincaré that Sophus Lie was with him in Leipzig and was planning to come to Paris, and that he had heard from Mittag-Leffler that Poincaré was busy “with elaborating substantial treatises” for the new journal Acta Mathematica. Klein published his results as “Neue Beiträge zur Riemann’schen Functionentheorie” [New Contributions to Riemannian Function Theory].291 He had completed this article on October 2, 1882, and he sent an offprint also to Poincaré, but he never heard back from him. In this article, Klein made sure to express his appreciation of Poincaré’s work: One is familiar with the long series of brilliant publications with which Mr. Poincaré has drawn general attention to these functions. For my part, I have already been working on similar ideas for a long time, and I supplemented Poincaré’s publications with two notes, in which I formulated certain theorems that may be of considerable importance to the applications of these new functions. In the following, I will be concerned with demonstrating, in a coherent and complete form, the general course of ideas that led me to these theorems. To this end, I will examine in the third section a relatively comprehensive class of single-valued functions with linear transformations into themselves. I will explain at length their manner of existence and provide the means to construct the related linear substitutions from independent items of determination. Then, in my fourth section, I will formulate a general theorem, which I have designated a fundamental theorem on account of its importance, and which contains in itself the results of my two aforementioned notes as special cases.292

290 KLEIN 1923 (GMA III), pp. 618–19 (a letter from Klein to Poincaré dated September 19, 1882). 291 KLEIN 1883; reprinted in KLEIN 1923 (GMA III), pp. 630–710. 292 KLEIN 1883, pp. 142–43 (emphasis original). In a footnote (p. 142), Klein refers to Poincaré’s publications and remarks: “As I have heard, Poincaré has further refined his ideas, and they will appear shortly in Mittag-Leffler’s new journal.”

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In his fourth section, Klein asked “[…] on which Riemann surfaces of the genus p with n given branch points of a particular index normal functions η of a certain type can exist.”293 With the following explanations, he arrived at his fundamental theorem: The type is defined by connecting – on our surface – certain pairs of branch points through cuts Q, then by adding any loop-cuts R that do not disconnect the surface, and finally by constructing so many cut systems (π, ν) that the dissected surface can be transferred in an everywhere one-to one way to a piece of the plane. For the sake of brevity, I consider those functions η which depend on one another linearly, to be identical. Our fundamental theorem then states: That on every Riemannian surface (p, n, l, k) there exists always one and only one normal function of any given type.294

Depending on how the Riemann surface is dissected, it is possible to derive from this fundamental theorem a series of special uniformization theorems, including, as mentioned above, the loop-cut theorem and the limit-circle theorem as the most important cases. With this article, Klein thought that he had reached a high point in his mathematical productivity – in retrospect, in fact, he felt that it represented the peak. He also believed, however, that there was additional work to be done on the topic. Thus, knowing the situation well, he predicted in his article that Poincaré would soon provide necessary addenda. According to Klein, Poincaré was not only influenced, like himself, by the “geometric manner of thinking” and by a desire to “apply the continuity concept.” Beyond that, he [Poincaré] had also “wrestled successfully with the analytic formation law of the new functions and had also attempted, in the case of given algebraic irrationalities, to actually produce related single-valued functions with algebraic transformations into themselves by means of convergent processes.”295 As in the case of his other work, Klein hoped that younger mathematicians would take up the topic and advance it further: “By discussing the important different directions in which our fundamental theorem might be taken, I hope to induce other – perhaps younger – mathematicians to devote their energy to this promising area of research.”296 The encouragement that Klein provided to young mathematicians will come up repeatedly throughout the remainder of this book. Here I should only remark that, beginning in 1907, Klein cooperated with Emil Hilb (then Gordan’s and M. Noether’s assistant in Erlangen) and traced his fundamental theorems from the theory of automorphic functions back to his oscillation theorem of linear differential equations.297 Klein also created the conditions for others to devise the final, rigorous proofs of his fundamental theorems.

293 294 295 296 297

KLEIN 1883, p. 206. Ibid. Ibid., p. 144. Ibid. See KLEIN 1923 (GMA III), pp. 770–74.

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5.5.4.4 Remarks on the Proofs Robert Fricke made the following comments in 1919: A quarter century would have to pass, however, until all of the theorems formulated by Klein found unobjectionable proofs. The methods of proof at the beginning of the 1880s were still insufficient for this. Not until 1907 did P. Koebe begin to succeed in gradually providing rigorous proofs for all of Klein’s theorems.298

Because Fricke, who wrote this shortly after the First World War, mentioned only Paul Koebe’s rigorous proofs, it should be mentioned that, in 1907, Koebe and Poincaré published (analytic) proofs around the same time and that, in subsequent years, the Dutch mathematician Luitzen E.J. Brouwer also played a significant role in further developments.299 When Felix Klein was to celebrate his sixtieth birthday on April 25, 1909, in Göttingen, Hilbert arranged for Poincaré to be invited to give a series of lectures from April 22nd to the 28th,300 and numerous additional researchers from Germany and abroad participated. With Poincaré in attendance, Hilbert stressed, in his birthday speech in Klein’s honor, Poincaré’s great contributions as well: If […] I were to single out a special area of mathematics – now that we hear the names Poincaré and Klein together – which mathematician would not think of automorphic functions, the theory of which Poincaré first founded, but the rich design of which is to your [Klein’s] credit. This is the most profound matter that you [Klein] predicted – praesagiente animo – and for which you also produced ideas for proofs. Even today, you await their completion.301

The early, incomplete proofs by Klein and Poincaré, which were based on the idea of continuity, have already been analyzed in detail by Erhard Scholz in 1980. In the 1880s, Klein and Poincaré discovered partial answers that were correct in some respects and incorrect in others. Scholz compared their respective approaches, which were hampered by the inadequate topological concepts at the time, and concluded that the historical and methodological significance of their work did not lie in “their valid solution but rather in the problems that they posed.”302 That this is true more generally is indicated by the fact that Hilbert made this topic the subject of his aforementioned twenty-second problem. Klein supported the solution of this problem (for two variables) not only by repeatedly integrating the topic into his lectures, but also by conducting four seminars on it together with Hilbert and Minkowski (from 1905/06 to 1907).303

298 FRICKE 1919, p. 276. – See also TOBIES 2021a. 299 Poincaré’s proofs of the limit-circle theorem are also cited in the preface to FRICKE/KLEIN 2017 [1912], p. xxvi. – Regarding Brouwer, see VAN DALEN 2013. 300 For the titles of Poincaré’s lectures, see Jahresbericht der DMV 18 (1909) Abt. 2, pp. 27, 39, 78–79. 301 The complete original German text of Hilbert’s speech is printed in TOBIES 2019b, pp. 513– 14. The English translation here is from ROWE 2018a, pp. 198–99. 302 SCHOLZ 1980, p. 215. 303 These seminars are documented in [Protocols] vols. 23–26.

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These seminars focused on analyzing works by Klein, Poincaré, and also more recent results by E. Papperitz, R. Fricke, W. Wirtinger, Osgood, and others. Klein gave the majority of presentations himself; in 1906/07, Hilbert, Paul Koebe, and others also gave presentations. Koebe had earned his doctoral degree in 1905 under H.A. Schwarz in Berlin, after which he went to Göttingen, where he was able to finish his Habilitation in 1907. Koebe participated in Klein’s seminar in the summer of 1906, and he gave his first presentation during the following semester on November 14, 1906: “Über die conforme Abbildung der Halbebene auf ein Grenzkreispolygon” [On the Conformal Mapping of the Half-Space Onto a Limit-Circle Polygon].304 Not long thereafter, he published articles in the Göttinger Nachrichten and in Mathematische Annalen that contained (analytic) proofs for the limit-circle theorem. Here he used a deformation theorem that has since been named after him: the Koebe quarter theorem (Viertelsatz).305 Klein still hoped, however, for someone to demonstrate that his original idea of a continuity proof was viable. He also induced the Dutch mathematician Brouwer to devote attention to the topic.306 Brouwer had been a contributor to Mathematische Annalen since 1909 (vol. 67), and he presented his first results on the continuity issue in September of 1911 at the annual conference of the German Mathematical Society in Karlsruhe, where Klein presided over a special symposium on the theory of automorphic functions. On May 21, 1912, Brouwer was invited to give a lecture at the Göttingen Mathematical Society. In the Jahresbericht der DMV, it was announced that Brouwer intended to speak about how, according to the proof of his theorem about the invariance of domain, Klein’s and Poincaré’s continuity proof of the existence of linearly polymorphic functions on Riemann surfaces can be executed perfectly by means of his method of expanding the modular manifold. In addition to Fricke’s Würfelsatz [cube theorem], which plays a fundamental role throughout, the classical results by Klein and Poincaré will be used in the case of the limitcircle, whereas, in the remaining cases, an idea by Koebe will be used instead.307

SCHOLZ (1980) has analyzed Brouwer’s topological formulation of the proof structure and shown that the latter could lead to a complete continuity proof of the limit-circle theorem. All that was missing in the complex logical structure of the proof was a sixth, final step. Koebe ultimately borrowed Brouwer’s approach and used it to formulate continuity proofs for all three of Klein’s theorems. 304 In this presentation, Koebe concluded: “With this I have now proven the existence of those transcendentals, the knowledge of which forms the foundation for the uniformization of the solution of given linear homogeneous differential equations with rational functions as coefficients and with exclusively real branch points.” Quoted from [Protocols] vol. 25, p. 62. 305 Koebe published four articles with strict proofs for the uniformization of real algebraic curves and finally also for arbitrary analytic curves in Göttinger Nachrichten: Math.-physikal. Klasse, in 1907 (pp. 177–90, 191–210, 410–14, and 633–69). They were alternately submitted to the journal by Hilbert and Klein. The corresponding study by Poincaré is his article “Sur l’uniformisation des fonctions analytiques,” Acta Mathematica 31 (1907), pp. 1–63. 306 See VAN DALEN 2003; and also MEHRTENS 1990, pp. 257–99. 307 Quoted from Jahresbericht DMV 21 (1912) Abt. 2, p. 142.

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Klein was pleased that his original idea for a proof of continuity could be realised. The results were incorporated into the second volume of Fricke and Klein’s book on automorphic functions (FRICKE/KLEIN 1912). For the second volume of the ENCYKLOPÄDIE, Fricke (1913) and Leon Lichtenstein (1918/19) wrote informative summary articles about them. Lichtenstein illustrated Poincaré’s method with an example and he emphasized how Klein’s approach was different: “In contrast to H. Poincaré, F. Klein sought to draw stringent conclusions from the behavior of the mapping inside the two manifolds alone (the method of the open continuum). Klein’s method refrained from investigating the margin of the two entities – something that Poincaré had considered indispensable – and it was confirmed and perfected in articles by Paul Koebe (1912).”308 Koebe had written to Klein on March 30, 1912: Poincaré’s view was based on the opinion that the bounding polygons must form a necessary constituent in the proof of continuity. That this opinion is erroneous, that it is actually possible to give a precise justification of the continuity method in the sense of your approach, for which, in contrast to Poincaré’s, the property of the continua not to be closed [Ungeschlossenheit] seems to me to be the most characteristic, was the essential content of my Karlsruhe communication. It is precisely in this non-closedness, of which Poincaré expressly accuses you in bold type on p.236, Acta IV, that the lifegiving moment in the proof of continuity lies. […]309

In the third volume of his collected works, Klein included a detailed report on his and Fricke’s monograph on the theory of automorphic functions as well as a commentary on the proofs, which Klein had coauthored with Erich Bessel-Hagen.310 They attempted to demonstrate that Klein’s first ideas concerning a possibly rigorous proof for the fundamental theorem were essentially correct. It still bothered Klein that his first idea of a proof had been criticized by Poincaré and that, because of this criticism (as he thought), progress in this research direction had long been impeded (until 1912). Klein regarded his work on this subject as his most important result, thus that, on the occasion of celebrating the fiftieth anniversary of the day when he had earned his doctoral degree, he chose Koebe to be the main speaker at the event (see Section 9.2.3). To this day, Hilbert’s twenty-second problem, which concerns analytic relations with more than two complex variables, has yet to be fully solved.

308 Leon Lichtenstein, “Neuere Entwicklungen der Potentialtheorie: Konforme Abbildungen,” in ENCYKLOPÄDIE, vol. II.3.1 (1918/19), pp. 177–377, esp. 348–51. 309 [UBG] Cod. MS. F. Klein 10: 506 (Koebe to Klein on March 30, 1912). The related article is POINCARÉ (1884, p. 236): “Ainsi, il n’est pas évident que S est une multiplicité fermée et il est nécessaire de le démontrer, par une discussion spéciale à chaque cas particulier, avant d’affirmer qu’à tout point de S’ correspond un point de S. C’est ce que M. Klein a négligé de faire. Il y a là une difficulté dont on ne peut triompher en quelques lignes.” 310 See KLEIN 1923 (GMA III), pp. 731–41, 742–47.

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5.5.5 The Polemic about and with Lazarus Fuchs Klein instigated a heated polemic over the fact that Poincaré had named the class of functions he investigated after Lazarus Fuchs. This topic has already been discussed in detail.311 The context of this dispute needs to be outlined here, however, because several later events in Klein’s biography cannot be fully understood without some familiarity with how his feud with Fuchs began and unfolded. Poincaré’s approach to this particular class of functions differed from Klein’s. The Frenchman was inspired by differential equations that Lazarus Fuchs had analyzed.312 Poincaré requested and received permission from Fuchs to honor him with the term fonctions fuchsiennes (for the solution functions of the respective differential equations). Klein made the mistake of not realizing (or not recognizing at once) that a purely analytic starting point could also lead to the results that he himself had achieved by means of geometric methods. He obviously saw himself deprived of a field of research that he himself had developed from Riemann’s geometric approach. Concerned about his priority, Klein called Fuchs’s approach “ungeometric” and “flawed.” On June 19, 1881, Klein wrote to Poincaré: I reject the term “fonctions fuchsiennes,” though I understand well that Fuchs’s studies inspired your ideas. Essentially, all of such research is based on Riemann. Regarding my own development, the closely related work by Schwarz was […] of considerable importance. The work by Mr. Dedekind on elliptic modular functions in vol. 83 of Borchardt’s Journal [i.e., Crelle’s Journal] appeared at a time when I was already familiar with the geometric representation of modular functions (the fall of 1877). By virtue of their ungeometric form, Fuchs’s articles stand in deliberate contrast to the works named above. I do not dispute the great merits of Fuchs’s work on other aspects of the theory of differential equations, but in this particular case, his articles are not relevant. He only dealt once with elliptic modular functions, in a letter to Hermite (Borchardt’s Journal, vol. 83 [1876/77]) […], and here he made a fundamental error, which Dedekind then only too gently criticized.313

When Klein suggested to Poincaré on December 4, 1881 that he should submit his results to Mathematische Annalen, Klein also told him that he would like to add a footnote to Poincaré’s text, “in which I will explain how the whole matter appears from my perspective and how the research program that you are still engaged in was the guiding principle underlying my work on modular functions.”314 Poincaré agreed to this in his prompt response (December 8, 1881), so that the article was quickly ready to be printed. On January 13, 1882, Klein sent the proof sheets of

311 See GRAY 1984, 2000, and 2013; ROWE 1992b and 2018a, pp. 111–33; see also KING 2019. 312 See GRAY 2013, pp. 207–24; and SCHOLZ 1980, pp. 180–81, 198–99, 357–59. 313 KLEIN 1923 (GMA III), pp. 591–92 (a letter from Klein to Poincaré dated June 19, 1881). See Richard Dedekind, “Schreiben an Herrn Borchardt über die Theorie der elliptischen ModulFunctionen,” Crelle’s Journal 83 (1877), pp. 265–92. Here, Dedekind explained an error that Fuchs had made, but he added that this mistake did not substantially affect the main point of Fuchs’s interesting article. In a letter to Dedekind, Heinrich Weber commented that Dedekind had treated Fuchs very decently. See SCHEEL 2014, p. 169. 314 KLEIN 1923 (GMA III), p. 602.

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the article to Poincaré with his note, and he explained that, in this note, he protested “against the two terms fuchsiennes and kleinéennes; regarding the latter, one should instead cite Schottky and mention Riemann as the man to whom all of these investigations go back.”315 Although Poincaré had agreed to allow Klein to insert his note, he nevertheless wanted to stick with the terms that he had already used. In his Annalen article, he spoke of Fuchsian groups: “Je vais chercher d’abord à former tous les groupes discontinus formés de substitutions Si où les coëfficients αi, βi, γi, δi sont réels. Je les appelle groupes Fuchsiens.” He also spoke of Kleinian groups: “Reste à examiner le cas le plus général, celui où l’on ne fait aucune hypothèse sur les substitutions Si. Dans ce cas il y a encore des groupes discontinus que j’ai appelés Kleinéens et dont j’ai démontré l’existence et étudié le mode de génération par des procédés empruntés à la géométrie non-euclidienne à trois dimensions.” Moreover, he also spoke of Fuchsian and Kleinian functions or theta functions: “ϴ (ζ) s’appellera une fonction thétafuchsienne ou thétakleinéenne selon que le groupe correspondant sera Fuchsien ou Kleinéen.” Poincaré defined the properties of these functions and added: “Nous appellerons fonction Fuchsienne (ou Kleinéenne) toute fonction jouisant [sic] de ces deux propriétés.”316 In his footnote, Klein thanked Poincaré for the article and stressed the special role of “functions that seem fit to compete, in the theory of algebraic irrationalities, with Abelian functions, and which furthermore provide entirely new insight into those dependencies which are defined by linear differential equations with algebraic coefficients.” After this, Klein justified why he considered it “premature” to name the terms in question after Fuchs and himself: On the one hand, all of the studies that Mr. Schwarz and I have published on the topic at hand are concerned with “fonctions fuchsiennes,” about which Mr. Fuchs himself has never published anything. On the other hand, I have not yet published anything about the more general functions that Mr. Poincaré associates with my name; I merely brought the existence of these functions to Mr. Poincaré’s attention in our personal correspondence.317

Later, both Poincaré’s and Klein’s terminology would become established. For instance, the English mathematician William Burnside used Klein’s “automorphic functions” early on.318 Stanisław Kępiński, who studied under Klein and was the first Polish mathematician to publish in Mathematische Annalen (vol. 4, 1896, pp. 573–75), explained “Fuchsian functions of two variables” as follows: “In contrast to Poincaré’s Fuchsian functions (of one variable), which, according to Klein’s terminology, are a special case of general automorphic functions of one variable […].”

315 KLEIN 1923 (GMA III), p. 605. 316 Henri Poincaré, “Sur les Fonctions Uniformes qui se reproduisent par des Substitutions Linéaires,” Math. Ann. 19 (1882), pp. 553–64, quoted here from pp. 554, 557, 558. 317 Math. Ann. 19 (1882), p. 564. 318 See ADELMANN/GERBRACHT 2009. As already mentioned above, Klein introduced the concept of “automorphic functions” in an article of 1890, see Section 5.5.1.1.

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Lazarus Fuchs felt that he was being attacked. In a short note – “Ueber Functionen, welche durch lineare Substitutionen unverändert bleiben” [On Functions that Remain Unchanged by Linear Substitutions] – he objected to Klein’s statement that he had never published anything on fonctions fuchsiennes. Fuchs mentioned two of his articles and claimed that Klein had used his results.319 Surprised by Fuchs’s attack, Klein informed Hurwitz: “In the Göttinger Nachrichten from March 4th, Fuchs targeted me in a note.”320 Hurwitz responded: “Prof. Fuchs would certainly not be pleased at all to learn that I did not cite him in my dissertation. About this I have nothing more to say than that I did not study his work in detail at the time and that it has therefore had no influence on my own intellectual development.”321 Shortly thereafter, Poincaré explained his terminology yet again, and Klein printed this explanation as a letter to the readers of Mathematische Annalen.322 At the same time, Klein straightened out Fuchs’s arguments in a personal letter to Poincaré: I only asserted that Fuchs has never published anything about “fonctions fuchsiennes.” In this regard, the second of the articles that he mentions (a copy of which I have requested in order to study it more closely) is irrelevant. The first is admittedly concerned with “fonctions fuchsiennes” to the extent that it deals with modular functions, but Fuchs, owing to his lack of geometric intuition, failed to recognize the true character of the latter, which lies in the nature of the singular line – this was the error that Dedekind already pointed out in vol. 83 of Borchardt’s Journal (1877). Finally, his insinuation toward the end of his note that my work was somehow inspired by his own research is simply historically incorrect. I began my investigations in 1874 by determining all infinite groups of linear transformations of one variable. In the year 1876, I demonstrated that the problem then posed by Fuchs – the problem of determining all algebraically integrable linear differential equations of the second order – was thereby settled. The matter is therefore the opposite of what Fuchs claims it to be. It was not the case that I took any ideas from his work; rather, I showed that his topic has to be treated with my ideas.323

In his article “Neue Beiträge zur Riemann’schen Functionentheorie,” moreover, Klein responded to Fuchs with a long footnote. He described Fuchs’s developments as “indeterminate ideas […] because the results that Mr. Fuchs expresses in a very definite form are incorrect as such. Throughout, he confuses the concepts of unbranched and single-valued functions.” Klein went on about Fuchs’s publication in the Göttinger Nachrichten and remarked: “Although personal discussions are hardly useful in general, I feel compelled to respond here with a few lines.”324 This polemic would still be a matter of discussion when the occasion arose to hire Klein in Göttingen (Section 5.8.2). In the reprint of “Neue Beiträge” in Klein’s collected works, he omitted the polemical footnote. Instead, his preface 319 320 321 322

Göttinger Nachrichten (1882), pp. 81–84. [UBG] Math. Arch. 77: 67 (a postcard from Klein to Hurwitz). [UBG] Cod. MS. F. Klein 9: 911/3 (a letter from Hurwitz to Klein dated March 15, 1882). Henri Poincaré, “Sur les Fonctions Uniformes qui se reproduisent par des Substitutions Linéaires,” Math. Ann. 20 (1882), pp. 52–53. 323 KLEIN 1923 (GMA III), p. 608 (Klein to Poincaré on April 3, 1882), emphasis original. 324 KLEIN 1883, pp. 214–16.

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to the article even contains a few positive words about Fuchs’s work, which had pointed Poincaré’s attention toward modular functions.325 For a long time, the dispute between the two men remained heated. In his seminars and lectures, Klein repeatedly criticized Fuchs’s publications or had his students scrutinize them for errors.326 Hurwitz wrote the following to Klein on April 6, 1882: “Regarding Mr. Fuchs, he must be up in arms over your note on Poincaré’s work; it must upset him to hear you sing: ‘Fuchs, you stole the function. Give it back!’ We used to sing that at school. […] The result of your last article outshines everything that you’ve done before.”327 A few years later, Hurwitz would identify a grave error in Fuchs’s work (Section 6.4.3 and Appendix 5). Sophus Lie, who thought (as mentioned above) that Klein had spoken “harsh truths” to Fuchs, commented again: “When the occasion arises, please tell me if Fuchs has answered your recent remarks and where. I have no doubt that mathematicians will come to value your essential preliminary work to Poincaré’s discoveries. In all your students, you have an army that represents a great force.”328 With his comment about Klein’s students, Sophus Lie hit upon an important point. Klein worked and lived with and for his students. In his long article on Riemann’s function theory, he made sure to stress the importance of articles written by Dyck, Gierster, and Hurwitz. Hurwitz wrote to him from Göttingen: Thank you for sending me your “Contributions to Riemannian Function Theory.” The article evoked in me a sort of moral despair, for I feel as though I have not made as much progress as I should have. I can’t even tell you how much I miss the stimulating hours that I spent with you; the true value of what one has is never really appreciated until it has been lost.329

Both as a Privatdozent in Göttingen and as an associate professor in Königsberg, Hurwitz continued to participate in and contribute to Klein’s research program – with his own work, as an advisor and reviewer, and with his critical view of Lazarus Fuchs. Despite Lie’s opinion of the matter,330 the ongoing polemic with Fuchs was hardly reasonable, and it ultimately did Klein more harm than good.

325 See KLEIN 1923 (GMA III), p. 580. – Max Noether had already written to Klein on January 18, 1883, that the “style” of his polemics against Fuchs was unsuitable and that he considered it quite possible that the theory of the functions in question could be developed in another way, based on the linear differential equations. [UBG] Cod. MS. F. Klein 11: 64. 326 Francesco Gerbaldi’s presentation on linear differential equations was discussed above in Section 5.4.2.3. In a presentation given on July 31, 1886, Mellen Woodman Haskell also analyzed work by Fuchs, particularly his article “Zur Theorie der linearen Differentialgleichungen mit veränderlichen Coefficienten,” Crelle’s Journal 66 (1866), pp. 121–60. See [Protocols] vol. 8, pp. 79–81. 327 [UBG] Cod. MS. F. Klein 9: 913. Here, Hurwitz is rephrasing a children’s song with the line “Fuchs, du hast die Gans gestohlen,” that is, “Fox, you stole the goose.” The note that he refers to is Klein’s article on the limit-circle theorem. 328 Ibid. 10: 687 (an undated letter from Lie to Klein, sent in late 1882 or early 1883). 329 Ibid. 9: 936 (a letter from Hurwitz to Klein dated December 31, 1882). 330 Ibid. 10: 692 (an undated letter from Lie to Klein).

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5.5.6 The Icosahedron Book Whereas Klein encouraged Hurwitz to forge ahead with his work, he himself returned to his idea of writing textbooks, which he had expressed in his 1880 inaugural address in Leipzig. In other words, his aim was to summarize his results from the 1870s, which he had begun to systematize in his lectures. Hurwitz, who spent the Easter holiday of 1883 with Klein’s family, was one of the first to hear about this. After the beginning of the semester, Klein informed him: “Since the beginning of the week, I have been reading about equations of degree 5 in order to finish by June 15th and then perhaps go to Spiekeroog.”331 Klein did indeed travel with his wife and son Otto to the East Frisian island of Spiekeroog. His daughter Luise remained behind with her aunt Sophie Hegel. At the beginning of August, they then went to Grafenberg near Düsseldorf. When Otto fell ill with diphtheria, Anna Klein went back to Leipzig with him. Felix Klein remained behind at his parents’ house and continued to work on his icosahedron book. It took longer than he expected. On March 28, 1884, he announced: “I am concentrating on editing my book, and I hope to finish it over the break, though I can’t make any promises.”332 On June 20, 1884, Klein finally reported: Little new here. My book on modular functions lies further in the distance. In contrast, my book about the icosahedron is complete except for a few details, and it will come out in some weeks. Regarding all of this, I have a guilty conscience about “going backwards,” which I in fact had a vivid dream about last night.333

We see that Klein was already thinking about his next book (on elliptic modular functions); however, he was somewhat discontent summarizing his old material and not producing new results. All the more, then, he spurred on Hurwitz’s productivity. The latter reported his new results from Königsberg and thanked Klein for sending him a copy of the icosahedron book: I was just preparing for a trip when I received your book and your letter about my work. Thank you very much for the copy of the book, which will accompany me on my trip and which I very much look forward to studying. I have also notified two of our best students in Königsberg about its publication. One of them, Mr. Hilbert, just finished his dissertation: “Über Kugelfunctionen vom Standpuncte der Invariantentheorie betrachtet” [On Spherical Functions, Considered from the Standpoint of Invariant Theory]. Hilbert is very passionate about invariants; after taking his doctoral exams, he intends to go to Leipzig for a year to enjoy your stimulation. He has an ardent, speculative mind, and I’m sure you’ll like him.334

Both Hilbert and Minkowski (the other Königsberg student referred to) studied the book. In 1900, Hilbert still stressed how important it was that Klein had recognized the broad applicability of the “problem of regular polyhedra.”335 331 332 333 334 335

[UBG] Math. Arch. 77: 94 (a postcard from Klein to Hurwitz dated April 28, 1883). Ibid. 77: 119 (a letter from Klein to Hurwitz dated March 28, 1884), emphasis original. Ibid. 77: 116 (a letter from Klein to Hurwitz dated June 20, 1884). [UBG] Cod. MS. F. Klein 9: 968 (a letter from Hurwitz to Klein dated August 4, 1884). HILBERT 1900, p. 256.

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Sophus Lie wrote to Klein: “I have received your book on the icosahedron, as I wrote to you. I feel very flattered that you mentioned me in such an honorable way in your preface. However, I am quite sure that I am only half-way deserving of this.”336 In his preface, Klein had thanked Lie for working together with him on “investigating geometric or analytic forms susceptible of transformation by means of groups of changes.”337 Lie explained his reservations about acknowledging others as follows: I am ashamed that in my recent works I only quoted you with a certain reserve. I have learned that I have to be careful. When I quote somebody, one believes the person quoted has done everything. I do not understand why this is: probably, it is simply assumed that my ideas are proportional to my ability to express them.338

Lie regarded Klein’s icosahedron book as being “very well written,” and he thought it would grant him renewed access to Klein’s ideas. Lie decided to adopt Klein’s term isomorph [isomorphic], for which he had previously used the term gleichzusammengesetzt [equally composed].339 At the same time, Lie also felt inspired by Klein’s book to start planning – “with Friedrich Engel’s help” – his own future book project, in which he hoped to unify his “collected investigations on transformation groups into a single work.”340 Klein’s preface to Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade (the book was translated into English as Lectures on the Ikosahedron [sic], and the Solution of Equations of the Fifth Degree in 1888) begins with a forecast of his comprehensive book program: The theory of the Ikosahedron has during the last few years obtained a place of such importance for nearly all departments of modern analysis that it seemed expedient to publish a systematic exposition of the same. Should this prove acceptable, I propose to continue in the same course and to treat in a similar manner the subject of Elliptic Modular Functions and the general investigations newly made of Single-valued Functions with linear transformations into themselves [as of 1890: automorphic functions]. Thus a treatise of several volumes would grow, in which I should expect to promote science, at least in so far as it might introduce many to the realms of modern mathematics rich in far-stretching vistas.341

The icosahedron book consists of two main parts, the first of which is titled “Theory of the Ikosahedron Itself.” The second part – “Theory of Equations of the Fifth Degree” – is concerned with proving that the solution of an arbitrary equation of degree five can be reduced to the solution of an icosahedron equation. Klein’s overarching goal was to present those theories about equations of degree five that had been created with the construction of transcendental solutions

336 337 338 339 340

[UBG] Cod. MS. F. Klein 10: 704 (an undated letter from Lie to Klein, 1884). KLEIN 1888 [1884], p. viii. For the German original, see KLEIN 1884, p. iv. [UBG] Cod. MS. F. Klein 10: 704 (Lie to Klein, 1884). Ibid. Ibid. 10: 704 and 706/1. This letter from Lie to Klein, which was written toward the end of 1884, includes a detailed table of contents for the book that Lie had in mind. 341 KLEIN 1888 [1884], p. vii (preface written on May 24, 1884).

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since the middle of the 19th century. He had deliberately chosen an inductive approach for the arrangement of the material, as he later explained to a reviewer: “I believe the reader learns more with inductive presentation.”342 With this textbook, Klein wanted to provide tools that he had needed in his own efforts to reduce the equation of degree five to an icosahedron equation. The book covers a wide range of topics, including geometric, algebraic, and functiontheoretical methods; the Platonic (regular) solids, group theory, and the Riemann sphere; approaches from the theory of differential equations, series developments, and the theory of equations; all significant contributions made by mathematicians such as Gordan, Brioschi, Jacobi, Hermite, Kronecker, and others. The book culminates in the proof of Kronecker’s theorem, which Klein had achieved in 1876 (see Section 4.2.1). In the icosahedron book, the theorem is stated as follows: “[I]t is impossible, in the case of an arbitrarily proposed equation of the fifth degree, even after adjunction of the square root of the discriminant, to construct a rational resolvent which contains only one parameter.”343 Klein’s successful proof would still be important to him in 1905, when he revisited the subject in the only essay that he ever published in Crelle’s Journal. The journal’s editor at the time, Kurt Hensel, had requested a contribution from Klein for a special volume on Dirichlet. Klein summarized his own results and discussed further developments in the matter of solving general equations of degree five and degree six.344 He stressed in particular a new proof of Kronecker’s theorem by Gordan (1877) and the results produced by Jordan, Wiman, and Valentiner. Klein produced a novel result by synthesizing the ideas of the latter three scholars with the theory of cubic plane curves.345 Klein also used the occasion of this publication in 1905 to formulate his views on the nature of discovering and proving theorems, two processes that do not necessarily coincide: It happened that Kronecker, in his investigation, did not quite arrive at the icosahedron substitutions (which he had come so close to figuring out), and thus never found a sufficient proof for his main theorem! This, I think, is a very remarkable fact, even from a general point of view. For it confirms, as one especially interesting example, what Gauß so often stressed: that the discovery of important mathematical theorems is more a matter of intuition than deduction, and that the formulation of proofs is an entirely different affair from the discovery of theorems. Later, I never had a chance to discuss this subject with Kronecker, but I heard a few years ago that, after the publication of my “icosahedron book,” Kronecker devoted some time to the icosahedron theory in his course on solving equations of degree five.346

342 [UBG] Cod. MS F. Klein 11: 565B, fol. 5v (Klein to H. Schaeffer, February 3, 1885). 343 KLEIN 1888 [1884], pp. 286–87. 344 Felix Klein, “Über die Auflösung der allgemeinen Gleichungen fünften und sechsten Grades,” Crelle’s Journal 129 (1905), pp. 151–74; reprinted in Math. Ann. 61 (1905), pp. 50– 71; and KLEIN 1922 (GMA II), pp. 481–504. See also Felix Klein, “Sulla risoluzione delle equazioni di sesto grado,” Rendiconti della Reale Accademia dei Lincei: Classe di scienze fisiche, matematiche e naturali 8 (1899), p. 324. 345 KLEIN 1922 (GMA II), pp. 495–502. 346 Ibid., p. 491.

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When Klein prepared this article for the edition in his collected works (GMA II, 1922), however, he had just learned from Kurt Hensel (who edited Kronecker’s five-volume collected works) that Kronecker disregarded his findings.347 Thus, Klein felt impelled to describe once more how he had arrived at his proof(s). He also criticized Kronecker’s general attitude toward new theories, and he posed two rhetorical questions that demonstrated his own openness to novel ideas: When new phenomena arise (in this case, the efficiency of the accessory irrationalities), should one impede further development in favor of a previously conceived systematic idea or, instead, reject this thinking as too narrow and approach the new problems in an unbiased manner? Should one be dogmatic or, like a natural scientist, always be willing to learn something new from the objects of investigation themselves?348

The icosahedron book contained some notions that would later be named after Klein, such as the “Klein four-group” (Kleinsche Vierergruppe) and “Klein’s form problem” (das Kleinsche Formenproblem). The latter was taken up in 1935 by Richard Brauer, who used the theory of hypercomplex quantities to demonstrate that Klein’s form problem could be generalized.349 When a finite group G of linear transformations or more generally of collineations is given, Klein’s form problem for G was understood as the task of computing the coordinates of an ndimensional point when the values of the invariants of G are known for it. The primary task was to investigate which equations could be reduced to a form problem for a given group G. Klein was concerned with the question of whether every fifth-degree equation can be traced back to an icosahedron equation. The answer is easy for a group of entire linear homogeneous transformations. In the case of a group of fractional linear transformations, which is of greater interest with respect to applications, things were more complicated. Here, Klein managed to reduce the equation of degree five to an icosahedron equation only by adding an irrationality to the basic field (Grundkörper) that could not be represented as a rational function of the roots of the equation with coefficients from the field itself. Before Brauer’s new approach, proofs could be given only for special cases. Klein had systematized the results of the research area, and he had encouraged the discovery of new results as well. The dissertation of Klein’s doctoral student Otto Fischer, “Konforme Abbildung sphärischer Dreiecke auf einander mittelst algebraischer Funktionen” [Conformal Mapping of Spherical Triangles Onto One Another by Means of Algebraic Functions] (submitted November 24, 1884), is notable. Klein wrote that Fischer developed “versatile methods to map the elementary triangle of the icosahedron conformally onto the other spherical triangles bounded by the symmetry arcs of the same configuration. This must be achieved each time by the means of algebraic functions.”350 Fischer thus solved a problem 347 348 349 350

See PETRI/SCHAPPACHER 2002, pp. 261–62. KLEIN 1922 (GMA II), p. 504. See BRAUER 1935. See KLEIN 1923 (GMA III), p. 282.

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formulated by Klein and corrected a mathematical error discovered by Cayley. Later, Klein described Otto Fischer’s methods for treating the hypergeometric functions belonging to the icosahedron as being “of general significance.”351 In the 1990s, Gert-Martin Greuel likewise stressed that Klein’s work did much to stimulate further research on the topic of singularities because, in his efforts to solve the equation of degree five, Klein had discovered equations of simple singularities (as invariants of finite subgroups). Singularities of this sort have become fundamental in various branches of mathematics. Greuel’s exposition of the “deformation and classification of singularities and moduli” cites Klein’s book; it contains an overview of simple singularities, and it elucidates Peter Kronheimer’s approach to demonstrating direct relationships between simple Lie groups and finite Klein groups.352 In a new edition of Klein’s book, with an introduction and commentary by Peter Slodowy (1993), Slodowy refers to the continuing importance of “icosahedron mathematics,” in which the geometry and symmetry of the icosahedron and other regular polyhedra are relevant. In his preface, Slodowy mentions several areas of mathematical research that have been derived from it. At the same time, he provides an assessment of Klein’s goals, approach, and argumentation from the perspective of modern mathematics.353 Slodowy regards Klein’s book as a “quarry” that can still be mined for treasures by contemporary mathematicians. The English translation of Klein’s icosahedron book (London 1888) was undertaken by George Gavin Morrice, a member of the London Mathematical Society. His translation was informed by a detailed review of the German edition that was written in English by Klein’s doctoral student Frank Nelson Cole. In addition, Morrice sought advice from Arthur Cayley about how to translate certain technical terms.354 A letter from Klein to Fricke in 1891, however, reveals that Klein was less than satisfied with the English version.355 In 2019, Morrice’s English translation was reissued by the Higher Education Press in Beijing and was supplemented with Slodowy’s introduction and commentary from 1993.

351 352 353 354

KLEIN 1923 (GMA III), p. 63. See also KLEIN 1922 (GMA II), pp. 317, 346, 582. See GREUEL 1992, pp. 178, 185–87. KLEIN 1884, ed. by SLODOWY 1993, p. viii. [UBG] Cod. MS. F. Klein 10: 1317 (a letter from Morrice to Klein dated February 24, 1888). Cole’s review was published in the American Journal of Mathematics 9/1 (1886), pp. 45–61. From this review, for instance, Morrice borrowed the term “self-conjugate subgroup” to translate ausgezeichnete Untergruppe. 355 [UA Braunschweig] A letter from Klein to Fricke dated December 14, 1891. In this letter, Klein recommended that Fabian Franklin should be chosen to translate the next book: KLEIN/ FRICKE (1890/92). This two-volume work did not appear in English translation until 2017 (see the next Section).

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5.5.7 A Book on the Theory of Elliptic Modular Functions Felix Klein had not even written the preface to his icosahedron book when, in June of 1884, he was already looking ahead to his next book project: Today I made a thorough plan for the book that I would like to write on modular functions. Whether I will ever finish it is another question. It is still sorely lacking in terms of theory. Yet even aside from that, writing books involves a great deal of work that I find extremely unpleasant. I am forcing myself to do it because, at the moment, I consider it the most useful thing that I can do.356

Ultimately, Robert Fricke would become the collaborator on this project: Vorlesungen über die Theorie der elliptischen Modulfunktionen [Lectures on the Theory of Elliptic Modular Functions].357 It has, however, an interesting prehistory that was largely unknown until recently. First, Klein knew that the theory he had developed in Munich in the 1870s needed to be supplemented. To this end, he worked in Leipzig with numerous students and cooperated mainly with Adolf Hurwitz. Second, Klein was looking for a suitable editor for the book, because he did not want to write it alone. Before working with Fricke, he had tested out another candidate for this role. 5.5.7.1 Supplementing the Theory Klein treated the topic of elliptic modular functions in his lecture courses during the winter semester of 1883/84, the summer semester of 1884, and in his seminar from November 1884 to January 1885.358 Moreover, he presented reports about new results in this research area to the Saxon Academy of Sciences. His correspondence with Adolf Hurwitz shows how Klein and his students supplemented the theoretical approaches to these functions. Hurwitz himself was always interested in collaborating further with Klein. As a Privatdozent in Göttingen, he missed the stimulating work that they had done together. He thus sent regular letters to Klein and willingly accepted the latter’s invitations to meet in person: “Having a discussion with you is one of my favorite things to do.”359 Hurwitz helped Klein to complete his theory, and he filled in gaps in Klein’s proofs. As early as September 14, 1883, for instance, he had written about new ideas for a proof: “You see from this that your theory of modular correspondences can essentially be completed by introducing Weierstrass’s Efunction at a certain point.”360 In his reply, Klein expressed his continued skepticism about using Weierstrass’s prime functions in a more general context: 356 357 358 359 360

[UBG] Math. Arch. 77: 115 (a letter from Klein to Hurwitz dated June 3, 1884). KLEIN/FRICKE 1890/92. For an English translation, see KLEIN/FRICKE 2017 [1890/92]. [Protocols] vol. 6, p. 155. [UBG] Cod. MS. F. Klein 9: 962 (a letter from Hurwitz to Klein dated March 20, 1884). Ibid. 9: 951/3 (Hurwitz to Klein, September 14, 1883).

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5 A Professorship for Geometry in Leipzig To your letter I can respond at once, even though I do not have much to say. The question is whether real progress can be achieved both in the theory of modular functions and, beyond that, in the theory of algebraic structures in general by introducing prime functions. I must admit that I have so far doubted that this is the case; I believed that the prime function is only an expression for another formulation of the already well-known theory. If I may express myself even more directly, I regarded Kronecker’s assertion (that we should leave behind algebraic functions and only concern ourselves with prime functions) as no more than a trendy phrase. I would be happy to be convinced to the contrary. Bring me a class relation that cannot otherwise be derived with the same ease, and then I will be convinced.361

They agreed about Klein’s statement that “the actual instrument with which one must work is and remains Abel’s theorem.”362 Hurwitz worked further on the topic of modular correspondences. Klein was counting on Hurwitz’s results for his planned book, for which he envisaged: “A systematic study of nth-level moduli will have to form the conclusion of my projected work.”363 Klein repeatedly suggested topics for Hurwitz to investigate, for example: “Regarding moduli of the nth-level, you should attempt to construct appropriate moduli from the partial values of the σ-function […].” Klein went on to offer longer explanations of the moduli of different levels. During his North Sea vacation in 1884, Klein edited a small work of his own, in which he summarized his “old results on the n-part σ-products.”364 He revised several of his older works on the topic for publication in Mathematische Annalen, and he repeatedly polished the outline of his book on elliptic modular functions. His urgent desire to meet with Hurwitz was fulfilled in Düsseldorf (at Klein’s parents’ house) in mid-September 1884, and also during the summer vacation in 1885. Their correspondence testifies to their mutual stimulation and appreciation. Hurwitz concluded a letter dated September 12, 1884 with the words: “Your theory of modular functions has opened up access to a large and fertile field of research.”365 In his reply, Klein explained his own new results, formulated open questions, and stressed: “I don’t have to tell you how stimulating your visit was. We must see to it that we get together next year in a similar fashion.”366 Klein’s cooperation with Hurwitz left him feeling intellectually refreshed, and now he planned a trip to Paris for the end of September in 1884. He prepared for this trip by writing to Picard, Hermite, and Darboux,367 and Hurwitz brought 361 [UBG] Math. Arch. 77: 102 (a letter from Klein to Hurwitz dated September 16, 1883. 362 Ibid. 77: 104 (a postcard from Klein to Hurwitz dated September 26, 1883). For an explanation of Abel’s theorem, Weierstrass’s prime function, and Klein’s prime forms for representing algebraic functions as a product of factors, each of which becomes zero or infinite only at one point, see W. Wirtinger’s article in the ENCYKLOPÄDIE, vol. II.2 (1901), pp. 155–68. 363 [UBG] Math. Arch. 77: 120 (a letter from Klein to Hurwitz dated August 17, 1884). 364 Ibid. 77: 119 (a letter from Klein to Hurwitz dated August 29, 1884). 365 [UBG] Cod. MS. F. Klein 9: 972 (a letter from Hurwitz to Klein dated September 12, 1884). 366 [UBG] Math. Arch. 77: 122 (a letter from Klein to Hurwitz dated September 23, 1884). Their next meeting took place in August and September of 1885, and by then their discussion had already moved on to hyperelliptic functions. 367 See ibid. and a postcard to Hurwitz dated September 25, 1884 (No. 103).

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Klein up to date by sending him his latest results concerning the nth-level problem. Hurwitz also provided an assessment of Hermite’s recent work. According to Hurwitz, Hermite had returned to his class number relations “without making any essential modifications to his earlier ideas.”368 Furthermore: “In Hermite’s work the development of σ-quotients in Fourier series is crucial.”369 The trip to Paris fell through, however, because Klein was suffering from an upset stomach.370 Klein continued to work with his seminar participants, and he asked Hurwitz to review the results. Regarding the work of Giacinto Morera, Hurwitz believed that it was “closely related to my recent investigations into integrals of the first kind,” and he thought that Theodor Molien’s work was useful.371 Klein attached particular importance to Hurwitz’s judgment of Georg Pick’s first work on the complex multiplication of elliptic functions: The work is so important that I must have every guarantee regarding its accuracy before I accept it for publication in the Annalen. This work is very interesting for your own research, so any time spent on it will be useful. This would be Pick’s first publication (I praised him to you in one of our conversations in the fall). Who knows how things will develop?372

Hurwitz responded by classifying Pick’s work in Klein’s research program: “The execution of your general program – based on your level theory [Klein’s grading of the different kinds of elliptic functions in 1879] – will also certainly bring to light very beautiful results for complex multiplication. Mr. Pick has taken the first step.” Three days later, Hurwitz reported to Klein that his impression of Pick’s study was that “everything is correct; however, I am too unfamiliar with the composition of quadratic forms to evaluate all of the details of his investigation.”373 On January 1, 1885, Klein sent New Year’s greetings to Hurwitz along with mathematical ideas divided into six points. Hurwitz replied two days later: “Your friendly messages about the σgh might perhaps be useful to me for the integrals. Already some time ago, I had also made a recalculation of Kronecker’s Λ. It would be of the utmost importance to me if the idea you suggested on p. 95 of your recent publication on elliptic functions of the nth level were to be pursued further. […] Good luck with your book!”374 Klein sent Hurwitz a report about elliptic functions and modular functions including results from his doctoral students (Ernst Fiedler, Georg Friedrich, Robert Fricke, Paul Nimsch, Paul Biedermann). He sent the report with the wish “that, for your part, you may perhaps also work in the direction presented here.”375 Hur368 369 370 371 372 373

[UBG] Cod. MS. F. Klein 9: 974 (a letter from Hurwitz to Klein dated September 28, 1884). Ibid. 9: 975 (a letter dated October 16, 1884). [UBG] Math. Arch. 77: 123 (a postcard from Klein to Hurwitz dated October 20, 1884). [UBG] Cod. MS. F. Klein 9: 991 (a postcard from Hurwitz to Klein dated February 11, 1885). [UBG] Math. Arch. 77: 127 (a letter from Klein to Hurwitz dated November 29, 1884). [UBG] Cod. MS. F. Klein 9: 981, 980 (letters from Hurwitz to Klein dated December 1, 1884 and December 4, 1884; wrongly arranged in the archive). 374 Ibid. 9: 985 (Hurwitz to Klein on 3 January 1885). Klein had written to Hurwitz earlier about Kronecker’s “analytic invariant Λ for solving Pell’s equation.” [UBG] Math. Arch. 77: 130. 375 [UBG] Math. Arch. 77: 132 (Klein to Hurwitz on January 29, 1885).

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witz supplemented this work, corrected it, referred to unanswered problems, and offered the following assessment: “The results of Messrs. [Georg] Friedrich and [Ernst] Fiedler are especially interesting to me. The fact that it is now possible to explicitly specify modular equations for such high degrees of transformation is indeed a major advance over the methods of the Jacobi and Weierstrass school.”376 Klein divided his report in two parts, submitted the first part to be published in the Sitzungsberichte of the Saxon Academy of Sciences (dated February 2, 1885), and he published the second part in Mathematische Annalen (dated September 17, 1885).377 With this publication, Klein wanted to document how his “program for a pure theory of elliptic modular functions” (Math. Ann., vols. 14 and 15), which he had first developed in Munich, had been advanced by himself and his students. In particular, he wanted to show that these new investigations had become more closely connected “with the actual theory of elliptic functions – especially with the basic forms that Weierstrass used to give in his lectures.” Klein noted that, in his study published by the Munich Academy (1879), he had limited himself to “modular functions in the strict sense.” Further investigations, however, would require an analysis of modular forms. He explained: One can imagine this connection in such a way that Riemann’s methods (the construction and discussion of fundamental polygons etc.), which I prioritized at that time, provided the first necessary preliminary results. A refined approach and the treatment of more complicated cases requires the use of form-theoretical methods. The supreme principle of classification remains the group-theoretical approach (the level graduation etc.).378

By the end of the winter semester of 1884/85, Klein had made so much progress that he was seriously thinking about organizing this material into a book. Regarding his research seminar, he remarked: “My seminar instruction is now turning away from modular functions, because we have all had enough of the subject. Also, a new generation of students is coming up.”379 5.5.7.2 Who Should Be the Editor? – Georg Pick When searching for an appropriate editor for the book, Klein had thought at first of Hurwitz, but he concluded that this creative mind should rather be left free to engage in its own investigations.380 Robert Fricke, who would later become the actual editor of the project, was not yet part of the discussion, although Klein mentioned, in a letter dated August 29, 1884, that Fricke had given a seminar presentation on moduli of the sixteenth 376 377 378 379 380

[UBG] Cod. MS. F. Klein 9: 992; 995 (Hurwitz to Klein, Febr 27, 1885; March 19, 1885). The full report is reprinted in KLEIN 1923 (GMA III), pp. 255–82. Math. Ann. 26 (1886), p. 457; reprinted in KLEIN 1923 (GMA III), p. 275. [UBG] Math. Arch. 77: 134 (a letter from Klein to Hurwitz dated February 10, 1885). [UBG] Math. Arch. 77: 136 (a letter from Klein to Hurwitz dated March 10, 1885).

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level.381 However, Fricke’s dissertation – “Ueber Systeme elliptischer Modulfunctionen von niederer Stufenzahl” [On Systems of Elliptic Modular Functions of a Low Level Number] (submitted on September 13, 1885) – was not yet complete. Klein was able to persuade Georg Pick to take on the role.382 Pick was a doctoral student of Leo Koenigsberger, who had supervised his thesis “Über eine Klasse Abelscher Integrale” [On a Class of Abel’s Integrals] in Vienna. Already in 1881, Pick completed his Habilitation with the thesis “Über die Integration hyperelliptischer Differentiale durch Logarithmen” [On the Integration of Hyperelliptic Differentials by Logarithms] at the University of Prague. Since the fall of 1883, Pick numbered among the new participants in Klein’s seminar (see Table 6) who worked together on Klein’s research goal of developing the theory of elliptic modular functions. During the winter semester of 1883/84, Pick gave three seminar reports on the fundamental concepts of the theory of complex-valued functions (Gauß, Cauchy, Riemann, Weierstrass, Pringsheim, Mittag-Leffler, Jacobi, Schwarz, Cantor, etc.). Pick analyzed Poincaré’s articles in volume 1 of Acta Mathematica and explained why one of Poincaré’s proofs was “incomplete,” so that it could not be decided with certainty whether the series stated by Poincaré “really represent the functions, which have to be constructed, in their entire domain.”383 Klein tested Pick further by having him talk about the main results of his lecture course on algebra and number theory (1879, elaborated by Gierster), and finally Klein directed Pick toward the topic mentioned above: the complex multiplication of elliptic functions.384 Even before Pick spoke about this in the seminar (on June 16, 1884), Klein was so enthusiastic about his abilities that he wrote to his old friend Otto Stolz in Innsbruck as follows: Dear Stolz! Though it is normally against my principles to do such a thing, I have sat down and am writing to you in order to recommend a job candidate in the event that certain developments take place. Here, one expects that your colleague [Leopold] Gegenbauer will be offered a position in Vienna, in which case his position in Innsbruck will be vacant. Now, I happen to know an Austrian mathematician who as yet hasn’t published very much, so that you might not be aware of him, but who possesses such excellent qualities that I would like to help him advance his career. I would like to help him, because there are not many mathematicians with such outstanding talents in Austria and there is a danger that such talents will have to take second place to less qualified people. The candidate I have in mind is Privatdozent Dr. Pick in Prague. Originally a student of [Leo] Koenigsberger, he completed his Habilitation three or four years ago and was then an assistant under [Ernst] Mach, until he came here in the fall of 1883 and has since been a participant in my seminar. Pick has, above all, an aptitude for function theory, and especially for number theory. He has a very clear and profound understanding of things, so that it is always a pleasure to speak with him; he also has an outstanding style of teaching and lecturing – something that I have only observed in a few of my

381 Ibid. 77: 119 (a letter from Klein to Hurwitz dated August 29, 1884). Regarding Fricke’s presentation, see [Protocols] vol. 6, pp. 77–83 (the presentation was given on June 30, 1884). 382 [UBG] Cod. MS. F. Klein 11: 229 and 230 (Pick to Klein, February 11 and 16, 1885). 383 [Protocols] vol. 5, pp. 159–69, 260–67, 268–79 (quoted here from p. 279). 384 [Protocols] vol. 6, pp. 9–23, 49–61 (Pick’s seminar presentations, which were given on May 9, 1884 and June 16, 1884).

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5 A Professorship for Geometry in Leipzig students (Dyck and Harnack, for instance). That he has published so little is attributable to the fact that, as with so many others, he had not received the necessary encouragement early on. He became excessively self-critical, and this reduced his productivity. I expect that he will overcome this before the end of the semester; I have him working on complex multiplication, where he is to justify and continue Kronecker’s results. Of course, as his name suggests, Pick is a Jew, but he is one of the exceptions with whom it is pleasant to interact, as is clear from his overall popularity in my seminar. The younger mathematicians all turn to him for advice. But enough! If you need further information, I will send it to you immediately. For the rest, please forgive the initiative that I have taken in this letter. I must stress that it was not prompted by Dr. Pick but is rather entirely self-motivated. With kind regards, Your F. Klein385

Klein’s remark about Pick’s Jewishness stemmed from his awareness of the widespread anti-Semitism that also existed in Austria at the time.386 Klein himself did much to oppose anti-Semitism, as is evident from his support for Max and Emmy Noether, Hurwitz, Schoenflies and others. Leopold Gegenbauer did not leave Innsbruck until 1893. In 1888, Gegenbauer and Otto Stolz supported Georg Pick’s candidacy for an associate professorship at the German University in Prague (then part of Austria-Hungary).387 This also required, however, the influence of Klein, who sent a letter at Pick’s request to Heinrich Durège, a member of the hiring committee in Prague. Furthermore, Klein arranged for H.A. Schwarz to send his opinion of the candidates to the committee member Ernst Mach.388 In 1892, Pick was promoted to full professor. Before Pick received these positions, Klein had been able to print his results in Mathematische Annalen, where, up to the year 1901, Pick would publish eleven articles.389 Klein and Pick first met to discuss the book project during the Easter break of 1885. Before this meeting, however, Klein wrote to Hurwitz that he was still somewhat unsure about his theoretical approach: I have omitted the passage about the scope of your method so as not to say anything incorrect. In my opinion, the issue is this: You define the modular correspondence by the zeroes of a function and not by a system of intersection points, as I always imagined this earlier. This distinction is more interesting to me than you might suppose, because I have recently come across exactly these conflicting views on other questions. The notion that one has to define the modular correspondence by a system of intersection points comes from geometry. The question is whether, in the case of higher problems, geometry really has to give way to function theory, as I tend to believe more and more. This would mean that I would have to scrap many of my favorite beliefs! In general, things look very bleak as far as the systematic 385 [Innsbruck] A letter from Klein to Otto Stolz dated May 28, 1884. 386 See SIEGMUND-SCHULTZE 2009; BERGMANN/EPPLE/RUTI 2012; BEČVÁŘOVÁ 2016. – Pick became a victim of the Nazi regime. He died in the Theresienstadt concentration camp. 387 [UBG] Cod. MS. F. Klein 11 (Otto Stolz to Klein, May 26, 1888). 388 [UBG] Cod. MS. F. Klein 11: 319; 323 (Pick to Klein, January 26, 1888; July 31, 1888). 389 See Georg Pick, “Ueber die complexe Multiplication der elliptischen Functionen I,” Math. Ann. 25 (1885), pp. 433–47; and idem, “Ueber die complexe Multiplication der elliptischen Functionen II,” Math. Ann. 26 (1886), pp. 219–30. See also TOBIES/ROWE 1990, p. 39.

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completion of my views is concerned, and I almost fear that my discussions with Pick, which will begin on Friday, will not yield any results.390

Klein commissioned Georg Pick to prepare the first five chapters of the first section of the book by September of 1885. Pick was expected to use Klein’s lecture courses and recent results. Regarding the latter, Hurwitz provided him with summaries; he had himself made so much progress concerning the “class-number relations of prime levels”391 that he was able to assert: “At least, however, the existence of the relations for an arbitrary level has now been established, and their general form is known.”392 Yet in November, Klein stated that “Pick […] has not made as much progress with the modular functions as I expected.”393 Klein nevertheless invited Pick to Leipzig for further consultations and, as of May of 1886, he invited him to Göttingen. On May 15, 1886, Klein wrote to Hurwitz: Then, however, for the last eight days I have begun to draft, on the basis of Pick’s preliminary work, the first section of my book on elliptic modular functions! In the meantime, Pick has moved on to the second section (the main section), which we discussed at length at the end of the Easter holiday. May it all end well. I oscillate between confidence and uncertainty [Misstrauen]. I also regret the time that I am still devoting to these old stories, but it will probably prove to have been a reasonable thing to do.394

Approximately one year later, Klein’s cooperation with Pick came to an end. In July of 1887, Robert Fricke briefly met Georg Pick in Vienna,395 and in September of 1887, Klein informed Hurwitz: “My plan to edit the book on elliptic modular functions together with Pick was abandoned some time ago (by mutual agreement).” Klein then posed the rhetorical question of whether Hurwitz himself might perhaps be interested in taking over Pick’s role.396 Wiggling out of a difficult position, Hurwitz replied: “The book will only have the proper intellectual freshness and stimulating force if you write it yourself. This has always been my opinion on the matter […].”397 It must be acknowledged that the main reason for Pick’s failure was that the theoretical foundations of elliptic modular functions were still insufficiently developed at the time when he was preparing Klein’s book. Klein himself, Hurwitz, and ultimately Robert Fricke would spend the next five years working on this very topic. In a letter to Hurwitz dated August 5, 1888, Klein reported about the state of Pick’s career – “Pick has now finally become […] an associate professor” – and he also wrote about his new collaborator, Robert Fricke:

390 [UBG] Math. Arch. 77: 137 (Klein to Hurwitz on March 17, 1885), emphasis original. 391 See, for example, Hurwitz’s article “Ueber Relationen zwischen Classenanzahlen binärer quadratischer Formen von negativer Determinante,” Math. Ann. 25 (1885), pp. 157–96. 392 [UBG] Cod. MS. F. Klein 9: 998 (a letter from Hurwitz to Klein dated May 1, 1885). 393 [UBG] Math. Arch. 77: 142, 149 (Klein to Hurwitz on June 20 and November 27, 1885). 394 Ibid. 77: 156 (a letter from Klein to Hurwitz dated May 15, 1886). 395 [UBG] Cod. MS. F. Klein 11: 312 (Pick to Klein, July 12, 1887). 396 [UBG] Math. Arch. 77: 190 (a letter from Klein to Hurwitz dated September 21, 1887). 397 [UBG] Cod. MS. F. Klein 9: 1060 (a letter from Hurwitz to Klein dated September 24, 1887).

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5 A Professorship for Geometry in Leipzig Now, before anything else, I have to say that Fricke is really working on the modular functions; at any rate, he is doing so with more consistency and also with more energy than Pick during his time. I am letting him work as independently as possible. We have only discussed matters briefly during the Christmas and Easter breaks, and now I expect to see him again over the next few days.398

Klein and Fricke finished this book project in 1892 (see Section 6.3.3). 5.5.8 Hyperelliptic and Abelian Functions In 1885, moreover, I began to work more intensively on a problem that I had already had my eyes on for a long time and that I would concentrate on for several years to come, namely the problem of transferring the new formulations, which I successfully constructed in the case of elliptic functions, to hyperelliptic and Abelian functions.399

Klein spent parts of August and September in 1885 with Hurwitz on the island of Borkum. They had a copy of Weierstrass’s lecture course on hyperelliptic functions in their luggage. After working further on the topic, Klein informed Hurwitz on December 13, 1885 about his new results concerning “completely independent moduli […], whereby a new system of hyperelliptic ‘main moduli’ is given.”400 Hurwitz responded with these words: “I read your other theorems about the geometric integration of hyperelliptic differential equations with interest. Hopefully, your configuration article will be published soon. I am very eager to study it, for I have already forgotten much of what I had learned in Borkum.”401 In the “configuration article,”402 Klein further advanced an old topic that involved combining the Kummer surface with hyperelliptic integrals. Karl Rohn (1879) had already worked on this, and Klein’s doctoral student Willibald Reichert took it further with his dissertation “Über die Darstellung der Kummerschen Fläche durch hyperelliptische Funktionen” [On the Representation of the Kummer Surface by Hyperelliptic Functions] (1887). Klein involved Hurwitz in this study: Later, when you are again feeling refreshed and are able to take up the δ-questions that we discussed in Borkum, I believe this could lead to very promising results. I am convinced that by following through with our speculations – that is, essentially by constructing and thoroughly discussing the normal configuration corresponding to the Kummer surface with a higher p – it will also be possible to solve the question of principles in the case of p = 4. What would you think if the Jablonowski Society, which now again must present a mathematical topic for its prize, were to formulate a question along these lines?403

398 399 400 401 402

[UBG] Math. Arch. 77: 196 (a letter from Klein to Hurwitz dated August 5, 1888). KLEIN 1922 (GMA II), p. 259. [UBG] Math. Arch. 77: 150 (a letter from Klein to Hurwitz dated December 13, 1885). [UBG] Cod. MS. F. Klein 9: 1014 (a letter from Hurwitz to Klein dated January 11, 1886). Felix Klein, “Ueber Configurationen, welche der Kummer’schen Fläche zugleich eingeschrieben und umgeschrieben sind,” Math. Ann. 27 (1886), pp. 106–42; reprinted in KLEIN 1921 (GMA I), pp. 164–99. 403 [UBG] Math. Arch. 77: 152 (a letter from Klein to Hurwitz dated January 8, 1886).

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The problem posed by the Societas Jablonoviana (see Section 5.7.2), which was formulated by Klein, read as follows in an abbreviated form: The Society invites a detailed investigation of the more general double integrals of the form

∫∫

f ( xy ) dx dy , R( xy )

where f is a rational function, in its relation to the theta functions of two variables.404

The task set for this prize was part of Klein’s broader research program, which, based on the work of Clebsch and Gordan, involved “bringing the theory of hyperelliptic functions and Abelian functions into a proper connection with the theory of forms or invariant theory.” Klein used Weierstrass’s theory of elliptic functions as a model, which he combined more closely with approaches from invariant theory. In this respect, he defined his main goal as follows: “to construct, systematically, the 22p theta functions of a given Riemannian cross-cut system for an arbitrary algebraic entity of the genus p.”405 Klein published his first results in April of 1886,406 and continued this subject in Göttingen (see Section 6.3.2). He constructed his own special sigma functions and thus he obtained an overview of this field, which combines function theory, algebra and geometry. From his correspondence with Hurwitz, we learn how Klein progressed. In January of 1886, Klein informed Hurwitz: “I am currently very busy with the hyperelliptic σ, with which I am making good progress (drawing a connection to invariant theory); next week I will resume my special lectures.”407 He hoped to collaborate further with Hurwitz: “Now that [Karl] Rohn, Franz Meyer, and Dyck have all ‘committed themselves’ [i.e. become engaged to be married], will you still remain faithful to me for a little while, even though I strongly advised you against this in Borkum?”408 Regarding Klein’s latest article from April of 1886, Hurwitz predicted: Your new treatise on hyperelliptic modular functions will undoubtedly have an epoch-making effect; it paves the way for a great number of interesting and important investigations, and I hope that you will prove to be the first to have found the correct and important generalization of your level theory.409

404 Math. Ann. 27 (1886), pp. 471–72. The term “theta function,” which denotes a special class of functions of several complex variables, comes from Jacobi (1829). 405 KLEIN 1923 (GMA III), p. 317. 406 Felix Klein, “Ueber hyperelliptische Sigmafunctionen,” Math. Ann. 27 (1886), pp. 431–64; reprinted in KLEIN 1923 (GMA III), pp. 323–56. 407 [UBG] Math. Arch. 77: 153 (a letter from Klein to Hurwitz dated January 29, 1886). 408 Ibid. (Klein’s postcard to Hurwitz on February 2, 1886). Regarding the engagements mentioned here, see Hurwitz’s letter (No. 1017) to Klein dated February 5, 1886. 409 [UBG] Cod. MS. F. Klein 9: 1021 (a letter from Hurwitz to Klein dated April 7, 1886).

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5.6 FELIX KLEIN AND ALFRED ACKERMANN-TEUBNER In their brief biographical study, ACKERMANN/WEISS (2016) present an impressive portrait of Alfred Ackermann-Teubner,410 who was in charge of the publishing house’s mathematical division from the 1880s until his departure from this position in 1916. The authors also mention Felix Klein’s contribution to the rising success of the press’s mathematical publications during this period.411 The B.G. Teubner publishing house had been founded by Benedictus Gotthelf Teubner in 1811. Klein had first introduced himself to this press on May 27, 1868 as “F. Klein, student of mathematics, Plücker’s former assistant,” because he had been commissioned to edit the second volume of Plücker’s book on line geometry.412 As of 1876, moreover, he was closely connected to the publishing house on account of his editorial role at Mathematische Annalen. In the year 1868, Alfred Ackermann-Teubner, a grandson of the company’s founder, was only eleven years old. After years of apprenticeship at the Teubner press and internships in London and Paris, he was made a co-owner of the firm in 1882, while Klein was working as a professor in Leipzig. Alfred AckermannTeubner had attended lectures at the University of Leipzig (the natural sciences, economics) and was ultimately made responsible for the division of the publishing house devoted to mathematics, the natural sciences, and technology. In this role, he benefited from Klein’s interests and personal network. Klein’s book Über Riemanns Theorie der algebraischen Funktionen und ihre Integrale (1882) was the first short monograph that Klein submitted to the press. In a book celebrating B.G. Teubner’s one hundredth anniversary, the complete table of contents of Klein’s monograph is reproduced, and Klein’s vision for establishing a monograph series at the press, which he had expressed in a letter from August 24, 1883, is also recounted: Let me further specify the general intention behind my decision to take on this work. It has long been my principle that the results of longer journal articles have to be summarized in a more editorially consistent fashion, namely in the form of monographs.413

In the following years, Klein pursued his vision for monographs in such a way that he not only fulfilled his personal publishing agenda but also ensured that the German Mathematical Society would take up the objective of writing reports on mathematical subjects (see Section 6.4.4). Promoted by this society, the reports

410 Alfred Ackermann-Teubner’s father, Albin Ackermann, who was active in the firm B.G. Teubner since 1850, had married Anna Teubner and from then on went by the name Ackermann-Teubner. See ACKERMANN/WEISS 2016, pp. 14–15. 411 Ibid., pp. 31–33, 39. – In their innovative study of the relationship between mathematicians and their publishers, REMMERT/SCHNEIDER 2010 failed to mention the special relationship between Klein and Ackermann-Teubner. THIELE 2011 contains an article on Felix Klein’s time in Leipzig and pays tribute to his importance to the Teubner press. 412 See SCHULZE 1911, p. 297. 413 Ibid., pp. 305–06 (Klein’s quotation appears on p. 306).

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were first published first in its Jahresbericht, and later they were incorporated into the ENCYKLOPÄDIE. This, in turn, benefited the Teubner press, which published the project’s many fascicles from 1898 to 1935 (see Section 7.8). From the ENCYKLOPÄDIE project, Ackermann-Teubner gained many outstanding authors for his publishing division, and later he sponsored a prize to honor ENCYKLOPÄDIE contributors. Klein’s relationship with Ackermann-Teubner and the B.G. Teubner Press can be described in ten points: First. While in Leipzig, Klein took advantage of the publishing house’s proximity to make many personal introductions there, and this benefited a number of students, colleagues, and contributors to Mathematische Annalen. An account by Otto Hölder offers a typical example of this: Prof. [Aurel] Voss was here from Dresden over the Easter break. Because I was back from Göttingen, Klein invited me to visit him in order to meet Mr. Voss. The mathematical Privatdozent [Friedrich] Schur also joined us. We then went together to B.G. Teubner, where I picked up my manuscript. It was very interesting to see this magnificent press from top to bottom. We were introduced to all of the company’s directors as contributors to Mathematische Annalen.414

Second. During his period of cooperation with Ackermann-Teubner, Klein’s publications included his own monographs, works written with his students or colleagues, and lithographic reproductions of his lecture courses. The large-scale projects that Klein edited and published with Teubner included, in addition to the ENCYKLOPÄDIE, a volume on mathematics within the framework of the series Kultur der Gegenwart [Culture of the Present], and the five-volume Abhandlungen über den mathematischen Unterricht [Treatises on Mathematical Instruction], which Klein initiated as president of the International Commission on Mathematical Instruction (ICMI) (see also Sections 7.8 and 8.3). Third. Whereas the collected works of August Ferdinand Möbius were published by S. Hirzel in Leipzig, Klein worked to ensure that Hermann Graßmann’s collected works would be contracted by B.G. Teubner. Thanks to Klein, Julius Plücker’s collected works (PLÜCKER 1895/96) were also printed by Teubner. Regarding Gauß’s works, Klein arranged for vol. 8 (1900) and vol. 7 (new edition 1906) through vol. 10.1 (1917) to be published “on a commission basis by B.G. Teubner in Leipzig.” Not until 1923, when Max Born succeeded Klein as the chief editor of Gauß’s works, did their publication change hands from Teubner to the Julius Springer publishing house.415 Fourth. Klein motivated Teubner to publish German translations of books written in English, French, Italian, and Russian. The latter included A.A. Markov’s Differenzenrechnung [Calculus of Finite Differences] (1896) and Wahrscheinlichkeitsrechnung [Probability Calculus] (1912).416 Klein inspired his doc414 Quoted from HILDEBRANDT et al. 2014, p. 152. 415 [AdW Göttingen] Scient. 105,2: 9 and 107,5: 6b; REICH/ROUSSANOVA 2013, pp. 226–27. 416 Markov’s student Theophil Friesendorff was responsible for translating his book on the calculus of finite differences into German; in Friesendorff’s presentations in Klein’s seminar  

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toral student Friedrich Schilling (see Section 7.1) to write a book-length German summary of Maurice d’Ocagne’s Traité de Nomographie,417 and he incited Paul Stäckel to edit and translate the school textbooks by Émile Borel, which appeared in German as Die Elemente der Mathematik (1908/09). Klein likewise instigated translations of Horace Lamb’s Treatise on the Mathematical Theory of the Motion of Fluids (3rd ed., 1879), which was published in German as Lehrbuch der Hydrodynamik (1907), and Edward Hough Love’s A Treatise on the Mathematical Theory of Elasticity (1892/93), which was translated by Klein’s doctoral student Aloys Timpe as Lehrbuch der Elastizitätstheorie (1907).418 In his work on educational reform, Klein repeatedly cited John Perry’s book The Calculus for Engineers (2nd ed., 1897), and he instigated its translation into German, which sold through several Teubner editions (the first edition was published in 1902; see also Sections 3.6.3 and 8.3.4.2). Grace Chisholm Young would take Klein’s advice and write an introductory book on geometry,419 and Klein arranged for it to be translated by Felix Bernstein (and his mother Sophie). It appeared in German as Der kleine Geometer (1908). On the basis of an analysis of Benchara Branford’s A Study of Mathematical Education, Including the Teaching of Arithmetic that was presented in Klein’s seminar (on January 26, 1910), Rudolf Schimmack and Hermann Weinreich decided to produce a German version of the book (1913). Fifth. Klein himself wrote a preface or introduction for quite a number of books published by Teubner, including, for example, the revised dissertations of Friedrich Pockels (1891) and Maxime Bôcher (1894),420 Edward John Routh’s Die Dynamik der Systeme starrer Körper [The Dynamics of Systems of Rigid Bodies] (2 vols, 1898), Federigo Enriques’s Vorlesungen über projektive Geometrie [Lectures on Projective Geometry] (1903),421 Jules Tannery’s Elemente der Mathematik [Elements of Mathematics] (1909; 2nd ed., 1921), and Jan A. Schouten’s Grundlagen der Vektor- und Affinoranalysis [The Foundations of  

417 418

419 420 421

(given on May 1 and May 31, 1895), he used the Russian book. At the time, the calculus of finite differences was becoming an increasingly important field of applied mathematics. At Klein’s request, Rudolf Mehmke wrote a preface to the book and emphasized Klein’s farsighted initiative. Markov’s book on probability calculus was translated by Heinrich Liebmann, who was Klein’s assistant in 1897 and 1898. See TOBIES 2018; KRENGEL 1990; and SCHNEIDER 1989. See Friedrich Schilling, Über die Nomographie von M. d’Ocagne: Eine Einführung in dieses Gebiet (Leipzig: B.G. Teubner, 1900). See KLEIN 1922 (GMA II), p. 508. Both Love and Lamb contributed to Vol. IV of the ENCYKLOPÄDIE, Love with an article entitled “Hydrodynamik” [Hydrodynamics] (1901) and Lamb with an article entitled “Schwingungen elastischer Systeme – insbesondere Akustik” [Oscillations of Elastic Systems – Especially Acoustics] (1906). See Section 7.8. Although she wrote the book on her own, her husband’s name also appears on the title page: Grace Chisholm Young and W.H. Young, The First Book of Geometry (London: J.M. Dent, 1905); see also MÜHLHAUSEN 1993, p. 206; GRATTAN-GUINNESS 1972. For further discussion of Pockels’s and Bôcher’s work, see Section 6.3.5 below. It was Klein himself who had instigated the German translation of Enriques’s Lezioni di geometria proiettiva. See L. Giacardi’s article in COEN 2012, p. 225.

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Vector and Affinor Analysis] (1914). Klein used these introductions and prefaces to classify the respective books in light of his own research projects and to refer to additional (Teubner) books. Klein was also consulted by the publishers when it was unclear whether a given book should be published or not. Sixth. Over time, in order to reduce his own workload, Klein recommended additional consultants for special projects or research fields: Walther Dyck for the ENCYKLOPÄDIE, for instance, and Arnold Sommerfeld for physics. Friedrich Engel, who already had a good relationship with Teubner on account of his edition of Sophus Lie’s work, became the main contact person throughout the publication process of Graßmann’s collected works. Seventh. Klein endorsed Alfred Ackermann-Teubner to become not only a member of the German Mathematical Society (in 1894), but also its treasurer (1905–1919).422 Following the Society’s annual meeting in Hamburg in 1901, it transferred its accounts to Leipzig, where they were managed by the B.G. Teubner publishing house. The new board position of treasurer was introduced when the German Mathematical Society was registered as an association in Leipzig. In the interest of the press, Ackermann-Teubner also became a member of several foreign mathematical societies: the Société Mathématique de France, the London Mathematical Society, the Circolo matematico di Palermo, and the American Mathematical Society.423 Klein was also able to gain him as a member of the Göttingen Association for the Promotion of Applied Physics and Mathematics (see Section 8.1.1). All told, Ackermann-Teubner would donate 22,500 Mark to the Göttingen Association.424 Eighth. In October of 1899, Ackermann-Teubner turned to Klein for advice about how his publishing house should proceed with its mathematical journals. The latter had begun to overlap in content, and they were no longer meeting all the needs in the field. On New Year’s Eve in 1899, Klein found the time to write down a number of programmatic ideas on the topic (see Fig. 26). Regarding Mathematische Annalen, Klein’s proposal to leave his position as a main editor and join the advisory board, in order for Hilbert to take his place (Point 1 in Fig. 26), was rejected by the journal’s other editors. Instead, Adolph Mayer voluntarily stepped down to join the advisory board so that Hilbert, as of vol. 55 (1902), could become one of the main editors alongside Klein and Dyck. As of vol. 46 (1901), the Zeitschrift für Mathematik und Physik [Journal for Mathematics and Physics] (Point 2 in Fig. 26), which Oscar Schlömilch had founded in 1856 as Teubner’s first mathematical journal, was published with the subtitle “Organ für angewandte Mathematik” [Organ for Applied Mathematics] under the editorship of Rudolf Mehmke and Carl Runge (not Arnold Sommerfeld, whom Klein had first had in mind for the position). At Klein’s recommendation, Mehmke had been an editor of the journal (with Moritz Cantor) since vol. 42 (1897), but 422 See Jahresbericht der DMV 14 (1905) I, p. 525; and 29 (1920) Abt. 2, p. 42. 423 See Jahresbericht der DMV 11 (1902) I, p. 10; ibid. 21 (1912) Abt. 2, p. 4. 424 The donation amounts are listed in [StA Berlin] Rep. 92 Nachlass Schmidt-Ott C55, fol. 109.

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at first he had only limited success in carrying out its transition to applied mathematics.425 In 1901, Klein joined the editorial team as a member of the advisory board, and in this capacity he was able to appoint additional mathematicians (Guido Hauck, Heinrich Weber), prominent physicists (Hendrik Antoon Lorentz), geodesists (Robert Helmert), astronomers (Hugo Seeliger), and engineers (Carl von Linde, Carl von Bach, Heinrich Müller-Breslau) as advisory board members. This journal was discontinued as a consequence of the First World War, its last issue appearing in 1917 (vol. 64). In 1921, when Richard von Mises founded the Zeitschrift für angewandte Mathematik und Mechanik [Journal for Applied Mathematics and Mechanics],426 Klein saw this as a welcome continuation of the old tradition (see also Section 9.5).

Figure 26: An excerpt of Klein’s drafted letter to A. Ackermann-Teubner, December 31, 1899 ([UBG] Cod. MS. F. Klein).

In the nineteenth century, the history of mathematics (Point 3) developed into an independent field of research, and by this point it was in need of a journal of its own. In France, a Bulletin de bibliographie, d’histoire et de biographie was published from 1855 to 1862 under the editorship of Olry Terquem as an appendix to the Nouvelles annales de mathématiques. Beginning in 1859, Moritz Cantor had 425 [UBG] Cod. MS. F. Klein 10: 1138–40 (Mehmke’s letters to Klein dated December 5, 1896; July 4, 1897; and April 10, 1899). 426 See SIEGMUND/SCHULTZE 2020a, 2020b.

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published articles on the history of mathematics in Teubner’s Zeitschrift für Mathematik und Physik. As of 1875, the journal devoted a special section to this topic, and supplementary volumes on the subject were published as of 1877. Klein thought it would be a good time to create a journal devoted exclusively to the topic, and he recommended Gustaf Eneström and Paul Stäckel as potential editors. For a long time, Eneström was an active contributor to Mittag-Leffler’s Acta Mathematica,427 and in 1884 he had visited Klein in Leipzig (see Section 5.4) right around the time when he had founded the series Bibliotheca Mathematica in Stockholm (published by the F. & G. Beijer förlag). In 1900, Eneström ended his relationship with F. & G. Beijer and switched to Teubner, which enabled him to expand the annual volume from eight to thirty-five printed sheets (Druckbogen). Teubner’s publication of Bibliotheca Mathematica came to an end with vol. 14 (1914) on account of the First World War. Klein thought that a “journal for elementary mathematics” (Point 4 in Fig. 26) would be a fitting supplement to Teubner’s spectrum of publications. In 1900, following Klein’s advice, Teubner therefore purchased the Archiv der Mathematik und Physik [Archive of Mathematics and Physics], which had existed since 1841, from a smaller Leipzig-based press (C.A. Koch’s publishing house). What Klein had in mind was a journal for students, and his model for this was the French journal Nouvelles annales de mathématiques, journal des candidats aux écoles polytechnique et normale. Edited by Charles-Ange Laisant since 1896, the latter journal had published a number of Klein’s articles in French translation during the 1890s. After consulting with Klein, Ackermann-Teubner appointed Franz Meyer and the Berlin-based mathematicians Emil Lampe and Eugen Jahnke as editors for the Archiv der Mathematik und Physik. Unlike the French journal, the Archiv appeared with the subtitle “Mit besonderer Rücksicht auf die Bedürfnisse der Lehrer an höheren Unterrichtsanstalten” [With Special Consideration for the Needs of Teachers at Institutions of Higher Learning]. The Sitzungsberichte der Berliner Mathematischen Gesellschaft [Proceedings of the Berlin Mathematical Society] – the Society had just been founded in 1901 – were published as an appendix to this journal. Lampe died in 1918, and Jahnke in 1921. The journal ceased publication with vol. 28 (1920); see also Section 9.4.1. Klein’s idea to establish a publication for “mathematical notices” (Point 5 in Fig. 26) has had lasting effects. That is, today’s Mitteilungen der DMV [Notices of the German Mathematical Society] can be traced back to Klein’s initiative. As early as 1896, after the first four volumes of the Jahresbericht der DMV [Annual Report of the German Mathematical Society] had been published (with some delays) by the Georg Reimer press in Berlin, the Society decided to switch publishers to Teubner. In his plans to reorganize this publication, Klein followed the model of the Bulletin of the American Mathematical Society428 and the Physikalische Zeitschrift, which had been founded by Eduard Riecke and Hermann Theodor 427 See STUBHAUG 2010, p. 289 (Eneström was not a member of the journal’s editorial board). 428 [BStBibl] Klein’s letter to Dyck on December 23, 1899.

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Simon in 1899. In 1901, Klein and Ackermann-Teubner instituted a new program for the Jahresbericht der DMV that included monthly notices (Mitteilungen), later published as an independent section (2. Abteilung of the Jahresbericht). August Gutzmer, with whom Klein had a good working relationship,429 was named the editor as of vol. 11 (1902). This issue contained, for the first time, news about universities, inaugural addresses, presentations, lectures, personnel changes, and information about national and international conferences, events, etc.430 Ninth. Teubner’s publishing program also benefited from the “Kleinian educational reform” (Section 8.3.4). It published not only new and reform-oriented series of mathematical textbooks. Klein also relied on Teubner to publish suggestions for reform, lectures on mathematical instruction, and the aforementioned five-volume work Abhandlungen über den mathematischen Unterricht in Deutschland. Klein himself furnished such works with prefaces or postfaces. He also arranged for the first German-language book on the didactics of mathematics by the Viennese pedagogue Alois Höfler to be published with Teubner.431 Tenth. In 1908, when B.G. Teubner presented a comprehensive catalog at the Fourth International Congress of Mathematicians in Rome, Ackermann-Teubner made sure that it included “images of some of the main representatives of my mathematical and natural-scientific publishing house.” The only mathematicians pictured here were Felix Klein, Carl Neumann, and the historian of mathematics Moritz Cantor.432 In 1911, when the Teubner press celebrated its hundredth anniversary, Ackermann-Teubner donated 20,000 Marks for the establishment of the aforementioned prize, an “Alfred Ackermann-Teubner Memorial Prize for the Promotion of the Mathematical Sciences.” The founding documents stipulated that the jury for the prize had to be composed of three professors from the University of Leipzig (to be determined by the University senate) and an additional two members to be appointed by the board of the German Mathematical Society. Ackermann-Teubner expressly associated the prize with the ENCYKLOPÄDIE project: The prize is to be awarded retrospectively to a representative of the mathematical sciences who has produced significant works in one of the areas of research that fall within the purview of the Encyklopädie der Mathematischen Wissenschaften (Leipzig, 1898ff.), which was inaugurated by the Deutsche Mathematiker-Vereinigung [German Mathematical Society], and is edited under the aegis of the Academies of Science in Göttingen, Leipzig, Munich, and Vienna. The works under consideration, which may be published as monographs, articles, or in another format, must stand out for their great contribution to scientific or pedagogical progress […].

429 See TOBIES 1988b. 430 For detailed discussions of this journal’s publishing agenda, see TOBIES 1986b; and TOBIES 1987a. 431 Höfler, a student of Boltzmann, followed Klein’s suggestions for instruction reform and began to write this book in the summer of 1904. See HÖFLER 1910, p. xiii. 432 TEUBNER 1908, p. vi.

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Alfred Ackermann-Teubner then enumerated the research areas that would be eligible for the prize and noted that, over time, this list might be expanded to include newly established fields of research: 1. History, philosophy, didactics, education; 2. Mathematics (primarily arithmetic and algebra); 3. Mechanics; 4. Mathematical physics; 5. Mathematics, primarily analysis; 6. Astronomy, fitting methods [Ausgleichsrechnung], and error theory; 7. Applied mathematics, primarily geometry; and 8. Applied mathematics, primarily geodesy and geophysics, to the extent that the research in question does not already belong to one of the fields mentioned above.433

Ackermann-Teubner chose the first winner himself, and he took this opportunity to honor his main collaborative partner, Felix Klein, who was awarded the prize and a sum of 1,000 Mark in 1914. Afterwards, Klein served as one of the members of the prize jury appointed by the German Mathematical Society (he would step down from this role in 1922).434 During the First World War, the mathematical and natural-scientific division of the press was seen as no longer profitable. Ackermann-Teubner resigned from his position as a liable partner in the firm. No longer involved in the press’s operations, he now performed merely ceremonial duties for the company, as he explained to Klein (and Hilbert) in personal letters.435 He did, however, fulfill one last promise to Klein, which was to guide the ENCYKLOPÄDIE project to its conclusion (1935) – even though Klein would not live to see that final accomplishment. In any event, Ackermann-Teubner’s resignation from his position was one of the reasons why the journal Mathematische Annalen left Teubner for the Julius Springer press in 1920. Furthermore, Klein’s collected works – Gesammelte Mathematische Abhandlungen (GMA, 1921–23) – and his other mathematical books were no longer published by Teubner but by Springer as well (see Section 9.2). 5.7 FELIX KLEIN IN LEIPZIG’S INTELLECTUAL COMMUNITIES Felix Klein was actively engaged in several Leipzig communities; he used them for important scientific goals and thus quickly took on a recognized role. First of all, a so-called “Professorium” should be mentioned: this included the entire body of university professors, who came together at irregular intervals on special occasions. Klein integrated himself here, but not much is known about his role.436 Nevertheless, this association is noteworthy because, in 1886, Klein would initiate a similar arrangement at the University of Göttingen (see Section 6.4.1). In

433 [UA Leipzig] Rep. III/II/I, No. 93, vol. 4, p. 20. 434 The prize was awarded every two years. The early winners after Klein were Ernst Zermelo (1916), Ludwig Prandtl (1918), Gustav Mie (1920), and Paul Koebe (1922). The winners were announced in Mathematische Annalen. 435 See ACKERMANN/WEISS 2016, pp. 31–35. 436 See Klein’s remarks in JACOBS 1977 (“Vorläufiges über Leipzig”), p. 1.

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Leipzig, the founding of this body can be traced back to the physicist and physiologist Ernst Heinrich Weber, who had earned a Habilitation there from both the Philosophical Faculty and the Faculty of Medicine, and who was hired as a professor of anatomy in 1821. Weber is also considered the founder of the Polytechnical Society (1825–44) and the Royal Saxon Society of Sciences (1846).437 5.7.1 A Mathematicians’ Circle Klein created a mathematicians’ circle (Kränzchen) as a venue in which matters of teaching, research, and organization could be discussed. This group first convened under Klein’s leadership in the winter semester of 1880/81, and its original participants were Walther Dyck, Adolf Hurwitz, Ernst Lange, Karl Rohn, and Friedrich Schur.438 After Hurwitz had moved to Göttingen, he recalled the circle fondly (and somewhat enviously), for he missed “the invigorating stimulation that I always had over the course of my student years and which encouraged my work to such a great extent.”439 In later semesters, the circle was joined by the other professors of mathematics in Leipzig, including the astronomer Heinrich Bruns. At Klein’s initiative, they discussed the curriculum, determined the course schedule for each semester, and reached an understanding concerning when and in which rooms each of them preferred to teach, etc. A letter from Mayer reveals that Klein’s efforts as a coordinator were sorely missed after his departure from Leipzig: Recently, we met for the first time at the Lies’ home for an expanded circle [Kränzchen] with women, and I was pleased by what fine hosts they were. A noticeable shortcoming, which never would have happened under your management, is that there will be no lecture courses for beginners this semester: One prerequisite for Scheibner’s course on differential and integral calculus is algebraic analysis, which hasn’t been taught here for an eternity.440

This group also provided a venue for the mathematicians to discuss the Jablonoviana society’s upcoming prize challenge and how it ought to be formulated. 5.7.2 The Societas Jablonoviana In our context, this German-Polish scholarly society, which was initiated in 1769 for the sake of promoting science and culture (and which still exists today), is relevant above all because of its prizes. The statutes of the Societas Jablonoviana stipulated that an annual prize (700 to 1,000 Mark) was to be awarded for work in 437 See Carl Rabl, Geschichte der Anatomie an der Universität Leipzig (Leipzig: J.A. Barth, 1909), pp. 82–84. 438 See Klein’s remarks in JACOBS 1977 (“Vorläufiges über Leipzig”), p. 1. 439 [UBG] Cod. MS. F. Klein 9: 932 (a letter from Hurwitz to Klein dated June 21, 1882). 440 Quoted from TOBIES/ROWE 1990, p. 160 (Mayer to Klein on December 4, 1886).

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mathematics, physics, economics, or history. Applications for this award could be written in German, Latin, or French. On June 21, 1882, Hurwitz wrote to Klein and asked: “Have you set the prize task for the Societas Jablonoviana?”441 Formulated in March of 1882, the mathematical prize task was announced in Mathematische Annalen 20 (1882), p. 146. Hurwitz rightly suspected that it was Klein who had come up with the problem, which consisted of executing, by the year 1885, “an investigation of general surfaces of the fourth order.” It was stressed that the theory of surfaces of the third order had been completed to some extent in the works of Schläfli, Klein, Zeuthen, and Rodenberg, and that preliminary work relevant to the challenge at hand could be found in Plücker’s Neue Geometrie des Raumes [New Geometry of Space], in Karl Rohn’s studies of Kummer surfaces, and in the works of Zeuthen and others on surfaces of the fourth order with a double conic section. This challenge from 1882 is also noteworthy because, while still in Munich, Klein had repeatedly formulated a similar problem for a prize. The problem in question, which Klein posed for the year 1876/77, read as follows: In the case of curves of the fourth order with two double points, an analysis of the systems of intersection points leads to elliptic integrals. By using these integrals, applicants are invited to discuss and numerically determine the reality and position of the different configurations that the curve can assume and also those of the non-adjunct curves of contact, at least in individual cases.442

Klein strove to classify curves of the fourth order by using Abelian integrals,443 and he hoped to encourage progress in this area of research. The same problem was presented yet again in 1877/78, but no one rose to the task.444 In order to develop the theory of rational curves of the fourth order, the topic for 1879/80 was reformulated, explained in greater detail, and repeated in the same form for the 1880/81 challenge in Munich: Among the plane algebraic curves, those rational curves whose coordinates can be uniquely assigned to the values of a variable parameter are the most accessible to a detailed treatment. In particular, the properties of the designated curves of the third and fourth order can be regarded as determined. In order to produce a general representation of these curves, however, it is necessary to proceed from starting points that differ from those that have been previously used. What is desired now is the development of a theory of rational curves of the fourth order that is based on equations for the parameters of the inflection points and cuspidal points in such a way that possibly all relationships are expressed in the coefficients of these equations, whereby a connection to the theory of binary forms should be sought.445

Of the two prize-winning papers from the year 1881, one came from Friedrich Dingeldey, who had attended Klein’s lectures in Munich. He received half the

441 442 443 444 445

[UBG] Cod. MS. F. Klein 9: 932 (a letter from Hurwitz to Klein dated June 21, 1882). BERICHT 1877, p. 27. See KLEIN 1922 (GMA II), pp. 99–169. BERICHT 1878, p. 17. BERICHT 1881, p. 19.

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prize,446 and he completed his doctoral degree in 1885 in Leipzig with a dissertation titled “Über die Erzeugung von Kurven 4. Ordnung durch Bewegungsmechanismen” [On the Generation of Curves of the Fourth Order by Means of Motion Mechanisms]. This research topic was also treated in the aforementioned doctoral thesis by Klein’s student Paul Domsch (see Section 5.5.2.3). The problems that Klein formulated for prizes are indicative of the research areas that were especially important to him at particular points in time (see also Sections 5.5.8 and 8.3.2). 5.7.3 The Royal Saxon Society of Sciences in Leipzig On July 24, 1882, Felix Klein was elected as a full member of the mathematicalphysical class of the Royal Saxon Society of Sciences in Leipzig.447 Wilhelm Wundt, who was seventeen years Klein’s senior and who had been a professor in Leipzig since 1874, was named a member on the same day, as was Christian Braune, who had been a professor of topographical anatomy at the university since 1872. The Society did not have a special class for the medical disciplines. Klein cooperated with both of his newly elected fellow members. When he left Leipzig, he was made a non-resident full member of this Saxon academy.448 Although most of the historical records pertaining to this institution were lost in the Second World War, it is still possible to ascertain Klein’s high level of engagement in it. About the year 1884, Klein remarked: “Reshuffling the Leipzig Society of Sciences. The edition of Möbius’s works.”449 His initiatives included the following: First, Klein worked together with Carl Ludwig, a professor of physiology and the secretary of the mathematical-physical class, to propose three new addenda for the Society’s statutes. The suggested addenda were sent to the Ministry of Culture in Dresden for approval: a) The limitation of the number of full local members to forty is to be repealed. b) Only the votes cast by full local members who are present in the relevant meeting will be considered valid. c) Regarding non-resident members, each class can decide on its own whether they may deliver lectures, participate in meetings, and submit work for publication.450

The Saxon Ministry of Culture approved these addenda, so that two of Klein’s students who attained professorships in Saxony were able to become full members

446 BERICHT 1881, p. 20. To this day, the topic still has a few open problems; see, for instance, https://www.iaz.uni-stuttgart.de/AbDartheo/ehemalige/Oehms/zula.html. 447 [AdW Leipzig]. The Society was founded on June 23, 1846 (on the 200th anniversary of Leibniz’s birth). Since July 1, 1919, it has been known as the Saxon Academy of Sciences. 448 See https://www.saw-leipzig.de/de/mitglieder/kleinf; and WIEMERS/FISCHER 1996. 449 Quoted from JACOBS 1977 (“Vorläufiges über Leipzig”), p. 3. 450 [StA Dresden] 10272/4, fol. 246.

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of the Society relatively quickly: Axel Harnack on July 18, 1885, and Karl Rohn on December 2, 1889.451 Second, Klein’s efforts to initiate an exchange of publications between the Saxon Society of Sciences and other similar organizations are also noteworthy. Because his colleagues shied away from the effort, Klein wrote to Darboux on May 3, 1881 (before he was even a member) and asked – in the names of Wilhelm Scheibner, Carl Neumann, and Adolph Mayer – whether Darboux might be interested in establishing an “exchange […] between the publications of the mathematical-physical class of the Royal Saxon Society of Sciences here and the Bulletin de la Société mathématique in Paris.”452 Third, Klein was not only able to publish his own results quickly in the Society’s proceedings (Berichte über die Verhandlungen der Kgl. Sächsischen Gesellschaft der Wissenschaften, Mathematisch-physische Classe); in 1884, he also received permission “to submit works by non-members that are not too long and have scientific content.” He communicated this to Hurwitz, and he encouraged him and Lindemann to send him suitable articles.453 In addition, Klein presented works by some of his students (Biedermann, Dyck, Otto Fischer, Gierster, Hurwitz, Rohn, Staude, Willibald Reichardt) and seminar participants (Molien, Morera, Pick, Hilbert) at the class meetings of the Society.454 Fourth, Klein took care to ensure that the edition of August Ferdinand Möbius’s collected works would be sponsored as an “Academy project.” Klein’s interest in Möbius was based above all on the fact that, early on, he had noticed points of contact between Möbius’s work and his own.455 Even in the aforementioned letter to Darboux from May 3, 1881, Klein had inquired about a work on the “question des polyèdres” (1861), with which Möbius had applied for a prize from the Paris Academy.456 Klein added: “For many years there has been talk of editing Möbius’s works; however, I am afraid that a long time will pass before anything comes of it, because those most closely involved do not have the proper initiative.”457 Soon thereafter, Klein learned that Richard Baltzer, who had been a 451 Klein’s vote also counted in hiring processes at the Technische Hochschule in Dresden. On January 5, 1885, he wrote to Axel Harnack (who was already a professor there): “In my opinion, Rohn is superior to the other four candidates in terms of his independence and academic personality.” The (ranked) list of suggested candidates for the second professorship for mathematics and analytic mechanics comprised: 1. Karl Rohn, 2. Otto Staude, 3. Friedrich Schur and Hans von Mangoldt. See [StA Dresden] 15547. 452 [Paris] 72: Klein to Darboux, May 3, 1881. 453 [UBG] Math. Arch. 77: 126 (a postcard from Klein to Hurwitz dated November 19, 1884). 454 See REGISTER 1889. Among other studies by Klein’s students, the Society published David Hilbert’s article “Über eine allgemeine Gattung irrationaler Invarianten und Covarianten für eine binäre Grundform geraden Grades,” Berichte über die Verhandlungen der Kgl. Sächsischen Gesellschaft der Wissenschaften: Math.-physische Classe 37 (1885), pp. 427–38. 455 See KLEIN 1979 [1926], pp. 105–08; PLUMP 2014; and DOMBROWSKI 1990, p. 330. 456 In 1858, the prize challenge posed by the Académie des Sciences in Paris read as follows: “Perfectionner en quelque point important la théorie géométrique des polyèdres.” 457 [Paris] 72: Klein to Darboux, May 3, 1881.

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member of the Saxon Society since 1864 and was a professor at the University of Gießen, was likewise interested in the project.458 In short order, Baltzer, Klein, and Scheibner were commissioned by the Society to produce the edition. Baltzer edited the first volume, which contains Möbius’s work on barycentric calculus. He wrote to Klein: “I have anew the opportunity to admire your resolution and efficiency: while I am starting with the first volume, you have already finished the second, and have in the meantime produced an enormous book [Klein 1884], for the completion of which I congratulate you!”459 Klein was responsible for the second volume (additional geometric studies) and the third (statics). In his prefaces to volumes II (October 1885) and III (February 1886), Klein thanked Privatdozent Otto Staude, Dr. C. Reinhardt,460 and his French colleagues Bertrand and Darboux for their support and assistance. Shortly before moving to Göttingen, Klein informed Hurwitz: “The third volume of Möbius’s works is still being completed, and all sorts of half-measures, from whose incompleteness I suffered, are still being clarified.”461 While in Göttingen, Klein also contributed to the fourth volume (astronomy, etc.), for which Wilhelm Scheibner was the main editor. On October 8, 1887, Carl Ludwig, the already mentioned secretary of the Society’s mathematical-physical class – who was thirty years Klein’s senior – invited Klein to Leipzig for a meeting of the “Möbius commission,” which was scheduled to take place on October 22nd at 3 o’clock in the afternoon. Klein responded to this invitation by suggesting that his presence would be unnecessary, and this prompted Ludwig to pen, just two days later, the following informative reply: My esteemed friend! Are you suggesting that we should proceed without you on October 22nd? That would be impossible. There are still many obstacles that need to be overcome before the edition of Möbius’s works can be completed, and because you have been the soul of this undertaking, you will surely be eager to ensure that it is brought to a flawless and unimpeachable conclusion.462

Ludwig enclosed with this letter a newly discovered article by Möbius (on geometric addition and multiplication), which Klein then included in the fourth volume; he wrote an additional explanatory preface for it. Fifth, before it began in earnest, the edition of Hermann Graßmann’s collected works likewise needed a boost of initiative from Felix Klein. Carl Ludwig wrote: Dear esteemed colleague, your lightning bolt has struck. Today, after the lecture by [Friedrich] Engel, the class elected a commission that includes, in alphabetical order, Klein, Lie, A. Mayer, and Scheibner (Engel will serve as the record keeper). We hope that you will honor the request of our board and be willing to grant us your advice as often as it is needed.463

458 459 460 461 462 463

[Paris] 74: Klein to Darboux, May 28, 1881. [UBG] Cod.MS F.Klein 8:54 (Baltzer to Klein on September 13. 1884). Curt Reinhardt had completed his doctorate under W. Scheibner and C. Neumann in 1882. [UBG] Math. Arch. 77: 154 (a letter from Klein to Hurwitz dated March 2, 1886). [UBG] Cod. MS. F. Klein 10: 883, 884 (Carl Ludwig to Klein on October 8, 1887). Ibid. 10: 887 (a letter from Carl Ludwig to Klein dated December 5, 1892).

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Before the Society decided to go ahead with this project, Victor Schlegel had already published two volumes based on Graßmann’s works.464 Klein had written a rather critical review of Schlegel’s first volume, System der Raumlehre [The System of Spatial Theory] (1872).465 In 1881, Klein nevertheless made it possible for Schlegel, who had been Graßmann’s colleague and was six years Klein’s senior, to be awarded a doctoral degree in Leipzig by accepting the second volume of this work as his dissertation: Die Elemente der modernen Geometrie und Algebra [The Elements of Modern Geometry and Algebra] (1875).466 For the edition of Hermann Graßmann’s collected works, however, Klein preferred to have editors whom he could trust to produce a sophisticated scientific commentary. With this in mind, he reached an understanding with Graßmann’s family,467 and, as already mentioned, he secured a commitment from Friedrich Engel. Later, on October 17, 1892, Klein attended a meeting of the mathematicalphysical class in Leipzig to set the project in motion. Besides Engel, Klein recommended additional collaborators (Jacob Lüroth), and he drafted conditions for a publishing contract with B.G. Teubner. Engel, who was put in charge of the edition as a whole, described in his preface to the first volume (1894) how Klein had been the impetus behind the project and how, after providing this initial impulse, he largely stepped into the background. Graßmann’s works appeared in three volumes (each of them divided into two parts) between 1894 and 1911. Sixth, within the framework of the Society’s broad mathematical-physical class, Klein used his contacts specifically to promote the career of his doctoral student Otto Fischer. Klein put Fischer in touch with the anatomist Christian Braune and with the physiologist Carl Ludwig. As early as 1886, Fischer expressed that he was pleased in his “dual role as a mathematical and anatomical assistant, on the one hand, and as a probationary teaching candidate at the Realgymnasium in Leipzig on the other.”468 In 1893, he was able to complete his Habilitation in physiological physics, and in 1896 he was made an associate professor in the Faculty of Medicine. Fischer went on to become a recognized biophysicist. The Royal Saxon Society of Sciences elected him as an associate member in 1893 and named him a full member in 1905. Before that, Fischer had contributed the article on physiological mechanics to Vol. IV of the ENCYKLOPÄDIE. Wilhelm Lorey described Otto Fischer’s career as “an especially fitting example […] of how Klein, with his sharp eye for talent, always understood how to point everyone in the most suitable direction.”469 464 Victor Schlegel, System der Raumlehre: Nach den Principien der Grassmann’schen Ausdehnungslehre und als Einleitung in dieselbe dargestellt, 2 vols. (B.G.Teubner, 1872; 1875). 465 See ROWE 1996; and ROWE 2018a, pp. 95–103. 466 Schlegel submitted this work as his dissertation on Nov. 14, 1881. Klein served as the first reviewer, and C. Neumann as the second. See KÖNIG 1882, A6-2. Klein also took two of Schlegel’s papers with him on his later trip to Chicago. See MOORE et al. 1896, pp. 331–40. 467 [UBG] Cod. MS. F. Klein 9: 486, 487 (H. Graßmann Jr. to Klein, Sept. 18 and 26, 1892). 468 Ibid. 9: 40 (a letter from Fischer to Klein dated May 12, 1886). 469 LOREY 1926, p. 141.

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5.8 TURNING HIS BACK ON LEIPZIG Similar to Klein’s move from Munich to Leipzig, his departure from Leipzig was based on his desire to leave coupled with lengthy decision processes. His desire to leave was based above all on his dissatisfaction with his own mathematical productivity and on the fact that he had taken on too many ancillary duties. On March 10, 1885, for instance, Klein wrote the following to Hurwitz, who was then his most intensive academic correspondent: “Regarding my own affairs, I have become so deeply embroiled in business of a subordinate sort that I have almost no time at all for independent scientific thought.”470 By changing institutions, Klein hoped that he would once again have more time for his own research. He entertained offers from Great Britain, the United States (Section 5.8.1), and finally from Göttingen (Section 5.8.2). The conditions surrounding the job offer from Johns Hopkins in Baltimore have been described in detail by Parshall and Rowe.471 Klein’s path to the University of Göttingen, however, has at times been inaccurately represented in scholarly literature.472 Both of these processes will be examined here in light of new sources and correspondence. Before departing from Erlangen and Munich, Klein had been successful in determining his own successor. Here it will be shown how Klein was able to secure Sophus Lie as his successor in Leipzig and how the mathematical community reacted to this (Section 5.8.3). 5.8.1 Weighing Offers from Oxford and Johns Hopkins Felix Klein wrote to the Saxon Ministry of Culture on December 9, 1883: Having been personally acquainted with Prof. Cayley in Cambridge (England’s top mathematician) for ten years now, I received from him during the last Easter break an invitation to apply for the professorship for higher geometry in Oxford, which has become vacant on account of the death of [Henry] Stephen Smith. I did not do so at the time because, upon further inquiry, the conditions in Oxford did not seem favorable. The issue was not of a material nature; rather, it did not seem possible there to execute a plan that might lead to a truly high level of mathematical research.473 Now, however, Cayley informed me on October 5th that Prof. Sylvester in Baltimore has decided to retire from his position at Johns Hopkins University and that he has been unofficially requested to ask whether I might be inclined to become Sylvester’s successor. I must say that there is much about this invitation that is enticing to me: all of the conditions in Baltimore are new and in an early stage of development; there is great potential there to initiate thoroughly independent and perhaps very successful activity.474

470 471 472 473 474

[UBG] Math. Arch. 77: 136 (a letter from Klein to Hurwitz dated March 10, 1885). See PARSHALL/ROWE 1994. See, in particular, FREI 1984, which has served as the basis for other historians. Regarding mathematicians at the University of Oxford (UK), see FAUVEL et al. 2013. [StA Dresden] 10281/184, fols. 25–25v (Klein to the Saxon Ministry, December 9, 1883).

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Invited by Sylvester, Cayley had worked as a guest professor from January to May of 1882 at Johns Hopkins, a private university founded in 1876, and he reported to Klein about the conditions there. Klein was excited about the job offer; he informed Paul Gordan and other colleagues about the prospect of succeeding Sylvester far before he notified the Ministry in Dresden about this possibility. As early as October 22, 1883, Gordan congratulated him: “Best of luck regarding Baltimore; no matter how the matter turns out, something good will come of it for you.”475 This, however, was a premature conclusion. When Klein sent the official job offer from President Gilman of Johns Hopkins to Dresden on December 12, 1883, the Ministry responded “discourteously.”476 Although Minister Carl von Gerber expressed that he did not want the University of Leipzig to lose Klein’s “outstanding teaching abilities,” he did not propose a counter-offer of any sort. Instead, he pointed out that Klein’s many wishes had always been fulfilled and that he was now expected “to reject, of his own accord and of his own free will, the appointment to a non-European educational institution.”477 This lack of goodwill on the part of the Ministry was enough to make Klein want to accept the position in Baltimore. To seek advice, he engaged in a whirlwind of conversations and correspondence with his family, friends, and colleagues. Sophus Lie, on the one hand, recommended him to accept the offer unconditionally, also to escape the rivalry with the Berliners; on the other hand, Lie added that he would then have fewer chances to see his friend and that the journal Mathematische Annalen would suffer in Klein’s absence: “Indeed, you are the journal’s heart and soul.”478 Even at this early point, the two of them discussed the question of whether Lie might want to succeed Klein in Leipzig. Lie seemed to be interested; he weighed the pros and cons, and he stressed: even if an offer from Leipzig would never come to pass, he would nevertheless “live for many years with feelings of appreciation for the tribute that you have paid to me in your letter. And I will never forget it.”479 In December of 1883, Klein wrote a long draft of the ideas that he would like to express in his response to Johns Hopkins, including: Let me say in advance that, in principle, I am in favor of accepting the offer. I am excited by the novelty of the task and the great potential that it promises: I am even still young enough to find something invigorating in the very act of taking on a new position. However, this enthusiasm is tempered by the uncertainty of success, the difficulty of the undertaking, and above all by the circumstance that, for three years here in Leipzig, I have held a first-rate position, which the university has lavishly endowed with all the attributes that are needed to support my success and productivity. My enthusiasm is also tempered by the fact that, having taught in Germany now for thirteen years, I have developed close relationships with the younger ge-

475 476 477 478

[UBG] Cod. MS. F. Klein 9: 434, fol. 39 (Gordan to Klein on October 22, 1883). See Klein’s comment in JACOBS 1977 (“Vorläufiges über Leipzig”), p. 3. [UBG] Cod. MS. F. Klein 22 L. fols. 31–31v (Minister Gerber to Klein, December 21, 1883). Ibid. 10: 690/1 (Lie to Klein, December 1883). Although Lie played a role in the establishment of Mittag-Leffler’s journal Acta Mathematica, he did not publish anything in it. 479 Ibid. 10: 691–696 (an undated letter from Lie to Klein); also STUBHAUG 2003, pp. 216–17.

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5 A Professorship for Geometry in Leipzig neration of German (and, I may add, European) mathematicians, so much so that my friends and relatives have strongly implored me not to leave them.480

Klein was not silent about his health problems, noting that “my nervous system was exhausted by overwork in earlier years, so that I tend to suffer from asthma, stomach ailments, etc.” He emphasized that he was maintaining a strict diet and that, for years, he had “not socialized in the evenings.” All in all, he indicated that he could accept the position if his arrangement were equal to Sylvester’s, that is, if he only had to give a few special lectures and if he received the same salary: “Instead of $5,000 per year, as you have proposed, I would settle for $6,000 per year plus an additional allowance for living expenses. Regarding my acceptance of the position, I consider it a necessary condition that my benefits will be no less than Sylvester’s in either respect.” As an additional condition, Klein mentioned providing security for his family: “If I die, my wife is to receive […] an annual pension of 1,400 to 1,600 Mark, or approximately $400.” Klein also wanted guarantees that his salary would continue to be paid in the event that he suffered from a prolonged illness. Even though Klein would omit this line from the final version of his letter, it is clear that the matter seemed important to him. In his draft, he stated that his salary in Germany “would not be diminished in the event of a longlasting illness” and that “a younger colleague would be ready to serve as a substitute” should such circumstances arise. An arrangement of this sort did not exist in the United States. In his closing sentence, Klein expressed in no uncertain terms that he would “consider the negotiations to be over between us” if he did not receive a response within six weeks (by the end of January 1884). When Hurwitz asked him how things were unfolding, Klein replied: “The situation with Baltimore is still unclear; every day, I expect an answer to the preliminary questions that I asked them. In any case, I do not intend to make a rash decision.”481 Because Klein felt that Johns Hopkins had failed to meet his conditions, he remained in Leipzig. He informed the Saxon Ministry of Culture of his decision on February 1, 1884.482 When, later on, another opportunity to leave Saxony presented itself, Klein was all the more pleased to take advantage of it. 5.8.2 The Physicist Eduard Riecke Arranges Klein’s Move to Göttingen On March 13, 1885, Felix Klein notified the Ministry of Culture in Dresden “[…] that, on January 18, 1885, the Philosophical Faculty in Göttingen had nominated [him] to the Prussian Ministry as the leading candidate to succeed Professor Stern.”483 This process has a lengthy prehistory; in fact, a considerable amount of time would pass before Klein received an official offer from Prussia. 480 481 482 483

Ibid. 22 L, fols. 23–25v, quoted here from fols. 23–23v (Klein’s draft, dated Dec. 18, 1883). [UBG] Math. Arch. 77: 110 (a letter from Klein to Hurwitz dated January 19, 1884). [StA Dresden] 10281, fol. 31. Ibid. fol. 33. – Regarding this nomination, see Appendix 4.1.

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On April 19, 1884, the seventy-six-year-old Moritz Abraham Stern had requested “dispensation from the obligation of giving lecture courses.” He wanted to relocate to Bern (where his son Alfred was living at the time) because he had lost his only daughter in Göttingen.484 An imperial decree issued on September 12, 1884, approved this request, effective October 1, 1884.485 Eduard Riecke sought Klein’s advice: You know the difficulties that arise from the personalities of Schwarz and Schering and from the consideration we owe to Enneper, and you can better assess what we need in scientific terms. It would be best if we could get you back to Göttingen, but there is hardly any hope for that. 486

When Klein indicated that he would like to turn his back on Leipzig, Riecke did everything he could to overcome existing hurdles. Riecke informed Klein that Schwarz and Schering intended to offer the professorship to Georg Hettner,487 who had completed his Habilitation under Schwarz in 1876 and had been working as an associate professor in Berlin since 1882. A hiring committee was formed on November 13, 1884, for which Wilhelm Müller,488 a Germanist and then the dean of the Philosophical Faculty, was responsible for keeping the minutes: The committee is composed of the following experts in the field at hand: Messrs. Schering, Schwarz, Weber (possibly), Riecke, Voigt. A vote will be taken concerning the inclusion of an additional member, for it may be necessary to replace Mr. Weber. The number of votes is 22. Mr. G.E. Müller receives 12 votes; Mr. [Carl] Klein 10 votes.489 Accordingly, Mr. G.E. Müller is elected. As the member further removed from the field [i.e., mathematics], Privy Councillor H. Sauppe is elected with 16 votes.490

Riecke persuaded several committee members to accept his suggestions, namely the theoretical physicist Woldemar Voigt; the psychologist Georg Elias Müller, who had been chosen to replace the ailing physicist Wilhelm Weber and had already had contact with Klein in the past; and the philologist Hermann Sauppe, who, as the secretary of the Royal Society of Sciences in Göttingen, was hoping for Klein’s active involvement. On December 3, 1884, the committee finalized its (ranked) list of potential candidates for the position: 1. Klein, 2. Voß, 3. Enneper. 484 See Ferdinand Rudio, “Erinnerungen an Moritz Abraham Stern,” Jahresbericht der DMV 4 (1897), p. 35. 485 [UAG] Kur. 5846, pp. 80–86. Stern kept his full salary and used a portion of it to found the Stern Foundation, the mission of which was to support widows, orphans, janitors, and caretakers associated with the University of Göttingen. 486 [UBG] Cod. MS. F. Klein 111: 505 (a letter from Riecke to Klein dated September 19, 1884). 487 Ibid., 111: 506 (a letter from Riecke to Klein dated October 4, 1884). 488 W. Müller had served as a member of Klein’s Habilitation committee (see Section 2.7.2). 489 In 1887, the mineralogist Carl Klein accepted a professorship in Berlin. He was succeeded in Göttingen by Theodor Liebisch. 490 [UAG] Phil. Fak. 170a, No. 39a, 39b.

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This proposal was signed by the dean Wilhelm Müller, as well as by Sauppe, Voigt, and Georg Elias Müller. The full professors of mathematics Ernst Schering and H.A. Schwarz added the following remark to their signatures: “Subject to a separate opinion [Separatvotum].”491 On December 4, 1884, Riecke informed Klein of the result: he and Voss had been recommended for the professorship by the majority of the committee members, and Georg Hettner had been rejected.492 In January of 1885, Schwarz and Schering each wrote a dissenting separate vote, as announced (see Appendix 4.2, and 4.3). Both were in favor of Georg Hettner’s appointment. Schering pleaded additionally for Enneper, while Schwarz was not willing to support him in this. The faculty’s hiring proposal and both of the separate votes were sent to the university’s Kurator, who forwarded everything, along with his own cover letter, to the Ministry of Culture in Berlin. In his cover letter, the Kurator Adolf von Warnstedt formulated his own ranking of the proposed candidates: 1. Hettner, 2. Klein, 3. Voß.493 It is thus clear that he valued the mathematicians’ separate opinions more than he valued the suggestion from the faculty. In addition, the faculty records also contain a denunciation prompted by H.A. Schwarz: During a chance encounter on the street, Schwarz had informed the orientalist Paul de Lagarde about the academic dispute between Klein and Lazarus Fuchs (see Section 5.5.5). Having learned about this, Lagarde thought that it would be appropriate to make this information known by a letter to the faculty at large. This letter – written on the same day as Schwarz’s separate opinion (see Appendix 4.3) – contains references to sources (supplied to him by Schwarz): “Göttinger Nachrichten from March 4, 1882” (Fuchs) and “Mathematische Annalen, vol. 19, p. 564; vol. 20, p. 52; vol. 21, pp. 143, 214–16” (Klein’s replies). Lagarde referred to the scientific dispute contained in the articles and concluded that there was good reason to take it into consideration.494 This may have had some influence on the Kurator’s decision. Having received the Kurator’s vote, the ministerial director Friedrich Althoff asked Georg Hettner whether he would accept the offer of the professorship in Göttingen. Much to the disappointment of Weierstrass, Schwarz, and Schering, however, Hettner modestly turned down the offer with the remark that “he had not yet accomplished enough with his publications,” as Althoff noted.495 Schwarz’s behavior was Janus-faced. On the one hand, he had written laudatory words about Klein in his separate vote: “If the appointment of Prof. Klein is successful, an outstanding teacher and an important scholar will be gained for our University and for our Prussian fatherland” (Appendix 4.3). On the other hand, he 491 [UAG] Phil. Fak. 170a, No. 41Z, pp. 41ff. This document can also be found in the records of the dean’s office (pp. 41gg–41kk) and in Klein’s personnel file ([UAG] Kur. 5956). 492 [UBG] Cod. MS. F. Klein 11: 509 (a letter from Riecke to Klein dated December 4, 1884). 493 [StA Berlin] Abt. Merseburg, Rep. 75 Va Sekt. 6, Tit. IV, No. 1, vol. 11, fols. 195–98v. 494 [UAG] Phil. Fak. 170a, No. 41uu–41vv (de Lagarde’s letter, dated January 25, 1885). 495 [StA Berlin] Abt. Merseburg, Rep. 76 Va Sekt. 6, Tit. IV, No. 1, vol. 11, fol. 195v. See also TOBIES 1991, pp. 90–93. Regarding Weierstrass’s reaction, see BIERMANN 1988, p. 144.

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had initiated the denunciation mentioned above. Schwarz was somewhat fearful of Klein’s appeal to students and younger colleagues. After Klein had informed Schwarz, on August 13, 1885, about his successful negotiations with the ministerial official Althoff in Berlin, Schwarz wrote to Weierstrass: That which I considered so improbable has indeed come to pass! […] Who can predict how my teaching activity can be organized in the near future? Perhaps, as a result of your decision no longer to hold lectures, a greater number of ambitious students of mathematics will turn to Göttingen, given that genuine function theory will no longer be taught in Berlin. I worry, however, that if your prophecy about the difficulty of working with Mr. Klein is fulfilled, my wish for a “change of scenery” will become all the more urgent.496

Before deciding to offer the position to Klein in August, Althoff had let the matter rest for some time. This is because it had been suggested to him that Felix Klein and Aurel Voss merely wanted to improve the conditions of their current positions, and that they would be too expensive in any case. Not until Althoff visited Göttingen in July of 1885 to resolve a hiring issue in astronomy did Riecke definitively persuade him to extend an offer to Klein (see Appendix 10.1). On July 23, 1885, Riecke informed Klein: “Now I have very reliable news that the Ministry would be willing to offer you the position, so long as the opinion does not spread that it would be entirely hopeless to try to bring you here.”497 Through Riecke, Althoff inquired about Klein’s conditions: a 9,000 Mark annual salary and a reading room for students. In early August, Klein was invited to Berlin to meet with Althoff, and he informed the Saxon Ministry in Dresden about this. There, the cultural minister Carl von Gerber was startled by the news, and he had the following telegram sent to Klein: “His Excellency, to keep you in Leipzig, is offering you a yearly salary of 9,000 Mark.”498 Klein, however, was eager to leave, and he accepted Althoff’s offer of 9,040 Mark (an 8,500 Mark annual salary plus a yearly allowance of 540 Mark for living expenses). Until then, the maximum salary for a full professor in Göttingen had been 7,200 Mark.499 In a letter to the Saxon Ministry, Klein explained that “the character and the scope of my scientific activity” had determined his decision. He added: “Within mathematics, the composition of the student body has developed in such a partisan way that, throughout my 10 semesters of teaching in Leipzig and among the ca. 100 participants in my advanced seminars, there has not even been one Prussian candidate and there has been only one from northern Germany (from Braunschweig).”500 Klein was now thirty-six years old, and he envisioned having a broader influence by working in Prussia, where he might also be able to diminish the dominance of the mathematicians in Berlin. On September 10, 1885, after his

496 497 498 499 500

[BBAW] NL Schwarz 1254, pp. 195–96 (Schwarz to Weierstrass on August 22, 1885). [UBG] Cod. MS. F. Klein 11: 511 (a letter from Riecke to Klein dated July 23, 1885). [StA Dresden] 10281/184, fol. 41 (the telegram dated August 11, 1885). [StA Berlin] Abt. Merseburg, Rep. 76 Va Sect. 6, Tit. IV, No. 1, vol. 11, fols. 313v, 314. [StA Dresden] 10281/184, fols. 42–43. Klein’s remark that approximately one hundred students had participated in his Leipzig seminars is an exaggeration (see Table 6).

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summer vacation, much of which he spent in the company of Hurwitz (see Section 5.5.8), Klein traveled with his wife to Göttingen, where they stayed with Eduard Riecke and found an apartment to rent in October. H.A. Schwarz accepted an offer to visit Klein in Leipzig,501 where he thought it might be possible for him to become Klein’s successor. Klein, however, was interested above all in showing Schwarz the institutions that he had helped to develop. In December, after “His Majesty the Emperor and King [Wilhelm I]” had finally signed the official letter of appointment for Klein’s full professorship in Prussia,502 Klein sent an application to the University of Göttingen’s administration in which he requested funding for a “reading and work room.” Soon thereafter, he informed Hurwitz that his new position was scheduled to begin on April 1, 1886.503 In an earlier letter to Hurwitz, Klein had dreamed about spending another fall break with him, and he ruminated about the possibilities that lay ahead: “Will I ever again enjoy the tranquility of the genuine scholarly life? The possibility for that exists in Göttingen; however, it seems highly unlikely to me that this will become a reality.”504 Tranquility, however, was not Klein’s thing, and he already suspected that he would soon be as active as ever – just as Riecke expected that he would (see Appendix 10.1). 5.8.3 The Appointment of Sophus Lie as Klein’s Successor – and the Reactions Klein wanted Sophus Lie to be his successor in Leipzig. Lie alone, he thought, could be trusted “to establish an independent geometric school.” This was the main argument in the extensive application that Klein wrote in his own hand to nominate his favored candidate.505 Their friendship aside, Klein expected that Lie would be able to perpetuate the high international reputation of the mathematical institution in Leipzig, and he also expected that Lie would gladly accept the position. Lie had already signaled his willingness to take the job when Klein was still considering the offer from Johns Hopkins, and he had expressed (even before Friedrich Engel came to Christiania [Oslo] as his assistant): “The next time I come to Germany, I hope to stay there for a longer period of time. May this soon be possible for me! It is lonely, terribly lonely here in Christiania, where no one understands my work and interests!”506 The first draft of Klein’s proposal to hire Lie as his successor is dated October 28, 1885.507 In a letter to Hurwitz written a few days later, however, Klein was less than certain that his proposal would be accepted: “It has been a long time 501 502 503 504 505 506 507

See Klein’s notes in JACOBS 1977 (“Vorläufiges über Leipzig”), p. 4. [UAG] Phil. Fak. 171a, No. 19a. [UBG] Math. Arch. 77: 150, 151/2 (Klein to Hurwitz, Dec. 13, 1885 and January 3, 1886). Ibid. 77: 147 (a letter from Klein to Hurwitz dated September 24, 1885), emphasis original. [UA Leipzig] PA 693, p. 31. [UBG] Cod. MS. F. Klein 10: 689 (a letter from Lie to Klein, September 1883). [UA Leipzig] PA 693, pp. 29–36.

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since I’ve written to you because the hiring process here has drained my time and humor. I am fully at odds with [Carl] Neumann, and even if I am able to win over the faculty, it is still unclear what the Ministry will do, given that a separate opinion will be submitted alongside the faculty’s vote.”508 Klein’s disagreement with Carl Neumann was based on the fact that Neumann wanted to promote Adolph Mayer to full professor. However, Mayer, who was then an honorary professor, was financially secure on account of his family’s wealth, and he allowed Klein to explain in the faculty meeting that “he would take no offense if a geometrician were hired to fill the vacant position.”509 At the end of November, after Klein had revised his proposal once more and the majority of the faculty had voted in favor of it, Carl Neumann, Wilhelm Scheibner, and the physicist Wilhelm G. Hankel nevertheless composed a separate vote. Interestingly, however, they did not turn against Sophus Lie. As H.A. Schwarz informed Weierstrass, Neumann even considered Sophus Lie an acceptable choice, especially “because he [Lie] would not compete with him [Neumann] in the area of function theory.”510 The separate vote was in fact directed against Ferdinand Lindemann, Aurel Voss, and Axel Harnack, who were the additional mathematicians mentioned on the list of possible candidates. It was argued that Lindemann was no longer a geometrician, and that the other two were no better than the Privatdozenten who were already working in Leipzig.511 Klein, however, had the further success of “his” institute in mind, and he noted in his application that the latter would need a director “who, through his earlier activity, has demonstrated an understanding and interest in the tasks that the role demands: Voß and Lindemann satisfy this requirement and, in the unfortunate event that Lie should reject the offer, the faculty believes that, in the interest of the institute, the Ministry should not hesitate to offer the position to either one of them.”512 After the faculty had submitted its proposal to the Saxon Ministry of Culture on December 12, 1885, the offer went straight to Norway. During the New Year’s celebration, which Klein spent together with Georg Pick and David Hilbert, he was able to report that Sophus Lie had responded positively. Lie wrote: “It is more than remarkable that you pulled this off. I only hope that you never regret it!”513 After a great deal of correspondence, and after settling his affairs in Norway, Lie accepted the offer with a letter to the Ministry dated January 16, 1886. He came to Germany in February, and Klein accompanied him to Dresden.

508 [UBG] Math. Arch. 77: 149 (a letter from Klein to Hurwitz dated November 7, 1885). 509 [StA Dresden] 10281/212, fols. 7–7v. – On July 7, 1890, the Ministry agreed to a new application to appoint Adolph Mayer as full professor after all. [UA Leipzig] PA 725, fol. 22. 510 [BBAW] NL Schwarz 1254, p. 223 (Schwarz’s letter to Weierstrass, April 3, 1886). 511 [StA Dresden] 10281/212, fols. 2–9 (hiring proposal), 10–15 (separate vote). 512 [UA Leipzig] PA 693, fol. 35R. 513 [UBG] Cod. MS. F. Klein 10: 712/1 (Lie’s letter arrived in Leipzig on New Year’s Eve, 1885).

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Klein had made his intentions known early on, for instance in a letter that he sent to Darboux on November 21, 1885: Perhaps you have already heard that I am about to move to Göttingen (that is, on April 1, 1886). Here I have taken on more work than is good for me, and yet I have not been able to actualize my ideas as much as I would like. There are also some strictly private reasons for me to move: family considerations, etc. What would you say if I were to succeed in bringing Lie to Leipzig? I am working on that, but I am not sure whether I will prevail.514

In a letter dated January 2, 1886, Darboux replied that he had since heard about the job offer to Sophus Lie, but that Lie was held up in Christiania for one reason or another. Addressing Klein, he added: “J’espère que nous nous verrons un de ces jours […].”515 In Germany, there were colleagues who misunderstood Klein’s information, and this misunderstanding was caused by the similar way in which the words “Lie” and “Sie” (you) appeared in his handwriting. On December 7, 1885, H.A. Schwarz516 reported to Weierstrass that Klein had written him about Leipzig’s hiring proposal: “1. you (Sie) [or: Lie], 2. Lindemann, 3. Voss, in addition to which there is a separate vote in which only you are [or: Lie is] nominated.” Schwarz commented further: Now, if I am right to assume that the word following 1. is Sie [you] and not Lie, then I have the distinction of being the first candidate on the list, and this is very gratifying to me. In Klein’s letter, which is not very legible, the reading Lie is possible in both places, but a proposal to hire Sophus Lie does not seem very probable to me, though it is possible. Perhaps you already know more details about this.517

Weierstrass responded on December 20, 1885: I am deeply sorry that Klein’s careless handwriting has caused such a terrible disappointment for you. I became familiar with the situation a day or so before you, and from a reliable source. Kronecker did not want to believe it at first. If Leipzig were a Prussian university, I would have felt obligated to express my opinion to the proper authority regarding the outrageous procedure approved by the Leipzig faculty, which is an insult to every German mathematician who is now in the prime of his life. Although I cannot deny that Lie has produced some valuable works, he is not of such significance as a researcher and teacher that he – a foreigner – should be given preference over all possible German candidates. Now it will be said that he is a second [Niels Henrik] Abel, who had to be won at all costs. A lovely beginning to a new era, which is set to begin under Klein’s presidency! P.[aul] Dubois[-Reymond] hit the nail on the head when, already years ago, he referred to the trefoil of Klein – Lie – Mayer as the “société thuriféraire.”518

Weierstrass’s influence, however, did not extend to Saxony. 514 [Paris] 76: Klein to Darboux, November 21, 1885. 515 [UBG] Cod. MS. F. Klein 9: 503 (Darboux to Klein). They would next meet in 1887. 516 Schwarz was not a “rival candidate” to be Klein’s successor, as GRAY suspected (2013, p. 490). 517 [BBAW] NL Schwarz 1254: 203 (Schwarz to Weierstrass, December 7, 1885). 518 Ibid. 1175: 314–15. Here, “société thuriféraire” (literally: incense-carriers’ society) is a derogatory reference to an “association for mutual admirers.” See also STUBHAUG 2002, p. 341.

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Yet Schwarz was not the only person to misinterpret Klein’s letters about the hiring process in Leipzig; Lindemann and Hurwitz in Königsberg misread his handwriting as well. On December 14, 1885, Hurwitz informed Klein: Lindemann received your letter and has shared its content with me. He was unsure, however, whether at one point you wrote “which Lie accepts” or “which you [Sie] accept.” If Lindemann should receive the offer, I am convinced that he would accept it. An aggravating factor for Lie, at any rate, is the current trend among his opponents to make such a fuss over his nationality.519

Klein’s Erlangen friends were also not very enthusiastic about Klein’s list of candidates. In a letter to him, Max Noether, who was still waiting for a full professorship, wrote: “I don’t think Lie is a suitable candidate for the Leipzig position; and 3) [Voss] will not be able to replace you.”520 Klein helped Sophus Lie cope with the many difficulties that he faced in the beginning. On Klein’s advice, Lie visited H.A. Schwarz in Göttingen, where he found the conversation difficult because of their different ways of mathematical thinking. Thus, in a subsequent letter to Klein, Lie stated that “there is perhaps no one in Germany whom I better understand than you.”521 During the summer semester of 1886, Lie made use of Klein’s lectures on projective geometry, and he also requested more of his lecture notes. Earlier, Klein had also acquired copies of Weierstrass’s lecture notes for him.522 Following Klein’s example, Lie now attempted to collaborate with talented students. In his course offerings, he also adhered to Klein’s previous plan.523 Klein helped him with smaller matters, such as choosing appropriate exam topics for future teachers of mathematics. In September of 1886, Klein traveled together with Lie to the annual conference of natural scientists and physicians (GDNÄ) in Berlin, he edited Lie’s talk (on the problem with Helmholtz, see Section 6.3.6), and he made further attempts to establish Lie’s reputation.524 Lie sought Klein’s advice about how he might best promote the careers of Eduard Study and Friedrich Schur, and they discussed which lecture courses these Privatdozenten should offer. Lie repeatedly asked Klein: “Does all of this accord with your plan?”525 Lie faced a number of tasks that were new to him, and he wanted to overcome his self-diagnosed difficulty of dealing with others. He wanted to learn how to be more approachable and to improve his communication skills with colleagues and students. He grew frustrated with Carl Neumann, who tried to entice away his students and his assistant (Famulus), who “outdid himself with his great lack of

519 520 521 522 523

[UBG] Cod. MS. F. Klein 9: 1010 (a letter from Hurwitz to Klein dated December 14, 1885). [UBG] Cod. MS. F. Klein 11: 88 (Max Noether’s letter to Klein on November 5, 1885). Ibid. 10: 718 (an undated letter from Lie to Klein). Ibid. 10: 694/1, 695 (undated letters from Lie to Klein, 1884). Ibid. 10: 721 (a letter from Lie to Klein, July 1886), 730 (a letter dated December 8, 1886), 730 (a letter dated March 3, 1887), and 733 (an undated letter sent to Klein in 1887). 524 Ibid. 10: 722, 723, 724, 725. 525 Ibid. 10: 730 (a letter from Lie to Klein dated December 8, 1886).

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consideration,” and who even announced that he would be offering lecture courses on geometry.526 Lie felt as though he was being poorly treated by his colleagues and by the Ministry of Culture, and he was irritated that he had been excluded from examination committees. Klein heard about all of these problems, and he was asked for advice. Furthermore, Lie feared that other mathematicians did not sufficiently appreciate his work and that someone else might achieve results in “his” research area. When Wilhelm Killing’s studies were published in Mathematische Annalen, Lie’s paranoia reached new heights, and this affected Klein as well (who, of course, was the editor of the journal).527 Their relationship became strained, even though Klein repeatedly tried to resolve their differences. In 1889, Sophus Lie fell so ill in Leipzig that the neurologist Paul Flechsig recommended that he recover in the private sanatorium of Dr. Ferdinand Wahrendorff near Hanover, which had been founded in 1862 as an asylum for people with psychiatric illnesses.528 There was much speculation about the specific causes of Lie’s illness. Klein kept abreast of Lie’s condition and regularly inquired about his status with Adolph Mayer.529 After Lie had begun to recover, Klein tried again to include him in various projects. However, he was met with suspicion, mistrust, and signs of paranoia. Later, Lie’s illness was diagnosed as pernicious anemia,530 associated with a mental illness, as Elling Holst – who also felt badly treated by Sophus Lie – informed Klein.531 Lie worked for several fruitful years in Leipzig, and he produced a number of prominent students. His relationship with Klein remained tense for a long time, even though Klein repeatedly lauded his academic achievements. In an undated letter of 1898, Lie wrote again to Klein and thanked him for his “Report on the Lobachevsky Prize.”532 In the same year, when Lie was on his way back to Norway, he visited Klein one more time in Göttingen and reconcile with him.533 Lie died shortly thereafter in Christiania, on February 18, 1899. On April 29, 1899, Klein gave a memorial address in Lie’s honor in Göttingen. In this speech, he referred to Lie as “the greatest talent in the area of geometry that the second half of this century has seen.”534

526 [UBG] Cod. MS. F. Klein 10: 727 (Lie to Klein, November 2, 1886), 735 (an undated letter, sent after Easter of 1887), 736 (an undated letter, sent in the fall of 1887). 527 Ibid. 10: 741 (an undated letter from Lie to Klein, sent in 1888). See also ROWE 1988. 528 Today, the Wahrendorff clinic exists as a psychiatric and psychotherapeutic hospital. 529 See TOBIES/ROWE 1990, pp. 178–86. 530 See CZICHOWSKI/FRITZSCHE 1993, pp. 191–93; and FRITZSCHE 1991. David Hilbert would later suffer from the same illness ([UBG] Cod. MS. D. Hilbert 749, a report on the status of Hilbert’s health from 1929 to 1936). 531 [UBG] Cod. MS. F. Klein 2G (Holst to Klein, May 12, 1899); see also JONASSEN 2004. 532 [UBG] Cod. MS. F. Klein 10: 769. – For this report, see Section 6.3.6. 533 See YOUNG 1928, p. xiii (Young’s account is based on a letter written by Anna Klein). 534 [UBG] Cod. MS. F. Klein 22 G (Klein’s address, April 29, 1899). See also ROWE 1988, p. 45.

6 THE START OF KLEIN’S PROFESSORSHIP IN GÖTTINGEN, 1886–1892 On April 1, 1886, Felix Klein began working as a full professor at the University of Göttingen in Prussia. Prussia was by far the largest German state with the most universities, which were then in Berlin, Bonn, Breslau, Göttingen, Greifswald, Halle, Kiel (Prussian since 1867), Königsberg, Marburg (Prussian since 1866), and Strasbourg (Prussian from 1871 to 1917). There was also a Catholic Academy in Münster;1 polytechnical schools (later named Technische Hochschulen) in Aachen, Berlin, and Hanover; and a mining academy in Clausthal. The Prussian Ministry of Culture in Berlin was responsible for hiring the faculty and other staff members at these institutions. The mathematics professorships were predominantly held by men who had studied in Berlin. This would gradually change over the course of Klein’s tenure in Göttingen. Klein’s first six years as a full professor in Göttingen can be regarded as a preliminary phase. During this period, he did not have to concern himself much with institutional matters. The Royal Mathematical-Physical Seminar already existed, belonging to the Philosophical Faculty of the university (see 2.8.1). Klein became one of its co-directors on April 28, 1886 (the other directors were Riecke, Schering, H.A. Schwarz, and W. Voigt). Klein moved ahead immediately to establish the student reading room that the administration had approved.2 The collection of models and the Seminar library were under the direction of Hermann Amandus Schwarz. There was no assistant yet at the mathematical institute. According to his own accounts, Klein concentrated on his mathematical work and on his research-oriented teaching. In addition, he was also developing further plans. By this point, Klein was responsible for a five-member family, which would continue to grow and for which he wanted to provide spacious accommodations (Section 6.1). He hoped to collaborate on good terms with his colleagues, but some of them felt as though his presence was dictatorial (6.2). He continued to build upon his earlier fields of research, he cooperated with previous collaborators, and he found new collaborators for specific fields. The latter included a growing number of students from Germany and abroad, whose careers he would influence in decisive ways (6.3).

1 2

This Catholic Academy had a philosophical faculty including mathematics, and it had the right to grant doctoral degrees. In 1902, it acquired the status of a university. Klein had newly requested this reading room in order to make it part of the Seminar’s facilities ([UAG] Math. Nat. 0012). See also FREWER-SAUVIGNY 1985. – In the fall of 1886, the astronomer Wilhelm Schur was added as another co-director of the Seminar.

© Springer Nature Switzerland AG 2021 R. Tobies, Felix Klein, Vita Mathematica 20, https://doi.org/10.1007/978-3-030-75785-4_6

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With a sense of responsibility for the university’s overall goals, Klein attempted (as before) to reform existing committees, establish new committees, and use them to achieve his aims. He revived his general ambition to connect mathematics with other fields, particularly technical disciplines. During these years, however, the success of these efforts was limited (6.4). Not until H.A. Schwarz had accepted a professorship in Berlin and Klein had declined an offer from the University of Munich in 1892 did the proper conditions arise for Klein to pursue his goals in Göttingen without having to face much resistance (6.5). 6.1 FAMILY CONSIDERATIONS In Klein’s view, the following were motivating factors for his move to Göttingen: “A house with a garden. Fewer administrative duties. Prussia.” He went on: “A concentrated academic existence on the basis of a reasonable family life.”3 He purchased an apartment at Weender Chaussee 6,4 and he invited his parents there in May of 1886, so that they could “see for themselves how much more comfortable my family’s situation is compared to Leipzig.”5 When Anna and Felix Klein announced the birth of their fourth child, Elisabeth (b. May 21, 1888), the family decided to build a house of their own: Wilhelm-Weber-Straße 3. They moved into it on May 22, 1889.6 It is located near the botanical garden, which Klein could easily cross to be at the university building where the mathematicians taught.

Figure 27: Felix Klein’s home in Göttingen, Wilhelm-Weber-Straße 3 (photographs courtesy of Dr. W. Mahler, May 31, 2014).

3 4 5 6

Quoted from JACOBS 1977 (“Vorläufiges über Leipzig”), pp. 4–5. On May 24, 1923, this street was renamed as Weender Landstraße; see TAMKE/DRIEVER 2012, p. 213. [UBG] Math. Arch. 77: 156 (a letter from Klein to Hurwitz dated May 15, 1886). See Klein’s remarks in JACOBS 1977 (“Personalia”), 22 L, fol. 3.

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In the new house, there was not only enough room for the family but also for domestic help, Anna’s youngest sister Sophie, and guests. Adolf Hurwitz often accepted Klein’s invitations to stay with him. David Hilbert (Wilhelm-Weber-Straße 29) and Carl Runge (Wilhelm-Weber-Straße 21) would later also choose to settle on this quiet and centrally located street. Finally, Klein’s house would serve as the starting point of the famous walks that the mathematicians in Göttingen would take on Thursday afternoons. Hermann Minkowski also took part in these outings when, from 1902 to 1909, he lived nearby at Planckstraße 15.7 Initiated by Klein, this tradition was still alive when Peter Debye came to Göttingen as a professor of physics in 1913. Debye reported that Klein was the first to leave his house and he was joined one after the other by Hilbert (with his dog Pussy), Carl Runge, and Debye himself. Carathéodory and Ludwig Prandtl waited at the corner, and they would be joined by Edmund Landau, who lived on Herzberger Straße. They would walk together to Rohns tavern on the Hainberg, where – in Debye’s words – “all the faculty business was decided, independent of all the other people in the faculty!” Klein had introduced this independent decision-making process on account of his experiences with other members of the Philosophical Faculty.8

Figure 28: Rohns Tavern on the Hainberg (a historical postcard).

7 8

On January 1, 1898, this street was named after the judge Gottlieb Planck, an uncle of the physicist Max Planck. [Debye] 1962.

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6.2 DEALING WITH COLLEAGUES, TEACHING, AND CURRICULUM PLANNING Here it will be shown how Klein attempted to get along with the colleagues who had disapproved of his appointment to Göttingen, how he sought to win over the Privatdozenten there, and that he developed new ideas for reorganizing the curriculum. 6.2.1 The Relationship Between Klein and Schwarz At first, Klein anticipated no problems dealing with Hermann Amandus Schwarz, who was six years Klein’s senior. Thus he wrote to Adolf Hurwitz: I am getting along with Schwarz quite well and better than expected. I have really gained a good deal from interacting with him, and I find that we are on the same page when it comes to practical matters. This is indeed someone who lives entirely for his job! From Leipzig, I am used to people who put everything else first and only then care about advancing mathematical knowledge and teaching, so that I gratefully appreciate anything that lies in this direction.9

Klein’s letter indicates his efforts to reach an understanding with H.A. Schwarz, even though he had not been fully aware of his unfriendly behavior. In March of 1886, Klein had sent Sophus Lie to Schwarz so that the two of them might form an amicable relationship. Outwardly, at least, Schwarz was warm and agreeable during this meeting. He welcomed Lie and had positive words to say about Klein: Overall, his opinion of you was correct. Like you, he clearly has the best intentions. He grumbled about Mittag-Leffler and Kronecker. It’s really an odd situation: he wants to join forces with us against Kronecker. Like [Georg] Cantor, he mentioned that Weierstrass had become so irritated with Kronecker that he refused to give any more lecture courses in Berlin.10

Otto Hölder was also present at this meeting. In a letter to his parents, Hölder mentioned that he was impressed with Lie, and he made critical comments about Schwarz’s “annoying self-importance.”11 In his letters to Weierstrass, Schwarz changed his tone about Klein: Recently, Prof. F. Klein told me that Dirichlet’s writing is boring! That is quite a beautiful new perspective that he has opened up on the matter. On top of this, there is his boundless self-satisfaction! Mr. Klein informed me that he has recently worked out a study – especially for me – regarding hyperelliptic σ-functions and their development according to moduli.12

9 10 11 12

[UBG] Math. Arch. 77: 156 (a letter from Klein to Hurwitz dated May 15, 1886). [UBG] Cod. MS. F. Klein 10: 718 (an undated letter from Lie to Klein, ca. March 1886). Quoted from HILDEBRANDT et al. 2014, p. 215 (Hölder to his parents, March 24, 1886). [BBAW] NL Schwarz 1254: 222–23 (Schwarz to Weierstrass, April 3, 1886).

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For the sake of context, it should be mentioned that Klein, in his article “Ueber hyperelliptische Sigmafunctionen” [On Hyperelliptic Sigma Functions] (see Section 5.5.8), had cited Weierstrass’s theory of elliptic functions from H.A. Schwarz’s book Formeln und Lehrsätze zum Gebrauche der elliptischen Functionen [Formulae and Theorems on the Use of Elliptic Functions] (1885), because Weierstrass’s lecture courses were unpublished. On May 7, 1886, Schwarz wrote to his former teacher Karl Weierstrass that Felix Klein had become a member of the Royal Society of London and that Klein had invited him to republish his older articles from the Monatsberichte [Monthly Reports] of the Berlin Academy in Mathematische Annalen.13 Klein thus wanted Schwarz to become a contributor to his journal, and he wanted Schwarz’s work to reach a wider international audience. Legally, nothing stood in the way of reprinting Schwarz’s articles; after two years, publication rights reverted to the author. Schwarz, who had never published anything in the Annalen, seemed willing at first to accept Klein’s offer. Weierstrass suggested, however, that Schwarz should instead collect his articles from the Monatsberichte and his studies on minimal surfaces and print them in the form of a book.14 Consequently, Schwarz declined Klein’s offer to cooperate and never published in Mathematische Annalen. This reinforced the tensions that already existed between them (see also Section 6.2.2). At this point, it should be mentioned that professors in Germany did not have to be politically neutral like Klein, who never joined a party. Schwarz, for example, was outspokenly political; during the German parliamentary elections of 1887, he vehemently supported the National Liberal Party, which backed Bismarck. He wrote to Weierstrass about this. Moreover, Otto Hölder was aware that Schwarz went from town to town with a jar of glue in order to hang election posters.15 6.2.2 The Göttingen Privatdozenten Hölder and Schoenflies When Klein came to Göttingen, there were two Privatdozenten whom he invited to collaborate: Otto Hölder, whom he had already supported in Leipzig (see Section 5.4.1), and Arthur Schoenflies.16 After earning a doctoral degree in Berlin,17 Schoenflies completed his Habilitation in Göttingen in November of 1884 with a thesis evaluated by M.A. Stern. 13 [BBAW] NL Schwarz 1254: 225–26. 14 Ibid. 1175: 323 (Weierstrass to Schwarz, May 15, 1886). The result of this undertaking was Schwarz’s Gesammelte mathematische Abhandlungen, 2 vols. (Berlin: J. Springer, 1890). 15 Ibid. 1254: 249 (Schwarz to Weierstrass, March 15, 1887); HILDEBRANDT et al. 2014, p. 238. 16 Regarding Schoenflies’s biography, see KAEMMEL/SONNTAG 2006. 17 The title of Schoenflies’s doctoral thesis was “Synthetisch-geometrische Untersuchungen über Flächen zweiten Grades und eine aus ihnen abgeleitete Regelfläche” [Synthetic-Geometric Investigations of Surfaces of the Second Degree, and a Ruled Surface Derived from Them] (1887). See BIERMANN 1988, p. 354.

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The probationary lecture that Schoenflies gave as part of his Habilitation procedure – “Darstellung der Lehre der Zusammensetzung der Kräfte und der Bewegung eines festen Körpers im Anschlusse an die bezüglichen Untersuchungen von Plücker und Ball” [A Representation of the Theory of the Composition of Forces and the Motion of a Rigid Body in Light of the Relevant Investigations by Plücker and [Robert Stawell] Ball]18 – contained several points of contact with Felix Klein’s work. Erhard Scholz has also attributed Schoenflies’s interest in studying proper (eigentliche) discontinuous motion groups to Klein.19 H.A. Schwarz, however, wrote to Weierstrass that he had been the first person to inspire and encourage Schoenflies’s work in this research area.20 This was yet another matter of contention between Klein and Schwarz. On the first Sunday after the beginning of the semester, Klein invited both of these Göttingen Privatdozenten to his home, where they were joined by Georg Pick (a Privatdozent in Prague) and Eduard Study (a Privatdozent in Leipzig). As in Leipzig, Klein wanted his mathematical colloquium in Göttingen to be a point of attraction for students. He therefore invited Hölder and Schoenflies to participate, but he was surprised by their resistance to the idea. While a Privatdozent himself, Klein had led his first research seminar together with Clebsch, and as a professor he had grown accustomed to Privatdozenten willingly participating in his colloquium. Hölder, however, feared being caught in between Schwarz and Klein, and he also persuaded Schoenflies to decline Klein’s offer. During a trip to Berlin in the fall of 1885, Hölder had met with his doctoral supervisor Paul du Bois-Reymond, who told him that the relationship between Schwarz and Klein might be strained: Du Bois-Reymond wanted me to know that the disagreements between Schwarz and Klein that I anticipated have already begun, at least when they are not in each other’s presence. If he is to be believed, Klein has already behaved in a characteristically immodest manner in his negotiations concerning the Seminar. For this reason, Schwarz supposedly wants to leave Göttingen. Whenever I meet him, he puts his arm around me and explains in a sentimental tone that he still wants to see me quite often this winter; you don’t know what the future will hold.21

Schwarz was clearly anxious about Klein’s influence. Hölder remained cautious, even though he was invited to socialize with both of the professors and their wives together. On May 9, 1886, he reported to his parents that he had been invited to the Schwarz family’s home, and that Anna and Felix Klein were also there: “Klein in a tailcoat and his wife in a very elegant red gown.” Hölder nevertheless wrote that he wanted to avoid having a dependent relationship with either of them:

18 [UAG] Phil. Fak. 170a, No. 39a. 19 See SCHOLZ 1989, pp. 121, 290. 20 [BBAW] NL Schwarz 1254: 291–93 (Schwarz to Weierstrass, November 20, 1888): “Two or three years ago, I prompted Dr. Schoenflies to conduct a geometric study of discontinuous motion groups.” 21 Quoted from HILDEBRANDT et al. 2014 (Hölder to his parents, November 17, 1885).

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Such a relationship could be disastrous for me if Schwarz and Klein have a falling out, which is possible any day. Klein has to manage absolutely everything and everybody, and that can’t be easy to tolerate. In fact, I find it somewhat harsh when he says that participating in his colloquium (which, in the end, is just a seminar for students) is the only opportunity we might have to work closely with him. At least he bid us farewell in an outwardly very friendly manner.22

In 1886, Klein found another way to cooperate with Hölder, as the latter informed his parents. Klein was not resentful, he wrote, and “my relationship with Klein is now very good.” In another letter: “I’m now interacting more with Klein. Tomorrow I’ve been invited to have another scientific discussion with him. He wants to publish the work [in Mathematische Annalen] that I brought him as soon as possible, which is very pleasing to me.” Around a month later, he commented: “Poor Klein […] is now suffering from rather severe asthma attacks.”23 In the fall of 1892, keeping in mind that other Privatdozenten might have similar feelings about participating in his colloquium, Klein created a new forum for communication: the Göttingen Mathematical Society (see Section 7.2). By 1894/95, however, Klein again found Privatdozenten (Burkhardt, Sommerfeld, Ritter) and professor colleagues (Hilbert since April of 1895, later Minkowski and others) to cooperate in his seminars.24 Arthur Schoenflies followed Klein’s advice more closely and had more success as a teacher than Hölder.25 When Klein was preparing to teach mechanics for the winter semester of 1886/87, Schoenflies coordinated with him and offered “An Introduction to the Geometric Aspects of Mechanics.”26 In subsequent semesters, Schoenflies took over the geometry lectures for beginning students. When a general lecture course on mathematics was newly introduced for natural scientists – “A General Introduction to Higher Mathematics” – Klein turned to Schoenflies to teach it.27 Nevertheless, when an associate professorship became vacant on account of Alfred Enneper’s death, it was offered to Hölder and not to Schoenflies. In 1899, when Hölder left for a professorship in Tübingen, Klein was still unable to convince the Philosophical Faculty to offer Hölder’s position to Schoenflies. Klein tried again in 1891, explaining to Friedrich Althoff at the Prussian Ministry of Culture: “I have no doubt that the reasons for this dismissive behavior has much more to do with me than it does with Dr. Schoenflies. Because Dr. Schoenflies had adopted my approaches in his teaching and in his research, the faculty fears that hiring him would strengthen my position.”28 In the same letter, Klein argued 22 23 24 25

Ibid., p. 217 (a letter from Hölder to his parents dated May 9, 1886). Ibid., pp. 219–26 (Hölder to his parents, June 15, June 27, and July 24, 1886). [Protocols] vol. 12, pp. 371–76; and Section 8.2.4. Hölder himself wrote to his parents about how few students he had compared to Schoenflies. See ibid., p. 254 (a letter from Hölder to his parents dated November 13, 1887). 26 See Göttinger Nachrichten (1886), p. 471. 27 See ibid. (1887), p. 355. 28 [UBG] Cod. MS. F. Klein 1C: 2, fol. 32 (Klein’s draft letter to Althoff, March 28, 1891).

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6 The Start of Klein’s Professorship in Göttingen, 1886–1892 […] that in Göttingen we will not be able to maintain the previous high level of our mathematical instruction if, as regards the needs of beginners and students of the natural sciences (etc.), we cannot count on the support of our younger teachers. Indeed, simply leafing through the course catalogues from the last few years reveals that, in this respect, we have continuously relied on two mathematical assistants to do the most important tasks (first Hölder and Schoenflies, then Schoenflies and Burkhardt).

Furthermore, Klein emphasized why Schoenflies (at 37 years old) should absolutely receive the associate professorship: “Besides, he is Jewish and thus it will be difficult for him to advance further in any case.” In vain, Klein attempted to influence Althoff with the following sentence: “If it proves to be impossible to promote Dr. Schoenflies, then instruction for beginners, as I envision it and as it has recently been carried out with the help of Privatdozenten, will be disrupted for many years to come.”29 Klein would first have to strengthen his position in Göttingen before he would be able to realize this and other plans (see Section 6.5.2). 6.2.3 Klein’s Teaching in Context Regarding his early years as a professor in Göttingen, Klein explained how his teaching program related to that of Hermann Amandus Schwarz: Because Schwarz insisted on teaching the main component of the curriculum, I was able to revisit old ideas from my student years in Bonn and return to teaching physics, which I did by offering general lectures on mechanics, potential theory, etc. In addition, I tried to complete all the purely mathematical investigations I had begun in my previous special lectures. However, as far as this seemed possible, I assigned individual problems to the students who were best suited to solve them.30

Klein began the “special lectures” mentioned here in the summer semester of 1886 (April 28th to August 15th) with a four-hour lecture course on algebra: “On the Solution of Algebraic Equations,” which had twenty-three students. He also devoted his research seminar (Wednesdays, 11–1 o’clock) to this area of research: “On Regular Solids and Triangular Functions.”31 In addition, Klein offered a twohour special lecture on elliptic modular functions (9 students). With his current monograph project in mind, he saw these three courses as “consistent preparation for a large lecture course for the summer of 1887 on single-valued functions with linear transformations into themselves,” as he informed Hurwitz.32 As of the winter semester of 1886/87, Klein began to offer lecture courses on mechanics and mathematical physics. In doing so, he fulfilled a wish of the physicists at the university, which Eduard Riecke had conveyed to him as early as No-

29 30 31 32

Ibid., fol. 32v. – Regarding Klein’s opinion of Schoenflies, see also Appendix 6.2. KLEIN 1922 (GMA II), pp. 259–60. For an analysis of Klein’s algebra course and seminar during this semester, see HELLER 2015. [UBG] Math. Arch. 77: 156 (a letter from Klein to Hurwitz dated May 15, 1886).

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vember 9, 1884.33 At the same time, this also corresponded to Klein’s own “mechanical program,” which he had first mentioned explicitly in the fall of 1881 (see Section 5.5). At the beginning of his lecture course “An Introduction to Analytical Mechanics” from 1886/87,34 he made the following notes about his goals: “Expand my horizons and my own knowledge.”35 Klein had already taught analytic mechanics while working in Erlangen and Munich (in Leipzig, this was Adolph Mayer’s domain). Now he saw the opportunity to combine mechanical topics with recent work on geometry and function theory. On March 8, 1887, he wrote about this to Sophus Lie in Leipzig: Perhaps Mayer has told you that I worked diligently on mechanics this winter. For the time being, I have not extended myself to general theories; rather, I have intuitively treated special examples, relying on studies from France and (in part) from England. I can’t even tell you how much these matters interest me and how much I regret that I have had to wait until now to come to them, now that my youth has passed and I struggle somewhat to understand and formulate new ideas. If I am successful, I hope to continue to work on similar subjects, so that in 2 or 3 years I will perhaps be at home with those as well. None of this should conflict with my earlier work on geometry and function theory; in fact, all of these subjects can be integrated organically.36

The number of enrolled students was low in all of Prussia at the time. Hurwitz reported from Königsberg that he, Lindemann, and Hilbert had just two to four students in their classes. Klein replied on November 9, 1886: We are now deeply bogged down in the semester’s work. Of course, my enrollment numbers are not as good as they could be: I have 17 students in mechanics and only 5 in my seminar and special lecture course (on higher equations). Not one of them is from Göttingen! Luckily, two Frenchmen have now shown up, students of Picard who seem good and are only hindered by their poor German; this will provide us with some life. By the way, we will gratefully accept every older mathematics student who is sent to us; otherwise, it could really happen that I would one day have to cancel my advanced lecture courses altogether!37

The French students whom Klein mentions here, and who came to him with recommendations by Darboux, were Paul Painlevé and Nicolas Cor. They participated in Klein’s 1886/87 seminar on group theory and algebraic equations, without giving a presentation.38 Hölder was repeatedly astounded by all the “young 33 34 35 36 37

[UBG] Cod. MS. F. Klein 11: 507. For an edition of these lectures, see KLEIN 1991. Quoted from JACOBS 1977 (“Personalia”), 22 L, p. 1. [Oslo] A letter from Klein to Lie dated March 8, 1887. [UBG] Math. Arch. 77: 166 (Klein to Hurwitz, November 9, 1886). In 1887, Hurwitz sent his doctoral student Felix Klitzkowski to work with Klein (K.’s Königsberg thesis was titled “Ueber die Integration der mten Wurzel aus einer rationalen Function” [On the Integration of the mth Root from a Rational Function], 1887), and in 1893 Hurwitz sent the Swiss Charles Jaccottet to Göttingen to complete his doctoral research under Klein. 38 [Protocols] vol. 8, p. 264. Later, Painlevé and his student Auguste H.L. Boulanger applied Klein’s methods, and Painlevé had fond memories of his time studying under Klein [UBG] Cod. MS. F. Klein 11: 159 (Painlevé to Klein, February 19, 1896). Painlevé would also write an article on ordinary differential equations for the ENCYKLOPÄDIE, vol. II (1900).

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mathematicians from this and the other side of the ocean” who came to sit at Klein’s feet, whose work Klein encouraged, and whom Klein invited to his home: The company at Klein’s home was very animated; the occasion was to celebrate his wife’s birthday. Schwarz was the one man to attend wearing a tailcoat. […] Then there was the crème de la crème of Klein’s seminar, which at the moment is very American. This time around, all of these exotic plants also paid visits to us lowly Privatdozenten, even though we made no efforts to approach them.39

Otto Hölder obviously found it difficult to interact with foreign mathematicians. Klein’s renown was especially widespread in the United States. Harry W. Tyler, who studied under Klein in the winter semester of 1887/88 and the summer semester of 1888 (along with Haskell, Osgood, Thompson, H.S. White, and others), reported to his parents after attending his first lecture by Klein: Not till Thursday did I hear and see the great Klein (so to speak), whose fame as the greatest mathematical teacher in Germany (consequently in the world) has attracted me to Göttingen. He is a tall slender man of about 40, his hair is light brown, his eyes blue, keen and alert; the strength of his face lies chiefly in his large nose and high forehead. He speaks rather quickly and with a somewhat high voice, but clearly enough, and methodically, enunciating frequently statements to be taken down verbatim. He lays much stress upon the notes taken, and has one student write up the lectures, which after his own revision are put in the reading room for general reference. His subject was Potential – a subject of mathematical physics, in which I have no interest. In spite of my first disinclination, I am gradually concluding to take this course – 4 lectures a week – partly for the sake of the Mathematics involved, mainly to hear the man.40

Klein prepared Tyler to collaborate with Paul Gordan by giving him a suitable seminar topic,41 and Tyler completed his doctorate in 1889 under Gordan at the University of Erlangen. Osgood followed the same procedure: after studying with Klein for four semesters, he went to Erlangen and relied on Max Noether’s assistance to complete his thesis there in 1890. At the same time, Klein supervised additional doctoral theses in Göttingen, seven from 1887 to 1893, five of which were by Americans: Haskell, Bôcher, White, Thompson, and E.B. Van Vleck. During these years, further students came to study under Klein from the United States (Fabian Franklin, Frederick S. Woods, James Harrington Boyd),42 from Great Britain (Arthur Berry),43 France (Henri Padé), Italy (Ernesto Pascal), Russia (B. Młodziejewski, a mathematician of Polish origin), the Polishmen

39 HILDEBRANDT et al. 2014, pp. 242–43 (Hölder to his parents, May 4 and May 23, 1887). 40 Quoted from BATTERSON 2009, p. 921. 41 Tyler’s presentation was titled “Referat über das erste Kapitel von Clebsch und Gordan’s Abelsche Functionen unter Bezugnahme auf unser Colleg.” [Protocols] vol. 9, pp. 65–81. 42 The successes and failures of these students are described in detail in PARSHALL/ROWE 1994. 43 On April 9, 1887, Cayley had written to Klein: “I would be very much obliged for anything you may be able to do for him [Berry].”[UBG] Cod. MS. F. Klein 8: 381. Berry attended Klein’s course in the summer semester of 1887 (hyperelliptic functions), and in July of 1887 he gave a presentation in Klein’s colloquium titled “Differentialinvarianten, insbesondere Reciprocanten” [Differential Invariants, Particularly Reciprocants]. [Protocols] vol. 8, p. 272.

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Stanisław Kępiński, K. Żorawski, and others,44 from Greece (A. Karagiannides45), and elsewhere. Franklin already held a professorship, and some of the others had already completed their doctoral degrees – Młodziejewski in Moscow (1889), Żorawski under Sophus Lie in Leipzig (1891), Kępiński in Krakow (1891). Klein felt as though he had achieved a sought-after goal, about which he wrote to Althoff at the Ministry of Culture in October of 1891: I may add, as you have perhaps already heard from other sources, that the quality of our students in Göttingen has increased considerably (though their numbers have not), and that in recent years we have finally achieved for the first time our goal of having a full and prestigious international student body, as I had in mind in advance before moving to Göttingen.46

Klein thus felt sufficiently confident to coordinate the teaching program with his colleagues: the full professors of mathematics Ernst Schering (see Section 2.8.1) and H.A. Schwarz, the Privatdozenten Otto Hölder and Arthur Schoenflies, the theoretical physicist Woldemar Voigt, and the astronomer Wilhelm Schur.47 Klein attempted to accomplish four things in particular: First, as in Leipzig, Klein strove to coordinate the times and themes of courses to best serve the interest of students. In June of 1887, he invited his colleagues to participate in a novel “conference for the purpose of discussing lecture courses” in order to organize the course offerings for the coming winter semester. That there had never been a coordinated planning effort of this sort is clear from an earlier letter by the Privatdozent Otto Hölder, who made the following complaints to his parents before the beginning of the summer semester in 1886: This time, it took a great deal of effort and second thoughts to schedule my course for next semester. It is very unpleasant that we don’t discuss the matter together in advance. Now, most of us don’t learn when and what the others are teaching until we see the final proofs of the course listings. My previous course time has been taken away by Klein.48

The older full professors Schwarz and Schering found it difficult to agree with one another. In 1887/88, they taught the same subject.49 Klein’s attempt to coordinate the teaching schedule was only temporarily successful. In 1891, he reported to Althoff in despair: 44 On further Polish students in Göttingen, see CIESIELSKA et al. 2019. Danuta Ciesielska, Lech Maligranda, and Joanna Zwierzyńska are currently working on a project titled “Studies and Scientific Research of Polish Mathematicians, Physicists, and Astronomers at the University of Göttingen.” I would like to thank Danuta Ciesielska for informing me that, as she discovered in an archive, Felix Klein was awarded an honorary doctorate in 1900 from the Jagiellonian University in Krakow (Fig. 42). K. Zorawski had been a professor there since 1898. 45 Karagiannides refers to his “esteemed teacher Klein” in his little booklet on non-Euclidean geometry (Die Nichteuklidische Geometrie vom Alterthum bis zur Gegenwart. Berlin: Mayer & Müller, 1893, p. 34). – The author thanks M. Mattmüller for this reference. 46 [StA Berlin] Rep. 92 Althoff A I No. 84, pp. 82–83. 47 See https://gdz.sub.uni-goettingen.de/id/PPN6654655340_1886_SS. 48 Quoted from HILDEBRANDT et al. 2014, p. 209 (a letter dated February 14, 1886). 49 See ibid., p. 247. Both Schwarz and Schering planned to teach “An Introduction to the Theory of Analytic Functions.” See Göttinger Nachrichten (1887), p. 355.

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6 The Start of Klein’s Professorship in Göttingen, 1886–1892 Ever since I came here, as you know, I have worked with the idea of coodinating the efforts of all available colleagues. As I have already indicated to you, however, this system has become more and more difficult to implement. Things took a turn for the worse in the summer of 1889 precisely because I was honored in one way or another by the Royal Ministry.50 […] During the last winter semester, I followed the general Göttingen tradition by teaching and working for myself alone, but I wonder whether this is really the right way to conduct oneself here.51

Second, following the model established in Leipzig, Klein successfully reduced the number of courses required of those studying mathematics as a minor field. To this end, he established introductory lecture courses for natural scientists (taught by Schoenflies). He thus wanted to remedy the “dilemma faced by mathematical Dozenten: fragmentation,”52 and he wanted to be able to concentrate on his own research-oriented teaching. Third, as in Leipzig, Klein succeeded in introducing a new course on descriptive geometry. H.A. Schwarz was willing to teach this subject, as Klein noted in 1888: “Descriptive geometry, offered by Schwarz with the assistance of Hölder and Schoenflies.”53 Even though the title chosen for the course was not “Descriptive Geometry,” Schwarz taught “On Curved Surfaces and Curves of Double Curvature” during the summer semester of 1888/89. In December of 1888, Klein and Schwarz successfully applied for 3,000 Mark in order to expand the model collection and to offer exercises in “Constructive Geometry.”54 In the summer of 1890, Schwarz directed “Exercises in Geometric Construction”; Hölder offered “On the Possibility of Ruler-and-Compass Construction,” and Schoenflies taught “On the Regular Division of Space and Its Applications, Particularly to Crystallography” (see also Section 6.3.7.2).55 Fourth, together with Eduard Riecke, Klein drafted a document titled “Advice and Clarifications for Students of Mathematics and Physics,” which was distributed to incoming students upon their matriculation. In a report to Althoff dated June 10, 1890, Klein sought support from the Ministry of Culture for this plan. A few years would pass, however, before Althoff wrote back to him about it: “Your curriculum for teaching candidates in mathematics and physics […] has been favorably received here, and we are considering whether the same plan ought to be recommended as a model for other universities.”56 Within the context of his research-oriented teaching, Klein inspired talented students to pursue their own ideas in various directions.

50 In 1889, Klein received the Order of the Red Eagle, 4th Class (Roter Adler-Orden vierter Klasse) because he had been pressured to decline a visiting professorship in the USA (see Section 6.3.7.1). Klein felt that the envy of some colleagues made his life more difficult. 51 [UBG] Cod. MS. F. Klein 1C: 2, fols. 31–32 (Klein to Althoff, March 28, 1891). 52 Klein’s comment is quoted from JACOBS 1977 (“Personalia”), 22 L, p. 1. 53 Ibid., p. 2. 54 [UBG] Cod. MS. F. Klein 2E: fol. 22. 55 Göttinger Nachrichten (1888/89), p. 7; (1889), p. 75; and (1890), p. 48. 56 [UBG] Cod. MS. Klein II, A, p. 3 (a letter from Althoff to Klein dated January 15, 1894).

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6.3 INDEPENDENT AND COLLABORATIVE RESEARCH As before, Klein’s research-oriented teaching was an important springboard for creative ideas, which he increasingly passed along to younger collaborators to work on in greater depth. The following sections will focus on the trends that are reflected in Klein’s publications at the time and in the work of his research partners and students. 6.3.1 The Theory of Finite Groups of Linear Substitutions: The Theory of Solving Equations of Higher Degree As a professor in Göttingen, Klein took this area of research, which he had been working on since his time in Erlangen, in two new directions: First, in October of 1886, he submitted his article “Zur Theorie der allgemeinen Gleichungen sechsten und siebenten Grades” [On the Theory of General Equations of the Sixth and Seventh Degree] to Mathematische Annalen.57 This work had originated as a result of his first algebra seminar in Göttingen. Here, Klein used his theory of equations of the fifth degree, as he had presented it in his book on the icosahedron, to extend the problem to (general) equations of higher degree. In the footnotes to this article, Klein referred to the work of his doctoral students Willibald Reichardt and Frank Nelson Cole (see Section 5.4.2.2). When Klein edited this article for his collected works, he also made sure to include references to studies by Heinrich Maschke, who, together with Oskar Bolza, had participated in Klein’s algebra seminars beginning in the fall of 1886.58 Klein submitted Maschke’s and Bolza’s results to the Royal Society of Sciences in Göttingen. Especially noteworthy was Maschke’s work, which Klein presented at the Society’s session on July 2, 1887, under the title “Über das Formensystem einer gewissen endlichen Gruppe quaternärer linearer Substitutionen” [On the System of Forms of a Certain Finite Group of Quaternary Linear Substitutions]. In order to make the results more apparent and create a reference to Berlin mathematics, Klein induced Maschke to publish the article with a slightly different title: “Ueber die quaternäre, endliche, lineare Substitutionsgruppe der Borchardt’schen Moduln” [On the Quaternary, Finite, Linear Substitution Group of Borchardt’s Moduli].59 In a longer version of this article, which was published in Mathematische Annalen, Maschke explained: 57 Math. Ann. 28 (1886), pp. 499–532; reprinted in KLEIN 1922 (GMA II), pp. 439–72. – HELLER 2020 gives a detailed analysis of this work by Felix Klein. 58 See HELLER 2015. For two semesters, Klein held weekly private meetings with Bolza and Maschke in his apartment. Both Bolza and Maschke went on to have careers in the United States. Bolza’s doctoral research was not highly regarded in Berlin, so Klein accepted his dissertation in Göttingen (the degree was awarded on June 28, 1886). See BOLZA 1936, pp. 15– 20; and PARSHALL/ROWE 1994, pp. 197–202. 59 Göttinger Nachrichten (1887), pp. 421–24.

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6 The Start of Klein’s Professorship in Göttingen, 1886–1892 The group named in the title was initially derived by Mr. Klein completely from line geometry, without any connection to hyperelliptic functions. Later, Mr. Klein showed that the socalled Borchardt moduli of the hyperelliptic functions of genus p = 2 admit the same group of linear substitutions, whereby the group, which so far only had the advantage of being one of the few groups of linear substitutions of finite ordinal number – and indeed, apart from the simplest cases, the first which was known in the quaternary domain – gained much more interest. This interest will be of prime importance for many mathematicians if in the following I can establish, at Professor Klein’s instigation, the full system of invariant forms for this group.60

Klein’s final opinion on the matter was that “the transcendental problem here can be solved without recourse to Borchardt’s moduli and directly with hyperelliptic functions”61 (see also Section 6.3.2). Second, closely related to this was the investigation of the relationship between the problem of the trisection of hyperelliptic functions and that of the 27 lines of the cubic surface. It was known from the work of Camille Jordan (1869) that both problems have isomorphic groups. Klein outlined how the one problem can be reduced to the other in a lecture that he delivered on April 13, 1887 to the Société Mathématique de France in Paris. At Jordan’s request, Klein elaborated his lecture for publication: “Sur la résolution, par les fonctions hyperelliptiques, de l’équation du vingt-septième degré, de laquelle dépend la détermination des vingt-sept droites d’une surface cubique.”62 Klein showed that a group-theoretical treatment of the cubic surface with 27 straight lines is isomorphic with the trisection of hyperelliptic functions of genus 2.63 His results had arisen from the cooperative work in his algebra seminar of 1886/87. Klein referred explicitly to the seminar participants Witting and Maschke. Maschke had given three presentations with the title “Ueber die Gruppe derjenigen Gleichung, von welcher die 27 Geraden der Fläche dritter Ordnung abhängen” [On the Group of that Equation on which the 27 Lines of the Surface of the Third Order Depend], and he later published on the topic as well.64 Burkhardt expanded upon Klein’s outline and discovered in particular that, “in the case of an equation of the 27th degree of the group under consideration, one can specify explicit linear combinations of roots, which immediately admit

60 Math. Ann. 30 (1887), pp. 496–515, at p. 496 (emphasis original). 61 KLEIN 1922 (GMA II), p. 440. See also Anders Wiman, “Endliche Gruppen linearer Substitutionen,” in ENCYKLOPÄDIE, vo. I.1 (1899). 62 Felix Klein, “Extrait d’une lettre adressée à M. C. Jordan,” Journal de mathématiques pures et appliquées 4 (1888), pp. 169–76; reprinted in KLEIN 1922 (GMA II), pp. 473–79. – See also [Paris-ÉP] 114 (Klein to C. Jordan, August 17, 1887) and BRECHENMACHER 2016c. 63 On the term genus, which Klein borrowed from Riemann and Clebsch for the purpose of classifying surfaces, see Section 3.1.3.1; also DEHN/HEEGAARD 1907, p. 200. 64 [Protocols] vol. 8, pp. 88–97, 115–19 (Nov. 3, and 17, 1886; Febr. 26, 1887); and Heinrich Maschke, “Aufstellung des vollen Formensystems einer quaternären Gruppe von 51840 linearen Substitutionen,” Math. Ann. 33 (1889), pp. 317–44.

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the same substitutions as the coordinates aik of a linear complex.”65 This problem was taken up with new methods by the American mathematician Arthur B. Coble, who was influenced by the mathematician Eduard Study.66 6.3.2 Hyperelliptic and Abelian Functions Based on his earlier work (see Section 5.5.8), Klein taught a lecture course on general hyperelliptic functions from the Easter of 1887 to the Easter of 1888. He published results in the Göttinger Nachrichten and in Mathematische Annalen: “Über hyperelliptische Sigmafunctionen (Zweite Abhandlung)” [On Hyperelliptic Sigma Functions (Part Two)].67 Klein especially answered the question: “How do I express by a single algebraic condition that a curve p = 3 becomes hyperelliptic?” He provided an overview of his lectures and demonstrated how recent developments can be extended to hyperelliptic functions of an arbitrary genus. Hurwitz recognized Klein’s fundamental idea: The remarkable circumstance that the hyperelliptic case p = 3 can be characterized by a single algebraic condition is new evidence for the correctness of your approach in comparison with Weierstrass’s. If I understand the matter correctly, the essential issue – in the algebraic formulation of the question – is the transition to line coordinates [Linienkoordinaten].68

Klein later explained that vague conclusions by analogy had led him to these results.69 He inspired Heinrich Burkhardt to work out his ideas in greater detail: The necessary complement that my presentation requires in its details is supplied in large part by Mr. Burkhardt’s article published in the following, which I repeatedly have occasion to cite […]. I should not refrain from mentioning that my scientific interaction with Mr. Burkhardt was also conducive in many respects to the ideas that I will develop below.70

Recommended by his teachers Dyck and Voss in Munich, Burkhardt had come to Klein in 1887 after completing his doctoral degree with a thesis titled “Beziehungen zwischen der Invariantentheorie und der Theorie algebraischer Integrale und ihrer Umkehrungen” [Relationships between Invariant Theory and the Theory of Algebraic Integrals and Their Inverses].71 Klein used his seminar of 1887/88 – attended by Burkhardt, Johannes Schröder, M.W. Haskell, and H.D. Thompson – to augment his lecture course material (on “Select Chapters of Hyperelliptic Func-

65 KLEIN 1922 (GMA II), p. 479. – Heinrich Burkhardt, “Untersuchungen aus dem Gebiete der hyperelliptischen Modulfunctionen,” Math. Ann. 41 (1893), pp. 313–43. 66 See Arthur B. Coble, “Point Sets and Allied Cremona Groups (Part III),” Transactions of the American Mathematical Society 18 (1917), pp. 331–72. 67 See Felix Klein in Göttinger Nachrichten (1887), pp. 515–21; Math. Ann. 32 (1888), pp. 351– 80; and KLEIN 1923 (GMA III), pp. 357–87. 68 [UBG] Cod. MS. F. Klein 9: 1063 (a letter from Hurwitz to Klein dated February 21, 1888). 69 [Protocols] vol. 12, p. 10. 70 Klein in Math. Ann. 32 (1888), pp. 351–52 (reprinted in KLEIN 1923 (GMA III), pp. 357-58). 71 H. Liebmann, “Zur Erinnerung an H. Burkhardt,” Jahresbericht DMV 15 (1915), pp. 185–95.

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tions”) with presentations of his own.72 At the same time, Klein guided Schröder, Haskell, and Thompson to the completion of their doctoral degrees. In his Habilitation thesis on the topic of hyperelliptic modular functions,73 Burkhardt followed Klein’s example and combined the methods of Weierstrass and Riemann, as he expressed in his article “Beiträge zur Theorie der hyperelliptischen Sigmafunktionen” [Contributions to the Theory of Hyperelliptic Sigma Functions].74 Burkhardt’s work “Grundzüge einer allgemeinen Systematik der hyperelliptischen Functionen I. Ordnung” [Fundamentals of a General Systematic of Hyperelliptic Functions of the First Order]75 used Klein’s lectures from 1887/88 and provided, on the basis of Klein’s classification of elliptic functions (his “level theory”), a classification of hyperelliptic functions. Recommended by Klein, Burkhardt spent the winter of 1893/94 in Paris. He attended lectures by Picard (differential equations), Poincaré (partial differential equations), and others. Afterward, he published additional works on function theory, wrote a well-received book – Funktionentheoretische Vorlesungen [FunctionTheoretical Lectures] (Leipzig: B.G. Teubner, 1897) – and served as the primary editor for the second volume (analysis) of the ENCYKLOPÄDIE. Burkhardt received a titular professorship in Göttingen in 1894; he was hired as a full professor by the University of Zurich in 1897, and he accepted a full professorship in 1908 at the Technische Hochschule in Munich, where he worked alongside Walther Dyck. From Easter of 1888 to the fall of 1889, the theory of Abelian functions was at the heart of Klein’s teaching (see also Section 5.5.2.3). Klein published on the subject,76 and Hurwitz wrote enthusiastically that the Academy in Naples had offered a prize for summarizing Klein’s results on the theory of Abelian functions.77 Klein also inspired younger mathematicians in this area of research. Wilhelm Wirtinger, who had completed his doctoral studies in Vienna with Emil Weyr, belonged to this group; he described Klein’s close supervision as follows: In the summer semester of 1889, I went to Göttingen to study under Klein. The latter lectured on Abelian functions and differential equations of physics; he involved the advanced students in collaborative work as far as possible. At the time, the participants in the seminar were Burkhardt, Haskell, Osgood, White, and myself. Klein encouraged and stimulated me immediately. He understood extremely well how to clarify and work through budding ideas, and he devoted a great deal of his time and energy to having periodic discussions with each of us.78

72 73 74 75 76 77

[Protocols] vol. 9, p. 271. [UAG] Kur. 6238 (Burkhardt’s personnel file). Math. Ann. 32 (1888), pp. 381–442, at p. 381. Math. Ann. 35 (1889), pp. 189–296. See Math. Ann. 36 (1890), pp. 1–83; reprinted in KLEIN 1923 (GMA III), pp. 388–473. [UBG] Cod. MS. F. Klein 9: 1083 (Hurwitz to Klein, September 8, 1890). – On the occasion of the 100th birthday of N.H. Abel (in 1902), Klein received “le grade de Docteur-ès-mathématiques (doctor mathematicae)” from the University of Christiania [Oslo]. 78 [AdW Wien] Wirtinger 1939, p. 5. His first articles inspired by Klein were “Ueber das Analogon der Kummer’schen Fläche für p = 3,” Göttinger Nachrichten, 1889, pp. 474–89; and

 

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Under Klein’s guidance, Wirtinger made a seamless transition “into the epistemic world of mathematical modernity,” as Moritz Epple wrote in his 1999 Habilitation thesis on the history of knot theory. Epple discerned Klein’s “intellectual hegemony.”79 This intellectual circle also included the Italian mathematician Ernesto Pascal and Eduard Wiltheiss, who died young.80 Two decades later, Klein stated: “When I was a student, Abelian functions were, as an effect of Jacobi’s tradition, considered the uncontested summit of mathematics, and each of us was ambitious to make progress in this field. And now? The younger generation hardly knows Abelian functions.”81 Despite the open questions in this research area, mathematics advanced in new directions. For this reason, Klein believed that thorough survey articles about old topics might help to attract fresh interest and lead to further progress. This was one of his motivations for undertaking the ENCYKLOPÄDIE project. Klein’s collaborators at the time – Wirtinger, Burkhardt, and Osgood82 – later became professors and contributors to the ENCYKLOPÄDIE. His doctoral students at the time helped with other tasks: Haskell, as mentioned above, was the first to translate Klein’s Erlangen Program into English,83 and Henry S. White organized the Evanston Colloquium for Klein in 1893 (see Section 7.4.2). 6.3.3 The Theory of Elliptic Modular Functions (Monograph) Klein requested further rounds of talks with Adolf Hurwitz to advance the monograph that he had begun in Leipzig with Georg Pick (see Section 5.5.7).84 With Hurwitz, Klein considered whether they might be able to learn more about ideal theory from Dedekind’s work. Together, they contemplated formulating “promising problems” in this research area.85 Klein informed Hurwitz about the status of his cooperation with Georg Pick and about his new collaborator Robert Fricke, who would finally bring the project to a successful conclusion in 1892.86  

79 80

81 82

83 84 85 86

“Untersuchungen über Abel’sche Functionen vom Geschlechte 3,” Math. Ann. 40 (1892), pp. 261–312, see also Wirtinger’s articles in Monatshefte für Mathematik und Physik (Vienna). See EPPLE 1999, pp. 240–58. See KLEIN 1923 (GMA III), p. 322; and Wilhelm Wirtinger, “Eduard Wiltheiß,” Jahresbericht DMV 9 (1901) I, pp. 59–63. Regarding Ernesto Pascal, see his book Repertorio di matematiche superiori (Milan: U. Hoepli, 1898) and its German translation: Repertorium der höheren Mathematik, ed. A. Schepp (Leipzig: B.G. Teubner, 1900). KLEIN 1979 [1926], p. 294. Osgood, as mentioned above, submitted his dissertation – “Zur Theorie der zum algebraischen Gebilde ym = R(x) gehörigen Abelschen Functionen” [On the Theory of the Abelian Functions Pertaining to the Algebraic Entity ym = R(x)] – to Max Noether at the University of Erlangen (see Section 6.2.3, and also 7.4.2). See the Bulletin of the New York Mathematical Society 2 (1893), pp. 215–49. [UBG] Math. Arch. 77: 155 (a letter from Klein to Hurwitz dated April 3, 1886). Ibid., 77: 168, 190 (letters from Klein to Hurwitz, December 31, 1886; September 21, 1887). KLEIN/FRICKE 1890/92. In English translation: KLEIN/FRICKE 2017 [1890/92].

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From Klein’s correspondence with Robert Fricke, it is clear that they valued Pick’s preliminary work but also thought that the monograph should be reorganized on the basis of Klein’s lecture course from the summer of 1886. Persuaded by Fricke, Klein set aside his original intention “of underscoring that my ideas had come before those of the French mathematicians.” “Of course,” he admitted, “this concern is merely of a subordinate nature.”87 Klein gave further lecture courses with a view toward his plan, and he forwarded his latest findings to Fricke. In June of 1889, despite his regular consultations with Fricke about the text of the monograph, Klein felt as though it still left “a somewhat incomplete impression,” and this led him to a general remark: The more I think about the text, the more I would like to summarize it in the following way: Regarding the theory of elliptic functions, one should bring to bear modern mathematical disciplines that have been developed in the meantime […] This would explain earlier obscurities and immediately enable new and further developments. This general program should now be implemented in three directions: 1) Riemannian function theory in the discussion of modular functions, 2) invariant theory, and last but most importantly 3) group theory. However, other modern disciplines could have been drawn on, if only we commanded them as well as we do 1), 2), and 3). For instance: 4) Modern formation laws of transcendent functions (Weierstrass – Mittag-Leffler …); 5) Modern number theory (Dedekind – Kronecker). The limits within which we have treated our subject matter are therefore essentially subjective. We can’t change that, and we can also be content with it. When explaining the nature of our work, I would thus like to avoid the expression “the theory of modular functions.” On the one hand, it is too narrow, because we have not limited ourselves to modular functions; on the other hand, it is too pretentious, because we remain quite far from presenting a systematic theory.88

Klein and Fricke received the proofs for the monograph in November of 1889, and they were able to add references to the latest scholarly literature. On New Year’s Eve, Klein wrote: “Our book should remain relevant not only for a few years, but for decades to come!”89 In March of 1890, before the final printing of the book was set to begin, Fricke suggested that the material should be divided into two volumes. Klein and the publishing house agreed; for the second volume (1892), they were thus able to take into account works by Poincaré, H.A. Schwarz, Lazarus Fuchs, and others that had yet to be published when their first volume was released (1890).90 In September of 1890, Klein and Fricke sent copies of the first volume to mathematicians in Germany and abroad. Hermite’s son-in-law Picard wrote to Klein on June 24, 1892 that he and Hermite were eagerly awaiting the publication of their second volume on modular functions. When Picard acknowledged receipt of the second volume on October 17, 1892, he expressed his gratitude that his own

87 88 89 90

[UA Braunschweig] A letter from Klein to Fricke dated October 11, 1887. Ibid. (Klein to Fricke, June 11, 1889). Ibid. (Klein to Fricke, December 31, 1889). [UA Braunschweig] Letters from Klein to Fricke, March 18, 1890 to October 17, 1892.

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results had received such positive attention. Picard praised Klein’s “belle illustration électrique des principaux problèmes,” and he emphasized that he shared Klein’s opinion of Riemann’s significance. However, Picard also defended Hermite and others (H.A. Schwarz, L. Fuchs), referring to a few critical passages in the book and cautiously noting “que la polémique dans un cours nous étonne un peu.”91 Despite Klein’s occasionally polemical remarks, Hermite became increasingly interested in his work and ultimately arranged for many of his articles to be translated into French. In London, Alfred George Greenhill consulted the first volume of the Klein and Fricke monograph (1890) when preparing his famous book The Applications of Elliptic Functions.92 Greenhill, who became one of the world’s foremost experts on the applications of elliptic integrals to electromagnetic theory, cultivated a close relationship with Klein over the years (see also Section 8.3.4). The English translation of KLEIN/FRICKE 1890/92, which was published in Beijing in 2017 and announced by the American Mathematical Society, is evidence of the lasting interest in the subject. 6.3.4 The Theory of Automorphic Functions (Monograph) On March 1, 1891, even before volume 2 of their Vorlesungen über die Theorie der elliptischen Modulfunktionen (KLEIN/FRICKE 1892) was published, Klein sent an “automorphic plan” for their next book to Fricke. Klein wrote that he “would like to write a truly highly scientific book in which we can present the totality of our convictions about geometric function theory.” For it, Klein considered his “article in vol. 21 of Mathematische Annalen as a sort of program that I would only like to supplement with one point (no. V below).”93 In his letter to Fricke, Klein formulated the following five sections: I. The general Riemannian theory. Arbitrary closed surfaces, or also open surfaces with related boundaries (fundamental domains) … developed up to form theory, including η-functions. II. Automorphic groups. The construction of all useable fundamental domains based consistently on nonEuclidean geometry of the x + iy sphere. III. Related functions (automorphic, homomorphic) Their nature, their laws of formation. IV. The fundamental theorems. “Every Riemann surface can be presented by automorphic functions of any given type.”

91 [UBG] Cod. MS. F. Klein 11: 209, p. 7 (Picard to Klein, October 17, 1892). 92 See GREENHILL 1892, p. ix. – Klein and A.G. Greenhill (Royal Artillery Institution, Woolwich) first began to correspond on January 25, 1886 ([UBG] Cod. MS. F. Klein 9: 490–99). 93 For this and the following quotations, see [UA Braunschweig] Klein to Fricke, March 1, 1891. Klein is referring here to his article “Neue Beiträge zur Riemann’schen Functionentheorie” [New Contributions to Riemann’s Function Theory]; see KLEIN 1883.

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6 The Start of Klein’s Professorship in Göttingen, 1886–1892 V. Embedding the fundamental theorems within the general theory of linear differential equations of the second order.

Regarding his fifth point, Klein explained in the same letter: “These are the things I am currently pursuing by generally studying the representation of Riemann surfaces by related η-functions. I find that a Riemann surface can be mapped each time by a related η onto a polygon where certain defining elements can be prescribed ad libitum.94 The fundamental theorems are thus a special case of this, which is especially interesting because it leads to automorphic functions. (The matter is going to be quite good, but I still need more time to develop it clearly).” In this letter, Klein linked his previous and future lectures to the five planned book sections mentioned above: for I, his lectures on Abelian functions; for II, his lectures on non-Euclidean geometry.95 He regarded the lecture course that he was currently offering at the time (on linear differential equations) as the foundation for III, IV, and V, which would be limited to Riemann surfaces where p = 0. Klein wanted to devote future lectures to the case of p > 0. In the same letter, Klein estimated that he would need three semesters to prepare the monograph. As it turned out, this would not be enough time. In light of the critique that certain aspects of the monograph on elliptic modular functions received (KLEIN/FRICKE 1890/92), Klein took it upon himself to delve deeper into certain research areas for the new book on automorphic functions.96 The areas in question included number theory, as Dedekind had criticized the authors for treating it in an elementary manner, and the theory of linear differential equations: here, Klein wanted to address the objections of Ludwig Schlesinger (a student and son-in-law of Lazarus Fuchs), whose works he came to appreciate more and more, especially Schlesinger’s Handbuch der Theorie der linearen Differentialgleichungen [Handbook of the Theory of Linear Differential Equations] (B.G. Teubner, 1895–98). As early as April of 1892, Klein mentioned to Fricke that he wanted to do what he could to help before “handing over this research field to you and the younger generation.”97 He could not have guessed that his efforts to prepare the monograph on the theory of automorphic functions would continue for decades to come (FRICKE/KLEIN 1897 to 1912; see also 8.2.1). 6.3.5 The Theory of Lamé Functions and Potential Theory From 1888 to 1890, Klein continued to pursue these research areas, which he had begun to work on in Leipzig (see Section 5.5.1.1). He offered lecture courses on potential theory, differential equations of physics, and Lamé functions. On this 94 See Klein’s lecture courses on Riemann surfaces from 1891/92 and 1892: https://gdz.sub.unigoettingen.de/id/PPN592520080. 95 See https://gdz.sub.uni-goettingen.de/id/PN516972847. 96 [UA Braunschweig] Klein’s letters to Fricke dated March 25, April 4, and April 14, 1892. 97 Ibid. (a letter from Klein to Fricke dated April 23, 1892).

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basis, Friedrich Pockels and Klein’s American doctoral student Maxime Bôcher made important contributions to the theory of boundary value problems. Friedrich Pockels had completed his doctoral studies in 1888 under the supervision of the physicist Woldemar Voigt and had been influenced by Klein’s lecture courses mentioned above.98 He was a brother of Agnes Pockels, an autodidact whose pioneering research on surface tension provided the foundations of the discipline of surface science. Her work was recognized by Rayleigh and published in the journal Nature. Klein inspired F. Pockels to write the monograph Über die partiellen Differentialgleichungen Δu + k2u = 0 und deren Auftreten in der mathematischen Physik [On the Partial Differential Equations Δu + k2u = 0 and Their Occurrence in Mathematical Physics].99 In his introduction to this book, Pockels wrote that his own results arose from Klein’s lectures and from the insights that he had gained from Rayleigh’s Theory of Sound.100 Pockels went on to have a career as a professor of physics. Motivated by Klein, he also edited the physics volume of Julius Plücker’s collected works.101 When George A. Campbell (who later became an important industrial researcher in the United States) studied in Göttingen in 1893/94, he gave a seminar presentation – proposed by Klein – in which he offered a critical analysis of Pockels’s monograph from 1891.102 Bôcher’s book Über die Reihenentwickelungen der Potentialtheorie [On the Developments of the Potential Function Into Series] (B.G. Teubner, 1894) was based on his prize-winning doctoral thesis of the same name. The Philosophical Faculty had formulated the problem for its annual prize on June 4, 1890, and in 1891 Bôcher was awarded for successfully representing the orthogonal systems used in potential theory as special cases of the system of confocal cyclides. Bôcher had used Klein’s oscillation theorem in order to determine the constants in the differential equations that define the series.103 Bôcher became a professor at Harvard, and Klein recruited him to write the article on boundary value problems in ordinary differential equations for the ENCYKLOPÄDIE.104 Klein continued to conduct research in this field, and he followed international developments closely. Thus, for instance, he informed the Russian mathematician Andrey A. Markov about his own progress and that of his students: 98 Pockels’s thesis was published as “Ueber den Einfluss elastischer Deformationen, speciell einseitigen Druckes, auf das optische Verhalten krystallinischer Körper,” Annalen der Physik 273/5 (1889), pp. 144–72. 99 POCKELS 1891. 100 John William Strutt Rayleigh’s Theory of Sound, 2 vols. (London: Macmillan & Co., 1877– 78) was translated into German by Friedrich Neesen as Die Theorie des Schalles (Braunschweig: Vieweg, 1880). Arnold Sommerfeld also made use of Rayleigh’s book in his article “Randwertaufgaben in der Theorie der partiellen Differentialgleichungen,” in ENCYKLOPÄDIE, vol. II.A.7.c (1900), pp. 504–70. 101 PLÜCKER 1895/96. 102 [Protocols] vol. 11, p. 80. 103 On the validity of the oscillation theorem, see KLEIN 1922 (GMA II), pp. 592–600. 104 Maxime Bôcher, “Randwertaufgaben bei gewöhnlichen Differentialgleichungen,” in ENCYKLOPÄDIE, vol. II.1.1 (1900), pp. 437–63.

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6 The Start of Klein’s Professorship in Göttingen, 1886–1892 Let me tell you that I have meanwhile further pursued, from various perspectives, the ideas upon which I had based my theorem in vol. 37 [of Mathematische Annalen]. In particular, I created an application to Hermite’s form of Lamé’s equation. Here, the relationships are very similar to those in your work, to the extent that the linear differential equation of the second order has two particular solutions whose product is identical to a rational polynomial. For this latter polynomial, I have determined the number and position of the real roots.105

In this letter, Klein added: “Mr. Van Vleck has taken up the further development of the considerations that I myself could only sketch in my lecture course. In doing so, he has analyzed in particular the questions that you had taken into account in Annalen 27 (“Sur les racines de certaines équations” [1886], pp. 143–50, 177–82).” Besides, Klein emphasized that he himself “was able to penetrate, in a geometric manner, more deeply into the essence of linear equations of the second order (i.e., into the functions defined by these differential equations).”106 Supervised by Klein, the American Edward Burr Van Vleck completed his doctoral degree in 1893 with a dissertation titled “Zur Kettenbruchentwicklung Lamé’scher und ähnlicher Integrale” [On the Development of the Continued Fractions of Lamé’s and Similar Integrals]. Finally, Klein also encouraged the Swiss student Charles Jaccottet (who had been sent to him by Hurwitz)107 and his American student Mary F. Winston (see Section 7.5) to write dissertations on this topic. Emil Hilb later judged that Bôcher’s exposition may have provided a formal law for the formation of series, but a convergence proof was still lacking. Jaccottet made a first attempt to fill this gap with his thesis “Über die allgemeine Reihenentwicklung nach Laméschen Produkten” [On the General Development of Series According to Lamé’s Products] (1895). In 1906, Hilb himself would ultimately achieve, on the basis of Hilbert’s integral equation theory and with Klein’s inspiration, a rigid proof of Klein’s “continuity principle”.108 Looking ahead even further, I should note that Klein’s seminar during the winter semester of 1906/07, which he co-directed along with Hilbert, Minkowski, and Herglotz, also focused on the oscillation theorem.109 After presentations by Hilbert (“Oscillation Theorems”) and Klein (“On the Oscillation Theorem and Conformal Mapping”), Otto Toeplitz, Robert König,110 Ernst Hellinger (who provided yet another discussion of Bôcher’s work), and others analyzed the scholarly literature on the topic. The Canadian-American mathematician Roland G.D. Richardson, who studied in Göttingen in 1908/09, was also inspired to achieve new 105 [Archiv St. Petersburg] (8), pp. 12–13 (Klein to Markov, Febr. 1, 1892); F. Klein, “Ueber die Nullstellen der hypergeometrischen Reihe,” Math. Ann. 37 (1890), pp. 573–90; “Ueber den Hermite’schen Fall der Lamé’schen Differentialgleichung,” Math.Ann. 40 (1892), pp.125–29. 106 [Archiv St. Petersburg] (8), p. 13 (Klein to Markov, February 1, 1892). 107 [UBG] Cod. MS. F. Klein 9: 1123 (a letter from Hurwitz to Klein, November 29, 1893). 108 See Emil Hilb, “Die Reihenentwicklungen der Potentialtheorie,” Math. Ann. 63 (1906), pp. 38–53. Here, the continuity principle states that no eigenvalue is lost when the occurring parameters are continuously changed. 109 [Protocols] vol. 25. 110 The Hungarian mathematician Robert König spoke about “The Oscillation Theorem and the Calculus of Variation”; however, the results that he presented proved to be erroneous.

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results in this research area. As of 1910, he published a number of papers on Klein’s oscillation theorem, including an article in “Klein’s” Annalen: “Über die notwendigen und hinreichenden Bedingungen für das Bestehen eines Kleinschen Oszillationstheorems” [On the Necessary and Sufficient Conditions for the Existence of a Kleinian Oscillation Theorem]. Klein’s oscillation theorem for periodic boundary conditions remained an important area of research.111 6.3.6 Refreshing His Work on Geometry As early as 1884, Sophus Lie had signaled to Klein that the time had now come in which his Erlangen Program would be better understood (see Section 3.1.1). It was not until this 1872 Program, which was an attempt to systematize various approaches to geometry, had captured the interest of Italian and French mathematicians that Klein felt now (twenty years later) that he ought to reprint it in “his” journal Mathematische Annalen. But he had other plans as well. In order to combine his early geometric research and that of Sophus Lie with more recent results, Klein held two cycles of lecture courses, one on non-Euclidean geometry (1889/90, 1890) and one on higher geometry (1892/93, 1893). At the same time, he also intended to reprint Lie’s early studies along with a report on their collaborative work from the years 1870 to 1872. During the Easter break of 1891, Klein traveled to Leipzig, where Lie consented to his publication plan. In August of 1892, while vacationing on the island of Borkum, Klein wrote a first draft about their collaborative work, which he revised yet again after receiving Lie’s feedback (this latter version is dated November 1, 1892).112 This plan never materialized, however, because Klein and Lie had conflicting views about what and how much each of them had contributed in their earlier collaboration. For this reason, Klein reprinted only his Erlangen Program, with a few added notes, in the Annalen.113 As mentioned above (see Section 5.8.3), Lie had grown increasingly paranoid and felt as though people were systematically working against him. When Lie had a chance to read Klein’s first cycle of lectures, he was indignant that Klein had not sufficiently discussed his recent critiques of Helmholtz’s axioms. In response to the draft of Klein’s report mentioned above, Lie wrote numerous (partially contradictory) notes on the text, which he never sent back to Klein but which are preserved in his Nachlass. Here one reads, for instance: “Your investigations of non-Euclidean geometry from 1871 […] certainly have their own independent value. It is because of me, however, that you 111 Math. Ann. 73 (1913), pp. 289–304, with a correction in vol. 74 (1913), p. 312. See also R.C. Archibald, “R.G.D. Richardson, 1878–1929,” Bulletin of the American Mathematical Society 56/3 (1950), pp. 256–65; and A. Howe, “Klein’s Oscillation Theorem for Periodic Boundary Conditions,” Canadian Journal of Mathematics 23 (1971), pp. 699–703. 112 [Oslo] II (Klein’s second draft, dated November 1, 1892). This text is printed as an appendix in ROWE 1992a, pp. 588–604. See also ROWE 1988, 1989a, and 2019b. 113 Math. Ann. 43 (1893), pp. 63–100; reprinted again in KLEIN 1921 (GMA I), pp. 460–97.

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were able to operate there with the concept of a group and with infinitesimal transformation” […] “You are responsible for developing the theory of discontinuous transformation groups” […] “As much as I recognize the value of your Erlangen Program, I have never felt that you could lay any claim to the theory of continuous groups” […] “You realized that the methods of geometry are related to a group.”114 In the time that followed, Lie’s behavior became excessively offensive, and he demeaned Klein (and others). In the preface to the third volume of his Theorie der Transformationsgruppen (1893), Sophus Lie wrote: A simple image of a manifold with constant negative Riemannian curvature was provided by Mr. Beltrami, who showed that, for n = 3, such a manifold can be mapped onto the interior of a real surface of the second degree in R3. In this regard, I also point to Cayley’s projective metric. As far as I can see, the merit of the related studies by Mr. F. Klein on non-Euclidean geometry lies essentially in the fact that they have served to popularize the results of his predecessors. Here, Klein also makes use of my concepts of infinitesimal transformation and one-parameter group. […] The studies of the fundamentals of geometry by Messrs. v. Helmholtz, de Tilly, F. Klein, Lindemann, and Killing contain a series of crude errors, which are ultimately due to the fact that the authors of these investigations possessed either no knowledge at all or very inadequate knowledge of group theory.115

Lie asserted further that he had already developed the concept of a finite continuous group during the years 1870 to 1872: F. Klein, to whom I have communicated my ideas in the course of these years, was therefore inspired to develop similar conceptions about discontinuous groups. In his Erlangen program, where he describes his own and my ideas, he further discusses groups which, according to my terminology, are neither continuous nor discontinuous. For example, there is mention both of the group consisting of all Cremona transformations, and of distortion groups. That there is a distinct difference between these kinds of groups and the groups described by me as continuous, namely that my continuous groups can be defined on the basis of differential equations, while such is not the case for the discontinuous groups, had obviously completely passed him by. Nor does Klein’s programme include any trace of the theory of the very important differential invariant concept.116

Lie’s pronouncements in this preface culminated in the sentence: “I am no pupil of Klein, and neither is the opposite the case, although this might be nearer to the truth.” Klein was less concerned about himself than about his Annalen, and he wrote to Hurwitz on November 12, 1893: “I have some concern about the Math. 114 [Oslo] IV, Nos. 3–5, pp. 14–15, 18. 115 LIE 1893, pp. xii–xiii. For a discussion of this preface, see, for instance, JI/PAPADOPOULOS 2015, pp. 16–17. The paper that Klein had sent to Lie in 1892 was not, however, a revised version of his Erlangen Program, as Ji and Papadopoulos claim (ibid., p. 18). Rather, it was Klein’s unpublished account of his and Lie’s collaborative work from 1870 to 1872 (“Ueber Lie’s und meine Arbeiten aus dem Jahren 1870–72”). See [Oslo] II. 116 LIE 1893, pp. xvi–xvii. The English translation here is taken from STRØM 1992, p. 12. According to Thomas Hawkins, “Lie’s prodigious mathematical research activity between 1869 and the winter of 1873–74 was not dominated by group-related considerations but involved a diverse spectrum of mathematical ideas” (HAWKINS 1989, p. 276). See also HAWKINS 1984, 2000; ROWE 2019c; and Sections 2.6.2.1 and 2.8.2 above.

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Ann. Lie’s behavior, which cannot be further described here, deprives me of a number of co-workers who naturally belong to us; on the other hand, I have less and less support from my contemporaries the more they rust.”117 Hurwitz, who had meanwhile become a professor in Zurich, recommended that Paul Stäckel, who could be counted on to do good works, should be brought in more as an author, and continued: As far as Lie is concerned, the entire world is in agreement that he suffers from excessive delusions of grandeur and that his unfathomable behavior toward you is to be attributed to that. You should be indifferent to what he says, because no one takes Lie’s remarks seriously. However, I can imagine that Lie’s conduct must be painful for you on account of your old friendship with him.118

Wilhelm Fiedler, who was likewise a professor in Zurich, wrote to Klein a multipage commentary on Sophus Lie’s pronouncements, at the end of which he asked: But why did he do this, really? I see only one reason: “Because Klein’s students and friends have repeatedly misrepresented the reciprocal relationship between Klein’s and my own [Lie’s] work.” I don’t remember where and how this should have happened, only that you and Prof. Mayer, from the very beginning, have gone out of your way on every occasion to stress the importance of Lie’s ideas.119

Fiedler also remarked that he had compared Klein’s Erlangen Program from 1872 with the new edition and found “only painstaking references to Lie and his work.” Fiedler rightly predicted: “Knowing how you operate, my esteemed colleague, I of course have not doubted for a minute that you will do everything within your power to bridge and close this divide instead of letting it grow even wider.” David Hilbert, who had just become a full professor at the University of Königsberg (as Lindemann’s successor), held Lie’s collaborator Friedrich Engel partially responsible for the “inexplicable and entirely useless personal spitefulness that pervades the third volume of Lie’s work on transformation groups,” as he expressed in a letter to Klein dated November 15, 1893. Hilbert also wrote: Even earlier, as I have heard, Lie wanted to publish these matters in Crelle’s Journal, but he was rejected by the editors. In any case, his megalomania shines forth like a bright flame from the pages of this third volume. I will have to delve into the matter itself next semester, when I plan to teach non-Euclidean geometry. As it seems to me, Lie always approaches the subject with a preconceived and one-sided analytic viewpoint.120

Hilbert taught his lecture course on non-Euclidean geometry in the summer of 1894, and he had also studied Klein’s lectures on geometry beforehand. In a letter to Klein, he referred to these lecture courses as “convincingly clear,” and he also

117 118 119 120

[UBG] Math.Arch. 249 (Klein to Hurwitz, November 12, 1893). [UBG] Cod. MS. F. Klein 9: 1123 (Hurwitz to Klein, November 29, 1893). Ibid. 9: 24 (a letter from W. Fiedler to Klein dated November 8, 1893). Quoted from FREI 1985, p. 101 (a letter from Hilbert to Klein dated November 15, 1893).

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cited some of his own results in this context.121 The course ultimately led to his 1899 book Grundlagen der Geometrie [The Foundations of Geometry].122 Here it is worth mentioning, too, that in August of 1893, before Klein saw Lie’s third volume (with the comments cited above), he had given lectures in the United States in which he praised Lie’s work (see Section 7.4.2). After Klein had seen Lie’s volume, he reacted with composure, for instance in his report in Mathematische Annalen about his aforementioned lectures on higher geometry.123 There, he focused on the concepts of transformation, group, and system of coordinates, and he referred to the latest relevant results by Sophus Lie. To Hermann von Helmholtz, whom Klein would get to know better during his trip to the United States, Klein sent copies of his work, and he also commented on Lie’s remarks from the volume in question: You will understand if I refrain from going into the subjective remarks that the new volume contains in such large number. Regarding the factual exaggerations and the one-sidedness of Lie’s exposition, I would like to think that I have already countered such things in my [American] lecture. Nevertheless, I hope that this same lecture succeeded in emphasizing the great importance of Lie’s theory, which has too often been overlooked in Germany. The fact that Lie’s work could be ignored by so many people for all of 20 years is the counter-image to the pathological self-regard from which he is now suffering.124

Klein’s decency would come to expression yet again when the Russian mathematician A.V. Vasilev (see Section 5.4.2.5) asked him to write an evaluation of Lie’s third volume on transformation groups (the book with the humiliating remarks), which Lie had submitted to be considered for the inaugural Lobachevsky Prize announced by the Physical-Mathematical Society of the Imperial University in Kazan. Vasilev had written to Klein on December 18, 1896, and Klein responded positively after a short delay.125 In his evaluation, Klein stressed: “The volume submitted by Prof. Lie stands so far above all the other works to which it might be compared that there can hardly be any possible doubt that it should be granted the prize.”126 Klein added further positive remarks on Lie’s book and referred once more to the lectures that he had given as part of the Evanston Colloquium.127 121 122 123 124 125

See ibid. (Hilbert to Klein, February 14, 1894). See also TOEPELL 1986. Math. Ann. 45 (1894), pp. 145–49. [BBAW] NL H. v. Helmholtz, No. 233, pp. 6 –7 (Klein to Helmholtz, December 1, 1893). [UBG] Cod. MS F. Klein 12 (Vasilev to Klein, and Klein’s draft in response). Vasilev and Klein had met on several occasions, for instance at the 1894 meeting of the German Mathematical Society in Vienna, where Vasilev [Wassiljef] gave a talk titled “Lobatschefski’s Ansichten über die Theorie der Parallellinien vor dem Jahre 1826” [Lobachevsky’s Views on the Theory of Parallel Lines Before the Year 1826]; see Jahresbericht DMV 4 (1897) II, pp. 88– 90. Vasilev had a strong command of German, French, and English, and he participated (with L. Laugel) in translating some of Klein’s articles into French, including Felix Klein, “Sur la géométrie dite non euclidienne,” Annales de la Faculté des sciences de Toulouse: Mathématiques 11/4 (1897), pp. G1–G62. 126 See KLEIN 1921 (GMA I), pp. 384–401, at p. 385. 127 Regarding the Evanston Colloquium, see KLEIN 1894a and Section 7.4.2.

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Klein used this evaluation as an opportunity to present his own views about the fundamentals and axioms of geometry and to point out the contributions of recently published scholarly literature.128 He concluded his review with a few comments about Helmholtz, whose general ideas he considered brilliant, even though some of their details were unsatisfactory. Later, when Leo Koenigsberger asked him to contribute to writing Helmholtz’s biography, Klein expressed his opinions yet again on the topic of Helmholtz, axioms, and Sophus Lie: I remember in particular that Helmholtz asked me one day why Lie had attacked him so vehemently in the Comptes rendus and whether he had been incited to do so by Bertrand. I answered that Bertrand certainly had nothing to do with it, but rather that the tone of Lie’s critique could be explained by his own fierce and almost pathological temperament; Lie felt offended by the fact that the mathematicians in Berlin constantly disregarded him. We then spoke about the monodromy axiom of spatial geometry (among other topics);129 here Helmholtz stated that he was especially satisfied with the explanations that I had given in Math. Ann., vol. 37, p. 565, the relation of which to Lie’s own developments I had explained shortly before in Part II of my lectures on higher geometry, which have since been printed in lithographic copies.130

It is remarkable that Eduard Study – a vocal critic of Klein – also condemned Lie’s behavior towards Helmholtz and Klein. Study wrote to Rudolf Lipschitz: “In any case, Helmholtz’s conclusion from the finite to the infinitesimal has not been a gross error, and no worse than numerous mistakes Lie made himself before he was able to systematically work on his theory through [Friedrich] Engel’s involvement. [...] Lie’s entire stance against Helmholtz, as well as against Klein, whom he also treated unfairly, can, I believe, be explained by his emotional disorder, which was a remnant of an illness that he had had, I believe in 1890.”131 Lie and Klein had attempted to axiomatize geometry with group-theoretical methods. The development of modern axiomatics and algebraic topology did not fully take off until after this time. In 1894, Klein formulated a question for a philosophical prize in order to elucidate the topic further (see Section 8.3.2). Hilbert primarily conceived his well-received book on The Foundations of Geometry (1899) as a renewal of the fundamental structures of classical Euclidean geometry.132 In 1900, his fifth problem concerned the axioms of geometry. Here, he referred to Lie’s attempt to set up and prove “a system of geometric axioms with the aid of the concept of continuous groups of transformations.” Hilbert noted that this system of axioms was sufficient for geometry; Lie had assumed, however, that the functions defining the groups could all be differentiated. Hilbert therefore posed the question of “the extent to which Lie’s concept of continuous groups of 128 In particular, Klein mentioned the revised German edition (1894) of Giuseppe Veronese’s Fondamenti di geometria a più dimensioni e a più specie di unità rettilinee, esposti in forma elementare (Padova: Tipografia del Seminario, 1891). 129 On this topic, see M. NOETHER 1900, pp. 38–39; and the sixth chapter of BIAGIOLI 2016. 130 KOENIGSBERGER 1903, vol. 3, p. 81. 131 Quoted from SCHARLAU 1986, p. 203 (a letter from Study to Lipschitz, December 15, 1898). 132 On the origins of this book, see TOEPELL 1986. See also ROWE 1997.

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transformation is approachable even without the assumption of the differentiability of the functions.”133 Later, Hilbert’s fifth problem was reformulated in modern terms, and today it is considered to have been solved. In order to ensure that the topic of the principles of geometry would be properly elucidated in the ENCYKLOPÄDIE, Klein chose the Italian mathematician Federigo Enriques to write the article on it. Klein discussed the subject with him in Italy in 1899 and also in 1903 during Enriques’s stay in Göttingen.134 In a section on groups of motion, Enriques described the contributions of Helmholtz, Sophus Lie, Poincaré, and Hilbert. Klein’s own contributions and viewpoints are also discussed.135 The revised French version of the ENCYKLOPÄDIE (vol. 3) contains this contribution as well – “Principes de la géométrie” – and Enriques is credited as its sole author (in most of the French chapters, a French mathematician is added as a coauthor). 6.3.7 Visions: Internationality, Crystallography, Hilbert’s Invariant Theory This section will focus on aspects of Klein’s research that shed light on his general modus operandi: his interest in developments abroad, his recognition of new ideas, and his enthusiasm for promoting the development of new theories. 6.3.7.1 An Eye on Developments Abroad In Göttingen, Klein found time to follow international developments more closely. He traveled again to France and Great Britain. From the United States, a job offer came to Klein, and he fostered additional contacts with Russian mathematicians. For health reasons, the trip that Klein had planned to take to Paris in 1884 had not taken place (see 5.5.7.1). When he finally refreshed his old contacts there in 1887 (see also 6.3.1), he was able to develop a close relationship with Charles Hermite, who still dominated the scene despite his old age.136 Thus, two years later, Klein sent his latest results to Hermite, who presented them to the Académie des Sciences (on January 21 and February 11, 1889) and had them published in the Comptes rendus: “Formes principales sur les surfaces de Riemann” and “Des fonctions théta sur la surface générale de Riemann.”137 Klein’s new personal contacts also led again to the publication of articles by French authors (P. Appell, E. Picard, C. Hermite) in Mathematische Annalen. 133 Hilbert 1900, p. 269. 134 See Livia Giacardi’s article in COEN 2012, p. 224. 135 F. Enriques, “Prinzipien der Geometrie,” in ENCYKLOPÄDIE, vol. III.1.1 (1907), pp. 107–12. – See also the survey from Klein-Lie-Helmholtz to Weyl and Cartan in SCHOLZ 2016. 136 See Hermite’s letter to Klein dated September 11, 1887 (with greetings from Madame Hermite and Émile Picard) in [UBG] Cod. MS. F. Klein 9: 687. See also TOBIES 2016. 137 Published in Comptes rendus 108 (1889), pp. 134–36, 277–80.

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Klein had written to Hurwitz about the advances that had been made in France: “My stay in Paris was very interesting. In recent years there, in any case, they have done an amazing amount of work on function theory […]. The joy over the new results is almost outweighed by the envy that one feels towards the lucky discoverers. If we meet in the fall, we will have to speak about this at length.”138 Since 1888, Klein had been planning to travel in Great Britain with his old friend William Robertson Smith (see Section 3.3), who had now “a small chair of Arabic” at Trinity College in Cambridge.139 Before his trip, Klein had sent his latest results to the London Mathematical Society, where his contribution “Ueber die constanten Factoren der Thetareihen im allgemeinen Falle p = 3” [On the Constant Factors of the Theta Series in the General Case of p = 3] was presented (in German) at a meeting on April 11, 1889.140 He traveled in August and September of 1889, during which time he renewed his contacts and visited an ailing Arthur Cayley in Keswick. Klein took note of the results that had been made in Great Britain in the field of mechanics, which he would later use himself and help to disseminate. At the same time, he promoted David Hilbert’s work, who had not only discovered an error in Klein’s article “Zur Theorie hyperelliptischer Functionen beliebig vieler Argumente” [On the Theory of Hyperelliptic Functions of Arbitrarily Many Arguments]; he had identified an error in one of Cayley’s articles as well.141 Klein thus wrote Hilbert to tell him how well his work had been received there, and he informed Hurwitz: “I rushed through England, then I hastily revised my work on Abelian functions, and now I still have to plan my winter lectures as quickly as possible.”142 When Klein’s close friends W.R. Smith and Arthur Cayley died in 1894 and 1895, respectively, it was above all Cayley’s former Cambridge student Andrew Russell Forsyth who cultivated a relationship with Klein and kept him informed of intellectual developments in Great Britain.143 Forsyth began working as a lecturer at Trinity College in 1884. In his several textbooks, he regularly quoted Klein’s results.144 In a letter from February 10, 1895, Forsyth asked Klein to support his application to succeed Cayley with a “letter or testimonial” for the electors (who included G.G. Stokes, G.H. Darwin, and R.S. Ball). Klein wrote this letter of

138 139 140 141

[UBG] Math. Arch. 77: 184 (a letter from Klein to Hurwitz dated April 24, 1887). [UBG] Cod. MS. F. Klein 11: 1035, 1036 (Smith to Klein, May 22 and October 6, 1888). See Proceedings of the London Mathematical Society (1889), pp. 235–37. It was about Hilbert’s finitude theorem (see 6.3.7.3). Cayley had thought to have found an abbreviation, but wrongly assumed that for every invariant and covariant the degree is equal to the weight. ([UBG] Cod. MS. F. Klein 9: 1071, Hurwitz to Klein, February 6, 1889). 142 [UBG] Math. Arch. 77: 202 (Klein to Hurwitz, Dec. 31, 1889); and FREI 1985, pp. 21, 46–51. 143 [UBG] Cod. MS. F. Klein 9: 61–75 (Forsyth’s letters to Klein, from November 24, 1889, to November 24, 1912). – After Cayley’s death, Forsyth completed the edition of his teacher’s Collected Mathematical Papers (12 vols., 1889–1897). 144 See, for example, A.R. Forsyth, A Treatise on Differential Equations, 6th ed. (London: Macmillan & Co., 1929), p. v; and Forsyth’s Theory of a Complex Variable, 2nd ed. (Cambridge: Cambridge University Press, 1900), p. vii.

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recommendation at once, and Forsyth became the Sadleirian Professor of Pure Mathematics at the University of Cambridge in 1895. Klein asked Forsyth to send him biographical information about Cayley, and he immediately assigned Max Noether the task of writing an obituary for this long-time contributor to Mathematische Annalen.145 Forsyth’s letters to Klein contain commending words about the female mathematician Frances Hardcastle, who translated Klein’s book On Riemann’s Theory of Algebraic Functions (see Section 5.5.1.2 and Fig. 25). Forsyth also advised further women to study under Klein; the latter included Grace Chisholm in 1893, and Ada M. Johnson in 1894 (see also Section 7.5). Forsyth’s letters document the great extent to which Klein’s work was admired in Great Britain; thus he conveyed, for instance, on November 10, 1893: “Greenhill has undertaken to forward the [De Morgan] Medal to you.”146 Forsyth notified Klein that Weierstrass had been awarded the Royal Society’s Copley Medal in 1895, and he informed Klein on May 18, 1896 that he had been chosen to receive an honorary Doctorate of Science from the University of Cambridge – “among the highest honours which lies in the power of the University to bestow.” Klein made time to receive this honor in person; the occasion was celebrated on March 11, 1897, and he attended the ceremony with his wife, his wife’s sister Sophie, and one of his daughters.147 Because Klein had reoriented his research agenda toward applied mathematics, he also used his stay to establish further contacts, to which end he organized meetings with Osborne Reynolds, Lord Kelvin, and others.148 In the final letter that survives from Forsyth to Klein (November 24, 1912), Forsyth happily reported: “It is a real pleasure to know that you are this year to receive the Copley Medal, the highest honour which the Royal Society has to bestow. All your friends will rejoice at this recognition of the work you have done in our science.”149 Back to the year 1889. In February, Klein had received an offer to teach as a guest professor in the United States. Stanley Hall – a psychologist and the first president of Clark University, which had been founded in 1887 in Worcester, Massachusetts – asked Klein whether he might be interested in giving lectures there during the upcoming semester. Klein would have gladly accepted this guest professorship, as he expressed in clear terms to the Ministry of Culture: “If the ministry can see and recognize how this undertaking might serve the public interest, then I am ready to carry it out. Otherwise, I will not pursue it.”150 However, just as in 1883 the Saxon Ministry of Culture had not supported the possibility of Klein working abroad, the Prussian Ministry of Culture in 1889 was likewise uninterested in supporting such a prospect. Althoff recommended that

145 146 147 148 149 150

See Max NOETHER 1895. [UBG] Cod. MS. F. Klein 9: 66 (Forsyth to Klein, November 10, 1893). As of 1910, Sophie Hegel would work as a teacher at a private school in Malvern, England. [UBG] Cod. MS. F. Klein 9: 74 (a letter from Forsyth to Klein, January 7, 1897). Ibid., 9: 75 (Forsyth to Klein, November 24, 1912). [StA Berlin] Rep. 92 Althoff B, No. 92, p. 63 (Klein to Althoff, February 23, 1889).

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Klein “should reject the offer, for we have no interest in you carrying out this mission.”151 At the time, Prussia was pursuing certain colonial and political interests in negotiations with the United States;152 an organized scientific exchange was not yet part of the plan. Klein therefore turned down the offer to teach in the United States. In response, Althoff considered creating a new professorship for Klein at the University of Berlin, but Klein rejected this idea.153 On June 20, 1889, he was instead honored, by imperial decree, with the Order of the Red Eagle (Fourth Class) – as already mentioned. This Prussian order of merit had existed since November 17, 1705, when it was founded as the Ordre de la Sincérité. Klein’s efforts to expand his international relationships also involved seeking further contacts with Russia and Eastern Europe, where he had already reached out to a number of mathematicians during his time in Leipzig, in the interest of finding new authors to Mathematische Annalen (see Section 5.4.2.5). On January 16, 1891, he wrote to Adolph Mayer, his coeditor at the journal: “To a certain degree, I have faith in the future of Russian mathematics, and I believe that now would be a good time to make contact with the mathematicians there.”154 With some authors, such as A.A. Markov in St. Petersburg, Klein had maintained regular correspondence in the meantime (he also arranged for Markov’s books to be translated into German). Back when he was in Munich, he had Hurwitz report about the works of Chebyshev, the head of the St. Petersburg school. Later, Klein himself gave a lecture to the Göttingen Mathematical Society on Chebyshev’s numerical methods (concerning interpolation by polynomials).155 Regarding the Moscow school, which for many years competed with the school in St. Petersburg,156 it was above all P.S. Nekrasov with whom Klein cooperated; he organized an exchange of journals with the Mathematical Society there. Nekrasov’s contributions to Mathematische Annalen supported Klein in his smoldering academic dispute with Lazarus Fuchs (see Appendix 5). Nekrasov also put Klein in touch with mathematicians in Kiev and Kharkiv.157 In Kazan, Klein’s contact person was the aforementioned A.V. Vasilev. Klein’s interest in Russian scholarship would later lead to him argue on behalf of establishing a professorship for Slavic philology in Göttingen (this was part of his program of “universalism”; see Section 8.3.2). In January of 1914, moreover, he joined the newly founded German Society for the Study of Russia (Deutsche Gesellschaft zum Studium Rußlands; see Section 9.1.2). 151 [UBG] Cod. MS. F. Klein 1, B4 (Althoff to Klein, February 25, 1889). 152 See SIEGMUND-SCHULTZE 1997a, p. 25. 153 Klein regretted that his alternative idea of only spending the fall semester at Clark University was likewise “inhibited by Althoff’s diplomatic delays.” JACOBS 1977 (“Personalia”), p. 3. 154 Quoted from TOBIES/ROWE 1990, p. 188. 155 [UBG] Math. Arch. 49, pp. 133–36 (Klein’s lecture to the Göttingen Mathematical Society, May 21, 1895). See also SINAI 2003. 156 Regarding this rivalry, see DEMIDOV 2015. 157 [UBG] Cod. MS. F. Klein 11: 370 (a letter from Pokrovsky to Klein dated December 2 and 14, 1891). P.M. Pokrovky’s field of research was the theory of hyperelliptic functions.

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6.3.7.2 Arthur Schoenflies and Crystallography Klein’s ability to evaluate new theories in mathematics and the natural sciences led him to recognize work that was ahead of its time. One example of this was the field of crystallography. Arthur Schoenflies had investigated “planar configurations and related groups of substitutions,” and in doing so he had come across research on the structures of crystals.158 After being introduced to this line of inquiry by H.A. Schwarz, who presented Schoenflies’s first important study – “Ueber reguläre Gebietstheilungen des Raumes” [On Regular Area Divisions of Space] – to the Göttingen Society of Sciences,159 Schoenflies then made a major breakthrough after consulting with Klein. In a subsequent article published in Mathematische Annalen, Schoenflies acknowledged Klein’s inspiration as follows: “In a conversation, Mr. Klein drew my attention to the useful idea of expanding the theory of motion groups with reference to the concept of symmetry.”160 In another article by Schoenflies – “Ueber das gegenseitige Verhältniß der Theorien über die Struktur der Krystalle” [On the Reciprocal Relationship of Theories About the Structure of Crystals], dated June 7, 1890 – we read: Three different theories come into question, namely 1) the theory of Bravais and Wulff, 2) the theory of [Christian] Wiener and Sohncke, and 3) the theory that I recently presented here in the pages of the Göttinger Nachrichten. I should remark that the need to further develop the theory in accordance with the latter publication was first emphasized to me by Mr. Klein.161

Schoenflies was able to demonstrate the existence of 230 crystallographic space groups, a finding that he summarized in his book Kristallsysteme und Kristallstruktur [Crystal Systems and the Structure of Crystals] (Leipzig: B.G. Teubner, 1891). Correspondence with the Russian mineralogist E.S. Fedorov had supported this result, for the latter had reached the same conclusion in another way as early as 1885. Schoenflies would ultimately be the main author of the ENCYKLOPÄDIE article on crystallography (1905), to which the mineralogists Theodor Liebisch and Otto Mügge also contributed. With characteristic foresight, Klein believed in the validity of Schoenflies’s theory about the structure of crystals, even though it was rejected at the time by many experts in the field. He later remarked: “About 1890 the experts were still thoroughly accustomed to considering a crystal, or better a crystalline medium, as a space-filling continuum having the same properties at all its points!”162 Convinced by Schoenflies’s work, Klein presented the latter’s theory in 1893 in Chicago as well (see Section 7.4.1). Finally, in 1912, Klein was enthusiastic to learn that a 158 For a detailed discussion of Schoenflies’s work, see SCHOLZ 1989, pp. 110 – 48. 159 Published in Göttinger Nachrichten (1888), pp. 223–87. 160 Arthur Schoenflies, “Ueber Gruppen von Transformationen des Raumes in sich,” Math. Ann. 34 (1889), pp. 172–203, at p. 172. 161 Göttinger Nachrichten (1890), pp. 239–50, at p. 239. 162 KLEIN 1979 [1926], p. 325.

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crystal lattice can be analyzed by X-ray diffraction – a finding that earned Max von Laue the Nobel Prize and confirmed Schoenflies’s theory.163 6.3.7.3 Felix Klein and Hilbert’s Invariant Theory This would be a good place to finally introduce David Hilbert in greater depth. During these years, Klein clearly recognized Hilbert’s talent; he encouraged him and supported his invariant-theoretical approaches. Klein even entertained the idea of traveling to Königsberg himself to seek inspiration for his mathematical research. In August 1884, Klein had learned from Hurwitz that Hilbert had completed his dissertation – “Über invariante Eigenschaften specieller binärer Formen, insbesondere der Kugelfunctionen” [On the Invariant Properties of Special Binary Forms, Particularly Those of Spherical Functions], supervised by Lindemann – and that he intended to come to Leipzig in 1885 (see Table 6).164 In November of 1885 in Leipzig, Hilbert summarized the results of his dissertation for publication in Mathematische Annalen (vol. 27, 1886), and shortly thereafter he discovered new findings, which Klein submitted to the Sitzungsberichte der Sächsischen Gesellschaft der Wissenschaften (on December 7, 1885). These latter results culminated in Hilbert’s Habilitation thesis.165 Toward the end of Klein’s final semester in Leipzig (1885/86), Hilbert gave his first presentations in Klein’s seminar. The works that Hilbert discussed suggest that they were meant to prepare him for a trip to Paris, which Klein recommended that he should take. Hilbert spoke about Picard’s work on integrals of the first kind and algebraic surfaces (Journal des mathématiques pures et appliquées, 1885), and he analyzed studies by Riemann, Weierstrass, Poincaré, Picard, and Frobenius in a presentation titled “Ueber periodische Funktionen zweier Variabler” [On Periodic Functions of Two Variables].166 In March of 1886, when Hilbert traveled to Paris with a letter of recommendation from Klein, Eduard Study was already there, and both of them continued to receive letters from Klein with further advice. This included tips about mastering the “art” of making contacts with people by attempting to understand their interests. At the same time, Klein also recommended that, when back in Germany, they should make an effort to have further interactions with Paul Gordan and Max Noether.167 Klein subsumed Hilbert and Eduard Study in his own research direction when he wrote to them about his start in Göttingen: “My relationship with 163 See ibid. – A summary of the latest results at the time is Max Born’s article “Atomtheorie des festen Zustandes – Dynamik der Kristallgitter,” in ENCYKLOPÄDIE, vol. V (1922). 164 [UBG] Cod. MS. F. Klein 9: 1000 (a postcard from Hurwitz to Klein, May 27, 1885). 165 Published in Math. Ann. 28 (1886). 166 [Protocols] vol. 7, pp. 218–25, 274–83 (Hilbert’s presentations, given on January 11, 1886 and February 15, 1886). Regarding the work of Georg Frobenius, see HAWKINS 2013. 167 See FREI 1985, pp. 7–8 (an undated letter from Klein to Hilbert and Study, April of 1886).

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Schwarz has gotten off on the right foot; it remains to be seen, however, whether it will really be possible to create an intimate mutual relationship between the function theory of the Berliners and our own way of seeing things.”168 The surviving correspondence between Klein and Hilbert following this trip to Paris documents a period of intensive exchange. Up until September 1, 1896, Hilbert uniformly addressed Klein as “Esteemed Professor,” whereas Klein had been addressing Hilbert since 1892 as “Dear Colleague” or “Dear Friend.” Following Klein’s request, Hilbert sent him his latest results. Klein read these and responded with references to works by German and foreign mathematicians with relevance to the topic at hand. Hilbert studied these works immediately and contextualized them. From 1887 to 1889, Hilbert thus published seven articles in Mathematische Annalen, and others appeared during this time in the Göttinger Nachrichten. The latter included three notes “Zur Theorie der algebraischen Gebilde” [On the Theory of Algebraic Entities],169 which concerned the finitude theorem on which Paul Gordan had already done such excellent work (see Section 3.5). Gordan, however, now regarded Hilbert’s results with skepticism. It was Lindemann who had introduced Hilbert to Clebsch and Gordan’s approach to invariant theory. Since then, Hilbert had been led in new directions, especially by Hermite’s work.170 Klein appreciated Hilbert’s new approach, and he encouraged him to pursue it further: “I find that your understanding of the task of invariant theory is still too narrow – in the sense that it is still directly related to the Clebsch-Gordan tradition – whereas I believe that the research area in question has meanwhile been expanded and deepened in various ways.”171 In this letter to Hilbert, Klein listed five points in which he discussed the approaches of other mathematicians, and Hilbert indicated that Klein’s references had enabled him to improve his longer article for the Annalen.172 Gordan, who was then “the king of invariant theory” (see Section 2.4.1) and a member of Mathematische Annalen’s editorial board, was at first reluctant to accept the validity of the new approach that Hilbert presented in this work, which was titled “Ueber die Theorie der algebraischen Formen” [On the Theory of Algebraic Forms].173 In Klein’s opinion, it was “the most important work on general algebra […] which the Annalen has ever published.”174 When Klein, Hilbert, Minkowski, and others met in September of 1890 in Bremen (see Section 6.4.4), Klein was able to give Hilbert further motivation to

168 Ibid., p. 8. 169 See Göttinger Nachrichten (1888), pp. 450–57 (dated December 5, 1888); ibid. (1889), pp. 25–34 (dated January 30, 1889); and ibid. (1889), pp. 42–30 (dated July 31, 1889). 170 See FREI 1985, pp. 44–45 (a letter from Hilbert to Klein dated December 12, 1888). See also ROWE 2018a, pp. 160–67. 171 Quoted from FREI 1985, p. 53 (Klein to Hilbert, June 27, 1889). 172 See ibid., 54–55 (Hilbert to Klein, June 30, 1889). 173 Math. Ann. 36 (1890), pp. 473–534. 174 Quoted from FREI 1985, p. 62 (a letter from Klein to Hilbert dated February 18, 1890).

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take his work in a number-theoretical direction.175 Klein’s references to invariants in number theory and to “Kronecker’s module systems” are taken into account in Hilbert’s article “Ueber die vollen Invariantensysteme” [On the Complete Systems of Invariants].176 Hilbert reached out to Kronecker personally, he discussed the topic with Hurwitz, and he applied arithmetic methods to algebraic problems. Hilbert combined algebraic invariant theory with Kronecker’s theory of forms and Dedekind’s theory of ideals and moduli. He thereby became the actual founder of modern abstract algebra – a field to which Emmy Noether would later make important contributions.177 Klein was fully convinced of Hilbert’s abilities, so that when the next opportunity arose (a vacant professorship at the Catholic Academy in Münster) he wrote the following to Friedrich Althoff at the Prussian Ministry of Culture: Hilbert is “the rising man.” I have ranked him in last place here only because he is much younger than the other candidates. The articles that he has published in the last two years attest to his quite extraordinary talent for abstract thinking. After not seeing him for four years, I met him at the meeting of natural scientists [in Bremen] and was surprised by how much he has matured in the meantime and how he has since thought through all possible mathematical questions and raised new and important problems in every area of research. To me, whether Hilbert will now be considered for this position seems to come down to the essential question of whether preference will be given to someone with outstanding talent all around or to someone with many previous achievements. In the present case, perhaps the latter would be more suitable for the job than the former, and thus I would prefer a quiet man in Münster who might then remain there for many years.178

A year later, when Althoff suggested that Hilbert should transfer to Göttingen as a Privatdozent in Schoenflies’s place (by a “Umhabilitation”), Klein rejected the idea categorically. He emphasized the necessity of Schoenflies’s position in Göttingen and indicated that he had Hilbert’s long-term career development in mind:

175 On December 10, 1890, Klein wrote the following to Hurwitz: “For the Annalen, Bianchi has now revised his theory of linear substitutions with coefficients a+bi, a+bρ by treating the quadratic forms with complex coefficients by Dirichlet and Hermite exactly as I discussed the matter with Hilbert and the others in Bremen” ([UGB] Math. Arch. 77: 214). Bianchi began his article – “Geometrische Darstellung der Gruppen linearer Substitutionen mit ganzen complexen Coefficienten nebst Anwendungen auf die Zahlentheorie” [A Geometric Representation of Groups of Linear Substitutions with Complex Integer Coefficients, Along with Applications to Number Theory] – as follows: “The geometric method on which Professor Klein based the arithmetic theory of ordinary binary quadratic forms can be applied more broadly with the same success” (Math. Ann. 38 [1891], pp. 313–33, at p. 313). 176 Math. Ann. 42 (1893), pp. 313–73. 177 See the article by B.L. van der Waerden in HILBERT 1933, pp. 401–03. 178 [StA Berlin] Rep. 93 Althoff B, No. 92, pp. 76–77 (a letter from Klein to Althoff dated October 23, 1890). The words “the rising man” appear in English in the original. In 1891, Althoff ultimately offered the position (as an associate professorship) to Reinhold von Lilienthal, who had previously taught for two years in Santiago, Chile. Lilienthal was made a full professor in 1902, and he remained in Münster for the rest of his career.

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6 The Start of Klein’s Professorship in Göttingen, 1886–1892 Also, I cannot accept the counterproposal that you made to me regarding Hilbert transferring here. That would be a death sentence for Schoenflies, and I cannot allow that to happen. Furthermore, Hilbert is not what I need here, and he would not provide what Schoenflies already offers. Hilbert will go his own independent way; his rank is such that I would consider him when my own position becomes vacant or when another professorship becomes available alongside mine here. In the present case, however, what I need to find is someone to help me teach the beginning students, given that all my time is taken up by my mid-level and upper-level courses.179

In 1891/92, one of the participants in the “upper-level courses” mentioned here was Fabian Franklin, who had completed his doctoral degree under James J. Sylvester at Johns Hopkins University, where he then became a professor in the Department of Mathematics. While in Göttingen, Franklin attended Klein’s lecture course on algebraic equations and prepared a presentation on Hilbert’s article “Ueber die Theorie der algebraischen Formen,” which he would give in the research colloquium (during that semester, the colloquium was run by the Privatdozenten Burkhardt and Schoenflies).180 Klein regularly met with Franklin to discuss Hilbert’s work, which Klein himself wanted to penetrate in greater depth. Impressed by this work, Klein wrote to Hurwitz: Our pride and joy, of course, is Prof. Franklin, with whom I have been diligently studying Hilbert’s work. If Hilbert is now extending his ideas about finitude to questions of reality, that seems like a fine and important development to me.181 Indeed, tell Hilbert that, if he has any work that he considers fit for publication (especially concerning the application of Dirichlet’s methods), I am more than willing to have it published in the Göttinger Nachrichten, just as I have requested him not to withhold any of his more comprehensive studies from the Annalen. In general, I would like to create even closer connections between here and Königsberg. Above all, of course, I hope that you will be able to follow through with the plan that we discussed and stay with us here for a longer period of time over Easter. Then again, I am also seriously considering coming to Königsberg myself for a while. For, after concerning myself over the past few years with applied mathematics, I now need to revive and enhance my theoretical interests, and I cannot think of a better way to do so than to have extensive conversations in person with you and Hilbert.182

Fabian Franklin visited Hilbert and Hurwitz twice in Königsberg, where, in March of 1892, he completed a short note that would be published in Mathematische Annalen.183 Klein never made the trip himself, for he had to concentrate on a series of new personnel changes in Prussia (and Bavaria). It is no surprise, however, that when H.A. Schwarz’s position opened up in Göttingen in 1892, Klein wanted to hire Hilbert or Hurwitz to work alongside him as his immediate colleague (see Section 6.5.1.2). 179 [StA Berlin] Rep. 92 Althoff A I, No. 84, pp. 82–83 (Klein to Althoff, October 25, 1891). 180 [Protocols] vol. 10, p. 187. 181 Regarding these ideas, see Christian Tapp, An den Grenzen des Endlichen: Das Hilbertprogramm im Kontext von Formalismus und Finitismus (Berlin: Springer, 2013). 182 [UBG] Math. Arch. 77: 222 (a letter from Klein to Hurwitz dated November 6, 1891). 183 Fabian Franklin, “Bemerkungen über einen Punkt in Riemann’s ‘Theorie der Abel’schen Functionen’,” Math. Ann. 41 (1893), p. 308.

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6.4 BRINGING PEOPLE AND INSTITUTIONS TOGETHER Klein continued to pursue Clebsch’s intended program of uniting mathematicians. It is a lesser-known fact, however, that he was also instrumental in bringing together all the instructors at the University of Göttingen (see Section 6.4.1). With the overall interests of the university and its students in mind, Klein also came up with the novel idea of combining the University of Göttingen with the nearest Prussian Technische Hochschule, which was in Hanover (6.4.2). Although he had only recently been made a full member of the Royal Society of Sciences in Göttingen, Klein nevertheless thought of ways to reorganize this “Academy” (6.4.3). Shortly before the founding of the German Mathematical Society (Deutsche Mathematiker-Vereinigung – DMV), which was initiated by Georg Cantor, Klein became involved and worked in the background to ensure its coherent purpose, organizational structure, and leadership (6.4.4). 6.4.1 The Professorium in Göttingen Klein had learned to appreciate the Professorium in Leipzig, a professors’ association (see 5.7). Letters from Otto Hölder to his parents reveal that there had been no such organization in Göttingen, and that Klein succeeded in creating one: This “professors’ association” is a large organization that was brought into being by Klein and planned in such a way that, if possible, the entire university can be involved in it. This winter, it will cost me four additional evenings of dancing, on top of everything else. The matter is of course in the interest of those who otherwise have little interaction with one another, but for us it is just an added burden. It is hardly possible to avoid participating; I confirmed my membership, but tonight I’ll skip the constituent assembly.184

Otto Hölder was a staunch individualist; as it turned out, he drank beer with his compatriots from Württemberg instead of attending the founding of the Professorium. The aim was for university teachers and their wives to get together at cultural events. As a young professor in Erlangen, Klein had been a willing participant in similar cultural gatherings (see Section 3.6.1). The Privatdozent Hölder, however, felt that it was beneath his dignity to be involved in such things: “The professors’ association will begin its activities at the end of this month; I was expected to be an actor in a play, but of course I have rejected this imposition.”185 The first gathering of the Göttingen Professorium took place on Tuesday, November 30, 1886, and Klein felt that it was worth mentioning in a letter to Hurwitz: “Meanwhile, we have been living in a buzz of activity and with a lack of sleep: the first Professorium got together the day before yesterday, with 182 par-

184 Quoted from HILDEBRANDT et al. 2014, p. 227 (Hölder to his parents, October 21, 1886). 185 Ibid., p. 231 (a letter from Hölder to his parents dated November 16, 1886).382

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ticipants and three performances!”186 Hölder sent a more detailed report to his parents, and here it is possible to detect how much care and organizational effort Klein had devoted to the event: After a long period of deliberation, debate, and planning, yesterday was the first evening of the professors’ association. There were quite a few theater performances. First came [Emanuel] Geibel’s play “Pure Gold Is Revealed in Fire,” a terribly serious piece full of love and renunciation; really, it was not the right play for the occasion at hand because of its minimal plot and long speeches. This piece had been chosen because of the “prima donna,” who played the part very well; the same cannot be said of the other main role – that of a prince played by a Privatdozent. The second play – “Youthful Love” by Wilbrandt! – was excellent. It was truly funny, with well-acted roles; especially good were the characters of an old aunt, a sentimental teenager, and an idle student (this latter role was acted quite naturally by a friend of mine). After dinner, there was a third performance, a comedy set in a hospital […]. It was all very nice, but it was too much. There was an excess of gentlemen, and thus at first I did not have a dinner partner; then, however, [Felix] Klein came over to me, who had an extra lady to care for. Thus I had the honor of sitting next to the wife of Professor Benfey, the widow of the famous orientalist. […] I took part in two round dances and left before the end of the whole event.187

In a time without radio, television, or other overstimulating media, Klein sought to create a pleasant distraction in which he could include his wife. A grand ball was planned for Saturday, January 16, 1887; in Hölder’s words, it was “instigated by the professors’ association and will bring together nearly all of Göttingen’s society.”188 In June of 1887, the association organized an excursion with dinner and dancing in the evening. In October of 1887, Klein attempted to recruit Hölder to join the Professorium’s organizing committee, but Hölder successfully rejected the offer. This gesture was indicative, however, of Klein’s initiative. For Klein, the Professorium was also a way to increase his profile within the university as a whole. Almost twenty years later, his engagement on behalf of the institution’s overall interests would culminate in his election to the Upper House (Herrenhaus) of the Prussian State Parliament (Landtag) as the representative of the University of Göttingen (see Section 8.3.4.1). 6.4.2 A Proposal to Relocate the Technische Hochschule in Hanover to Göttingen The Polytechnical School in Hanover had belonged to Prussia since 1866, and it was reorganized into a Technische Hochschule in 1879. From early on, Klein had recognized the innovative achievements of the institution’s director, the engineer Wilhelm Launhardt, and he especially admired Launhardt’s book Mathematische Begründung der Volkswirtschaftslehre [The Mathematical Foundation of Econo-

186 [UBG] Math. Arch. 77: 166 (a letter from Klein to Hurwitz dated December 2, 1886). 187 Quoted from HILDEBRANDT et al. 2014, p. 232 (Hölder to his parents, December 1, 1886). 188 Ibid., p. 234 (a letter from Hölder to his parents dated January 12, 1887). For Hölder’s further remarks about the Professorium, see ibid., pp. 239, 246, 252.

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mics] (1885). In May of 1887, when all the faculty members in Göttingen were asked to nominate worthy candidates to be awarded an honorary doctoral degree (the occasion was the 150th anniversary of the university, which was celebrated on August 7, 1887), Klein argued vehemently for Launhardt. He persuaded a number of colleagues – including H.A. Schwarz, E. Schering, E. Riecke, W. Schur, and W. Voigt – to cosign his application. Klein knew from his time as a student in Bonn that the granting of honorary doctorates could bring needed recognition to newly established research directions (see Section 2.3.2). Within the university as a whole, however, Klein’s proposal remained “in the minority.”189 Even before the twenty-nine-year-old Wilhelm II was crowned German Emperor and King of Prussia (June 15, 1888) and announced his initiative to reform education, Klein had felt that the general objective of the university was insufficient. Since the summer of 1887, Klein had been serving on the examination committee for teaching candidates, which led him to reflect even more on “the general task of mathematical education” at the university level. He believed that this task could be better achieved by relocating the nearby Technische Hochschule in Hanover, which in his view was more practically oriented, to Göttingen.190 In order to meet the practical needs of students, Klein argued that the University of Göttingen and the Technische Schule should be combined. In the fall of 1887, Klein spent “a considerable amount of time in Hanover with Launhardt in order to get to know the facilities there and the relevant literature.” On May 27, 1888, he wrote to Friedrich Althoff that he was “proposing to relocate the Technische Hochschule to Göttingen.” He enumerated the goals that this move would achieve as follows: 1. 2. 3. 4.

A more comprehensive education of our students. A healthier development of our disciplines. The greater effectiveness of our institutions. General cultural interest: A truly modern education.191

The ministry requested a more detailed proposal, so Klein composed a lengthy memorandum and sent it to Berlin on October 6, 1888. This eighteen-page text reflects the breadth of Klein’s thinking, including his concern for “all questions of modern culture,” his idea that doctoral degrees should be allowed to be granted in technical disciplines, and his wish to combine mathematics with the latest developments in the natural and technical sciences. Regarding mathematicians, he wrote:

189 A total of nine proposed candidates for this award were rejected. Recipients of the honor included, among others, the Russian chemist D.I. Mendeleev (who created a version of the periodic table of elements) and Johann A. Repsold (the director of a company in Hamburg that produced astronomical and optical instruments). [UAG] Phil. Fak. 172a, No. 107k. 190 See Klein’s remarks in JACOBS 1977 (“Personalia”), p. 1. 191 Ibid., p. 2. Later, Klein would work together with Launhardt at the Prussian educational conference in Berlin (1900) and also in the Prussian Parliament (Landtag).

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6 The Start of Klein’s Professorship in Göttingen, 1886–1892 One must say that we have long neglected to keep up with the advances of neighboring disciplines. Let me mention only that area of our science whose general importance is immediately obvious even to non-experts: theoretical mechanics. Where is the university mathematician who has absorbed the ideas that are involved in the new physical discipline of the mechanical theory of heat192? Who has noticed that the theory of the motion of rigid bodies (kinematics) has gained new content in the hands of mechanical engineers? Or that, in the field of statics, the task of constructing bridges has led to the development of original and far-reaching graphical methods?

Turning his attention to engineers and natural scientists, he went on: Are those to whom such things should be a concern even aware of our profound theorems, our brilliant conceptions? I maintain that, with respect to their education in the exact sciences, German engineers are lagging behind their contemporaries in Italy and France. I maintain that, as opposed to earlier, there has been a complete deterioration of mathematical education among our physicists and astronomers. I maintain that German chemistry has fallen behind because its representatives, on account of their lack of preliminary mathematical education, are unable to follow the advances that have been made by others.193

Althoff responded to Klein’s suggestions with reservations, and the idea of combining a university and a Technische Hochschule in Germany remained an unfulfilled wish. At this same time, however, there were already (private) American universities with associated technical institutes, and Klein would have a chance to acquaint himself with such arrangements in 1893 (see Section 7.4.3). 6.4.3 The Idea of Reorganizing the Göttingen Society of Sciences Klein had belonged to the Royal Society of Sciences in Göttingen as a corresponding member since 1872 (see Section 2.8). In 1886, he was listed as an associate member. In order to be named a full member of this academy,194 there had to be a vacancy. As the secretary of the academy, the classical philologist Hermann Sauppe, who had supported hiring Klein in Göttingen (see 5.8.2), informed him on December 28, 1886: “We would like to accept you into our academy, because we all know how much more effective our community will become on account of your creative energy.”195 Sauppe told Klein that the number of twenty-four members for the existing three classes had long been established and that Klein could not yet be elected. The reason was that Moritz Abraham Stern – whose professorship at the university Klein held – had been assured that he could remain a full member of the academy after his retirement. Sauppe did offer to allow Klein to

192 Klein was aware of the work on statistical mechanics and thermodynamics by Maxwell, Boltzmann, and Josiah Williard Gibbs, and he supported their new theories. In a letter dated January 17, 1889, Gibbs thanked Klein for trying to have his work on mechanical heat theory published in book form; see KÖRBER 1961, p. 107. 193 [UAG] Kuratorialakten, 4 I, No. 88a, pp. 2–10 (quoted here from pp. 4–5). 194 On all members of this academy from 1751 to 2001, see KRAHNKE 2001. 195 [UBG] Cod. MS. F. Klein 11: 656 (a letter from Sauppe to Klein, December 28, 1886).

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participate in the society’s work immediately, attend its academic meetings, and submit work for publication in the Göttinger Nachrichten. Klein was still barred, however, from participating in the society’s administrative deliberations.196 Sauppe had been right about Klein’s “creative energy.” As early as New Year’s Eve in 1886, Klein asked Hurwitz and his colleagues in Königsberg to send him appropriate (shorter) articles for publication in the Göttinger Nachrichten.197 In the society’s meeting on February 5, 1887, Klein presented Hurwitz’s first work, which was especially momentous on account of the ongoing polemic with Lazarus Fuchs (see Section 5.5.5 and Appendix 5). Hurwitz corrected one of Fuchs’s errors and he included his relevant correspondence with Fuchs in an appendix.198 This was orchestrated with Klein and took place with H.A. Schwarz’s support, who later explained Fuchs’s error to Weierstrass in a letter.199 At the meeting held on July 2, 1887, Klein introduced three additional (algebraic) articles – by Bolza, Maschke, and Aurel Voß.200 On November 5, 1887, Klein presented his own new results on hyperelliptic functions, and he was finally elected as a full member of the mathematical class on November 12, 1887, with one dissenting vote.201 Shortly thereafter, Klein submitted his plans for reorganizing the society. The Göttingen “Academy” had little power to act, because it was not a “legal entity.” For this reason, the orientalist Paul de Lagarde, who had earlier acted at Schwarz’s instigation to thwart Klein’s appointment (see Section 5.8.2), had already composed a report of his own with ideas for reorganization the academy.202 Lagarde stated that universities were regressing and that the University of Göttingen must cease to be a “provincial university.” His hope was that it could become an institution with relevance throughout Europe. In addition to reforming education in general, he believed that an overall reorganization of the Province of Göttingen should be initiated by the Society of Sciences: “Göttingen must begin to be what only Göttingen can be […].” He also proposed that members of the “Academy” should be chosen according to a new set of criteria: it would not benefit, he thought, from electing “former luminaries” but rather from appointing “men who can still become something.” The (new) objectives that Lagarde suggested were restricted just to the philological-historical class.

196 [AdW Göttingen] Chron 4, 6: 41 (minutes from a meeting held on December 18, 1886). 197 [UBG] Math. Arch. 77: 168 (a letter from Klein to Hurwitz dated December 31, 1886). 198 Adolf Hurwitz, “Über diejenigen algebraischen Gebilde, welche eindeutige Transformationen in sich zulassen,” Göttinger Nachrichten (1887), pp. 85–107. 199 [BBAW] 1254 (a letter from Schwarz to Weierstrass dated March 15, 1887). 200 See Oskar Bolza, “Darstellung der rationalen ganzen Invarianten der Binärform sechsten Grades durch die Nullwerte der zugehörigen δ-Functionen,” Göttinger Nachrichten (1887), pp. 418–21; Heinrich Maschke, “Ueber die quaternäre, endliche, lineare Substitutionsgruppe der Borchardt’schen Moduln,” ibid., pp. 421–24; and Aurel Voß, “Ueber bilineare Formen,” ibid., pp. 424–33. 201 [AdW Göttingen] Chron 4, 6: 51. On that same day, Wilhelm Weber was made an honorary member, and Ludwig Boltzmann was elected as an external member. 202 See LAGARDE 1894, pp. 162–77 (this report is dated January 24, 1887).

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Although Klein did not agree with Lagarde’s political views, he did espouse a few of the points made in Lagarde’s proposal.203 In his “Major Report on the Reorganization and Expansion of the Society,” which was completed in December of 1888, Klein likewise argued in the general interest of improving education and enhancing the status of the university. At the same time, Klein argued in favor of combining “pure research” with technical applications. In this respect, he won the support of the mineralogist Theodor Liebisch, the psychologist Georg-Elias Müller, the physicist Eduard Riecke, and the classical philologist Ulrich von Wilamowitz-Moellendorff. Together, they suggested that representatives of technical disciplines should be accepted as members, and they gave some specific names: Wilhelm Launhardt (the engineer in Hanover), Johann A. Repsold (an expert in precision mechanics in Hamburg), and a mechanical engineer to be named later.204 The matter had to be approved by the Ministry of Culture, and Althoff, who was skeptical of the idea, sent Klein’s report to the classical scholar and historian Theodor Mommsen for further review.205 The concept of this new type of membership was ahead of its time, and it was never realized at any academy during Klein’s life. Klein was one of the most active members of Göttingen’s Royal Society. The 1889 volume of the Göttinger Nachrichten contains two articles by him (on the theory of Abelian functions); moreover, he presented nine studies by his students or colleagues at the society’s sessions: by Burkhardt (on a hyperelliptic multiplier equation); Heinrich Maschke; Ernesto Pascal (one work on the theory of odd Abelian sigma functions, another on the theory of even sigma functions of three arguments); Eduard Wiltheiss (on the partial differential equations of Abelian theta functions of three arguments); Wilhelm Wirtinger (on the analog of the Kummer surface for p = 3); and three works by Hilbert. Klein was thus deeply dismayed when his application to edit Riemann’s lecture notes was rejected by the society. Klein had already recruited Heinrich Weber to examine the material presumably located in Göttingen.206 Klein had submitted an application to Schering and Sauppe, but during the meeting held on July 5, 1890, he received the answer that there was no unpublished Riemann material, a fact that was later contradicted by M. Noether and W. Wirtinger’s edition of these very notes. This edition was induced by Klein and published as a supplementary volume to Riemann’s collected works in 1902.207 The negative vote was influenced above all by Ernst Schering, whom the Royal Society of Sciences had commissioned to edit Gauß’s works.208 However, 203 See Klein’s remarks in JACOBS 1977 (“Personalia”), p. 2. Paul de Lagarde was a vocal antiSemite, an opponent of women’s rights, and a supporter of expansionist colonization. 204 [UBG] Cod. MS. F. Klein 1 B: 3, p. 91. 205 See REBENICH/FRANKE 2012 (a letter from Althoff to Mommsen dated June 25, 1889). 206 Along with Richard Dedekind, Heinrich Weber had been one of the editors of Riemann’s collected works (see RIEMANN 1876). 207 See Klein’s remarks in JACOBS 1977 (“Personalia”), p. 4. 208 See REICH/ROUSSANOVA 2013.

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no new volume of that edition had been published since volume 6 (1876). Deeply upset about his colleague Schering,209 Klein wrote to Robert Fricke: For four years now in Göttingen, I have been working to establish, on the basis of our common academic interests, a positive relationship with my colleagues, from whom I had been kept apart by historical tradition, and yesterday I finally had to register the fact that everything was in vain, that my efforts have resulted in envy and hatred, and that I am more isolated than ever.210

These events gave Klein even more incentive to attempt to reorganize the Royal Society of Sciences in Göttingen by proposing new statutes. In 1892, when he received a job offer from the University of Munich, he made sure that the University of Göttingen’s counteroffer included terms that would also allow him to reform the “academy” (see Section 6.5.2). 6.4.4 Felix Klein and the Founding of the German Mathematical Society In 1899, when Georg Cantor took the initiative to bring together German mathematicians,211 Klein’s initial response was one of cautious reservation. When Leo Koenigsberger, speaking in 1899 at the annual meeting of the Society of German Natural Scientists and Physicians in Heidelberg, announced Cantor’s proposal – “It would be desirable to establish a more cohesive association of German mathematicians than has heretofore existed” – Felix Klein was in Great Britain and he was thinking about his failed opportunity to work as a visiting professor in the United States. Upon his return, his student Walther Dyck informed him about the meeting in Heidelberg, which would lead to the founding of the German Mathematical Society (DMV) at the next year’s meeting of the Society of German Natural Scientists and Physicians in Bremen. This history has already been investigated in detail.212 Klein’s role in it can be summarized in five points. First, Klein consciously decided to operate behind the scenes. He was aware that he represented a specific group and that his presence might therefore deter others from participating. Klein made his ideas known through Dyck and in letters to Georg Cantor. Second, Klein’s idea concerning the main role of the new association was that it should commission “reports to be written about the development of various branches of our science,” as Walther Dyck informed Cantor as early as October

209 Schering’s widow Maria Heliodora (née Malmstén) justified her husband’s negative vote, remarking that he was bitter about the fact that he had received too little money and recognition for his edition of Gauss’s works. [UBG] Cod. MS. F. Klein 11: 682: her letter to Klein on February 3, 1899. 210 [UA Braunschweig] A letter from Klein to Fricke dated July 6, 1890. 211 On Georg Cantor’s life and work, see DAUBEN 1979, PURKERT/ILGAUDS 1987. 212 TOBIES 1991a, TOBIES/VOLKERT 1998, pp. 125–57. On Walther Dyck’s involvement, see HASHAGEN 2003, pp. 403–37.

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12, 1889. This idea found its way into the statutes of the German Mathematical Society. From the beginning, Klein made efforts to ensure that reports were written on the areas of both “pure” and “applied” mathematics.213 These reports, many of which were quite comprehensive, were published in the first ten volumes of the Jahresbericht der DMV and were ultimately integrated into the ENCYKLOPÄDIE. Third, the most widely discussed organizational question was whether the new Society to be founded should hold its annual meetings in conjunction with the Society of German Natural Scientists and Physicians. Georg Cantor argued for separate meetings, following the model of the Société Mathématique de France. Klein, in contrast, wrote to Cantor on May 14, 1890: “For my part, I cannot understand why it would be necessary to separate them; this can always happen later if a majority of members is in favor of it and if the central leadership of the conference of natural scientists turns out to be too tyrannical.”214 Cantor allowed himself to be convinced when Weierstrass made a similar argument in a letter to him dated September 6, 1890.215 Until 1913, the German Mathematical Society and the Society of German Natural Scientists and Physicians thus met together at the same place and time. When the latter organization decided in 1920 to meet only every other year, the mathematical society (together with the physical societies) continued to have annual conferences.216 Fourth, Klein, more than others, made attempts to ensure that the proper personnel would be put in place to lead the society. Independent from one another, he and Weierstrass both wrote to Cantor that it would be desirable to elect a committee in Bremen to oversee the planning of the association’s annual meeting. Like most of the full professors in Berlin, however, Weierstrass did not participate in the foundational meeting in Bremen. Klein decided to participate, and he submitted suggestions to Cantor regarding the election of the committee: In Bremen itself, it will thus be essential to elect a perhaps four-member committee, which, taking into account any other wishes expressed in Bremen, will also be responsible (as independently as possible) for preparing for the meeting in 1891. The most important matter in this case is undoubtedly the question of suitable personalities, and about this question I would like to add as an aside that I am not standing for election (I would refuse being elected to the committee, because I am in fact so busy with other matters that it would be impossible for me to take on any further obligations). My stance on this issue is thus as objective as possible. In advance, our basic principle should probably be to elect only those mathematicians who will demonstrate their interest in the matter by being present in Bremen, and to give equal consideration to mathematicians in northern and southern Germany. As a potential representative from northern Germany, of course, you yourself would have to be considered first of all, otherwise we should ask Kronecker (if he is there). If he should reject this idea, [Hermann] Schubert would seem to be the most fitting. Among the southern

213 For example, Klein recruited Hilbert and Minkowski to write about number theory, Sebastian Finsterwalder to write about photogrammetry, and August Föppl to write a report on technical mechanics, to name just a few authors. See TOBIES 1989b. 214 [UA Freiburg] 36 (a letter from Klein to Cantor dated May 4, 1890). 215 For this letter, see TOBIES/VOLKERT 1998, p. 141. 216 See TOBIES 1996b.

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Germans, I would like Dyck to be considered, whom you now recognize, at any rate, for his abilities as an active administrator and bookkeeper. [Leo] Koenigsberger is another possibility, if he is interested. If he is not, then Lüroth immediately comes to mind. Lüroth is a man who, despite his personal modesty, is socially adept, and he possesses a very broad scientific perspective that also concerns disciplines adjacent to mathematics.217

Klein convinced further colleagues to participate in Bremen. Thus his long-time allies at Mathematische Annalen, Paul Gordan and Adolph Mayer, traveled there, as did several of his students and collaborators (Heinrich Burkhardt, Walther Dyck, Franz Meyer, Erwin Papperitz, Hermann Wiener, Eduard Wiltheiss). Hilbert and Minkowski made the long journey from Königsberg. Hurwitz, after an illness, had to go to a spa instead, and Lindemann also did not attend. The words written by the Privatdozent David Hilbert to Klein on July 24, 1890 were well-suited to increase even more Klein’s trust in this young, like-minded mathematician with a similarly broad horizon: As Professor Hurwitz has already told me, and as I am now able to see on the program, you will also travel to Bremen in the fall, and I am very pleased about this. For I am likewise firmly determined to make this trip, and I especially hope that other mathematicians – young and old – will be there in large numbers. In fact, I believe that a closer personal connection between mathematicians would be very desirable for our science, and I think that the suggestions contained in the Heidelberg protocol [in 1889] are very timely in general. As it seems to me, mathematicians today understand one another too little; they are not interested in one another enough and, as far as I can judge, they are too unfamiliar with the classic works in our discipline. Moreover, many of them are working arduously on branches that are already dead.218

In Bremen, the resolution to establish the German Mathematical Society was ratified on September 18, 1890 with thirty-three signatories.219 Two of these are not pictured in Figure 29 (p. 370): Friedrich Simon Archenhold, a researcher at the Berlin observatory directed by Wilhelm Förster, and Eduard Study, who was then a Privatdozent in Marburg. Among the founding members, six had been present in Heidelberg the year before when the original idea to create the society had first been proposed: Georg Cantor, Dyck, Lothar Heffter, Papperitz, Ernst Schröder, and Heinrich Weber. Nine had attended the conference of mathematicians that had been held back in 1873 in Göttingen: P. Gordan, R. Hoppe, L. Kiepert, F. Klein, E. Lampe, A. Mayer, E. Schröder, H. Schubert, and R. Sturm (see Section 2.8.3.4). Four of the founding members were mathematics teachers at secondary schools in Bremen: H. Kasten, F. Klemm, G. Meyer, and H. Wellmann.

217 Quoted from TOBIES/VOLKERT 1998, pp. 142–43 (a letter from Klein to Cantor dated June 12, 1890). Lüroth did not come to Bremen and thus he was not elected to the committee. 218 Quoted from FREI 1985, p. 68 (a letter from Hilbert to Klein dated July 24, 1890). 219 For a list of the founding members, see Jahresbericht der DMV 1 (1890/91), p. 7 (“Chronik”). A comparison of this list with the names under Figure 29 reveals that Roth and Ueltzen were not among the signatories and were not DMV members (see also TOEPELL 1991). The persons depicted could instead be the undersigned Enno Jürgens, professor of mathematics at the Technische Hochschule in Aachen, and the secondary school teacher G. Meyer.

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Figure 29: The founding members of the German Mathematical Society (Deutsche Mathematiker-Vereinigung, DMV), September 18, 1890 (a postcard)

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The election of the first advisory board went according to Klein’s plans: Georg Cantor as the chairman, Walther Dyck as the secretary, along with Hermann Schubert (Hamburg), Emil Lampe (Technische Hochschule Charlottenburg/ Berlin), and Theodor Reye (Strasbourg), who was not present in Bremen but had attended the meeting in Heidelberg. Klein described his impression of the event in a letter Hurwitz (October 19, 1890): I see the resolutions in Bremen […] as being entirely favorable and promising. Cantor knows (according to recent news) how to present the matter in such a way that Weierstrass, Kronecker, and now also [Carl] Neumann are uniformly sympathetic to it! Now the task will be to organize the conference in Halle to be as multifaceted as possible. I will also give a talk there, though I will otherwise keep a low profile in the interest of the organization, so that it doesn’t acquire a partisan character.220

The 1891 meeting of the German Mathematical Society took place (together with that of the Society of German Natural Scientists and Physicians) in Halle, where Klein used his talk to discuss British results in the field of mechanics, particularly Hamilton’s theory of integrating dynamic differential equations.221 By June of 1891, the German Mathematical Society had 205 registered members; the fee for membership was two Mark. The society’s president Georg Cantor, who held a professorship at the University of Halle, invited Leopold Kronecker to give the opening address at the meeting in September. Although Kronecker had to decline on account of his wife’s death, he accepted his election to the society’s board. Thus, as of January 1, 1892, the board would have consisted of Georg Cantor (president), W. Dyck, P. Gordan, L. Kronecker, E. Lampe, and H. Schubert.222 Kronecker died, however, on December 29, 1891. Fifth, Klein continued to influence the organization through its secretary Dyck, who sought Klein’s advice about drafting statutes, attracting additional members, and choosing authors to write reports about the development of mathematical research areas. This is documented, for instance, in the marginal notes that Klein made on Dyck’s letter from March 30, 1891: Consultation with [Heinrich] Hertz. Hilbert. Report on algebraic functions? Who else? Written requests from Dyck. Opening address? – Cantor, at last. (Cantor’s own ideas: Leipzig) Recruiting people who have yet to join. (Bruns, Lipschitz, Wirtinger. The mathematical physicists.)223

Numerous mathematicians also joined from abroad; by 1901, there were 150 members from outside of Germany (that is, slightly more than 30% of the total of 498), including two women. Among non-German-speaking countries, the most widely represented were the United States (32) and Russia (10). American mem220 221 222 223

[UBG] Math. Arch. 77: 210 (Klein to Hurwitz, October 19, 1890). See Jahresbericht der DMV 1 (1890/91), pp. 35–36. See ibid., pp. 4, 7, 11, 15–20 (“Chronik”). [UBG] Cod. MS. F. Klein 8: 702 (Klein’s marginalia on Dyck’s letter, March 30, 1891).

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bers included Klein’s students (Bôcher, Haskell, Snyder, E.B. Van Vleck, H.S. White) and Charlotte A. Scott, who in 1898 – with Klein’s support – became the first woman member (see Section 7.5). Among the Russians were Sintsov, Sonin, Vasilev, Zhukovsky, and Nadezhda von Gernet (who studied under Klein and Hilbert; she completed her doctoral degree under Hilbert in 1901 and became a member that same year). Russian mathematicians presented new results (in German) at the annual meetings of the German Mathematical Society from relatively early on – in Munich in 1893, Vienna in 1894, Lübeck in 1895.224 When Georg Cantor left the position as the society’s president of his own accord in 1893, every later president served for a term of one year. Those elected to the board could serve on it for a period of three years and serve as the chairman for one of these years. For the first ten years after Cantor, the presidents were P. Gordan (1894), H. Weber (1895), A. Brill (1896), F. Klein (1897), A. Voß (1898), M. Noether (1899), D. Hilbert (1900), W. Dyck (1901), Franz Meyer (1902), and F. Klein (1903). In 1908/09, Klein would serve as chair for the third time. Overall, he served as a board member from 1895 to 1898, from 1903 to 1905, and from October 1, 1907 to September 30, 1910.225 Although no Berlin mathematician took this position, this was not due to Klein. In 1893, Klein had attempted to involve Georg Frobenius – a full professor in Berlin since 1892 – as a board member. Hilbert had supported this,226 but Frobenius declined. Ulf Hashagen has shown that H.A. Schwarz, who had been in Berlin since 1892, was trying to stir up opposition against the German Mathematical Society.227 Frobenius had been a member since 1891, as had the Berlinbased professors Weierstrass and Fuchs, while Schwarz did not join until 1894.228 Klein remained an active participant in the German Mathematical Society. He initiated reports, section meetings, and committees, and he helped to prepare its members for international conferences (in Zurich, Paris, Heidelberg, Rome, Cambridge).229 He managed to secure the society’s support for projects that were important to him (the ENCYKLOPÄDIE, the edition of Gauß’s works, the international bibliography, educational reform, etc.). On the occasion of the fiftieth anniversary of his doctoral degree (December 12, 1918) and his seventieth birthday (April 25, 1919), the German Mathematical Society named him its honorary chairman for the fiscal year of 1918/19.230 In 1924, to mark the occasion of Klein’s seventyfifth birthday, he became the first honorary member of the society, which praised him for his universality, leadership skills, and idealism.231 224 This statement is based on an analysis of the journal Jahresbericht der DMV. 225 See the annual reports published in the Jahresbericht der DMV. The organization’s fiscal year at first began on January 1st, but this was later changed to October 1st. 226 See FREI 1985, pp. 94, 96. 227 See HASHAGEN 2003, p. 436. 228 See TOEPELL 1991, p. 352. 229 See also CURBERA 2009. 230 See Jahresbericht der DMV 27 (1918) Abt. 2, pp. 59–60. 231 See ibid. 33 (1925) Abt. 2, p. 4; and ibid. 34 (1926) Abt. 1, p. 89.

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6.5 THE PIVOTAL YEAR OF 1892 At the beginning of year 1892, Klein greeted Hurwitz with these words: Dear friend! Now, above all, accept my and my family’s best wishes for the new year. […] Kronecker’s death will have made no less of an impression on you than it did on me. What are all of our plans when a more powerful hand intervenes? And what should happen now? I grow numb when I think about all of the conceivable possibilities, and the new professorial appointments, which can happen one way or another, are not even the main issue. What is essential is that the Berlin school has fallen apart beyond repair and that we, the survivors, now have greater freedom but must also shoulder the great, great responsibility [for mathematics in Germany]. Extend my greetings to Hilbert and Lindemann. Sincerely yours, F. Klein232

Klein obviously felt responsible for the development of German mathematics. At the University of Berlin, there were then three mathematics professorships: Kronecker’s, Weierstrass’s, and Lazarus Fuchs’s. Because Kronecker had died and Weierstrass planned to retire, two vacancies opened up alongside Fuchs’s position. Moreover, when Heinrich Eduard Schroeter died on January 3, 1892, another full professorship had to be filled in Prussia at the University of Breslau. In Bavaria, too, a full professorship was available at the University of Munich because Ludwig Seidel had retired in 1891. Felix Klein saw himself at the center of events. Although he was unable to realize all of his ideas, both he and the University of Göttingen profited from the final decisions that were made. 6.5.1 Refilling Vacant Professorships in Prussia Friedrich Althoff had been the Senior Privy Counselor in the Ministry of Culture since 1890. As such, he was able to make hiring decisions more or less independently, but he increasingly relied on Klein’s judgment regarding the appointment of mathematics professors at Prussian universities and other institutions. The first hiring process in which Althoff turned to Klein for advice was the aforementioned open professorship at the Catholic Academy in Münster, for which Klein, in addition to other input, had provide his euphoric statement about the talents of David Hilbert (see Section 6.3.6.2). In what follows, I will discuss the hiring opportunities that arose in 1892 at the universities in Berlin, Breslau, and Göttingen. 6.5.1.1 Berlin, Breslau, and Klein’s System for Classifying Styles of Thought Regarding Berlin and Breslau, Althoff wrote to Klein on Sunday, January 3, 1892: “Under these circumstances, I would be very grateful if you would be so kind as to share your full opinion about the situation and about what should happen.”233 232 [UBG] Math. Arch. 77: 224 (an undated postcard from Klein to Hurwitz, January 1892). 233 [UBG] Cod. MS. F. Klein 1C: 2, fols. 1–1v. On Klein’s hiring policy, see TOBIES 1987b.

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Althoff also sent letters of this sort to other mathematicians. Klein’s replies seemed to convince him, however, that Klein had developed a useful system for classifying mathematical styles of thought, and that this system could be helpful in guiding his hiring decisions. As early as January 6, 1892, Klein sent his requested full report to Althoff. Here, he interwove his positive views of “universality” with a critique of Kronecker’s one-sided or biased approach: My only critique concerns the bias with which Kronecker opposed the philosophical viewpoints of scientific approaches that differed from his own, and which prevented his students from forming an accurate image of the scope and significance of today’s mathematics. This bias lay less in Kronecker’s innate talent than in his character. Over time, his goal became to acquire, as much as possible, an absolute command of all of German mathematics, and he pursued this goal with the full range of his intelligence and with the utmost tenacity of his will. Now that he is gone from the scene, it is little surprise that no one among the younger generation of mathematicians in Berlin can be called his equal. Lazarus Fuchs can certainly not be regarded as such.234

Lazarus Fuchs, whom Klein belittled here, had completed his doctorate in 1858 under Kummer and had been a professor in Berlin since 1884. Klein’s polemical relationship with Fuchs would influence the hiring process in Berlin. In his report, Klein developed a classification system for the three professorships in Berlin: Because the normal representation of mathematics at a large university is three full professorships […] the candidates for such a position should not be chosen according to the special field in which they work but rather according to the different nature of their mathematical thinking. Given the variety that exists among individuals, one should avoid being too schematic, but on the whole the following three types should be represented: 1) The philosophically minded mathematician who constructs from concepts; 2) The analyst, who operates essentially with the formula; 3) The geometrician, who proceeds from intuition [Anschauung].235

Using this system for classifying styles of thought, Klein categorized the following mathematicians (and theoretical physicists): Weierstrass (type 1), Kronecker (between 1 and 2), Kummer (2–3), Gustav Robert Kirchhoff (2), Helmholtz (3), L. Fuchs (1), Georg Cantor (1), H. Weber (2–3), Frobenius (2), H.A. Schwarz (3), Lindemann (3), Sophus Lie (3), Jacob Rosanes (2), and himself (3). Klein thus recommended that Friedrich Althoff should consider hiring the following mathematicians to work alongside Fuchs (type 1) in Berlin: as a type 2, Heinrich Weber (then a professor in Marburg) or Georg Frobenius (then a professor in Zurich); and as a type 3, Hermann Amandus Schwarz or Ferdinand Lindemann. Regarding Schwarz, however, Klein added the following jab: “On the basis of the clarity of his lectures and his enthusiasm for teaching, Schwarz would undoubtedly be at the top of this list, if only his personality were not so utterly prosaic and stodgy.”236

234 [StA Berlin] Rep. 92 Althoff A I No. 84, fol. 5 (Klein to Althoff, January 6, 1892). 235 Ibid., fol. 7v. 236 Ibid., fol. 8.

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Klein’s proposals for Breslau were formulated in the same report. At the University of Breslau, Jacob Rosanes (type 2) had held a professorship since 1877 alongside H.E. Schroeter. Klein recommended Hurwitz, Schottky, Lüroth, and Dyck. In this regard, he made a point to stress Hurwitz’s extraordinary creativity and that Hurwitz had a difficult time advancing his career on account of his Jewish background.237 Klein described Lüroth and Dyck, in contrast, as being less productive.238 Althoff, however, summoned Rudolf Sturm from Münster to Breslau in 1892 and hired Wilhelm Killing as a professor in Münster to work alongside Reinhold von Lilienthal. That is, Althoff did not follow Klein’s latest advice but rather considered what Klein had written in 1890 regarding the hiring opportunity in Münster and the most appropriate candidates for the position.239 At the University of Berlin, H.A. Schwarz and Frobenius were appointed, which corresponded to Klein’s classification system. It has often been discussed, however, whether Klein himself would have gone to Berlin (and worked alongside Lazarus Fuchs). A few of his comments suggest that he had at least expected an offer. Thus we also read in his report to Althoff: After the previous negotiations, it is obvious that I have recently thought about how I would respond to receiving an offer to become a professor in Berlin, and so there is no point in making a secret of it. […] My conclusion is that, in certain respects, I would be capable of being effective in Berlin in the sense of type 3), but on the other hand I am missing an essential characteristic: the hardiness of a big-city dweller […]. Thus, when I consider only my own contentment, there is no doubt in my mind that I have to stay in place [in Göttingen]. Yet, regardless, I could understand that those in my circle of friends might feel as though it would be my duty to accept the central position [in Berlin] if it were offered to me. Then I would ask you to alter the position in such a way that I would be able delegate more duties than I would have to perform myself. Thus, even today, I have to ask you not to regard me as being inactive, if I make few public appearances. May a kind fate steer the course ahead! F. Klein240

The sense of duty mentioned here was stressed by Robert Fricke, whose Habilitation had been rejected at the University of Berlin. Yet Klein replied: You claim that it would be my duty to accept a possible offer [from Berlin]. My ideas are rather moving in the direction of using this new situation to give new momentum and general importance to the mathematical school in Göttingen. If I had a more demanding position, do

237 When Klein recommended Hurwitz for a professorship in Münster, Rudolf Sturm replied to him: “I myself am far from being an anti-Semite, but we consider it pointless here to recommend a Jew. You know that the process of hiring Protestants here does not always run smoothly; the Ministry would simply ignore a Jewish candidate.” [UBG] Cod. MS. F. Klein 11: 1281 (a letter from Sturm to Klein dated August 13, 1890). 238 [StA Berlin] Rep. 92 Althoff A I No. 84, fol. 8v (Klein to Althoff, January 6, 1892). 239 In his letter to Althoff from October 23, 1890, Klein had written (in addition to his praise for Hilbert): “[Hermann] Kortum is unproductive; in his scientific work, Killing is too closely aligned with Sturm, though he is otherwise talented […]; von Lilienthal has yet to achieve very much academically. […] Hurwitz as well as [Eduard] Wiltheiss and [Otto] Hölder are suitable.” [StA Berlin Rep. 92 Althoff B, No. 92, fols. 76–77. 240 [StA Berlin] Rep. 92 Althoff A I No. 84, fol. 9v (Klein to Althoff, January 6, 1892).

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6 The Start of Klein’s Professorship in Göttingen, 1886–1892 you know that my health might not soon deteriorate as much as it did in 1882 in Leipzig? And what do you estimate would be the tiny amount of time that I would still have for my own scientific ideas? The issue is as double-edged as no other and, if necessary, it will have to be decided thoughtfully and dispassionately.241

Adolf Hurwitz also thought that Klein should go to the University of Berlin: You can imagine the deep impression that was made on us by Kronecker and Schroeter dying in quick succession. About this matter, it seems as though the entire impartial world of mathematics is in agreement that only you can fill that gap that has arisen in Berlin. The only question is whether the Ministry would be able to withstand the opposition [to your appointment]. And then: would you accept the possible offer?242

The hiring committee in Berlin consisted of the astronomers Wilhelm Förster and Friedrich Tietjen, the mathematicians Lazarus Fuchs and Karl Weierstrass, and the physicists Hermann von Helmholtz and August Kundt. Förster had taken it upon himself to seek Klein’s opinion. On January 15, 1892, Klein replied that he had suggested Heinrich Weber and Georg Frobenius for Kronecker’s position, as well as H.A. Schwarz and Lindemann as geometricians. In this letter, Klein also expressed himself at length about his conflict with Fuchs (see Appendix 5 for details). The Berlin committee members were aware that the Ministry of Culture valued Klein’s advice highly. In order to exclude him from the list of candidates, they downplayed his talents and achievements in their meeting.243 Weierstrass wanted to hire a “good analyst,” and he remarked: “For the public at large, Klein and Schwarz are the top candidates. And we must only refrain from considering them if they are absolutely not to be had.” Indeed, he preferred Schwarz: “Schwarz stays on point. Good lecture. Klein rambles more. Dazzler.” Fuchs commented in a similar vein: “I have to get along with him. Schwarz has made some really valuable achievements – Klein, quite the contrary (his icosahedron book is a compilation of Schwarz’s and Fuchs’s results in a journalistic style).” Kundt stressed, however: “Klein is a fascinating teacher.” Förster, who had asked for Klein’s opinion, summarized: “The general opinion is not much in Klein’s favor. We have to include Klein’s name for the Ministry, but we have to emphasize that it would be impossible to work with him here.” Helmholtz mentioned: “The late Kronecker considered Klein a busybody/doer [Faiseur].” Fuchs felt that it was necessary to add: “I should state that I have nothing to put forward against Klein’s personal characteristics, only against his deleterious approach to science. He does not work for the sake of the [common] cause but rather writes textbooks based on the work of others.” Ultimately, the committee members in Berlin suggested the same candidates whom Klein had proposed: H.A. Schwarz and Frobenius. In their application to the Minister of Culture, they justified their decision to exclude Klein as follows:

241 [UA Braunschweig] A letter from Klein to Fricke dated January 12, 1892. 242 [UBG] Cod. MS. F. Klein 9: 1101 (Hurwitz to Klein, January 14, 1892). 243 See BIERMANN 1988, pp. 305–07 (minutes of the committee meeting, January 22, 1892).

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Above all, however, it had to be taken into consideration that the chosen candidates would be suitable for guiding students toward the serious and selfless study of mathematical problems, as has been practiced at our university for generations. For this reason, it was necessary to reject such people as Professor Felix Klein in Göttingen (b. 1849). Opinions are very divided regarding his scientific achievements, and his overall effectiveness as a scholar and teacher conflicts with the tradition of our university mentioned above.244

Toward the end of February in 1892, Althoff traveled to Göttingen and informed Klein personally that he would not be offered the position in Berlin on account of his generally weak health and because of the faculty’s dismissive attitude toward his candidacy.245 In a letter to Althoff dated April 10, 1892, Klein let him know: “I personally greeted the news that I would not be offered the job in Berlin as a fortunate turn of events; however, this news had a depressing effect in many circles.”246 Even if Klein would not have accepted the offer, he had at least counted on it to improve his position in Göttingen. Paul Gordan wrote to him from Erlangen: “I am sorry that you did not come to Berlin; with your comprehensive intellect, you would have brought order to the mathematical conditions in Germany.”247 Had the offer been made (and had he accepted it), Klein would have faced a difficult problem because “his” journal, Mathematische Annalen, was based in Göttingen, while its main competitor, Crelle’s Journal, was based in Berlin. A year later, Klein acknowledged to his American student Henry S. White that such a situation would have put him in a quandary.248 6.5.1.2 Hiring a Successor for H.A. Schwarz in Göttingen On April 1, 1892, H.A. Schwarz was appointed to succeed Weierstrass at the University of Berlin. At the University of Göttingen, however, he remained a member, along with Klein and Schering, of the hiring committee in charge of seeking his own replacement. For some time, Klein had been corresponding directly with Friedrich Althoff (that is, outside of official channels), and on March 21, 1892 he wrote to Berlin to express his wishes regarding Schwarz’s successor. Again, his argument was based on his system for classifying styles of thought: If mathematics is to continue to grow on a healthy basis in Göttingen, I will need to be supplemented with someone along the lines of Kronecker and Weierstrass (whom I have always held in high regard, as little as I could approve of their exclusive prevalence). In this respect, as I have mentioned on various occasions, I have always thought of just Frobenius, Hurwitz, and Schottky. Only recently have I added the younger scholars Hilbert and Minkowski to this list.249

244 245 246 247 248 249

Quoted from BIERMANN 1988, pp. 307–08. [UA Braunschweig] A letter from Klein to Fricke dated February 26, 1892. [StA Berlin] Rep. 92 Althoff A I No. 84, fols. 33v–34 (Klein to Althoff, April 4, 1892). [UBG] Cod. MS. F. Klein 9: 464, p. 75 (a letter from Gordan to Klein dated April 16, 1892). See SIEGMUND-SCHULTZE 1996, p. 18. [StA Berlin] Rep. 92 Althoff A I No. 84, fol. 27 (Klein to Althoff, March 21, 1892).

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Klein’s preferred candidate was David Hilbert, who had meanwhile turned thirty years old (see Appendix 6.1). He saw no problem in the fact that Hilbert was just a Privatdozent. At the age of twenty-three, Klein himself had been promoted from the position of Privatdozent to full professor without ever having to hold an associate professorship. Because the majority of the hiring committee in Göttingen (Schering and Schwarz included) was opposed to hiring a Privatdozent, Klein fought successfully on behalf of Hurwitz. His name was placed first on the list of proposed candidates, so Klein was certain that he would be hired and even started to make plans with him.250 Because the university’s Kurator had deemed it necessary to name a third candidate, Heinrich Weber was added to the list by Klein’s “opponents” (Schwarz and Schering).251 Despite Klein’s insistence (see Appendix 6.2), Althoff disregarded his wish. Instead, Althoff first allowed Georg Frobenius to consider whether he might want to work in Göttingen or Berlin. In a postcard dated April 2, 1892, Klein informed Hurwitz: “Frobenius has turned down the offer, but now Althoff wants to negotiate with Weber!” Then, in a letter dated April 7, 1892, Klein made it known to Hurwitz that Heinrich Weber had accepted the position and would begin in the fall: For now, I am so agitated about this that I don’t know how organize my thoughts. That the two of us could advance our common goals with mutual encouragement and daily interaction, however, is obviously a thought for which the world is not yet ready and which we will have to bury because it is too beautiful. Accept my heartfelt greetings, and my wife’s as well (she would express herself much more vibrantly than I have done here), and also extend our wishes to your bride, although we still haven’t met. May both of you happily overcome the disappointment that you will understandably feel! Your old friend, F. Klein252

Through his conversations with Frobenius – whose career had begun in Berlin and brought him to Zurich in 1875 – Klein was able to pave the way for Hurwitz. In a telegram from Zurich sent on June 3, 1892, Hurwitz announced: “The Swiss Department of Education has selected me to be Frobenius’s successor” at the Eidgenössisches Polytechnikum there.253

250 See [UBG] Math. Arch. 77: 229 (a letter from Klein to Hurwitz dated March17, 1892): “Unfortunately, I was unable to include Hilbert on the list of possible candidates because Göttingen, which is so sure of itself, dismissed the idea that a Privatdozent could ever be hired as a full professor. Luckily, your appointment [to Göttingen] will indirectly benefit him. […] If my wife can be of any help arranging things for you here […], I hope that you will welcome this assistance. Of course, you will have to stay with us again during the early stages of your time in Göttingen.” Klein even coordinated his teaching with Hurwitz and told him that they should “march separately but do battle together.” 251 In his letter to Althoff from March 21, 1892 (cited above), Klein mentioned that this proposal had been made by his “opponents.” 252 [UBG] Math. Arch. 77: 234, 235 (letters from Klein to Hurwitz dated April 2 and 7, 1892). 253 [UBG] Cod. MS. F. Klein 9: 1115 (Hurwitz to Klein, June 5, 1892). The Eidgenössisches Polytechnikum, which was renamed as the ETH (Eidgenössische Technische Hochschule) Zurich in 1911, received the right to confer doctoral degrees in 1908.

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Klein let Althoff know that the ministry’s decision to hire Heinrich Weber (instead of Hurwitz) would be regarded as an effort to lower Klein’s status in the eyes of Göttingen’s Philosophical Faculty.254 However, Paul Gordan – who like Hurwitz came from a Jewish family, though he converted to Protestantism (see Section 3.5) – attempted to comfort Klein with the following words: It was right of you to recommend Hurwitz in Göttingen; he has earned this distinction. However, it is fortunate for you that your suggestion was not followed, and for this you cannot thank God enough. What would you have had with Hurwitz in Göttingen? You would have had to accept complete responsibility for this Jew; every noticeable or apparent mistake by Hurwitz would have been on your head, and everything that Hurwitz would ever said in the faculty and senate meetings would have been considered to have been influenced by you. Hurwitz would have been regarded as no more than an appendage of Klein. You can carry on your scientific interaction with him just as well in writing.

Regarding Heinrich Weber, Gordan also looked on the positive side: “He can make your teaching activity far easier; he will not take away the Americans and other foreigners from you, because in this respect you are so far above him. You will be able to get along with him very well; however, it would be good if you did not collaborate with him scientifically.”255 This advice is surprising, for Weber came from the tradition of the University of Königsberg, where he (like Clebsch) had learned about Riemann’s intuitive approach to geometry from Richelot.256 As early as April of 1892, Klein traveled to Cassel (Kassel today) to meet Heinrich Weber.257 The latter agreed with Klein’s ideas without any reservations. Together as colleagues in Göttingen, they would go on to achieve a number of important initiatives (see Chapter 7). 6.5.2 A Job Offer from the University of Munich and the Consequences In July of 1892, Felix Klein received an offer – “as the first and only candidate” – to succeed Ludwig Seidel at the University of Munich, with a salary of 12,000 Mark, the guarantee of an assistant with a salary of 1,500 Mark, and all of the resources needed to run a Mathematical Seminar. Moreover, the chemist Adolf Baeyer informed Klein in a letter from July 11, 1892 that the Bavarian Ministry of Culture would “create a position for you through which you would be able to influence the organization of mathematical education in all of Bavaria.”258 The theoretical physicist Ludwig Boltzmann wanted Klein by his side, and he described his “universality” and academic productivity with glowing words.259 254 255 256 257 258 259

[StA Berlin] Rep. 92 Althoff A I No. 84, fol. 33v. [UBG] Cod. MS. F. Klein 9: 464, pp. 75–76 (Gordan to Klein, April 16, 1892). See KOENIGSBERGER 2004, pp. 108–09 (a letter from Weber to Koenigsberger). [StA Berlin] Rep. 92 Althoff A I No. 84, fol. 35. [UAG] Kur. 5956, fols. 38, 40–41v. See HÖFLECHNER 1994, vol. 2, pp. 173–74 (a letter from Boltzmann to Paul von Groth). Boltzmann’s remarks in this letter are quoted at greater length in my preface.

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The offer from Munich was so generous that Althoff felt alarmed and did everything in his power to keep Klein in Göttingen. Klein had informed Althoff of the offer on July 9, 1892, to which the latter replied immediately on July 11th, saying that Munich had fired a “warning shot” and that he would soon be on his way to Göttingen.260 Already on July 15th, Althoff, Klein, and the University of Göttingen’s Kurator Ernst von Meier signed a (retention) contract:261 Göttingen – July 15, 1892 Professor Dr. Felix Klein will decline the offer from Munich […] provided that he is assured of the following, whereby the question of reorganizing the Royal Society of Sciences – to which Mr. Klein attaches the utmost importance – was already discussed yesterday at length and is on its way to being implemented.262 1. It will be requested emphatically and resolutely that Mr. Klein should receive a salary increase of 2000 (two thousand) Mark beginning in October of this year. Until this is officially ensured, Mr. Klein will receive this same sum in the form of an annual remuneration. 2. During this fiscal year and the next, an allowance of 3000 M. in total (either 1500 each year or 1000 this year and 2000 M. in 1893/94) will be granted for the purpose of improving the reading room of the Mathematical-Physical Seminar. 3. The University Library will receive 6000 M. in ca. 10 annual installments (the first in this fiscal year) to fill in the gaps of its mathematical holdings (including physics and astronomy) according to Mr. Klein’s requests.263 4. As of April 1, 1893, the remuneration for the assistant at the collection of mathematical instruments and models will be increased to 1200 M. According to Mr. Klein, this could also take place in the form of a Dozent stipendium. 5. As soon as possible, care will be taken to ensure the creation of a budgeted associate professorship for mathematics264 in Göttingen, whereas Schering’s full professorship can be abolished in the future or used in another way, e.g. by converting it into an associate professorship for geophysics. 6. The issue of changing the general rules concerning seniority in Göttingen (Weber and Wellhausen, for instance, fall under these) will be considered.265 Read and accepted Althoff (signed) F. Klein (signed) von Meier (signed)

260 [UBG] Cod. MS. F. Klein 1 C: 2, p. 47 (a letter from Althoff to Klein, July 11, 1892). 261 [UAG] Kur. 5956, pp. 42–43 (Klein’s retention contract). 262 See also Section 6.4.3. On June 21, 1893, the Emperor and King ratified a new set of statutes for the “Academy.” Its three previous classes would be converted into two (mathematicalphysical and philological-historical), each with 15 places for full members under the age of 75; 25 places for associate members, 75 places for corresponding members, and an unlimited number of places for honorary members. See Göttinger Nachrichten (1893), p. 516. 263 Klein regularly submitted lists of scholarly works and newly founded mathematical journals to be purchased, including works in the fields of technical physics, mechanics, pedagogy ([UAG] Kur. 5956, fols. 50–65: Klein to the Kurator). Klein was able to extend this source of funding beyond its initial period of ten years. As late as July 15, 1927, Richard Fick, the director of Göttingen’s University Library, still referred to this “Felix Klein Fund in the amount of 800–1000 M. annually.” [UAG] Math. Nat. 0047, No. 32. 264 Arthur Schoenflies received this associate professorship on April 1, 1893. 265 This point was not realized (JACOBS 1977,“Personalia”, p. 5). The older professors Heinrich Weber and Julius Wellhausen (a theologian and orientalist) received a lower salary than Felix Klein.

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During the negotiations, Althoff became convinced that Klein’s wishes were in the best interest of the University of Göttingen. For choosing to remain, Klein was honored with another order of merit, this time the Order of the Red Eagle (Third Class), with ribbon (November 14, 1892).266 Regarding Seidel’s successor at the University of Munich, Klein recommended his former student Ferdinand Lindemann, who had already threatened to end their friendship because Klein had not considered him as a possible candidate to replace H.A. Schwarz in Göttingen.267 The position in Munich had long been a topic of discussion in the correspondence between Klein and Lindemann, who wanted to leave Königsberg at all costs. Klein suggested that he might have a chance in Munich. Lindemann had his doubts about this, however, because he had learned from the University of Freiburg (Baden) that he had not been considered for a position there for the sole reason that Klein had recommended him.268 But in Bavaria, Klein’s word counted for something, and Lindemann was hired. In the years to come, Lindemann would indeed commit some mathematical errors, including an erroneous proof of Fermat’s Last Theorem in 1908.269 Nevertheless, Lindemann supervised a number of good doctoral students, among them Emil Hilb (who defended in 1903) and Arthur Rosenthal (1909). Both also participated in projects initiated by Felix Klein. Recommendations for professorships were often causes of friction. In December of 1892, for instance, Klein’s friend and colleague Paul Gordan in Erlangen (Bavaria), who was fifty-five years old at the time, was deeply disappointed that Klein had not recommended him for the position in Munich: “Instead of proposing me for the vacant position, you nominated two of your students, Lindemann and Dyck, who should only have been able to have their turn if I had rejected the offer. [...] Everyone here is talking about how insignificant Gordan must be if even his friend Klein doesn’t think him worthy to come to Munich.”270 In reality, however, Gordan was less active than he used to be, and he was less open to new developments (see also Section 6.3.7.3). 266 [UAG] Kur. 5956, fol. 52. 267 Overestimating himself, Lindemann had written the following to Klein on February 29, 1892: “I indeed have more important achievements to show for myself than Hurwitz. If you take Hurwitz […] without considering me, you will embarrass me to the whole world; you will inflict an insult on me that I did not deserve” ([UBG] Cod. MS. F. Klein 1C: 2, fols. 7v–8). 268 Ibid., fol. 16v (a letter from Lindemann to Klein dated March 12, 1892). 269 This theorem, noted by Fermat in 1637, states that no three positive integers a, b, and c satisfy the equation aⁿ + bⁿ = cⁿ for any integer value of n > 2. Paul Wolfskehl, who died in 1906, had bequeathed 100,000 Mark toward the cause of proving this theorem. (Funding from this bequest was used to cover the costs of series of lectures in Göttingen, for example Poincaré’s lectures in April of 1909). The proof of Fermat’s Last Theorem required substantial new ideas that were developed only in the second half of the 20th century and was finally provided by Andrew Wiles (1995). – On the announcement of the Wolfskehl Prize Foundation and Klein’s detailed commentary on the large amount of false evidence already submitted in advance and on the nature of the peer review, see Jahresbericht der DMV 17 (1908) Abt. 2, pp. 111–13. 270 [UBG] Cod. MS. F. Klein 9: 472 (A letter from Gordan to Klein, December 12, 1892).

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Figure 30: Felix Klein’s certification as a foreign member of the Società Italiana delle Scienze [Accademia nazionale delle scienze (detta dei Quaranta)] on Februar 12, 1896 ([UBG] Cod. MS. F. Klein 114: 20).271

271 Originally called the “Società Italiana” (the title “Academy” was then reserved only for associations operating within individual Italian states). It was composed of 40 members from the various Italian states and 12 foreigners. The name “Accademia nazionale dei XL,” which dates back to 1949, was changed to the present name in 1979, when the number of foreign members was also increased to 25.

7 SETTING THE COURSE, 1892/93–1895 During this period, Felix Klein laid the strategic groundwork for what would become the famous international center of mathematics, natural sciences, and technology in Göttingen. Klein described the years after 1892/93 as “characterized by a preponderance of organizational activity.”1 After the Prussian Ministry of Culture had informed Ernst von Meier, the Kurator of the University of Göttingen, that H.A. Schwarz had accepted a professorship in Berlin, Klein became the dominant professor of mathematics on Göttingen’s faculty. Von Meier turned to Klein for advice about how the Mathematical Institute should be organized. Thus, Klein first set the agenda for the future of his own position and requested to take over Schwarz’s previous duties as “director of the collection of mathematical instruments and models.”2 In addition, he suggested, in a long letter dated February 29, 1892, that the Institute’s hitherto separate facilities (the reading room, the Seminar library, the model collection) should be combined under one roof and that it would be necessary to hire a paid mathematical assistant and to create an associate professorship (for Arthur Schoenflies).3 Klein was able to hire, for the first time, a government-funded mathematical assistant at the University of Göttingen at the beginning of the summer semester on April 1, 1892 (Section 7.1). After the appointment of Heinrich Weber on October 1, 1892, he was able to realize further ideas. Together they founded the Göttingen Mathematical Society (7.2). At that same time, Klein also started an advanced mathematical training course (continuing education course) for active secondary school teachers (7.3). During the semester break in the summer of 1893, Klein traveled to the United States (7.4), and this trip served as the starting point for further new initiatives. The latter included the beginning of women studying mathematics at the university level (7.5); the introduction of actuarial mathematics as an official course of study (7.6); and the idea of establishing, with private funding, the field of technical physics at the University of Göttingen (7.7). It was during this period, too, that the initial ideas for the famous ENCYKLOPÄDIE project were formed (7.8). Because these programmatic undertakings would continue to guide Klein’s activity throughout the decades to come, I will both introduce them in this chapter and discuss their long-term implications. By convincing the administration and the Prussian Ministry of Culture to hire David Hilbert, effective April 1, 1895, Klein also ensured that pure mathematical

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KLEIN 1923a (autobiography), p. 23. [UAG] Kur. 5956, fol. 25. The Prussian Ministry of Culture officially granted Klein’s request on April 13, 1892 [UAG] Kur. 5691, fols. 15–15v. [UAG] Kur. 5691, fols. 1–8v. This document is published in TOBIES 2019b, pp. 510–13.

© Springer Nature Switzerland AG 2021 R. Tobies, Felix Klein, Vita Mathematica 20, https://doi.org/10.1007/978-3-030-75785-4_7

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research would continue to be conducted in Göttingen at a high level (see Section 7.9). By pushing through the associate professorship with his contract negotiations (see 6.5.2), Klein also paved the way for the institutional development of the applications of mathematics (see Section 8.1.2). After the hiring of Heinrich Weber had been finalized, Klein immediately reached out to him; as early as May of 1892, Klein wrote enthusiastically to Robert Fricke: “In the meantime, my friendship with Weber has only grown stronger. In the winter, we want to begin life anew under the banner of cheerful optimism!”4 Klein also informed Althoff at the Ministry of Culture that Weber would support his most important plans, which Schwarz had previously hindered: the new associate professorship for Schoenflies, the transfer of Robert Fricke’s Habilitation from Kiel to Göttingen, and the coordination of the curriculum. Although Klein believed that Weber, who was seven years his senior, had already passed the peak of his abilities, he regarded him highly and later claimed that Weber was perhaps the most versatile representative of his generation – a generation of mathematicians “which has more or less kept up the contacts between 1) invariant theory, 2) the theory of equations, 3) function theory, 4) geometry, and 5) number theory.”5 As early as the summer semester of 1892, Klein, who was eager to coordinate the course schedule to suit the interests of students and instructors alike, created a “conference of Seminar directors” to serve these ends. Schering took over the courses that H.A. Schwarz had announced for this semester. Because Heinrich Weber wanted to start by teaching number theory in the winter semester, Klein assigned the Privatdozenten Burkhardt and Fricke to teach courses on function theory. Schoenflies’s role was to teach descriptive geometry,6 offer beginners’ lectures, and direct exercises. Regarding his own broad and research-oriented teaching program, Klein planned, as he informed Robert Fricke, to devote his efforts first to higher geometry and then […] as soon as I see a new way ahead, I will revisit linear differential equations and publish my thoughts about them in a coherent way. Then, finally, I will be able to turn to number theory, probability theory, and mechanics and will thus be engaged in truly far-reaching mathematical activity!7

Klein’s zeal for his work was hard to interrupt. Toward the end of April in 1893, when he had to undergo a medical operation (which was successfully performed by F.J. Rosenbach, the director of surgery at the university’s polyclinic), he set aside his teaching duties for just a short while. Hurwitz learned that this procedure

4 5 6 7

[UA Braunschweig] A letter from Klein to Fricke dated May 19, 1892. KLEIN 1979 [1926], p. 308. See also Aurel Voß, “Heinrich Weber,” Jahresbericht der DMV 23 (1914) Abt. 1, pp. 431–44. Schoenflies started as an associate professor on April 1, 1893, with an annual salary of 2,000 Mark, plus a 540 Mark housing allowance. [StA Berlin] Rep. 76 Va Sekt.6 Tit. IV No.1, vol. 15, fols. 192–92v (Althoff to von Meier, May 18, 1893). [UA Braunschweig] A letter from Klein to Fricke dated August 23, 1892.

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concerned an “old chest defect, dating back to 1857, which had recently caused a disturbance.” Klein spent four and a half weeks recovering from this in bed, but even from there, he gave fifteen lectures on number theory in May (see Section 8.2.2).8 Shortly thereafter, in June of 1893, he taught his lecture course on higher geometry (II) and began his (first) seminar on probability theory. On May 4, 1894, Felix Klein was elected as the dean (Dekan) of the Philosophical Faculty at the University of Göttingen for one year. He received eighteen of the thirty-one votes (thirteen faculty members had voted for the mineralogist Theodor Liebisch).9 Klein used this position to enhance mathematics and the natural sciences at the university, which involved Walter Nernst’s appointment as a full professor of physical chemistry, hiring Theodor Des Courdes to teach applied electricity, promoting the field of technical physics, and hiring, as mentioned above, David Hilbert. Writing on June 24, 1894 to Hurwitz, who had accepted a professorship at the Polytechnikum in Zurich in 1892, Klein downplayed his many activities, referring to them as “all sorts of miscellaneous things” (allerlei Allotria):10 I have become a bad correspondent. This is probably due to the fact that, in addition to my ongoing scientific activity, I have also taken on all sorts of miscellaneous things. Once more, I am attempting (probably in vain) to create a closer connection between the university and technology, for which your institution in Zurich is an illuminating model. Then, I have devoted much time to getting closer to secondary school teachers: among other things, in May I was in Wiesbaden at a conference of such gentlemen, and in any case I managed to convince them to hold their next meeting in Göttingen. Lastly, as if my glass wasn’t already full enough, I also have to serve as dean of the Faculty for one year, beginning on July 1st. I’ll be happy if I succeed in carrying on with my own writing in the previous way and if I can also continue to encourage the young people to conduct investigations that are dear to my heart.11

Klein was aware that his own creative work would suffer if he were distracted by all these other activities. However, his decision to take on “all sorts of miscellaneous things” proved to be one of the necessary conditions for the creation of an international center of mathematics, natural sciences, and technology in Göttingen. 7.1 KLEIN’S ASSISTANTS AND HIS PRINCIPLES FOR CHOOSING THEM As mentioned above, Felix Klein was the first mathematician at a German university for whom a government-funded assistant was approved; this was in October of 1881, while he was working at the University of Leipzig (see Section 5.2). Although an assistant had been assigned to professors of descriptive geometry at some Technische Hochschulen, Klein had to struggle at the Technische Hoch8 9 10 11

[UBG] Cod. MS. F. Klein 9: 248 (Klein to Hurwitz, June 4, 1893). [UAG] Phil.Fak. Protokollbuch (1889–1905), p. 102. Liebisch was made the dean in 1896/97. The term that Klein uses here, Allotria, is Greek ἀλλότρια ‘miscellaneous things’. [UBG] Math. Arch. 77: 253.

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schule in Munich in order to receive one assistant for both himself and Brill in 1877, as neither of them held a professorship in that specific discipline (see 4.1.2). Klein’s main argument then for justifying this position was that he needed someone to oversee the collection of models. Thus, in April of 1892, when Klein became the director of the collection of mathematical instruments and models in Göttingen, he used this same argument in order to be granted an assistant there: I would not be able to take over as the director of the model collection, however, unless I have an assistant at my disposal who could perhaps also supervise the reading room, which is now being done by a student, and who might be instructed to assist me in preparing copies of my lectures (for the reading room). I have had to spend a disproportionate amount of time on this task, which has been carried out in part by a changing set of students and in part by myself, and I have often wished to have an assistant who could properly guide new students, especially foreigners, through the large number of copies of my lectures that are already arranged side by side in the reading room. Although this latter task is more closely related to the reading room, I would nevertheless ask you to employ this assistant especially for the model collection, since the reading room is a general Seminar facility and is thus, in principle, under the control of the general directors of the Mathematical-Physical Seminar.12

On April 19, 1892, the Ministry gave its approval for hiring an assistant, who, in accordance with Klein’s wishes, was assigned to the model collection and thus answered to him alone. Three days later, Klein informed the Kurator that he would like to appoint Fritz Schilling to this position. The Ministry then authorized the appointment (retroactively to April 1, 1892), with “an annual remuneration of 600 M, under the condition that it could be terminated at any time with six weeks’ notice.”13 The contract further stipulated that the holder of the position would also, in addition to managing the collection of models and instruments, assume the duty of supervising the reading room. The candidate’s curriculum vitae and letters of reference were to be sent to the ministry. All of the applications for this position from 1892 to 1927 are archived today at the University of Göttingen.14 In addition to managing the model collection and the reading room, Klein’s assistants also had to transcribe and edit his lecture courses. Klein would review and approve these texts, which were reproduced in their handwritten form by the B.G. Teubner press as so-called “autographs.” Klein explained: “This form of publication […] is rather uncommon in Germany, but it is used quite often in other countries, especially in France and Italy.”15 These copies of Klein’s courses were available in the reading room for students, and Klein also sent them to several colleagues in Germany and abroad. Since the 1920s, new editions of some of his lecture courses have been published as printed books (see Section 9.2.3). When Klein received the aforementioned job offer from the University of Munich in July of 1892 (see Section 6.5.2), the Bavarian government included 1,500 Mark for an assistant. On this basis, Klein was able to negotiate a salary of 12 13 14 15

[UAG] Kur. 5691, fols. 1–8 (emphasis original). [UAG] Kur. 5961, fols. 15–15v. [UAG] Kur. 7554 (Assistants of the Mathematical Institute, 1892–1927). KLEIN 1923a (autobiography), p. 23.

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1,200 Mark for his assistant in Göttingen, which means that he was able to double Friedrich Schilling’s yearly pay. Schilling had already transcribed Klein’s lectures on non-Euclidean geometry and now, with more funding, he performed the same task for Klein’s lectures on higher geometry during the winter semester of 1892/93 and the following summer semester.16 Schilling remained Klein’s assistant until September 30, 1893; he then completed his probational year as a secondary school teacher and earned his doctoral degree (under Klein’s supervision) in 1894.17 After holding positions in Aachen and Karlsruhe, he returned to Göttingen in 1899 to succeed Arthur Schoenflies as an associate professor of descriptive geometry (see also Section 8.1.2). Klein’s second assistant in Göttingen, Ernst Ritter, began on October 1, 1893. He is yet another example of how Klein integrated talented students into his field of research (which was then the theory of automorphic functions) and supported their careers. Ritter had completed his doctoral degree under Klein in 1891, after which he began an internship as a secondary school teacher and continued to collaborate with Klein. As early as April of 1893, Klein attempted to secure a scholarship for him: Besides my current assistant, Schilling, […] Dr. Ritter was my best student during the seven years that I have now been in Göttingen. Whereas Schilling is more geometrically talented, Ritter’s efforts lie in penetrating conceptual analysis; he thus represents a mathematical type that I value especially highly as a complement to my own exclusively intuition-oriented approach. […] Despite our usual guidelines, I have been able to publish the entirety of his extensive dissertation in Mathematische Annalen on account of its special significance, and I have repeatedly presented his further studies on the same subject to our Society. Now, because his time here is running out, I am about to lose this distinguished man. […] In this case, please do not leave me in the lurch, but rather gratify me with a favorable decision.18

In his letter to Althoff, Klein also made a more general argument. He claimed that it was in the interest of secondary schools “to keep younger people engaged in science for longer periods of time” and that otherwise “poor” Göttingen would hardly be able to remain “competitive” with Berlin. In Berlin, he added, there were several institutes – such as the Imperial Institute of Physics and Technology (Physikalisch-Technische Reichsanstalt), which had been founded in 1888 and whose president was Hermann von Helmholtz – where university graduates could pursue careers in science. Such opportunities did not exist in Göttingen. Because this scholarship application for Ritter was unsuccessful, Klein requested instead that Ritter should be made his next assistant. 16 See https://gdz.sub.uni-goettingen.de/id/PPN595921786. A second edition of this text appeared in 1907, and this formed the basis of KLEIN/BLASCHKE 1926. 17 For his dissertation, see Friedrich Schilling, “Beiträge zur geometrischen Theorie der Schwarz’schen s-Funktion,” Math. Ann. 44 (1894), pp. 161–260. 18 [UBG] Cod. MS. F. Klein 1C: 2, p. 71 (a draft of a letter from Klein to Althoff, April 1893). For Ritter’s doctoral thesis, see Ernst Ritter, “Die eindeutigen automorphen Formen vom Geschlechte Null, eine Revision und Erweiterung der Poincaré’schen Sätze,” Math. Ann. 41 (1893), pp. 1–82.

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In this latter application, Klein emphasized “that I hope to gain in him a collaborator who might be suitable for joining our teaching staff later.”19 Ritter transcribed and edited Klein’s lecture courses of 1893/94 (hypergeometric functions) and of the summer semester of 1894 (second-order linear differential equations).20 In addition, Ritter completed his Habilitation in the summer of 1894,21 and he received a stipend to work as a Privatdozent in October of that year, so that the assistant position became vacant again. With his doctoral dissertation, Habilitation thesis, and other studies, Ritter contributed to the development of the theory of automorphic functions. Klein also sought his advice concerning the monograph on automorphic functions that he was preparing with Fricke. In the preface to the first volume of this book, the authors praise Ritter for his “painstaking and insightful investigations” and thank him for the “valuable preparation” that he provided for the organization of the second volume.22 Recommended by Klein, Ernst Ritter was offered a professorship at Cornell University in 1895 (see Section 7.4.3). Shortly after arriving in the United States, however, he died of typhus in a New York hospital. This was painful news for Klein; Ritter was one of the few people for whom Klein himself wrote an obituary.23 On October 1, 1894, the assistant position was taken over by Arnold Sommerfeld. He had earned his doctoral degree under Ferdinand Lindemann in Königsberg with a dissertation titled “Die willkürlichen Functionen in der mathematischen Physik” [Arbitrary Functions in Mathematical Physics] (1891), and he had worked as an assistant for the mineralogist Theodor Liebisch in Göttingen. Sommerfeld, however, was more interested in mathematics and, while working for Liebisch, he studied Klein’s lectures on the partial differential equations of physics in the reading room. Klein had already involved Sommerfeld (one of Klein’s intellectual heirs via Lindemann) in his work even before he chose him as his assistant. At Klein’s instigation, Sommerfeld presented the results of his dissertation to the Göttingen Mathematical Society (on December 5, 1893); afterwards, as Sommerfeld reported to his mother, Klein directed him as follows: Next, I was asked to give a lecture again, this time on recent French works. Klein organizes everything around him; he doesn’t have the time to read all these things, so he wants to hear a lecture about them. He has very cleverly thought out a particular area of work for me. On the basis of my previous lecture, he wants me to write a short article for Mathematische Annalen as soon as possible.24

19 [UAG] Kur. 7554, fol. 10. 20 Klein published summaries of these lectures in Math. Ann., and these were reprinted in KLEIN 1922 (GMA II), pp. 578–97. See also Klein’s remarks in KLEIN 1923 (GMA III), p. 741. 21 See Ernst Ritter, “Die multiplicativen Formen auf algebraischen Gebilden beliebigen Geschlechtes mit Anwendung auf die Theorie der automorphen Formen,” Math. Ann. 44 (1894), pp. 261–374; and “Die Stetigkeit der automorphen Functionen bei stetiger Abänderung des Fundamentalbereiches,” Math. Ann. 45 (1894), pp. 473–544; 46 (1895), pp. 200–48. 22 FRICKE/KLEIN 2017 [1897], p. xxviii. See also FRICKE/KLEIN 2017 [1912], p. xxvii. 23 Felix Klein, “Ernst Ritter †,” Jahresbericht der DMV 4 (1897) I, pp. 52–54 24 Quoted from ECKERT 2013, p. 80 (Sommerfeld to his mother, January 5, 1894).

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Whereas Sommerfeld soon regarded his earlier work with Liebisch as a waste of time, Klein appealed to him. Klein, as Sommerfeld told his parents, was “witty, knowledgeable, open, and honest.”25 It was through his discussions with Klein (even before he began as Klein’s assistant) that Sommerfeld discovered a topic for his Habilitation thesis, and he presented his preliminary results to the Mathematical Society in August of 1894. Klein submitted some of these early findings (on the integration of the partial differential equation Δu + k2u = 0 on a Riemann surface) to the Göttingen Society of Sciences for publication in the Göttinger Nachrichten.26 Sommerfeld completed his Habilitation in mathematics on March 11, 1895, while Klein was recovering his health in Montreux, Switzerland. That is, Klein trusted that his assistant Sommerfeld would be able to manage his Habilitation procedure without him. Klein also trusted Sommerfeld to oversee the renovation of the mathematical reading room (on the third floor of the lecture building [Auditorium]). This involved taking into account the expectation that the number of people who used the room would grow considerably, and it did (in the summer semester of 1895, it was used by approximately 35 people, and this number of users would increase to around 245 by the summer of 1905). In the case of Sommerfeld, Klein deviated from his “basic principle of keeping the same assistant for no longer than one year.”27 Sommerfeld remained in this position for two years. He prepared copies of Klein’s lecture courses on the theory of the spinning top and on number theory.28 About Klein’s courses, Sommerfeld later reported: “Painstakingly prepared, vividly presented, and a stylistically wellrounded masterpiece – every ten minutes, he summarized his thoughts in a concise form.”29 Although Sommerfeld also acknowledged that Klein could be a demanding taskmaster, he was nevertheless willing to sacrifice much of his free time for Klein’s projects. In addition to the project on the spinning top (see Section 8.2.3), there was also the ENCYKLOPÄDIE (see Section 7.8) and the task of preparing an index for the first fifty volumes of Mathematische Annalen. For this latter job, Sommerfeld was able to enlist the help of his wife Johanna, who was the daughter of Ernst Höpfner, the newly appointed Kurator of the University of Göttingen.30 Johanna Sommerfeld coined the telling term “Felix duty” (Felix-Dienst) to describe all this work.31 While collaborating on Klein’s projects, Sommerfeld transformed from a mathematician into a physicist: in 1897, he became a professor of mathematics at the Mining Academy in Clausthal; in 1900, he accepted a new position as a professor of mechanics at the Technische Hochschule in

25 Quoted from ECKERT 2013, p. 72 (Sommerfeld to his parents, June 27, 1894). 26 See Arnold Sommerfeld, “Zur mathematischen Theorie der Beugungserscheinungen,” Göttinger Nachrichten (1894), pp. 338–42. 27 [UAG] Kur. 7554, fol. 15. 28 This work on number theory was also supported by Klein’s student Philipp Furtwängler. 29 SOMMERFELD 1949, p. 289. 30 Höpfner took over this position on April 1, 1894. 31 See ECKERT 2013, p. 132.

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Aachen; and in 1906, he was ultimately hired as a professor of theoretical physics at the University of Munich. The Greek mathematician Constantin Carathéodory, who would become Klein’s successor in 1913, wrote the following about Klein’s assistants: Most of Klein’s assistants, who in principle changed every year, have become respected and eminent researchers. What was remarkable was the certainty with which Klein was able to choose someone from among his students to serve in this trusted position. Even more remarkable, however, was Klein’s skill in getting the most out of these young men, each according to the nature of his respective talents, and how in doing so he never impeded the development of the personality in question but rather promoted it. This extremely high level of worldly wisdom, which never failed in any of the cases known to me, is the key to understanding the unique influence that Klein exerted on the teaching, cultivation, and further development of mathematics in Germany.32

A few exceptions aside, Carathéodory’s assessment was correct. Among the eighteen assistants whom Klein hired from October 1, 1896 to September 30, 1921,33 there were a few who held the position for more than one year: C.H. Müller, R. Schimmack, A. Timpe, Ludwig Föppl,34 E. Hellinger, and a few others after the beginning of the First World War, when there was a shortage of available personnel. This lack of suitable candidates was probably responsible for Klein’s sole error in this regard: offering the position to Walther Graefe (see Section 9.2, Table 10). In general, Klein’s view was as follows: I choose the candidate who would most diligently and effectively fulfill the duties of being my assistant. Completing a doctoral degree in mathematics is something that can only be accomplished by concentrating on a single topic. The duties associated with serving as my assistant (a role which I always limit to a short time, usually one year) stand in contrast with this goal: given the way that things are currently done at our university, such duties prevent the candidate from working towards his doctorate. They promote the development of his general intellectual personality but not his scientific qualifications in a special area of research.35

Klein supported the careers of his assistants, but his letters of recommendation were not always panegyrics, as is evidenced from his evaluation of Moritz Weber. Weber was enthusiastic about working with Klein,36 but Klein was critical of his abilities. After completing his degree in architectural engineering at the Technische Hochschule in Hanover, Weber served as Klein’s assistant during the winter semester of 1896/97, after which he worked on such projects as creating an electric streetcar system in Berlin and building the water supply system at the Charlottenburg train station. In 1901, when Weber was being considered for a professorship in mechanics, Klein wrote:

32 33 34 35

CARATHÉODORY 1925, p. 2. For a list of Klein’s assistants, see KLEIN 1923 (GMA III) Appendix, p. 14. Ludwig Föppl, who was August Föppl’s son, completed his doctorate under Hilbert in 1912. [UAG] Kur. 7554, fol. 93 (a letter from Klein to the Kurator of the University of Göttingen dated September 6, 1906), emphasis original. 36 Moritz Weber, “Felix Klein,” Zeitschrift des VDI 69 (1925), p. 1118.

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Moritz Weber is a very pleasant and reliable man whom I would like to support whenever possible. I must say, however, that I do not consider him to be highly talented (despite or perhaps precisely because of the fact that, in his day, he had passed his exams in Hanover with such distinction). As my assistant, he had prepared my elementary lectures on integral calculus and differential equations, which I was giving to second-semester and third-semester students, for the reading room. He worked on this task with unbelievable diligence (4–5 hours a day!) without, however, being able to describe the essence of simple mathematical ideas in a clear and precise manner.

Klein proposed the following solution for hiring professors of mechanics: I see only one way that will slowly lead to improvements in this state of affairs, and this is for Technische Hochschulen to hire, when possible, professors of mechanics who are mathematicians with an open mind and an interest in technology, and for universities to hire, from time to time, engineers who are theoretically inclined.37

Moritz Weber was nevertheless hired as a professor of mechanics (by the Technische Hochschule in Hanover in 1904 and by the Technische Hochschule in Berlin-Charlottenburg in 1913). Yet Klein’s hiring guidelines would be followed in other cases. Examples include his former assistants Arnold Sommerfeld and Karl Wieghardt (see Section 8.2.4), who were hired as university-trained mathematicians by Technische Hochschulen, and the appointment of the engineer Ludwig Prandtl as a professor at the University of Göttingen (see Section 8.1.2). Felix Klein was not only the first, but also, for a long time, the only professor of mathematics at a German university to have an assistant. At first, Hilbert only had private assistants.38 In 1904, Hilbert received state funding to pay for a parttime (50%) assistant, while the remaining funding was allotted to H. Minkowski to cover half the costs of their common assistant at the time, Ernst Hellinger. Hellinger’s salary then was 600 Mark, but it was soon increased to 900 Mark. After Hellinger had completed his doctoral studies under Hilbert’s supervision, he then worked as Klein’s assistant from October 1, 1907 to March 31, 1909. This was a better-paying position, but it was more demanding as well.39 Carl Runge, who began working as a full professor of applied mathematics in Göttingen in 1904, had to make do with a part-time assistant as well, as did the aforementioned associate professor Friedrich Schilling. Only after holding a successful visiting professorship in New York (1909/10) did Runge apply for a full-time assistant, a request that was finally granted in 1912 with annual funding of 1,500 Mark.40 Internationally, too, the position of a mathematical assistant was still uncommon then at most universities.41

37 [UBG] Cod. MS. F. Klein 1D, fols. 7–8 (a letter from Klein to Otto Naumann at the Prussian Ministry of Culture, June 26, 1901), emphasis original. 38 See BORN 1978, p. 89. 39 Regarding Klein’s reputation as a demanding employer, it was humorously reported that he put his assistants “in fetters.” See TOBIES 2012, p. 77. 40 [UBG] Cod. MS. Hilbert 93, p. 5; and [UAG] Kur. 7554, fols. 112, 131, 151. 41 See SIEGMUND-SCHULTZE 2001, pp. 123, 158, 161.

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7.2 THE GÖTTINGEN MATHEMATICAL SOCIETY The Göttingen Mathematical Society, which Klein established with Heinrich Weber in the fall of 1892, brought together the professors of mathematics, Privatdozenten, assistants, and doctoral students for weekly lectures (at first usually on Saturdays) with discussions about recent research results.42 During the winter semester of 1892/93, the registered members of this society included (in addition to Klein and Weber): the full professor Ernst Schering, the Privatdozenten Heinrich Burkhardt, Robert Fricke, and Arthur Schoenflies, and Friedrich Diestel, who had completed his doctoral degree in 1890 under Schering with a dissertation on the calculus of interpolations.43 Franz Meyer gave one lecture as a guest from the Mining Academy in Clausthal (see Table 7). Table 7: Lectures at the Göttingen Mathematical Society, 1892/93 Prof. Heinrich Weber: 1) On the Theory of Abelian Functions of Genus p = 3. 2) Notes on Elliptic Modular Equations and on Invariants of Binary Biquadratic Forms. 3) Number-Theoretical Investigations from the Field of Elliptic Functions. Prof. Felix Klein: 1) On Number-Theoretical and Geometric Developments (Especially with Reference to Hermite and Selling) and on Lattice Theory. 2) On [Sophus] Lie’s Sphere Geometry and on the Geometric Developments Based on the Transformation of Higher Spatial Elements. Prof. Franz Meyer (Mining Academy Clausthal): On the Discriminants of Singularity Equations. Dr. Arthur Schoenflies: 1) Geometric Theory of Rectilinear Triangles. 2) A Report on Hilbert’s Invariant Theory. Dr. Heinrich Burkhardt: 1) A Contribution to the Theory of Vector Functions. 2) A Report on Schottky’s Book Die Abel’schen Functionen vom Geschlechte 3 [Abelian Functions of Genus 3]. Dr. Robert Fricke: 1) On Arithmetical and Group-Theoretical Developments in the Theory of Automorphic Functions. 2) On Form-Theoretical Methods in the Theory of Modular Equations.

As of the summer semester of 1893, two senior teachers at the Göttingen Gymnasium, Eduard Götting and Otto Behrendsen, also registered as members of the society, though they did not give any lectures themselves. Klein had an opportunity to build a good relationship with these two when he organized his first continuing education course for secondary school teachers in 1892 (see Section 7.3). 42 [UBG] Math. Arch. 191, fols. 171–86; and [UBG] Cod. MS. F. Klein 20H, 20L, 21B, 21G (Mathematical Society, 1907–1911, Klein’s notes). As of 1901, reports on the lectures given at these meetings appeared in the Jahresbericht der DMV under the section (Abt. 2) “Mitteilungen und Nachrichten” [News and Notices]. 43 Diestel later worked as a librarian in Göttingen and Hanover; see TOEPELL 1991, p. 84.

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They were to become important allies in Klein’s efforts to enact educational reform. Both were qualified to teach mathematics, physics, and the natural sciences, Behrendsen since 1878 and Götting (who was ten years younger) since 1884.44 Götting had earned a doctoral degree in 1887 under H.A. Schwarz – his dissertation was titled “Bestimmung einer speziellen Gruppe nicht-algebraischer Minimalflächen, welche eine Schar von reellen algebraischen Kurven enthalten” [Determining a Special Class of Non-Algebraic Minimal Surfaces that Contain a Family of Real Algebraic Curves] – and he had also attended Klein’s lectures on hyperelliptic functions (1887) and Riemann surfaces (1891/92).45 Additional members of the Society included Klein’s first assistants (Schilling, Ritter, Sommerfeld) and advanced students from abroad, who also attended Klein’s lectures and seminars (the American Bôcher; the Hungarian Emanuel [Manó] Beke; Alfred Loewy, who had been born in Rawicz near Poznań;46 Gino Fano from Italy; Poul Heegaard from Denmark; Charles Jaccottet from Switzerland; and the Ukrainian Vladimir P. Alekseyevsky).47 There were other participants as well: Friedrich von Dalwigk, a doctoral student of Heinrich Weber who worked for some time as Walther Dyck’s assistant at the Technische Hochschule in Munich; and Eduard von Weber, who in 1893 completed his doctoral degree under Dyck with a dissertation on the theory of differential equations.48 By the summer semester of 1895 – when Hilbert arrived – the society had thirteen members, including Georg Bohlmann, whom Klein had recommended to complete his Habilitation at the University of Göttingen in 1894 and to teach actuarial mathematics (see Section 7.6). By the summer of 1899, there were twentyeight registered members, now including Klein’s first female doctoral student Grace Chisholm Young and Hilbert’s first female doctoral student Lucy Bosworth. The circle continued to expand. Within this loose organization, which did not have any official statutes, Klein was able to have scientific discussions together with his fellow teachers in Göttingen, something which he had earlier tried to achieve in vain. He also used the society as a venue in which to coordinate common objectives and projects. A few of the aspects of Klein’s activity in this organization will now be described. First, Klein’s own lectures document his interests at the time: his turn toward number theory in 1892/93 (see Sections 6.3.4 and 8.2.2), his further analysis of Sophus Lie’s works, the results from his research seminars, and his ideas on a broad spectrum of mathematical applications (see Section 8.2.4). Klein also used this forum to discuss psychology (see Section 8.3.3) and mathematical education (see Section 9.3.2). 44 [BBF] Personalbögen [Personnel Files]. 45 [UBG] Cod. MS. F. Klein 7E. 46 Loewy continued his studies in 1894 and 1895 with Klein after he had completed his doctoral degree under Lindemann in Munich. 47 See V.P. Alekseyevsky, “On the Reciprocity Law of Prime Numbers” [in Russian], Proceedings of the Kharkov Mathematical Society 6/2 (1898), pp. 200–02. 48 See HASHAGEN 2003, p. 247.

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Figure 31: The Göttingen Mathematical Society, 1902 ([UBG] Cod. MS. K. Schwarzschild 23:1).

Second, Klein presided over meetings, prepared reports on scientific activities at the beginning of every semester, announced changes in the teaching corps at the university, and discussed the latest scholarly literature. In the meeting held on July 7, 1908, for instance, he drew attention to the “dissertation by Miss [Emmy] Noether,” which had been published in Crelle’s Journal.49 At the beginning of each semester, Klein asked: “Which topics should we discuss?” He stimulated numerous debates: in 1905, on Poincaré’s work (see Section 10.1); in 1908, on the “problem of flying”; in 1911, on the “relativity principle” (see Section 8.2.4); in 1918, on “Einstein’s theory” (see Section 9.2.2). Max Born was not the only participant to remark that presenters had to be prepared for critical comments.50 Third, Klein promoted younger mathematicians by inviting them to give presentations. Beneficiaries of this encouragement included Burkhardt, Fricke, and 49 [UBG] Cod. MS. F. Klein 20H, fol. 28. – See TOBIES 2006, p. 247; TOBIES/KOREUBER 2002. 50 See BORN 1978 [1975], p. 104.

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Sommerfeld. Tellingly, the Danish student Poul Heegaard reported that he had been left uninspired by a trip to Paris (Zeuthen had sent him there), but that he was subsequently greatly encouraged by Klein. Heegaard attended Klein’s lectures in the summer semester of 1894 (differential equations, elementary geometry), and he gave his first presentation in the context of Klein’s research seminar, reviewing works by Maxwell and Sylvester on spherical functions.51 Klein invited him to give lectures at meetings of the Göttingen Mathematical Society, and Klein ultimately led Heegaard to his dissertation topic: Klein had me give two lectures in the “Mathematische Gesellschaft” with a summary of Zeuthen’s work on enumerative geometry. He also discussed with me the idea that would later form the basis for my dissertation. Altogether, there was a scientific atmosphere which stimulated me very much – stronger than anything I have ever met again.52

Heegard’s doctoral thesis, which he defended in Copenhagen in 1898, turned out to be an important contribution to modern knot theory.53 Fourth, the Society proved to be an excellent venue for guest lectures. The first guest speaker was Klein’s “student” Franz Meyer from the Mining Academy in Clausthal, which was about 60 km away from Göttingen (see Table 7 and Section 4.2.4.2). Meyer proposed the idea of writing a joint book with Klein – “The Spirit of Modern Geometry, by F. Klein and F. Meyer”54 – and this in fact became the inspiration behind the ENCYKLOPÄDIE project (see Section 7.8). For special guests, Klein organized ceremonial meetings (Festsitzungen); thus Klein’s assistant Sommerfeld wrote in the society’s minutes book on June 10, 1895: “Festsitzung in honor of [Henri] Poincaré’s presence in Göttingen.”55 During this event, Klein commented in his introductory remarks on the successful Göttingen meeting of the teachers’ association (see Section 7.3). Next spoke Poincaré – “Über den Existenzbeweis des räumlichen regulären Potentials, wenn die Werte des Potentials auf einer Fläche S vorgeschrieben sind” [On the Existence Proof of a Regular Spatial Potential when the Values of the Potential on a Surface S are Prescribed] – and the recently appointed David Hilbert: “Die Grundzüge der Diskriminante des Galois’schen Zahlkörpers” [Fundamentals of the Discriminant of the Galois Number Field]. Klein planned additional ceremonial meetings (and speaking opportunities for himself) when eminent researchers spent time in Göttingen. The latter included the Dutch physicist H.A. Lorentz (Nobel Prize, 1902), who gave a series of lectures in October of 1910 in Göttingen that was funded by the aforementioned Wolfskehl Foundation (see 6.5.2). While the Nobel laureate Albert A. Michelson was spending a semester as a guest professor in Göttingen as part of a German51 [Protocols] vol. 12, pp. 1–4 (Heegaard’s presentation took place on May 1, 1894). 52 Quoted from http://www-groups.dcs.st-and.ac.uk/history/Biographies/Heegaard.html. Later, Heegaard would also contribute to the ENCYKLOPÄDIE (see DEHN/HEEGAARD 1907). 53 See EPPLE 1999. 54 [UBG] Cod. MS. F. Klein 10: 1245 (a letter from F. Meyer to Klein dated May 31, 1893). 55 [UBG] Math. Arch. 79: 1 (see also ECKERT 2013, p. 99).

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American exchange program, Klein organized a session of the Mathematical Society for May 2, 1911, as well as more casual gatherings: “This evening: Michelson at the Rohns Tavern” (August 1, 1911). Albert Einstein (see Section 9.2.2) and other renowned scholars were also welcomed as guest speakers by the Society. The minutes from the sessions reveal that Klein himself formally introduced many of the special guests. Among these were David Eugene Smith from the United States (on July 7, 1908) and Nadezhda N. Gernet from Russia (on July 21, 1908; August 9, 1909; and July 26, 1910). Smith was one of the translators of Klein’s book Famous Problems of Elementary Mathematics (see Section 7.3), and he became his close collaborator on the International Commission on Mathematical Instruction (ICMI; see Section 8.3.4). Gernet became a Dozent in St. Petersburg after she had attended Klein’s lecture course on function theory (1898/99), gave a lecture in his research seminar,56 and completed her doctoral studies under Hilbert (1901).57 She was a member of the German Mathematical Society. Fifth, beginning in the fall of 1895, this form of organization, which Klein had initiated, was carried on by Heinrich Weber at the University of Straßburg (as of 1918/19 Université de Strasbourg). There, Weber and Adolf Krazer formed a mathematical society “based on the Göttingen model,” as Weber informed Klein.58 This tradition is still alive at German mathematical institutes today. Even as a professor emeritus, Klein was still recognized as the driving force behind the Göttingen Mathematical Society, as is clear from a speech that Hilbert gave in 1918 on the occasion of the 50th anniversary of Klein’s doctoral degree: Indeed, we have the good fortune of having the best part of you, namely your entire personality. You are our society’s founder, chairman, intellectual heart, and main source of energy. We have also had the good fortune of seeing your scientific work emerge and reach maturity. […] The welfare of mathematics depends – unfortunately – not only on its scientific advancement but also and essentially on the non-scientific activity of its representatives. […] In this respect, you have now rendered services to mathematics such as no mathematician in Germany has ever done before, and you have brought mathematics to prominence by the splendor of your name and your personal prestige. As far as the Mathematical Society is concerned, I would like to thank you today and urgently ask you to wield your influence wherever it may benefit mathematics.59

Since the beginning of the twentieth century, a number of students considered Klein to be an “unapproachably great scholar” (unnahbar grossen Gelehrten), and they referred to him as “Felix Augustus.”60

56 57 58 59 60

[Protocols] vol. 15, pp. 113–31. See TOBIES 1999b; ABELE/NEUNZERT/TOBIES 2004, pp. 139–40. [UBG] Cod. MS. F. Klein 12: 210 (a letter from Weber to Klein, June 13, 1895). [UBG] Cod. MS. Hilbert 575: No. 2 (Hilbert’s notes, dated December 12, 1918). See LIETZMANN 1925, p. 257. The reference is to the first Roman emperor.

7.3 Turning to Secondary School Teachers

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7.3 TURNING TO SECONDARY SCHOOL TEACHERS It was an international trend at the time for teachers at secondary schools to form associations in order to promote their interests. In England, an Association for the Improvement of Geometrical Teaching had existed since 1871 (it was renamed the Mathematical Association in 1894). Later, Mathesis was formed in Italy (1901), the Association of Teachers of Mathematics (Vereinigung der Mathematiklehrer) in Switzerland (1902), the Association of Teachers of Mathematics in New England (1903), etc. The German organization was created in 1890: the Association for the Promotion of Mathematical and Natural-Scientific Instruction (Verein zur Förderung des mathematischen und naturwissenschaftlichen Unterrichts).61 Its members strove for an equal position to that of their colleagues teaching philological-historical subjects and for a practical and up-to-date curriculum. Early on, Klein had expressed his interest in the development of secondary education (see Section 3.2). To bring new findings into schools, he organized the first mathematical teacher continuing education course (Fortbildungskurs) to take place in October of 1892 during the fall break in Göttingen. He modeled it along the lines of similar courses in natural-scientific subjects that had already taken place in 1890 in Berlin and Frankfurt am Main. Klein’s course in 1892 was focused on “models and the theory of the top.” The interest in models does not need to be explained. The inclusion of the theory of the spinning top as material for further education can be explained by the fact that Klein had become aware, while in Paris in 1887, of Darboux’s edition of Théodore Despeyroux’s Cours de Mécanique (1884). Klein had used this book in his first seminar on top theory (1887) and now, in view of his contacts with teachers and engineers, he returned to this subject to examine a practical topic in more detail. In agreement with the Ministry, the mathematical and scientific training courses for secondary school teachers were held every two years. The fact that Klein decided to spend some time in the teachers’ classrooms shows how intensively he prepared himself. He did this in March of 1893 in Hanover, with ministerial approval. As he informed Robert Fricke, Klein learned that, in a continuing education course, it would be best to present only those things “which are of direct importance to secondary school teachers (and yet lie outside the scope of their school lessons).” Klein explained: “The motion of spinning tops, now the transcendence of π, later the basic concepts of geometry, especially parallel theory and the model collection,” and he stressed: “For teachers, a course on the axioms of geometry has more direct educational value than one on algebraic curves, and a course on probability theory has more than one on determinants.”62 In the second continuing education course in Göttingen, which took place during the Easter vacation in 1894, Klein lectured on “select questions of elementary geometry.” The focus was on the famous three classical geometric prob61 For further discussion of Klein’s role in this association, see TOBIES 2000. 62 [UA Braunschweig] A letter from Klein to Fricke dated May 10, 1894.

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lems that cannot be solved with a compass and a straight edge alone: the duplication of the cube; the trisection of an arbitrary angle; and the quadrature of the circle, which is not possible on account of the transcendence of π. Klein presented historical results on these topics as well as the latest research approaches. Later, in the spring of 1894, Klein traveled to Wiesbaden to attend the conference of the teachers’ Association for the Promotion of Mathematical and NaturalScientific Instruction. There he gave a report on his continuing education courses and, in coordination with the teachers Otto Behrendsen and Eduard Götting – whom he had already integrated into the Göttingen Mathematical Society (see 7.2) – he successfully arranged for the association’s next annual meeting to be held in Göttingen (June 3–6, 1895). Klein participated on the planning committee and in large part determined the conference’s program, for which he planned to give a plenary lecture and to prepare a commemorative text (Festschrift).63 This Festschrift, a small book – Vorträge über ausgewählte Fragen der Elementargeometrie (66 pages), which was given the English title Famous Problems of Elementary Geometry – received rapidly international recognition. The way it came about is interesting because it demonstrates Klein’s great skill at cooperative management. A senior teacher named Friedrich Tägert had participated in Klein’s continuing education course in early 1894, and even at that early stage he announced that he would be willing to edit the work. In preparation for this, Klein gave a lecture course on the topic during the summer semester of 1894 and asked the participants in the course to produce transcripts of its content. Klein sent these transcripts to Tägert in Bad Ems (Hesse-Nassau).64 Hurwitz later discovered that the proof of the transcendence of π which Klein had presented in this work (using Lindemann’s approach from 1882) was incomplete.65 Nevertheless, the topic attracted a great deal of interest. In his preface, Klein emphasized: The more precise definitions and more rigorous methods of demonstrations developed by modern mathematics are looked upon by the mass of gymnasium professors as abstruse and excessively abstract, and accordingly as of importance only for the small circle of specialists. With a view to counteracting this tendency it gave me pleasure to set forth last summer in a brief course of lectures before a larger audience than usual what modern science has to say regarding the possibility of elementary geometric constructions.66

The comment here about “a larger audience than usual” is a bit of an exaggeration, for this lecture course had consisted of just fourteen people – most of them

63 KLEIN 1895b. In English translation: KLEIN 1897 [1895]. 64 Friedrich Tägert was qualified to teach mathematics, physics, chemistry, minerology, botany, and zoology ([BBF] Personnel Files). 65 Hurwitz notified him “that your proof according to Gordan in the lectures edited by Tägert is not fully complete,” and he explained the details in [UBG] Cod. MS. F. Klein 9: 1131 (Hurwitz to Klein, Jan. 3, 1896). Klein reacted: “Your remark concerning π terrified me, and right now I don’t have the time to think through the details again.” [UBG] Math. Arch. 77: 262 (Klein to Hurwitz, Jan. 26, 1896). There were other, more elegant proofs by then. 66 Klein 1897 [1895], p. iv.

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from abroad, and two of them women.67 They clearly grasped the importance of making mathematics accessible at the secondary school level. One of the women involved was Klein’s British doctoral student Grace Chisholm, who already here internalized the idea that it would be valuable to produce books for elementary instruction (see also Section 5.6). Another participant was the Italian Gino Fano, who had translated Klein’s Erlangen Program and had already begun, in 1894, to propagate Klein’s pedagogical ideas in Italy.68 Likewise in Italy, Federigo Enriques was inspired by Klein’s small book to write a series of monographs on elementary geometry in collaboration with his friends and followers.69 The aforementioned Poul Heegaard (see 7.2) was also in attendance; while later working in Denmark and Norway, he became actively involved in international educational reforms.70 The American Charles A. Noble would return to Germany in 1926 to analyze the Prussian school system and specifically explain mathematics instruction for American teachers (see 9.3.2). A few years later, he translated (with Earle Raymond Hedrick) the first two volumes of Klein’s Elementary Mathematics from an Advanced Standpoint, though the numerous shortcomings of their work would necessitate the production of a new translation.71 Another participant, Wilhelm Lorey, was also a firm supporter of Klein’s pedagogical projects.72 Klein’s Festschrift was soon translated into Italian (1896), French (1896), English (1897), and Japanese (1897). The Italian translation was instigated by Gino Loria. The impetus behind the French translation was a letter to Klein from Jean Griess, a graduate of the École Normale Supérieure in Paris and a teacher at a Lycée in Algiers. He requested Klein’s permission to translate the book because he was “delighted by its content and clarity.”73 In coordination with Klein, Griess modified certain sections of the text for French readers. This French version served as the basis for the English translation, which was undertaken by Wooster W. Beman and David Eugene Smith.74 Its Japanese translator, Tsuruichi Hayashi, was regarded in his homeland as “the Felix Klein of Japan.”75 Loria, Beman, Smith, and Hayashi also became members of the German Mathematical Society. 67 The attendees were C.A. Noble, V. Snyder, W. Lorey, G. Fano, J. Wigger, C. Jaccottet, P. Heegaard, G.F. Metzler, J. Ehlers, L. Schütz, Miss G. Chisholm, Miss M.F. Winston, G.A. Campbell, and H. Siedentopf. [UBG] Cod. MS. F. Klein 7E. 68 See COEN 2012, pp. 210–45, esp. p. 214; and GIACARDI 2013. 69 See Livia Giacardi’s article in COEN 2012, esp. p. 225. 70 See https://www.icmihistory.unito.it/portrait/heegaard.php. 71 As G. Schubring explains in the new edition – Elementary Mathematics from a Higher Standpoint (KLEIN 2016; see the preface to vol. 1) – even the title of the original translation (… from an Advanced Standpoint) is somewhat misleading. Both Noble and Hedrick completed their doctorates under Hilbert in 1901; their dissertations concerned the Dirichlet principle. 72 See LOREY 1916. 73 [UBG] Cod. MS. F. Klein 11: 499D–E (a letter from Griess to Klein dated July 30, 1895). 74 Beman took part in the Mathematical Congress in Chicago in 1893; see MOORE et al. 1896, p. ix; KLEIN 1894, p. vii; and also Section 7.4. Regarding D.E. Smith, see also Section 8.3.4. 75 See OGURA 1956, pp. 145–47. Hayashi founded the Tōhoku Mathematical Journal, and he cofounded a mathematical institute on the basis of the Göttingen model together with M.  

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Klein’s activity in this field led to him being made a member of the advisory board (comité de patronage) of the first international journal for mathematical instruction: L’enseignement mathématique, founded in 1899 (see Section 8.3.4). Back to the meeting of the teachers’ association in Göttingen in June of 1895. Klein’s plenary lecture was entitled “On Mathematical Instruction at the University of Göttingen, Especially in Light of the Needs of Teaching Candidates.” Klein used this speech to underscore the benefits of studying in Göttingen. He spoke highly of the working conditions there, the curriculum, and the collaborative spirit of the faculty. He also stressed that the following things should be achieved when training future teachers: “1) A uniform foundation in elementary matters […]; 2) A scientific concentration in one specific field […]; 3) An overview of the importance of higher mathematics to secondary education.”76 Klein referred to the effort in Southern Germany, which had been ongoing for twentyfive years, to organize secondary education in a more intuitive way. In order to promote a similar undertaking in Prussia, he sent his report and his inaugural address from Leipzig, which had meanwhile been published,77 to Robert Bosse, who was then the Prussian Minister of Culture in Berlin.78 Following Klein’s suggestion, the Association for the Promotion of Mathematical and Natural-Scientific Instruction chose as the theme of its 1896 conference “the relation of mathematical instruction to the training of engineers,” which was then a widely discussed topic (see Section 7.7). In 1898, in fact, such discussions led to new examination regulations for teaching candidates (see Section 8.1.2). As of 1900, and in connection with these regulations, Klein focused the continuing education courses more strongly on the applications of mathematics. The lectures that were given in these courses in 1900 and 1904 were edited by Klein and the physicist Eduard Riecke as books.79 In the year 1909, Klein also initiated the first continuing education course for mathematics teachers at the newly established secondary schools for girls (see Section 8.3.4.1). Klein’s cooperation with the Association for the Promotion of Mathematical and Natural-Scientific Instruction turned out to be an influential factor in the educational reform that would be implemented, even though the organization’s longstanding chairman, Friedrich Pietzker (who died in 1916), opposed Klein’s wish for differential and integral calculus to be taught at secondary schools (see Section 8.3.4.1). On April 25, 1917, Klein was named an honorary member of this association. Not until 1924 was differential and integral calculus made a component of the Prussian curriculum, a cause for which Klein was still an active proponent at the time (see Section 9.3.2).  

76 77 78 79

Fujiwara, who attended Klein’s courses in 1909/10. See also KÜMMERLE 2018, KÜMMERLE 2021, and DAUBEN/SCRIBA 2002, pp. 289–95, 430–31, 440. His report was published in the Zeitschrift für mathematische und naturwissenschaftlichen Unterricht 26/5 (1895), pp. 3–8 (quoted here from p. 7). KLEIN 1895a. [UAG] Kur. 5956, fols. 77–90. See KLEIN/RIECKE 1900, and KLEIN/RIECKE 1904.

7.4 A Trip to the United States

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7.4 A TRIP TO THE UNITED STATES In 1883, Klein had turned down the opportunity to become Sylvester’s successor at Johns Hopkins University (see Section 5.8.1), and in 1889 he had to decline an offer to teach as a visiting professor at Clark University in Massachusetts (see Section 6.3.7.1). In 1884, three of Klein’s students – Dyck, Lindemann, and Wedekind – had tested out this foreign terrain.80 Finally, in 1893, the possibility presented itself for Klein to travel to the United States, on an official basis, to attend the World’s Fair in Chicago. With 3,000 Mark of funding from the Prussian Ministry, he was chosen to serve as one of two “commissioners” (Kommissare) tasked with “touring and reporting on the exhibition.”81 The state’s reoriented approach to policy would turn out to be a resource for Klein’s scientific goals. Under Wilhelm II, who had been the German Emperor and King of Prussia since 1888, the government’s cultural and educational policies became more closely attuned to developments abroad. Like Hermann von Helmholtz, Alois Riedler, and others, Klein was enlisted to present the results of German research in the New World. As early as January of 1893, Klein completed an article on mathematics to be included in a book titled Die deutschen Universitäten, which was commissioned by the Ministry of Culture (and edited by his Göttingen colleague Wilhelm Lexis) for the sake of being presented in Chicago.82 Klein was obviously emulating the preferred rhetoric of the day when, in retrospect, he would speak of America as the “greatest possible and most fortunate object of scientific colonization.” However, he was aware that Germany and the United States could mutually benefit from one another, as is clear when he wrote of “influencing the mathematical life of American universities increasingly from our side – certainly not to the detriment of our own effectiveness and vigor.”83 Klein’s former students in Chicago were eager for him to come. By placing him in the center of attention and by publishing his contributions, they increased his international stature even more. By observing the developments there with open eyes, Klein himself was able to adopt new ideas. 7.4.1 The World’s Fair in Chicago and the Mathematical Congress In Chicago, the World’s Fair was held from May 1 to October 30, 1893 to celebrate the four-hundredth anniversary of Christopher Columbus’s arrival in the New World in 1492 (the event was therefore called the World’s Columbian Expo-

80 [Lindemann] Memoirs, pp. 95–103; and HASHAGEN 2003, pp. 179–87. 81 Friedrich Schmidt (since 1920 Schmidt-Ott), an official at the Prussian Ministry of Culture, asked Klein whether he would like to accept this position. [UBG] Cod. MS. F. Klein 11: 721 (a letter to Klein dated January 13, 1893). 82 See KLEIN 1893. 83 SIEGMUND-SCHULTZE 1997b, p. 246 (a draft of Klein’s report to Althoff, October 11, 1893).

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sition). This was the first World’s Fair to include an (international) exhibition on all areas of education.84 Friedrich Schmidt (later named Schmidt-Ott), an official at the Prussian Ministry of Culture, had been placed in charge of creating an exhibit showcasing German education there. He soon sought the advice of Felix Klein, whom he had already learned to appreciate while sitting in for the ailing Kurator of the University of Göttingen from March to May in 1892.85 Together, Schmidt and Klein developed the idea that the German exhibit in Chicago should have a special focus on mathematics. Klein was largely able to delegate to others the organizational preparation leading up to the event. For an earlier exhibition in Munich, Walther Dyck had already created a catalog of mathematical models and instruments, and now he was (reliably but less enthusiastically) willing to come up with a new catalog concept for Chicago. For this, Dyck compiled representative excerpts from nearly all of German mathematical scholarship: textbooks, journal series, collected works, dissertations, and Habilitation theses.86 Wanting to emphasize both mathematics and its applications, Klein also initiated a Gauß-Weber exhibit with a presentation on their telegraph, among other things. Heinrich Maschke and Oskar Bolza, who were both professors at the recently established University of Chicago, fulfilled Klein’s wish of keeping all the materials in order on site. Klein focused mainly on preparing for the “International Mathematical Congress held in connection with the World’s Columbian Exposition,” which was being planned by E. Hastings Moore, Oskar Bolza, Heinrich Maschke, and Henry S. White. In May of 1893, while still recovering his health, Klein wrote personally on behalf of the Ministry to potential authors, asking if they would produce written contributions for Chicago, i.e., “short reports on recent and the latest developments in a given branch of mathematics,” which might attract particular interest in the United States.87 Thus, besides congress contributions by American authors, papers came from France (Ch. Hermite, É. Lemoine, M. d’Ocagne), Italy (A. Capelli, S. Pincherle), Russia (I.M. Pervushin from Kazan), Austria (M. Lerch, Eduard Weyr), and Germany. In addition to contributing himself, Klein had recruited sixteen German authors (Burkhardt, Dyck, Fricke, Heffter, Hilbert, Hurwitz, Krause, F. Meyer, Minkowski, Netto, M. Noether, Pringsheim, V. Schlegel, Schoenflies, E. Study, and H. Weber), who together made up a large percentage of the thirty-nine total authors represented in the Mathematical Papers Read at the International Mathematical Congress.88 However, only four foreigners participated in the congress in person: Klein, Eduard Study (who was there seeking a position; see Section 5.4.1), the Austrian astronomer Norbert Herz, and the Italian mathematician Bernardo Paladini. Neither Herz nor Paladini presented a paper.

84 85 86 87 88

See WERMUTH 1894, p. 969. See SCHMIDT-OTT 1952, p. 26. Regarding Schmidt-Ott, see also Section 9.4.1. DYCK 1892; DYCK 1893. See also HASHAGEN 2003, pp. 425–28. Quoted from FREI 1985, pp. 88–89 (a letter from Klein to Hilbert, May 19, 1893). MOORE et al. 1896.

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When Klein boarded the Lahn, a ship operated by Norddeutscher Lloyd,89 in Bremen on August 8, 1893, he had numerous manuscripts by his colleagues in his luggage. He reached New York on August 17th and arrived in Chicago one day later.90 Here, the International Mathematical Congress took place from the 21st to the 26th of August, and Klein was regaled as the “Imperial Commissioner” of the German government. He gave one of the opening addresses as well as the concluding presentation. On the second day, Klein was given the title of Honorary President and he was made a member of the Executive Committee. This allowed him to influence the further course of the conference, which involved the presentation of papers in person or in absentia.91 Moreover, on three afternoons (Tuesday, Wednesday, Friday), Klein was able to discuss and answer questions about the materials in the exhibit on German education. In the closing session, he was thanked “for his very valuable contributions to the proceedings of the Congress and for his interesting expositions of the mathematical material in the German University Exhibit at the Exposition.”92 The official German report, too, emphasized Klein’s enthusiastic commitment: “His demonstrations and lectures gave the mathematical exhibition real life. It was astonishing how many visitors were eager to see and study this part in detail.”93 The talks that Klein gave at the congress are noteworthy because they introduced so many visionary ideas to a new audience. In his opening address – “The Present State of Mathematics” – he commented on the stronger tendency of the time to unify branches of mathematics that had previously grown apart.94 He noted that this unifying trend was made possible by the general concepts of the function and the group, and he referred to the emergence of new sub-disciplines such as geometric number theory. He added that this same tendency also extended to the applications of mathematics, and in this regard he cited the examples of Schoenflies’s results in the field of crystallography, which were based on group theory (see Section 6.3.7.2), and Burkhardt’s results on the relations between astronomical problems and the theory of linear differential equations. Klein built a bridge between German and American mathematical achievements, and he proclaimed that mathematics was a global enterprise. He compared the tendency in Göttingen to “return to the general Gaussian programme,” to projects initiated by the German Mathematical Society, and to the unifying efforts in France that had been instigated “by the powerful influence of Poincaré.” In order for his global vision to be achieved, Klein recommended that mathematicians should join together to form “international unions.”

89 This shipping company became a paying member (28,000 Mark in all) of the Göttingen Association for the Promotion of Applied Physics and Mathematics (see Section 8.1.1). 90 Klein’s comments in JACOBS 1977 (“Personalia”), p. 6; and KLEIN 1922 (GMA II), p. 613. 91 For an overview of the conference program, see PARSHALL/ROWE 1994, pp. 328–30. 92 MOORE et al. 1896, p. xii. 93 WERMUTH 1894, p. 989. 94 MOORE et al. 1896, pp. 133–35; reprinted in KLEIN 1922 (GMA II), pp. 613–15.

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In his final talk, which was on the development of group theory over the previous twenty years, Klein commented yet again on the need for mathematics to take a practical turn: “For there is no question that, for the ongoing development of our culture, more and more of such men will be needed who are in full command of the scientific premises both on the technical side and on the mathematical-physical side.”95 This would serve as Klein’s program throughout his remaining years in Göttingen. 7.4.2 Twelve Lectures by Klein: The Evanston Colloquium Even before his trip to the United States, Klein had made arrangements with Henry Seely White to give an additional series of lectures while he was there. These lectures were published in 1894 as Lectures on Mathematics: The Evanston Colloquium.96 After completing his doctoral studies under Klein in Göttingen with a dissertation on Abelian integrals (1891), White was hired as an assistant professor at Northwestern University in Evanston, Illinois. While in Evanston, Klein stayed at White’s home, and his lecture series took place from August 24 to September 9, 1893. Three aspects of this event are noteworthy here. First, for two straight weeks – Monday to Saturday – Klein gave a lecture every day in English from 9 to 11 o’clock in the morning. The twenty-four people in attendance included, in addition to White, Oskar Bolza, Fabian Franklin, Heinrich Maschke, James E. Oliver,97 Eduard Study, Harry W. Tyler, and Edward Burr Van Vleck. The only woman in the audience was Mary F. Winston, who would become Klein’s doctoral student (see Section 7.5). Another participant in the colloquium, Edwin M. Blake, came to Göttingen in 1896 and attended Klein’s courses on technical mechanics and number theory.98 The mathematician Alexander Ziwet, born in Breslau (Prussia; now Wrocław, Poland), who was then a professor at the University of Michigan, edited Klein’s lectures in Evanston and discussed his work on them with Klein in the evenings after they were given. Second, these lectures were meant to provide basic overviews of various subjects, and they did not go into great detail. Klein discussed the developments of mathematics that he himself had experienced and partially determined. He used the occasion to emphasize, classify, and set apart his own research and that of his students, and to underscore his research methods. In his first lecture (“Clebsch”), he used his teacher as an important point of departure and then distanced himself from the Clebsch-Gordan approach to invariant theory, for he had Hilbert’s new approach in mind (see Section 6.3.7.3). Klein’s second and third lectures were both titled “Sophus Lie,” and here he elucidated Lie’s early work in order to draw

95 96 97 98

MOORE et al. 1896, p. 136. KLEIN 1894a. – See also PARSHALL/ROWE 1994, pp. 333–54. On the relationship between Klein and James E. Oliver, see COCHELL 1998. [UBG] Cod. MS. F. Klein 7E.

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a connection to his own work and that of his students, all the way up to Bôcher’s. The fourth lecture was “On the Real Shape of Algebraic Curves and Surfaces,” and on this topic he stressed not only earlier results but also Hilbert’s new approach to algebraic geometry.99 The focus of the fifth lecture, “Theory of Functions and Geometry,” was the hypergeometric function and its applications in astronomy and mathematical physics. In his sixth lecture, “On the Mathematical Character of Space-Intuition, and the Relation of Pure Mathematics to the Applied Sciences,” Klein expressed his vision for approaching research in a particular way.100 Years later, he still considered the vision outlined here to be relevant, for this was the only lecture from the Evanston Colloquium that he chose to reprint in his collected works. In the seventh lecture, he recommended that the proofs of the transcendence of the numbers e and π should become common knowledge, and he would make the same case not long thereafter in his continuing education courses for secondary school teachers in Germany (see Section 7.3). Klein used his eighth lecture, “Ideal Numbers,” to discuss the ideas of Kummer, Kronecker, and Dedekind and to highlight his own approach to geometric number theory (see Section 8.1.2). In his ninth, tenth, and eleventh lectures – “The Solution of Higher Algebraic Equations,” “On Some Recent Advance in Hyperelliptic and Abelian Functions,” and “The Most Recent Research in Non-Euclidean Geometry,” respectively – he also concentrated on his own fields of research. In this latter lecture on non-Euclidean geometry, he discussed Sophus Lie’s latest work and its relation to Helmholtz at greater length than he had done in his own lectures in Göttingen during 1889/90 and in his 1890 article in the Annalen. At that time, Klein was unaware of comments that Lie would make in the third volume of his book on transformation groups, which was published in the fall of 1893 (see Section 6.3.6). In his final, twelfth lecture – “The Study of Mathematics at Göttingen” – Klein explained the systematic approach to teaching that had been implemented at his home institution. He remarked that his own lectures often have “an encyclopedic character, conformable to the general tendency of my programme,” and that he regarded his students “not merely as hearers or pupils, but as collaborators.” He thus stated that he would be pleased to welcome more American students in Göttingen so long as they were sufficiently prepared to take an active part in his own research initiatives. Third, around seventeen years later, William F. Osgood, who had not attended Klein’s Evanston Colloquium, still considered these lectures to be useful for introducing young students to mathematics. For this reason, he persuaded the American Mathematical Society to republish the book in 1911. In his preface, he asked: “What is important in the development of mathematics?” Responding to his own question, Osgood referred to Klein’s instinct “for that which is vital to mathematics,” and he stressed that “the light with which his treatment illumines 99 In particular, he cited Hilbert’s article “Ueber die reellen Züge algebraischer Curven,” Math. Ann. 39 (1891), pp. 115–38. See KLEIN 1894a, p. 28. 100 KLEIN 1922 (GMA II), pp. 225–31. See also Section 8.3.2.

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the problems here considered may well serve as a guide for the youth who is approaching the study of the problems of a later day.”101 Osgood had a very strong command of the German language. He had not only completed his doctorate with Max Noether after being prepared by Klein, but he was also the sole author of the article on the general theory of analytic functions of one or several variables in the ENCYKLOPÄDIE (vol. II, 1901). His work on this subject resulted in his Lehrbuch der Funktionentheorie [Textbook on Function Theory], three editions of which were published by B.G. Teubner in Leipzig (1907, 1912, 1921).102 7.4.3 Traveling from University to University After his lecture series in Evanston, Klein remained in the United Stated for approximately four more weeks. Instead of spending this extra time sight-seeing, he used it to familiarize himself with the institutions and faculty members at a number of universities. His former students and colleagues made it possible for him to visit some of the most prestigious private universities in the country.103 Having left Illinois, Klein made his first visit to Cornell University in Ithaca, New York, where he had been invited by James E. Oliver. In 1889–90, while already a professor, Oliver (twenty years older than Klein) had attended his courses on non-Euclidean geometry in Göttingen, and later he sent a few male and female students to study under Klein.104 It was Klein’s relationship with Oliver that led to Klein’s student Ernst Ritter being hired by Cornell, as I mentioned earlier (see Section 7.1). While in Ithaca, however, Klein was especially impressed by Cornell’s Sibley College of Mechanical Engineering, which, since 1889, housed the world’s first department of electrical engineering.105 Accompanied by his student Edward Burr Van Vleck, who had just completed his doctoral degree in 1893 in Göttingen, Klein next traveled to Clark University in Worcester, Massachusetts, where he had been offered (but had to decline) a visiting professorship in 1889. His itinerary then took him to Harvard University in Cambridge, Massachusetts, where he toured the imposing Harvard College Observatory directed by Edward Charles Pickering.106 His next stop was the nearby Massachusetts Institute of Technology. In the state of Connecticut, Klein visited Yale University in New Haven, where he met the physicist Josiah Willard

101 KLEIN 1894a (quoted here from the 1911 reprint, p. v). 102 Osgood’s textbook can still be found today in the reading room of the Mathematical Institute at the University of Göttingen. Klein’s influence on this book is clear to see, for instance, in its section on Riemann surfaces (Ch. 3, § 8). 103 For an outline of Klein’s full itinerary, see PARSHALL/ROWE 1994, pp, 355–57. 104 These included Virgil Snyder, who earned a doctorate under Klein in 1895, and the Canadian Annie L. MacKinnon. See TOBIES 2020a, pp. 12, 13, 32, and also COCHELL 1998. 105 See SIEGMUND-SCHULTZE 1997b, p. 36. 106 Later, Klein would use this institution as a model for the Göttingen Association (see 8.1.1).

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Gibbs107 (and where Klein would be offered a professorship in 1896). From there he went to Wesleyan University in Middletown, where the father of his companion, John Monroe Van Vleck, worked as a professor of astronomy. In New York City, where newspapers reported enthusiastically about his visit, the Mathematical Society organized a meeting in Klein’s honor at Columbia College.108 The meeting was chaired by the actuarial mathematician Emory McClintock, and it was here that Klein was inspired to establish actuarial mathematics as a course of study in Göttingen (see Section 7.6). In addition, Simon Newcomb arranged for Klein to meet with Daniel Gilman, the president of Johns Hopkins University in Baltimore, who, ten years earlier, had invited Klein to become Sylvester’s successor and who had presented Klein with a bronze copy of the gold medal with which Johns Hopkins had honored Sylvester upon his departure.109 The final stop on his tour was the College of New Jersey (Princeton University as of 1896), where Klein was hosted by his Leipzig doctoral student Henry Burchard Fine, who was then a professor there, and Henry Dallas Thompson, who in 1892 had completed his doctoral degree under Klein in Göttingen.110 Already during this meeting in 1893, they agreed that Klein should return in 1896 and give lectures on the occasion of the university’s 150th anniversary.111 7.4.4 Repercussions From the 7th to the 17th of October in 1893, Klein returned from New York to Bremerhaven aboard the Saale (another ship in Norddeutscher Lloyd’s fleet). As it turned out, his trip to the United States had a variety of effects. First, Klein’s stature increased in the international scientific sphere. It should be said here that Klein himself was pleased with how things went at the Congress and the Evanston Colloquium. On September 12, 1893, he wrote to Friedrich Althoff (Prussian Ministry of Culture in Berlin) from Illinois: So far, I have achieved everything here that I could hope to achieve. The Mathematical Congress under the German flag went very well; I have enclosed a copy of the program. Then I held the planned colloquium here in Evanston for fourteen days, and it was well attended by scholars from all over the country. By speaking for more than two hours a day, I had the

107 Klein celebrated J.W. Gibbs’s use of Lagrangian formalism in his work on physical chemistry (see Klein 1979 [1926], p. 226). Gibbs also made contributions to the theory of Fourier series; and he made a breakthrough in vector analysis by drawing a connection between Graßmann’s extension theory and Hamilton’s approaches (see KLEIN 1927, vol. 2, pp. 38, 45–47). 108 Founded in 1754, Columbia College is the oldest undergraduate college of Columbia University. It is located on the university’s main campus in Manhattan. 109 See HASHAGEN 2003, p. 182. 110 For his doctoral thesis, see Henry Dallas Thompson, “Hyperelliptische Schnittsysteme und Zusammenordnung der algebraischen und transzendenten Thetacharakteristiken,” American Journal of Mathematics 15 (1893), pp. 91–123. 111 See PARSHALL/ROWE 1994, p. 357; KLEIN 1897; and Section 8.2.3 below.

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7 Setting the Course, 1892/93–1895 opportunity to develop a complete program of my scientific views, as I had wanted to do for many years. These lectures will be published, and I hope to be able present finished copies soon after my return.112

News about the Congress spread quickly. Hurwitz expressed his happiness about the latest issue of the Bulletin of the New York Mathematical Society, from which he gathered that Klein had been the “intellectual focus of the Congress.”113 With the event in Chicago, Klein saw that a shift was beginning to take place in German-American cultural policy. Whereas, previously, American students had come to Germany without any special initiative from the German side, his appearance in the States had actively promoted this kind of exchange. Already planned in 1893, his trip to Princeton University in 1896, where he would give five lectures on the theory of the top (see Section 8.2.3), happened in the same way. Bernhard vom Brocke has already made the case that Klein’s trips to the United States served as a starting point for a joint German-American scientific policy that ultimately led to an official professorial exchange program.114 An outward expression of Klein’s recognition in the United States came in the form of an honorary doctorate from Princeton in 1896. In that year, too, he was made a member of the New York Academy of Sciences. In 1898, Klein was elected as a foreign associate member of the National Academy of Sciences of the United States of America in Washington; the chemist Oliver Wolcott Gibbs was the Academy’s president from 1895 to 1900 (see Fig. 44). Moreover, in 1904, John Throwbridge, a physicist at Harvard University and vice-president of the American Academy of Arts and Sciences, founded in Cambridge in 1780, wrote to Klein to inform him that he had been chosen to replace the late Cremona as a Foreign Honorary Member in the Academy’s Mathematics and Astronomy Section.115 The Russian mathematician A.V. Vasilev, a member of the German Mathematical Society since 1893, wrote to Klein in March of 1894 to tell him that he had received Klein’s “Lectures on Mathematics from America and read it with pleasure.”116 This was followed by Vasilev’s participation at the annual meeting of the German Mathematical Society in Vienna (September 1894), by further cooperation between Klein and Vasilev (see Section 6.3.6), and by Klein receiving the Golden Lobachevsky Medal in 1897. Klein’s visit to the United States had an especially strong effect in Paris. Hermite, whose paper “Sur quelques propositions fondamentales de la théorie des fonctions elliptiques” was presented in Chicago and published in the Congress’s 112 113 114 115

[UBG] Cod. MS. F. Klein 1B: 2, p. 80 (draft letter from Klein to Althoff, Sept. 12, 1893). Ibid., 9: 1123 (a letter from Hurwitz to Klein dated November 29, 1893). See VOM BROCKE 1981. In a letter expressing thanks (dated May 28, 1904), Klein stressed that he “had been indebted and grateful to Cremona since his early years” and that he “had always regarded Cremona as a model because he devoted his creative energy just as strongly to his scientific research as he did to administrative duties in the service of science.” [UBG] Cod. MS. F. Klein 114: 36. 116 Ibid., 12: 199 (a letter from Vasilev to Klein dated March 14/26, 1894).

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proceedings (pp. 105–15), took notice of Klein’s Evanston Colloquium and immediately had his seventh lecture (on the transcendence of e and π) translated into French. Enthusiastic about this work, Hermite then arranged for the French translation of the entire Evanston Colloquium and additional works by Klein.117 The translator, Léonce Laugel, corresponded with Klein on Hermite’s behalf, received Klein’s permission to have the works translated, and kept Klein informed of Hermite’s reactions. Most of these translations appeared in the Nouvelles annales de mathématiques, journal des candidats aux écoles polytechnique et normale, after Hermite had reviewed them. Having read a lecture that Klein had given in Vienna in 1894118 – “Riemann und seine Bedeutung für die Entwicklung der modernen Mathematik” [Riemann and His Significance for the Development of Modern Mathematics]119 – Hermite remarked that it was like spending an hour in heaven.120 Hermite inspired the French translation of Riemann’s collected works, and Klein agreed to the request that his Vienna speech on Riemann may be printed as an introduction.121 On December 26, 1895, Hermite congratulated Klein on his election to the Academy of Sciences in St. Petersburg and sent the message (via Laugel) to Klein that the Academy in Paris intended to do the same at the next available opportunity. Klein was elected on May 17, 1897, taking the place of the late J.J. Sylvester.122 All told, Klein would become a member of fifty-one academies and societies. Second, because of his international reputation, Klein was also able to achieve an exalted position within the Society of German Natural Scientists and Physicians (GDNÄ). In 1894, he was invited to give the aforementioned plenary lecture 117 For further discussion of this development, see TOBIES 2016, pp. 116–23. 118 This 1894 conference in Vienna played an important role in several contexts, and it will be mentioned on several occasions below. For this reason, it should be noted here that Klein gave two lectures there. Klein’s talk on Riemann was delivered on Wednesday, September 26, 1894 at the plenary session of the GDNÄ. Within the framework of the German Mathematical Society, he lectured on second-order linear homogeneous differential equations between two variables, published in the Jahresbericht der DMV 4 (1897) II, pp. 91–92. 119 [KLEIN 1894b]. Jahresbericht der DMV 4 (1897) II, pp. 71–87; reprinted in KLEIN 1923 (GMA I), pp. 482–97; and translated into English, Italian, and French. 120 [UBG] Cod. MS. F. Klein 10: 627 (Laugel to Klein, Oct. 1, 1895): “Monsieur et très honoré Professeur, Je prends la liberté de solliciter votre autorisation, nécessaire à la publication dans les Nouvelles Annales de Mathématiques d’une traduction de ‘Transcendance des nombres e et π’. Evanston Colloquium. C’est M. Hermite pour l’usage personnel duquel j’ai traduit tout le Colloquium et pour qui je traduis en ce moment avec le plus grand plaisir et intérêt Riemannsche Flächen et Hypergeometrische function [sic] qui m’a suggeré cette idée, dans le vif désir de porter les études sur ces admirables productions, trop peu étudiées en France. Je lui ai traduit aussi le discours ‘Über Riemann und seine Bedeutung’ dont il m’a écrit qu’en le lisant ‘il avait passé une heure comme dans le ciel’. En autoriseriez vous aussi la publication. […].” 121 Œuvres mathématiques de Riemann, traduites par L. Laugel, préface de Charles Hermite et discours de Felix Klein (Paris: Gauthier-Villars, 1898); see also Figure 35 (p. 464). 122 “Nous sommes heurex d’avoir à vous annoncer que, dans la Séance de se jour, l’Académie vous a nommé à la place de Correspondant, devenue Vacante dans la Section de Géométrie, par suite du décès de M. Sylvester.” ([UBG] Cod. MS. F. Klein 114: 25).

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(on Riemann’s significance) at the organization’s annual meeting, which was held in Vienna. He informed Dyck on August 12, 1894: “I have been asked to give a lecture in Helmholtz’s place, and I have agreed to do so. I intend to speak about Riemann’s significance.”123 This represented a passing of the torch from Helmholtz to Klein, both formally and in terms of content. In 1893, the two men had traveled to the United States and back on the same ship, though they attended different conferences in Chicago.124 In his report to Leo Koenigsberger, Klein mentioned that his conversations with Helmholtz on their return journey, which had taken place with the ship’s captain in the smoking room, were pleasant enough but that their content was somewhat backward-looking. Klein had tested out various topics of discussion – including the problem of axiomatics (see Section 6.3.6) – and he noticed that Helmholtz had taken hardly any interest in his Evanston Colloquium (Klein already had the proofs of this publication with him on the ship). Furthermore, even though Helmholtz was the president of the Imperial Institute of Physics and Technology, he did not deem it necessary for technical physics to be taught at German universities, an idea that Klein had brought back with him from the United States.125 Helmholtz suffered an accident on the ship. One year later, after suffering a series of heart attacks, he died on September 8, 1894. Klein used his talk in Vienna to spread his vision of unifying disciplines. The topic of Riemann allowed him to paint a broad picture of the development of mathematics and its applications. In particular, Klein compared Riemann’s and Weierstrass’s different approaches to function theory, and he stressed that these approaches complemented one another. Because Riemann’s approach was based on physics, Klein was able to build a bridge between Riemann and theoretical physics, including the results which had been achieved by British researchers (Faraday, Green, Maxwell), French scholars (Cauchy, Hermite, Picard, Poincaré), and others.126 At the meeting in Vienna, the Society of German Natural Scientists and Physicians appointed Klein to its board, as the successor of Koenigsberger.127 Third, Klein’s trip to the United States in 1893 also influenced the measures that would be taken to develop the University of Göttingen. On December 10, 1893, Klein had sent a comprehensive report on his travels to the Ministry of Culture. Even from Althoff’s short response to this report, we can gain a clear picture of Klein’s closely interconnected goals: Berlin – December 12, 1893 Esteemed Professor! Your valuable report from the 10th of this month was extraordinarily interesting to me. Today, however, I can only respond to it in brief […]. What you have written about women studying 123 124 125 126 127

[BStBibl] Dyckiania (a letter from Klein to Dyck dated August 12, 1894). Helmholtz attended the International Electrical Congress in Chicago (August 21–25, 1893). See KOENIGSBERGER 1903, vol. 3, pp. 93–94. See Jahresbericht der DMV 4 (1897) II, pp. 71–87; and KLEIN 1923 (GMA III), pp. 482–97. See the Tageblatt [Newsletter] of the Society’s meeting in Lübeck (September 16–21, 1895), p. 9: https://epub.ub.uni-muenchen.de/11458/1/4H.lit.2300_67.pdf; and FREI 1985, p. 111.

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at universities is fully in accordance with my views. Regarding teaching candidates, I will speak with the gentlemen in our secondary-education department. At first glance, your ideas about building relationships to technology seem plausible to me, and Göttingen seems like an entirely appropriate place to make such an attempt in the direction that you have endorsed. As you yourself remarked, however, all of this will require further consideration and can only be implemented if the financial conditions will allow for it. It would therefore be better if personnel issues were not raised in connection with any of this. Apart from that, I will wait to revisit the matter in a personal conversation with you. With the utmost respect, Sincerely yours, Althoff128

Althoff did not revisit the matter of his own accord. Instead, Klein had to work actively to achieve his vision. In order to receive a green light to pursue his goals – ambitions that were strengthened by his time in the United States – he regularly had to write to Althoff and call attention to his main points, which involved the right of women to study at universities, establishing relationships between secondary and higher education, and focusing on technical areas of research. 7.5 THE BEGINNINGS OF WOMEN STUDYING MATHEMATICS129 After Sofya Kovalevskaya, with Weierstrass’s support, had earned her doctoral degree in 1874 in Göttingen with the highest distinction (summa cum laude), twenty-one years passed before the next woman, thanks to Klein, would be awarded a doctorate in mathematics in Germany, incidentally from this same institution. It is a little-known fact that Klein had already admired Kovalevskaya’s results as a young man. When he had read her dissertation,130 he recommended to Sophus Lie, whose work he had just accepted for publication in Mathematische Annalen,131 that a reference should be included to Kovalevskaya’s work: What do you think about the work of Sophie Kowalevsky [Sofya Kovalevskaya] in Borchardt’s journal? By a direct series development, she proves the existence of integrals as well as their determination within certain boundaries. Indeed, it is always irritating that an integral M, thought to be unlimited, runs through the whole R [Raum = space]. Could a remark not be made about this in the revision of your article?132

Sophus Lie did not add any reference to Kovalevskaya, whose results went beyond what Lie and also Darboux published on the topic in the same year. Weierstrass recognized the precedence of Kovalevskaya’s (his student’s) ideas,133 and 128 [UBG] Cod. MS F. Klein 2 A, pp. 1–2. 129 For further discussion of this topic, see TOBIES 1991b; 1999b; 2020a, and 2021a. 130 Sophie von Kowalevsky, “Zur Theorie der partiellen Differentialgleichungen,” Journal für die reine und angewandte Mathematik 80 (1875), pp. 1–32. – See also TOLLMIEN 1997. 131 Sophus Lie, “Allgemeine Theorie der partiellen Differentialgleichungen erster Ordnung,” Math. Ann. 9 (1875), pp. 245–96. 132 [Oslo] A letter from Klein to Lie dated July 8, 1875. 133 See BÖLLING 1993, p. 198.

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her innovation is still reflected today in the name of the Cauchy-Kovalevskaya theorem. Meanwhile in Göttingen, there had in fact been a few scattered attempts to promote the higher education of women. At the beginning of the 1880s, H.A. Schwarz, following the model of his teacher Weierstrass, considered training an American woman on a private basis. Hurwitz informed Klein: Not much new is happening here in Göttingen. Regarding mathematics, perhaps the most interesting thing is that we now have a female colleague here who wants to further her education with Prof. Schwarz. She is an American woman named Miss [Ella C.] Williams; I met her yesterday evening at Schwarz’s home, and I was quite surprised to encounter a young lady of about 22 to 26 years old who leaves an impression that is nothing short of emancipated. Then I was personally very ashamed when I heard that Miss Williams reads Latin treatises with ease – something which I still have some trouble doing myself.134

Whereas women in the United States had already won access to certain universities,135 and women in other German states had been permitted to attend university courses as auditors, women in Prussia, the largest German state with the most universities, were not allowed to attend any university. Beginning in 1869, for instance, Kovalevskaya had studied in Heidelberg (Baden) under Weierstrass’s former student Koenigsberger, but Weierstrass was not allowed to let her enroll in Berlin (Prussia).136 In July of 1891, Klein had to turn away the American Ruth Gentry when she asked whether she might be able to study under him. Likewise, Christine LaddFranklin, who had been Sylvester’s student at Johns Hopkins, was not allowed to attend Klein’s courses. In the fall of 1891, she had come to Göttingen with her husband Fabian Franklin (see Section 6.3.7.3). The university’s conservative Kurator at the time, the legal scholar Ernst von Meier, had brusquely rejected this idea with the following remark to Klein: “This is worse than social democracy, which only wants to do away with differences in property. They want to abolish the difference between the sexes!”137 On April 8, 1893, when Heinrich Maschke wrote to Klein from Chicago to ask whether Klein might be able to obtain permission for one of his talented female students to enroll in Göttingen, Klein addressed the request directly to the Prussian Ministry of Culture to evaluate, during his planned trip to the United States, the situation of women studying mathematics. The Ministry responded quickly, for in 1891 the topic women’s access to higher education had been discussed in the Reichstag on account of the growing influence of the women’s movement. On May 20, 1892, Althoff had created a new dossier with the title 134 [UBG] Cod. MS. F. Klein 9: 937/3 (a letter from Hurwitz to Klein dated January 18, 1883). On Ella C. Williams, see PARSHALL 2015, p. 75. 135 See ROSSITER 1984; FENSTER/PARSHALL 1994; and PARSHALL 2015. 136 For a detailed discussion of Kovalevskaya’s graduate studies, see TOLLMIEN 1997. 137 Quoted from JACOBS 1977 (“Personalia”), p. 6: “Das ist schlimmer als die Sozialdemokratie, die nur den Unterschied des Besitzes abschaffen will. Sie wollen den Unterschied der Geschlechter abschaffen!”

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“The Request on the Part of Women for Permission to Matriculate and Attend Lectures at the Royal State Universities.” Before Klein’s departure to the United States, Friedrich Schmidt(-Ott) informed him (in a letter dated July 30, 1893): With respect to women studying, I can confidentially say that, as I know from Privy Councillor Althoff, the Ministry will not hinder the matter, although it will not especially encourage such questions. Regarding their [women’s] participation in lectures, this custom will also become more entrenched than limited; and if American women come to study in Germany, they will not have difficulties here. Mr. Althoff is of the opinion that, without asking, you could just arrange for your numerous female American admirers to come over.138

The student recommended by Maschke, Mary Frances Winston, who was then an A.B. honorary fellow of mathematics at the University of Chicago, attended both the 1893 International Mathematical Congress and Klein’s Evanston Colloquium.139 When Klein learned in Chicago that Winston had full financial support for her studies,140 he immediately wrote to Althoff hoping that the Prussian Ministry of Culture would do everything in its power – “despite the legal regulations still in place” – to allow Winston to be admitted as a visiting student during the winter semester of 1893/94. Klein stressed that the other directors of the Mathematical-Physical Seminar in Göttingen would support him and that Bolza and Maschke regarded Winston as “the best student of mathematics at the University of Chicago.” Anticipating a negative response from the university’s Kurator, Ernst von Meier, Klein added: “I would like to ask you, if possible, to present the matter […] in such a way that Mr. von Meier, the Kurator, has the opportunity to express his dissenting view at an early stage and is in no way left with the impression that I have tried to circumvent him.”141 Klein was still on his way home when three women applied to study at the University of Göttingen: Mary F. Winston; Grace E. Chisholm, who was recommended by Forsyth in England; and the American Margaret E. Maltby, who would earn a doctoral degree in physics under the supervision of Walther Nernst. Felix Klein’s wife Anna took these students under her wing, welcomed them into her family (see Fig. 32), and introduced them to the city and the university, where Klein’s assistant at the time, Ernst Ritter, took care of them. Having returned from the United States, Felix Klein immediately accepted the official applications from these women and asked to have them forwarded to the Ministry in Berlin. Although the Kurator Ernst von Meier appended a cover letter to these applications (dated October 21, 1893), in which he stated that he considered it “very worrisome to defy the existing regulations in favor of three foreign

138 [UBG] Cod. Ms. F. Klein 11: 726 (Schmidt(-Ott) to Klein, July 3, 1893). 139 Another woman who participated in the Congress in Chicago was Charlotte C. Barnum, who, in 1895, became the first woman to earn a doctoral degree in mathematics from Yale University. See MOORE et al. 1896, p. ix. 140 Winston was financially supported by her own family and by Christine Ladd-Franklin. See PARSHALL/ROWE 1994, p. 243. 141 Quoted from TOBIES 2020a, p. 26 (a draft of a letter from Klein to Althoff, Sept. 12, 1893).

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women,” the Prussian Ministry of Culture nevertheless granted its approval within five days.142 This was just one of several instances in which the university Kurator felt that his opinion had been overruled. In February of 1894, he resigned from his position because he was “disgruntled by the regiment enforced by the shadow Kuratoren [Wilhelm] Lexis and Klein, who are supported by Althoff.”143 With the new university Kurator, the already mentioned Ernst Höpfner, Klein finally had an ally who supported his initiatives.

Figure 32: Grace Chisholm and Luise Klein [Hillebrand]

A good deal has already been written about the participation of women in Klein’s seminars, their growing numbers from various countries (the United States, Great Britain, Russia, etc.), the career paths of academically trained women, and Klein’s encouragement of their work.144 Here I would like to underscore six aspects of Klein’s general attitude toward women studying at the university level. First, until 1908, women studying in Prussia only had the status of auditors. That meant that every woman had to ask each professor individually for permission to attend and then had to apply for admission from the Ministry of Culture. In many cases, Klein personally helped to formulate applications. During his time serving as dean in 1894/95 the number of women applying to study finally increased to such an extent that the Ministry ceded admission decisions to the universities and simply requested lists of the women who had enrolled. In 1893, Klein had expressed his support for the English system (with extra colleges for women within universities). Yet after he had taught women in his own courses, he endorsed their equal right to participate in all university classes. In part because of Klein’s support for this development, the government decided, 142 See TOBIES 1991b, pp. 154–55. 143 Quoted from BROCKE 1980, p. 70. 144 For recent articles in English on this topic, see TOBIES 2020a and TOBIES 2021b.

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on August 18, 1908, to allow women to matriculate as official students at Prussian universities. After all, such legislation had already been passed in the states of Baden (1900), Bavaria (1903), Württemberg (1904), Saxony (1906), Saxon-Ernestine duchies (1907),145 and Hesse (May 29, 1908). The last German state to follow was Mecklenburg in 1909.146 Second, Klein regarded mathematical achievement independently of gender. Although there were still many people in Germany who opposed the right of women to study, few of these opponents were mathematicians, and this is because achievements in this discipline can be evaluated rather objectively. In 1897, Arthur Kirchhoff published an anthology titled Die akademische Frau [The Academic Woman], in which he collected opinions about women’s access to higher education from supporters, opponents, and from those who argued on behalf of making special exemptions. The latter camp included Max Planck, whose oftencited comment – “In the intellectual realm, too, Amazons are contrary to nature” – implies that it is just as unnatural for women to participate in higher education as it is for them to bear arms. Klein, on the contrary, was unambiguously supportive on this issue: I am all the more pleased to answer this question as the opinion prevailing in Germany, which is that the study of mathematics must be virtually inaccessible to women, essentially blocks all efforts directed toward the development of women’s higher education. In this regard, I am not referring to extraordinary cases, which as such do not prove very much, but rather to our average experiences in Göttingen. Though this is not the place to enter more deeply into the matter, I would simply like to point out that during this semester, for instance, no fewer than six women have participated in our higher mathematics courses and practica and have continually proven themselves to be equal to their male classmates in every respect. The nature of the situation is that, for the time being, these women have been exclusively foreigners: two Americans, an Englishwoman, and three Russians, but certainly no one would wish to assert that these foreign nations possess some inherent and specific talent that we are lacking, and thus that, with suitable preparation, our German women should not be able to accomplish the same thing.147

The lack of “suitable preparation” mentioned here would later incite Klein to induce reforms for girls’ secondary schools in Germany (see Section 8.3.4.1). Third, foreign women paved the way for German women; they had access to education preparatory for university before this was offered at German schools for girls. Klein supervised more than fifty doctoral students, and around thirteen of these came from abroad. This latter group included Grace E. Chisholm from England and Mary F. Winston from the United States. Winston’s first presentation in Klein’s seminar took place on December 13, 1893, when she discussed the results of Oskar Bolza (her former teacher in Chicago) on “The Connecting

145 This region, known as Thuringia since 1920, consisted then of four Saxon-Ernestine states, which together funded the University of Jena. 146 See TOBIES 2008c, pp. 25–26. 147 Klein in KIRCHHOFF 1897, p. 241 (cf. the German original also in TOBIES 2019, p. 371). For Max Planck’s comment, see KIRCHHOFF 1897, p. 256.

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Formulae for the Main Branches of the P-Function.” Chisholm spoke on January 31, 1894 about “Spherical Trigonometry,”148 which would ultimately be the topic of her dissertation.149 In the summer of 1895, Klein was quoted in an article on women’s study at the university in the local newspaper: “On June 22, 1895, under the government of our most merciful Emperor and Lord, Wilhelm II, etc., and under the prorectorate of H. Schultze, etc., I, Felix Klein, the current dean of the Philosophical Faculty and the lawfully appointed promoter, bestowed upon the learned Miss Grace Emily Chisholm from London, who demonstrated her knowledge of mathematics, physics, and astronomy with her published dissertation (“Algebraic, Group-Theoretical Investigations of Spherical Trigonometry”) and her doctoral examination, which she passed with distinction, the degree of Doctor of Philosophy and Master of the Liberal Arts, and I certified this diploma with the seal of the Philosophical Faculty.” The academic success of [Miss Chisholm], as announced here, has also been attested by other university instructors. Fourteen women are presently studying here during the summer semester, in contrast to five during the previous winter semester.150

Fourth, Klein supported women and women’s education as the principal editor of the journal Mathematische Annalen, as president of the German Mathematical Society, and later as a member of the Prussian Parliament and other bodies. Klein’s doctoral student Mary F. Winston became the first female author to publish in Mathematische Annalen. Even before she completed her doctorate, Klein had accepted her article “Eine Bemerkung zur Theorie der hypergeometrischen Funktion” [A Remark on the Theory of the Hypergeometric Function] (dated October 1894) for publication in vol. 46 (1895).151 She submitted her dissertation – “Ueber den Hermite’schen Fall der Lamé’schen Differentialgleichung” [On Hermite’s Case of Lamé’s Differential Equation] – in Göttingen on July 17, 1896, and she received her doctoral diploma on June 30, 1897. When Klein held a series of lectures in Princeton in October of 1896, he made a point to emphasize Winston’s results (see Section 8.2.3). Not coincidentally, it was under Klein’s presidency that the German Mathematical Society accepted, in 1898, its first female member: Charlotte Angas Scott, who was already a member of the London Mathematical Society and a board member of the American Mathematical Society.152 As of 1885, after completing her doctorate in London under Arthur Cayley (she was the first Englishwoman to earn this degree), Scott chaired the mathematics department at the famous women’s college Bryn Mawr in Pennsylvania.

148 [Protocols] vol. 11, pp. 297–302 and 317–22, respectively. 149 The original title of Chisholm’s doctoral thesis was “Algebraisch-gruppentheoretische Untersuchungen zur sphärischen Trigonometrie.” Her defense took place on April 26, 1895 in the subjects of mathematics, astronomy, and physics. Klein’s evaluations of the dissertations by Chisholm and Winston are printed in TOBIES 1999b. 150 Quoted from Elisabeth MÜHLHAUSEN 1993a, p. 196 (Göttinger Zeitung, August 2, 1895). 151 [Protocols] vol. 12, pp. 29–32 (Winston’s related presentation, on June 6, 1895). 152 See OAKES/PEARS/RICE 2005.

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At Bryn Mawr, Scott supervised female doctoral students.153 She also supported Frances Hardcastle’s translation of Klein’s short book on Riemann’s theory of algebraic functions (see 5.5.1.2). A remarkable partnership developed between Bryn Mawr and Göttingen. Scott and her British-born colleague James Harkness recommended some of their female students to continue their studies in Göttingen. On July 25, 1894, Harkness wrote to Klein: “Miss Isabel Maddison informs me that she wishes me to write to you in support of her application for admission to Göttingen University,” and he explained her qualifications in detail, mentioning that, in addition to attending Bryn Mawr, she had also studied successfully in Cambridge (U.K.).154 Klein, who admired Harkness and Frank Morley’s book A Treatise on the Theory of Functions,155 supported Maddison’s studies in Göttingen for two semesters, during which she attended his courses and gave two lectures in his seminars.156 Moreover, Maddison took on the task of translating Klein’s famous talk on the arithmetization of mathematics.157 One year later, she completed her doctorate under Scott’s supervision at Bryn Mawr. On March 19, 1897, Scott wrote to Klein: “I expect to send two of my best students to Göttingen next year. Both have been awarded a College Fellowship, and both are eager to study under your direction for a year, if this is agreeable to you.”158 Klein and Hilbert supported the students who were sent to them. Klein appreciated Scott’s work on algebraic geometry. Her article, which included the first geometric proof for an important theorem by Max Noether (up to then, only algebraic proofs existed), became the second paper by a woman to be published in Mathematische Annalen.159 It is noteworthy, too, that Felix Klein’s daughter Elisabeth spent one semester at Bryn Mawr, from the fall of 1910 to Easter in 1911. As a result, the third subject of her teaching examination, besides mathematics and physics, was English.160 Up until the 1930s, Bryn Mawr remained the only women’s college in the United States that offered a PhD program in mathematics. From 1933 and until the end of the war, Emmy Noether and other Jewish women from German-speaking countries (Olga Taussky, Hilda Geiringer) were also able to find refuge there and a place to continue their academic work. Fifth, Klein served as an example to his colleagues and students. Hilbert followed in his footsteps by supervising sixty-nine doctoral students, six of them women. Adolf Hurwitz and Heinrich Burkhardt in Zurich, Wilhelm Wirtinger and Philipp Furtwängler in Vienna, George Pick in Prague, Virgil Snyder at Cornell, 153 See https://www.genealogy.math.ndsu.nodak.edu/id.php?id=6965; and PARSHALL 2015. 154 [UBG] Cod. MS. F. Klein 9: 547 A. 155 HARKNESS/MORLEY 1893. Robert Fricke’s ENCYKLOPÄDIE article on elliptic functions (1913) was based on Harkness’s preparatory work. 156 [Protocols] vol. 12, pp. 123–26, 279–83 (Maddison’s lectures, Dec. 19, 1894, July 10, 1895). 157 See KLEIN 1895c. On this talk, see Section 8.2.3. 158 [UBG] Cod. MS. F. Klein 9: 947. For more on Scott’s life and work, see LORENAT 2020. 159 Charlotte Angas Scott, “A Proof of Noether’s Fundamental Theorem,” Math. Ann. 52 (1899), pp. 593–97. 160 [BBF] Personnel Files.

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and Max Winkelmann at the University of Jena were the first to supervise female doctoral students at their respective institutions. Klein’s former doctoral student Johannes Schröder, who was a secondary school teacher in Hamburg, taught mathematics to young women and was explicitly unbiased. He stressed: Earlier, and for a long time, the prevailing prejudice was that women utterly lacked the predisposition for mathematical thinking and that their feminine nature attracted them more towards engaging with literary, linguistic, historical, and ethical questions than toward strict logical thought, which mathematics has always required. Appropriately, [Felix] Klein pointed out how unjustified and untenable the opinion is that mathematics is unsuitable for women.161

Sixth, Klein helped to make it possible for women to overcome other hurdles as well. Until 1905, women were ineligible to take the teaching examination for secondary schools. When Klein asked his former student Wilhelm Lorey to write an article on the topic of “Die mathematischen Wissenschaften und die Frauen” [Women and the Mathematical Sciences] (1909), he advised him to contact the women who had meanwhile become senior teachers of mathematics. Klein wrote explicitly that Thekla Freytag had been “the first to struggle through all of these difficulties (in Berlin).” She had studied successfully in Berlin, Munich, and Zurich, but only after repeatedly applying to the Prussian Ministry of Culture did she finally receive permission, in 1905, to take the examination.162 Freytag paved the way for Elisabeth Klein, Iris Runge, and many others. Women were not officially allowed to complete a Habilitation, the prerequisite for a professorship in Germany, until a regulation was passed on February 21, 1920.163 In this regard, attempts by the Göttingen mathematician Emmy Noether to be treated as an exceptional case failed in 1915 and 1917.164 A third attempt was needed, which Klein had largely been responsible for setting in motion, before Emmy Noether became, in 1919, the first woman to be granted this qualification in Germany (see Section 9.2.2). 7.6 ACTUARIAL MATHEMATICS AS A COURSE OF STUDY165 Wilhelm Lorey has reported that Klein was impressed by the business operations of the Mutual Life Insurance Company, which had been founded in New York in 1843, and also by Emory McClintock’s work on actuarial mathematics.166 161 J. Schröder, Die neuzeitliche Entwicklung des mathematischen Unterrichts an den höheren Mädchenschulen Deutschlands (Leipzig: B.G. Teubner, 1913), p. 89. Schröder had completed his doctoral thesis – “Über den Zusammenhang der hyperelliptischen σ- und ϑ-Functionen” [On the Connection Between Hyperelliptic σ and ϑ Functions] – with Klein in 1890. 162 See, for further details, TOBIES 2017b. 163 [StA Berlin] Rep. 76 Va Sekt. 1, Tit. VIII No. 8, Adh. III, p. 162; TOBIES 1991a. 164 See in detail TOLLMIEN 1990 and 2021. 165 For further discussion of this topic, see TOBIES 1990b, 1992b. 166 See LOREY 1950, p. 45. McClintock was the author of the short book On the Effects of Selection: An Actuarial Essay (New York: Mutual Life Insurance Company, 1892).

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McClintock worked for this company from 1889 to 1911 and, as the president of the New York Mathematical Society, he had given Felix Klein an especially warm welcome in 1893 (see Section 7.4.3). Since 1886, this insurance company also had agencies in Europe (Berlin, Hamburg, London). Having returned from the United States, Klein discovered that the topic of actuarial mathematics was the order of the day. Both in Germany and in Austria, government officials and a growing number of insurance companies were interested in regulating the formal education of actuarial specialists. In 1893, the Ministry of Education in Vienna commissioned the mathematician Leopold Gegenbauer to prepare a report on the subject. Gegenbauer invited Ludwig Kiepert – who, in addition to his professorship in Hanover, was also the director of an insurance company – to give a talk at the aforementioned 1894 meeting in Vienna on the mathematical education of insurance underwriters.167 Here, Kiepert explained why a university education was necessary for such work. While still in Vienna, Klein spoke with his colleague Albert Wangerin (University of Halle), and together they arranged for the topic to be discussed in the Prussian Parliament. Funding was made available in surprisingly short order, so that Althoff, Klein, Kiepert, and the university Kurator Ernst Höpfner were able to meet in Göttingen in September of 1895 in order to establish an Actuarial Science Seminar (Seminar Versicherungstechnik).168 The Seminar started on October 1, 1895 under the direction of Wilhelm Lexis, who, in addition to being a professor of economics at the University of Göttingen (since 1887), had also been working for Althoff at the Ministry of Culture since 1893. Klein described Lexis as the “reinventor of mathematical statistics.”169 Lexis was known above all for his dispersion theory concerning the fluctuations of statistical time series.170 The Seminar, which included a mathematical and an administrative class, was intended to train mathematicians and upper-level administrators for careers in the public and private insurance industry.171 The candidates had to pass examinations in “actuarial mathematics, actuarial economics and statistics, and practical economics.” Graduates of the mathematical class had to pass an additional test in mathematics, while those of the administrative class had to take an additional examination in insurance law. Klein recruited the Privatdozent Georg Bohlmann to teach actuarial mathematics, and, as of February 29, 1896, Klein himself took over the examinations in “pure mathematics” for these candidates.172

167 See Jahresbericht der DMV 4 (1897) II, pp. 116–21. 168 The second Seminar of this sort in all of Germany was created in 1896 at the Technische Hochschule in Dresden, see VOSS 2003. 169 KLEIN 1914a, p. 315 (Klein’s obituary for Lexis). 170 Ladislaus von Bortkiewicz (Lexis’s doctoral student) described this method in his article on the applications of probability theory to statistics, in vol. 1.2 of the ENCYKLOPÄDIE (1901). 171 For the founding statutes of the Seminar, see TOBIES 1990b. 172 [UBG] Cod. MS. F. Klein A 1: 10, No. 867; and JACOBS 1977 (“Personalia”), p. 7.

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Bohlmann had studied in Berlin, but because he used [Sophus] Lie’s group theory in his dissertation, he chose to submit his thesis to the University of Halle. In 1894, he accepted Klein’s offer to complete his Habilitation in Göttingen. At first, the enrollment in Bohlmann’s courses on actuarial mathematics was minimal. Klein supported these courses, however, by steering some of his own students toward this area of research. One of such students was Wilhelm Lorey, who went on to have a great deal of success in the field.173 During the Easter break of 1900, Klein included a lecture on actuarial mathematics by Bohlmann in the continuing education course in order to draw greater attention to this new course of study.174 Bohlmann also contributed to the ENCYKLOPÄDIE, for which he wrote the article on the mathematics of life insurance (vol. I, 1901). Klein arranged for Bohlmann to receive the title of professor. In 1897, there was a danger that the aforementioned Berlin branch of the Mutual Life Insurance Company would lose its license. Because the Prussian Ministry of the Interior was interested in keeping the business in place, Klein was asked to prepare a report on the matter. He agreed to do so in a letter dated June 27, 1897, and he was able to recruit Wilhelm Lexis and Georg Bohlmann to work on it with him.175 Their efforts resulted in the branch staying in place, and for this Klein was awarded “the [Prussian] Royal Order of the Crown (Kgl. Kronenorden), 2nd Class, in recognition of his valuable service in reporting on the state of the insurance industry.”176 Bohlmann, who had done most of the work on the report, accepted a position at this insurance company in 1903. After Bohlmann’s departure, the astronomer Martin Brendel took over his courses in Göttingen. In 1904, Klein supported the program by directing a seminar (with the astronomers Brendel and Karl Schwarzschild) on probability theory, for which he stressed that he had particularly considered “the interests of actuarialscience candidates when choosing the topics.”177 Thanks to Klein, an official position was finally created for this subject. Klein’s first move was to entice Felix Bernstein to transfer as a Privatdozent from Halle to Göttingen (1907), then he ensured that Bernstein was made an associate professor of actuarial mathematics, mathematical statistics, and probability theory (1909). The status of the subject areas was increased even further by its integration into the examination for teaching candidates (as an elective for applied mathematics). Klein continued to support this field of research and instruction by directing a seminar with Bernstein in the summer semester of 1911 on actuarial 173 See TOBIES 1990b; SCHNEIDER 1989, p. 355; and Ulrich Krengel, “On the Contributions of Georg Bohlmann to Probability Theory,” Electronic Journal for the History of Probability and Statistics 7/1 (2011), pp. 1–13. 174 See KLEIN/RIECKE 1900. 175 [UBG] Cod. MS. F. Klein 7K. 176 [UAG] Kur. 5956 (Klein’s personnel files), p. 143. – The Royal Order of the Crown was a Prussian order of chivalry, instituted in 1861. 177 [Protocols] vol. 20, p. 177. There were eighteen participants in this seminar who gave lectures, and Klein gave three lectures of his own.

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mathematics.178 Despite Bernstein’s occasionally idiosyncratic behavior over the subsequent years, an independent institute for mathematical statistics was created in Göttingen in 1918. On June 7, 1919, Bernstein himself applied to be promoted to the rank of full professor. The mathematical and natural-scientific division of the Philosophical Faculty rejected his application, political reasons playing a role in the background. Nevertheless, Felix Klein and David Hilbert, with the support of Richard Courant and Carl Runge, wrote personal letters to the Ministry on Bernstein’s behalf, so that he was ultimately made a full professor on October 13, 1921, under the new political conditions of the Weimar Republic. Klein argued that, “for students of mathematics, and especially under the current circumstances, all paths leading to liberal professions that might suit them should be kept open (or should be reopened).”179 7.7 CONTACTING ENGINEERS AND INDUSTRIALISTS Klein had repeatedly referred to the need for universities to incorporate technical disciplines into their teaching and research. While in the United States, he witnessed how the study of engineering could function within a university setting and how this could be supported with private funding. Thus, in Göttingen as well, he wanted to enliven physical research with the problems of technical praxis and, vice versa, he hoped to infuse practical “mechanical engineering with the mathematical-physical spirit of the trained theoretician.”180 As a short-term goal, he imagined the establishment of an institute for technical physics, an objective which he had already discussed with Helmholtz during their voyage home from America on the same ship. Klein went into greater detail about this plan in his travel report to Althoff, who responded by saying that the financial means were lacking for such an undertaking. At that time, there were no such technical programs at German universities. Even at Technische Hochschulen, the first largescale laboratories for technical physics would not be established until later.181 Klein dutifully claimed that Friedrich Althoff had played a significant role in the development of the center for mathematics, natural sciences, and technology in Göttingen.182 However, the sources document that Klein himself was the engine behind the structural changes that took place in German higher education at the time. Limited government resources, low taxes, high military spending (on the 178 [Protocols] vol. 28. Seven presentations were given in this seminar, including one by the Polish mathematician Stefan Mazurkiewicz (on the further development of dispersion theory) and one by Arthur Rosenthal (on biometrics). 179 Quoted from TOBIES 1992b, pp. 23–24 (Klein to the Ministry, August 18, 1919). 180 KLEIN 1923a (autobiography), pp. 26–27. 181 Carl von LINDE (1984, p. 126) reported that the foundation of a laboratory for technical physics at the TH Munich, which deliberately combined theory and experimentation, had been based on the movement initiated by Klein. See also HASHAGEN 2003, pp. 301–16. 182 See KLEIN 1923a (autobiography), pp. 24–25.

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one hand), and the growing costs of equipping scientific-technical institutes (on the other) forced the Prussian Ministry of Culture to adopt a policy of fostering specific research concentrations at specific universities.183 In this regard, it was Klein’s initiatives that steered developments at the University of Göttingen. The bright idea for how he should proceed came to him in Düsseldorf on New Year’s Day in 1894, while he was taking a walk with his brother Alfred Klein, a lawyer with contacts to the industrialist Emil Schrödter.184 On March 24, 1894, Klein informed Althoff: I was in my hometown for a few days, and there I began to rethink things. I will have to report to you about this at length. Proceeding from the conviction that industry itself must be highly interested in the matter, I have established contact with distinguished experts in this sector, and I have succeeded in assembling a committee whose stated purpose is to provide us with material support.185

The members of the committee included Dr.-Ing. Emil Schrödter (Düsseldorf), the chief executive of the Association of German Steel Manufacturers; Henry Theodore Böttinger (Elberfeld), who became the most important patron for Klein’s projects in Göttingen (see Section 8.1.1); Dr. Wilhelm Beumer (Düsseldorf), the general secretary of the Association for the Protection of Common Economic Interests in the Rhineland and Westphalia; Prof. Otto Intze (Technische Hochschule in Aachen); Adolph Kirdorf, the director of the “Rothe Erde” steel mill in Aachen (who, like Klein, had attended secondary school in Düsseldorf);186 and Fritz Asthöwer, the technical director of the Krupp Company. Klein’s goal was to find donors to fund specific research initiatives: The first project will concern the physical properties (elasticity, strength, etc.) of crystals and then of solid bodies in general, in their mutual dependence and as regards their chemical constitution. For this I would like to receive a donation of 100,000 Mark from these gentlemen, apportioned over five years. Of course, this would only be the beginning; additional funding would have to follow as soon as our initial success is achieved.

Klein had developed these ideas with Walther Nernst, an associate professor of physical chemistry and electrochemistry in Göttingen since 1891. Nernst had published an outstanding textbook on theoretical chemistry in 1893 and, at Klein’s instigation, was writing a textbook with Arthur Schoenflies: Einführung in die mathematische Behandlung der Naturwissenschaften [Introduction to the Mathematical Treatment of the Natural Sciences],187 when he received an offer for a full professorship from Munich in 1894. As dean at the time, Klein traveled to see

183 184 185 186

See BROCKE 1980, 1991. [UBG] Math. Arch. 51, p. 45. Quoted from TOBIES 1991c, pp. 98–99 (a letter from Klein to Althoff dated March 24, 1894). Kirdorf’s approach to business has been described as scientifically grounded (he was the first to use the [Sidney Gilchrist] Thomas process in steel production) and as socially oriented. In 1912, he was awarded an honorary doctorate from the Technische Hochschule in Aachen. 187 NERNST/SCHOENFLIES 1895. This book was translated into English as The Elements of Differential and Integral Calculus (New York: D. Appleton and Co., 1900).

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Althoff in Berlin in order to underscore Nernst’s importance for Göttingen. Now it was the aforementioned patron Henry Theodore Böttinger, the business director of the Elberfeld chemical dyestuffs factories (Elberfelder Farbenfabriken), who offered his support. He knew Nernst from the newly founded Society for Physical Chemistry. Althoff secured a donation from Böttinger for the purpose of creating a full professorship for Nernst and establishing an institute for physical chemistry in Göttingen. Nernst was thus able to remain in Göttingen for another ten years. Josef-Wilhelm Knoke has shown that Böttinger’s generosity was not entirely altruistic: through his donation, he gained the exclusive rights to the commercial use of Nernst’s future inventions (Nernst won the Nobel Prize in 1920). When he wanted to sell the patent for his “Nernst lamp” (1897) on the open market, he was only able to do so after engaging in an arduous legal battle with Böttinger.188 Further progress was slow at first. At Althoff’s request, Klein prepared further memoranda in which he argued on behalf of establishing a university institute for technical physics. Althoff supported Klein’s continuing meetings with industrialists in Berlin, but he was cautious enough not to be too optimistic about their outcomes. He forwarded Klein’s memoranda to his colleagues at the ministry and to engineers, and among the latter group Klein’s plans sparked vehement opposition. Klein’s choice of words – he wanted the “general staff officers of technology” to be educated at universities and the “infantry of engineers” to be trained at Technische Hochschulen – was regarded as insulting by representatives at the latter institutions, who viewed the plans in Göttingen as competition. The heated debates that followed have already been analyzed in detail by Karl-Heinz Manegold and Susann Hensel.189 Although Eduard Riecke, Klein’s friend at the University of Göttingen, was quick to support his initiatives, others needed more time to dispel their fears that the university could become dependent on industry and that its devotion to science might be compromised. With the help of Carl Linde (see Section 4.3.1), Klein was ultimately able to set the course of things to come. In September of 1894, he wrote to Linde: Indeed, I have a major plan that I would like to discuss with you. It concerns the establishment, at the university here, of a laboratory for applied physics, in which experiments would be carried out in a similar way to those conducted in the various laboratories at the Munich Polytechnikum, but for which a connection would have to be made to our lectures on mathematics and mathematical physics. To achieve this, I am not only seeking direct contact with industry, but also its material support. Just these few lines will suffice to tell you that this is a project whose roots go back to the time that I spent with you in Munich, and that its aim is to overcome, in the interest of everyone involved, certain inherent biases in the way that universities are organized today.190

Linde immediately expressed his support for Klein’s plan, and he put Klein in touch with the Association of German Engineers (Verein Deutscher Ingenieure,

188 See KNOKE 2016, pp. 148–50. – On Nernst, see also KORMOS BARKAN 1999. 189 See MANEGOLD 1970 and HENSEL et al. 1989. 190 Quoted from LINDE 1984, p. 134 (Klein’s letter to Linde, dated September 1894).

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VDI), of which Linde would serve as a board member in 1895 and 1896. In August of 1895, Klein traveled to Aachen to attend the association’s annual meeting. There he became a member of the association himself and found further support for his plans. In Aachen, Linde helped to orchestrate a “truce” between Klein and the Association. As part of this agreement, however, Klein had to set aside his vision of training engineers at the university and limit his efforts “to providing future teachers of mathematics and physics […] with insight into the applications of these disciplines to technical areas, and to offering chemists, lawyers, and agriculturalists a basis for understanding the tasks involved with these careers that have arisen from the importance of today’s industry.”191 On December 6, 1896, Klein also presented his plans to the local branch of the Association of German Engineers in Hanover.192 Before an even closer alliance between science, industry, and the government would be forged in Göttingen (see Section 8.1.1), Böttinger, Linde, and the Munich-based locomotive producer Georg Krauß donated enough start-up funding to get the new Institute for Technical Physics off the ground.193 Böttinger and Linde were awarded honorary doctorates from the University of Göttingen in June of 1896.194 Professors of engineering at the Technische Hochschule in Berlin, especially the influential Alois Riedler (an Austrian who had joined the faculty there as a professor of mechanical engineering in 1888) and Adolf Slaby (who became the first professor of electro-technology there in 1886), were strongly opposed to this development. Riedler wrote to Klein that his actions would be detrimental, “so that I will have to stand up against your Göttingen institute at every opportunity.”195 It would take several more years before Klein was able to make peace with these opponents as well (see Section 8.1.1). Klein wanted Carl Linde to serve as the director of the Göttingen institute. Linde promised his further support, but he explained that he was tied to Munich.196 Toward the end of 1896, Richard Mollier, a Privatdozent of mechanics in Munich, was hired as the first director of this Institute for Technical Physics and as an associate professor of (agricultural) technology. After just one year, however, he left for a professorship at the Technische Hochschule in Dresden. Mollier’s successor in Göttingen, Eugen Meyer, likewise stayed for just a short while; in 1900, he accepted an attractive offer to work in Berlin. Things went similarly in the field of applied electricity, for which Klein, as dean, likewise set the course by enticing Theodor Des Coudres to transfer his Habilitation from Leipzig to Göttingen in January of 1895. Although Des Courdres 191 Ibid., p. 135. See also LUDWIG/KÖNIG 1981, pp. 148–49. 192 Felix Klein, “Über den Plan eines physikalisch-technischen Instituts an der Universität Göttingen,” Zeitschrift des Vereins deutscher Ingenieure 40 (1896), pp. 102–07; reprinted in KLEIN/RIECKE 1900. 193 Böttinger donated 10,000 Mark, Linde 5,000, and Krauß 5,000. 194 See LINDE 1984, p. 137; and TOBIES 1991c, pp. 101–02. 195 [UBG] Cod. MS. F. Klein 4 C, p. 94 (a letter from Riedler to Klein dated March 16, 1896). 196 See LINDE 1984, Appendix (a letter from Linde to Klein dated July 7, 1895).

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was made an associate professor in 1897, he moved on to a full professorship in Würzburg in 1901.197 In Göttingen, the new applied disciplines would not have a secure footing until Klein established new funding and hiring policies that enabled full professors to be hired in these fields (see Section 8.1.2). 7.8 THE ENCYKLOPÄDIE PROJECT198 When all our names are no longer, or perhaps one or another have some historical interest, distant generations will remain grateful to you for the magnificent work of the Encyklopädie, whose realization required exactly a man like you with so much selflessness and self-sacrifice.199

David Hilbert made this statement in his speech commemorating Felix Klein’s sixtieth birthday. Hilbert contributed as an author to the first volume of the ENCYKLOPÄDIE.200 As late as 1909, he still regarded Klein as the head of this unfinished project, which he admired. Of course, we know today that an encyclopedia quickly reaches its limits or that, after its publication, it can soon be made obsolete by the discovery of new knowledge. Nevertheless, undertakings of this sort are not only of historical value; by showcasing earlier and forgotten results, they can also lead to something new. With the ENCYKLOPÄDIE, Klein wanted to create a research tool that would make it easier to access previous knowledge: In mathematics, as in other sciences, the same processes can be observed again and again. First, new questions arise, for internal or external reasons, and draw the younger researchers away from the old questions. And the old questions, just because they have been worked on so much, need ever more comprehensive study for their mastery. This is unpleasant, and so one is glad to turn to problems that have been less developed and therefore require less foreknowledge – whether the topic is formal axiomatics, or set theory, or some such thing! And so there is nothing for it but to collect together the old subjects in good references – in the Jahresberichte [the journal of the German Mathematical Society], the Encyklopädie, etc. – or in monographs, so that later developments may tie in with them, if fate should so decree!201

The aim of the first six volumes of the ENCYKLOPÄDIE was not to provide a comprehensive historiographical account.202 Klein wanted above all to have the latest findings in mathematics presented systematically. The work was meant to “put a check on the ever-increasing fragmentation of science”203 In his introductory

197 [UAG] Phil. Fak. 4/Vb 248. 198 For further details on the development of this project, see TOBIES 1994a; HASHAGEN 2003, pp. 439–70; GISPERT 1999; and GISPERT/VERLEY 2000. 199 Quoted from ROWE 2018a, p. 198; on the German orginal, see TOBIES 2019, p. 514. 200 See David Hilbert, “Theorie der algebraischen Zahlkörper, Theorie des Kreiskörpers,” in ENCYKLOPÄDIE, vol. I.2 (1900), pp. 675–732. 201 KLEIN 1979 [1926], p. 294 (on the German original, see also TOBIES 2019, p. 353). 202 Hélène GISPERT (1999) raised this question, comparing the German with the French edition. 203 [AdW Göttingen] Scient. 305, 1 No. 2a.

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report for the first volume of the ENCYKLOPÄDIE, Walther Dyck stressed that its goal was to provide an overall picture of the position that “mathematics occupies in present-day culture” – a goal that Klein pursued on many fronts (see Section 8.3). In 1928, the mathematician Karl Boehm (a student of Leo Koenigsberger) stated that “scientific research in all areas of mathematics has been facilitated and strongly supported by the Encyklopädie.”204 As mentioned earlier, the starting point of the ENCYKLOPÄDIE was Franz Meyer’s idea of writing a book with Klein: “The Spirit of Modern Geometry, by F. Klein and F. Meyer,” as he envisioned it in a letter to Klein dated May 31, 1893.205 In early September, 1894, Klein and Heinrich Weber went to visit Meyer in Clausthal, a town in the Harz mountains.206 They stayed there for three days, during which they hiked together and reshaped Meyer’s idea into a “mathematical lexicon.” At the 1894 meeting of the German Mathematical Society in Vienna, Franz Meyer was officially commissioned by the society to draft a plan for the lexicon. The minutes of the society’s board meeting that year state that the project “seemed like a fitting undertaking to be supported materially by the consortium of academies created in 1893.”207 From his father-in-law, Klein was already familiar with this sort of academy-based funding model for large-scale projects (see Section 3.6.2). A short time later, when disagreements arose about Meyer’s concept, Klein explained: “It was not Franz Meyer but rather I myself who, with [Heinrich] Weber in Clausthal, came up with the idea for creating a lexicon to be funded by academies.”208 Representatives of this “consortium” of learned societies (the academies in Vienna, Munich, Göttingen, and Leipzig) were present at the 1894 conference in Vienna, and they took on the task of persuading their respective institutions to support this endeavor. The Austrians Gustav von Escherich (who was elected to the board of the German Mathematical Society in 1894) and Ludwig Boltzmann won approval from the Academy of Sciences in Vienna. Walther Dyck succeeded to do the same in Munich, as did Klein in Göttingen. The first ENCYKLOPÄDIE committee consisted of the following people: Dyck (chairman and the representative from the academy of Munich), Klein (academy of Göttingen), Escherich (academy of Vienna), Boltzmann209 (as an advisor for specific scientific questions), and Heinrich Weber (as a representative from the German Mathematical

204 205 206 207 208 209

Karl Boehm, “Adolf Krazer,” Jahresbericht der DMV 37 (1928) Abt. 1, p. 23. [UBG] Cod. MS. F. Klein 10: 1246 (a letter from Meyer to Klein dated May 31, 1893). [UA Braunschweig] A letter from Klein to Fricke dated September 13, 1894. Jahresbericht der DMV 4 (1897) I, p. 5 (“Chronik”). [BStBibl] A letter from Klein to Dyck dated January 24, 1896. Klein numbered among the staunch advocates of Boltzmann’s statistical concept of nature, a theory that was attacked by proponents of energetics (Georg Helm, Wilhelm Ostwald) at the annual meeting of the Society of German Natural Scientists and Physicians in 1895. See Arnold Sommerfeld, “Ludwig Boltzmann zum Gedächtnis,” Wiener Chemiker Zeitung 47/3-4 (1944), p. 25. In the end, Klein ensured that Boltzmann’s theory received its due in vol. IV of the ENCYKLOPÄDIE (on mechanics); see EHRENFEST/EHRENFEST 1909–1911.

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Society). The title that was ultimately chosen for the project – Encyklopädie der mathematischen Wissenschaften mit Einschluß ihrer Anwendungen [Encyclopedia of the Mathematical Sciences, Including Their Applications] – was based on a suggestion by Gustav von Escherich.210 The Royal Saxon Society of Sciences in Leipzig (see Section 5.7.3) was part of this group of academies, but Sophus Lie discouraged it from supporting the ENCYKLOPÄDIE undertaking. Shortly after publishing his disparaging remarks about Klein (see Section 6.3.6), Lie wrote about his decision to Adolph Mayer and spewed even more vitriol: “I liken Klein to an actress who, in her youth, was enchanting with her brilliant appearance but who gradually had to resort more and more to reprehensible means to achieve success on third-rate stages.”211 Klein convinced his colleagues to go ahead with the project even without support from Leipzig. Thus, in May and June of 1896, the academies of science in Vienna, Munich, and Göttingen signed a contract with B.G. Teubner in Leipzig to publish the ENCYKLOPÄDIE.212 The Leipzig academy did not commit its support to the project until 1904, when it was persuaded to do so by Otto Hölder (see Section 5.4.1), who had been named Sophus Lie’s successor at the university there in 1899. In 1904, Hölder also completed an article on Galois theory and its applications for vol. I of the ENCYKLOPÄDIE (arithmetic and algebra). He now became Leipzig’s representative on the project’s committee. Founded in 1909, the Academy of Sciences in Heidelberg joined the consortium of German-speaking academies in 1911 and likewise lent its support to the ENCYKLOPÄDIE. Paul Stäckel represented Heidelberg on the committee. Support was still lacking from the Berlin academy, even though the latter had joined the consortium of academies in 1906.213 Klein avoided involving the Berlin academy in the ENCYKLOPÄDIE project, however, because he had been told that Frobenius dismissed the undertaking as “senescent [greisenhaft] science.”214 Klein responded to this in his course on the ENCYKLOPÄDIE (1902/03) by paraphrasing the words of the historian Leopold von Ranke: “We nevertheless have to try it!”215 This comment was inspired by Ranke’s preface to the first volume of his

210 Thanks to Escherich’s nomination (dated May 16, 1900), Felix Klein was made a foreign member of the Imperial Academy of Sciences in Vienna. In his letter supporting this nomination, Escherich emphasized Klein’s broad knowledge of both pure and applied mathematics, technical mechanics included. See TOBIES 1994, p. 2. 211 Quoted from TOBIES 1994a, pp. 10–11. 212 The contract and its later addenda are published in ibid., pp. 69–75. Remarkably, the terms of the contract include an “editor and author honorarium of 100 Mark per page,” 30 Mark of which was to be paid by the publishing house, 25 Mark each by the academies in Munich and Vienna, and 20 Mark by the Royal Society of Sciences (academy) in Göttingen. 213 [AdW Wien] Kartell, I 153; and LAITKO 1999. 214 Georg Frobenius made this remark in 1901 at a meeting of the Mathematical Society of Berlin; see HASHAGEN 2003, pp. 469, 457. 215 [UBG] Cod. MS. F. Klein 19 B, fol. 32v: “Wir müssen es trotzdem versuchen!”

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Universal History, in which he discussed the usefulness and the dangers of encyclopedic works.216 Klein continued to dwell on this matter, and he addressed Frobenius’s comment yet again in a course that he gave in 1910/11: Regarding this matter [the ENCYKLOPÄDIE], someone made the rather unfortunate analogy with antiquity: Great encyclopedias were produced in Alexandria, and science began to be canonized when productivity ceased. Thus the opinion was expressed about our encyclopedia, too, that the idea of it was a sign that mathematical productivity was coming to an end and that there was nothing else to do but bring together the knowledge that already existed. Well, we never had such a thought in our plan, and the development of science since 1894 has also taken a different path. The Encyclopedia involves more than just gathering old ideas; it also involves processing large amounts of heterogeneous material in a uniform way.217

In the year 1913, something that had been prophesied for Klein when he was a young professor came to pass: “You will be made a member of every notable academy on earth, and last of all by the Berliners!”218 Frobenius, Schottky, H.A. Schwarz, and Planck nominated Klein to be elected as a corresponding member of the Academy of Sciences in Berlin. In their nomination letter, they also mentioned: “It is to his credit […] that the great work of the mathematical encyclopedia was begun and has been carried out energetically” (see Appendix 9). When, in 1918, the Berlin Academy honored Klein yet again on the occasion of the fiftieth anniversary of his doctoral degree, Klein responded by inviting this organization to participate in the ENCYKLOPÄDIE project.219 In 1921, when all of the academies involved signed a fifth addendum to the ENCYKLOPÄDIE’s publishing contract with B.G. Teubner, the academy in Berlin was included. In the meantime, the University of Berlin had hired mathematicians who had already supported the project as contributing authors: Richard von Mises, who wrote the article on dynamic problems in mechanical engineering (1911), and Ludwig Bieberbach, who supplied an article on the latest investigations into functions of complex variables (1920). From the beginning, the ENCYKLOPÄDIE was in high demand. By just the second year of its production, orders had been placed for nine hundred copies of the first edition, for which a run of one thousand copies had been planned. The Teubner publishing house thought about increasing the size of the first printing, and it also considered possible translations. In France, Darboux indicated that he would be in favor of such an undertaking. Thus, in a third addendum to the original publishing contact (signed in June and July of 1900), the academies authorized the production of a “French or English edition in collaboration with a different pub216 See Leopold von Ranke, Universal History: The Oldest Historical Group of Nations and the Greeks, ed. G.W. Prothero (New York: Charles Scribner’s Sons, 1884), p. xiii: “We came to the conclusion that perfection was not to be attained, but that it was none the less necessary to make the attempt.” 217 These comments were made in a course that Klein taught on the modern development of mathematical instruction during the winter semester of 1910/11 ([Hecke] p. 272). 218 This prediction was made by Wilhelm Ahrens, a heir of Klein, who completed his doctorate with Otto Staude and also wrote an ENCYKLOPÄDIE article ([UBG] Cod. MS. F. Klein 117). 219 [BBAW] Bestand PAW, III b 137: 129 (Klein’s letter to the Academy, December 28, 1918).

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lishing house.” This addendum entitled the academies, the ENCYKLOPÄDIE committee, and the authors of articles to choose the editors and translators of the foreign editions and to review the translated contributions. Large portions of an (expanded) French edition were published; an English version never came to be.220 Alfred Ackermann-Teubner (see Section 5.6), who had meanwhile become the treasurer of the German Mathematical Society and a member of the Société Mathématique de France, cultivated contacts with Jules Molk, “the editor of the French edition,” and with the Parisian publisher Albert Gauthier-Villars. Both of these Frenchmen joined the German Mathematical Society in 1900. Klein had been asked whether he considered Molk, a professor of mathematics at the Université de Nancy, a suitable leader for the project, and he had given his approval. He consulted with Molk over the course of several days, and together, at the ENCYKLOPÄDIE conference held in Leipzig in 1902, they revealed the first list of collaborators for the French edition of volumes I and II. In 1904, at the third International Congress of Mathematicians in Heidelberg, Klein and Molk presented the first parts of the German and French editions. Molk (who died on May 14, 1914) and Gauthier-Villars participated in further ENCYKLOPÄDIE meetings in Germany, and up until 1914 they sent regular reports on the status of the French project.221 Klein’s way of working on the ENCYKLOPÄDIE can be summarized as follows: First, Klein’s scientific role in the project was to provide upper-level management. He tried as much as possible to delegate organizational tasks to others, in which case Walther Dyck served as a loyal partner. Klein chose the editors (Redakteure) for the individual volumes, while the academies themselves functioned as the publishers (Herausgeber). He discussed the structure of the volumes with the editors, and he discussed the structure and content of the articles with numerous authors. Wilhelm Wirtinger and Heinrich Burkhardt were chosen to edit vol. II (analysis). Wirtinger compared Klein’s activity to that of “a field commander storming ahead.”222 That is, Klein controlled the project’s editors and authors and urged them on. When he was unable to find a suitable editor for vol. IV (mechanics), he took on the job himself (until Conrad H. Müller offered his services). Second, Klein would recognize some relevant open problems in the area of engineering mathematics through his work on the applied volumes of the ENCYKLOPÄDIE (see Section 8.2.4). On August, 1, 1898, he wrote to Hurwitz (at the Polytechnikum in Zurich) about his idea of “engineering mathematics” (Ingenieurmathematik) and about a planned trip to penetrate deeper into the area: [Heinrich] Burkhardt may have told you that, in the interest of the “applied” volumes, I plan to take a long trip in order to get to know people and things. If we use the term “engineering

220 ENCYCLOPÉDIE (1904–16), reprint 1991–95; GISPERT/VERLEY 2000. – In 1901, the married couple Grace Chisholm and William Henry Young were consulted regarding the original plan for an English edition that was ultimately not realized. Regarding the sections on applied mathematics, Klein also had thought of collaborating with Edward Hough Love. 221 These reports are archived in [AdW Wien]. See TOBIES 1994a, pp. 24–25. 222 Quoted from HASHAGEN 2003, p. 467.

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7 Setting the Course, 1892/93–1895 mathematics” to describe a certain complex of disciplines, my question is: What is the state of engineering mathematics in France, Italy, etc. from the mathematician’s point of view? No one in Germany seems to know clearly.223

Third, as the quotation above shows, Klein wanted the ENCYKLOPÄDIE to have an international profile. He sought to supplement Franz Meyer’s choice of (mostly German) authors for vols. I and III. In his original design, Meyer estimated that thirty-five collaborators would be needed for the whole project. In the end, ninetytwo authors would contribute to just the first three volumes (arithmetic and algebra; analysis; geometry), thirty-three of whom were from outside of Germany. Klein ensured that representatives of the Italian algebraic-geometric school contributed; that French mathematicians were represented in the German version of the volume on analysis and that contributions to the French edition were translated for the German project (by Arthur Rosenthal);224 that British and Dutch work on mechanics was written about by the experts themselves; and that American, Russian, Scandinavian authors (etc.) took part in the project. Klein realized that he would need to rely on foreign authors especially for the ENCYKLOPÄDIE volumes on the applications of mathematics (vols. IV–VI: mechanics, physics; geodesy, geophysics, astronomy) and for the planned seventh volume of the project (on the didactics, philosophy, and history of mathematics). In his opinion, his German colleagues did not have sufficient knowledge of the relevant scholarly literature in these fields: Our German technical colleagues seem to have insufficient knowledge in this respect. Foreign scientific literature – names such as Greenhill, Boussinesq – is known at best by hearsay, even among the most distinguished representatives of these disciplines here, with whom I have recently held repeated negotiations. Things are similar regarding our knowledge of didactic literature. That, too, is limited by national boundaries; it will be quite an achievement if we succeed in breaking through these limitations.225

This statement served as a successful argument for Klein to receive travel funding from the academies and from the Prussian Ministry of Culture for the purpose of recruiting potential authors. When Klein was invited to accept an honorary doctorate from the University of Cambridge, which was celebrated on March 11, 1897 (see Section 6.3.7.1), he planned his first (three-week) trip, which took him to Manchester, Glasgow, and London in the interest of the ENCYKLOPÄDIE. During these travels, he gained an overview of the state of “pure” and “applied” mathe223 [UBG] Math.Arch. 276 (Klein to Hurwitz, August 1, 1898). – Klein’s co-worker Heinrich Burkhardt had been full professor at the University of Zurich since 1897. 224 A. Rosenthal, who in 1909 completed his doctorate under F. Lindemann in Munich, gave two presentations in Klein’s research seminar in 1911. For the ENCYKLOPÄDIE (vol. II/3.2 [1923], pp. 851–1187), Rosenthal translated and revised articles from the French Encyclopédie on “Recherches contemporaines de la théorie des fonctions” (edited by Émile Borel). 225 Quoted from TOBIES 1994a, p. 22 (a letter from Klein to Dyck dated June 13, 1896). On Greenhill, see Section 6.3.3 and 6.3.7.1. Regarding Joseph Boussinesq, Klein had the participants in his research seminar analyze the Frenchman’s fundamental equations for turbulent hydrodynamics (see Section 8.2.4).

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matics, and he considered this new-found information so important that, on April 27, 1897 he assembled his Göttingen colleagues to give a lecture on it.226 In 1898, Klein embarked on another trip, this time with Arnold Sommerfeld, who had agreed to take over the responsibility of editing of vol. V (physics).227 Before their visit to the Netherlands, Klein explained to Hendrik A. Lorentz, a professor of theoretical physics at the University of Leiden: I have now taken it upon myself to establish necessary personal contacts abroad. Regarding mathematical physics, of course, Holland deserves special attention. […] My wish would be, before everything else, to discuss the entire mathematical-physical section with you yourself and then, with the help of your mediation, to become more familiar with the Dutch circle of mathematical physicists.228

Lorentz, who would win the Nobel Prize in 1902, agreed with this plan, and he discussed the structure of the physics volume with Klein and Sommerfeld. Lorentz also wrote three important articles for the volume.229 In Amsterdam, Klein and Sommerfeld met with Johannes Diderik van der Waals (Nobel Prize, 1910), who is known for his work on the equation of state for gases and liquids. They were able to recruit Heike Kamerlingh Onnes (Nobel Prize, 1913) and his student Willem Hendrik Keesom to write the ENCYKLOPÄDIE article on this topic (1911). In October of 1898, Klein traveled on his own to Paris in order to meet further potential authors.230 In March of 1899, Klein began a longer ENCYKLOPÄDIE trip to Italy (4–5 weeks). He gained an overview of the developments in mathematics there, recruited a number of authors for the project, and lectured about his findings upon returning to Göttingen.231 In the fall of 1899, Klein’s next joint trip with Sommerfeld was to Great Britain, where Klein had already made connections in 1897. While in Cambridge, they consulted with Joseph John Thomson, Joseph Larmor, Edward Routh, and the eighty-year-old Sir George Gabriel Stokes. Lord Rayleigh invited them to his estate in Essex. The first German edition of Routh’s book Dynamics of a System of Rigid Bodies, with a preface by Klein (dated October 11, 1897), had just been published by B.G. Teubner (see Section 5.6). Even though these distinguished scholars did not write articles of their own for the ENCYKLOPÄDIE, they helped to broaden the circle of contributors to include other younger authors. Gilbert Walker, for instance, was recruited to write an article on the mathematical aspects of

226 227 228 229

[UBG] Cod. MS. F. Klein 22F, Bl. 92–95 (Klein’s concept of the lecture). See ECKERT 2013, 137–47. Quoted from TOBIES 1994a, pp. 21–22 (Klein to Lorentz, September 5, 1898). H.A. Lorentz, “Maxwells elektromagnetische Theorie” (1903), “Weiterbildung der Maxwellschen Theorie – Elektronentheorie” (1903), and “Theorie der magnetooptischen Phänomene” (1909). See ENCYKLOPÄDIE, vol. V.2, pp. 63–144, 145–280; vol. V.3, pp. 199–281. 230 [Deutsches Museum] HS 1977-28 (a letter from Klein to Sommerfeld, October 20, 1898). 231 [UBG] Cod. MS. F. Klein 22F (a draft of Klein’s lecture). Klein persuaded the following Italian mathematicians to contribute to volumes III and IV of the ENCYKLOPÄDIE: F. Enriques, G. Fano, L. Berzolari, G. Loria, G. Castelnuovo, C. Segre, G. Jung.

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sport and games,232 and contributions to vol. IV (mechanics) were made by Augustus Edward Hough Love (1901) and Horace Lamb (1906). Klein also arranged for a textbook by Love and one by Lamb to be translated into German (see Section 5.6). When Klein and Sommerfeld visited George Hartley Bryan in Bangor, Wales (see Fig. 33), they were quite familiar with Bryan’s work on applied mathematics. Together with Joseph Larmor,233 Bryan had already published summary reports on thermodynamics (1891) and statistical mechanics (1894) for the British Association for the Advancement of Science. Klein and Sommerfeld persuaded him to write the article on the general foundations of thermodynamics for the ENCYKLOPÄDIE (1910).

Figure 33: An ENCYKLOPÄDIE trip to Wales. Felix Klein (seated in the middle) and Arnold Sommerfeld (left) with George Hartley Bryan (standing in the middle) and Bryan’s family ([Deutsches Museum] CD 66310).

Through correspondence, Klein endeavored to recruit additional authors, including Diederik Korteweg in Amsterdam, whom Klein and Sommerfeld had met in 1898.234 In 1895, Korteweg and his doctoral student Gustav de Vries had devel232 Gilbert Walker, “Spiel und Sport,” in ENCYKLOPÄDIE, vol. IV.2 (1900). 233 Regarding Joseph Larmor and Felix Klein, see also Section 9.3.1. 234 [UBG] Cod. MS. F. Klein 10: 536–38 (correspondence between Korteweg and Klein from October 14, 1899 to May 4, 1900).

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oped a non-linear partial differential equation (the Korteweg-de Vries equation) for describing waves on shallow water surfaces. Klein wanted Korteweg himself to write an article on this topic, but Korteweg tried to pass the task on to a student, and the article never materialized. Fourth, Klein did not write any articles himself. What he brought to the table was his vast overview of knowledge, which he continued to expand through his teaching. In his lecture course titled “The Encyclopedia of Mathematics” (winter semester, 1902/03), he discussed arithmetic, algebra, and analysis. The fifty-six students enrolled in this course included Paul Ehrenfest and Tatyana Afanasyeva, who had already attended Klein’s lecture course on mechanics in 1902 and had presented papers in his seminar on the principles of mechanics (1902/03). Klein was quick to recognize their talent, and he recruited them to contribute as authors to the ENCYKLOPÄDIE.235 In his lecture course on the “Encyclopedia of Geometry” (summer semester, 1903), Klein reorganized the project’s third volume (geometry) because he was dissatisfied with Franz Meyer’s plan for it. Later supplements were produced for this volume to take into account new research developments in the field. Working with Hans Mohrmann and Wilhelm Blaschke in 1915 and 1916, Klein thus designed the concept for Part 4 of the geometry volume. Although the final volume III ultimately did not include this part, the original plan is interesting;236 it included the following eight articles: Volume III (Geometry), Part 4 Art. 1: Developments concerning the spherical circle: [Franz] Meyer Art. 2: Tetrahedron geometry: [Franz] Meyer – Zacharias. Art. 3: Analysis situs: Tietze. Art. 4: General theory of shapes: Hjelmslev. Art. 5: Algebraic curves and surfaces according to the theory of shapes: Mohrmann. Art. 6: Complex geometry: Dyck. Art. 7: Invariant theory of particular geometric groups: Weitzenböck. Art. 8: Infinite-dimensional spaces: [Gerhard] Kowalewski.237

Fifth, in 1896, Klein already outlined the structure of the planned final volume of the ENCYKLOPÄDIE, which was intended to cover the history, philosophy, psychology, and didactics of mathematics. Work on this volume came to a halt on account of the outbreak of the First World War, which thwarted Klein’s ambitions to collaborate with international authors.238 Nevertheless, Klein’s outline of this volume would later inform his own theoretical work on these topics (see 8.3).

235 See Paul Ehrenfest, and Tatjana Ehrenfest (1909–11), “Begriffliche Grundlagen der statistischen Auffassung der Mechanik,” in: ENCYKLOPÄDIE, vol. IV.4, pp. 1–90. 236 Some of the planned articles were included in vol. III: Roland Weitzenböck, “Neuere Arbeiten der algebraischen Invariantentheorie: Differentialinvarianten” (1921); Heinrich Tietze and Leopold Vietoris, “Beziehungen zwischen den verschiedenen Zweigen der Topologie” (1929); Karl Rohn, “Algebraische Raumkurven und abwickelbare Flächen” (1926). 237 [Blaschke] A letter from Klein to Blaschke, 1915/16 (a meeting concerning the ENCYKLOPÄDIE took place in Göttingen, Dec. 13 and 14, 1916). 238 On the attempts to realize this volume and their failure, see TOBIES 1994a, pp. 56–69.

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7.9 KLEIN SUCCEEDS IN HIRING DAVID HILBERT At his home university in Königsberg, Hilbert had become an associate professor in 1892 (succeeding Hurwitz) and a full professor in 1893 (succeeding Lindemann). Klein indicated his approval of these developments to Friedrich Althoff in the Prussian Ministry of culture. Klein continued to regard Hilbert as the up-andcoming mathematician and to publish Hilbert’s latest results in Mathematische Annalen or the Göttinger Nachrichten. Hilbert also studied Klein’s lectures on higher geometry; he used them in his own courses, and he wrote to Klein about corrigenda that he happened to notice. In his letters to Klein, Hilbert still addressed him as “Highly Esteemed Professor” (Hochgeehrter Herr Professor).239 When Heinrich Weber accepted an offer to succeed Elwin Bruno Christoffel at the University of Strasbourg in 1895, the path to Göttingen was cleared for David Hilbert. Already on December 6, 1894, Felix Klein, who was then the dean of the Philosophical Faculty, wrote the following to Hilbert: You may not know this yet, but Weber is leaving for Strasbourg. This is why the faculty is meeting this evening, and although I can’t predict the final decision of the hiring committee, which still has to be appointed, I want to let you know that I will make every effort to ensure that you are hired here. You are the man whom I need here to complement my research: by virtue of the nature of your work and the potency of your mathematical thinking, and in light of the fact that you are still in your productive period. The mathematical school here is developing well and will continue to grow. I expect you to bring a new inner content, and that you might perhaps have a rejuvenating effect on myself. […] It is impossible for me to know whether I can enforce my own view on the Faculty, and it is even less clear whether the Ministry in Berlin will realize the appointment. However, the one thing that you have to promise me today is that you will not reject the offer if it comes to you!240

Hilbert promised to accept the offer “without a moment’s hesitation and with great pleasure.”241 The hiring committee consisted of Felix Klein, Heinrich Weber, Ernst Schering, the physicists Woldemar Voigt and Eduard Riecke, the astronomer Wilhelm Schur, and the theologian Rudolf Smend (as a representative of the philological-historical division of the Faculty).242 On December 13, 1894, the committee unanimously agreed on two top candidates: Hilbert and Hermann Minkowski. Althoff settled the matter quickly. Klein learned from Hilbert on December 19, 1894 that he had promptly accepted Althoff’s offer, which included a yearly salary of 4,600 Mark plus a housing allowance and moving costs.243 The salary offered to Hilbert was less than half of what Klein was earning at the time. Minkowski was hired to replace Hilbert as a full professor in Königsberg, and a year later he accepted a professorship at the Polytechnikum in Zurich alongside Hurwitz. The newly vacated position in Königsberg was taken by Franz Meyer 239 240 241 242 243

See FREI 1985, pp. 106–08 (a letter from Hilbert to Klein dated February 14, 1894). Quoted from FREI 1985 (a letter from Klein to Hilbert dated December 6, 1894). Ibid., p. 116 (a letter from Hilbert to Klein dated December 8, 1894). [UAG] Phil. Fak. Protokollbuch (July 1, 1889–June 30, 1905), p. 112. See FREI 1985, p. 117 (a letter from Hilbert to Klein dated December 19, 1894).

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(from Clausthal), so that Meyer’s former professorship of mathematics at the Mining Academy could serve as the springboard for Arnold Sommerfeld’s career. Klein had preferred Hilbert because he regarded him as the most creative mathematician at the time in the entire German-speaking part of the world. Even before Klein had written his letter to Hilbert (quoted above), he had tried to explain the matter to Hurwitz, who had replied: Regarding the hiring opportunity in Göttingen, I immediately thought that you would consider Hilbert. I approve of Hilbert’s appointment with all my heart, and I have no doubt that you have made a good choice. My only concern is that the remarks in your letter make me wonder whether it was hopeless from the beginning to think that I might be offered the position.244

As is apparent from Klein’s response to Hurwitz, Klein clearly wanted to shy away from the conflicts that might arise from the Faculty’s anti-Semitism; moreover, he saw Hurwitz as being too closely aligned to his own approach to mathematics, and he felt that Hurwitz and his family were in a good position in Zurich.245 Klein expected that Hilbert’s presence would provide a stronger spark to his own research, even though he was aware of their differing views. In early January of 1895, after Hilbert had been in Göttingen for just a few days, Klein informed Hurwitz: We differ in many respects: I love the applications of mathematics, to which he is indifferent, and I demand less of ordinary teaching candidates, whereas he might want to start off with the strictest definitions, if possible in the first semester. I am eager to see how we will find common ground in this matter. I am all the more optimistic about how we work together scientifically.246

Regarding the coordination of the teaching schedule, Hilbert would prove to be a congenial partner.247 On May 1, 1895, Klein and Hilbert started their first research seminar together.248 As early as June 22, 1895, Klein managed to have Hilbert elected (unanimously) as a full member of the mathematical-physical class of the Royal Society of Sciences.249 Thus the road was paved for a new prince of mathematics in Göttingen. Hilbert would remain faithful to Göttingen and to Klein despite receiving enticing job offers from Leipzig, Munich, Berlin (twice), Heidelberg, and Bern. When Hilbert received an offer from Leipzig (to succeed Sophus Lie), Klein wrote extensively to the Prussian Ministry of Culture in Berlin on June 27, 1898,

244 [UBG] Cod. MS. F. Klein 9: 1129 (Hurwitz to Klein, December 4, 1894). 245 [UBG] Math. Arch. 77: 255 (a letter from Klein to Hurwitz dated December 6, 1894). – Regrading this context, see also Appendix 6.1 and 6.2 in this book. 246 Ibid., 77: 257 (a letter from Klein to Hurwitz dated January 5, 1895). 247 See FREI 1985, p. 122 (a letter from Hilbert to Anna Klein dated March 6, 1895). 248 [Protocols] vol. 12. This seminar was devoted to differential calculus. Directed by Klein, it was conducted with the assistance of Hilbert, Ernst Ritter, and Arnold Sommerfeld. Of its seventeen participants, six were women. 249 [AdW Göttingen] Pers 16: 131; Chro 4, 6: 200.

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and in this letter he underscored the differences between himself and Hilbert to bolster his argument: I insist and confidently hope that everything will be done to keep Hilbert at our university. Hilbert is without doubt the most outstanding representative of our science in Germany among the younger generation. [...] I would like to add that Hilbert’s mathematics is quite different from mine. He lacks interest in the applications of mathematics, as well as in the organizational questions to which I am primarily devoting myself at the moment. Instead, he is completely focused on fundamental abstract investigations. I have always regarded this contrast to be a very fortunate addition […].250

When Hilbert refused to succeed Lazarus Fuchs at the University of Berlin in June 1902, he was assured an annual salary of 10,000 Mark in Göttingen (Klein’s was then 11,000 Mark). At the same time, Klein was also supportive of Hilbert’s close friend Hermann Minkowski receiving a so-called “personal ordinariate” position alongside them. Klein’s wife Anna was privy to all events. When Hilbert rejected an offer from Heidelberg in 1904, she wrote to her husband: I just received your letter, and I am quite pleased by Hilbert being offered yet another position. He is indeed an honorable and reliable man to have resisted this strong temptation, and he certainly did so in the interest of preserving Göttingen’s reputation. […] Now he will probably be safely in place with us [in Göttingen].251

In Königsberg, Hilbert had not had any doctoral students. While in Göttingen, he would supervise the doctoral procedures of sixty-nine mathematicians, including six women. The first of these students, Otto Blumenthal, described the professors Hilbert, Klein, and Heinrich Weber as follows: “[Hilbert], a man of middling stature who looked not at all like a professor, dressed unassumingly, and had a broad reddish beard […], stood out so oddly against Heinrich Weber’s venerable, stooped figure and against Klein’s commanding presence and beaming gaze.”252 Paul Kirchberger, who also studied with Klein and Hilbert, completed his doctoral thesis “Über Tchebychefsche Annäherungsmethoden” [On Chebyshev’s Approximation methods]253 in 1902, and later he compared both mathematicians as well. He described Klein’s tall, slender figure in tasteful clothing and his friendly, self-confident appearance. With his Rhenish humor, according to Kirchberger, Klein stood out above everyone else in the Mathematical Institute. For three decades, Klein would work harmoniously and amicably with the equally prominent Hilbert. Hilbert’s nature, however, was diametrically opposed to Klein’s in every respect.254

250 251 252 253

[StA Berlin] Rep.76 Va Sekt.6 Tit.IV Nr.1 Vol.XVII, fols. 97–97v. [UBG] Cod. MS. F. Klein 10: 284 (a letter from Anna Klein to Felix Klein, August 15, 1904). Quoted from HILBERT 1935 (GA III), p. 399. For an extract of this thesis, which was inspired by Hilbert, see Math. Ann. 57 (1903), pp. 509–40. 254 See KIRCHBERGER 1925, pp. 2–3.

8 THE FRUITS OF KLEIN’S EFFORTS, 1895–1913 Felix Klein had set the course for numerous developments, and this chapter will examine the results of his efforts up until the time of his early retirement. In 1912, the final volume of Robert Fricke and Klein’s monograph on automorphic functions was published. In the preface, Fricke remarked that Klein’s “harmonic creations” (harmonische Schöpfungen), which he called the “aestheticizing branch of present-day mathematics,” had resulted from “working out the inner relationships of many different disciplines.”1 Klein responded to Fricke’s comments: The fact that I myself left these “aestheticizing” developments behind is because I had already become fed up with them (by the time I came to Göttingen in 1886). Perhaps the possibility of being productive, with which nature had endowed me in this respect, had diminished by this point, but in fact I was piqued by other interests – by applied mathematics, by the encyclopedia, and by coordinating educational reforms. These pursuits were no less inherent to me and, given my advanced age, I was of course only able to promote them in an organizational manner.2

At the University of Göttingen, Klein pursued his often-expressed vision (see Sections 6.4.2 and 7.5.1) of establishing the fields – once represented by Gauß – of mathematics, the natural sciences, and technology on a higher level. To this end, he made use of government support and the backing of industry (Section 8.1). Klein struggled still to be received as a creative mathematician, and he formulated open problems for the applications of mathematics (8.2). At the same time, he increasingly devoted his time to the history, philosophy, psychology, and education of mathematics – the topics which, in 1896, he had planned to be addressed in the final volume of the ENCYKLOPÄDIE. This theoretical program was closely related to the initiative of reforming all of mathematical education, from kindergarten to the university level (8.3). Klein participated in further international projects, and he stood up against nationalistic tendencies (8.4). When the number of his obligations increased to such an extent that they began to occupy even his semester breaks, he decided to retire early (8.5). On April 16, 1896, “His Majesty the Emperor and King” Wilhelm II signed an official document that bestowed upon Felix Klein, who was then forty-seven years old, the status of a privy government councilor (Geheimer Regierungs-Rat).3 Not until six months later, however, when Klein turned down a professorship at

1 2 3

FRICKE/KLEIN 2017, p. xxvi (German original FRICKE/KLEIN 1912, p. V). [UA Braunschweig] A letter from Klein to Fricke dated December 4, 1911. [UAG] 5956, fols. 104–07 (submitted by the dean Wilamowitz-Moellendorff, Jan. 13, 1896). Norbert Wiener compared: “In German science, I may say, the social position of a Geheimrat was like that of a scientist in England who had been knighted” (WIENER 2017 [1956], p. 292).

© Springer Nature Switzerland AG 2021 R. Tobies, Felix Klein, Vita Mathematica 20, https://doi.org/10.1007/978-3-030-75785-4_8

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Yale University, did he feel as though his efforts were fully supported by the Ministry. Only then did he rise to become an important advisor to the Ministry’s influential director Friedrich Althoff.4 During his time in the ministry, Althoff saw many Prussian ministers of culture come and go, and he himself possessed the right to report directly to the emperor. He was authorized to make many decisions against the wishes of university faculties. About his own situation, Klein remarked: “The fact that I rejected this offer [to work at Yale University] without any further negotiations – because I felt bound by the undertakings and plans that I had initiated in Göttingen – ultimately tore Althoff away from his previous reservations […] and prompted him to provide active assistance.”5 8.1 A CENTER FOR MATHEMATICS, NATURAL SCIENCES, AND TECHNOLOGY The development of our cultural conditions is increasingly pushing toward the need for a certain number of people who are capable of using, in technical areas, their university education in mathematics and physics. […] You all know that Siemens has benefited from employing theoretical physicists in its major ventures. Another especially interesting example in this respect is the Zeiss Optical Institute in Jena, whose ever new and surprising achievements are the object of admiration abroad. This success has only been achieved because such an outstanding mathematician and physicist as Prof. [Ernst] Abbe has put all his theoretical knowledge at the service of the company.6

Klein expressed these thoughts in a talk that he gave on December 6, 1896, after which he sent the text to Robert Bosse, who was the Prussian Minister of Culture from 1892 to 1898. Despite the limitations that had been placed on him by his “truce” with the engineers’ association in Aachen (see Section 7.7), Klein forged ahead in his efforts to combine theory and technology. He was aware of the success of Ernst Abbe, who in 1889 had founded the Carl Zeiss Foundation in Jena with private resources and who used this foundation to support (state) institutions such as the University of Jena.7 Klein was also familiar with the technical institutes that had been established at private universities in the United States (see Section 7.4.3), and on this basis he developed a funding model that suited the government-run University of Göttingen. He brought together university researchers and scientifically inclined industrialists to form the “Göttingen Association for the Promotion of Applied Physics and Mathematics” (the Göttingen Association, for short), and this will be the topic of Section 8.1.1 below. The alliance that Klein organized among science, the state, and industry led to the creation of new examination regulations for applied mathematics along with new teaching assignments, professors, and institutes (Section 8.1.2). 4 5 6 7

On the so-called “Althoff system,” see BROCKE 1980. KLEIN 1923a (autobiography), p. 29. [UAG] Kur. 5956, fols. 99–100 (Klein’s speech, on December 6, 1896). See TOBIES 1984 and 2020b. There were no private universities in Germany at the time.

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Klein was always early to recognize and promote the potential of new research fields in applied mathematics. His own special engagement on behalf of aeronautical research offers new insights into his approach (Section 8.1.3). 8.1.1 The Göttingen Association Ever since the time of Gauß and W. Weber, it has been a tradition in Göttingen for mathematics and physics to make advancements together, not independently. Klein safeguarded this position with particular vigor, and he expanded it by including the technical sciences.8

In the article quoted here, Max Born compared Klein with Hilbert. Born was aware that the technical sciences were necessarily more expensive and he acknowledged (as did Hilbert) Klein’s ability to raise funding from industrialists (see also Section 9.4.2), who in turn expected to benefit from new research results and from the improvement of general education. Industrial production in Germany had received a considerable boost from the unification of the Empire in 1871. After the Franco-Prussian war, five billion gold francs flowed into the German economy as war reparations, and this led to the establishment of numerous scientific and technical companies. More than nine hundred new joint-stock companies (Aktiengesellschaften) were formed during the boomtime of the so-called “founding years” (Gründerjahre) from 1871 to 1873, and an additional 1,860 companies were created during the period from 1874 to 1889.9 Representatives from these companies participated in the founding of chemical and electro-technical associations and were also won over to support Klein’s goals in Göttingen.10 Founded on February 28, 1898, the “Göttingen Association for the Promotion of Applied Physics” expanded its scope to include applied mathematics on December 17, 1900. The following figures participated in the extension meeting: the chairman of the Göttingen Association, Henry Theodore Böttinger (a board member of the Eberfeld dyestuffs factories); Felix Klein, deputy and secretary of the minutes; Tonio Bödiker, chairman of the board at Siemens & Halske;11 Anton Rieppel, who became the director of the Machine Works of Augsburg and Nuremberg (MAN) in 1898; the Kurator of the University of Göttingen, Dr. Ernst Höpfner; and the Göttingen professors Theodor Des Coudres (applied electricity), Hans Lorenz (technical physics), Eduard Riecke (experimental physics), Woldemar Voigt (theoretical physics), Otto Wallach (organic chemistry),12 and Wilhelm Lexis (economics). Two members of the Association were listed as excused: Dr.

8 9 10 11

Max Born, “Hilbert und die Physik,” Die Naturwissenschaften 10 (1922), pp. 88–93, at p. 93. See Meyers Neues Lexikon, vol. 5 (Leipzig: Bibliographisches Institut, 1973), p. 701. See TOBIES 2002a, MANEGOLD 1968 and 1970, and also MEHRTENS 1990, pp. 377–401. Bödiker served as the president of the Imperial Insurance Office from 1884 to 1897. From 1897 to 1903, he was the chairman of the board at Siemens & Halske. 12 Wallach was awarded the Nobel Prize in 1910.

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Emil Ehrensberger, a representative of Krupp Cast Steel Works in Essen,13 and Theodor von Guilleaume, the general director of the Carlswerk Co. in Cologne,14 a wires and cables factory; it was here, in 1904, where the first telephone cable would be manufactured that connected Europe and America. In the session on December 17, 1900, Böttinger, Klein, and Rieppel had argued that applied mathematics and physics were complementary disciplines and that the funding required to support mathematics was considerably lower than in the case of electrical engineering and technical physics.15 The associate professors Friedrich Schilling (descriptive geometry and graphical statics) and E. Wiechert (geodesy, geophysics) were made university members of the Association. The Göttingen Association was a novel institution designed to promote applied mathematics, the natural sciences, and technology research at the University of Göttingen. Klein’s influence and role in this organization can be summarized in the following six points. First, Klein and Böttinger laid the groundwork for a scientist-driven research organization with the Göttingen Association, which would later become typical with the foundation, in January of 1911, of the Kaiser Wilhelm Society for the Advancement of Science (since 1948, the Max Planck Society). The Göttingen Association, however, functioned without any binding statute.16 Böttinger managed the finances; he himself made the largest personal donation to the organization (128,000 Goldmark), and he contributed another 900,000 Goldmark from his chemical company. As a successful entrepreneur, he recognized the value of innovation, and as a member of various business associations, he recruited additional industrialists to join the organization in Göttingen. All told, the Göttingen Association was joined by approximately fifty representatives from the chemical, electrical engineering, optical, and steel industries. By the time of Böttinger’s death in 1920 and the absorption of the Göttingen Association by the Helmholtz Society (see Section 9.4.2), these members had donated a sum of 2,318,900 Goldmark to promote its initiatives. Second, in their efforts to develop institutes at the University of Göttingen, Klein and Böttinger secured the collaboration of government authorities. Since 1889, Böttinger held a seat in the Second Chamber, the Lower House of Representatives (Abgeordnetenhaus), of the Prussian Parliament (Landtag)17 as a delegate of the National Liberal Party. In 1909, Böttinger became a member of the 13 When the promotion of aeronautical research became an urgent matter in 1909, Gustav Krupp von Bohlen und Halbach himself became member of the Göttingen Association ([UGB] Math. Arch. 5022: 10). 14 [StA Berlin] Nachlass Althoff, AI, No. 138, fol. 179. For a list of the fifty industrial members of the Göttingen Association and their financial contributions, see TOBIES 1986a, pp. 130–32. 15 [StA Berlin] Nachlass Althoff, AI, No. 138, fols. 179–84. 16 See KNOKE 2016. Efforts to establish a statute for the Association failed on account of the opposition of Anton Rieppel, who wanted to limit membership to the (founding) six to eight members. [UBG] Cod. MS. F. Klein 4F, fols. 78–79, 106–12. 17 For a list of delegates, see https://en.wikipedia.org/wiki/Landtag_of_Prussia.

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First Chamber, the Upper House (Herrenhaus), of this governing body, and as such he acquired state funding for projects in Göttingen. Friedrich Althoff, the director of the Prussian Ministry of Culture, was made an honorary member of the Göttingen Association on February 22, 1908, the year of his death (see Fig. 34). Even after Althoff’s death, Klein was able to maintain good relations with the Ministry because the University of Göttingen elected him as its representative in the Upper House of the Prussian Parliament in Berlin, where he served from 1908 to 1918 (see Sections 8.3.4, and 9.1.2).

Figure 34: The Göttingen Association for the Promotion of Applied Physics and Mathematics. An invitation to the celebration of its tenth anniversary, February 22, 1908. Chairman: Henry Theodore Böttinger (the “crowned moon” in the upper right) Deputy: Felix Klein (represented as the sun) Honorary member: Friedrich Althoff (represented as Zeus giving his blessing) ([UBG] Cod. MS. F. Klein 4E)

Third, it was Klein who gave reports about scientific advancements at the Association’s annual meetings. Here he also discussed his book projects on the applications of mathematics (the theory of the top) and on mathematical instruction (the ICMI monograph series), the latest results in theoretical physics, etc. The professors working in “applied” research areas had to prepare their own special research reports, and Klein encouraged them to apply to the Göttingen Association to receive funding for books, scientific instruments, and other resources that they might need to expand or renovate their institutes. It should be mentioned that Klein also succeeded in including the “pure” mathematicians Hilbert, Minkowski, and Edmund Landau (an expert in analytic number theory) as members of

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the Association.18 Landau’s participation in this organization is especially noteworthy given that he (reportedly) regarded applied mathematics as “lube oil mathematics” (Schmierölmathematik).19 Fourth, beginning in 1903, the industrial supporters of the Göttingen Association invited the members of this group to attend an additional annual meeting at one of their places of operation, and these meetings included tours of factories, museums, etc. The first of such gatherings took place in Elberfeld (at the dyestuffs factories); in 1904, it was held in Essen (Krupp); in 1905, in Berlin (Siemens & Halske); in 1907, in Nuremberg (MAN); and in 1908, at the B.G. Teubner publishhing house in Leipzig (see Section 5.6), for example. In May of 1912, Klein arranged for such a meeting to take place in Jena at the Carl Zeiss optical company,20 but, for the first time, he was unable to attend for health reasons. In 1914, he did attend the last of these meetings in Dessau, hosted by the German Continental Gas Company (Deutsche Kontinental-Gas-Gesellschaft).21 Fifth, Klein sought, with the support of the Göttingen Association, to reach an understanding with the engineers teaching at Technische Hochschulen who had sparked an “anti-mathematics engineering movement.” The issue was the relationship between theory and laboratory instruction in engineering education, in particular the extent and nature of the mathematical knowledge required of engineers. For a long time in Germany, theoretical mathematical education had been dominant, and some influential engineering professors opposed this.22 Klein reacted to this with a lecture titled “Universität und Technische Hochschule,”23 which he delivered in 1898 at the seventieth annual meeting of the Society of German Natural Scientists and Physicians in Düsseldorf. Here, he recognized the importance of technical professions, but he argued for the necessity of their mathematical basis. Klein based his argument on the example of the École Polytechnique in Paris, where mathematics traditionally possessed a high status. Klein had closely followed the developments taking place in technical disciplines. In his memorandum from 1888 (see Section 6.4.2), he had already made the case that engineers should be able to earn doctoral degrees (with the title “Dr.-Ing.”) from Technische Hochschulen. Now, at Althoff’s request, he was asked to review the new regula-

18 Replacing the recently deceased Minkowski, Edmund Landau was made a member of the Göttingen Association during its meeting in July of 1909 ([UBG] Math. Arch. 5022, fol. 9). For a list of all the university-affiliated members of the Association, see TOBIES 2012, pp. 58– 59. Regarding Hilbert’s and Minkowski’s participation in the Association’s meeting in 1904 and Landau’s involvement in its meeting in 1913, see KNOKE 2016, pp. 155–56. Hilbert’s participation is also mentioned in the Association’s records from the First World War, during which a meeting was held on November 16 and 17, 1917. [UBG] Math. Arch. 5030, fol. 15. 19 See OSTROWSKI 1996, p. 105; and ECKERT 2013, p. 170. 20 [UBG] Math. Arch. 5028. 21 See KLEIN 1918, p. 227. 22 For details on this movement, see HENSEL et al. 1989, pp. 52–82. 23 Klein’s lecture was published in the Jahresbericht der DMV 7 (1899) II, pp. 39–50. It can also be viewed online at https://gdz.sub.uni-goettingen.de/ed/PPN517154005.

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tions governing the conferral of such a degree.24 On October 11, 1899, to mark the centenary celebration of the Technische Hochschule in Charlottenburg (Berlin), the Emperor of Germany and King of Prussia Wilhelm II ultimately bestowed upon Prussian Technische Hochschulen the right to grant doctoral degrees.25 Klein and Böttinger were able to persuade Adolf Slaby and Alois Riedler, the most vocal opponents against the efforts to develop the applied sciences in Göttingen (see Section 7.7), to set aside their grievances. In July of 1900, Slaby and Althoff came to Göttingen to sign a (new) “peace agreement” with Klein, and this truce was communicated to the members of the Göttingen Association.26 Finally, the Association of German Engineers welcomed Klein’s suggestions for educational reform “in light of the growing importance of commercial issues” (see Section 8.3.4),27 and it invited him to discuss these matters with its members in September of 1904. In 1905, this organization, whose chairman at the time was Carl Linde, became a member of the Göttingen Association, and altogether it donated 18,500 Goldmark to institutions at the University of Göttingen. This coincided with a shift in attitude at the Association of German Engineers, which now regarded mathematics as an (important) basis of engineering education. This shift is reflected in the reports by the German Committee for Technical Education (Deutscher Ausschuss für technisches Schulwesen), which was formed in 1908 by the Association of German Engineers and the Association of German Mechanical Engineers.28 The mathematician Paul Stäckel, who was then a professor at the Technische Hochschule in Karlsruhe, represented the domain of mathematics on this committee. As Gert Schubring has shown, Stäckel coordinated closely with Klein and was able to implement a number of his suggestions.29 Sixth, unlike the later Kaiser Wilhelm Society, which concentrated exclusively on research, the Göttingen Association was a strong supporter of the broader educational reforms that were being proposed at the time.30 Klein was personally engaged in persuading the industrial members of the Göttingen Association to provide financial support for the development of new curricula, for the establishment of a new institute (Fachschule) for precision mechanics, for continuing education courses, for the creation of national and international education committees,31 for the development of scientific and technical courses for lawyers and administrators (the first of these took place in Göttingen and Hanover in the

24 [StA Berlin] Rep. 92 Althoff AI, No. 179, fols. 51–52v (Klein’s review, January 18, 1899). 25 Since 1900, this degree has also been granted in Saxony by the Technische Hochschule in Dresden (“Dr.-Ing.”), and since 1901 it has been conferred in Bavaria by the Technische Hochschule in Munich (there the title bestowed is “Doctor rerum technicarum”). 26 [StA Berlin] Rep. 92 Althoff AI, No. 138, fols. 179v–80. Regarding the earlier “Aachen truce” in 1895, see Section 7.7. 27 Quoted from the Zeitschrift des VDI 48 (1904), p. 1473. 28 On this committee, see LUDWIG/KÖNIG 1981, pp. 256–59. 29 See SCHUBRING 1989a, pp. 190–91. 30 See SCHÜTTE 2007. 31 [UBG] Math. Arch. 5031, fol. 24.

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summer of 1911), and for the establishment of a “Böttinger – Studienhaus” for exchange students, etc.32 On September 1, 1908, moreover, Böttinger donated 100,000 Mark to found a Wilhelm Foundation for Scholars to support members of the scientific academies in Berlin and Göttingen, the instructors at Prussian universities and Technische Hochschulen, and librarians. After Althoff’s death, Wilhelm II decreed on December 21, 1908 that this foundation, which had been named in his honor, would henceforth be known as the “Friedrich Althoff Foundation.” Klein and Böttinger were appointed to the board of this foundation.33 In a report to the members of the Göttingen Association written in April of 1911, Klein emphasized that the purpose of the organization was “to promote scientific research in the areas of applied mathematics and physics as well as education in the broadest sense.”34 The engineer and industrialist Anton von Rieppel even described the improvement of pedagogical training as the decisive motivation behind the foundation of the Göttingen Association: The founders of the association who came from industry were engineers. These included, in addition to Mr. [Henry] von Böttinger, Director [Wilhelm] Schmitz from the Krupp Company, Professor von [Carl] Linde, Commerce Councilor [Georg] Krauß, Commerce Councilor [Ernst] Kuhn,35 and myself. In his lectures, Privy Councilor [Felix] Klein established that we should aim to achieve the following goals: 1. above all, to work toward improving the training of future teachers; 2. to promote increased research in the field of applied sciences; and 3. to steer university policy down a path that was more in touch with practical life than was the case at the time. Above all, we agreed that the first point was the most important one, because we had repeatedly been confronted with the fact that young engineers, because of the inadequate and impractical education that they had received at secondary school, had to waste their time at university in order to make up for what, in our opinion, secondary schools could have provided them quite well […]. The founding idea of the Göttingen Association was to help improve these conditions.36

When Klein was formulating proposals for reforming mathematical education (see 8.3.4.1), he had also sent his suggestions to Rieppel, who responded approvingly: If you succeed in having mathematics – analytic geometry, elementary differential and integral calculus, the foundations of descriptive geometry – taught at secondary schools to the extent that the needs of architects and chemists are met, your current opponents would surely consider this a commendable success, and I would regard it as the fulfillment of many years of wishes.37

32 [UBG] Math. Arch. 5015–5024. 33 The board consisted of sixteen members, included Adolf von Harnack, Friedrich Schmidt-Ott, Hermann Diels, and Alfred Hillebrandt. [UBG] Cod. MS. F. Klein 2E, fols. 33–26. 34 [UBG] Math. Arch. 5026, fol. 22. 35 Ernst Kuhn was the owner of the G. Kuhn machine factory in Stuttgart. In 1895 and 1896, he was the chairman of the Association of German Engineers. See LUDWIG 1981, pp. 566, 576. 36 [UBG] Math. Arch., No. 5029, fols. 20–21 (a speech given by Rieppel on November 30, 1912). 37 [UBG] Cod. MS. F. Klein 2F, fol. 40 (a letter from Rieppel to Klein dated May 16, 1900).

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Rieppel donated 31,500 Goldmark to the Göttingen Association. At the organization’s meeting in Nuremberg in 1907, he donated an additional 2,000 Mark with the stipulation: “I request in particular that this sum be left at the disposal of Privy Councilor Klein for the mathematical division.” This funding was used to construct a new harmonic analyzer and to create mathematical tables (Tabellenwerke).38 These instruments were used by Carl Runge, who, owing to the establishment of new examination regulations, had meanwhile been hired in Göttingen as a full professor of applied mathematics. 8.1.2 Applied Mathematics in the New Examination Regulations and the Consequences Within the German Mathematical Society, the idea of new examination regulations for teaching candidates had been a topic of discussion since the 1890s. On June 2, 1897, Klein wrote to Althoff about his new ideas for these regulations: We are of the opinion that now might be the right time to redress the complaints of engineers (etc.) about the insufficient training that mathematics teaching candidates receive in the applied sciences – insofar as these complaints seem justified. In this respect, we have introduced three innovations to our plan, namely 1) the recognition of a certain number of semesters spent at a Technische Hochschule, 2) the inclusion of professors from Technische Hochschulen on the examination board, and 3) an appropriate definition of academic requirements […]. We have […] come up with the idea of establishing a new Facultas [teaching qualification for school teachers] for applied mathematics. I have written down the “requirements” on the back of the next page.39

The idea of establishing a “new Facultas for applied mathematics” for secondary school teachers was incited not only by the complaints of engineers but also by the thought that, as a result, “the Ministry will no longer be able to avoid granting appropriate teaching assignments at all universities and thereby creating an open path for the necessary development [of applied mathematics].”40 Klein outlined the following plan on the back of his letter to Althoff: Requirements for a teaching certificate in applied mathematics [i.e. the Facultas]: 1. Lower level. Elements of analytic geometry and of differential and integral calculus. The usual projection methods of descriptive geometry and the elementary aspects of technical mechanics, basic geodesy. 2. Upper level. A command of differential and integral calculus and its geometric applications. Projective geometry. Analytic mechanics. Advanced geodesy and probability theory.41

38 39 40 41

[UBG] Math. Arch. 5019 (the source of the quotation here) and 5022, fols. 5–8. [StA Berlin] Rep. 92 Nachlass Althoff B, No. 92, fols. 182v–183. KLEIN 1914a, p. 317. [StA Berlin] Rep. 92 Nachlass Althoff B, No. 92, fol. 185v.

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On June 27, 1898, Klein insisted to Althoff that now would be the time to introduce new examination regulations in order to support the intentions of the Göttingen Association.42 As a consequence, these regulations – the “Ordnung der Prüfung für das Lehramt an höheren Schulen in Preußen” [Examination Regulations for Teachers at Upper Schools in Prussia] – were decreed on September 12, 1898 and came into force on April 1, 1899. Klein remarked with certainty: “As soon as we have organized instruction in all of these areas, then, in accordance with the tradition of the university, new research clusters will be created everywhere.”43 Table 8: Applied Mathematics in the Prussian Examination Regulation for Teaching Candidates at Secondary Schools 1898 1. Descriptive geometry up to the theory of central projection and skills in graphical methods. 2. Technical Mechanics: Mathematical methods, esp. graphical statics. 3. Basic geodesy and elements of advanced geodesy alongside the theory of the adjustment of observation errors.

1917 Mastery of graphical and numerical methods (descriptive geometry, graphical calculation, approximate calculation) and their application to at least one of the following fields: 1. Astronomy 2. Geodesy 3. Meteorology and geophysics 4. Applied mechanics 5. Applied physics 6. Mathematical statistics and actuarial mathematics.

1921 Familiarity with the aspects of analysis most important to applications – particularly its computational, graphical, and instrumental methods – with descriptive geometry, mechanics (including graphical statics and kinematics), probability and approximation theory. More in-depth theoretical and practical studies in at least one of the following areas of application: 1. Astronomy, 2. Surveying, 3. Meteorology and geophysics, 4. Applied mechanics, 5. Applied physics, 6. Financial mathematics, mathematical statistics and actuarial mathematics, 7. Technical sciences (e.g. electrical engineering, heat engineering, aeronautical engineering, or the statics of structures).

By the year 1910, 178 students in Prussia had earned the new qualification to teach applied mathematics.44 Most of the latter went on to teach at secondary schools, though some of them found positions at special technical schools (Fachschulen). Because of this educational background, some of these graduates also became important experts in industrial research. The list of possible elective subjects that the new examination regulations made available (see Table 8) were developed relatively early on at the University of Göttingen and the University of Jena (the only university within the Ernestine duchies), both of which received strong financial support from industry to create professorships for applied mathematics and applied/technical mechanics. At most of the other German universities, there was at first only a teaching position for descriptive geometry.

42 Ibid., fol. 206v (a letter from Klein to Althoff dated June 27, 1898). 43 Quoted from TOBIES 1988a, p. 563. 44 See SCHIMMACK 1911, p. 62.

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The mathematician August Gutzmer was a professor in Jena, and he ensured (in close cooperation with Klein) that the Prussian examination regulations would be adopted there on January 17, 1900, and that the Carl Zeiss Foundation would finance the development of new institutions and the appointment of new professors. Klein’s former student Walther Dyck influenced similar developments in Bavaria, where he was a member of the state’s highest Board of Education (Oberster Schulrat).45 In a letter to Gutzmer, Klein expressed his vision of establishing full professorships for all applied areas of research: Indeed, I would like the representatives of applied subjects to become full professors. Then, however, we would need different representatives for applied mathematics and applied physics, at least two. In Göttingen, even if I leave aside Brendel and Bohlmann, we in fact still have four (on the one hand, Schilling and Wiechert; on the other hand, Lorenz and Simon).46

Klein would need five years to achieve this vision. Descriptive geometry / applied mathematics. The associate professorship that Klein had secured for Arthur Schoenflies in 1892 had to be refilled in 1899 when Schoenflies was made a full professor in Königsberg. Klein’s former doctoral student (and his first assistant) Friedrich Schilling received this position, but he had not been the first choice. Klein had formulated the following criteria for Schoenflies’s successor: “1. He must be a scientific man. 2. Within mathematics, he must represent the geometric approach and possibly also know descriptive geometry. 3. He must possess talent and experience as a teacher.” Regarding Schilling, Klein had remarked that he “fulfills points 2) and 3) in an outstanding way, but there are reservations about point 1).” At first, the top candidate for the position had been Georg Scheffers, who had been influenced by Sophus Lie, but Scheffers did not accept the offer. Thus a second shortlist of candidates was made, which Klein, as the director of the Mathematical-Physical Seminar, forwarded to the Ministry in Berlin: 1) Dr. Gino Fano, a lecturer at the University of Rome; and 2) Friedrich Schilling, a professor at the Technische Hochschule in Karlsruhe.47 Because the Italian mathematician Fano turned down the offer (see Section 3.4), only Schilling remained. Klein’s initial idea, which was to lobby for Schilling to be named a full professor of applied mathematics just one year later, was thwarted by Hilbert.48 In 1902, Klein was able to improve the position of the associate professors of applied mathematics by implementing a few innovations. First, Klein stepped down from his position on the examination committee as the person in charge of applied mathematics and nominated Schilling and Wiechert to take his place. Second, Klein made it possible for Schilling and Wiechert to participate as directors 45 On the developments in Jena, see TOBIES 2020b; in Bavaria, HASHAGEN 2003, pp. 347–60. 46 Quoted from TOBIES 1988b, p. 44 (Klein to Gutzmer, February 6, 1902). Martin Brendel, an astronomer, became a professor in Frankfurt/Main in 1907. On Bohlmann, see Section 7.6. 47 [StA Berlin] Rep. 76 Va Sekt. 6, Tit. 4, No. 1, Vol. XVII, fols. 167–70v (hiring recommendations in Klein’s handwriting, dated January 20 and February 11, 1899). 48 See SCHIRRMACHER 2019, p. 9.

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in the administration of the Mathematical-Physical Seminar. Third, Klein arranged for the former “collection of mathematical instruments and models” to be split into two sections. He himself remained the director of Section A (mathematical models), with which his own assistant was engaged, while Section B (graphical exercises and mathematical instruments) was put in the hands of Schilling, with a part-time assistant.49 In 1904, Schilling ultimately accepted a professorship at the newly founded Technische Hochschule in Danzig (today: Politechnika Gdańska in Poland). Now the path was clear in Göttingen for Klein to create the first full professorship for applied mathematics in all of Germany. This position was offered to and accepted by Carl Runge – “the founder of modern numerical mathematics,” in the words of the numerical analyst Lothar Collatz.50 Klein had kept Runge in his sights for a long time. As early as 1894, Runge had sent to Klein his programmatic article “Ueber Anwendungen der Mathematik” [On the Applications of Mathematics] and had explained, in a letter, his new iterative methods for the approximate solution of ordinary differential equations (known today as the Runge-Kutta methods).51 Runge changed Klein’s (initially disciplinary) understanding of applied mathematics: “Ever since our colleague Runge has been here, we have understood applied mathematics as the theory of finalizing mathematics [die Lehre von der mathematischen Exekutive], i.e., numerical and graphical methods, of which descriptive geometry forms a sidebranch.”52 With Runge as a full professor, Klein was able to come closer to his goal of developing “empirical sciences” into “calculating sciences.”53 In addition to courses on descriptive geometry, technical mechanics and geodesy, and astronomy, which were available from the beginning as possible elective areas in the exams for applied mathematics, other courses were gradually introduced: meteorology and geophysics, financial mathematics, and further technical sciences, including electrical engineering, heat engineering, aeronautical engineering, and the statics of structures. These new additions are reflected in the later revisions that were made to the examination regulations (see Table 8). Geodesy / geophysics. Emil Wiechert had already achieved outstanding results while working as a Privatdozent in Königsberg. In 1897, around the same time as the Scottish physicist Joseph John Thomson, he discovered the particle

49 [StA Berlin] Rep.76 Va Sekt. 6, Tit.4, No.1, Vol. XVII, fol. 167; and Vol. IX, fols. 164–72. 50 See COLLATZ 1990. 51 [UBG] Cod. MS. F. Klein 11: 646 (Runge to Klein, April 6, 1894). Wilhelm Kutta developed Runge’s methods further in his doctoral thesis at the Technische Hochschule in Munich, which he wrote under Walther Dyck and defended in 1901. See also TOBIES 2012, pp. 60–79; RICHENHAGEN 1985; and HENTSCHEL/TOBIES 2003, pp. 32–41. 52 [UBG] Math. Arch. 5029, fol. 25 (1908). 53 Carl Runge’s students, who included his own children Iris Runge and Wilhelm Runge, would go on to develop branches of applied mathematics at Technische Hochschulen and in the field of industrial research. See my own analysis of these developments in TOBIES 2012.

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that is known today as the electron.54 As an associate professor in Göttingen, Wiechert made advancements in the fields of surveying and photogrammetry. He also earned a reputation for his seismological research (for the Wiechert seismograph, the Wiechert seismological station, and as the founder of the Association Internationale de Séismologie in 1903). With strong support from Klein, Wiechert received a so-called personal full professorship for geophysics.55 Thus one of Gauß’s research fields was secured with an independent professorship. Applied electricity. Hermann Th. Simon had come to Göttingen in 1901 to succeed Th. Des Coudres as an associate professor of applied electricity (electrical engineering). He became known for his invention of a so-called “singing arc lamp,” which was a type of radiophonic instrument, and he founded a research facility for wireless telegraphy at the university. In 1907, when he was made a personal full professor in Göttingen, he thanked Klein for this promotion.56 Technical physics. Klein also intended to arrange for Hans Lorenz to receive a personal professorship. In 1900, Lorenz had become an associate professor and the director of the Institute of Technical Physics, which had been founded in 1895 (see Section 7.7). He was, however, still not the most fitting solution for this research area in Göttingen. On the one hand, he proved to be rather uncooperative; for instance, he turned down Klein’s offers to teach joint seminars with him. On the other hand, Lorenz lacked mathematical sophistication, as is evident from his own notes and from a critique of his papers by Richard von Mises.57 The university records reveal that Klein and Böttinger worked together to ensure that Lorenz would be lured away by a favorable offer from the newly founded Technische Hochschule in Danzig (Gdańsk) in 1904 (much like Friedrich Schilling).58 In October of 1904, Lorenz’s position in Göttingen was filled by Ludwig Prandtl, who came at the same time as Carl Runge, and also from the Technische Hochschule in Hanover. Prandtl was a student of August Föppl, whose own doctoral research Klein had supported in Leipzig (see Section 5.3.1). In comparison to Hans Lorenz, Prandtl possessed a deeper understanding of mathematics.59 Even though Prandtl’s mathematical abilities would be surpassed by younger scholars such as Theodore von Kármán and Richard von Mises,60 Lothar Collatz was right

54 In 1906, Joseph John Thomson was awarded the Nobel Prize for this discovery. 55 A personal professorship was tied to a given individual; the salary for this position was similar to that of an associate professorship (Extraordinariat). On August 11, 1904, Anna Klein wrote to her husband: “I’m curious when and how the decision about Wiechert will be made” ([UBG] Cod. MS. F. Klein 10: 283). 56 [UBG] Cod. MS. F. Klein 11: 1003 (H.Th. Simon to Klein, January 13, 1907). 57 [Deutsches Museum] HS 1993-001 (Lorenz’s memoirs). Regarding Richard von Mises’s criticism, see SIEGMUND-SCHULTZE 2018, pp. 483–85. 58 See TOBIES 1988b, pp. 44–45; and TOBIES 1988a, p. 265. Both Klein and Böttinger accepted an invitation from the Ministry of Culture to attend the inaugurating ceremony of the Technische Hochschule in Danzig, which took place in October of 1904. 59 Regarding Prandtl’s biography, see ECKERT 2019a [2017]. 60 See SIEGMUND-SCHULTZE 2018, pp. 483–85.

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to emphasize that Prandtl not only knew how to treat difficult hydrodynamic processes numerically but also that he contributed in a significant way to the development of numerical methods.61 When the threat arose of losing Ludwig Prandtl to the Technische Hochschule in Stuttgart, the University of Göttingen was able to retain him by offering him a personal full professorship. In addition, the Göttingen Association contributed to increase Prandtl’s salary substantially.62 From 1898 to 1908, the number of full professorships in Göttingen for physics and mathematics grew from five to ten.63 Klein conducted joint seminars with most of these new hires (see especially Section 8.2.4), and he paved the way for the creation of new institutes for his young colleagues. Institutes were established in Göttingen for technical physics (completed in 1897), applied electricity (1897), geophysics (1898, renovated in 1901), inorganic chemistry (1903), and applied mathematics and mechanics (1905, for Runge and Prandtl). The Institute for Technical Physics was expanded. The Institute for Physical Chemistry was extended (1898–1900); facilities were constructed for an Institute of Agricultural Bacteriology (1901);64 renovations were made to the Physical Institute and to its applied-electricity division (1905); an experimental facility was built for systematically measuring air resistance (1907/08); and a research facility was constructed for studying wireless telegraphy (1909). Klein’s dream of a new building for the Institute of Mathematics, for which plans were made in 1909 and funding was secured in 1914, would not be fulfilled until 1929 (see also Section 9.4.2). In the following section, Klein’s engagement on behalf of these technical institutions will be demonstrated with the example of aeronautical research. 8.1.3 Aeronautical Research Since the 1890s, there was little doubt that it would soon be possible to build a navigable aircraft.65 In his courses, Klein had already focused on questions of applied hydrodynamics (see 8.2.4), and he initiated a funding proposal with the recently founded International Association of Academies. Beginning in 1900, Eduard Riecke and Emil Wiechert were thus able to direct fundamental research in this subject (at first on the topic of atmospheric electricity), and this work was further supported with government funding (4,400 Mark).66 In 1903, the Wright brothers in the United States flew the first motor-operated airplane, and this occurred around the same time that Wilhelm Kutta and the Russian mathematician and engineer Nikolay Y. Zhukovsky published their theories 61 62 63 64 65

See COLLATZ 1990, pp. 271–72. [StA Berlin] Rep. 92, Nachlass Schmidt-Ott B 43, p. 16; and C 55, fol. 126. See KLEIN 1908a. This was the first institute for microbiology and genetics in Germany. Ludwig Boltzmann, “Über Luftschifffahrt,” Verhandlungen der Gesellschaft deutscher Naturforscher u. Ärzte, 65. Versammlung, Nürnberg 1893, vol. 1 (Leipzig: Vogel, 1894), p. 90. 66 See KLEIN 1909, p. 131.

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of aerodynamic lift.67 During the same time, in 1903, Emperor Wilhelm II enacted the establishment of the Telefunken Company for Wireless Telegraphy, LLC (Telefunken Gesellschaft für drahtlose Telegraphie mbH) in order to equip the army and navy with the latest technology (without any patent disputes between the companies involved). In the fall of 1905, he proclaimed the foundation of a Company for the Study of Motorized Aircraft, LLC (Motorluftschiff-Studiengesellschaft mbH), with a capital investment of one million Mark. Scientists, aviation pioneers, and representatives from companies and the military were brought together to coordinate research projects. At Althoff’s request, Klein – enthusiatic about the possibility of flying – received an invitation to participate in the inaugural meeting of this organization, which took place in Berlin on October 28, 1906. The meeting’s agenda was as follows: 1. 2. 3.

Formation of a technical committee. Proposal for the formation of groups. Report by the managing director on the status of the issue of the motorized balloon and on the question of whether the Zeppelin airship is worthy of support.68

The technical committee consisted of four groups: a meteorology group (spokesman: Richard Assmann), a dynamics group (Felix Klein), a construction group (Director Otto Krell),69 and a machine group (Adolf Slaby).70 Klein saw an opportunity to attract additional funding for research on hydrodynamics and aerodynamics in Göttingen. He immediately incorporated Prandtl and Wiechert into the dynamics group under his charge and he had them formulate a research proposal, which he sent to Berlin on December 15, 1906. They applied for resources to construct a facility for testing airship models (LuftschiffModellversuchsanstalt), with which they planned to investigate air resistance, measure the distribution of pressure on model balloons, study the distribution of airflow velocities on models to determine the ideal placement of propellers, test the stability of various balloon shapes, etc. At the next meeting in Berlin on January 7, 1907, Klein was promised 5,000 Mark to undertake preliminary work in Göttingen. He won a new member to join the dynamics group: Sebastian Finsterwalder (A. Brill’s former doctoral student) of the Technische Hochschule in Munich, who had written the article on aerodynamics (1902) for vol. IV of the ENCYKLOPÄDIE and who had developed one of the first methods for reconstructing spatial objects from survey photographs. At this same meeting, Klein also successfully applied for Prandtl to become an additional member of the construction group (headed by Otto Krell, see above). According to the minutes, Klein had presented some of Prandtl’s recent 67 See BLOOR 2001; ECKERT 2019a [2017]; and SIEGMUND-SCHULTZE 2018. 68 For this quotation and the information below, see [UBG] Cod. MS. F. Klein 7C: 1–106. 69 Otto Krell was the director of the division of military and shipbuilding technology at the Siemens-Schuckert Company in Berlin. 70 F. Althoff and some industrial members of the Göttingen Association (H. Th. von Böttinger, among others) were shareholders of the Company for the Study of Motorized Aircraft.

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work71 – published before the Göttingen wind tunnel was built – in order to justify this membership. The work in question concerned flows with considerable density fluctuations (gas dynamics) as well as supersonic flows and the resulting shock waves, which had already been theoretically predicted in 1858 by the Göttingen mathematician Bernhard Riemann.72 At a meeting on September 14, 1907, Klein reported that Henry Theodore von Böttinger had acquired a plot of land in Göttingen and that this would allow the university to move ahead quickly with the construction of the planned testing facility with the wind tunnel (Luftschiff-Modellversuchsanstalt). In 1907, at Klein’s instigation, Carl Runge was also made a member of Klein’s dynamics group. Initiated by Klein, Carl Runge, with his wife Aimée and their daughter Iris, had recently begun to translate the first volume of Frederick William Lanchester’s book Aerial Flight (1906).73 In 1908, Richard Assmann recommended studying the progress that was being made abroad, and Klein arranged for Prandtl, Runge, and Wiechert to travel to France.74 In Paris, this domain was being advanced by Paul Painlevé, who had spent a semester studying under Klein in Göttingen (see Section 6.2.3). Together with Émile Borel, Painlevé would publish a book on aeronautics.75 As early as 1908, Painlevé flew in one of the motorized airplanes designed by the Wright brothers, and in 1909 he introduced aeronautics as a university subject (Prandtl’s courses in Göttingen were thus not unique). Thanks to Klein’s initiative, Prandtl’s teaching appointment was expanded to include “the entire area of scientific aeronautics,” for which, as of April 1, 1909, he received 4,000 Mark in additional salary during each of the next three academic years.76 When, on October 11, 1910, Wilhelm II declared the establishment of the aforementioned Kaiser Wilhelm Society for the Advancement of Science, Klein expressed the idea, only six days later, of applying for the creation of a Kaiser Wilhelm Institute to conduct basic research in the fields of hydro- and aerodynamics in Göttingen.77 This idea was motivated in part by Prandtl’s argument that Göttingen, “as a mathematics university par excellence,” would be the ideal location for such an institution.78 Klein noted in 1913 that this institute had been approved 71 See Ludwig Prandtl, “Zur Theorie des Verdichtungsstoßes,” Zeitschrift für das gesamte Turbinenwesen 3 (1906) pp. 242–45; and “Neue Untersuchungen über die strömende Bewegung der Gase und Dämpfe,” Physikalische Zeitschrift 8 (1907), pp. 23–30. 72 Regarding the context, see KLUWICK 2000. 73 See TOBIES 2012, pp. 71–74. 74 [UBG] Cod. MS. F. Klein 1 D, fol. 145; and https://gdz.sub.uni-goettingen.de/id/DE-611-HS3214512 (p.14). For Runge’s report, see HENTSCHEL/TOBIES 2003, pp. 172–73. 75 Paul Painlevé and Émile Borel, L’aviation (Paris: Alcan, 1910). 76 [StA Berlin] Rep. 76 Va Sekt. 6, Tit. 4, No. 1, Vol. XXII, fols. 2–4. During the summer semester of 1909, Prandtl gave a lecture course on the scientific foundations of airship travel, and in the winter semester of 1909/10 he organized a seminar on the applications of aerodynamics. 77 See ECKERT 2019b, p. 71. 78 [StA Potsdam] RMdI Nr. 89, 70/1, fols. 173–74.

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with the support of Böttinger, who had a seat on the Kaiser Wilhelm Society’s board. On account of the First World War, however, only an expanded testing facility was built as a Research Institute for the Navy and Army. The final establishment of the Kaiser Wilhelm Institute for Fluid Dynamics in July of 1925 also required the engagement of Felix Klein, who used its creation as leverage to ensure that Prandtl was not hired away by the Technische Hochschule in Munich.79 In October of 1908, the Göttingen Association discussed the topic of aeronautical research. From the 3rd to the 5th of November in 1911, an “Aeronautical Research Congress” (Flugwissenschaftlicher Kongress) took place in Göttingen. This led to the founding of a “Scientific Society for Aeronautics” (Wissenschaftliche Gesellschaft für Flugtechnik) on April 3, 1912, which brought together scientists, engineers, industrialists, and government authorities who were active in this domain. Its first board of directors consisted of Böttinger, Prandtl, and the airship designer August von Parseval. The following mathematicians were original members of the society: Otto Blumenthal (Aachen), August Gutzmer (Halle), Felix Klein, Wilhelm Kutta (Stuttgart), Carl Runge, Horst von Sanden (Göttingen), Aurel Voß (Munich), and N.Y. Zhukovsky (Moscow).80 At this point, the Company for the Study of Motorized Aircraft had used up its capital, and its task was considered to be complete. The company was dissolved, and the model testing facility in Göttingen became the property of the Göttingen Association.81 Around the year 1900, not everyone had been pleased with Klein’s turn toward organizing aspects of applied disciplines. Engineers had viewed this as an encroachment into their territory. Some mathematicians, too, frowned upon this activity, as can be seen from the following remarks by Otto Blumenthal: Klein has now turned completely to the scientific-astronomical side. He is no longer a mathematician; he is a general exact-scientific organizing angel, a change which can be very pleasant for you and which I find very distasteful. Personal stimulation in the mathematical field can no longer be expected from him.82

Blumenthal would first need to work at a Technische Hochschule (Aachen) – a professorship for which he had been recommended by Klein and Sommerfeld – before he likewise realized “that the inner content and outer appreciation of mathematics would gain much if our most capable forces were to concentrate on engineering mathematics and contribute to it.”83 Later, when Blumenthal dedicated a

79 It was Klein who garnered the necessary support for the establishment of this institute (see TOLLMIEN 1987, p. 465). Regarding the support provided by the Ministry of Culture for this undertaking, see SCHMIDT-OTT 1952, p. 27. 80 [StA Berlin] Rep. 92, Nachlass Schmidt-Ott, B 43, fols. 14–19. 81 [UBG] Math. Arch. 5029, fols. 10–12. 82 Blumenthal to the astronomer Karl Schwarzschild, August 15, 1898 (this English translation is quoted from ROWE 2018b, p. 89). 83 [StB Berlin] Sammlung Darmstaedter, Blumenthal, H 1910 (15).

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commemorative plaque at Klein’s birthplace, he ultimately extolled Klein and his “unprejudiced, broad vision […].”84 Already in 1899, while a professor at the Polytechnikum in Zurich, Hermann Minkowski had written approvingly about Klein’s applied program. Even though he had a rather condescending view of the theory of the spinning top (see Section 8.2.3) and he believed that he himself would be able to do much better, he was prepared to be supportive of Klein, as he informed Hilbert: Otherwise, I still work a great deal on applications [of mathematics]. From thermodynamics, I came to chemistry. I always think that, one day, I will defend Klein against his many opponents and show that mathematicians really can do something for practice (and something better than merely determining the movements of the spinning top).85

When Hilbert turned down the aforementioned offer from the University of Berlin in 1902 (to succeed Lazarus Fuchs), the Ministry of Culture agreed to hire Hermann Minkowski at the University of Göttingen on October 1, 1902, and thus further strengthened the center for mathematical and scientific research there.86 8.2 MAINTAINING HIS SCIENTIFIC REPUTATION I am of course very pleased that you have taken up the automorphic functions again, and also for general reasons: We can only be effective in our organizational plans if, at the same time, we maintain a scientific reputation among our colleagues in the field [of pure mathematics]!87

“When kings build, the carters have plenty to do.”88 Leopold Kronecker was of the opinion, however, that this aphorism by Friedrich Schiller did not apply to mathematicians, “for, among us, every researcher must be a king and a laborer at the same time.”89 Nevertheless, Klein was able to find more than a few students to take his sparkling ideas further and present them systematically. Of course, the task of transcribing and editing his lectures, which many of his students and assistants undertook, can also be described as drudgery. One of Klein’s long-term collaborators of this sort – someone who benefitted from the “king” but also did much of his hard labor – was Robert Fricke, who 84 BLUMENTHAL 1928, p. 3 (a speech on behalf of the German Math. Society, Oct. 12, 1927). 85 Quoted from MINKOWSKI 1973, p. 113 (Minkowski to Hilbert, February 11, 1899). 86 This third professorship for mathematics came about quickly because, upon Wilhelm Lexis’s suggestion, the university was able to create it out of a “still vacant full professorship for inorganic chemistry” (see TOBIES 1991c, p. 104). 87 [UA Braunschweig] A letter from Klein to Fricke dated October 13, 1903. 88 Schiller’s full distich, which concerns I. Kant and his interpreters, reads: “Wie doch ein einziger Reicher so viele Bettler in Nahrung / Setzt! Wenn die Könige bau’n, haben die Kärrner zu thun” [“See how a single rich man gives bread to an army of beggars! / When the sovereigns build, carters have plenty to do.”]. The translation is by M. Mattmüller (see also C.D. Warner, et al., Library of World’s Best Literature. New York 1897, vol. XXIII, p. 12905). 89 A letter from Kronecker to Georg Cantor (September 18, 1891), published in the Jahresbericht der DMV 1 (1892) II, p. 23; and in KRONECKER 1930, p. 497.

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completed their joint book project on automorphic functions during these years (Section 8.2.1). In the interest of this monograph on automorphic functions, Klein turned his attention yet again to number theory. His results in this field, which he derived geometrically, were held in high regard by Charles Hermite. However, Klein recognized the limitations of his geometric approach, and he allowed Hilbert to direct their joint seminar on number theory (8.2.2). Arnold Sommerfeld can be regarded as another mathematical “laborer” who worked under Klein’s direction (see Section 7.1), but he managed to produce a multi-volume work based on Klein’s lectures on the theory of the top (8.2.3). Klein’s broad interest in promoting the applications of mathematics led him to recognize a number of open or unsolved problems, and thus with these in mind he inspired innovative research in the domains of fluid dynamics, the statics of structures, friction theory, and the theory of relativity (8.2.4). 8.2.1 Automorphic Functions (Monograph) Certain aspects of this project have already been discussed above in Section 5.5.4. The subject of the book was still in a state of flux, and Klein’s first plan for it (see Section 6.3.4) was constantly being refined. Klein recognized the yeoman’s work that Fricke was putting into it, and in a letter from September 13, 1894, he suggested that Fricke’s name should come first on the title page. Yet soon thereafter, when Fricke demanded a greater financial share in the book’s proceeds, Klein reclaimed the project for himself. He noted that he had indeed allowed Pockels and Bôcher to receive all the proceeds from their editions of his work,90 but the latter did not have permanent positions. Fricke, however, had now become a professor at the Technische Hochschule in Braunschweig. Klein wrote decisively: Dear Robert! […] The fact of the matter is that this project is my own plan. It is part of my scientific life’s work, which I have been preparing for five years now in all possible ways through my lecture courses and through the work of others that I have instigated. And I have not stopped doing so recently. Rather, I wrote to you in the summer that my current lecture on number theory should directly benefit the planned first volume (by classifying and clarifying the material from a different angle), and that I intend to give a major lecture on the subject of automorphic functions in the coming years.91

Klein proposed to his nephew (by marriage) that they should “part in peace” and that he only wanted to concentrate on his lecture courses. This prompted Fricke to come around, and three years later the first volume of the book was finally published with the subtitle Die gruppentheoretischen Grundlagen [The Group-Theoretical Foundations]. In his preface, Fricke explicitly mentioned the valuable pre90 On Pockels’s and Bôcher’s editions of Klein’s lectures, see Section 6.3.5. 91 [UA Braunschweig] A letter from Klein to Fricke dated November 18, 1894.

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liminary work of H.A. Schwarz (on hypergeometric series) and Lazarus Fuchs (on linear differential equations), he referred to Klein and Poincaré as “the real founders of the theory of automorphic functions,” and he listed the lecture courses (Klein’s and his own) upon which the monograph was based.92 On December 9, 1900, by which point Klein’s time had increasingly been taken up by other projects, he now offered to split the proceeds of the second volume of the monograph two-to-one in Fricke’s favor.93 With a watchful eye, however, Klein continued to follow and promote every new research approach. For example, he wrote to Fricke on September 4, 1901: You have touched right on the central difficulty that subjectively stands in the way of our second volume on automorphic functions: the existence methods on which we have to base our work have essentially only been created in the last fifteen years by the new French school (Picard, Poincaré); neither of us worked on this at first, and yet now we are supposed to provide a full report on it! The situation is even worse when I add that the methods may have been greatly simplified and generalized by Hilbert over the last two years, but his contributions are only partially available in a fully worked-out form. On the other hand, there is the objective consideration that there is no better and more fruitful evidence for the new methods (including all the ideas of set theory) than the theory of automorphic functions.94

Vol. II of the Lectures on the Theory of Automorphic Functions was given the subtitle Die functionentheoretischen Ausführungen und die Anwendungen [The Function-Theoretical Explanations and Applications], and it was published in multiple fascicles. Klein repeatedly advised Fricke about how to proceed: As far as the specific consideration of set theory is concerned, there is probably no essential difference between your current view and the one that I advocate. This is in no way a matter of improving the actual central idea of continuity, as it appears in my work and Poincaré’s, but only of making this idea acceptable to such people who are less inclined to perceive continuity. This can be done by breaking it down into small steps in the language of set theory with which they are familiar.95

Klein continued to work on the task of “making this idea acceptable,” and such work included his aforementioned four-semester seminar with Hilbert and Minkowski, which ultimately resulted in Koebe’s proofs of Klein’s theorems (see Section 5.5.4). Supervised by Klein, Wilhelm Ihlenburg also completed his doctoral thesis “Über die geometrischen Eigenschaften der Kreisbogenvierecke” [On the Geometric Properties of Circular-Arc Quadrangles] (1909) in this area. When Poincaré came to Göttingen to give a series of lectures in April of 1909, Klein’s lecture on new developments in the area of automorphic functions was included in the week’s program.96 In 1911, when a session on automorphic func92 93 94 95 96

FRICKE/KLEIN 2017 [1897], pp. xxv–xxviii [UA Braunschweig] A letter from Klein to Fricke dated December 9, 1900. Ibid., a letter from Klein to Fricke dated September 4, 1901 (emphasis original). Ibid., a letter from Klein to Fricke dated October 13, 1903. During Poincaré’s presence, the Göttingen Mathematical Society met additionally on Thursday, April 23, with lectures by Hilbert and Klein; and on Tuesday, April 27, with lectures by Landau and Zermelo; see Jahresbericht der DMV 18 (1909) Abt. 2, p. 79.

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tions was held at the annual meeting of the German Mathematical Society in Karlsruhe, it was Klein who gave the first talk on the panel, followed by Brouwer, Koebe, Bieberbach, and Emil Hilb.97 The last fascicle of vol. II of the monograph, which was published in 1912, included the latest research findings and culminated in the continuity proofs of Klein’s theorems.98 In 1913, when Klein was nominated to become a member of the Berlin Academy, the professors H.A. Schwarz, G. Frobenius, F. Schottky, and M. Planck wrote in their nomination letter that Klein had produced a large, multi-volume textbook on analysis that was full of significant geometric methods (see Appendix 9). After a long delay – and after the deaths of Weierstrass, Kronecker, and L. Fuchs – they finally paid tribute to Klein’s geometric style of thinking. Conrad H. Müller, who would complete a Habilitation under Klein on the history of mathematics (see 8.3.1), classified Klein’s books published with Teubner: In 1882, F. Klein published a book titled On Riemann’s Theory of Algebraic Functions and Their Integrals, which, on an intuitive and geometric-physical basis, provides an exposition of this theory and, at the same time, served as a foundation for his additional publications on function theory. That book was followed in 1884 by his Lectures on the Icosahedron, the first of several further books on the comprehensive theory of automorphic functions of one variable. Here, the algebraic cases of these functions are settled in the simplest way with a function-theoretical treatment of the geometric theory of regular polygons. The first important special case of transcendental automorphic functions is treated in Klein’s Lectures on Elliptic Modular Functions (1890/92), which he wrote with Fricke’s editorial assistance, while Klein continued to discuss the “general theory of automorphic functions” in his ongoing lecture courses (1897 etc.). Riemann’s theta functions and the theory of characteristics are addressed in the works of [Friedrich] Prym, [Adolf] Krazer – who published a book of his own on theta functions in 1903 – and then further by [Georg] Rost, whereas the other side of this research area is represented by [Friedrich] Schottky’s Theory of Abelian Functions (1880) and [Wilhelm] Wirtinger’s Studies of Theta Functions (1895). Finally, [Hermann] Stahl wrote a textbook on this area of study: The Theory of Algebraic Functions (1896) […].99

8.2.2 Geometric Number Theory In fact, I am often pessimistic regarding my scientific performance. I have too many general concerns and no time at all to concentrate on individual issues. Thus I am very much afraid that you will be disappointed in my number theory. Perhaps closer interaction with Hilbert will rejuvenate me!100

Klein was lecturing for four hours a week on number theory when he wrote this to Hurwitz. In his lecture course, Klein had just finished discussing “Lagrange’s theory of the continued-fraction development of quadratic irrationalities based on the idea of the lattice,” but he was still uncertain about his own scientific output. 97 98 99 100

Reports in Jahresbericht der DMV 21 (1912) Abt. 1, pp. 153–66. See FRICKE/KLEIN 2017 [1912], pp. 229–448; and Section 5.5.4 above. In TEUBNER 1908, pp. xiv–xv. Like Krazer, Georg Rost was a student of Prym in Würzburg. [UBG] Math. Arch. 77: 255 (a letter from Klein to Hurwitz, December 6, 1894).

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Klein had been excited when he discovered Gauß’s geometric approach to the subject, according to which “any positive binary quadratic form ax2 + bxy + cy2 can be interpreted geometrically by a parallelogram lattice [Gitter].” He had already presented on this topic to the Göttingen Mathematical Society (see 7.2), published his preliminary findings in the Göttinger Nachrichten, and lectured on ideal numbers in his Evanston Colloquium (lecture 8). Going beyond Gauß, Klein had emphasized that his approach, with the right choice of components, “could produce theorems on the composition of forms directly by means of geometry.” 101 Hilbert spent the beginning of 1895 in Göttingen to prepare for his move there from Königsberg. During this stay, Klein discussed with Hilbert his approach to the theory of continued fractions, and he added in a letter: “Please tell me what Minkowski has to say about my idea regarding continued fractions; I will probably have to follow this matter more closely these days.”102 Hilbert replied: I spoke with Minkowski at length about your geometric interpretation of the ordinary continued fraction for quadratic irrationalities. Like me, he considers it to be new, and he only brought to my attention Poincaré’s two notes in the Journal de l’École polytechnique (1880) and the Comptes rendus (1884), which also present a geometric but less simple and applicable conceptualization of continued fractions.103

In Minkowski’s Geometrie der Zahlen [Geometry of Numbers], we read that Poincaré’s “geometric conceptualization of normal continued fractions (1880, 1884) is less applicable to the true nature of approximate fractions.”104 Encouraged by Hilbert’s letter, Klein chose the topic “Zur Theorie der gewöhnlichen Kettenbrüche” [On the Theory of Ordinary Continued Fractions] for his upcoming lecture at the 1895 meeting of the German Mathematical Society in Lübeck. Here he mentioned, among other things, that new aspects pertaining to higher forms would result from his approach, which his doctoral student Philipp Furtwängler was developing further, and that the subject as a whole would be included in his monograph on automorphic functions.105 After Klein had presented his results again in Göttingen and published them,106 Hermite immediately arranged for Klein’s article to be translated,107 and he expressed his enthusiasm about Klein’s results in flowery language in a letter to the translator Léonce Laugel: L’article de M. Klein sur la représentation géométrique des développ.[ements] en fract.[ion] cont.[inue] dont vous m’avez envoyé la traduction m’a intéressé au plus haut point. Il est l’aurore ; il contient l’annonce d’une conquête mathématique glorieuse à laquelle personne n’applaudira plus que moi; vous savez que je l’ai entreprise vainement et n’ai jamais cessé de 101 102 103 104 105 106

See Göttinger Nachrichten (1893), pp. 106–09; repr. in KLEIN 1923 (GMA III), pp. 283–86. Quoted from FREI 1985, p. 120 (a letter from Klein to Hilbert dated January 10, 1895). Quoted from ibid., p. 120 (Hilbert to Klein, January 16, 1895). MINKOWSKI 1910, p. 162. See Jahresbericht der DMV 4 (1897) III, pp. 153–54. Göttinger Nachrichten: Math.-physik. Klasse (1895), pp. 357–59; reprinted in KLEIN 1922 (GMA II), pp. 209–11. 107 Felix Klein, “Sur une représentation géométrique du développement en fraction continue ordinaire,” Nouvelles annales des mathématiques 15/3 (1896), pp. 327–31.

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regretter mon insuccès. […] L’observation joue, je le crois, un rôle capital dans les recherches sur les nombres; […] elle peut révéler les mystères qui se jouent de nos efforts, tant leur nature est cachée. Les nombres s’offrent d’eux mêmes et tout d’abord à l’étude; ils se trouvent comme en première ligne bien avant les transcendantes de l’analyse et ils se dérobent comme jaloux de garder des secrets qui sont pour nous d’un si grand prix. […] Je serai heureux de voir […] l’illustre Klein entrer comme un nouveau Josué dans la terre promise […].108

On account of Hermite’s enthusiasm about his work, Klein would soon be made a member of the Section de Géométrie of the Académie des Sciences in Paris (see Section 7.4.4); Hermite and Picard congratulated him in a telegram.109 After Hilbert had permanently resettled in Göttingen in 1895, Klein conducted joint research seminars with him. In their seminar on number theory during the winter semester of 1895/96, Klein let Hilbert direct the discussions concerning more abstract methods. He explained this decision to Hurwitz as follows: In the seminar that I am conducting with Hilbert, we have discussed the ideal theory of the quadratic field under Hilbert’s direction, and I also intend to comment on this topic in the summer semester. For me, Hilbert’s representation is too abstract. Of course, I understand all the details, but the whole does not interest me in this form. Thus I hardly believe that I will pursue my number-theoretical studies any further. My entire goal is to provide a clearer representation of the theory of binary forms, which will involve making extensive use of my new representation of the development of the continued fraction (in the lattice) and of your article in Mathematische Annalen 45, etc., etc. This will again result in an autograph publication of my lecture course.110

The publication of Klein’s lecture courses on number theory (1895/96 and 1896), which were prepared by Sommerfeld and Furtwängler, contains his geometric interpretation of continued fractions, his “theory of singular elliptic entities,” and his “theory of singular values related to the icosahedron irrationality.” Klein stressed in these courses that his geometric methods could certainly be useful for higher areas of number theory, and he referred to the “approaches that Minkowski has provided in the first part of his promising work (The Geometry of Numbers, 1896)”.111 In comparison to Minkowski’s theory of space lattices (Raumgitter), however, Klein admitted that his own approach was less ambitious: “I have limited myself to clarifying already known fundamentals geometrically, whereas Minkowski undertook the discovery of new things.”112 It was Klein who had incited Minkowski and Hilbert to write the report on number theory for the German Mathematical Society, and they had already presented a provisional plan for it in 1895 at the annual meeting in Lübeck. This work had resulted in the aforemen108 109 110 111

[UBG] Cod. MS. F. Klein 10: 631/Anl. (Hermite to Laugel, January 6, 1896). Ibid. 114: 25. [UBG] Math. Arch 77: 262 (Klein to Hurwitz, January 29, 1896); [Protocols] vol. 12. Felix Klein, “Autographirte Vorlesungshefte III (Ausgewählte Kapitel zur Zahlentheorie),” Math. Ann. 48 (1897), pp. 562–88; reprinted in KLEIN 1923 (GMA III), pp. 287–314 (the quotations here are from pp. 287–88). – Minkowski’s complete book – Geometrie der Zahlen (B.G. Teubner, 1910) – was published posthumously from his estate by Hilbert and the Swiss mathematician Andreas Speiser, who had completed his doctorate in Göttingen in 1909. 112 KLEIN 1976 [1926], p. 309.

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tioned first part of Minkowski’s book and in Hilbert’s book-length “Zahlbericht” [Number Report] (1897).113 Klein continued to refine and explain his own methodological approach,114 which proved to be inspiring to younger mathematicians. Among those who were stimulated by Klein’s approach was Georg Pick, who, in 1899, first described his famous theorem (“Pick’s theorem”) for determining the area of polygons and who referred to Klein’s geometric conceptualization of continued fractions in his proof.115 Klein’s doctoral student Furtwängler expanded this topic to include any number of dimensions; in doing so, he classified Klein’s approach as follows: An especially intuitive representation of ideal theory is obtained by examining the decomposable forms belonging to the field and by interpreting them through spatial point lattices. The quantities which are to be adjugated to the given field for the purpose of establishing unique decomposability into prime factors then appear directly as entire algebraic numbers of a higher number field and, what is more, are represented by the point lattices in a way that is just as simple as it is intuitive. This idea was first proposed by Mr. F. Klein; the purpose of the following article is to develop Klein’s lattice figure for any given algebraic number field.116

Furtwängler achieved outstanding results in number theory, some of which have been regarded as approaches to the solution of Hilbert’s twelfth problem.117 As a professor at the University of Vienna, Furtwängler supervised numerous students who would rise to prominence, Olga Taussky among them. Algebraic, analytical, and geometric approaches to number theory all have their legitimate place today. Some textbooks still refer to Klein’s results.118 The more recent area of algebraic arithmetic geometry – Gert Faltings and Peter Scholze were awarded the Fields Medal119 for their work on this topic – is based on new epistemic foundations.120 Although Felix Klein concentrated on geometric approaches, he appreciated Hilbert’s algebraic number theory and also held Edmund Landau’s analytical methods in high esteem.121 Landau came to Göttingen on April 1, 1909 to replace Minkowski, who had died on January 12th of the same year. Landau had written a thesis on a self-cho113 Hilbert, “Die Theorie der algebraischen Zahlkörper,” Jahresbericht DMV 4 (1897), pp. 175– 546; trans. I.T. Adamson, The Theory of Algebraic Number Fields, Berlin: Springer, 1998. 114 See KLEIN 2016 [31924], pp. 42–46. 115 Georg Pick, “Geometrisches zur Zahlenlehre,” Sitzungsberichte des deutschen naturwissenschaftlich-medicinischen Vereines für Böhmen 19 (1899), pp. 311–19 (see esp. p. 318). 116 P. Furtwängler, “Punktgitter und Idealtheorie,” Math. Ann. 82 (1921), pp. 256–79, at p. 256. 117 See MASAHITO 1994. 118 See, for instance, Harold M. Stark, An Introduction to Number Theory (Cambridge, MA: MIT Press, 1978); and Oleg Karpenkov, Geometry of Continued Fractions (Berlin: Springer, 2013), pp. 215–34, 249–79. – Many thanks to Nicola Oswald for this information. 119 The Canadian mathematician John Charles Fields devised $47,000 for a Fields Medal fund; in 1894/95, he had studied with Klein in Göttingen, see [Protocols] vol. 12, p. 371. 120 See Allyn Jackson, “The Work of Peter Scholze,” Notices of the American Mathematical Society 65 (2018), pp. 1286–87; GRUBER 1990; and SCHAPPACHER 2007, 2010, 2015. 121 See KLEIN 1979 [1926], pp. 312–14.

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sen topic and completed his doctorate under Frobenius at the University of Berlin in 1899. As a Privatdozent there (since 1901), he had already offered lecture courses on the theory of Riemann’s zeta function and its application to number theory, and on the distribution of prime numbers.122 Whereas Landau’s career did not advance in Berlin, he was recognized in Göttingen as a rising star. Even though he was listed third in the hiring proposal (behind Hurwitz and Blumenthal), he was also deemed to be “the scientifically most multifaceted and effective mathematician of his generation.” Among other things, the hiring committee remarked: “His most important works on number theory concern the question of the density [Dichtigkeit] of prime numbers, and he has succeeded in transferring his results, obtained here according to Riemann’s method, to prime numbers in arithmetic progression and to the prime numbers of algebraic number fields.”123 On the day that he accepted the offer, Landau wrote to Klein, “[…] I am well aware how much I owe this opportunity to your supportive advocacy on my behalf.”124 The hiring committee had consisted of Hilbert, Klein, Karl Schwarzschild, and Woldemar Voigt.125 The hiring process confirmed Klein’s statement: “When I served on hiring committees for pure mathematics, which were necessary at the University of Göttingen, my primary aim was to add depth to the faculty by attracting an excellent candidate whose expertise differed from my own.”126 8.2.3 A Monograph on the Theory of the Spinning Top In the same letter from January 29, 1896, in which Klein had informed Hurwitz that he had no interest in Hilbert’s abstract number theory, he also wrote enthusiastically about another one of his ongoing projects: I am all the more pleased with my course on the motion of the top. In a curious way, I have made good progress there beyond Hermite in that I represent the rotations by ζ = αZ + β and γZ +δ define α, β, γ, δ as a function of T [= time]; these will namely become simple Θ quotients (with only one Θ in the numerator and denominator). But this is only incidental; my actual purpose is to convey to my students, with the example of the top, a complete understanding of the way in which movements occur and how these can be represented with formulae, so that engineers and physicists might really benefit from this. The whole rivalry with Technische Hochschulen aside, I hope to make the matter useful if I continue to go in this direction.127

Klein was inspired to work on the theory of the spinning top during his trip to Paris in 1887 (see Section 7.3). He later revisited the subject when, with an eye 122 See BIERMANN 1988, pp. 175–77. 123 [StA] Rep.76 Va Sekt.6, Tit.4, No.1, vol.XXII, fols. 21–27v (at fol. 26v). See also Landaus’s two-volume Handbook on the Theory of the Distribution of Prime Numbers (Teubner, 1909). 124 [UBG] Cod. MS. F. Klein 10: 609 (Landau to Klein, February 15, 1909). 125 [UAG] Phil. Fak. III, vol. 5, fol. 50. 126 KLEIN 1923a (autobiography), p. 32. 127 [UBG] Math. Arch. 77: 262 (Klein to Hurwitz, January 29, 1896).

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toward his contacts with teachers and engineers, he wanted to elucidate a practical topic in greater detail. This idea was supported by Gustav Holzmüller, the director of the Royal Provincial Trade School (Provinzial-Gewerbeschule) in Hagen, whom Klein had visited in 1895.128 In his lecture courses of 1895/96, Klein discussed the rotation problem on the basis of the theory of functions of complex variables and on the results presented in his icosahedron book. The coefficients α, β, γ, δ (see the quotation above) were represented as appropriate functions of time (T). In a short note submitted to the Göttinger Nachrichten on January 11, 1896, Klein emphasized “that α, β, γ, δ (in Hermite’s notation) become such elliptic functions of the second kind that contain only a single theta function in the nominator and denominator.”129 As early as January 27, 1896, Hermite reacted positively to the fact that Klein had attempted to approach the equations of motions of the top with Hermite-Lamé differential equations.130 Putting in many hours of “Felix duty” (see Section 7.1), Sommerfeld worked further on this topic and wrote their coauthored monograph Theorie des Kreisels [The Theory of the Top] in four parts (1897, 1898, 1903, and 1910) – 966 pages in all. Fritz Noether, a son of Max Noether, contributed extensively to the fourth volume. A new edition of the first three parts was issued in 1921, and an English translation of the entire book was published from 2008 to 2014. Parts III and IV, which were devoted to applications, went beyond what Klein himself had prepared in his courses, talks, and prior publications. There he had worked out the theoretical foundations of the subject. The four lectures that he gave in 1896 at Princeton University under the general title “The Mathematical Theory of the Top” (October 12–15, 1896),131 and another lecture, “The Stability of the Sleeping Top,” which he delivered at the conference of the American Mathematical Society on October 17, 1896, quickly gained international recognition. The latter lecture was immediately translated into French.132 In his third lecture at Princeton – “Concerning the Multiplicative Elliptic Curves” – Klein was sure to draw attention to the dissertation of his student Mary F. Winston: “Über den Hermiteschen Fall der Laméschen Differentialgleichung” 128 See KLEIN 1922 (GMA II), p. 509, 658–59. 129 Felix Klein, “Über die Bewegung des Kreisels,” Göttinger Nachrichten (1896), pp. 3–4; reprinted in KLEIN 1922 (GMA II), pp. 616–17. 130 Laugel had translated this article for Hermite: Felix Klein, “Sur le mouvement d’un corps grave de révolution suspendu par un point de son axe (der Kreisel),” Nouvelles annales de mathématiques 15 (1896), pp. 218–22. Hermite had written about this work (on Jan. 27, 1896): “Vous pensez combien j’ai été sensible à la communication extrêmement bienveillante dont M. Klein vous a fait l’intermédiaire. Le résultat concernant les formules pour le mouvement d’un corps pesant de révolution est d’une bien haute importance. Veuillez présenter tous mes compliments les plus cordiaux à M. Klein” ([UBG] Cod. MS. F. Klein 10: 632/Anl.). 131 KLEIN 1897; reprinted in KLEIN 1922 (GMA II), pp. 618–54. See also the Memorial Book of the Sesquicentennial Celebration of the Founding of the College of New Jersey and of the Ceremonies Inaugurating Princeton University (New York: Charles Scribner’s Sons, 1898). 132 Felix Klein, “Sur la stabilité d’une toupie qui dort (sleeping),” Nouvelles annales de mathématiques 16 (1897), pp. 323–28. – A Russian translation was published in Moscow in 2003.

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[On Hermite’s Case of Lamé’s Differential Equation]. Hermann Liebmann, Klein’s assistant in 1897 and 1898, simplified the approaches of the Italian mathematician Tullio Levi-Civita, who had used Sophus Lie’s theory of transformation groups to treat the differential equations of the motion of tops in order to identify integrable cases.133 In his dissertation “Zur Theorie des Maxwell’schen Kreisels” [On the Theory of Maxwell’s Top] (1904), Klein’s doctoral student Max Winkelmann investigated the motions of a nearly symmetrical top (this study is cited in the addenda to Klein and Sommerfeld’s monograph).134 In a section in the second part of Klein and Sommerfeld’s book – “On the Motion of the Heavy Asymmetric Top” – mention is made of Sofya Kovalevskaya’s work on the subject.135 In 1888, Kovalevskaya had discovered in the field of classical mechanics the third integrable special case of solid bodies (solvable by theta functions), which is known as the “Kovalevskaya top.”136 This case is theoretically significant, but a technical application has still not been found for it. In Moscow, the Russian mathematician and engineer Nikolay Y. Zhukovsky, one of the founding fathers of aerodynamics (see Section 8.1.3), had arranged for models to be constructed to illustrate the movements of tops, the Kovalevskaya top included. He gave talks on this subject at the meetings of the German Mathematical Society in Munich (1892) and Lübeck (1895).137 Klein ordered reproductions of these models and received them from Moscow.138 When Fritz Kötter criticized Klein and Sommerfeld’s exposition of the Kovalevskaya top as insufficient, they responded in the addenda to their fourth volume by remarking that a “full discussion of this case would have required too long of an analytical exposition.”139 In the second part of the book, Klein and Sommerfeld had only briefly summarized the representation of the motion of tops by elliptic functions, referring to the book Einführung in die Theorie der analytischen Funktionen [An Introduction to the Theory of Analytic Functions] (1897), whose author was Klein’s collaborator Heinrich Burkhardt.

133 H. Liebmann, “Classification der Kreiselprobleme nach der Art der zugehörigen Parametergruppe,” Math. Ann. 50 (1898), pp. 51–67 (cited in KLEIN/SOMMERFELD 1897, p. 161). 134 See KLEIN/SOMMERFELD 1910, p. 952. 135 See KLEIN/SOMMERFELD 1898, pp. 376–77. In English: KLEIN/SOMMERFELD 2010 [1898], pp. 376–77 (the pagination is the same in the German and English editions). 136 See Sophie Kowalevski, “Sur le problème de la rotation d’un corps solide autour d’un point fixe,” Acta Mathematica 12 (1889), pp. 177–232. For this work, Kovalevskaya was awarded the Prix Bordin by the French Academy of Sciences. 137 See Jahresbericht der DMV 3 (1893) II, pp. 62–70; and 4 (1897) III, p. 144–50. 138 [UBG] Cod. MS. F. Klein 10: 19, 19A (letters from Zhukovsky to Klein dated December 8, 1895 and October 24, 1896). 139 KLEIN/SOMMERFELD 1910, p. 950.

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Figure 35: The title page of the French edition of Riemann’s Collected Works, including Felix Klein’s speech (discours) on Riemann, KLEIN 1894. (https://archive.org/details/oeuvresmathmat00riem)

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8.2.4 Inspiring Ideas in the Fields of Mathematical Physics and Technology Now that I see that Hilbert is turning his attention to physics and that Prandtl and his students are making great strides in hydrodynamics, I can believe that a new era is dawning for mathematical physics.140

In October of 1881, Klein had suspected that now he would only be able to come up with original ideas in the fields of mechanics and mathematical physics (see Section 5.5, p. 252). Since the late 1890s, he had defined the field of applications for the ENCYKLOPÄDIE project, planned the new curriculum in applied mathematics, focused his own teaching on applied subjects, and devoted his research to formulating open-ended problems. Perhaps it is not too bold to suggest that Klein’s unsolved problems in applied mathematics were almost as inspiring as Hilbert’s famous unsolved problems in pure mathematics (see Section 10.1). In 1904, Klein presented a report to the Göttingen Mathematical Society about his seminars on applied mathematics: “In 1899/1900, we discussed the motion of ships, then descriptive geometry, geodesy, various branches of mechanics, including celestial mechanics [astronomische Mechanik]. Now we are focusing on hydrodynamics, and the next topic will be probability theory.” He went on: Complete insight can only be gained by taking practical activity and experimentation into account. What is accomplished in the seminars is more like preliminary work, in that we discuss original scholarly literature and classify it from a mathematical point of view. In doing so, I hope to be of equal service to applications and to the science of mathematics. If I continue to receive support from my younger colleagues, as I have done so far, then I think that I will always be able to include new areas of research, such as electrical engineering, when the opportunity arises.141

There was no lack of support from his younger colleagues. Klein arranged to cooperate with them by including newly appointed professors in his research seminars on applied mathematics. With Hilbert, he had already conducted seminars on mechanics in the winter semester of 1897/98 and the summer semester of 1898. In 1900, he conducted a seminar with Max Abraham on the technical applications of elasticity theory. With the astronomer Karl Schwarzschild (hired in 1901), he directed seminars on astronomy (1902), the principles of mechanics (1902/03), graphical statics and the strength of materials (1903),142 and hydrodynamics (1903/04). For the seminar on probability theory in 1904, Klein also included Martin Brendel and Constantin Carathéodory.143 After Carl Runge and Ludwig 140 [UA Braunschweig] A letter from Klein to Fricke dated December 4, 1911. 141 [Protocols] vol. 20, pp. 133–42, at p. 133 (Klein’s report, February 9, 1904). 142 In the course listings for the summer semester of 1903, this seminar was announced with the title “Statik der Baukonstruktionen” [The Statics of Structures]. 143 Raised in Brussels, the Greek mathematician C. Carathéodory earned his doctorate in Göttingen with Minkowski (October 1, 1904); recommended by Klein, he already completed his Habilitation in 1905 ([UAG] Phil. Fak. 190a, V 31–43). After being appointed to professorships at the Technische Hochschule (TH) in Hanover (1909) and the TH in Breslau (1910), Carathéodory was made Klein’s successor in Göttingen (see Section 8.5.3).

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Prandtl had been appointed, Klein relied primarily on them to co-direct his seminars, but he involved other experts as well: “Select Chapters in the Theory of Elasticity” (1904/05, conducted by Klein, Carl Runge, Prandtl, and Woldemar Voigt); “Electrical Engineering” (1905, conducted by Klein, Runge, Prandtl, and Hermann Theodor Simon);144 “Hydrodynamics” (1907/08, Klein, Runge, Prandtl); “Ship Theory and Dynamic Meteorology” (1908, Klein, Runge, Prandtl, and Wiechert); “Statics and Structures” (1908/09, Klein, Runge, Prandtl); “The Strength of Materials” (1909, Klein, Runge, Prandtl). Hilbert followed Klein’s example. In parallel with Klein’s seminar on electrical engineering, he thus offered a seminar of his own on electron theory together with Minkowski, Wiechert, and Herglotz.145 The interdisciplinary research seminars in Göttingen, which were unique at the time, helped to dispense with the idea that German universities should focus exclusively on educating future school teachers (see Section 7.7). The training offered here produced a number of prominent engineers, physicists, and industrial researchers, including notable examples such as Heinrich Barkhausen, George A. Campbell, Erwin Madelung, Reinhold Rüdenberg, Henry Siedentopf, Stephen P. Timoshenko, and Iris Runge.146 Klein’s efforts to formulate open-ended problems in applied disciplines can be shown with a few examples. 8.2.4.1 Hydrodynamics / Hydraulics Klein had already assigned his students to analyze the literature on this topic when he was a professor in Munich (see Section 4.1.2), and he was familiar with the classic works by Helmholtz, Kirchhoff, and Zhukovsky on the relations between function theory and hydrodynamics. Even though the importance of automorphic functions to hydrodynamics – for modeling currents around a polygon to produce an airfoil profile, for instance147 – had yet to be investigated systematically, Klein attempted early on to classify such problems from a mathematical point of view. After teaching a course on “The Mechanics of Deformable Bodies (Especially in Hydrodynamics)” (1899/1900), Klein conducted a seminar of his own titled “Select Chapters of Hydrodynamics” during the winter semester of 1903/04. In his introductory lecture in the seminar, he underscored the relationship between theory and experiments and stressed that many technical problems could not be solved mathematically without a sufficient amount of empirical data.148 After eight presentations by participants in the seminar, Klein gave his own lecture about how the total course of a flow curve, the outflow problem, and the

144 145 146 147 148

For an analysis of this seminar on electrical engineering, see TOBIES 2014. See the Vorlesungsverzeichnis (summer semester, 1905), p. 16. Regarding Iris Runge in particular, see TOBIES 2012. See MATTHIEU 1949. [Protocols] vol. 20, pp. 1–6 (Klein’s introductory lecture, October 28, 1903).

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problem of conduits could be better understood mathematically. Regarding the outflow problem, he said: […] a simplification of the related potential problem suggested by Mr. Hilbert could be used. This involves the assumption that the cross-section outflow opening is infinitesimally small. And the treatment of this simplified potential problem would then naturally be followed by one in which the cross-section is finite but very small (Poincaré’s perturbation theory […]).149

In another interim report, Klein categorized the problems of hydrodynamics and hydraulics into three groups: those that are mathematically well defined, those that are somewhat poorly defined, and those that very poorly defined: In the first group belong the problems of outflow, overflows, and permanent waves on standing bodies of water. Theoretically, these are problems of potential theory, but they have not been treated much so far because they do not involve fixed boundary surfaces with linear boundary conditions. For these cases, numerous experimental observations have already been made. It certainly seems possible to apply these observations and make progress with the desired theory. The second category includes the flow of water in pipes and channels, and also the channel waves (in flowing water). Whenever the movement is not very slow, the phenomenon of turbulence appears. The latter, in any case, depends to a great extent on the nature of the walls. One can ask whether, in very wide pipes, the turbulence fills its entire cross-section. Experiences from balloon travel on the turbulence of airflow do not seem to support this. The question of how the onset of turbulence should be explained theoretically still seems to be unexplained; we will soon hear more about the previous approaches to this matter in the presentations by Schwarzschild, Herglotz, and Hahn.150 The third category includes currents in natural rivers and the motion of groundwater (both of which will likewise be discussed in further presentations). The course of rivers can hardly be addressed theoretically, and this is because the flow forms the riverbed itself and modifies its details while moving ahead. A natural riverbed always bends. Experiments have further shown that the speeds in the cross-section are distributed very unequally, that the maximum speed lies beneath the surface, etc. etc. Recently, researchers have proceeded more and more by constructing their own river-engineering laboratories (Engels in Dresden […]), where extensive studies are being done on, among other things, the influence of railway bridges and other structures on the water beneath the surface.151

Klein ended his report by returning to the topic of turbulence: In conclusion, the speaker [Felix Klein] used the sink in the collection room to demonstrate a) the reduction of the turbulence of the stream flowing from the faucet by inserting a thin screen (“stream regulator”), b) the hydraulic jump caused by this stream hitting the sink with the appropriate force. (The idea of reducing the resistance of flow through a conduit by inserting a screen from time to time and thereby eliminating turbulence. Analogous to Pupin’s telephone?)152

149 [Protocols] vol. 20, pp. 63–65, at p. 64 (Klein’s interim report, Februry 9, 1904). 150 In their presentations, Karl Schwarzschild, Gustav Herglotz, and Hans Hahn analyzed turbulent fluid motion on the basis of equations formulated by Boussinesq ([Protocols] vol. 20). 151 [Protocols] vol. 20, pp. 134–36. Hubert Engels, a professor of hydraulic engineering at the TH Dresden, built the first experimental facility for river engineering in 1897. 152 [Protocols] vol. 20, pp. 140–41. – On M.I. Pupin, see also TOBIES 2012, p. 137.

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Here, Felix Klein used his analogic thinking and compared the possible reduction of turbulent flow with the possible prevention of signal distortion in long-distance telephone wires by inserting a “Pupin coil,” a device patented by the SerbianAmerican physicist Mihajlo Pupin in 1894.153 Michael Eckert has confirmed that Klein had formulated in advance many of the important research questions that defined later work in the domain of hydrodynamics by Ludwig Prandtl.154 Klein also recognized and fostered a number of young talents in this area, including the Hungarians Győző Zemplén and Theodore von Kármán. Zemplén developed a new mathematical approach to the theory of shock waves, based on lectures that he gave in Klein’s seminar in 1904/05.155 Kármán reported, among other things, that Klein was enthusiastic about his calculation of a swirling vortex that formed after the flow of a fluid was disrupted by an object (his results were published in 1911; the phenomenon is known today as the Kármán vortex street). Having seen Kármán’s work, Klein prophesied: “I guarantee that you’ll get the next chair in your line, just as soon as it becomes vacant.”156 This prophecy was fulfilled in 1912, when Kármán was offered a professorship at the Technische Hochschule in Aachen as the successor to Hans Reissner (Kármán accepted the position in 1913).157 By this point, both Zemplén and Kármán had already contributed as authors to vol. IV (mechanics) of the ENCYKLOPÄDIE. 8.2.4.2 Statics Klein had enthusiastically mentioned the newer graphical methods of statics in 1888 (see Section 6.4.2). Klein had read and reviewed books of this field,158 he had commissioned Lebrecht Henneberg (see Section 4.3.3) to write the ENCYKLOPÄDIE article on the graphical statics of a rigid body (1903), and he had recruited Heinrich Müller-Breslau, a pioneer in this area, to join the editorial board of the Zeitschrift für Mathematik und Physik in 1901 (see Section 5.6). Beginning in 1900, Klein had the participants in his seminars analyze scholarly literature on statics (graphical and numerical methods by Culmann, Cremona, Maxwell, Castigliano,159 August Föppl, and others). Klein categorized various topics of this domain and promoted new approaches. On April 29, 1903, he opened his seminar with these words: 153 154 155 156

Pupin had obtained his PhD degree under Hermann von Helmholtz in Berlin. See ECKERT 2019b. [Protocols] vol. 21, pp. 35–47, 73–74. KÁRMÁN/EDSON 1967, p. 73. Kármán, who came from a Hungarian Jewish family, was able to continue his career as a pioneering aerospace engineer in the United States. 157 See also SIEGMUND-SCHULTZE 2018, pp. 493–94. 158 On the history of this field, see KURRER 2018, and regarding the reviews, see KLEIN 1889. 159 Klein himself spoke about Carlo Alberto Castigliano’s book Théorie de l’équilibre des systèmes élastiques et ses applications (1879) in his seminar. [Protocols] vol. 27, pp. 277–79.

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The statics of structures is concerned with the problem of determining tensile stress, compression stress, and other “tensions” that appear in parts of a structure when their static equilibrium is influenced by external forces (“loads”). A simple example of this is the horizontal, load-bearing beam whose individual cross-sections are “stressed” in a certain way (which needs to be determined more closely) as a result of the occurring deformation (it is “sagging,” for instance). Iron structures provide further examples. Here in Göttingen, for instance: the train station, the gasometer braces, the gas plant. Distinguished by its magnificence: the Müngsten bridge (between Remscheid and Solingen).160 A third group of examples consists of “arches” and “domes.” Their weight and the wind pressure “stress” parts of the dome. How great are the pressures that are consequently transferred onto the walls and pillars on which the dome is based, and in which directions do they act? Think of the Cologne Cathedral.161

Klein discussed how deformations associated with tensions could be represented mathematically, and he referred to Edward Hough Love’s book A Treatise on the Mathematical Theory of Elasticity (2 vols. 1892–93). The main purpose of this seminar was to consider the simpler methods that engineers had developed to achieve practical goals: “a) They are predominantly graphical, and b) they contain error assumptions [Näherungsannahmen].” Klein explained these graphical methods by discussing, for instance, the problem “of finding, for an empirically given curve, the enclosed surface area, the static moments, and the inertia moments, and thus determining the numerical values of the integrals: ∫∫ dxdy; ∫∫ xdxdy; ∫∫ ydxdy; ∫∫ x2dxdy; ∫∫ xydxdy; ∫∫ y2dxdy.”162 He stressed that applied mathematics required not only knowledge but also practical skills; that is, mathematical methods necessarily had to be put into practice. He thus paved the way for practical exercises of this sort to be taught at the university level and for Carl Runge’s development of new graphical methods.163 Klein’s contributions to statics are also reflected in the work of his assistants. Karl Wieghardt, assistant in 1899/1900, was the first student in Göttingen to earn the teaching qualification (Facultas) in applied mathematics (1901), and in 1902 he completed a dissertation under Klein’s supervision – “Über die Statik ebener Fachwerke mit schlaffen Stäben” [On the Statics of Planar Frameworks with Unstrained Beams] – in which he treated a technical question with mathematical rigor. Wieghardt continued this work with Klein; he edited the lecture that Klein delivered at the Göttingen Mathematical Society (July 7, 1903)164 to underscore the importance of Maxwell’s work, to which Henneberg had devoted insufficient

160 Here, Klein is referring to an achievement by Anton Rieppel, who oversaw, from 1894 to 1897, the construction of the highest railroad bridge in Germany. 161 [Protocols] vol. 10, pp. 101–02 (Klein’s introductory lecture, April 29, 1903). 162 Ibid., pp. 103, and 104–05. 163 See Carl Runge, Graphical Methods (New York: Columbia University Press, 1912). 164 See Felix Klein and Karl Wieghardt, “Über Spannungsflächen und reziproke Diagramme, mit besonderer Berücksichtigung der Maxwellschen Arbeiten,” Archiv der Mathematik und Physik 8 (1904), pp. 1–10, 95–119 (KLEIN 1922 [GMA II], pp. 660–91).

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attention in his ENCYKLOPÄDIE article on statics (1903). Klein combined Maxwell’s approaches with the statics of load-bearing columns and beams, and he underscored the central role of the “Airy function” (a stress function), with which it is possible to solve analytically boundary value problems of linear planar electrostatics.165 Wieghardt completed his Habilitation in 1904 at the Technische Hochschule in Aachen with a thesis on highly indeterminate frameworks, and he went on to teach statics as a professor at various Technische Hochschulen (Braunschweig, Hanover, Vienna, Dresden). He contributed the ENCYKLOPÄDIE article “Theory der Baukonstruktionen, I und II” [Structural Theory, I and II] (1914).166 Aloys Timpe, who was Klein’s assistant in 1905 and 1906, continued along these lines with his dissertation “Probleme der Spannungsverteilung in ebenen Systemen, einfach gelöst mit Hilfe der Airyschen Funktion” [Problems of Stress Distribution in Planar Systems, Simply Solved by Means of the Airy Function] (1905).167 He referred to the practical uses of beam and arch problems, and he demonstrated how such problems could be reduced to the integration of a single linear differential equation. Timpe likewise proved to be a reliable collaborator on the ENCYKLOPÄDIE project. With the Italian mathematician Orazio Tedone, he wrote the articles “Allgemeine Theoreme der mathematischen Elastizitätslehre (Integrationstheorie)” [General Theorems of the Mathematical Theory of Elasticity (Integration Theory)] (1900) and “Spezielle Ausführungen zur Statik elastischer Körper” [Special Explications on the Statics of Elastic Bodies] (1906) for vol. IV; and with Conrad Heinrich Müller, he wrote the ENCYKLOPÄDIE article “Die Grundgleichungen der mathematischen Elastizitätstheorie” [The Basic Equations of the Mathematical Theory of Elasticity] (1906). He also translated the book by Edward Hough Love mentioned above (see also Section 5.6). Klein revisited the theory of beam problems and tension systems yet again in his seminar “Statik der Baukonstruktionen” [The Statics of Structures] in 1908/09. He induced his assistant Ernst Hellinger to write the ENCYKLOPÄDIE article on the mechanics of continua;168 and Klein’s next assistant, Friedrich Pfeiffer, wrote the article “Zur Statik ebener Fachwerke” [On the Statics of Planar Frameworks].169 Klein himself spoke three times in this seminar on Airy’s strained surfaces,170 and on May 11, 1909, he gave a presentation to the Göttingen Mathematical Society on “planar frameworks that are a projection of one-sided spatial polyhedra, whereby he [Klein] resolved, by introducing double polyhedra, the contradictions that initially result when determining the self-tensions of the framework.”171 165 See George Biddell Airy, “On the Strains in the Interior of Beams,” Philosophical Transactions of the Royal Society of London 153 (1863), pp. 49–80; see also KURRER 2018, p. 851. 166 Part I of this article was written with the assistance of Martin Grüning. 167 Published in Zeitschrift für Mathematik und Physik 52 (1905), pp. 348–83. 168 On Hellinger’s article, see ENCYKLOPÄDIE, vol. IV, pp. 601–94, and KURRER 2018, p. 892. 169 Zeitschrift für Mathematik und Physik 58 (1909/10), pp. 262–72. 170 [Protocols] vol. 27, pp. 236–52, 288–90. 171 Quoted from the summary of Klein’s lecture in Jahresbericht DMV 18 (1909) Abt. 2, p. 79; see also Math. Ann. 67 (1909), pp. 433–44; reprinted in KLEIN 1922 (GMA II), pp. 692–703.

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Stephen P. Timoshenko developed a beam theory that contains the classical Euler-Bernoulli theory as a special case. For inspiring his mathematical approach, Timoshenko thanked Felix Klein.172 He had attended Klein’s course on mechanics in 1909 and had given a presentation in Klein’s seminar that year as well.173 Years later, Klein’s influence in this field could still be felt in Dorothea Starke’s dissertation “Die Maximalmomentenfläche eines Gerberschen Balkens” [The Maximum Momentum Surface of a Gerber Beam]. Starke completed her doctoral degree under Klein’s former student Max Winkelmann in Jena, and she became an assistant at Winkelmann’s Institute for Applied Mathematics; her position was funded by the Carl Zeiss Foundation.174 8.2.4.3 The Theory of Friction175 In 1909, Klein published two articles on this subject, the first of which sparked a few new impulses in the field: “Zu Painlevés Kritik der Coulombschen Reibungsgesetze” [On Painlevé’s Critique of Coulomb’s Laws of Friction] and “Über die Bildung von Wirbeln in reibungslosen Flüssigkeiten” [On the Formation of Vortices in Frictionless Fluids].176 Klein had advised Paul Stäckel while the latter was writing his comprehensive ENCYKLOPÄDIE article “Elementare Dynamik der Punktsysteme und starren Körper” [The Elementary Dynamics of Point Systems and Rigid Bodies], in which he analyzed “friction” as a fundamental concept and referred to Paul Painlevé’s surprising finding that Coulomb’s laws of friction lead to logical contradictions.177 Because Klein felt that Painlevé’s results were unsatisfactory, he prompted Ludwig Prandtl to conduct experimental analyses and he addressed the topic himself in his lecture courses on mechanics in 1908/09. In his article, Klein formulated the farsighted idea that Painlevé’s works “could become the starting point for the development of a new branch of technical mechanics.”178 This thought stimulated further discussion from Richard von Mises, Georg Hamel, and Ludwig Prandtl, and Klein’s assistant Friedrich Pfeiffer included the topic in his detailed study mentioned above.179

172 See TIMOSHENKO 1968; and C.R. Soderberg, “Stephen P. Timoshenko,” in Biographical Memoirs, vol. 53 (Washington, DC: National Academy Press, 1982), pp. 323–49. 173 [Protocols] vol. 27, pp. 338–46. 174 See BISCHOF 2014. The term “Gerber beam” goes back to Heinrich Gerber, who founded, with Anton Rieppel, the “Gustavsburger school” of steel bridge construction. 175 I am indebted to Reinhard Siegmund-Schultze and Rolf Nossum, Kristiansand (Norway), for calling my attention to Klein’s notable contributions in this domain. 176 Both articles appeared in vol. 58 of the Zeitschrift für Mathematik und Physik. 177 Stäckel in ENCYKLOPÄDIE Vol. IV.6 (1908), pp. 435–684, in particular p. 472. See Painlevé’s publications in Comptes rendus 80 (1895), p. 596; 81 (1895), p. 112; 140 (1905), p. 702; 141 (1905), pp. 635, 847; and his book Leçons sur le frottement (Paris: A. Herman, 1895). 178 Quoted from KLEIN 1922 (GMA II), p. 708. 179 These contributions are published in Zeitschrift für Mathematik und Physik 58 (1909/10).

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Writing in retrospect, Klein claimed that he wanted these contributions to give expression to his creed: “In discussions of whatever sort of mechanical or physical occurrences, if a certain idea is placed front and center, one should draw from this idea all the conclusions that logically follow from it or are compatible with it, even if the idea itself might not seem entirely suitable to physics.”180 What Klein meant was that the numerical model is only an approximation, and even Coulomb’s laws are only “an approximation of the conditions that appear in reality,”181 as he and Sommerfeld had already discussed in their book on the theory of the top.182 The inconsistency that Painlevé discovered in Coulomb’s laws of friction is known today as the “Painlevé paradox,” and it has since been the subject of numerous further studies. In a recent book, Le Xuan Anh even speaks of “The Painlevé-Klein extended scheme” and of the “Painlevé-Klein problem.”183 8.2.4.4 The Special Theory of Relativity Minkowski had introduced his colleagues to Einstein’s theory during one of their walks together, and he himself first presented his famous work “Raum und Zeit” [Space and Time] at a session of the Göttingen Mathematical Society on November 5, 1907. Klein reported enthusiastically about this at the annual meeting of the Göttingen Association for the Promotion of Applied Physics and Mathematics, which took place on October 16–17, 1908 in Leipzig: Yet also pure mathematics has produced a result that does not seem unworthy of attention on the part of the members of the Göttingen Association. The enormous significance of the theory of electrons to physics and chemistry needs no explanation here. It was Mr. A. Lorentz (the great Dutch physicist), who, in his relevant investigations from some years ago, had come to the highly remarkable assumptions that every electron passing through space contracts slightly in its line of motion as its speed approaches the speed of light. Now, Mr. Minkowski has succeeded in reducing the common features of these assumptions to the simplest mathematical expression, according to which the fundamental laws of electrodynamics generally remain unchanged in such linear substitutions of the space coordinates X Y Z and of the time parameter T if they leave the quadratic form x2 + y2 + z2 – c2t2 unchanged (c is the speed of light). It can only be suggested here that, on the basis of this observation, the hitherto insufficiently explained laws of the motion of electricity in moving conductors now seem to take on a completely transparent form. Pure mathematicians are thus experiencing the triumph that their investigations of quadratic forms, which they undertook decades ago and which were then regarded from many sides as exaggerated, might now be of the utmost significance to mathematical physics and, relatedly, to our natural-scientific conceptions of space and time.184

180 181 182 183

KLEIN 1922 (GMA II), pp. 704–13, at p. 13 (Klein’s commentary on his article). Ibid., p. 708. See KLEIN/SOMMERFELD 2010 [1903], pp. 537–83. Le Xuan Anh, Dynamics of Mechanical Systems with Coulomb Friction (Berlin: Springer, 2003), pp. 70, 141, 176. 184 [UBG] Math. Arch. 5021, fols. 14–15 (Klein’s remarks at the meeting in Leipzig).

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When Minkowski died unexpectedly, Klein, Hilbert, and others took it upon themselves to carry on his work. In the winter semester of 1909/10, Klein taught a course, which he had prearranged with Minkowski, on projective geometry to 109 students. Klein wanted to demonstrate how “the theory of the Lorentz group – or the modern principle of relativity espoused by physicists (which is the same thing) – can be classified as part of the general theory of the projective metric.”185 In connection with this course, he gave a talk on May 10, 1910 to the Göttingen Mathematical Society on the geometric foundations of the Lorentz group (“Über die geometrischen Grundlagen der Lorentzgruppe”). Here he provided a detailed historical overview based on Cayley’s invariant theory; he emphasized the role of projective geometry, which he thought the current course program seemed to neglect; and he made the following comparison: What modern physicists call the theory of relativity is the invariant theory of the four-dimensional space-time domain x, y, z, t (the Minkowskian “universe”) with respect to a particular group of collineations – namely, the “Lorentz group.” Or, more generally, and leaning toward the other side, one could say […] that the term “invariant theory relative to a group of transformations” could be replaced by the term “the theory of relativity with respect to a group.”186

At the same time, Klein formulated a set of problems to be solved, a program: It will be essential to produce a systematic invariant theory of the affine “universe,” for which all of the elements in the multi-dimensional investigations by mathematicians are already available, and on the basis of this theory it will be necessary to treat both types of mechanics side by side, the old and the new. It will then become clear whether the old mechanics is a boundary case of the new and to what extent it should thus be regarded as an approximation of the latter. Who will carry out this program?187

In May of 1911, Klein initiated a debate on the principle of relativity at a meeting of the Göttingen Mathematical Society, in which Hilbert, Runge, and Max Born were the main participants.188 At the 1911 conference of the Society of German Natural Scientists and Physicians in Karlsruhe, there was a special session on mechanics. The papers given in this session corresponded to Klein’s program: Karl Heun spoke about various approaches to expanding classical mechanics (“Ansätze zur Erweiterung der klassischen Mechanik”); Vladimir Varićak gave a talk on the non-Euclidean interpretation of the theory of relativity (“Über die nichteuklidische Interpretation der Relativitätstheorie”); and Lothar Heffter delivered a lecture in which he introduced Minkowski’s concept of the four-dimensional universe (“Zur Einführung der vierdimensionalen Welt Minkowskis”).189 In 1911, the experimental physicist Eduard Riecke described the situation with these words: “The

185 KLEIN 1921 (GMA I), p. 533. 186 KLEIN 1921 (GMA I), p. 539 (emphasis original); first published in Jahresbericht der DMV 19 (1910) Abt. 1, pp. 281–300, and reprinted in the Physikalische Zeitschrift 12 (1911). 187 KLEIN 1921 (GMA I), pp. 550–51. 188 [UBG] Cod. MS. F. Klein 21G, fol. 32v. 189 See Jahresbericht der DMV 20 (1911) Abt. 2, pp. 83, 166; and 21 (1912) Abt. 1, pp. 1–8, 103–27.

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mathematicians are hypnotized by the elegance of the rules of calculation, the physicists are critical.”190 In subsequent years, additional mathematicians participated in the search for equations to describe spacetime geometry. After the field equations for the general theory of relativity had been formulated, Klein made yet another foray into this research area (see Section 9.2.2). 8.3 PROGRAM: THE HISTORY, PHILOSOPHY, PSYCHOLOGY, AND INSTRUCTION OF MATHEMATICS Felix Klein’s program of these mathematical border areas had its origins in the ENCYCLOPÄDIE project (see Section 7.8), and over the following years it flowed into further book projects and into a mathematical and scientific reform movement that covered all areas of education and assumed broadly international dimensions. When the publishing contract for the ENCYKLOPÄDIE was signed in May of 1896, the first plan for its contents also contained an outline for a final volume (volume VII), which had been drafted by Klein and which looked as follows: Final Volume. A. History, Philosophy, Didactics. 1. History, i.e., an overview of the advances that have been made over the century in our knowledge and understanding of previous periods of development. 2. Logic and Epistemology: a) A critique of the fundamental mathematical concepts and of the mathematical methods of proof. (See IA 1.3; II A1; II A1; III B1) b) The applicability of mathematics to physical and psychic quantities. 3. Psychology: a) The psychological conceptions of numbers, time, and space; b) The psychology of mathematical thought: specific differences among individuals and the resulting consequences for didactic issues. 4. The Calculus of Logic (symbolic systems of logical operations and their application to mathematics). 5. The Didactics of Mathematics at educational institutions of all levels and purposes, in Germany and abroad. B. General overview of the development of mathematical sciences in the nineteenth century.191

The concept of this volume served as a program for Klein’s own occupation with the history, philosophy, psychology, and didactics of mathematics and for his efforts to impel others to promote these fields. He sought out and vetted potential authors and editors, among them Gustaf Eneström (Sweden) and Jules Tannery (France).192 In order to make preparations for the philosophical section, Klein himself traveled to Paris yet again in 1906.193 He initiated preliminary work and 190 191 192 193

Quoted from TOBIES 1994b, p. 345 (Eduard Riecke to Johannes Stark, October 13, 1911). [AdW Wien] I 170 (Math. Encyklopädie). See TOBIES 1994a, pp. 56–69. For Klein’s report on this trip, see Jahresbericht der DMV 15 (1906) IV, p. 331.

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recruited students, colleagues, and secondary school teachers to collaborate on the project. Klein made use of prize competitions, conferences, and committees to prompt discussions on these topics. His work to this end began with his own research and teaching. The completion of additional book projects seemed important to him, as the following remarks in the minutes of the 1910 meeting of the ENCYKLOPÄDIE committee show: Regarding volume VII, the committee approves the statement from Mr. F. Klein that, on the one hand, the work of the “International Commission on Mathematical Instruction” and, on the other hand, the essays planned for the project devoted to the “Culture of the Present” on the psychological and epistemological foundations of mathematics and on the history of mathematics could serve as preparations for this volume.194

I should first discuss the book series project Die Kultur der Gegenwart [The Culture of the Present],195 which is mentioned here, because its volume on mathematics was intended to summarize (for the general public) possibly all of the aspects of the planned vol. VII of the ENCYKLOPÄDIE (history, epistemology, didactics). The overall project, which had been initiated by the historian Paul Hinneberg, a proponent of Leopold von Ranke’s historicism, was originally planned to consist of sixty-two volumes. In the end, only twenty-five were produced.196 In 1906, when the first volume appeared, Klein expressed his dissatisfaction with the overall plan and felt as though he had to intervene: It must be called a misconception when, in numerous classifications of the sciences (thus also, for example, in the new encyclopedic work “The Culture of the Present”), mathematics is essentially thrown together with the natural sciences. […] At secondary schools, mathematics and the natural sciences are indeed natural allies and are set apart from the other subjects. Yet the science of mathematics has a great deal of significance even independent of every other field of human knowledge; it has relations on the widest variety of fronts and, viewed philosophically, it is not at all connected to any of the natural sciences: in itself, mathematics is a pure human science [Geisteswissenschaft].197

The position of mathematics as a science of order and structure, as we would say today, was something that Klein had hinted upon as early as his inaugural address in Erlangen (see Section 3.2). He considered it important for this status to be made clear within the framework of the large-scale cultural project. With Walther Dyck’s support, he thus incited a flurry of activity beginning in August and September of 1908. He recruited authors for the mathematics volume and also for the volumes on the natural sciences and technology. In coordination with the Prussian Ministry of Culture, he also initiated and chaired a conference that took place in Berlin from the 17th to the 19th of December in 1909. Just three years later, the project had advanced to such an extent that Klein was able to plan the content, authors, and even the page distribution for the contributions on mathematics, 194 195 196 197

[AdW Wien] I 170 (minutes from the meetings in Munich on April 15–16, 1910), fol. 2. KLEIN 1912–14. See TOBIES 2008b, which also informs the discussion below. KLEIN 1907, pp. 136–37.

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which were now arranged to be published as the first section of “Part III: The Mathematical and Natural-Scientific Cultural Spheres.” From Klein’s different plans (one proposed in June of 1912 and the other in August of the same year), it is possible to see how his views developed. His first plan differed from the second for the period of “modern mathematics”; he wanted “the development of pure mathematics in the nineteenth century” to be presented separately from applications. As for who should write the history of pure mathematics in the nineteenth century, he noted: “possibly Klein”. Regarding the history of applied mathematics, Klein’s plan mentioned the potential authors Carl Runge and Heinrich Weber, and he enumerated five sections: “1) Numerical and Graphical Calculation, Descriptive Geometry, 2) Probability Theory, Statistics, Actuarial Mathematics, 3) Measurement, 4) Mechanics and Astronomy, 5) Mathematical Physics.” Everything, Klein added, should be written “exclusively from the mathematician’s point of view.”198 In August of 1912, the Teubner press published the second plan, an announcement in which no author is mentioned for the contribution on modern mathematics [Neuzeit] (there one simply reads “N.N.” = nomen nominandum ‘author to be named’; see Fig. 36). Klein was then recovering in a sanatorium (see Section 8.5.1), and he had gained two insights: first, that he would not be able to complete such a contribution quickly because of his poor health; second, that it would be difficult, in writing such an overview, to separate “pure” and “applied” mathematics. On August 2, 1912, he explained to Heinrich Weber […] that I would also, however, leave the mathematics of the modern era unseparated. Everything that is to be said about applied mathematics can be included at the appropriate place within the historical presentation of pure mathematics. This is recommendable because, in many questions, it is hardly possible to draw a line between them.199

Figure 36: Klein’s updated plan for the volume “Die mathematischen Wissenschaften” of the book series project Die Kultur der Gegenwart [The Culture of the Present], August 1912 (B.G. Teubner’s publication announcement).

198 [UBG] Cod. MS. F. Klein 7M: fol. 14 (Klein’s provisional plan for the volume “Die mathematischen Wissenschaften” as part of the Kultur der Gegenwart project, June 9, 1912). 199 Ibid.: fol. 15 (A draft of a letter from Klein to Heinrich Weber, August 2, 1912).

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The contributions on mathematics in the Kultur der Gegenwart project appeared in three volumes. The first published volume (1912) contains the essay by H.G. Zeuthen, which was based on his book Geschichte der Mathematik im Altertum und Mittelalter [The History of Mathematics in Antiquity and the Middle Ages].200 Paul Stäckel, who was involved in many of Klein’s other projects, never completed the planned contribution on early-modern mathematics. The second published volume (1914) contains Aurel Voß’s article “Die Beziehungen der Mathematik zur allgemeinen Kultur” [The Relations of Mathematics to General Culture] (pp. 1–49) and Heinrich E. Timerding’s article “Die Verbreitung mathematischen Wissens und mathematischer Auffassung” [The Dissemination of Mathematical Knowledge and Mathematical Understanding] (pp. 50–161). The result of the fifth article noted in the announcement was Klein’s book Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert [Lectures on the Development of Mathematics in the Nineteenth Century], which was edited posthumously from Klein’s Nachlass and was based on lectures that he had given in Göttingen from 1914 to 1918 (see Sections 8.3.1 and 9.2).201 8.3.1 The History of Mathematics202 My way of doing history: posing specific questions, then comparing the sources.203

Klein, too, had been influenced by Leopold von Ranke’s historicism. This approach to historiography was source-based; it sought to unearth the causes behind the course of history, and it aimed to present the past as objectively as possible. Klein had learned to appreciate this method back when he was a Privatdozent in Göttingen (see Section 2.8.3.3). In 1871, while assisting Clebsch in his preparation of Plücker’s obituary, Klein thought it would be best to focus on the history of science to contextualize Plücker’s achievements, and Clebsch was of the same mind. In a letter to Plücker’s widow, Klein had explained: “The aim of the obituary would not be to provide a detailed history of his life but rather to represent in broad strokes how Plücker influenced the development of science; it would thus be a piece on the history of science, with Plücker foregrounded as the protagonist.”204 In the published obituary, we read in greater detail: The history of science has […] the task of tracing the ideas that develop over the course of generations and explaining general processes. As part of these processes, the discoveries of a given individual represent symptoms rather than driving causes. According to this understanding, there will be fewer opportunities to say that a discovery anticipated its time or that an individual exclusively defined the spirit of his era. Instead, the totality of science takes on

200 201 202 203 204

The original Danish version of this book was published in 1893, the German edition in 1896. KLEIN 1926/27. The first volume was translated into English: KLEIN 1979 [1926]. See Pieper/Tobies 1988; Dauben/Scriba 2002; and Tobies 2002b. [UBG] Cod. MS. F. Klein 20H. [Canada] A letter from Klein to Antonie Plücker dated November 10, 1871.

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8 The Fruits of Klein’s Efforts, 1895–1913 an organic nature. Regarding certain details, of course, and especially in the case of phenomena that were discovered at around the same time, it is still necessary to investigate the extent to which one discovery causally influenced the other. It would be a mistake, however, simply to confuse this temporal sequence with the causal effect as such.205

In the obituary, Clebsch and Klein focused on the oppositional pair of “abstract” and “intuitive” mathematics as the defining feature of historical development, an idea that Klein would return to again and again and that would ultimately influence his universal thinking: As a significant moment with consequences for further developments, it ought to be stressed that there have always been two directions whose opposition, whether more or less pronounced, has accompanied all epochs of mathematical research, and which could be called the abstract direction and the intuitive direction. Together, complementing and supplementing one another, they encompass the whole of mathematical research, and neither can long do without the accompaniment and influence of the other without doing severe damage to its own inherent essence.206

The extent to which Klein valued sources is reflected in his engagement on behalf of the collected works of Plücker, Clebsch, Möbius, Grassmann, Riemann, and Gauss. It was Klein who, in 1894, initiated the edition of Plücker’s works as a project of the Göttingen Academy, edited by F. Pockels and Schoenflies.207 After Schering’s death (November 2, 1897), Klein took over the direction of Gauss’s collected works, and as early as November 22, 1897 he expressed his program for how things should proceed: with the “immediate edition of the works related to non-Euclidean geometry” by P. Stäckel and F. Engel, and with a commitment from the Teubner publishing house.208 Klein involved further experts in the project and, with Martin Brendel and Ludwig Schlesinger, he prepared a scientific biography of Gauss. He was still participating in this project in 1919, when he recruited A. Fraenkel to edit Gauss’s works on algebra and the concept of number.209 Klein’s edition of Gauss’s scientific diary has been regarded as the beginning of a “shift in Gauss research, in that it now became possible to investigate, on a secure basis, the problem of the genesis of Gauss’s discoveries.”210 Despite Schering’s objectionable behavior (see Section 6.4.3 and Appendix 4.2), Klein wrote a brief obituary for him211 and supported the edition of Scher-

205 206 207 208 209

CLEBSCH 1872, p. 6. Ibid., p. 2 (= Obituary on Plücker, prepared by Klein and Clebsch). PLÜCKER 1895/96. [AdW Göttingen] Chro 4,6:1894; Scient 105,2; 105,3; 107,5. See REICH/ROUSSANOVA 2013. Materialien für eine wissenschaftliche Biographie von Gauß, collected by F. Klein, M. Brendel, and L. Schlesinger. Vol. VIII: Zahlbegriff und Algebra bei Gauß, ed. A. Fraenkel. With an appendix by A. Ostrowski: “Zum ersten und vierten Gaußschen Beweis des Fundamentalsatzes der Algebra” (Leipzig: B.G. Teubner, 1920), 58 pp. 210 K.-R. Biermann et al., eds., Mathematisches Tagebuch, 1796–1814, von C.F. Gauß (Leipzig: Akademische Verlagsgesellschaft, 1979), p. 10. – Klein’s editio princeps was published as a Festschrift in 1901 and reprinted in Math. Ann. 57 (1903), pp. 1–34. 211 Felix Klein, “Ernst Schering,” Jahresbericht der DMV 6 (1899) I, pp. 25–27.

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ing’s collected works by securing Robert Haussner, who had taken part in Klein’s courses on elliptic functions in 1886 and had completed his doctorate under Schering, as one of the editors. With Klein as its chairman, the German Mathematical Society decided in 1908 to support the edition of Euler’s collected works, which had been initiated by Ferdinand Rudio in Basel, and to establish an “archive of mathematicians” (Mathematiker-Archiv) at the university library in Göttingen.212 Thanks to Klein’s historical interests, moreover, commemorative plaques were placed on the former residences of Clebsch, Dirichlet, Tobias Mayer, and Riemann in Göttingen, and a Gauß-Weber memorial was installed there (it was unveiled on June 17, 1899).213 In order to elevate the level of general mathematical education, Klein recommended that teaching candidates should study classical sources (Euclid, Archimedes, Apollonius, and so on).214 He acquired mathematical-historical literature (by Moritz Cantor, Zeuthen, and others) for the reading room, and he promoted the publication of Gustaf Eneström’s historical journal Bibliotheca Mathematica with the Teubner press (see Section 5.6), even though he did not accept Eneström’s plan for the historical volume of the ENCYKLOPÄDIE.215 Klein inspired Conrad Heinrich Müller to practice a new type of historiography. After studying mathematics, natural sciences, and Indic philology, Müller completed his doctoral degree under Klein with a dissertation on Studies on the History of Mathematics, Especially of Mathematical Instruction at the University of Göttingen in the 18th Century in 1903. Müller’s stated goal in this work was “to study the influence and stimuli that pure mathematics received from applied mathematics.”216 He investigated the role of mathematics in “culture” (in a broad sense) and the organization of scientific work. Three years later, Müller planned his Habilitation in consultation with Klein.217 It has been overlooked until now that this would result in the first venia legendi awarded in Göttingen for “mathematics, in particular the history of mathematics.”218 Müller’s contributions to the

212 See Jahresbericht der DMV 17 (1908), p. 133; and GRENZEBACH/HABERMANN 2016. 213 [UBG] Cod. MS. F. Klein 1B, fols. 145–46 (letters from Klein to the trustees of the University of Göttingen and to the mayor of Göttingen, Dr. Merkel, December 1889 and January 1890); and ibid. 9: 498 (a letter from Alfred George Greenhill to Klein dated July 4, 1892). Greenhill had donated money to the cause and expressed to Klein his wish for the “success of the combined Gauss-Weber Memorial.” 214 [UBG] Cod. MS. F. Klein 19C (Klein’s lecture notes, summer 1903). 215 [StB Berlin] Sammlung Darmstaedter (letters from Eneström to Klein dated September 13, 1899; September 28, 1899; and October 29, 1899). Eneström played a more prominent role in the French edition of the ENCYKLOPÄDIE (see GISPERT 1999). 216 Conrad Müller, “Studien zur Geschichte der Mathematik insbesondere des mathematischen Unterrichts an der Universität Göttingen im 18. Jahrhundert,” Abhandlungen zur Geschichte der math. Wissenschaften mit Einschluß ihrer Anwendungen 18 (1904), pp. 50–143, at p. 59. 217 [UBG] Cod. MS. F. Klein (Klein’s notes from June 10, 1906). 218 In German: “Mathematik, namentlich Geschichte der Mathematik.” This information is lacking in DAUBEN/SCRIBA 2002, p. 493.

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ENCYKLOPÄDIE (with A. Timpe and A.N. Krylov)219 and the historical catalogue that he produced for the Teubner publishing house on the occasion of the 1908 International Congress of Mathematicians in Rome220 were accepted as his Habilitation thesis.221 As a Privatdozent in Göttingen, Conrad Heinrich Müller offered courses on the history of mechanics since Lagrange (1909/10) and on mathematics in the work of Archimedes (1910).222 For the winter semester of 1910/11, Müller had already announced a lecture course on the history of the discovery of infinitesimal calculus, but he left Göttingen to become a professor at the Technische Hochschule in Hanover (as Carathéodory’s successor). There he continued his research on the history of mathematics and continued to work for and with Felix Klein on the ENCYKLOPÄDIE project. Since his time working for the Kultur der Gegenwart,223 Klein had aimed to conceptualize the history of mathematics and to familiarize himself with the available literature on the subject. During the summer of 1908, Klein studied the chapter on the development of mathematics in John Theodore Merz’s book A History of European Thought in the Nineteenth Century.224 Klein’s notes on this chapter indicate that he was searching for the causes that had influenced the course of the history of mathematics. In the context of Merz’s discussion of geometry, for instance, Klein noted: “The practical stimulus.” Elsewhere we read: “Number theory: here again there are practical and theoretical interests.” Hermann Weyl supported Klein’s efforts by analyzing the development of more recent areas of research such as set theory, differential geometry, analysis, etc.225 For both the winter semester of 1910/11 and that of 1912/13, Klein had announced that he would be offering a lecture course on the development of mathematics in the nineteenth century, but it turned out that he was too busy (or unwell) at the time to teach them. In the summer semester of 1914, he finally started a colloquium on the history of mathematics, in which he was able to involve Carathéodory, Courant, Debye, and others.226 The lectures that he gave on this subject during later semesters culminated in his aforementioned book Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert [Lectures on the Development of Mathematics in the Nineteenth Century] (see Section 9.2.1). 219 Krylov’s pioneering theory of the oscillating motions of ships became internationally known in the 1890s, and Klein had asked this Russian scientist to write the contribution “Die Theorie des Schiffes” (1906/07) for vol. IV of the ENCYKLOPÄDIE (mechanics), supported by Müller. 220 TEUBNER 1908. 221 [UAG] Kur. 6289 (Habilitation Conrad Müller). – In 1927, Otto Neugebauer would become the next person to complete a habilitation on the history of mathematics in Göttingen ([UAG] Math. Nat. 0047, fol. 32). 222 Even before this, Klein had arranged for Felix Bernstein to offer mathematics courses on historical topics (in 1908 and 1908/09). 223 KLEIN 1912–14. 224 MERZ 1904–12, vol. 2, ch. 13. 225 [UBG] Cod. MS. F. Klein 7M, fols. 16–42, esp. 16–29 (Klein’s notes on Merz’s chapter; Weyl’s notes and letters to Klein, dated June 9, 1912 and August 16, 1912). 226 [UBG] Cod. MS. F. Klein 22C, fol. 63.

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Under Klein’s aegis, Wilhelm Lorey made further contributions to the history of mathematics by focusing on its development at German universities.227 As the editor of Lorey’s book, Klein supported the search for sources on the topic; he demanded verifiable information, read the text critically, recommended additional resources, and insisted that Lorey should leave out some caustic remarks.228 With a similar level of engagement, Klein edited the other volumes of the German ICMI-series on the history of mathematical education.229 Following Gert Schubring, we would be right to call Klein one of the founders of the “social history of mathematics.”230 8.3.2 Philosophical Aspects I consider philosophy a beautiful thing, but its methods seem highly uncertain to me, given that the best philosophers manage to reach precisely opposite opinions. I therefore draw no conclusions about the philosophical questions that attract me until I have made a more thorough philosophical investigation into these matters.231

Klein had written these words to Max Noether in 1877 to imply that this had been his basic attitude while writing his Erlangen Program. Since then, he had had to concern himself more closely with a number of philosophical questions. While a member of the Philosophical Faculty in Göttingen, Klein had participated in formulating mathematical-philosophical prize challenges. When the matter of hiring professors of philosophy arose, he and Hilbert attempted to support mathematically inclined philosophers who seemed most suitable to them.232 In a letter of support for the critical and mathematically talented philosopher Leonard Nelson,233 Klein argued that universities “should allow the individual philosopher to have the utmost freedom in his productivity but should not allow him to have dominion over others.”234 With this comment, Klein clearly had the great Georg Wilhelm Friedrich Hegel in mind, for he added “Cf. Hegel.” The grandfather of Klein’s wife Anna, Hegel had once managed to have the Privatdozent Friedrich Eduard Beneke expelled from the University of Berlin, presumably on account of the latter’s espousal of materialism. Beneke had then taught in Göttingen and later received an associate professorship in Berlin after Hegel’s death. All of this is worth mentioning because the consistorial councilor and preacher Carl Gustav Beneke established, in memory of his brother Eduard, a 227 228 229 230 231 232 233 234

See LOREY 1916. [UBG] Cod. MS. Philos. 182: F. Klein (Klein’s letters to Lorey). KLEIN 1909–16. See SCHUBRING 1986b. [UBG] Cod. MS. F. Klein 12: 581 (a letter from Klein to Max Noether, August 16, 1877). On this topic, see PECKHAUS 1990. On Leonard Nelson, see ibid. and TOBIES 2012, pp. 92–97. [UBG] Cod. MS. F. Klein 2G, fol. 34 (a draft of a letter from Klein to the university trustee, March 23, 1917).

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Beneke Prize Fund to promote the study of philosophy. The University of Berlin was unwilling to administer this prize, so it was taken over by the Philosophical Faculty at the University of Göttingen. Because the Philosophical Faculty was still undivided at the time, this made it possible for the Beneke Prize Fund to be used to sponsor prize competitions in pure mathematics as well. Both Clebsch (in April of 1872) and Klein (in 1892) had taken advantage of this opportunity.235 Now Klein preferred to formulate mathematical-philosophical challenges with an eye toward the philosophical section of the final volume of the ENCYKLOPÄDIE. Klein was primarily concerned with the relationship between exactness (precision) and approximation in connection with space intuition (Raumanschauung) and the applications of mathematics.236 In a lecture titled “Grenzfragen der Mathematik und Philosophie” [Questions on the Border of Mathematics and Philosophy], which he delivered on October 15, 1905 to the Philosophical Society at the University of Vienna, he began with these remarks: Although I received this invitation just three days ago, I am all the more willing to be here because I number among those mathematicians who would like to have closer relations with philosophical circles, for I am convinced that a great number questions should concern philosophers and mathematicians alike. Of course, I have nothing new to say to the mathematicians present here today. For I was invited to present some of the ideas that I have already discussed elsewhere about the inaccuracy of our conceptions of space.237

Already in his article “Über den allgemeinen Funktionsbegriff und dessen Darstellung durch eine willkürliche Kurve” [On the General Concept of the Function and Its Representation by an Arbitrary Curve] (1873), Klein had written about the limited accuracy of perception or intuition (see Section 3.1.3). In 1883, Klein decided to reprint this work, which had first appeared in the Sitzungsberichte der physikalisch-medizinischen Sozietät zu Erlangen (December 8, 1873),238 after Moritz Pasch had expressed his admiration of it (in 1882).239 Klein discussed this problem further in his lecture courses,240 in his Evanston Colloquium (1893; lecture 6); and in his presentation “Über Arithmetisierung der Mathematik” [On the Arithmetization of Mathematics], which he delivered at the Royal Society of Sciences in Göttingen on November 2, 1895, and which was subsequently translated into English, Italian, and French.241 He also included it in his lecture courses titled “Anwendung der Differential- und Integralrechnung auf Geometrie (Eine Revision der Prinzipien)” [The Application of Differential and Integral Calculus to 235 [UAG] II Phil. 13, vol. I, II (Beneke Prize Fund). 236 On this topic, see also Klaus VOLKERT 1986, esp. pp. 226–42. 237 KLEIN 1922 (GMA II), p. 247. The honorary president of the Philosophical Society, Alois Höfler, had invited Klein to speak while Klein already happened to be in Vienna. It is known that, among others, Wilhelm Wirtinger and Ludwig Boltzmann were in attendance. 238 Reprint in Math. Ann. 22 (1883), pp. 249–59; and KLEIN 1922 (GMA II), pp. 214–24. 239 See SCHLIMM 2013. 240 See, for example, a long quotation from his lecture course on the theory of linear differential equations (summer 1891), which is printed in TOBIES 1992a, p. 761–62. 241 KLEIN 1895c.

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Geometry (A Revision of Principles)] (1901), which served as the basis for the third volume of his book Elementary Mathematics from a Higher Standpoint.242 Klein’s position can be summarized in five points: First, in his article from 1873, Klein explained […] that one does not at all possess the ability to conceive even simpler examples of function theory and infinitesimal calculus both precisely and intuitively at the same time, and that space intuition [Raumanschauung] even fails when it concerns the details of those curves which are represented by polynomials.243

However, Klein stressed the heuristic value of intuition (Anschauung). What he meant was that, in the majority of cases, a sufficient degree of precision could be achieved instinctively, and he added in a later commentary that his own intuitive geometric approach “had often led to correct theorems of precision mathematics” and to “the discovery of relations and their essential proofs.”244 David Rowe has underscored that Klein “brought this anschauliche approach to bear on other branches of mathematics – algebra, function theory, potential theory, etc.”245 In space intuition or perception (Raumanschauung), Klein saw one possible source for new analytical concepts: We picture before us in space the infinite number of points and forms composed of them; from this idea have sprung the fundamental investigations on sets of points and transfinite numbers with which G. Cantor has opened up new spheres of thought to arithmetic science.246

In his lecture “Über Arithmetisierung der Mathematik” [On the Arithmetization of Mathematics] (1895), Klein classified “Weierstrass’s rigor,” “Kronecker’s refusal to employ irrational numbers,” and the work of Giuseppe Peano as instances of this arithmetizing tendency.247 On the one hand, Klein used his lecture to underscore the “extraordinary importance” of these developments; on the other hand, he rejected the view that the content of mathematics was exhaustively contained in this arithmetized science (arithmetisierte Wissenschaft). It was important for him to emphasize that, in addition to formal methods, intuition (Anschauung) still plays a crucial role (and, as an aside, he remarked that there is also an algorithmic side to mathematics).

242 243 244 245 246 247

KLEIN 31928. In English: KLEIN 2016 [1928]. Quoted from KLEIN 1922 (GMA II), p. 248. Ibid., p. 213. ROWE 1994, p. 191. KLEIN 1896 [1895c], p. 244 (trans. Isabel Maddison); repr. in KLEIN 1922 (GMA II), p. 235. In coordination with Adolph Mayer, Klein had recruited Peano to contribute articles to Mathematische Annalen, and some of his work was published there in volumes 32 (1888), 36 (1890), and 37 (1890). At first, Klein had a skeptical opinion of Peano’s logical-axiomatic studies (on this matter, see SEGRE 1997). Klein overcame this skepticism – and his skeptical view of Hilbert’s work – around the year 1908. In his Elementary Mathematics from a Higher Standpoint (KLEIN 2016 [1924], p. 15), Klein expressed his belief that Peano’s book Arithmetices principia nova method exposita (1889) was a more precise refinement of Hermann Graßmann’s Lehrbuch der Algebra [Textbook on Algebra] (1861).

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Hurwitz wrote to Klein: “As far as your remarks on arithmetization are concerned, I can subscribe to them in their entirety, though I am inclined to assign an even higher scientific value to arithmetization than you do. Regarding the pedagogical side of the question, it is also my conviction that it would be wrong to impose complete rigor when introducing the topic to students.”248 This pedagogical side will be discussed below in Section 8.3.4.2. Here it should be stressed that Klein ultimately came to appreciate the higher scientific value of arithmetization. In 1895, however, he remained highly skeptical of Hilbert’s abstract approach, in particular. This attitude may seem surprising, but it is clearly expressed in Klein’s notes from 1908 on the aforementioned book by John Theodore Merz: “In Merz’s work, arithmetization still seems like a completion, whereas now we know that it was only a beginning. (This was the subjective reason why I wrote my article [KLEIN 1895c]. Room for applied mathematics! Orientation against Hilbert).”249 In the meantime (that is, by 1908), Hilbert himself had turned part of his attention to applications, and Klein had come to accept and promote more recent and more abstractly oriented approaches to mathematics, though he himself was not an active researcher in this area. Second, regarding an issue closely related to space intuition and non-Euclidean geometry, Klein discussed the question of the epistemological value of axioms. He had studied Helmholtz’s writing on the topic and had quarreled about it with Sophus Lie (see Section 6.3.6). Together with Georg Elias Müller, Klein formulated the following challenge for a prize competition in 1894: What is sought is a representation “of the historical development of the basic concepts and axioms that underlie recent mathematical analysis (number theory, differential and integral calculus, function theory).” Candidates for the prize were expected to focus on “the importance of this development to further work in mathematics and possibly also on its logical and epistemological significance.” None of the submissions were worthy of the prize, and therefore Klein withdrew the challenge in 1902. Nevertheless, he himself often returned to the theme. His thoughts about it culminated in the following statement: “We ultimately perceive that space intuition is an inexact conception, and that in order that we may subject it to mathematical treatment, we idealize it by means of the so-called axioms, which actually serve as postulates.”250 Klein rejected the view that axioms were merely arbitrary propositions, which were agreed upon on the basis of conventions. In a discussion of the “foundations of geometry,” he stated: Many authors express themselves much more one-sidedly, however, so that in recent years, in the modern theory of axioms, we have frequently ended up back in the direction of that philosophy which has long been called nominalism. Here, interest in things themselves and

248 [UBG] Cod. MS. F. Klein 9: 1131 (a letter from Hurwitz to Klein, January 3, 1896). 249 Ibid., 7M, fol. 4. 250 KLEIN 1896 [1895c], p. 243–44 (trans. Isabel Maddison); KLEIN 1922 (GMA II), p. 235.

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their properties is entirely lost. What is discussed is the way things are named, and the logical scheme according to which one operates with the names.251

It should be noted that medieval and modern nominalism are not the same. Klein’s critique was directed against conventionalism, a concept that Henri Poincaré had introduced to mathematical thinking.252 According to this idea, mathematical axioms and theorems are arbitrary conventions without objective content, and they are only chosen on the basis of their convenience and “economy of thought.”253 Klein believed that this would lead to a loss of interest in things themselves and their properties, thereby bringing about the death of all scientific inquiry: “The axioms of geometry are – according to my way of thinking – not arbitrary, but sensible statements, which are, in general, induced by space intuition and are determined as to their precise content by expediency.”254 Klein ultimately accepted Hilbert’s mode of expression, noting: “[…] one posits the axioms of ordering, connection, and continuity, and erects a geometry upon them.”255 David Rowe has discussed how both Klein and Hilbert used the keyword “intuition” (Anschauung) in their geometric work and how both mathematicians were deeply indebted to axiomatic thinking.256 It should be mentioned yet again that Klein had shown that the consistency of non-Euclidean geometry could be reduced to proving the consistency of the axioms of Euclidean geometry. Third, on account of the acknowledged imprecision of space intuition, Klein called for “precise approximation mathematics”: “For graphical methods, etc., I asked for an error theory similar to the one that has been used since Gauß for all exact measurements.”257 In the sixth lecture of his Evanston Colloquium, we read: All this suggests the question of whether it would not be possible to create a, let us say, abridged system of mathematics adapted to the needs of the applied sciences, without passing through the whole realm of abstract mathematics. Such a system would have to include, for example, the researches of Gauß on the accuracy of astronomical calculations, or the more recent and highly interesting investigations of Chebyshev on interpolation.258

One of the problems formulated by Klein, Hilbert, and G.E. Müller for the 1901 Beneke Prize (approved by the Philosophical Faculty on February 24, 1898) asked for a mathematical treatment of natural phenomena on the basis of the principle of continuity. Klein’s old friend, the philosopher Carl Stumpf, reacted to this problem with the following words:

251 KLEIN 2016 [1925], p. 213 (the translation has been slightly modified). 252 See Henri Poincaré, Science and Method, trans. Francis Maitland (New York: Cosimo, 2009; originally published in 1908); and BIAGIOLI 2016, pp. 166–88. 253 Klein also discussed this problem in his research seminar: [Protocols] vol. 29, pp. 25–27. 254 KLEIN 2016 [1925], p. 213. 255 KLEIN 1979 [1926], p. 139. 256 See ROWE 1994, p. 197. 257 KLEIN 1922 (GMA II), pp. 212–13, 248 (emphasis original). 258 Ibid., p. 230 (emphasis original).

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8 The Fruits of Klein’s Efforts, 1895–1913 I read the short “Beneke” essay with the greatest interest. Here lies, without doubt, a cardinal point of all of natural philosophy (in a lecture on the idea of evolution, I also once referred to the difficulties of the continuity principle). But who can be expected to solve this problem? You will certainly have to do so yourself.259

Once again, none of the submissions was worthy of the prize, so that Klein, in his report, clarified that the judges had hoped for a more precise explanation of “the inherent significance of continuity requirements, so as to know whether they are more than just an aid to simplify the execution of mathematical investigations and to what extent the results derived from these requirements have a claim to objective validity.”260 Formerly, according to Klein, mathematicians had not considered this, while physicists and philosophers pushed the uncomfortable question aside, but it needed to be clarified in order to achieve “a critical understanding of the modern development of mathematics.” The report explained: 1. 2. 3.

The tendency of arithmetization as a basis for the newer developments should be maintained. If, however, the external world is to be investigated quantitatively, it must be asked which simplifications are to be allowed, given that only a limited degree of accuracy can be expected from the results. In Klein’s view, the solution to this problem requires “mathematicians and empiricists to find common ground.” For mathematicians, the task will be to “develop, on the basis of arithmetic science, a comprehensive theory of approximation methods and to cultivate a special discipline, which Mr. Heun has recently and fittingly referred to as approximation mathematics.” For empiricists, the task will be to “establish the degree of accuracy within which their (external or internal) observations can be regarded as correct or within which they hope to achieve reliable results.261

Klein stressed that his program was not really new. He referred repeatedly to Gauß, Chebyshev, and to Weierstrass’s (famous) approximation theorem, which states that “every continuous function defined on a closed interval can be uniformly approximated as closely as desired by a rational function of finite degree.”262 What was new, according to Klein, was the need to make this question the focus of applied mathematics and “to recognize that the approximative conception of quantitative relationships pervades our entire thinking far more than has previously been supposed.”263 In order to spread his vision of precise approximation mathematics, Klein gave a course titled “Anwendung der Differential- und Integralrechnung auf Ge259 [UBG] Cod. MS. F. Klein 11: 1251 (Stumpf to Klein, June 4, 1901); Carl Stumpf, Leib und Seele: Der Entwicklungsgedanke in der gegenwärtigen Philosophie. (Leipzig: Barth, 1909). 260 Klein’s report was first published in the Göttinger Nachrichten (April 1901), pp. 40–47. It was reprinted in Math. Ann. 55 (1902), pp. 143–48; and in KLEIN 1922 (GMA II), pp. 241–46 (quoted here from p. 243). 261 Ibid., pp. 244–45. Regarding the term “approximation mathematics” (Approximationsmathematik) referred to here, see Karl Heun, “Die kinetischen Probleme der wissenschaftlichen Technik,” Jahresbericht der DMV 9/2 (1901), pp. 1–123. 262 On this topic, see especially SIEGMUND-SCHULTZE 1988 and 2016b. 263 KLEIN 1922 (GMA II), p. 245 (a quotation of Klein’s report from 1901).

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ometrie (Eine Revision der Prinzipien)” [The Application of Differential and Integral Calculus to Geometry (A Revision of Principles)] in 1901. As mentioned above, this lecture course served as the basis of the third volume of Klein’s book Elementary Mathematics from a Higher Standpoint. The eighty-six students who attended the course included Mary Esther Trueblood from the United States, Takuji Yoshiye from Japan, Felix Bernstein, Georg Hamel, and Walther Lietzmann. Later, when Klein was preparing these lectures for publication (see Section 9.2.3), he decided to give the book the subtitle “Präzisions- und Approximationsmathematik” [Precision Mathematics and Approximation Mathematics], for these were the topics that best encapsulated the overall theme of his lectures.264 In his course, Klein referred to the scientific and pedagogical value of the points of contact between the calculus of differences and differential calculus (Differenzen- und Differentialrechnung), and he demonstrated, among other things, how Taylor’s theorem had been formulated on the basis of the calculus of differences. Here, Klein also devoted special attention to the works of Chebyshev, of which the St. Petersburg mathematicians Markov and Sonin had prepared a French edition.265 Klein had the participants in his seminars analyze related works on the interpretation and approximation of functions of one variable, and he instigated German translations of textbooks from the St. Petersburg school (see Section 5.6). Around the year 1900, Klein was expressly emphasizing research areas such as the calculus of differences and probability theory, which, more than twenty years later, proved to be indispensable to quantum physics and were therefore developed further.266 There was nothing “anti-modern” about Klein’s promotion of modern numerical mathematics (as compared to the more abstract and axiomatically oriented “modern” mathematics); instead, it was an effort to advance a different type of modernity, which, on the basis of the computer, would later develop into the areas of mathematical economics and techno-mathematics.267 Fourth: Universalism. On January 27, 1904, Klein gave a public lecture whose primary intention was to preserve the unity of the Philosophical Faculty at the University of Göttingen: “In that they have developed alongside one another but are also built upon one another, the different elements of science should fit together to form a comprehensive whole. If you would like to have a particular term for this, I could propose the word universalism.”268 In this speech, Klein commented on and welcomed the new and better position that the natural sciences and technology had achieved in many parts of Germany on account of changes that had taken place around 1900 (the right of Technische Hochschulen to confer 264 See KLEIN 2016 [1928], p. xiii. 265 Ibid., pp. 87–89. 266 See Alwin Walther, “Über die neuere Entwicklung der Differenzenrechnung,” Jahresbericht der DMV 34 (1926), pp. 118–31. 267 See NEUNZERT/PRÄTZEL-WOLTERS 2015. Regarding the categories “modern” and “anti-modern,” see MEHRTENS 1990. 268 KLEIN 1904a, p. 271. This lecture was delivered in Göttingen on the occasion of Kaiser Wilhelm II’s birthday, which was a common custom at the time at all German universities.

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doctoral degrees; the equal status granted to the three types of secondary schools, etc.). He also stressed the considerable expansion of mathematical, natural-scientific, and technical areas at the University of Göttingen and responded to the wishes of the historical-philological division to separate from the Philosophical Faculty. To justify the unity that he deemed necessary, Klein borrowed the argumentation set forth at the Congress of Arts and Science in Saint Louis (1904), which had been organized to underscore the unity and the mutual relations of the sciences, and where the following classifications were made: Philosophy and mathematics were categorized as normative sciences, followed by historical sciences, physical sciences, and mental sciences (psychology and sociology).269 At the same time, Klein specifically suggested that new professorships should be established within the philological-historical branch of the Faculty: professorships for Slavic languages (especially Russian), East Asian languages, the economics of world trade, etc.270 Klein’s goal in this case was to increase the international nature of philological and historical instruction, for in the meantime he had taught mathematics to students from Russia, Poland, Japan, and elsewhere.271 Two years later, when Althoff requested Klein’s opinion yet again on the unity of the Philosophical Faculty, he replied: My view is in fact entirely different from that which has now emerged on the philological side; I would like, to the extent that this is possible, a closer relationship to develop between the two camps. We could learn an immense amount from each other, and we should support one another in the tasks that we have in common (e.g., the education of teachers) by offering mutual advice and comprehensive references (even though there may be all sorts of differences of opinion). Whenever it has seemed necessary to strengthen such support, I have tried to do so in recent years, for instance by participating in the conference of German philologists and educators in Hamburg. Moreover, I have also begun to study the relationship between mathematics and philosophy.272

Klein was able to postpone the division of the Philosophical Faculty at the University of Göttingen, but he was ultimately unable to prevent it. On June 16, 1910, the Faculty decided to create a mathematical-scientific section and a historical-

269 See Howard J. Rogers, ed., Congress of Arts and Science: Universal Exposition – St. Louis, 1904, vol. 1 (Boston: Houghton, Mifflin & Co., 1905), p. 13. 270 KLEIN 1904a, p. 272. Regarding Klein’s demands regarding foreign language learning and studying abroad, see also Section 9.1.2. 271 Regarding Japanese students, it should be mentioned by way of example that Teiji Takagi and Takuji Yoshiye attended Klein’s course on projective geometry in 1900/01; a total of 98 students enrolled in this course ([UBG] Cod. MS. F. Klein 7E). – Takagi later worked on Hilbert’s twelfth problem and achieved notable results in class field theory. See Teiji Takagi, Collected Papers, ed. S. Iyanaga et al. (Tokyo: Springer, 1990); and Sasaki et al. 1994. –Takuji Yoshiye attended Klein’s courses for five semesters (1899–1902; see also Fig. 31). For an indication of Hilbert’s influence on his work, see T. Yoshiye, “Anwendungen der Variationsrechnung auf partielle Differentialgleichungen mit zwei unabhängigen Variablen,” Math. Ann. 57 (1902), pp. 185–94. See also SASAKI 2002 and KÜMMERLE 2021. 272 [UBG] Cod. MS. F. Klein 8: 8 Appendix (Klein’s letter draft to Althoff, February 19, 1906).

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scientific section.273 Even as a professor emeritus, Klein was still involved in this process, for, on January 29, 1921, “the committee for the separation of the Faculty into two Faculties […] convened in the home of our colleague Klein.”274 Fifth, Klein had a range of discerning opinions about various philosophers. He took the side of the Austrian Benno Kerry, who, in a review, had criticized Paul du Bois-Reymond’s ideas about the concept of a function.275 Klein himself criticized long-deceased Arthur Schopenhauer, who believed that mathematical truth could be derived directly from intuition (Anschauung). For, even though Klein deeply appreciated intuition as a heuristic principle, he claimed that “the last and the only decisive instance is the logical proof emanating from the premises.”276 Klein’s opinion of Immanuel Kant was ambivalent, for the latter expressed conflicting thoughts on the same topic. Klein rejected philosophers, however, who relied on Kant’s “orthodox conception of space,” according to which three-dimensional space is a necessary precondition for mathematical thinking. Discussing Klein’s aforementioned lecture in Vienna, Boltzmann emphasized: “I completely agree with Privy Councilor Klein’s opposition to Kant’s view. I do not understand how one can speak of mathematical proof by intuition.”277 Klein (like Hilbert) promoted mathematically talented philosophers. The latter included not only the aforementioned Leonard Nelson but also the forty-two-yearold Edmund Husserl, who had earned a doctoral degree in mathematics and whom Klein’s good friend Carl Stumpf had recommended for an associate professorship in philosophy at the University of Göttingen.278 Like Max Planck and Carl Runge,279 Klein rejected Gustav Robert Kirchhoff’s idea that the goal of science was “not to explain natural phenomena but to describe them completely and in the simplest way.” About this, Klein commented: 273 [UAG] Phil. Fak. III, vol. 5, fols. 80–81. 274 Ibid., fol. 251. 275 See KLEIN 2016 [1928], p. 16 (note 28); Benno Kerry, review of “Paul du Bois-Reymond, Allgemeine Functionentheorie: Erster Theil, Tübingen 1882,” Vierteljahrsschrift für wissenschaftliche Philosophie 9 (1885), pp. 145–55. See also FISHER 1981. 276 KLEIN 2016 [1925], pp. 271–72. Regarding Schopenhauer, Klein was critical of the latter’s book The World as Will and Representation, trans. Judith Norman et al. (Cambridge: Cambridge University Press, 2010). The original German book was first published in a single volume in 1819. 277 Ludwig Boltzmann et al., “Anhang: Aus der Diskussion über vorstehenden Vortrag,” in Wissenschaftliche Beilage zum neunzehnten Jahresbericht (1906) der Philosophischen Gesellschaft an der Universität zu Wien (Leipzig: Barth, 1906), pp. 8–10, at p. 8. 278 [UBG] Cod. MS. F. Klein 11: 1251 (Stumpf to Klein, June 4, 1901). Husserl had completed his doctoral thesis – “Beiträge zur Theorie der Variationsrechnung” [Contributions to the Theory of the Calculus of Variations] – under Leo Koenigsberger’s supervision in Vienna. Husserl had been raised Jewish but, following Koenigsberger’s example, he converted to Protestantism in 1886. One year later, under Carl Stumpf’s supervision at the University of Halle, Husserl completed his Habilitation in the field of philosophy with a thesis titled “Über den Begriff der Zahl” [On the Concept of Number]. He had to wait until 1901, however, before receiving his first salaried position as an associate professor in Göttingen. 279 On Planck and Runge, see HENTSCHEL/TOBIES 2003, pp. 22–24.

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“I cannot conceal the fact that this conception of science is extremely antipathetic to me, because it thwarts the joy of learning and the drive to research.” In this context, Klein criticized the overwhelming approval of this idea “among positivist philosophers, for example Ernst Mach.”280 In 1913, Klein nevertheless joined Hilbert, Einstein, Mach, G.E. Müller, and twenty-nine other academics (in fields outside of mathematics) in signing an appeal for the establishment of a Society for Positivist Philosophy. The aim of this society was “to provide a comprehensive worldview on the basis of the empirical material that the individual sciences have accumulated.”281 Even though Weyl later denounced Klein’s involvement in this development – “In this case, [Klein] remained tethered to the dogmas of his time: to empiricism and to a type of psychology whose extreme representative is Mach and which seems increasingly questionable to us today from an unbiased empiricist point of view”282 – Klein’s opinion of Mach was far from uncritical. With respect to psychology, Klein’s ideas were in fact shaped by other thinkers and factors, and this will be the topic of the next section. 8.3.3 Psychological-Epistemological Classifications Even though I am treating the matter of psychology separately here, it should be stressed again that Klein typically thought about the philosophical, psychological, and pedagogical aspects of his work in an integrative way. This is also evident from his seminar “Über die psychologischen Grundlagen der Mathematik” [On the Psychological Foundations of Mathematics], which he conducted with Felix Bernstein and Leonard Nelson during the winter semester of 1909/10. The topics that Klein introduced at the beginning of this seminar overlap to a great extent with the overall plan for vol. VII of the ENCYKLOPÄDIE: -

The way in which creative mathematicians work How basic mathematical views (views of space and numbers) originate The creation and the epistemological value of axioms The different types of errors in mathematics Implications for education, from kindergarten to university The place of mathematics within the system of science.283

280 KLEIN 1979 [1926], pp. 206–07. 281 Quoted from “Aufruf!”, Zeitschrift für Mathematik und Physik 61 (1913), p. 13. The main initiator of this society was Joseph Petzoldt, who had completed his doctoral studies under G.E. Müller in Göttingen (see HENTSCHEL 1990, pp. 401–20). 282 WEYL 1930, p. 5. 283 [Protocols] vol. 29, pp. 1–72 (Klein’s introductory lecture, pp. 1–5). Klein kept a record of what took place in this seminar in his own hand, in contrast to his typical procedure. The participants in the seminar also included the Privatdozenten Otto Toeplitz and Ernst Zermelo. During this same semester, Zermelo offered a course titled “Über die logischen Grundlagen der Mathematik” [On the Logical Foundations of Mathematics], a topic that Klein included in  

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Some of these aspects have already been addressed, and the issues pertaining to education will be examined below in Section 8.3.4. Here I would like to discuss the first matter on Klein’s agenda: The way in which creative mathematicians work. Klein sought to identify the defining moments that led mathematicians to their approaches, creativity, and productivity. By cooperating with numerous mathematicians, he had come to recognize a variety of research styles. One result of this was his classification of different types of mathematicians (see 6.5.1.1). Klein’s lecture “On the Mathematical Character of Space Intuition and the Relation of Pure Mathematics to Applied Sciences,” which he delivered as part of his Evanston Colloquium in 1893, contains remarks on the research styles of mathematicians in different countries that could have been construed later as nationalistic or racially biased. After commenting on the differing research styles in the work of Euclid, Newton, G. Cantor, M. Pasch, and G. Peano, Klein conjectured: Finally, it must be said that the degree of exactness of intuition of space may be different in different individuals, perhaps even in different races. It would seem as if a strong naïve spaceintuition were an attribute pre-eminently of the Teutonic race, while the critical, purely logical sense is more fully developed in the Latin and Hebrew races.284

As others have already shown, Klein’s classification here was not intended to denigrate any mathematical approach or “race.”285 He assigned equal value to the different ways of achieving knowledge. Nevertheless, after his death, statements of this sort were politically appropriated and misconstrued for anti-French or antiSemitic purposes. The most flagrant example of this was the activity of the mathematician Ludwig Bieberbach during the Nazi era.286 Bieberbach was an adherent of the racial typologies proposed by the Nazi psychologist Erich Jaensch, who referred to Klein as “a champion of German-natured science.”287 In his psychology seminar (1909/10), Klein had the participants analyze international literature on experimental psychology and talent research (Begabungsforschung), and he also had them examine the ways in which preeminent mathematicians were productive. He was interested above all in how someone discovered a new result. Klein lectured here on his own research methods and those of Gauss (here, he made use of Gauss’s diary) and Sophus Lie. Leonard Nelson, who participated in the seminar, described Dirichlet’s methods; Felix Bernstein analyzed Georg Cantor; and Erwin Freundlich gave a presentation on famous mental calculators and chess players.  

284 285 286 287

the plan for the final volume of the ENCYKLOPÄDIE. Presentations in this seminar were also given by Hermann Weyl, Erwin Freundlich, the Belgian mathematician Alfred Errera and the philosopher Paul Decoster, Bernhard Uffrecht (later a progressive pedagogue), Klein’s doctoral student Wilhelm Behrens, and Fritz Steckel from Marienburg (Malbork today). KLEIN 1922 (GMA II), p. 228 (an original English quotation). – For current views on the then common unscientific term “race,” see FISCHER et.al. 2020. See ROWE 1986; and MEHRTENS 1990, pp. 215–19. See MEHRTENS 1987 and 2004. In German: “Vorkämpfer deutschgearteter Wissenschaft.” JAENSCH/ALTHOFF 1939. – See also Klein’s critical comment about a draft of Bieberbach’s doctoral thesis in Appendix 7.

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On November 16, 1909, Klein held a joint session of this seminar together with the Mathematical Society in order to incorporate the views of the other mathematics professors in Göttingen.288 Here, Klein put his classification (philosophers, intuitionists, and algorithmists) from 1892 (see Section 6.5.1.1) up for discussion, and also raised the issue of Wilhelm Ostwald’s division of great thinkers into classicists or romantics. One telling result from this meeting was the general consensus that, when analyzing the work of researchers, it was necessary to take into account not only their inherent talent but also environmental factors. In general, Klein was hesitant to draw firm conclusions about psychological questions. When, in a later seminar presentation (February 16, 1910), Felix Bernstein spoke about making a psychological distinction between different types of mathematicians, Klein commented: “The material that the individual mathematician works on is determined less by the nature of his talent than it is by his educational background.” This area of research would have to be studied in greater detail, Klein thought, before it would be possible “to write true-to-life biographies of mathematicians.”289 Ultimately, Klein realized that the nature of talent has nothing to do with nationality, religion, or gender but rather depends on educational context. For this reason, Klein supported the idea that all children should be given an equal opportunity to develop their individuality.290 Klein’s large editorial projects contain several contributions on the psychology of mathematics. Regarding the ENCYKLOPÄDIE project, in 1899 Klein had already discussed psychological questions with Federigo Enriques in Italy, who had associated his analysis of the genesis of postulates with psychological questions.291 They had continued this conversation in 1903 in Göttingen while preparing the article “Principien der Geometrie” for the ENCYKLOPÄDIE. Aurel Voß’s article “Über die mathematische Erkenntnis” [On Mathematical Knowledge] for the Kultur der Gegenwart (see Fig. 36) begins with a discussion of psychological and logical matters. As the editor of the monograph series Abhandlungen über den mathematischen Unterricht, Klein included a volume titled Psychologie und mathematischer Unterricht [Psychology and Mathematical Instruction]. He commissioned the Göttingen Privatdozent David Katz to write this volume, and Katz based his work not only on the research that he had conducted at the University of Göttingen, where Georg Elias Müller had founded the world’s second Institute for Psychology and had initiated, in 1904, a Society for Experimental Psychology. In order to broaden the scope of Katz’s report on the topic, Klein had also sent him to Wilhelm Wundt’s Institute for Experimental Psychology in Leipzig and to Wilhelm Rein’s experimental “Laboratory School” in Jena.292 288 289 290 291 292

[Protocols] vol. 29, pp. 15–19. Ibid., pp. 59–60. See ABELE/NEUNZERT/TOBIES 2004, p. 32; and TOBIES 2018b. See Livia Giacardi’s article in COEN 2012, esp. p. 224. See D. Katz, Psychologie und mathematischer Unterricht (Leipzig: B.G. Teubner, 1913). In 1933, Katz, who was Jewish, was forced to leave his professorship (University of Rostock); in 1937, he became the first professor of psychology in Sweden (University of Stockholm).

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8.3.4 The “Kleinian” Educational Reform By and large, we are all influenced by the movement that was inaugurated by Prof. Klein and further promoted by the German Commission.293

The Hungarian Emanuel (Manó) Beke spoke these words at the Fourth International Congress of Mathematicians held in Rome (1908). He had studied with Klein in 1893/94, and already then he had been enthusiastic about Klein’s outreach to secondary school teachers.294 Early on, Klein had acquired an international reputation in the domain of mathematical pedagogy through his book Famous Problems of Elementary Geometry, which was translated into several languages (see Section 7.3). He had been become a member of the editorial board of the first international journal for mathematical instruction, L’Enseignement mathématique (Fig. 37), founded in 1899.295 As the president of the German Mathematical Society at the time (see Section 6.4.4), Klein had made preparations for the conference in Rome, but he did not travel there himself for scheduling reasons.296 As early as February 11, 1908, he had entrusted Walther Dyck to give his promised lecture on the ENCYKLOPÄDIE, and August Gutzmer attended the congress as Klein’s proxy for educational matters.297 In Rome, David Eugene Smith, a mathematics instructor at Columbia College in New York City who had been one of the translators of Klein’s book Famous Problems of Elementary Geometry (see Section 7.3), proposed: Convinced by the importance of a comparative investigation into the methods and teaching plans of mathematical instruction at secondary schools in various countries, the Congress commissions Messrs. Klein, Greenhill, and Fehr to form an international committee to study these questions and to present a general report at the next Congress.298

This proposal was accepted in Rome on April 11, 1908; its mandate was set to be fulfilled by the time of the next International Congress of Mathematicians in Cambridge (UK) in 1912.299 Although the original assignment was restricted to secondary schools, the International Commission on Mathematical Instruction (ICMI)

293 Emanuel Beke, “Über den jetzigen Stand des mathematischen Unterrichts und die Reformbestrebungen in Ungarn,” in Atti del IV congresso internazionale dei matematici (Roma, 6–11 Aprile 1908), Roma: R. Accademia dei Lincei, 1909, vol. 3, p. 531. 294 [Protocols] vol. 11 (Beke gave three presentations in Klein’s seminar, all in January of 1894). He also published two articles in Math. Ann. 45 (1894), pp. 278–300; and [UBG] Cod. MS. F. Klein 8: 76A, 76B/1 (Beke’s letters to Klein, dated March 8 and August 21, 1895). 295 On the history of this journal, see CORAY et al. 2003; and GISPERT 2021. 296 Klein excused his absence in Rome because the congress coincided with his appointment to the Upper House of the Prussian parliament; see Jahresbericht der DMV 17 (1908), p. 130. 297 [UBG] Cod. MS. F. Klein 22H (Klein’s delegation of his plenary lecture to Dyck). – For Gutzmer’s lecture, see Atti del IV congresso internazionale dei matematici (Roma, 6–11 Aprile 1908), Roma: R. Accademia dei Lincei, 1909, vol. 3, p. 445. 298 FEHR 1909, p. 1. 299 See TOBIES 1979; and also MENGHINI et al. 2008.

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took all levels of education into account.300 Klein was thus able to follow the plan that he had imagined for vol. VII of the ENCYKLOPÄDIE, which was intended to include a section on “The Didactics of Mathematics at educational institutions of all levels and purposes in Germany and abroad” (see p. 474). Klein managed the ICMI after he had been elected in Rome (in absentia) to serve as its chairman. He had a good relationship with the deputy chairman George Greenhill (United Kingdom) and with the general secretary Henri Fehr (Switzerland), and he organized regular meetings to discuss the main topics of educational reform (see also Section 8.3.4.1). Over time, he was able to recruit representatives from seventeen countries to serve on sub-committees. By that time, Klein was a member of thirty-nine academies and scientific societies, so that he found it easy to choose fitting ICMI delegates to represent the individual countries involved. Walther Lietzmann, whom Klein had appointed as his secretary for ICMI-related work, marveled at Klein’s “fabulous ability to seek out the most appropriate man in every country.”301 These included the aforementioned Poul Heegaard as the Danish representative or Rikitaro Fujisawa, who was the chairman of the Japanese sub-committee and in this capacity was also the editor of the Summary Report on the Teaching of Mathematics in Japan (Tokyo, 1912), which was presented at the Congress in Cambridge.302 Fujisawa is noteworthy because he brought European mathematics to Japan after he had studied in London, Berlin, and Strasbourg and had earned a doctoral degree under Elwin Christoffel with a dissertation on partial differential equations. Fujisawa and his younger colleague Takuji Yoshiye were professors in the College of Science at Tokyo Imperial University. In 1912, Fujisawa and Yoshiye also donated money (as did Fehr, Greenhill, Heegaard, D.E. Smith, and many others), so that Max Liebermann could paint a portrait of Felix Klein (see 8.5.2). Klein was also largely responsible for choosing the members of the German ICMI sub-committee. This body consisted of delegates for the ICMI (Klein, Paul Stäckel, Peter Treutlein) and a national advisory board.303 For the latter board, Klein recruited the editors of journals: August Gutzmer (Jahresbericht der DMV), Heinrich Schotten (Zeitschrift für den mathematischen und naturwissenschaftlichen Unterricht), Friedrich Poske (Zeitschrift für physikalischen und chemischen Unterricht), and Albert Thaer (Unterrichtsblätter für Mathematik und Naturwissenschaften). This proved to be an adept way to disseminate ideas for reform within his own country. Gutzmer, Schotten, and Poske were already members of the Breslau Education Commission (Breslauer Unterrichtskommission). Stäckel, Treutlein, and Thaer later contributed books to the ICMI monograph series. 300 The English title “International Commission on Mathematical Instruction” was first introduced in 1952. First, the German and French titles were used: Internationale Mathematische Unterichtskommision and Commission Internationale de l’Enseignement Mathématique. 301 LIETZMANN 1960, p. 45. – Regarding Walther Lietzmann, see also HESKE 2018 and 2021. 302 Another member of the Japanese sub-committee was Takuji Yoshiye, who had studied under Klein and played a part in composing the Summary Report. See FUJISAWA 1912, pp. viii–x. 303 On Treutlein, see Ysette Weiss’s article in WEIGAND et al. 2019, pp. 107–16.

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Figure 37: The title page of the first issue of the journal L’Enseignement mathématique (1899), which has been the journal of the ICMI since 1908.

Modern mathematical and scientific education was also regarded at the time as a matter of international competition. In 1896, the Royal Commission on Technical Instruction in London had reported the following about a study tour in Germany:

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8 The Fruits of Klein’s Efforts, 1895–1913 They [our foreign rivals] are convinced that the nation which has the best schools is the best prepared for the great industrial warfare which lies before us, and no money appears to be grudged for the creation, equipment, and maintenance of educational institutions of all grades, and especially of the science laboratories which, as we have seen, are being multiplied in Germany.304

International cooperation of the sort mustered to study mathematical instruction did not exist then in any other discipline. In his public statements about educational issues, Klein’s arguments were based above all on ideals, and therefore he was able to convince an international band of mathematicians to join his side. He also recognized, however, that arguments based on the premise of international competition could be useful in his homeland: The great attention that is now being paid in England and America, for instance, to education in physics and chemistry (and also mathematics) is expressly based on the hope that such efforts will strengthen the position of these populations in the international competition for industrial and military power! If we want the leading authorities to heed our demands, this is certainly a more important and perhaps more appropriate reason than the idealistic considerations with which we have otherwise operated.305

Within the ICMI, Klein was able to concentrate on questions of mathematical education. As a member of German committees, Klein had to keep broader interests in mind in order to achieve his goals of educational reform. He wrote the following about the causes and beginnings of the reform in Germany: The driving impulse is coming from technology and the natural sciences. By way of example, the names of a few forerunners of the reform movement in Germany should be mentioned: there is the physiologist [Emil] du Bois-Reymond; also Gallenkamp,306 a scion of the Lower Rhine industrial area; and furthermore A. Wernicke from Braunschweig, who was influenced by his father’s instruction at the old trade school. In short, the shift in interest toward the natural sciences and technology has also created a shift in the public evaluation of mathematics. Whereas, formerly, it had been just a means of formal education – a whetstone of the intellect – people are now beginning to see it also as the basis for understanding life around us.307

In 1877, Emil du Bois-Reymond had formulated: “Under the banner: Conic sections! No more Greek script! I dare say that I could convene […] a formidable meeting for the purpose of reforming secondary education.”308 His aim was to bring analytic geometry to secondary schools, where he hoped it would replace synthetic Euclidean geometry (this demand was then being made internationally).

304 Philip Magnus et al., Report on a Visit to Germany, with a View of Ascertaining the Recent Progress of Technical Education in that Country (London: Eyre/Spottiswoode, 1896), p. 412. 305 KLEIN 1907b, p. 203 (F. Klein, “Bericht an die Breslauer Naturforschungsversammlung über den Stand des math. und physikalischen Unterrichts an den höheren Schulen”, 1904). 306 See Wilhelm Gallenkamp, Die Elemente der Mathematik: Ein Leitfaden für den mathematischen Unterricht an Gymnasien und Realschulen (Mülheim/Ruhr: Jul. Bagel 1851, 41875). 307 Hermann Weinreich, “Der mathematisch-physikalische Ferienkursus an der Universität Göttingen, Ostern 1914,” Zeitschr. für den math. und naturwiss. Unterricht 45 (1914), pp. 487– 510, at p. 493. This is a quotation from Klein’s lecture in the continuing education course. 308 BOIS-REYMOND 1877, p. 629.

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Figure 38: Committees (etc.) in which Klein discussed educational issues.

As early as 1890, numerous discussions concerning educational reforms had already been held at a School Conference in Berlin.309 In the German Mathematical Society, which had been founded in 1890, pedagogical issues had been a matter of discussion since 1893. In 1890, German mathematics and science teachers formed the Association for the Promotion of Mathematical and Natural-Scientific Instruction (see Section 7.3). After Klein had been named to the board of the Society of German Natural Scientists and Physicians (see Section 7.4.4), he “had a say in integrating their sessions.”310 With this in mind, he not only initiated regular joint sessions with mathematicians and physicists at the annual conferences but also established the teachers’ association mentioned above as the responsible party for the section on mathematical and scientific instruction within the Society of German Natural Scientists and Physicians. This move had been influenced by precedents abroad: “At the other large natural-scientific associations in England (British Association for the Advancement of Science) and France (Association française pour l’avancement des sciences), significantly more consideration is given to educational issues than is the case here in Germany.”311

309 The aim of the conference in 1890 (under the aegis of Wilhelm II) was primarily to counter the development of a “subversive educated proletariat.” The only present mathematician was the trade school director Gustav Holzmüller, who supported the emperor’s (unrealized) idea for education without Latin, but who later opposed the idea of modernizing the mathematics curriculum by including analytic geometry and analysis (see SCHUBRING 2000). 310 Quoted from FREI 1985, p. 111 (a letter from Klein to Hilbert, October 4, 1894). 311 [StA Berlin] Rep. 76 Vc, Sekt. 1, Tit. XI, No. 8, Vol. VI, fol. 243.

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At the next Prussian School Conference (June 6–8, 1900), where Klein proposed suggestions for reform (see Section 8.3.4.1), he became an authority figure to whom other experts also sent their desiderata and concerns. In the years that followed, Klein strove to coordinate these various interests, an effort that he clearly regarded as an urgent task: “The issue seems so essential to me that I would like to raise my voice loudly to generate discussion among all who are interested and knowledgeable in the matter.”312 Klein made this note in preparation for an advisory meeting, which was held in Göttingen on December 11, 1901, with his university colleagues and local secondary school teachers in order to coordinate the interests of everyone involved. The immediate occasion for this meeting was a letter from Karl Kraepelin to Klein containing arguments from biologists,313 who were seeking to revive the status of their discipline in secondary school curricula, from which the subject of biology had been banned (see Section 4.3.3). Klein’s notes document the participation of all the experts present at the meeting and provide information about their discussions concerning the content of lessons and the number of weekly hours of instruction devoted to individual subjects at secondary schools. In 1903, at the annual meeting of the Society of German Natural Scientists and Physicians in Kassel, Klein proposed the formation of an education commission, which first convened in 1904 at the society’s next conference in Breslau. This “Breslau Education Commission” formulated suggestions for reform that were presented at the society’s subsequent meetings in Meran (1905), Stuttgart (1906), and Dresden (1907).314 Within this twelve-member “Breslau Commission,” Klein served as the representative from the German Mathematical Society; he successfully recruited A. Gutzmer to be its chairman, K. Kraepelin to represent biology, Carl Duisberg to represent chemistry and the chemical industry,315 two physicians, a representative from the Association of German Engineers, and teachers from secondary schools. This commission succeeded in taking into account the interest of women’s associations (see Section 8.3.4.1), and it also managed to convince many representatives of the humanities to support the demands of their colleagues teaching mathematics and the natural sciences. Klein himself did much of the latter work by attending and presenting at the annual conferences of the Association of Philologists and Educators (Verband der Philologen und Schulmänner) in Hamburg (1905) and Basel (1907). With diplomatic skill, he was able to organize lecture series with this group.316 Klein developed a coordinated approach to educational reform using various committees and societies. 312 [UBG] Cod. MS. F. Klein 31, fols. 30–35v, at fol. 34v (Klein’s notes, 12 pp.) 313 Ibid. 31 (Kraepelin to Klein, November 23, 1901). – The reintroduction of biology into the upper levels of secondary schools in Prussia was ultimately brought about by a decree issued on March 19, 1908, [StA Berlin] Rep. 76 Vb Sekt. 1, Tit. 5, Abt. V. No. 12, Vol. I, fol. 33. 314 These reform suggestions were published in GUTZMER 1908. 315 On Duisberg, see KÜHLEM 2012. 316 See TOBIES 2000, pp. 35–37.

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At the meeting of the Society of German Natural Scientists and Physicians in Dresden (September 1907), the work of the Breslau Commission was considered complete, and a German Committee for Mathematical and Natural-Scientific Instruction (Deutscher Ausschuss für den mathematischen und naturwissenschaftlichen Unterricht = DAMNU) was created to put the commission’s recommended reforms into practice. Twenty-one mathematical and natural-scientific societies were represented on this new committee. Klein and Stäckel served as representatives from the German Mathematical Society, and Klein led DAMNU’s subcommittee for teacher training.317 Klein was involved in three education committees nearly simultaneously: DAMNU, ICMI, and a third committee in the Upper House (Herrenhaus) of the Prussian Parliament, after the University of Göttingen had elected him, on December 14, 1907, as its representative there (in this position, he succeeded the late Richard Wilhelm Dove; see Section 2.7.1). As was customary, the emperor Wilhelm II made this a lifetime appointment (on February 17, 1908),318 but Klein’s parliamentary position came to end in 1918 with the downfall of the German Empire. With Konrad von Studt, the Minister of Culture from 1899 to 1907, Klein formed a Commission for Education here as well (see Table 9). New elections for the members of this commission were held at every parliamentary session, and, as of 1908, Klein was repeatedly chosen to serve as its speaker. The Prussian Parliament, which met in Berlin, was bicameral. The first chamber (the Upper House) consisted of representatives from every Prussian university and from the larger Prussian cities, of members appointed personally by the Emperor himself, and of noblemen (with hereditary status). The second chamber (the House of Representatives) was based on a three-class franchise system (Dreiklassenwahlrecht). Both chambers possessed the right to pursue legislative initiatives. In France and Italy, it was not unusual for mathematicians to serve in parliament or engage in other political activity. In Germany, we know that H.A. Schwarz actively campaigned for Bismarck’s National Liberal Party during the parliamentary elections of 1887 and that he held a local political office (as a “Bürgervorsteher”) in Göttingen until 1892.319 Klein, who was not a member of any party, was the only mathematician to have a seat in the Upper House of the Prussian Parliament, but he was not the only university professor with such a position. During the First World War, he and the Sanskritologist Alfred Hillebrandt argued for the benefits of studying abroad (see Section 9.1.2). As early as 1898, Adolf Slaby, a pioneer of wireless telegraphy, had been appointed to the Upper House by Wilhelm II, who was keenly interested in technology.

317 See GUTZMER 1914. DAMNU’s work came to an end with Gutzmer’s death in 1924. 318 Stenographische Berichte über Verhandlungen im Herrenhaus (1907/08), Fifth Session, p. 46. 319 [BBAW] NL Schwarz 1254, fols. 249, 300.

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Table 9: Members of the Commission for Education in the Upper House (Herrenhaus) of the Prussian Parliament, formed on March 19, 1909 Member

Category of Membership

1. Dr. von Studt, Konrad (Minister of Culture, 1899–1907) Chairman 2. Dr. Klein, Felix Full Prof., University of Göttingen Deputy Chairman 3. Dr. Graf York v. Wartenburg, Heinrich District Commissioner (retired) 4. Knobloch, Alfred Mayor of Bromberg 5. Faber, Wilhelm Consistorial Councilor 6. Graf v. Haeseler, Gottlieb General Field Marshal (retired) 7. Dr. Hillebrandt, Alfred Full Prof., University of Breslau 8. Graf v. Königsmarck, Karl Castle Governor, Rheinsberg 9. Cardinal Dr. v. Kopp, Georg Bishop of Breslau 10. Graf. v. Kospoth, Karl August Majorat Holder 11. Graf v. der Osten, Leopold Major (retired) 12. Dr. Slaby, Adolf Full Prof., Technische Hochschule in Berlin 13. Christian Ernst Hermann Fürst zu StolbergWernigerode 14. Voigt, Georg Mayor of Barmen 15. Dr. Zorn, Philipp Full Prof., University of Bonn

Imperial Appointment University Hereditary City of Bromberg Imperial Appointment Imperial Appointment University Family Association Imperial Appointment Imperial Appointment Family Association Imperial Appointment Hereditary City of Barmen Imperial Appointment

8.3.4.1 Suggestions for Reform The School Conference in June of 1900 concerned the content and structure of education at secondary schools for boys. Konrad von Studt, then the Minister of Culture, and eight government officials (including Althoff) invited thirty-four experts to attend, among whom were Felix Klein and Henry Theodore Böttinger. In advance of the conference, printed reports were circulated on ten issues, including reports on mathematical and scientific education by Bernhard Schwalbe (a school principal), Adolf Slaby (a professor of electrical engineering at the Technische Hochschule in Berlin), Wilhelm Lexis (a professor of economics), Emil Lampe, and Guido Hauck (both professors of mathematics at the Technische Hochschule

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in Berlin). Klein, however, was not initially asked to write a report. A letter from Anna Klein to her husband from April 4, 1900 sheds some light on the context: I believe that you will find much to do in Berlin, for it now seems as though Althoff wants to make use of your ideas. This may be related to the fact that last week, when discussing the cultural budget in parliament, Slaby mentioned the institutions here in Göttingen and your name as the causes of problems, even condemning them as harmful. This was the old song about encroaching on foreign territory and about “general staff officers.” [Johannes] Reinke (Kiel) and [Gustav] Schmoller (Berlin) have defended your efforts. Now, today, Reinke received a request from Althoff to send him your lectures on this matter.320

Almost immediately (on April 11, 1900), Klein had the Teubner press print copies of the lectures in which he offered to cooperate with engineers, including a rejoinder that was addressed to Slaby in particular.321 After he had sent these documents to the ministry, Klein was then asked to prepare, on short notice, his own report on the issues to be discussed at the School Conference.322 Klein’s proposals were innovative, and in several respects they would prove to be influential in the process of educational reform. First, Klein argued that the diplomas awarded by the three different types of German secondary schools should be treated equally with respect to university admissions, an argument that was in step with international trends. In his introductory report, Minister von Studt made it known that he was aware of the reforms being made in other countries, especially in France, where secondary schools “without dead languages” had meanwhile been recognized as gateways to universities.323 Von Studt formulated the alternatives: either a uniform school system with an increased focus on practical subjects at the expense of classical languages or the equal treatment of diplomas from the existing school types. In the heated debate that followed, Klein underscored that, for mathematics, equal rights to admission already existed and that students from various educational backgrounds were being accepted.324 Adolf Harnack325 made a motion for the equal treatment of the different final examinations (Abitur). On November 26, 1900, the emperor issued a decree that, regarding university admissions, placed the examinations (Abitur) administered by Realgymnasien and Oberrealschulen on equal footing with those of humanistic Gymnasien.326

320 Quoted from TOBIES 1989a, p. 11 (Anna Klein to Felix Klein, April 4, 1900). For Klein’s remarks about how German universities should train the “general staff officers” of engineers, while Technische Hochschulen should educate the “infantry of engineers,” see Section 7.7. 321 See KLEIN 1900. 322 For Klein’s report from May 22, 1900, see SCHUBRING 1989 and 2000. 323 [StA Berlin] Rep. 76, Sekt. 1, Gen. Z, No. 165, fols. 8, 40–43, 53. 324 Klein’s speeches are published in VERHANDLUNGEN 21902, pp. 29–31, and with his own commentary in the Jahresbericht der DMV 11 (1902), pp. 128–41. 325 Klein cooperated with Adolf Harnack, a theologian and historian (the twin brother of Klein’s former, already late doctoral student Axel Harnack). From 1911 to 1930, Adolf Harnack also served as the president of the Kaiser Wilhelm Society. 326 Some of the federal states still maintained special rules for the study of theology and law.

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Second, on the conference topic of “elevating the level of instruction in various subjects,” Klein argued for a higher level of general mathematical education.327 This means that areas of higher mathematics – which had so far only been taught at universities or Technische Hochschulen – should be included in the curriculum of secondary schools. In a speech on this topic at the School Conference, Klein explained: Every expert will confirm that one can only understand the basis of the scientific explanation of nature if one is at least familiar with the rudiments of differential and integral calculus and with analytic geometry – that is, with the basic elements of higher mathematics. There have always been teachers, even at humanistic Gymnasien, who to some degree have introduced these rudiments to their pupils. The question should be whether room could be reserved for these subjects in the general curriculum, at least in Realanstalten.328

Slaby, Hauck, and Lexis endorsed Klein’s idea, and the conference participants recommended it unanimously to the government. Nevertheless, it was not incorporated into the new curricula for 1901.329 Thus, Klein continued to promote the teaching of “practical differential and integral calculus, limited to the simplest relations, and illustrating them by means of the natural processes with which students [at secondary schools] are already familiar.”330 Klein argued for this by referring to the examples of Nernst and Schoenflies’s textbook (1895, English trans. 1900) and to John Perry’s textbooks on practical mathematics (1897, 1899). Klein also cited recent developments in France, where differential and integral calculus were made a required part of the curriculum at all secondary schools in 1902 and where the concept of the function played a central role in mathematical instruction.331 Klein also invited a contribution on the same topic from the French mathematician Francisque Marotte, who had visited him in Göttingen in 1894/95332 and studied the reform movement in Germany in 1901 and 1903. Regarding the situation in France, Marotte stressed: “Ces importantes notions de fonctions et d’approximation, par lesquelles les mathématiques prennent contact avec le monde physique et avec la réalité, nous allons les retrouver au centre de l’enseignement mathématique.”333

327 At the same time, Klein supported Böttinger’s wishes for secondary schools to prepare graduates to study chemistry and for English to be introduced as a subject at humanistic Gymnasien on account of its “great importance in global trade” (VERHANDLUNGEN 21902, pp. 188, 198). Böttinger had grown up in England. Up until this point, chemistry had been the only subject that could be studied at universities and Technische Hochschulen without the Abitur. 328 Quoted from VERHANDLUNGEN 21902, p. 154. Realanstalten denote Oberrealschule and Realgymnasium, where more weekly lessons in mathematics and science were taught. 329 Lehrpläne und Lehraufgaben für die höheren Schulen in Preußen (Halle: Waisenhaus, 1901). 330 Jahresbericht der DMV 11 (1902), p. 137. 331 Felix Klein, “Zur Besprechung des mathematisch-naturwissenschaftlichen Unterrichts auf der nächsten Naturforscherversammlung,” Jahresbericht der DMV 13 (1904), p. 198. 332 See DECAILLOT 2011, p. 173. 333 F. Marotte, “Les récentes réformes de l’enseignement des mathématiques dans l’enseignement secondaire français,” Jahresbericht der DMV 13 (1904), pp. 450–56, at p. 451; idem,  

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The reformed German curriculum proposed in Meran by the Breslau Education Commission in 1905 (the so-called “Meraner Reform-Lehrplan”) likewise concentrated on the concept of the function and emphasized the importance of graphical methods and analytic geometry. It also contained a compromise: mathematical instruction should rise “up to the threshold (Schwelle) of infinitesimal calculus.” That is, it remained unclear whether this subject matter should be taught or not, as Klein pointed out years later in a Göttingen continuing education course in April of 1914.334 This compromise was a result of the opposition championed by Friedrich Pietzker, a teacher at a secondary school (Gymnasialprofessor) and the longtime chairman of the Association for the Promotion of Mathematical and Natural-Scientific Instruction. Pietzker insisted on preserving the nineteenth-century notion of what a general mathematical education should be. For many years in Prussia, the content of mathematical education at secondary schools had been defined by Adolph Tellkampf.335 A paradigm shift eventually took place, but not before a long and arduous struggle. In a parliamentary speech given on May 4, 1914, Pietzker was still, as a representative of his party (Fortschrittliche Volkspartei), polemicizing against Klein and his idea to “introduce infinitesimal calculus into the secondary school curriculum.”336 The Prussian Ministry of Culture therefore recommended that the matter should first be tested at reform schools and “be allowed to develop from the bottom up.” The reports show that teachers increasingly adopted Klein’s ideas for reform.337 As the president of the ICMI, Klein organized sessions on the topic of teaching infinitesimal calculus at secondary schools at this commission’s meetings in Brussels (1910) and Paris (1914).338 As Ulf Hashagen has shown in this regard, Klein’s former student Walther Dyck had enjoyed quicker success in Bavaria than Klein did in Prussia. Infinitesimal calculus was made a mandatory component of the curriculum in Bavarian secondary schools in 1914, whereas this would not be the case in Prussia until 1925 – an initiative for which Klein would remain actively engaged until shortly before his death (see Section 9.3.2).339 Third, Klein elevated the didactics of mathematics to an independent discipline. Inspired by Klein, the Austrian mathematician Alois Höfler wrote the first ever German-language textbook on this subject, which was published in 1910 by Teubner in Leipzig (see Section 5.6).340 Klein’s three-volume Elementary Mathematics from a Higher Standpoint resulted from his lecture courses since 1901 (see  

334 335 336 337 338 339 340

L’Enseignement des sciences mathématiques et physiques dans l’enseignement secondaire des garçons en Allemagne (Paris: Impr. nationale, 1905). See the report in Zeitschr. für den math. und naturwiss. Unterricht 45 (1914), pp. 493–96. See FOLKERTS/SCHUBRING 2020. See TOBIES 2000, pp. 30–32. See TOBIES 2012; and LIETZMANN 1960, pp. 47–53. Due to his health, Klein did not travel to Paris himself in 1914, but he had prepared everything in advance ([Paris] 83: A letter from Klein to Darboux dated March 25, 1914). See HASHAGEN 2003, p. 358. See HÖFLER 1910.

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8.3.4.2) and follow the aim of improving the “methodological education of mathematics teachers.”341 Here he stressed the importance of fusing different disciplines – arithmetic and geometry, planimetry and stereometry – with the goal of preventing the “one-sided teaching of planimetry while neglecting three-dimensional space intuition.”342 Such a fusion of plane and spatial geometry had already been treated quite well in Giulio Lazzari and Anselmo Bassani’s book Elementi di geometria (1891). A German translation of this book was published by Teubner in 1911, and in that same year Klein deliberately chose “Strenge und Fusion” [Strictness and Fusion] to be the main topic of the ICMI meeting in Milan. Supervised by Klein, Rudolf Schimmack became the first scholar in Germany to complete a Habilitation in “didactics of the mathematical sciences” in 1911. Schimmack continued to work on the fusion of disciplines in mathematical instruction at secondary schools. His introductory lecture in the context of his Habilitation procedure was “Über die Verschmelzung verschiedener Zweige des mathematischen Unterrichts” [On the Fusion of Different Branches of Mathematical Instruction].343 Schimmack’s treatise on the development of the reform movement in German mathematical education was accepted as his Habilitation thesis. Schimmack’s success with this Habilitation was based on many years of collaboration with Klein, whose assistant he was from 1903 to 1905, and whose lecture courses on mathematical instruction (winter semester, 1904/05) he prepared for publication.344 Already his dissertation topic had been inspired by Klein; as Schimmack noted, he had first come up with the idea “in connection with a historical-mathematical seminar by Privy Councilor Klein on the principles of mechanics (winter semester, 1902/03).”345 Schimmack had published an article on the subject in 1903,346 but initially he pursued a career as a secondary school teacher. Therefore, it was not until January 19, 1908 that he passed his oral examination with distinction (Klein was the main supervisor of the dissertation, and Hilbert was the co-supervisor). Previously, Schimmack had also passed his teaching examinations with distinction. As of 1908, he was employed as a senior teacher at the Gymnasium in Göttingen. After his Habilitation, Schimmack taught as a Privatdozent at the university – in addition to his permanent position as a schoolteacher. He took over Klein’s seminars on the history of infinitesimal mathematics (as of November of 1911) and on educational reforms in Germany, France, England, Austria, and the United States (summer semester, 1912).347 For the winter semester of 1912/13, Schim341 342 343 344 345

SCHUBRING 2016, p. 7. KLEIN 2016 [1925], p. 239. [UAG] Kur. 6308 (Habilitation records). See KLEIN 1907. Rudolf Schimmack, Axiomatische Untersuchungen über die Vektoraddition [Axiomatic Investigations of Vector Addition] (Halle: Erhardt Karras, 1908), p. 106. 346 Rudolf Schimmack, “Ueber die axiomatische Begründung der Vektoraddition,” Göttinger Nachrichten (1903), pp. 317–25. 347 [Protocols] vol. 29, pp. 157–452.

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mack announced his first lecture courses: “Ausgewählte Abschnitte der mathematischen Didaktik” [Select Chapters of Mathematical Didactics].348 However, he was not able to complete this course because, after a bout of scarlet fever, he suffered a fatal heart attack on December 2, 1912. “I could not have been dealt a heavier blow; I had based all my plans for the near future on his collaboration.” Klein wrote these words the day after Schimmack’s death to Walther Lietzmann, whom he now encouraged to teach the didactics of mathematics in Jena: “Once you have found your bearings, you would then be able to operate at the university in a capacity similar to Schimmack’s. It seems to be the nature of development that private approaches of this sort will gradually become part-time teaching appointments (Nebenamt).”349 On October 20, 1913, Klein stressed more generally: “Apart from or beyond [independent professorships for pedagogy], we need our own teaching appointments for the didactics of various groups of disciplines.”350 In 1914, Klein enticed Lietzmann to move from Jena to Göttingen, where, in addition to working as the director of a secondary school, he also received a teaching appointment for “the didactics of the exact sciences” at the university.351 Lietzmann published works in the ICMI monograph series, articles on educational reform, and, in 1916, a book titled Methodik des mathematischen Unterrichts [The Methodology of Mathematical Instruction] (Leipzig: Quelle & Meyer, 440 pp.). Klein did not live to see the book written by Lietzmann during the Nazi era to include Bieberbach’s racial typology.352 Klein steered further mathematicians toward didactics, among them Otto Staude’s student Friedrich Drenckhahn (see Section 9.2, Table 10) and Paul Zühlke (a contributor to the ICMI monograph series).353 Klein recommended, moreover, that new professorships in the field of university didactics (Hochschulkunde) should be established, and he was appreciated for his own didactic articles on academic instruction. In 1912, he joined a Society for University Didactics (Gesellschaft für Hochschuldidaktik).354 Fourth, in addition to engaging on behalf of the right for women to study mathematics at the university level (see Section 7.5), Klein also supported initiatives to reform girls’ schools. The Breslau Education Commission, mentioned above, cooperated with engaged women and women’s associations, as is documented in a letter by the microbiologist Lydia Rabinowitsch-Kempner: Verzeichnis der Vorlesungen, 1912/13 (Göttingen, 1912), p. 15. [UBG] Cod. MS. Lietzmann I:202, 212 (Klein to Lietzmann, Dec. 3, 1912 and Febr. 2, 1913). [UBG] Cod. MS. F. Klein 22L, Personalia, fol. 19 (Klein’s notes for W. Lorey). Ibid. F. Klein 2G, fols. 18–19, 23, 65. Walther Lietzmann (with the assistance of Ulrich Graf), Mathematik in Erziehung und Unterricht (Leipzig: Quelle & Meyer, 1941). In 1945, Lietzmann lost his position as a director. (See Tobies 1993b; HESKE 2018; HOLLINGS/SIEGMUND-SCHULTZE 2020, pp. 76–85, 271–84). 353 See TOEPELL 1991, pp. 89, 143. Regarding Drenckhahn’s work on didactics, see SCHUBRING 2016, pp. 10, 13–14. 354 [Hecke] Klein’s lecture course of 1910-11, pp. 318–20. 348 349 350 351 352

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8 The Fruits of Klein’s Efforts, 1895–1913 Esteemed Professor! I have allowed myself to send you […] 12 copies of the report undertaken by Miss Thekla Freytag (who has passed the examination pro facultate docendi) and myself […]. The work was carried out with the support of the Organization of Progressive Women’s Associations (Mrs. Minna Cauer, Chairwoman), which in its time has offered input to the Education Commission of the Society of German Natural Scientists and Physicians. The General German Women’s Association (Miss Helene Lange, Chairwoman), with which you are certainly familiar, is also in agreement with our report […]. In the hope that the Commission will keep us informed about its further deliberations, and with the sincerest thanks for the goodwill that the Commission has shown to the cause of women, I sign this letter with the utmost respect and also on behalf of Thekla Freytag, Yours sincerely, Lydia Rabinowitsch-Kempner355

A great deal of persistence was needed to implement the suggested reforms for girls’ schools. When Klein reported to his wife about a meeting with the Prussian Minister of Culture at the 1907 conference of the Society of German Natural Scientists and Physicians, she replied: “This probably means that the Ministry intends to take the issue of school reform seriously. Then your efforts will bear fruit! The other day I read that they want to undo everything that seemed to have been secured for the girls’ schools.”356 Their daughter Elisabeth was then attending Realgymnasium courses in Hanover, and there she had to take the Abitur examinations externally at a Realgymnasium for boys (February 17, 1908), because there were still no girls’ schools in Prussia that led to this qualification. In early 1908, she enrolled as an auditor at the University of Göttingen. In October of that year, she was finally able to matriculate as a regular student after Prussia had passed the aforementioned decree on August 18, 1908. This decree also brought about a new structure for girls’ schools (with three types based on the model of boys’ schools: one oriented toward the humanistic Gymnasium, the second toward the Realgymnasium, and the third toward the Oberrealschule). An Abitur from one of these schools enabled young women to attend universities. Klein’s first speech in the Upper House of the Prussian Parliament, delivered on May 21, 1909, concerned the reform of girls’ schools. He argued for the continuing education of the teachers who taught at these reorganized schools, and he planned the first course of this sort, which took place in Göttingen from the 4th to the 16th of October in 1909 (with 74 participants). Klein also involved his daughter Elisabeth and her friend Iris Runge (Carl Runge’s eldest daughter) in the organization of this course. These two daughters of professors were then dominant figures in the women’s student organization at the University of Göttingen, and both benefited from the reform of secondary schools for girls.357 355 [UBG] Cod. MS. F. Klein 35, fols. 165–165v (a letter from Rabinowitsch-Kempner to Gutzmer dated December 12, 1905; Gutzmer, who managed the correspondence of the Breslau Education Commission, forwarded the original letter to Klein). – Regarding Klein’s appreciation of Thekla Freytag’s state teaching examination in 1905, see TOBIES 2017b. 356 [UBG] Cod. MS. F. Klein 10: 322 (Anna to Felix Klein, September 17, 1907). 357 See TOBIES 2012, p. 71.

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Klein was dissatisfied with what had been achieved. He regarded the new girls’ curriculum for mathematics as being “uncritically copied, in its main principles, from obsolete precedents used at schools for boys, without any understanding for recent developments.”358 In subsequent parliamentary speeches (on March 15 and May 27, 1910), he argued that changes should be made and that preliminary courses should be offered for female candidates with insufficient educational backgrounds. In Göttingen, he also established courses of this sort in 1910. In the debate at the time about the central issue of coeducation in secondary schools, Felix Klein took the side of Adolf Harnack, who made the point that girls would have to be allowed to attend secondary schools for boys because not every small city would be able to institute an upper school for girls.359 Fifth, at least since his lecture course of 1904/05,360 and then later in the Prussian Parliament, Klein vehemently supported a better education at elementary schools and at newly established continuation schools and vocational schools. He was in favor of education for the masses, and he supported adult education along the lines of the Perry movement in England.361 In parliament, Klein and the elderly Graf Haeseler (see Table 9) introduced a legislative proposal (1909–10) to make education available to anyone up to eighteen years of age. This proposal was unanimously approved. Regarding the issue of elementary education, Klein explained in his speech on May 27, 1910: In the case of my proposals for elementary schools, I have faced one objection in particular. Competent parties have repeatedly told me that all of these needs undoubtedly exist, but it costs money to meet them. The Finance Minister is not much inclined to make large sums available for these matters. Yet, in this respect, I would like to request the educational administration not to be too cautious, for the sums that we have in mind are minuscule in comparison with the 160 million that is already devoted to the educational budget. In my view, it would be as if a machine manufacturer intentionally refrained from purchasing the proper tool machines, even though such machines would make his factory better and cheaper. No practical person would do this.362

Klein’s efforts to analyze, study, and represent mathematical education from kindergarten to higher education on a theoretical level were reflected in his lecture courses and seminars, in the five volumes of the Abhandlungen über den mathematischen Unterricht in Deutschland [Treatises on Mathematical Instruction in Germany] that he edited (1909–16), and in the numerous ICMI reports that he instigated (by 1920, these reports constituted 187 volumes, with contributions from eighteen different countries).363 358 Quoted from TOBIES 1989a, p. 7. 359 [UBG] Cod. MS. F. Klein 7D (a letter from Harnack to Klein dated January 6, 1909 and a draft of Klein’s reply, dated January 8, 1909). 360 See KLEIN 1907, pp. 10–17. 361 While a student at the University of Göttingen, Iris Runge taught such adult education courses. See TOBIES 2012, pp. 96–97. 362 Quoted from TOBIES 1989a, p. 8. 363 See FEHR 1920, p. 339; and TOBIES 1979, p. 26.

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Klein’s commitment to practical improvements is clear from his speeches on the cultural budget and from his contributions to proposed legislation. In a parliamentary speech on April 7, 1911, he presented the following justifications for his position in favor of increasing the cultural budget for “elementary education”: Now I have to offer a word of apology, so to speak, for expressing my opinion about such matters as a university professor. In my circles, the opinion is often held that we should only concern ourselves with abstract science. It would be regarded as an abnormality if we were to discuss the pedagogical approach at our own institutions, let alone the pedagogical approach used in secondary schools. I myself am concerned here even with the teaching at elementary schools [Volksschulen] and at related institutions. My right to speak about such matters, I would expressly like to say, derives not only from my parliamentary position but also from my position at the university. We university professors stand up for science in general, not only for its further development but also, to the best of our abilities, for its prestige, and the ideal that I have in mind is that we should regard education as a large whole, beginning with kindergarten (with its own particular and highly interesting problems) and extending all the way up to higher education and far beyond.364

Klein also explained his commitment to the improvement of the instruction at elementary schools to a circle of secondary school teachers. He stressed: “In light of the immense differences that divide our society, there is a social obligation for us professionals not to remain indifferent to the educational issues that ultimately affect 94% of our population.”365 In his lecture courses from 1910/11 on mathematical pedagogy, Klein incorporated the debates that were being held at the time in the Prussian Parliament, and he expressed his belief that the growing influence of the Social Democrats would gradually cause other parties to lose their “inhibitions” to act on behalf of elementary and continuing education.366 8.3.4.2 A Polemic about the Teaching of Analysis at the University In his article on the arithmetization of mathematics from November of 1895 (discussed in Section 8.3.2), Klein remarked: I must add a few words on mathematics from the point of view of pedagogy. We observe in Germany at the present day a very remarkable condition of affairs in this respect; two opposing currents run side by side without affecting one another appreciably. Among the teachers in our Gymnasia the need of mathematical instruction based on intuitive methods has now been so strongly and universally emphasized that one is compelled to enter a protest, and vigorously insist on the necessity for strict logical treatment. This is the central thought of a small pamphlet on elementary geometrical problems which I published last summer. Among the university professors of our subject exactly the reverse is the case; intuition is frequently

364 Quoted from TOBIES 2000, p. 33. 365 Felix Klein, “Aktuelle Probleme der Lehrerbildung,” Schriften des deutschen Ausschusses für den mathematischen und naturwissenschaftlichen Unterricht 10 (1911), p. 2. 366 [Hecke] Klein’s (unpublished) lecture courses on mathematical instruction (1910/11), transcribed by his assistant Erich Hecke, p. 58.

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not only undervalued, but as much as possible ignored. This is doubtless a consequence of the intrinsic importance of the arithmetizing tendency in modern mathematics.367

With respect to primary and secondary education, Klein adopted a “doctrine of the middle way,” and this was reflected in his plans for reform. However, his polemical argument that the arithmetizing tendency in mathematics implied “not only a false pedagogy but also a distorted view of the sciences”368 also found its share of opponents. Klein believed that courses for beginning students of mathematics and for future natural scientists and engineers should proceed on the basis of intuition. Secondary school teachers, natural scientists, and engineers enthusiastically agreed to this and promoted the idea of teaching such courses. One result of Klein’s approach was his three-volume textbook Elementary Mathematics from a Higher Standpoint (on arithmetic, algebra, analysis; geometry; precision mathematics and approximation mathematics) which was printed in multiple editions, has been translated into several languages, and remains popular today.369 Klein’s arguments against the arithmetizing tendency in mathematics courses were not embraced by every mathematician. A heated polemic developed in particular with Alfred Pringsheim, who published a contrary opinion about how analysis should be taught in first-year university courses. Pringsheim adhered to Weierstrass’s teaching methods and argued: “Once the arithmetic foundations […] have been established, only then can geometrical intuition be introduced.”370 Klein preferred the opposite approach so as not to discourage beginning students and because natural processes could not be explained with precision mathematics but rather with the type of mathematics that can account for “those relations which occur with limited accuracy!”371 With this approach, too, he wanted to take the wind out of the sails of the anti-mathematics movement among engineers (see Section 8.1.1), and his ideas found international confirmation in H.A. Lorentz’s Leerbook der differentiaal- en integraalrekening (1882) and John Perry’s Calculus for Engineers (2nd ed., 1897), which Klein arranged to be translated into German (see Section 5.6).

367 Klein 1896 [1895c], pp. 247–48. – The mentioned “pamphlet” is KLEIN 1895b (see 7.3). 368 Ibid., p. 248. 369 See KLEIN 31924, 31925, 31928. For the most recent English translation, see KLEIN 2016. On the international influence of Klein’s textbook and on its whole or partial translations into Spanish (1927, 1928), English (1932, 1939), Japanese (1959/60, 1961), Russian (1987), Portuguese (2009–14), and Chinese (1989; repr. 1996), see Gert Schubring’s article in KAISER 2019, pp. 330–33. – In a paper presented at the Thirteenth International Congress on Mathematical Education (Hamburg, 2016), Mary Silva da Silva and Diogo Franco Rios analyzed the positive reception that these books received in Brazil before the First World War. – For a comparison of the lectures on elementary mathematics by Klein, Heinrich Weber, and Max Simon, see JAHNKE 2018 and ALLMENDINGER 2019. – For the domain “precision mathematics and approximation mathematics” (Vol. III), see also Section 8.3.2. 370 Alfred Pringsheim, “Über den Zahl- und Grenzbegriff im Unterricht,” Jahresbericht der DMV 6 (1899), pp. 73–83, at p. 82. 371 KLEIN 1899, p. 137. – On similar views by Charles Hermite, see GOLDSTEIN 2011b.

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Klein and Alfred Pringsheim differed in their pedagogical intentions. Whereas Klein sought to take along all of his students, Pringsheim considered it “fortunate” if certain unsuitable “elements would fundamentally be frightened away from mathematics even in the introductory lectures.”372 Klein, in contrast, emphasized: “In my elementary lectures, I strive above all to impart to my students an interest in and understanding of the questions and the meaning and the purpose of the mathematical treatment.” He explained that he did not neglect the deeper foundations of mathematics but rather waited to teach them in his seminars for upper-level students, which he conducted with Hilbert. Moreover, he added that he had “enthusiastic students” and that “the real aim of all teaching […] is to help students think independently.”373 In 1913, Klein saw that this polarity in teaching methods had grown wider: Incidentally, Klein and Pringsheim are no longer the extreme poles in this opposition. To my left, for instance, there is [Carl] Runge with his practical exercises, and to the right of Pringsheim there is now a youthful swarm of set theorists who do not hesitate to regale even first-semester students with their most far-reaching abstractions.374

This polemic would be resurrected in the 1930s in order to portray Klein as a unilateral supporter of intuition and thus as an especially “German” mathematician and to defame Pringsheim as a “Jewish” mathematician.375 Otto Toeplitz, who came from a Jewish family, took cues from Klein’s lecture course on differential calculus (1911) and developed a “didactically oriented indirect genetic method” for his own introductory lectures. In 1927, however, Toeplitz remarked that nothing had changed regarding the polarized viewpoints toward teaching. On the one hand, there was the exact approach, which had existed since Weierstrass and which insisted on teaching mathematical strictness from the very beginning (and thereby alienated 95% of students); on the other hand, there was Klein’s intuitive approach, which sought to reach a wide range of students.376 8.4 INTERNATIONAL SCIENTIFIC COOPERATION Much has already been made in this book of Felix Klein’s interest in international scientific cooperation, an interest that can be traced back to the influence of Plücker and Clebsch. Through his travels and correspondence, Klein cultivated an international exchange of ideas that yielded benefits for his own research, the

372 Alfred Pringsheim, “Zur Frage der Universitäts-Vorlesungen über Infinitesimalrechnung,” Jahresbericht der DMV 7 (1899), pp. 138–45, at p. 142. 373 KLEIN 1899, pp. 132–33 (emphasis original). 374 [UBG] MS. Philos. 182, No. 4 (a letter from Klein to W. Lorey dated April 7, 1913). 375 See JAENSCH/ALTHOFF 1939 and, for criticism, MEHRTENS 1987; BERGMANN/EPPLE 2009. 376 See Otto Toeplitz, “Das Problem der Universitätsvorlesungen über Infinitesimalrechnung und ihrer Abgrenzung gegenüber der Infinitesimalrechnung an den höheren Schulen,” Jahresbericht der DMV 36 (1927), pp. 88–100. On this topic, see also FRIED/JAHNKE 2015.

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journal Mathematische Annalen, book projects such as the ENCYKLOPÄDIE, and reforms to mathematical education. After Klein had proposed the idea in 1893 of forming “international unions” (see Section 7.4.1), the topic was discussed in 1894 at the conference in Vienna and in a great deal of correspondence,377 so that finally the First International Congress of Mathematicians took place in Zurich in 1897. Felix Klein belonged to the international planning committee as a representative of Germany,378 and he coordinated his proposals with Adolf Hurwitz. The history of the International Congresses is well researched. Since 1900 in Paris – where Hilbert gave his famous lecture on unsolved mathematical problems – the mathematicians met every four years: 1904 in Heidelberg, where Klein was a member of the organizing committee, 1908 in Rome, 1912 in Cambridge (UK), then the interruption of the First World War.379 In this section, I will discuss some of the lesser-known aspects of Klein’s activity and his international attitude. Bernhard vom Brocke has described the Prussian Ministry of Culture’s science policy, which was driven by the ideas of peacekeeping and international understanding, as “a nearly forgotten alternative government policy of world peace on the eve of the First World War.”380 Klein’s international engagement fit neatly into this policy, which was shaped by Friedrich Althoff. By 1907, the international balance of power, which was partially a result of Bismarck’s elaborate foreign policy, had been destabilized to such an extent that France (with fresh memories of losing Alsace-Lorraine) was more closely aligned with Russia and England, while Germany’s remaining (weak) allies were Austria-Hungary and Italy. Among other things, this prompted Germany to reorient its cultural policy toward improving international relations. German nationalism was on the rise even despite the country’s internationally integrated corporations and despite its scientific and technological innovations.381 Klein reflected that active countermeasures needed to be taken to curb the surging nationalism of the post-Bismarck era: Particularly in our time, when national chauvinism is at its orgiastic peak, it must be close to the hearts of all truly educated people to counteract a movement that is increasingly intent on alienating even the best citizens of different nations from one another. It is as though the whole human spirit is vanishing along with the humanism that was once so rightly celebrated. […] Intellectual exchange between modern cultures does not entail the elimination of national differences; rather, it leads to a clearer understanding of their true character and values, and this higher knowledge creates a friendly relationship among nations.382

377 [UBG] Cod. MS. F. Klein 8:452 (G. Cantor to Klein, Sept. 16, 1895); ibid., 12:199 (Vasilev to Klein, April 2, 1895); [UBG] Math. Arch. 258 (Klein to Hurwitz, March 29, 1895, etc.). 378 The fact that there were still tensions between Klein and the Berlin mathematicians was expressed by Hermann Minkowski in a letter to Hilbert, in which he commented that probably no one from Berlin would participate in Zurich. See FREI/STAMMBACH 1994. 379 See also CUBERA 2009. 380 BROCKE 1991, p. 185. See also BROCKE 1981. 381 On German foreign policy at the time, see OSTERHAMMEL 2009. On the role of technology in European integration, see the book series edited by SCHOT/SCRANTON 2013–2017. 382 [UBG] Cod. MS. Math. Arch. 5021, fols. 54–57v, at 56–56v (Klein, March 11, 1908).

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Klein had written these words on March 11, 1908 as part of a draft proposal for establishing a Göttingen Institute for Foreigners (Göttinger Institut für Ausländer), which he sent to the Ministry of Culture in Berlin. Althoff, who would die on October 20, 1908, helped to implement this idea by persuading H.Th. von Böttinger to donate 100,000 Mark to support the undertaking. On November 28, 1908, Klein gave an inaugural address to commemorate the opening of the “Böttinger-Studienhaus,” which would function as an academic center for foreign students and provide information about language courses, lectures, summer programs, and excursions. When German nationalists criticized this institution, its director defended its goal to establish friendly international relations and added that its aims were analogous to those of the Alliance Française, the language courses organized in England, the America Institute, and also the German-American professorial exchange, for which Klein’s travels in 1893 and 1896 were interpreted as starting points.383 Klein himself, however, was no longer willing to take another trip overseas,384 and thus in 1904, for instance, no German mathematician had attended the International Exposition in St. Louis (see 8.3.2), whereas Darboux, Picard, Poincaré, and the Austrian physicist Ludwig Boltzmann all traveled there to participate. Klein’s unwillingness to travel to this event had upset Althoff, but Klein was able to compensate for this by collaborating on a number of international projects. The latter included the International Catalogue of Scientific Literature (published from 1902 to 1921), the formation of an International Association of Academies,385 a proposal for establishing Esperanto as an international language, and the creation of the Internationale Wochenschrift für Wissenschaft, Kunst und Technik [International Weekly Publication for Science, Art, and Technology], which was initiated by Althoff. While participating in these projects, Klein was concerned exclusively with their scientific results; he deliberately avoided the promotion of any strict national interests. In the mid-1890s, the Royal Society of London had begun the International Catalogue, and Klein stressed that Germany’s special interest in the project was not to have it translated into German but rather that the “core of the entire undertaking was international cooperation,” as he firmly stated in a letter to the Ministry of Culture in Berlin.386 To prepare for the work ahead, Klein traveled repeatedly to London, and he cooperated with the French council member Henri Poincaré to determine the organization of the mathematics volumes.387 383 BROCKE 1991, pp. 224–25. 384 [UBG] Cod. MS. F. Klein 7F, fols. 6–9, 35v; TOBIES 1990a, p. 39. In Klein’s place, Carl Runge spent the academic year of 1909/10 in New York as an exchange professor of mathematics. See HENTSCHEL/TOBIES 2003, pp. 173–79. 385 For a detailed discussion of this development, see HASHAGEN 2003, pp. 471–82. 386 [BStBibl] A letter from Klein to Dyck dated April 6, 1897, in which Klein discussed his correspondence with the Ministry of Culture. 387 [UBG] Cod. MS. F. Klein 7J: fols. 86v, 101. – International Catalogue of Scientific Literature: Fourth Annual Issue. A: Mathematics (London: Royal Society of London, 1905).

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Regarding Esperanto, Klein proposed that an Academy project could perhaps be devoted to the question of establishing a “global language.” This idea had been suggested to him by the Leipzig-based physical chemist Wilhelm Ostwald (Nobel Prize, 1909), who was actively involved in the movement to popularize the constructed language (invented in 1887). Klein’s proposal, however, was met with a “storm of indignation” from the philologists in Göttingen, and it was thus impossible to make any headway with such a project. These same colleagues rejected the idea yet again even after the first World Esperanto Congress had taken place in 1905 (in France).388 Founded in 1907 in Berlin, the Internationale Wochenschrift für Wissenschaft, Kunst und Technik had been conceived as a publishing platform for ideas related to international academic exchange.389 The fifty-eight members of its advisory board included prominent foreign scientists, Hendrik A. Lorenz and Henri Poincaré among them. As German mathematicians, the only board members in addition to Klein were Walther Dyck (Munich) and Emil Lampe (Berlin). The publication was supported by the Koppel Foundation for the Promotion of Germany’s Intellectual Relations Abroad (Koppel-Stiftung zur Förderung der geistigen Beziehungen Deutschlands zum Ausland),390 which had been created in 1905 with 1,000,000 Mark of initial capital (known as the Leopold Koppel Foundation as of 1913).391 The editor of the journal was the historian Paul Hinneberg, who, at the same time, was also the director of the Kultur der Gegenwart project (see Section 8.3). Hinneberg emphasized that the goal of the publication was to foster “international communication […] without any national bias.” The inaugural issue opened with an essay titled “Über die Einheitsbestrebungen der Wissenschaft” [On the Unifying Efforts of Science], in which the scientifically minded classical philologist Hermann Diels intoned that, “at least on the neutral ground of science, unifying love has become stronger than divisive hatred.”392 Two of Klein’s speeches were published in the journal’s second volume.393 The unifying sentiments behind this publication quickly disintegrated with the outbreak of war in August 1914. In October of that year, the signatures of thirteen of its German board members appeared on a nationalistic declaration (see 9.1).

388 [BBAW] NL W. Ostwald, No. 1500 (Klein’s letters to Ostwald dated July 23, 1904 and February 13, 1907). 389 This journal existed from 1907 to 1921, though it became a monthly (instead of weekly) publication in 1912. 390 The Koppel foundation also supported the German-American professors’ exchange and the aforementioned Kaiser Wilhelm Society (the Max Planck Society today). 391 [UBG] Cod. MS. F. Klein 1 E, fol. 36 (a letter from the Prussian Ministry of Culture concerning a donation of 600 Mark from the Koppel Foundation for Klein’s collaboration). The German-Jewish banker Leopold Koppel was socially engaged and was a strong supporter of scientific research. Among other things, his foundation subsidized Albert Einstein’s salary in Berlin. 392 Quoted from BROCKE 1991, p. 192. 393 See KLEIN 1908a and 1908b.

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8.5 EARLY RETIREMENT AND HONORS As early as the summer semester of 1910, Klein applied to be released from his teaching duties. For a long time, he had felt overworked: For me, the rest of the summer vacation (aside from the parliamentary sessions, which might recommence soon) will be taken up by a series of conferences from which I cannot excuse myself. In addition, and perhaps even more importantly, I have, despite my diligent efforts, fallen behind with my scientific work in recent years, and I have left some of my work unfinished […]. Over time, this yields a situation that is unbearable and unworthy of a scholar. If I could be relieved from teaching my lecture courses in the summer semester, I have the hope that I might in the main be able to overcome this unfortunate condition, whereas otherwise it will only worsen.394

At Klein’s request, the Privatdozent Paul Koebe took over Klein’s already announced four-hour lecture course on the applications of differential and integral calculus to geometry. A seminar announced by Klein and Zermelo on the intersections of mathematics and philosophy was canceled and not replaced, because Zermelo accepted a professorship at the University of Zurich. The summer vacation was filled with conference activity: Klein traveled to Brussels, for instance, to attend the International Exposition, where he chaired the annual ICMI meeting (August 9–10) and participated in the Fourth International Congress for Higher Education (as of August 15). At this latter event, Klein and Peter Treutlein gave a presentation on the German exhibit of mathematical models. Subsequently, Klein also requested a reduced teaching load for the winter semester of 1910/11: The reason for this request is that the pedagogical work I have undertaken – as a member of the German Commission, the International Commission on Mathematical Instruction, and also as a member of Parliament – now necessarily absorbs, on account of its urgent nature, the majority of my time.395

Klein reduced the scope of his original plan, which was to teach lecture courses on the development of mathematics in the nineteenth century (4 hours per week), and instead he taught a course on the development of mathematical instruction (2 hours per week). On this latter topic, Klein had already lectured and published.396 Now, in 1910/11, he incorporated current reform discussions about all levels of education in Germany and abroad.397 The topic of Klein’s seminar during this semester aligned with that of his lecture course: “Einführung in die neuere pädagogisch-mathematische Literatur” [An Introduction to Recent Pedagogical-Mathematical Literature]. During the subsequent summer semester of 1911, Klein’s activity piled up yet again: teaching, meetings of the Göttingen Association, speeches in the Prussian 394 [UAG] Kur. 5956, fol. 156 (Klein to the Ministry of Culture, on April 4, 1910). Klein’s request was approved on April 16, 1910 (see ibid., fol. 157). 395 Ibid., fol. 160. 396 See KLEIN 1907 397 [Hecke] Klein’s unpublished lecture course (1910/11).

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Parliament, activity for the ICMI, managing the ICMI’s monograph series, the ENCYKLOPÄDIE, the Kultur der Gegenwart project, and so on. He had appointments with different colleagues almost every day: with his assistant Erich Hecke, who was editing Klein’s lectures on mathematical instruction;398 with Paul Koebe on automorphic functions; with Conrad H. Müller concerning the ENCYKLOPÄDIE; with Rudolf Schimmack to discuss didactics – to list just a few examples. Thus, on August 4, 1911, Klein informed Walther Lietzmann: “Starting on Tuesday morning, my address will be in Hahnenklee, with Dr. Claus [sic].”399 The physician’s letterhead read: “Dr. Klaus, Neurologist, Hahnenklee Sanatorium for Neurology and Internal Medicine – Patients in Need of Recuperation and Convalescence.”400 8.5.1 Recovering and Working in the Hahnenklee Sanatorium As of August 8, 1911, Felix Klein was convalescing in Bockswiese-Hahnenklee, a village in the Harz mountains located approximately sixteen kilometers from Goslar and around nine kilometers from Clausthal.

Figure 39: Bockswiese-Hahnenklee in the Harz mountains, a view of the sanatorium (a historical postcard)

398 This already repeatly quoted unpublished lecture course of the winter semester 1910/11 is kept in Erich Hecke’s estate [Hecke] at the University of Hamburg. 399 [UBG] Cod. MS. W. Lietzmann I: 118, 178. 400 See also Appendix 8.

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The village of Hahnenklee had been founded as a mining settlement in the sixteenth century; at the beginning of the twentieth century, it had just five hundred residents. After the region’s natural resources (silver, lead, etc.) had largely been extracted, tourism became its most important source of revenue; in 1882, Hahnenklee became a health resort. The Gustav Adolf Stave Church, which was consecrated there in 1908 and is the only church of this kind in Germany, has since become a landmark. Today, the facilities of the former sanatorium, which are still mostly preserved, are used as a retirement home. Klein remained there at first for one month. Letters to his wife, to his daughter Elisabeth, and to numerous colleagues document that he continued to pull the strings in several ongoing projects. His wife Anna wrote to him on September 1, 1911: “So you still receive mathematical visits and go to see your colleagues in Clausthal yourself?401 [...] When are you actually coming back?” One day later, she commented: “And [Felix] Bernstein has a job offer! I don’t know where. [...] He wants to come to Hahnenklee tomorrow, too [...]. A pity that you have to give your blessing to absolutely everything that happens in Göttingen!”402 On September 6, 1911, Klein returned to Göttingen because he believed that his presence was needed at several forthcoming conferences. On September 16th, he traveled to Milan to chair the meeting of the ICMI. Lietzmann reported that Klein managed everything as best he could, but he needed sleeping pills in order to be fresh on the next day.403 This event was followed by the annual conference of the Society of German Natural Scientists and Physicians in Karlsruhe, where the mathematicians had organized a session on non-Euclidean geometry and the theory of relativity. In addition, on September 27, 1911, there was also an afternoon-long session on automorphic functions, which Klein headed (see 8.2.1). After this, Klein attended the Annual Meeting of the German Museum of Masterpieces of Science and Technology (Deutsches Museum von Meisterwerken der Naturwissenschaft und Technik) in Munich. He had been a member of the museum’s board since December 21, 1907, when he replaced Nernst as a representative from the Göttingen Society of Sciences.404 From 1910 to 1912, he acted as the nominal chairman of this board. After some hesitation, he accepted this post because he had been assured that Oskar von Miller would perform the actual duties of the office.405 On October 28, 1911, having returned from Munich to Göttin-

401 It is conceivable that Klein helped to ensure that his son-in-law Fritz Süchting would be hired by the Mining Academy as a professor of mechanics and electrical engineering in 1912. 402 [UBG] Cod. MS. F. Klein 10: 371, 372 (Anna to Felix Klein, Sept. 1 and 2, 1911). Klein wanted to retain Bernstein for the field of actuarial mathematics at the University of Göttingen, and he managed to ensure that Bernstein’s position was improved (see Section 7.6). 403 See LIETZMANN 1960, pp. 259–60. 404 [AdW Göttingen] Scient. 255 (Deutsches Museum): fol. 26. 405 Oskar von Miller was the director of the German Museum, which had been founded in 1903. The museum’s board members Walther Dyck and Carl Linde had invited Klein to give the ceremonial address at the main meeting of the museum’s stakeholders in 1908, which he delivered before “the top authorities of the entire Empire and representatives of science, technology,  

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gen, Klein honored the meeting of the Göttingen Royal Society of Sciences (Academy) there with his presence, and in the beginning of November, he took part in the Aeronautical Research Congress that had been organized by the Göttingen Association of Applied Physics and Mathematics (see Section 8.1.3). It is thus no surprise that, at the beginning of the semester, Klein lacked the energy to teach his scheduled courses: a lecture course on differential and integral calculus II (4 hours per week), and a seminar on the history of calculus, in which his daughter Elisabeth and Iris Runge had enrolled (among other students). Klein did not make it through the first month, and he remarked on November 22, 1911: “Over the last few weeks, my personal condition has not been great; I suffer from fatigue. Perhaps I should relax for some time. The preparations for Cambridge [the Congress in 1912], however, should not be affected by this.”406 Hermann Weyl and Klein’s assistant Wilhelm Behrens took over his lecture courses, and Schimmack conducted the seminar. Klein returned to the sanatorium on November 29, 1911, where he now stayed in the “Viktoria-Haus,” a villa for which he was charged 6 Mark per day. Klein suffered from insomnia due to his many unfinished projects. His wife Anna wrote to him on December 4, 1911: “The main issue is that you can sleep again; then your powers will return.”407 Klein felt obligated to prepare the ICMI report for the Fifth International Congress of Mathematicians, which was to be held in Cambridge (UK) in August of 1912. To assist him with this, Lietzmann regularly made the walk from the train station in Goslar to Hahnenklee. Klein spent Christmas in Göttingen, and he organized, for December 27, 1911, an ICMI discussion with Lietzmann and with Carl Runge, who was tasked with presenting a report in Cambridge.408 In July, and thus shortly before the Congress, the New York-based mathematician D.E. Smith (who, in Cambridge, would be named the ICMI’s vice president) spent time in Hahnenklee to visit with Klein, as did Henri Fehr and George Greenhill; the latter traveled with a copy of Heinrich Heine’s book The Harz Journey.409 Klein’s tireless efforts were rewarded on February 2, 1912 with the Star of the Royal Order of the Crown, 2nd Class (Prussia), a silver four-pointed star.410 Not even a month later, however, he had to respond by applying for another leave of absence.411 The medical reports by Dr. Klaus in Hahnenklee contained a diagnosis of neurasthenia and intestinal neurosis, caused by permanent overstrain (see  

406 407 408 409 410 411

and industry” (see KLEIN 1908b). On June 12, 1909, Klein had agreed to serve as chairman of the board; on January 10, 1912, he was named one of its permanent members. ([UBG] Cod. MS. F. Klein 7D; and 114: no. 43) [UBG] MS. W. Lietzmann I: 137/2 (Klein to Lietzmann, November 22, 1911). [UBG] Cod. MS. F. Klein 10: 378 (Anna Klein to her husband, December 4, 1911). On the details of this report, see HENTSCHEL/TOBIES 2003, pp. 48–51. [UBG] MS. W. Lietzmann I, 144, 260; and LIETZMANN 1925, p. 260. Klein had already been awarded this order of the crown without the star; see Section 7.6. [UAG] Univ.-Kuratorium, Personalakte F. Klein, 4Vb, no. 216 (a letter from Klein to the Kurator dated March 10, 1912).

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Appendix 9). Klein himself spoke of excessive overwork.412 This had been a recurring problem throughout his professional life. His need to convalesce had been caused by his outsized ambition, his penchant for perfectionism, his sense of being responsible for everything, combined with his inability to turn down an honorary office. As before, Klein’s solution to managing his health problems was to cut back on his teaching. On August 5, 1912 he wrote to the university’s Kurator about his ongoing but still incomplete projects: “I mention that here because my nervous illness was certainly also caused by the fact that I saw too many unfinished tasks ahead of me. Their completion has still not been reached, but it is getting closer.” In the same letter, Klein indicated that he would yet again request a leave of absence for the upcoming semester, but that he would then have to decide whether he should take early retirement, a possibility that he had already discussed with Hilbert and Landau. When Klein was unable to realize his “entirely individual” teaching program in the winter semester of 1912/13, the faculty did not organize a substitute because the absence of Klein’s courses would not create “a systematic gap in our instruction.”413 Thus, when Klein received the news from London that the Royal Society had awarded him the Copley Medal (“on the ground of his researches in mathematics”), he considered this honor “a symbolic conclusion to my past activity.”414 At the same time, in a letter dated December 31, 1912, he applied “for permanent release from the obligation of teaching and for a replacement full professor of mathematics to be hired at the university here.”415 He hoped that the position would be filled by April 1, 1913. Klein wanted to reserve the right to teach courses again in the future, and he also wanted to maintain his administrative oversight over the model collection and the reading room (together with Landau, the co-director). These requests were approved, whereby Klein was able to keep his full-time assistant (the assistant was contractually associated with these facilities; see Section 7.1). Not until May 2, 1922 – and at Klein’s own request – did the Prussian Ministry of Culture relieve him of “the duties of managing the mathematical reading room and the collection of mathematical instruments and models.”416

412 [StB Berlin] Sammlung Darmstaedter H 1872 (9), Klein to Ludwig Elster, Ministry of Culture, June 2, 1912. 413 [UAG] Kur. 5956, fol. 182 (H.Th. Simon to the Kurator, November 12, 1912). 414 Ibid., fols. 184–85, 189. The medal, which was awarded by the Royal Society on November 30, 1912, was received on Klein’s behalf by a member of the German embassy. 415 Ibid., fols. 188–188v 416 Ibid., fol. 197.

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8.5.2 Max Liebermann’s Portrait of Felix Klein On August 19, 1912, Klein celebrated the fortieth anniversary of his professorial career. In early 1912, Walther Dyck sent out a letter with seventy-one signatories from Germany and abroad in order to invite donations for Klein’s portrait to be painted. Even Gösta Mittag-Leffler and Henri Poincaré417 were among those who signed the letter. By July 7, 1912, 331 of Klein’s friends, colleagues, and students from all over the world donated a total of 7,060.34 Mark. The painter Max Liebermann, who was one of the most significant representatives of German Impressionism, typically charged 20,000 Mark for a portrait. He waived his normal fee, however, and was persuaded to paint Klein for about 6,000 Mark.418 “You are truly a wonder-worker for managing that with Liebermann,” wrote Klein to Hilbert, and he mentioned that the painter intended to come to Hahnenklee on August 10, 1912.419 Liebermann, who was two years older than Klein, painted the portrait in the late summer of 1912 (see Fig. 40). To his friend Alfred Lichtwark, Liebermann also related how Klein’s problems with sleep and time had prevented him from devoting any attention to belletristic literature: I painted a portrait of Privy Councilor Klein, the great mathematician, a few months ago in Hahnenklee. When he complained of sleeplessness, I suggested that he should read a book, and he responded that, for the last forty years, he has not held in his hand a single belletristic book. And his wife is the granddaughter of Hegel! And this man is the King of Göttingen; he appoints professors and rules not only over the University of Göttingen but over all Prussian universities. He was Althoff’s favorite and the cock of the roost in all the Wilhelm Academies! Tempora mutantur!420

The portrait was presented to Klein in a celebration in Göttingen on May 25, 1913. On June 1, 1913, Walther Dyck sent to everyone involved around the world a reproduction of the painting, a list of all the donors who had made it possible, a copy of Eduard Riecke’s speech on the occasion of the presentation of the portrait, and a copy of Klein’s response to the gift (see Appendix 10). From a letter by Anna Klein to Walther Dyck, we know that not everyone was equally pleased by the reproduction of the painting. In this letter, she also provided a rather impressive depiction of her husband: Göttingen – January 10, 1913 Dear Privy Councilor! I was very sorry to learn from your letter to my husband that you are dissatisfied with Liebermann’s portrait. I would have liked to have seen the reproduction, because I believe that this is to be blamed for your low opinion of the painting. It will probably be a while before the prints are mailed off, and so I would like to tell you today that I (and my children too) am thoroughly satisfied with the portrait of my husband. However, the image does indeed fail

417 418 419 420

Poincaré died on July 17, 1912. [UBG] Cod. MS. Hilbert 86: 11, 12, 13. Quoted from FREI 1985, p. 139 (Klein to Hilbert, July 25, 1912). Quoted from PFLUGMACHER 2001, pp. 393–94 (Liebermann to Lichtwark, November 24, 1912). Tempora mutantur = the times change.

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8 The Fruits of Klein’s Efforts, 1895–1913 to recreate the bright expression that my husband can have in stimulating conversations and when he’s in a festive mood. Perhaps an artist who had known and observed my husband for a long time would have been able to capture him more intimately and inwardly. But there is no such artist, and I have to say that Liebermann, who did not know my husband at all and was only able to see him in his reduced state at the sanatorium, captured him amazingly well and characteristically reproduced his image. I have now looked at the painting often and at length, and it has become ever dearer and more familiar to me, even though I am quite happy that, in reality, my husband does not always look so imposing and vigorous. It is a ruler in his realm who is seated there, and one is left with the impression that he is pleased to have left behind a life full of work, full of energetic will and consequential activity. Moreover, the painting does not depict him as though he’s striking a grand pose; rather, its composition is extraordinarily simple, gray on gray, without any color effect. And yet, upon longer inspection, the image almost physically leaves the frame and is strikingly true to life, especially in the posture of his head, the position of his hands, and the depiction of his gaze. In short, I am very satisfied with it, and I’m sorry that you, who put so much effort into the matter, should now be so unhappy with it. It is quite possible, however, that the reproduction has distorted the painting’s details. The image is painted in the modern style, with thick brushstrokes, and it is not meant for up-close viewing. It is also possible that the reproduction has made it look entirely different. Of course, it would be a pity if all the donors were now to receive a false and unfavorable impression of the picture, and this would also be highly regrettable for Liebermann’s sake. To my great pleasure, I can inform you that things are going steadily better for my husband; especially recently, having luckily overcome a bad bout of flu, he is vigorous and merry. He looks very good, has become stronger, and he keeps himself as fit as he was during his best years. Unfortunately, I have less good news to report about his productivity. He still tires quickly and does not take it well when something unforeseen has to be dealt with or when multiple things require his attention at the same time. Even if one could hope that his energy will increasingly be revived, it should not be assumed that my husband will be able to lecture in the summer. And because he does not want to take another leave of absence, there is no other choice than to obtain a permanent release from his teaching duties. My husband came to terms with this idea already at the beginning of his illness. Nevertheless, it was a difficult decision, and I have only slowly grown accustomed to the idea that his position will have to be filled by someone else. Of course, I have always anxiously wondered what would happen if his life were to carry on at the same swift pace with which it began. And thus one must ultimately regard it as a merciful solution to this question if it is only his professional life that has to come a premature end, and not his life itself. That said, there will be no lack of work and interesting activity for my husband as soon as he has recovered enough to travel and participate in conferences. These days, nothing happens without conferences, and he is still unable to attend them. He already did far too much of this in the past. Now, however, I finally have to bring this letter to a close, dear Professor. I hope that you, your wife, and your daughter, whom I have unfortunately yet to meet, are happy and well. Our youngest is now about to take her final university examination; may it go well! For me, the main value of her studies is that she has been able to help her dad in many ways and that she shares his interests as much as I always wished and thought that she would. With warm greetings from me and my husband to you and your wife, I remain yours truly, Anna Klein421

421 [Hillebrand] Anna Klein to Walther Dyck.

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Figure 40: Max Liebermann’s portrait of Felix Klein (1912), oil on canvas (112.5 x 90 cm), Mathematical Institute, Georg August University of Göttingen.

The painting was not really just “gray on gray” (grau in grau), as Anna Klein wrote. Leaders of the Berlin Secession, an impressionist art movement, asked to present the portrait in their summer exhibit, noting: “Professor Liebermann places great value on being represented by precisely this painting, which he considers one of his best works.”422

422 [BStBibl] Dyckiana (managers of the Berlin Secession to Dyck, April 11, 1913).

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William Henry Young, the husband of Klein’s former doctoral student Grace Chisholm Young, rounded out the depiction of Klein’s physical appearance by describing him as follows in the Proceedings of the Royal Society of London, a society to which Klein had belonged as a foreign member since 1885: “Klein was very tall, erect and slim, with rich brown wavy hair and characteristically sparkling light blue eyes, with a genial glance which has not been completely caught in the photograph here produced.”423 8.5.3 The Successors to Klein’s Professorship In a letter to Klein dated January 2, 1913, Edmund Landau explained that he found it appropriate and gratifying “that you will continue to direct the model collection and the reading room.” He wished him a full recovery, “so that then the five of us – you, Hilbert, Runge, X, and myself can work together in full force to maintain Göttingen’s status in mathematics.”424 Landau informed Klein that everyone was in agreement about the top candidate X to be appointed: the Greek mathematician Constantin Carathéodory, whom Klein had already described as a “talented geometrician” when evaluating his Habilitation thesis.425 There would only be differences of opinion about the other possible candidates. The administrative records reveal that Klein himself was a member of the hiring committee (along with Hilbert, Landau, Mügge, Prandtl, C. Runge, W. Voigt, Wallach, a representative from the other faculty division, and the current dean). At the committee meeting on February 28, 1913, the members unanimously ranked Weyl and Brouwer as equals in second place on the hiring list. The records also contain an explanation for why Paul Koebe was not suggested as a possible candidate.426 This decision was based primarily on Koebe’s character. Eduard Riecke reported Klein’s opinion that Paul Koebe had a high standing with respect to his mathematical work, but that he was otherwise a “blatant egoist who always wants to have something better and more special than what others have received.”427 Years later, Klein described Koebe as “somewhat difficult from a personal point of view, because he is not always tactful and scientifically agnostic.”428 Klein also may have been struck by Koebe’s insufficient knowledge of scholarly literature

423 W.H. YOUNG 1928, p. xix. 424 [UBG] Cod. MS. F. Klein 10: 610 (Landau to Klein, January 2, 1913). Shortly after writing this letter, Landau turned down an offer from the University of Heidelberg ([UBG] Math. Arch. 80 Nachlass Margarethe Goeb: 11e). 425 [UAG] Phil. Fak. 190a, V, 35 (Klein’s evaluation, February 20, 1905). 426 Ibid., III, vol. 5, fol. 126. 427 [StB Berlin] Sammlung Darmstaedter (Riecke to an unknown colleague, June 1910). 428 [UBG] Cod. MS. F. Klein 2G, fol. 13 (a letter from Klein to Nernst dated February 21, 1917). This letter concerned the replacement of H.A. Schwarz in Berlin. By saying that Koebe was not “agnostic,” Klein meant, in particular, that Koebe was often unwilling to acknowledge other people’s achievements.

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when reading a draft of Ludwig Bieberbach’s dissertation, which Koebe had supervised (see Appendix 7). Carathéodory, who received Klein’s professorship on April 1, 1913, was not, however, as devoted to the University of Göttingen as were Klein, Hilbert, Runge, and Landau. Only a few years later, Carathéodory accepted an offer to replace Georg Frobenius at the University of Berlin. Thus Klein experienced two further successors, and in both cases he served on the hiring committees as a professor emeritus. On December 13, 1917, the new list of candidates read: 1) Erich Hecke (Basel), 2) Brouwer (Amsterdam) and Weyl (Zurich) ex aequo, and 3) Wilhelm Blaschke (Königsberg).429 Hecke, the first candidate on the list, accepted the position in Göttingen. He had earned his doctoral degree under Hilbert (1910), worked as Klein’s assistant (1910/11), then as Hilbert’s, and had completed his Habilitation in 1912, also in Göttingen. During the war, Hecke worked as a professor in Basel. He too, however, did not remain long in Klein’s professorship; on October 1, 1919, he moved to the newly established University of Hamburg. Now, on October 30, 1919, there was a new list of recommended candidates: 1) Brouwer (Amsterdam), 2) Herglotz (Leipzig), and 3) Weyl (Zurich).430 The Ministry asked only Brouwer and Weyl whether they might be interested in the job; both declined (Weyl after six months time to think it over). The dean of the Philosophical Faculty at the University of Göttingen at the time informed the Ministry about this on July 14, 1920; he wrote that Herglotz had yet to be asked and that no consensus could be reached in the Faculty about his candidacy. He enclosed two separate votes. In one, Edmund Landau argued vehemently for his old friend Issai Schur and offered an extensive positive assessment of his mathematical achievements. In the other, David Hilbert and Felix Klein made the case for Richard Courant.431 The ministry followed Klein and Hilbert’s suggestion and came to an agreement with Courant as early as August 10, 1920.432 Courant had begun his studies in Göttingen in 1908, and he had also attended two of Klein’s lecture courses. For some time, he worked as Hilbert’s private assistant, and it was under Hilbert that he earned his doctorate in 1910 with a dissertation titled “Über die Anwendung des Dirichlet’schen Prinzipes auf die Probleme der konformen Abbildung” [On the Application of Dirichlet’s Principle to the Problems of Conformal Mapping]. He completed his Habilitation in 1912. As a Privatdozent in Göttingen, Courant involved himself in the coordinated curriculum, but he eventually had to perform military service. Like many intellectuals at the time, Courant was politically awakened by the First World War. In 1918, he joined a workers’ and soldiers’ council, and during the 1919 election he and his sister-in-law Iris Runge were active on behalf of the

429 430 431 432

[UAG] Phil. Fak. III, vol. 5, fol. 196. Ibid., fol. 220. [StA Berlin] Rep. 76 Va Sekt. 6, Tit. 4, No. 1, vol. xxvi, fols. 425–34. Ibid., fol. 436.

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Social Democrats.433 When Courant and other Göttingen mathematicians ran for office and were elected to the Göttingen City Parliament, they surely did so on the basis of their own political engagement and convictions. This has been called into question; according to oral history, “Courant was enjoined by Klein to run for office as a member of the Social Democratic Party” for pragmatic reason.434 Records from the dean’s office of Philosophical Faculty in Göttingen document that Courant was offered a full professorship in Münster on February 20, 1920 and that, after just one semester in Münster, he became the successor of Klein, Carathéodory, and Hecke in Göttingen on October 1, 1920.435 Courant reported that Klein was largely to thank for this arrangement.436 Courant embarked upon an intensive and successful period of activity. Springer’s “yellow book series,” which Courant founded, would include a number of Klein’s previously unedited lecture courses (see Section 9.2.3). When Courant’s father-in-law Carl Runge died in 1927, however, the domain of applied mathematics with numerical, graphical, and instrumental methods, which Runge (and Klein) had introduced to the university, was abandoned – a development for which Courant was largely responsible. Richard von Mises, who held a personal full professorship for applied mathematics at the University of Berlin from 1920 to 1933, and whom Alexander Ostrowski regarded as the founder of “the first mathematically serious German school of applied mathematics,”437 criticized the decision that had been made about Runge’s successor in Göttingen.438 Carl Runge’s students were able to develop his direction of applied mathematics in industrial research and at Technische Hochschulen.439 Both Richard von Mises and Richard Courant were forced to leave Germany by the Nazi regime and ultimately had successful careers in the United States. Courant took a position in the Department of Mathematics at New York University, where the Courant Institute of Mathematical Sciences is named after him to this day.

433 434 435 436 437

On this political engagement, see TOBIES 2012, pp. 105–21. ROWE 2018a, p. 377. [UAG] Phil. Fak. III, vol. 5, fols. 232, 255–56, 262. See REID 1976, p. 83; and REID 1979, p. 98. Alexander Ostrowski, “Zur Entwicklung der numerischen Analysis,” Jahresbericht der DMV 68 (1966), pp. 97–111, at p. 106. 438 See the debate between von Mises and Courant in the journal Die Naturwissenschaften (1927). In a letter to Ludwig Prandtl dated March 24, 1930, von Mises also stated in clearly exaggerated terms that “applied mathematics, under the leadership of Göttingen, will once again be driven out of German universities” ([MPI Archiv] No. 1082). For further discussion of these developments, see SIEGMUND-SCHULTZE 2018, pp. 502–03; and SIEGMUNDSCHULTZE 2021. 439 See in detail TOBIES 2012.

9 THE FIRST WORLD WAR AND THE POSTWAR PERIOD During the First World War, the November Revolution of 1918, and the period of inflation in the Weimar Republic, Klein remained academically active as an emeritus professor. After his period of recovery in the sanatorium (see Section 8.5.1), he resumed control over his ongoing affairs: book projects (the ENCYKLOPÄDIE, the edition of Gauß’s collected works, the ICMI monograph series, the edition of his own collected papers and lecture courses); teaching; and his work on university committees, for the Göttingen Association, in the Upper House of the Prussian Parliament (until 1918), and on various education committees. He remained the preeminent expert in matters related to hiring decisions and research funding. His broad vision and his diplomatic activity in the interest of mathematics and its neighboring disciplines had earned him the unofficial title “the foreign minister of German mathematics.”1 In the Göttingen Society (Academy) of Sciences and in the German Mathematical Society, he worked to ensure, in 1917, that Gaston Darboux was honored with an obituary,2 that Georg Cantor (previously a corresponding member) was made an external (auswärtiges) member, that Paul Koebe was made a corresponding member in the seat vacated by Cantor, and that Albert Einstein would be nominated as a corresponding member.3 Klein’s much-discussed signature on the “Manifesto of the Ninety-Three” and his engagement during the war to promote studying abroad in an effort to avoid future conflicts will be discussed in Section 9.1. Section 9.2 provides an overview of his academic activity, which primarily concerned the history of mathematics, substantial contributions to the theory of relativity, and the edition of his own works on mathematics. As a member of multiple education committees and as a recognized authority, Klein successfully influenced new developments in educational policy (9.3). In 1920, Klein agreed to serve as the first chairman of the committee on mathematics, astronomy, and geodesy within the framework of the Emergency Association of German Science (Notgemeinschaft der deutschen Wissenschaft), the future German Research Foundation. The Göttingen Association for the Promotion of Applied Physics and Mathematics (see 8.1.1), was integrated into the broader Helmholtz Society in 1920, and the latter no longer corresponded to Klein’s universal line of thinking. Together with Courant, however, Klein was able to secure some funding for mathematical projects in Göttingen (Section 9.4).

1 2 3

See Abraham FRAENKEL 2016 [1967], p. 138. In 1919, Klein had recruited Fraenkel to collaborate on the edition of Gauss’s works (see Section 8.3.1). See Section 9.1.1. [UBG] Cod. MS. F. Klein 3F, fols. 6, 9–11 (a text in Klein’s handwriting, July 1917).

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Klein received numerous honors on the occasion of his seventy-fifth birthday (April 25, 1924). Sitting at his desk, he remained devoted to his work ethic until his death on June 22, 1925 (Section 9.5). The number theorist Edmund Landau, who had been a professor in Göttingen since 1909, acknowledged Klein’s selfless activity in a special way when he wrote the following, on April 16, 1920, to the Prussian Ministry of Culture: […] Allow me to raise another issue with you today, though I would like to state explicitly that I am acting on my own initiative and that my colleague Mr. Klein may very well disapprove of my gesture (which I am making without his knowledge). Today I learned by chance, in a letter from Mr. Klein about acquisitions for the reading room, that as an emeritus professor he has not received any allowance for the increased cost of living to date (previously, he had purchased certain scientific journals for the reading room at his own expense and now he no longer sees himself in a position to do so). I visited Privy Councilor [Erich] Wende and expressed my opinion to him: “If an exception to the rule seems to be justified in any case, this is the case of Klein. For, although retired, he continues to give lectures (partly at home; they are courses on the history of mathematics and the like, which are of the greatest benefit to our senior students), and he participates intensively in all the senate and faculty affairs, as well as in the affairs of the Göttingen Association, which benefit our university. Moreover, it is to Klein’s initiative that we owe the flourishing of pure and applied mathematics and physics in Göttingen, not to mention the creation of several institutes.” Privy Councilor Wende said that you alone could approve my suggestion and that, without further ado, you could grant Klein cost-of-living allowances retroactively and in the future. […] Given your great interest in our discipline and my certainty that you appreciate the outstanding importance of Klein to science and to the University of Göttingen, I hope that you will intervene to make a permissible exception to the rule in this regard. Respectfully yours, Landau4

9.1 POLITICAL ACTIVITY DURING THE FIRST WORLD WAR In 1908, Felix Klein had spoken out firmly against national chauvinism (see Section 8.4). As a young man, and as the son of an official in a Prussian district government (Rhineland Province), Klein had fulfilled his military duties during the Franco-Prussian War, but he only served as a paramedic. This war experience had done nothing to alter his relationship with French mathematicians. Rather, he and Clebsch alike had been dismayed by the excessive nationalistic rhetoric of Camille Jordan and others (see Section 2.7.1). By the time the First World War broke out, Klein had already developed an extensive international network. Like many other senior academics, however, he

4

[UAG] Kur. 10750, vol. 1, fols. 59–60v. This request was granted within four days. – Wende had been an official at the Prussian Ministry of Culture since 1917; in 1920 he was made an “expert councilor” there (Vortragender Rat). The recipient of this letter was Otto Naumann, who had joined the Ministry of Culture as a government assistant in 1884 and, by 1920, was a department director serving under the (Social Democratic) Minister Konrad Haenisch.

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could not abstain from the campaign to mobilize morale on the home front (Section 9.1.1).5 Yet during the war, Klein was actively engaged in the Prussian Parliament to promote studying abroad and learning foreign languages in an effort to increase intercultural understanding and possibly avoid future conflicts (9.1.2). 9.1.1 The Vows of Allegiance of German Professors to Militarism Just when we were able to believe that our plans were all but certain to be executed, the terrible war in which Germany is now embroiled threw everything into question. Yet precisely because the future is as unclear as ever, I do not want to give up on the basic idea. […] Of course, the precondition for all the hopes that we hold dear is that, culturally, we are not entirely set back by the events to come.6

Klein wrote these words in August of 1914, shortly after the outbreak of the First World War, in an article titled “Bericht über den heutigen Zustand des mathematischen Unterrichts an der Universität Göttingen” [A Report on the Current State of Mathematical Instruction at the University of Göttingen], which was published in the Jahresbericht der DMV. His statement about plans that were close to being executed was a reference to the construction of “his” new Mathematical Institute, for which funding had already been secured (it would not be completed until 1929). Here, at the same time, Klein also expressed his concerns about the precarious state of international relations. He referred to the efforts of the ICMI, to Emanuel Beke’s 1914 report at the ICMI meeting over Easter in Paris, and to the journal L’Enseignement mathématique. Klein mentioned the great number of foreign students in Göttingen, of which he was always especially proud, and he stressed: “The ideal was to contribute in a recognizable way to the advancement of all-encompassing science, whose high arc vaults evenly over the differences between nations.” He expressed his hope that mathematics might be able to maintain a central position despite the pressing “social, economic, and political issues” of the time.7 In August of 1914, the nationwide frenzy for war had compelled many young Germans to enlist voluntarily in the military. On August 3, 1914, Klein reported to Wilhelm Lorey, with whom he worked to complete the German ICMI monograph series: “Now the first wave of commotion is behind me, because my son came yesterday to say farewell and, at the same time, my daughter got married in a great rush.”8 Klein’s son survived the war; his son-in-law Robert Staiger fell in battle after a few weeks (see Section 3.6.3). On October 14, 1914, more than three thousand German professors deemed it necessary to demonstrate their loyalty to the state by signing a “Declaration of the

5 6 7 8

For a detailed discussion of this topic, see BROCKE 1985. KLEIN 1914b, p. 427. Ibid., p. 428. [UB Frankfurt] B.I.1 (Klein to Lorey, August 3, 1914). – See KLEIN ed. 1909–16.

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University Professors of the German Empire” (Erklärung der Hochschullehrer des Deutschen Reiches). The philologist Ulrich von Wilamowitz-Moellendorff, with whom Klein had cooperated during his time in Göttingen (see Section 6.4.3) and who had been a professor at the University of Berlin since 1897, formulated the declaration, which began as follows: We teachers at German universities serve science and perform a work of peace. But it fills us with indignation that the enemies of Germany, England at the forefront, allege, ostensibly in our favor, that there is a contradiction between the spirit of German science and what they call Prussian militarism. In the German army there is no other spirit than the one in the German people, for both are one, and we also belong to it.

The declaration ends with this sentence: Our belief is that salvation for the culture of Europe as a whole depends on the victory that German ‘militarism’ will gain: manly discipline, loyalty, the courage to sacrifice found in a peacefully harmonious, free German people.9

Although they had cultivated a wide range of international relations, a number of even liberally minded professors – Klein, Hilbert, and Carl Runge among them – pledged to offer their services to the war effort. Hilbert and Carathéodory wrote a memorandum about the possible military uses of mathematics students: calculating ballistic tables, photogrammetry, teaching methods for determining the location of airplanes, strength problems, etc.10 Klein mentioned the wartime activity of the mathematical institute in a speech to the members of the Göttingen Association, but he warned: “The danger of isolating theoretical speculation must be avoided, as must sinking to a low level with respect to the use of applications.”11 He himself had turned his attention to the theory of relativity (see Section 9.2.2), and he continued to keep his eye on the big picture. There were other vows of allegiance to the war by public servants, and not only in Germany.12 Published on October 4, 1914 in German newspapers, the proclamation “To the Civilized World” created the biggest splash because it bore the signatures of ninety-three German scientists, artists, and Nobel Prize winners, including Wilhelm Förster, Ernst Haeckel, Max Liebermann, Walther Nernst, Wilhelm Ostwald, Max Planck, Ulrich von Wilamowitz-Moellendorff, and Wilhelm Wundt (this text has come to be known as the “Manifesto of the NinetyThree”). That Felix Klein appeared on this list, as the only Göttingen professor and as the only mathematician, was based on his fame at the time and on his frequent presence in Berlin as a member of the Prussian parliament.13 As Klein later 9 10 11 12 13

The translation is based on Robert E. Norton, “Wilamowitz at War,” International Journal of the Classical Tradition 15 (2008), pp. 74–97, at p. 97 (German in TOBIES 2019b, p. 451). [UBG] Cod. MS. F. Klein 7H (War Assistance [Kriegshilfsdienst]; a memorandum sent to Otto Naumann, December 8, 1916). [UBG] Cod. MS. F. Klein 4G, fols. 79–81, at fol. 80. See AUBIN/GOLDSTEIN 2014. [UB Frankfurt] Klein to Lorey, Oct. 30, 1914: “I always have to rest a lot in between and have to live with extreme caution, but I have survived two trips to Berlin under these rules.”

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explained to his former English doctoral student Grace Chisholm Young, however, he (like Max Planck and others) had been asked via a telegram to sign the proclamation. Cordula TOLLMIEN (1993) has shown that Klein had been unaware of what the text contained until it was published. He had (naively) assumed that its intention would be to calm the storm brewing abroad. The opposite was true, for the proclamation, which was mainly addressed to countries that were still neutral, contained statements such as the following: It is not true that Germany is guilty of starting this war. […] It is not true that we trespassed in neutral Belgium. […] It is not true that the combat against our so-called militarism is not a combat against our civil nation, as our enemies hypocritically pretend it is. Were it not for German militarism, German civilization would long since have been extirpated. For its protection, militarism arose in a land which for centuries had been plagued by bands of robbers as no other land had been. The German Army and the German people are one, and today this consciousness fraternized 70,000,000 of Germans, all ranks, positions, and parties being one.14

The proclamation “To the Civilized World” therefore denied the atrocities committed by Germans.15 The Académie des Sciences in Paris annulled the membership of the document’s signatories.16 In Great Britain, however, German scientists were not removed from academic bodies. Felix Klein and Max Planck made efforts in Göttingen and Berlin to ensure that the membership of French scholars to the academies in these cities would not be revoked.17 Émile Picard, a corresponding member of the Royal Göttingen Society of Sciences since 1884, withdrew from this Göttingen academy on his own accord in 1916/17.18 When Gaston Darboux (a corresponding member in Göttingen since 1883 and a foreign member since 1901; in the Bavarian academy in Munich as of 1899) died on February 23, 1917, he was honored with obituaries, in Göttingen, in Munich, and by the German Mathematical Society – where Klein served as honorary chairman in 1918/19.19 After the war, and especially in France, the proclamation “To the Civilized World” served as a main argument for excluding German scientists from international committees.20 It may seem contradictory, but – politically – Klein remained loyal to the state while – scientifically – he remained internationally oriented. In an obituary, Grace

14 Professors of Germany, “To the Civilized World,” The North American Review 210 (1919), pp. 284–87. For further discussion, see UNGERN-STERNBERG/UNGERN-STERNBERG 1996. Regarding Göttingen in particular, see TOLLMIEN 1993, esp. pp. 172–77. 15 See, for example, THE MARTYRDOM OF BELGIUM 1915. 16 See GRAU 1993, p. 286. This announcement was made on March 15, 1915. 17 See TOLLMIEN 1993, p. 195. 18 See Göttinger Nachrichten: Geschäftliche Mitteilungen (1917), p. 2. 19 See David Hilbert, “Gaston Darboux,” Göttinger Nachrichten: Geschäftliche Mitteilungen 1917, pp. 71–75; and Aurel Voß, “Gaston Darboux,” Jahresbericht der DMV 27 (1918), pp. 196–217 (first published in Jahrbuch der Kgl. Bayerischen Akademie der Wissenschaften 1917). [UBG] Cod. MS. F. Klein 12: 186E, fol. 220 (Aurel Voss to Klein, Dec. 25, 1917). 20 Regarding mathematicians, see especially SIEGMUND-SCHULTZE 2011.

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Chisholm Young wrote the following words about her doctoral supervisor: “The aim of his life was to knit together in unity of object and effort the world of science, without distinction of nationality.”21 On December 7, 1918, Klein had attempted to explain his attitude to her: “Everyone, in bright or dark days, will remain loyal to his country, but we have to free ourselves from extreme passions if, for everyone’s benefit, the international collaboration that we desire so dearly is to regain its validity.”22 To understand Klein’s attitude, we have to keep in mind that he held a seat in the Upper House of the Prussian Parliament until the demise of the German Empire in November of 1918 and that he also directed the Göttingen Association for the Promotion of Applied Physics and Mathematics. In order to bolster the position of mathematics and the natural sciences, Klein sought to build good relationships both with government authorities (regardless of their political affiliations) and with the industrialist members of the Göttingen Association (most of whom backed the war). On June 22, 1918, when the Göttingen Association met to celebrate its twentieth anniversary, Klein expressed his thanks to the industrialists and to the government officials for supporting research and education. The longtime officials in the Prussian Ministry of Culture, Otto Naumann and Friedrich Schmidt-Ott – the latter served as the Minister of Culture from August 6, 1917 to November of 1918 – were named honorary members of the association.23 Even after 1918, both served as contacts for Klein in the scientific administration. When the boycott against German scientists had begun to have concrete effects, Klein formulated the motto “keep quiet and work” in a letter to Grace Chisholm Young (July 15, 1919): In general, my sense of objectivity prevents me from expressing myself about things that I only know, if at all, from the subjective and contradictory reports by one newspaper or another. Thus my approach throughout this entire war has remained: keep quiet and work. For the few years that I have left, this will have to get me through. The world, however, will take its course, and one day the nations will find themselves together again. For the time being, as during the construction of the Tower of Babel, they no longer understand one another.24

Klein kept on working, but he hardly kept quiet. In order to smooth over the international discussions about the proclamation “To the Civilized World,” he and Max Planck25 in Berlin hoped to initiate a declaration together: Like others, I found it excellent, on your Academy’s Leibniz Day, how you [Sie] thought about international scientific relations and our duty to continue our scientific work, and I believe that you [Sie] could succeed, to the extent that this is at all possible, in steering this

21 Grace Chisholm Young, “Obituary: Professor Klein,” The Times (July 9, 1925). 22 [UBG] Cod. MS. F. Klein 3A, fol. 14 (a letter from Klein to G. Chisholm Young dated December 7, 1918). For further analysis, see also GRATTAN-GUINNESS 1972, esp. pp. 159–61. 23 [UBG] Math. Arch. 5038, fol. 4. 24 Quoted from TOLLMIEN 1993, p. 185 (Klein to Grace Chisholm Young, July 15, 1919). 25 Max Planck (Nobel Prize in Physics in 1918) had studied with Klein in 1877 (see 4.1.2).

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difficult matter down a better path. Let me indicate that I myself am prepared to collaborate on writing the text of the declaration that should be offered.26

An explanation of this sort, however, never materialized.27 Klein had to accept being excluded from things that he himself had once put in motion: mathematics conferences and his associations with international academies. As the president of the Conseil International de Recherches (founded in 1919), Émile Picard, who in 1912 had numbered among the donors who funded Max Liebermann’s portrait of Klein (see Appendix 10), was among the driving forces behind banning German scientist from participating in this organization.28 9.1.2 A Plea for Studying Abroad Felix Klein repeatedly used the word “universal” (allseitig). We can trace this back to his interest in the work of pedagogues such as Pestalozzi, Fröbel, Diesterweg, and others, whose motto was: “Teach everything to everyone!” In this respect, Klein made continued efforts not only to promote mathematical and scientific education but also to advocate foreign-language learning and studying abroad. The fact that he took on a special initiative to do this during the war is worthy of further explanation. Klein acted in coordination with Carl Heinrich Becker, who had been appointed in 1916 as a professor of oriental philology at the University of Berlin and who, in the same year, had become a (politically independent) advisor in the Prussian Ministry of Culture. Even after the November Revolution, Becker remained at this Ministry, and he also served as the Minister of Culture for a time. In 1916, he wrote a “Memorandum on the Future Expansion of Foreign Studies at Prussian Universities” (Denkschrift über den künftigen Ausbau der Auslandsstudien an den preußischen Universitäten). Becker was fully in favor of promoting a better understanding of other cultures, with the justification that such knowledge might make future conflicts avoidable. His memorandum was discussed in both chambers of the Prussian Parliament. Induced by the philologist Alfred Hillebrandt (see Table 9), Klein devoted himself extensively to this issue. Klein composed a twelve-page text titled “Bemerkungen über die Aufgaben der Universitäten nach dem Kriege, insbesondere in der Richtung auf ein verbessertes Studium der Verhältnisse des Auslandes” [Remarks on the Objectives of Universities After the War, Particularly Toward an Improved Study of the Situation in Foreign Countries], which he submitted to the Parliament’s academic committee. Klein had discussed this text with some col26 [UBG] Cod. MS. F. Klein 3A, fols. 2–4 (a letter from Klein to Planck dated September 8, 1919). “Leibniz Day” was a regularly held celebration on the occasion of the birthday of G.W. Leibniz, who had initiated the founding of the Berlin Academy of Sciences in 1700. 27 See TOLLMIEN 1993, pp. 186–96. 28 See SIEGMUND-SCHULTZE 2011.

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leagues (Constantin Carathéodory, August Gutzmer, Edmund Landau, Carl Runge). It also contains nationalistic undertones: In the great crucible [of war], the ethos and also the abilities of those trained at universities stood the test with flying colors. Wherever there are shortcomings, however, it remains necessary to repair them and, in the heightened competition among nations to be expected after the peace agreement, to preserve the superiority of German education and thus that of the German people.

This nationalistic rhetoric sprang apparently from the fear that the Ministry of Culture might reduce the available funding for science and education. Klein based his argument on historical precedent as well: Given that Prussia, in the midst of its deepest humiliation in 1809/10, mustered the courage and means to create the University of Berlin, it will possess enough insight and strength, after the hopefully victorious end to the terrible struggles for the status of Germany as a great power, to further honorably preserve the heritage of the past.29

Klein formulated concrete suggestions for modern language studies, which he enumerated in a parliamentary speech on July 8, 1916: Besides French and English, additional foreign languages should be taught, especially Russian, Polish, and Italian; but Hungarian, Scandinavian, Dutch, and oriental philology should also be established at universities. Education in modern languages should not be elective but rather obligatory. Travel stipends should be made available for language studies. The universities should be expanded to accommodate such studies.30

In this context, however, Klein also did not fail to mention his particular fields of interest: “the institutes for applied mathematics and physics in Göttingen and Jena, which were built by the governments with support from private donors,” and he stressed: “This course of development should resolutely be pushed ahead.”31 Overall, he argued that the objective of the university was to provide universal education, and he explained that new institutes, new professorships, and funding for scientific staff, libraries, and travel were indispensable. Like Becker, Klein and Hillebrandt believed that officials might be more receptive to their ideas during the war, and thus that they could implement certain initiatives that they had already wanted to accomplish for years. On April 12, 1913, for instance, the Slavicist Erich Berneker had informed the Indo-Europeanist Herman Lommel that, as early as 1901, Felix Klein had been the driving force behind the creation of an associate professorship for Slavic philology in Göttingen,32 and that Klein could certainly be recruited to join a “German Society for the Study of Russia” (Deutsche Gesellschaft zum Studium Rußlands). This 29 [UBG] Cod. MS. F. Klein 1A, fols. 1–58, at fol. 2. This text is initialed by Carathéodory and Landau, and also archived in [StB Berlin] Nachlass Runge – Du Bois-Reymond: 604. 30 See TOBIES 1989a, p. 9. 31 [UBG] Cod. MS. F. Klein 1A: 5, fol. 8. 32 See also Klein’s speech in Göttingen on the occasion of the emperor’s birthday (January 27, 1904), which is discussed above in Section 8.3.2.

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society had been founded on October 16, 1913, and its goal was “to promote the knowledge of Russia in Germany while maintaining a thoroughly unpolitical character.”33 Herman Lommel, a son of the physicist Eugen Lommel and thus a nephew of Klein (see Fig. 2), had turned to him on May 30, 1913. Klein immediately had a proposal ready: “To send fifty available teaching candidates to Russia for one semester,” with the aim of placing ten each in the foreign service, the chamber of commerce, secondary schools, libraries, and universities. In a letter dated January 10, 1914 to Otto Hoetzsch, the secretary of the new society and (since 1913) an associate professor of Eastern European history and geography in Berlin, Klein had expressed his willingness to join the society and had outlined additional ideas to round out the program that Hoetzsch had developed: 1) Attention should be paid to the importance of Russia’s scientific literature in the exact sciences. 2) The study of Russian should be linked to a “qualification,” i.e., a Facultas for Russian should be introduced to the examination regulations for teaching candidates, as is already the case for Danish, Polish, etc.34

The developments after the war, when Friedrich Schmidt-Ott not only took over the leadership of the Emergency Association of German Science (Notgemeinschaft der deutschen Wissenschaft; see 9.4.1), but also chaired the German Society for the Study of Russia, followed a direction that was oriented toward foreign policy.35 Germany sought to establish closer relations with Soviet Russia. During the 1920s, representatives from the government and science – experts in aeronautics, engineers, pedagogues, and also mathematicians – traveled there for various purposes. Klein renewed his correspondence with Russian mathematicians.36 Younger Russian mathematicians (P.S. Aleksandrov and P.S. Urysohn) came to him with a letter of recommendation by the Moscow mathematician Nikolai N. Luzin, who had studied in Götttingen from 1910 to 1914.37 In a report on its activity from 1922 to 1927, the Leningrad Physical-Mathematical Society listed Klein and Hilbert as its only foreign corresponding members.38 Klein had been consulting with Carl Heinrich Becker since 1916 on measures to be implemented after the war.39 Klein’s notes from June of 1918, which he made in preparation for a conversation with Becker, indicate that he had sought to identify causes for possible upheaval even before the November Revolution of 1918: 33 34 35 36 37

[UBG] Cod. MS. F. Klein 1A: 5, fols. 1–16. Ibid., fol. 14. Hoetzsch was defamed as pro-Bolshevist and forced into retirement in 1935. See KIRCHHOFF 2003, pp. 130–36. [UBG] Cod. MS. F. Klein 11: 202 (a letter from Vasilev to Klein, February 18, 1924). The author received this letter from the late mathematician Martin Kneser in Göttingen. See TOBIES 1990c and TOBIES 2003; see also DEMIDOV et al. 2016. 38 For this report, see www.mathsoc.spb.ru/rus/reporter.html (Klein was a member until 1925; see p. vii of the report). I would like to thank Danuta Ciesielska for refering me to this source. See also JUSCHKEWITSCH 1981. 39 [UBG] Cod. MS. F. Klein 4G: fol. 79 (Klein’s note from June 5, 1916).

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9 The First World War and the Postwar Period The driving forces: - Democratization (elementary school teachers). - Military requirements, economic concerns. - Governmental requirements, “as I understand them.”40

Regarding “governmental requirements,” Klein’s program of new ideas extended far beyond the promotion of studying abroad. He also wanted to discuss the following problems with Becker: the unity of pure and applied research; the establishment of areas such as mathematical statistics,41 photochemistry, radiology, history of mathematics, and the didactics of exact disciplines; the creation of professorships in the field of education,42 and the establishment of a professorship for applied mathematics at the University of Berlin43; academy projects and projects of the Kaiser Wilhelm Society44; the future of the International Commission on Mathematical Instruction; and advanced training courses for the expected influx of students after the war.45 9.2 HISTORY OF MATHEMATICS, THE “CRY FOR HELP OF MODERN PHYSICS,” AND EDITION PROJECTS In his aforementioned report from August of 1914, which appeared in the Jahresbericht der DMV, Klein remarked: Two decades ago, the theory of algebraic functions and the transcendental functions that can be immediately derived from them stood at the center of general interest. Algebraic curves and surfaces, elliptic functions and theta functions, linear differential equations in the complex domain – and perhaps also algebraic numbers – formed the armory within which the mathematician sought out his problems. Entirely different areas are in the foreground today. First there is the development of mathematics towards abstraction, as has emerged in modern axiomatics, set theory, the new function theory of real and complex variables, and in the refined analysis situs. Furthermore, and in apparent contradiction to this, there has also been a revival of long-forgotten practical questions: the calculus of variations and probability theory are once again being worked on diligently; the integral equations are a means to finally solve important old problems of mathematical physics. And, in addition to this, there is the cry for help of modern physics, which, in its turbulent and indeed revolutionary development, is calling for the support of mathematicians and threatens to consume a great deal of our working energy!46

40 Ibid. 5A: fols. 14–15 (Klein’s notes, June 23, 1918). 41 Felix Bernstein was appointed as a full professor of this discipline in 1921 (see Section 7.6). 42 In 1920, Herman Nohl became associate professor for practical philosophy at the University of Göttingen; in 1922, this position was turned into a full professorship for pedagogy. 43 In 1920, Richard von Mises became a personal full professor of applied mathematics at the University of Berlin. 44 Klein’s initiative was still required to ensure that Göttingen’s Kaiser Wilhelm Institute for Fluid Dynamics was finally established in 1925 (see Section 8.1.3). 45 [UBG] Cod. MS. F. Klein 5A: fols. 14–15 (Klein’s notes, June 23, 1918). 46 KLEIN 1914b, p. 422.

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This quotation from Klein’s 1914 article demonstrates that he maintained a comprehensive overview of the latest trends in mathematics. In the courses that he taught as of the winter semester of 1914/15, he concentrated first on the developments of mathematics in the nineteenth century (Section 9.2.1). He set this topic aside when the general theory of relativity began to capture his interest (Section 9.2.2). When he was encouraged to produce an edition of his own collected mathematical works, Klein devoted his attention to writing commentaries on his articles for the edition in question. After completing the three volumes of his works, he even started a new task: the production of new editions of his older lecture courses (Section 9.2.3). His assistant Hermann Vermeil, who had completed his doctorate in 1913 under Otto Hölder in Leipzig and was severely injured in the war, produced a tabular outline of Klein’s teaching and research activity during the First World War and up to the year 1922 (Table 10): Table 10: Felix Klein’s Courses and Other Activity, 1914–192247 Assistant Graefe48

Graefe Baade49 Baade Baade Baade Baade Baade

Semester Content 1914/15 Development of math. in the 19th c., Pt. 1: The first decades 1915 Development of math. in the 19th c., Pt. 2: Mathematics to ca. 1850; mathematical physics to ca. 1880. 1915/16 Development of math. in the 19th c., Pt. 3: The theory of functions, 1850ca. 1900 1916 An introduction to Einstein 1916/17 The special theory of relativity on an invariant basis Spring of 1917 (2 Notebooks) 1917 The general theory of relativity (foundations) 1917/18 Editing the latter (1 notebook) until New Year’s Day 1918 3 publ. on Einstein and Hilbert

Comments Ed. by Elisabeth Staiger (née Klein) (10 participants) Ed. by Elisabeth Staiger (née Klein) (24 participants) Ed. by Käthe Heinemann & Helene Stähelin (13 participants) (13 participants) Ed. by Walter Baade (7 participants)50

KLEIN 1921 (GMA I), 553–612.

47 [UBG] Cod. MS. F. Klein 22C, fol. 63 (a handwritten document by H. Vermeil). Klein’s courses from 1915 to 1920/21 were also announced in the course listings as the “Mathematical-Physical Seminar” (Wednesdays 11–1 o’clock, later without a specific time). On the participants from 1914 to 1916, see [UBG] Cod. MS. F. Klein 7E. 48 Walther Graefe was the one assistant with whom Klein was dissatisfied. The records reflect a suspicion of forgeries and indicate that Graefe produced copies of Klein’s lecture courses and sold them without his knowledge ([UAG] Kur. 7554, fol. 180). After his military service, Graefe became a teacher at secondary schools ([BBF]). 49 Walter Baade became a famous astronomer and worked in the United States from 1931 to 1959. Before he earned his doctorate (examination subjects: astronomy; mathematics, geophysics) in Göttingen in 1919, he served as Klein’s assistant, and in 1917 and 1918 he had to work for eight hours a day at the aerodynamics research facility in addition. 50 On the participants in 1916/17, see ROWE 2020, p. 6.

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Assistant Baade

Semester Content 1918/19 Lectures in the Math. and Phys. Soc.; and in my room. Ostrowski Lectures on reprinting Priv. Ass. my collected works, 1.2.1919Spring of 1919: Line geometry 31.10.1920 and Plücker’s models Baade; 1919 Non-Euclidean geometry, Ostrowski Erlangen Program Vermeil; Ostrowski Vermeil Vermeil

Vermeil

Priv. Ass: Vermeil; BesselHagen Vermeil; BesselHagen

Comments 50th anniversary of doctorate A.M. Ostrowski, W. Schmeidler, Gerda Laski, […] Ostrowski, Emmy Noether, Jakob Nielsen, F. Drenckhahn, W. Windau […]

1919/20 Before Christmas: Shapes of algebraic entities, Ostrowski, H. Vermeil, H. Kneser, Spring of 1920: Influenza Antonie Stern51, […] 1920 Klein: work on the ENCYKLOPÄDIE – article by Krazer (see 5.4.1) 1920/21 Late fall 1920 – Christmas: Variation principles of mechanics and the theory of relativity Klein: Work on the ENCYKLOPÄDIE article by W. Pauli Spring of 1921: Intuitive studies on algebra, groups of linear substitutions und algebraic equations 1921 As of March 1921: Vermeil takes over the edition of Vol. II of the collected works Late summer 1921: Discussions about algebra 1921/22 Oscillation theorems and linear differential equations February 1922: Influenza

Vermeil, E. Noether, P. Bernays, E. Bessel-Hagen52, H. Kneser, W. Windau […]

Vermeil, Bessel-Hagen, E. Noether, W. Krull, H. Kneser, W. Windau Vermeil, Bessel-Hagen, Ostrowski Vermeil, Bessel-Hagen, E. Noether, E. Artin, A. Bokowski, H. Kneser, W. Pauli, R. Minkowski, W. Windau

1922-23 From mid-March 1922: Edition of Vol. III

9.2.1 Remarks on Klein’s Historical Lectures It should be stressed yet again that Klein’s conception of the (unrealized) seventh volume of the ENCYKLOPÄDIE and the subsequent Kultur der Gegenwart project formed the starting point for his more detailed historical works. He had been working intensively on this subject since 1908, and he had involved numerous colleagues in it as well (see Section 8.3). As early as the winter semester of 1910/11, he 51 Antonie Stern earned her doctorate with Courant in 1925; she emigrated to Palestine in 1938. 52 Erich Bessel-Hagen had completed his doctoral thesis under Constantin Carathéodory at the University of Berlin, in 1920.

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had announced a lecture course on mathematics in the nineteenth century, but this plan had to be changed on account of his health (see Section 8.5). Klein was finally able to teach courses on this topic during the war (see Table 10). Klein’s seminar of 1914/15 was not officially announced in the university calendar; on his seminar from 1915, we read the following in the university’s course catalogue: “Seminar on the development of mathematics since 1850: Prof. Klein with Prof. Carathéodory, Wednesday 11–1, privatissime and free.”53 The university’s course catalogue for 1915/16 stated: “Continuation of the lectures on the development of mathematics in the 19th century: Prof. Klein, Wednesday 11–1, privatissime and free of charge.”54 The typewritten transcriptions of “Seminar Presentations on the History of Mathematics in the Nineteenth Century” were prepared by Klein’s widowed daughter Elisabeth Staiger (winter semester, 1914/15; summer semester, 1915) and by Käthe Heinemann and the Swiss mathematician Helene Stähelin (winter semester, 1915/16).55 These texts served as the foundation for the first volume of Klein’s posthumously published Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert (1926), which has been available in English since 1979 as Development of Mathematics in the 19th Century. Richard Courant and especially Otto Neugebauer prepared this text for publication in [Julius] Springer’s “yellow book series.”56 In this work, they were also able to rely on the support of Constantin Carathéodory, Dirk Struik, Conrad Heinrich Müller, and Erich BesselHagen. Nowhere in the book, however, is there any mention of the women who had prepared the original text.57 Published from Klein’s Nachlass, the book was highly acclaimed,58 even though Klein himself never considered it to be finished. Records in his estate show that some of the sections originally planned for the book never made it into the final publication, such as his reflections on the 1900 Congress of Mathematicians in Paris, on Hilbert’s unsolved problems (see Section

53 https://gdz.sub.uni-goettingen.de/id/PPN654655340_1915 (p. 14). 54 https://gdz.sub.uni-goettingen.de/id/PPN654655340_19151916 (p. 14) 55 Käthe Heinemann, who had passed teaching examinations in mathematics, physics, chemistry, and botany, earned a doctoral degree in botany on August 2, 1922 and became a schoolteacher (see [BBF]). Helene Stähelin completed her doctoral studies in 1924 in Basel (under Hans Mohrmann and Otto Spiess); she had completed the typewritten text of Klein’s 1915/16 lectures on October 17, 1918 in Basel, complete with figures drawn by Erwin Voellmy, who had just completed his own doctorate there under Erich Hecke. 56 Regarding Klein’s influence on Neugebauer, see PYENSON 1979 and PYENSON/RASHED 2012. 57 See also KLEIN 1923 (GMA III), Appendix, p. 11: “These lectures were edited (in part by Klein himself) and reproduced in multiple typescript copies.” 58 In 1937, a trans. into Russian was published: http://ilib.mccme.ru/djvu/klassik/razvitie.htm. – Iris Runge wrote to her mother, in a letter dated February 25, 1938: “[T]his is a truly magnificent book: Klein has surely been a remarkable personality. Perhaps it would be even more necessary to write his biography instead!” Iris Runge meant that it was perhaps even more important to write Klein’s biography than the biography of her father Carl Runge, which she herself had just begun to write; see TOBIES 2012, p. 331–35 (quotation on p. 333).

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10.1), and on set theory, among other topics.59 As it stands, the book can leave a false impression about what Klein held important and wanted to discuss. Regarding set theory, it should be noted once more that Klein and A. Mayer had accepted Georg Cantor’s seminal articles on the topic for publication in Mathematische Annalen and that Klein had instigated both Schoenflies’s comprehensive reports on set theory (the first of their kind in German) and the first English book on the subject: The Theory of Sets of Points (1906), which was written by the married couple Grace Chisholm Young and William Henry Young.60 Klein accepted set theory as an important fundamental subject of mathematics. He did not, however, regard it as a reasonable basis for elementary mathematical instruction. The third edition of his multi-volume book Elementary Mathematics from a Higher Standpoint (1924–28) includes lengthy passages about set theory, and he wrote: “But it is all the more admirable that, thanks to the conceptual definition of our point set, it is actually possible to assert something about it. It has been a particular merit of Georg Cantor (1845–1918) to make this possible; Cantor was the first to show how infinity could itself be the object of mathematical considerations.”61 In the third volume of this book, Klein’s collaborator Friedrich Seyfarth supplemented the footnotes with the latest results at the time on set-theoretical topology and dimension theory.62 9.2.2 Felix Klein and the General Theory of Relativity In 1918, as can be seen in Table 10 above, Klein published three articles related to Einstein and Hilbert. Einstein had accepted an invitation to give a series of lectures in Göttingen in June and July of 1915.63 Nearly simultaneously in November of 1915, Hilbert and Einstein published articles with gravitational field equations. In this regard, there is a mountain of literature about whose ideas came first.64 Klein did not enter the discussion until he had looked into the matter in greater detail. His Nachlass contains handwritten notes with the following titles: “Notizen aus Hilberts Vorlesung über die Grundlagen der Physik” [Notes from Hilbert’s Lecture on the Foundations of Physics] (August 26, 1916), “Die neuen Arbeiten von Einstein 1911 bis 1915” [The Recent Works by Einstein, 1911 to 1915] (September 24, 1916), “Weiterbildung der Theorie bei Hilbert 1915” [Further De59 60 61 62

[UBG] Cod. MS. F. Klein 22A, fols. 74–83 (Klein’s notes from the winter of 1915/16). See SCHOENFLIES 1898, and 1900/1908; YOUNG/YOUNG 1906; MÜHLHAUSEN 2020, p. 126. KLEIN 2016 [31928], pp. 117–70, at p. 123. See also KLEIN 2016 [31924], pp. 274–92. This resulted from the aforementioned study visits of Aleksandrov and Urysohn in Göttingen (see Section 9.1.2). 63 On June 29, 1915, Einstein delivered a lecture titled “Über Gravitation” to the Göttingen Mathematical Society (see Jahresbericht der DMV 24 [1915] Abt. 2, p. 68). In the same week, he gave six lectures, which were funded by the Wolfskehl Foundation (see EINSTEIN 1997, vol. 6, pp. 586–91). 64 See, in particular, SAUER 1999, SAUER/MAJER 2009, ROWE 1999, and ROWE 2020.

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velopment of Hilbert’s Theory, 1915], and “Notizen zu [Gustav] Mie: Grundlagen einer Theorie der Materie” [Notes on Mie’s Foundations of a Theory of Matter]. About Einstein, Klein noted: Einstein’s achievement. – Not bringing up arbitrary curvilinear coordinates – but rather new physical ideas. A general move toward unifying theoretical physics. The inner connection between gravitation and inertia. The Gμν are far away from matter – at a great spatial distance – as earlier. No longer Rn = 0 but more general, the universe = the gravitational field, whereby a distinction must be drawn between the field outside of matter and within matter. Matter is conceived as filling the universe: a phenomenological point of view.65

The lecture courses that Klein gave from 1916 to 1918 (see Table 10) were officially announced as “Lectures on Selected Aspects of Newer Mathematics,” to be held on Wednesdays at the same time. From 1917/18 on, no specific time was given; the course listings stated simply: “hours to be determined.” These lectures resulted in the second volume of Klein’s posthumous book Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert, which was published in 1927. The editors of this volume, Richard Courant and Stefan Cohn-Vossen, who did most of the work, furnished it with the subtitle “Die Grundbegriffe der Invariantentheorie und ihr Eindringen in die mathematische Physik” [The Fundamental Concepts of Invariant Theory and Their Infiltration into Mathematical Physics]. This edition was again supported by Dirk Struik, Otto Neugebauer, and others. It is evident, however, that there are many omissions, and also added material. Unlike the first volume, it has yet to be translated into English.66 When teaching his courses on Einstein’s theories, Klein had sought to collaborate with others on the topic; he sent his lectures to Sommerfeld and to Einstein himself almost as soon as they were ready. Together with Carl Runge, Klein arranged for the Göttingen Mathematical Society to discuss the wide-ranging literature on Einstein: Emmy Noether, Klein, Hilbert, and Runge gave presentations on this theme in January, May, June, and July of 1918.67 Moreover, Klein’s Nachlass in Göttingen contains thirty-eight postcards or letters that Klein and Einstein exchanged from March 26, 1917 to April 28, 1920, twenty-one by Einstein and seventeen (drafts) by Klein.68 Most of this correspondence took place in 1918, when Klein published the three articles mentioned above, and it concentrated on two topics above all: 1) the interpretation of the conservation of energy (the integral conservation laws); and 2) the question of the structure of space of constant curvature in cosmology. Much has already been written about this,69 and therefore the following remarks will concentrate on a few specific aspects that concern Klein in particular.

65 66 67 68 69

[UBG] Cod. MS. F. Klein 22B. There is a Russian translation from 2003. For these presentations, see Jahresbericht der DMV 27 (1918) Abt. 2, pp. 28, 42–47. For the printed letters, see EINSTEIN 1998. For discussion, see TOBIES 1994b and 2005; RÖHLE 2002; ROWE 1999, 2018a, and 2021.

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First, regarding Einstein’s and Hilbert’s works on gravitational equations, which appeared in November of 1915, Klein did not think that there should be any debate about whose ideas came first. When Klein included his essay “Zu Hilberts erster Note über die Grundlagen der Physik” [On Hilbert’s First Note About the Foundations of Physics] (dated January 25, 1919) in the first volume of his collected works, he supplemented it with the following commentary: Einstein’s “Zur allgemeinen Relativitätstheorie” was published in the Sitzungsberichte [Proceedings] of the Berlin Academy from Nov. 11 to Nov. 25, 1915 […], Hilbert’s first note (commented on above) […] appeared on Nov. 20, 1915. The question of precedence is not an issue, because both authors follow completely different trains of thought (so much so that their results did not seem compatible at first). Einstein proceeds inductively and immediately thinks of arbitrary material systems. Hilbert deduces […] from the aforementioned principles of variation. […] It was only in his […] communication to the Berlin Academy on October 29, 1916 that Einstein established the connection between the two approaches.70

Second, Klein cooperated in particular with Emmy Noether. As he wrote in the commentary quoted above, which was directed toward Hilbert: “You know that Miss Noether continues to advise me in my works and that it is only because of her that I was able to delve into the present material.”71 In his article “Über die Differentialgesetze für Erhaltung von Impuls und Energie in der Einsteinschen Gravitationstheorie” [On the Differential Laws for the Preservation of Momentum and Energy in Einstein’s Gravitational Theory] (dated July 19, 1918), Klein thanked Emmy Noether for her “supportive participation,” and he referred to her work “Invariante Variationsprobleme” [Invariant Variation Problems], which Klein himself presented to the Göttingen Society of Sciences on July 26, 1918.72 Emmy Noether continued to revise this article until September of 1918, and she dedicated it to “F. Klein on the fiftieth anniversary of his doctoral degree.” She also expressed that her work and Klein’s had “mutually influenced one another.” The article contains the two theorems that are named after her (the “Noether theorems”), which remain important in modern physics because they combine three great principles: symmetries, conservation laws, and extremal principles.73 Klein relied on Emmy Noether’s assistance when he was editing his article from July 19, 1918 for the first volume of his collected works (1921). Here, in his supplementary commentary, he stressed that his “main theorem” in § 2 was just “a special case […] of the wide-reaching theorem proved by Miss Noether,” and that E. Noether, in her work, had furthermore generalized and proved a theorem by Hilbert. In this commentary, Klein quoted both of Noether’s theorems in full.74 70 KLEIN 1921 (GMA I), p. 566. 71 Ibid., p. 559. 72 Felix Klein, “Über die Differentialgesetze für Erhaltung von Impuls und Energie in der Einsteinschen Gravitationstheorie,” Göttinger Nachrichten: Math.-physikal. Klasse (1918), pp. 171–89, at p. 189. See Emmy Noether, “Invariante Variationsprobleme,” in ibid., pp. 235–57. The latter article would become Emmy Noether’s Habilitation thesis. 73 See TOBIES 2004 and, especially, TOLLMIEN 2018. – See also ROWE 2021 (Chapter 3). 74 KLEIN 1921 (GMA I), p. 585.

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Shortly thereafter, Klein read the draft of Wolfgang Pauli’s ENCYKLOPÄDIE article on the theory of relativity (see Table 10). In response, he sent Pauli copies of his own lectures on the topic and referred him to works by mathematicians (Poincaré, Hilbert, “E. Noether’s theorems”) that Pauli’s draft failed to mention.75 In his final version, Pauli cited works by Poincaré, Hilbert, and Klein, but he made no reference to Emmy Noether’s theorems; there is also no mention of her in Kottler’s ENCYKLOPÄDIE article on gravitation and the theory of relativity.76 Klein made further efforts, however, to disseminate Emmy Noether’s results. Just a few weeks before his death, on April 13, 1925, Klein wrote to Max Planck: If I judge the matter correctly, there is now agreement between you and me, but not with our colleague [Max] von Laue. The situation is dealt with quite clearly in Miss Noether’s work in the Göttinger Nachrichten from 1918 […]. There, on p. 255, clear mathematical reasons are given for why the actual conservation laws apply in the case of the special theory of relativity but not in the general theory of relativity. Unfortunately, Miss Noether’s work is written very concisely, and its full scope is difficult to grasp on account of the generality of the presentation. This may be the reason why physicists have not read the work. – Incidentally, colleague [Max] von Laue comes very close to the facts of the matter on pp. 175–77 in the second volume of his book on the theory of relativity; he only interrupts the conclusive mathematical development with an example in which he draws upon ideas from traditional physics. Whether one wants to accept the general theory of relativity or not is a question of its own, and I have no firm opinion about it. Yet if one accepts it, its mathematical development is inevitable; only in this sense am I a “purist.”77

Einstein had written to Klein about Emmy Noether’s work as early as December 27, 1918: “Upon receiving the new paper by Miss Noether, I again feel that it was a great injustice that she be denied the venia legendi. I would very much support our taking an energetic step at the Ministry.”78 Klein reacted immediately, and on January 5, 1919, he turned to the aforementioned Otto Naumann: Your Excellency, Surely you remember the application, submitted by the faculty here, for Miss Noether to be allowed to complete a Habilitation in mathematics. Vigorously supported by the representatives of mathematics, this application was rejected at the time for general reasons, but an agreement has been reached whereby Miss Noether is nevertheless still able to be effective. At the time, of course, I understood the circumscribed decision of the Ministry very well, but I would like to ask whether it will continue to be upheld in all cases. If not, then I would like to arrange for the faculty here to address the matter yet again.

75 See HERMANN/V. MEYENN/WEISSKOPF 1979, p. 27 (Klein to W. Pauli, March 8, 1921). 76 See Wolfgang Pauli, “Relativitätstheorie,” in ENCYKLOPÄDIE, vol. V.2 (1920), pp. 593–775 (in English: Theory of Relativity, trans. G. Field [Oxford: Pergamon, 1952]); and Friedrich Kottler, “Gravitation und Relativitätstheorie,” in ENCYKLOPÄDIE, vol. VI.2.2 (1922), pp. 159– 237. On the transmission of E. Noether’s theorems, see also KOSMANN-SCHWARZBACH 2011. 77 [UA Frankfurt] A letter from Klein (written in his daughter Elisabeth’s handwriting) to Max Planck dated April 13, 1925. I would like to thank Dieter Hofmann, Berlin, for referring me to this source. See also LAUE 1921. 78 [UBG] Cod. MS. F. Klein 22B (Einstein to Klein, December 27, 1918). The English translation here is from EINSTEIN 1998, vol. 8, p. 714.

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9 The First World War and the Postwar Period In today’s circumstances, there can be no doubt that many people will regard Miss Noether’s current position as an inequitable restriction, especially because Miss Noether’s scientific achievements have far surpassed all of our expectations. In the past year, she has completed a series of theoretical investigations that are superior to the achievements accomplished by everyone else here during this same time (the work by full professors included). Through her discussions and presentations in the Mathematical Society, she has also had a positive influence on the collaborative efforts of like-minded mathematicians. The requirements for treating this as an exceptional case have thus been met to the fullest extent. In light of the recent decision to allow women to work in the broadest variety of state offices, however, it is perhaps no longer necessary at all to argue on the basis of exceptional achievements. A brief reply is all I request. Sincerely yours, Kln79

Klein’s intervention was successful. In May of 1919 (on her third attempt), Emmy Noether was finally able to habilitate, and in 1922 she received an unofficial (nichtbeamtete) associate professorship. The conservative establishment at the time did not allow any woman in Prussia to hold a full professorship.80 Third, on account of his knowledge of projective and non-Euclidean geometry, the theory of invariants, and group theory, Klein was able to embed Einstein’s theory into his Erlangen Program and make substantial contributions to it. In this respect, he differed from mathematicians who sought to replace Einstein’s mathematics with inadequate methods (Eduard Study),81 from theoretical physicists who outright rejected the theory of relativity (Max Abraham), and from Nobel-Prize-winning experimental physicists who did not understand the theory and who were blatantly anti-Semitic (Philipp Lenard, Johannes Stark). In order to clarify the relationship between his articles on the theory of relativity and his Erlangen Program, Klein made eleven points in the commentary that he added to the first volume of his collected works. These will not be discussed here in detail; Kleins concluding remark was: “It hardly needs to be said that [Hermann] Weyl’s further development of Einstein’s theory can also be made to fit just as well with the scheme of the Erlangen Program.”82 Fourth, in his discussions of the two cosmological models of the universe, proposed by Einstein and the Dutch astronomer Willem de Sitter, Klein was able to clarify a central point of dispute.83 Einstein and Weyl, who had engaged in a polemic with de Sitter, had argued that spacetime singularities had to exist in de Sitter’s model. They thought that this refuted de Sitter’s contention that this solution of the cosmological field equation was free of matter. Klein could show, however, that the apparent singularities (in de Sitter’s model) disappear under 79 80 81 82 83

Quoted from TOBIES 1991b, p. 172. See TOLLMIEN 1990 and 2021; TOBIES 2008c. – See also KOREUBER 2015; and ROWE 2021. See HARTWICH 2005, pp. 133–35. See KLEIN 1921 (GMA I), pp. 565-67, quotation on p. 567. – See also SCHOLZ 2001; 2016. Both models were understood as static back in 1917-18, when no one imagined an expanding universe. I am indebted to David E. Rowe for pointing this out to me. See also RÖHLE 2002.

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appropriate coordinate transformations.84 That is, de Sitter’s solution was indeed free of singularities. When Einstein received Klein’s article “Über die Integralform der Erhaltungssätze und die Theorie der räumlich-geschlossenen Welt” [On the Integral Form of Conservation Laws and the Theory of the Spatially Closed Universe], he responded positively to Klein’s work in a letter: I am thrilled with your new paper like a child who gets a piece of chocolate from his mother, What you plant squarely on its feet is exactly what is crookedly limping and lurching every which way with me. Now I am sending you the proofs of a new paper that relies far more on physical than on mathematical support […]. A brief note with your opinion of it would be of great interest to me.85

Einstein – thirty years younger than Klein – still found reasons to reject de Sitter’s alternative model, but Klein recognized in him someone whose methods were akin to his own. Einstein, as Klein expressed to him in a letter, was always receptive to new ideas and willing to put them to productive use: In this context I hope to succeed in giving a condensed presentation of your theories in particular, from my formal, mathematical point of view. From the outset I do feel that I am in agreement with you in principle, as far as the scope of the different appoaches is concerned: in contrast to the majority of your followers, who see the latest form of your theories as final and binding, you have maintained the freedom to look for increasingly refined formulations of the general foundations and simultaneously, in accordance with each individual problem under consideration, for specific assumptions that sufficiently approximate the relevant circumstances. In heartily concurring with you in my own way of thinking, I also welcome in particular your new speculations […].86

Fifth, it was mathematicians (Klein foremost among them) who proved to be Einstein’s allies when he had to defend himself against strident attacks from physicists and philosophers.87 Klein convinced Einstein to become a member of the German Mathematical Society in 1918. In 1920, when the journal Mathematische Annalen moved from the publishing house of B.G. Teubner to Julius Springer, Klein ensured that Albert Einstein succeeded Walther Dyck as one of the principal editors, alongside Hilbert, Blumenthal, and himself (see also Section 2.4.3). Klein explained to Einstein in a letter dated April 28, 1920:

84 See Felix Klein, “Über Einsteins kosmologische Ideen 1917,” Jahresbericht der DMV 27 (1918) Abt. 2, pp. 42–43, 44 (a lecture delivered at the Göttingen Mathematical Society on May 7, 1918); idem, “Bemerkungen über die Beziehungen des Sitter’schen Koordinatensystems B zu der allgemeinen Welt konstanter positiver Krümmung,” Koninklijke Akademie van Wetenschappen te Amsterdam: Proceedings 20 (1918), pp. 614–15; idem, “Über die Integralform der Erhaltungssätze und die Theorie der räumlich-geschlossenen Welt,” Göttinger Nachrichten (1918, December 6), pp. 394–493 (repr. KLEIN 1921 [GMA I], pp. 586–612). 85 [UBG] Cod. MS. F. Klein 22B (a letter from Einstein to Klein dated April 14, 1919). The English translation here is from EINSTEIN 2004, vol. 9, p. 19. 86 This English translation is from EINSTEIN 2004, vol. 9, p. 22 (a letter from Klein to Einstein dated April 22, 1919). See also TOBIES 1994b, p. 351. 87 See Klaus HENTSCHEL 1990.

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9 The First World War and the Postwar Period Today’s physical production, as presented for ex. in the Physikalische Zeitschrift, suffers from a restlessness which is hardly compatible with the depth that is necessary for mathematical works. Therefore, I would be particularly grateful if you would help assure the appearance of suitable papers for the Annalen.88

Beginning with volume 81 (1920) of Mathematische Annalen, further mathematicians and theoretical physicists were integrated into the extended editorial board of this journal, among them Max Born, Theodore von Kármán, and Arnold Sommerfeld. Max Born, who knew that Klein was open-minded about new theories, wrote to Klein […] that traditional mathematical physics – with its continuum as the basic idea for space, time, and the physical world – was on the wrong path. The method of partial differential equations does not correspond to the essence of the processes to be described. The more that physics and chemistry approximate one another and blend together, and the more that the atom is understood as the building block of all bodies, the more it becomes clear that we do not have adequate mathematical processes and methods, or at least, if they do exist somewhere hidden within the great realm of mathematics, that we do not recognize them in their significance.89

It was in Göttingen where Max Born – a full professor of theoretical physics there from 1921 to 1933 – would find the necessary mathematical background to make advancements in quantum physics.90 Regarding the theory of relativity, Robert Fricke, who was then the chairman of the German Mathematical Society, informed Klein about the much-discussed session at the Society of German Natural Scientists and Physicians’ 1920 meeting in Bad Nauheim (the first meeting of this organization to take place after the First World War): The sensational session on relativity went extraordinarily well and filled me with the utmost enthusiasm. The development became a triumph for Einstein, who really is a superior intellect. I was proud for having instigated the session, and I am pleased that, after it was over, I was able to express my sentiments personally to Einstein […]. In the discussion, Einstein’s superiority over Lenard was palpable even to a layperson.91

88 [UBG] Cod. MS. F. Klein 22B (Klein to Einstein, April 28, 1920), the English translation is modified from EINSTEIN 2004, vol. 9, p. 333 (German original in TOBIES 2019b, p. 463: “Die heutige physikalische Produktion, wie sie sich zum Beispiel in der Physikalischen Zeitschrift darstellt, leidet an einer Unrast, welche mit der für mathematische Arbeiten notwendigen Vertiefung schwer verträglich ist. Ich würde Ihnen besonders dankbar sein, wenn Sie sich demgegenüber für das Zustandekommen für die Annalen geeigneter Arbeiten einsetzen.”) – Einstein stayed on as an editor until vol. 100 (1928). Hilbert, who was less diplomatic than Klein, acted as the sole editor beginning with vol. 101 (1929), with Otto Blumenthal and Erich Hecke supporting him (“Unter Mitwirkung”, see, for example, Fig. 7). From 1925 to 1933, Blumenthal also served as an editor of the Jahresbericht der DMV. 89 [UBG] Cod. MS. F. Klein 5C, fol. 69v (Max Born to Felix Klein, July 15, 1920). 90 See also SCHIRRMACHER 2019. 91 [UBG] Cod. MS. F. Klein 9 (Fricke to Klein, September 28, 1920). For a survey of the key points discussed at this session, see Hermann Weyl, “Die Relativitätstheorie auf der Naturforscherversammlung in Bad Nauheim,” Jahresbericht der DMV 31 (1922), pp. 51–63.

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9.2.3 The Golden Anniversary of Klein’s Doctorate, and Edition Projects In honor of the fiftieth anniversary of Klein’s doctoral degree (December 12, 1918), his friends and colleagues donated money with the following wish in mind: “They would like to encourage your decision to produce a uniform edition of your collected works.”92 The result was three volumes (KLEIN 1921/22/23), which are probably unique in nature. Klein furnished his works with commentary and classified them. He wrote comments about how they originated, and he supplemented them with recent results in the research areas at hand. With these volumes, Klein created a picture of himself, his works, and many of his collaborators. It is well known that he attempted to present everything as objectively as possible. The extent to which he managed to do so in each individual case, however, is worthy of a special analysis of its own and will not be addressed here. Throughout the process of producing this edition, Klein was able to rely on a number of assistants. His first collaborator is especially noteworthy. Born in Kiev, Alexander Ostrowski was the primary editor of the first volume (1921). While doing this editorial work, he also completed his dissertation: “Über Dirichlet’sche Reihen und algebraische Differentialgleichungen” [On Dirichlet Series and Algebraic Differential Equations] (submitted on March 16, 1920). In the enclosed vita, he expressed his thanks above all to Klein, “with whom I have had discussions about his mathematical works almost every day over the last year and a half. Anyone who has had the good fortune of being close to this famous researcher will be able to judge the extent to which my education and inspiration are indebted to these conversations.”93 Together, Klein and Edmund Landau evaluated Ostrowski’s dissertation and declared that he passed with distinction; in their review, they wrote that Ostrowski had “solved a famous and hitherto unsolved problem from Hilbert’s lecture in Paris (1900)” (in fact, the dissertation made an essential contribution to solving Problem 13). In a later article, Ostrowski also mentioned that, while editing the first volume of Klein’s collected works, he was inspired by the Erlangen Program (which was reprinted in this volume) to engage with new sets of questions in the area of algebraic invariant theory.94 Klein’s responses during the anniversary celebrations make it clear that he was not surprised by the wish of his colleagues, friends, and doctoral students. By this point, in fact, he had already recruited his first important collaborators for the project. 92 The founding charter of this fund is printed in KLEIN 1921 (GMA I), pp. vii–x (quoted here from p. ix). Because of the period of high inflation after the war, the fund, which was held in a bank account in Göttingen and managed by Robert Fricke, was insufficient to cover the costs of the project, but additional funding from the Göttingen Association and from the Emergency Association of German Science (see Section 9.4) was provided to pay for assistants to work on volumes II and III. 93 [UAG] Phil. Fak. Prom. Vol II (1920–21), No. 2 (Ostrowski). 94 Alexander Ostrowski, “Eine neue Fragestellung in der algebraischen Invariantentheorie,” Jahresbericht der DMV 33 (1925), pp. 174–84.

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The celebrations themselves are noteworthy for two reasons: first because Klein planned, in advance, the day’s program and the exact words that he would like to say at various points (and we can thus see what was particularly important to him);95 and second because we have Heinrich Behnke’s first-hand account of Paul Koebe’s keynote speech. From Klein’s notes, we learn that a deputation was expected to arrive at noon: the rector, four deans (from the four university faculties), two secretaries from the Society of Sciences (Academy), and the university’s Kurator. In his first thankyou speech, Klein wanted to mention his “student years in Bonn”; his time as a Privatdozent in Göttingen with Clebsch and his interactions with “the like-minded [Aurel] Voß, [Max] Noether, [Eduard] Riecke”; the “seminar with [Georg] Waitz”; his “first presentation at the [Royal] Society of Sciences [in Göttingen], January 4, 1869,” his election to the position of assessor there, and his “first acquaintance with [Friedrich] Althoff.” Next on the agenda were congratulatory remarks on behalf of the Göttingen Mathematical Society (by Hilbert; see Section 7.2), the German Mathematical Society, and on behalf of the friends who donated money to fund the edition of Klein’s collected works (by Robert Fricke). Regarding his reply, Klein sketched the following notes: “Klein’s response: the joy of collaborating. The objective and goal of reprinting my collected works (Ostrowski). Personal thanks to Fricke for 35 years of work.” After a break for lunch, Klein invited “mathematical colleagues to tea” at 4:30 and intended to tell stories about his experiences in elementary school, where he learned to “make calculations in his head” and where he received his first “natural-scientific inspiration” from a student teacher (see Section 2.2.3). This was followed, at 6 o’clock, by a special session of the Mathematical Society, where Paul Koebe gave a keynote lecture about Klein’s work on the uniformization theory. From Heinrich Behnke, however, we learn that Koebe instead gave a panegyric in praise of himself: “The organizers and the many guests felt embarrassed, but Klein, in his thank-you speech, set everything right as though it was the most natural thing in the world, and everyone’s mood lightened.”96 Klein’s preparatory notes for this response indicate that he wanted to place himself in Wilhelm Ostwald’s system of categories: “[I am] a romantic, not a classic. Implementation of my universal [allseitig] program in Göttingen, from 1893 on by organizational means. Long live Göttingen!”97 At 4 o’clock on Wednesday, December 18th, there was then “coffee at the Rohns Tavern, organized by the student body.” Here, among other things, Iris Runge recited a poem, which is unfortunately lost. Klein spoke about “his” mathematical student unions (in Bonn, Göttingen, and Berlin), and he recommended that current unions should “once again be organized more freely, without compul95 [UBG] Cod. MS. F. Klein 1: 22 (4 pages of Klein’s notes). 96 BEHNKE 1978, p. 38. On Behnke, see HARTMANN 2009. 97 [UBG] COD. MS. F. KLEIN 1: 22. – Regarding Ostwald’s categories, see W. OSTWALD 1909.

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sory affiliations [Verbindungszwang].”98 He discussed the beginnings of the right of women to attend university, his doctoral students “Miss Winston” and “Miss Chisholm,” and concluded by stressing the “importance of the exact disciplines to the general public.” When the work on his three-volume Gesammelte Abhandlungen was coming to an end (see Table 10), Klein noted on June 9, 1922: “What to do when vol. 3 is completed? Will I still have assistants at my disposal, without whom it just won’t do?”99 Richard Courant found a new edition project for Klein: re-editing his lecture courses that existed only in autograph (handwritten) reproductions and publishing them as printed books in the yellow Springer series. Klein divided this edition program into two groups, and he started to prepare these works for publication with the help of suitable collaborators. The first group consisted of his mathematical special lecture courses: on nonEuclidean geometry, higher geometry, hypergeometric functions, linear differential equations, Riemann surfaces, and number theory. Klein was unable to complete everything himself, but he began working on non-Euclidean geometry with his private assistant Walther Rosemann, who in 1922 had earned his doctoral degree under Hilbert with a dissertation on geometry. It is highly remarkable that, in the university course listings, Klein still announced two four-hour lecture courses on related topics: “Elementary Projective Geometry with Non-Euclidean Geometry” for the winter semester of 1924/25, and “Line Geometry” for the summer semester of 1925 (Mo, Tu, Th, Fr, 8:00–9:00).100 By the time that Klein died on June 22, 1925, the corrected proofs of the first chapters of his Vorlesungen über nicht-euklidische Geometrie [Lectures on Non-Euclidean Geometry] had already been sent to him, as Rosemann mentions in the book’s preface.101 The new edition of Klein’s lectures on higher geometry was taken over by Wilhelm Blaschke,102 who had already published his own lectures on differential geometry, using Klein’s Erlangen Program as a guiding light. Hans Reichert, who published a new edition of Blaschke’s Einführung in die Differentialgeometrie (Berlin: Springer, 1960), wrote about Klein’s influence on Blaschke’s work: “Klein’s suggestion to divide not only geometry into different areas – depending on which transformation group the respective geometric entities are compatible with – but also, conversely, to proceed from any given transformation group of a manifold and to build the related theory of invariants geometrically was realized by Blaschke, together with an ever growing number of collaborators, in his affine

98 That is, Klein was in favor of scientific student unions and was against fraternity-like associations with (conservative) political agendas. 99 [UBG] Cod. MS. F. Klein 5E: fol. 159v. 100 [UBG] Cod. MS. F. Klein 22J (course preparations as of 1923). 101 KLEIN/ROSEMANN 1928. 102 KLEIN/BLASCHKE 1926.

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differential geometry.”103 Further lecture courses by Felix Klein from this group were not published as printed books until much later.104 The second group of lectures in Klein’s edition program – a project which he undertook at the same time – concerned his three-volume book Elementarmathematik vom höheren Standpunkte aus [Elementary Mathematics from a Higher Standpoint] (see Section 8.3.4.2). All three volumes were prepared for a third, printed edition; previously, only autograph copies existed. As a collaborator, Klein recruited Friedrich Seyfarth, who in 1916 had earned a doctoral degree under Max Winkelmann (Klein’s doctoral student) at the University of Jena and who, as of April 1, 1920, had been working as a teacher (his examination subjects were pure mathematics, applied mathematics, and physics) at the Oberrealschule in Göttingen, which is known today as the Felix-KleinGymnasium.105 In these volumes, Klein largely left his original text unchanged; he added notes and had Seyfarth add supplementary comments in the appendix. The first volume (on arithmetic, algebra, and analysis; dated Easter 1924) and the second volume (on geometry, dated May 1925) were published with prefaces by Klein himself. In the two months before his death, Klein was still giving directions about how to proceed with the third volume, even suggesting that the title should be changed.106 The translation of these volumes into numerous languages, including their recent (second) translation into English, has been mentioned above in Section 8.3.4.2. 9.3 MATHEMATICAL EDUCATION – INTERNATIONAL AND NATIONAL One of Klein’s main concerns was to lead the work of the International Commission on Mathematical Instruction (ICMI) to a good conclusion. (Regarding the foundation of the ICMI, see Section 8.3.4). In 1920, there was a break in this commission’s activity; however, the German subcommittee continued formally to exist within the framework of the German Mathematical Society (see 9.3.1). Because of the lost war, an anti-technology movement arose in Germany, and governmental efforts were made to reduce the number of hours devoted to mathematics and science lessons in schools. In cooperation with allies, Klein took decisive initiatives to counteract this development (Section 9.3.2). 103 Hans Reichardt, “Wilhelm Blaschke†,” Jahresbericht der DMV 69 (1966), pp. 1–8, at p. 6. 104 See KLEIN 1986, KLEIN 1987, and KLEIN 1991. 105 Seyfarth became the school’s director in 1947, and he succeeded in having its name changed to the Felix-Klein-Oberschule in April of 1948 (as of 1956: Felix-Klein-Gymnasium). 106 In the first and second editions, the title of this third volume was Anwendung der Differentialund Integralrechnung auf Geometrie (Eine Revision der Prinzipien) [The Application of Differential and Integral Calculus to Geometry (A Revision of Principles)]. In the third edition, it was changed to Precision Mathematics and Approximation Mathematics (see also Section 8.3.2). – There was no translation into English until 2016. The first translation of vol. III was into Chinese in 1989 (together with vols. I and II).

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9.3.1 The International Commission on Mathematical Instruction Klein succeeded in bringing the ICMI Abhandlungen [Treatises] on mathematical instruction in Germany to a conclusion in 1916.107 In the prefaces to these volumes, which were written during the war, Klein also made references to the military importance of mathematics. He had sent some volumes to the Ministry of Culture and received a green light to carry on the ICMI’s international activities in cooperation with neutral Switzerland and its representative Henri Fehr.108 In Cambridge in 1912, the mandate of the commission had been extended to the next (fifth) International Congress of Mathematicians. This Congress was planned to take place in Stockholm in 1916,109 and Klein intended to give a final report on the project there. The next international congress, however, did not take place until 1920 (September 22–30) in Strasbourg, a location chosen for political reasons (after the Franco-Prussian war, it had fallen into German hands, and now after the First World War it had been reclaimed by France). Even though German mathematicians had not been invited there, it is interesting to know that several of them were prominently mentioned. Because Einstein’s theory of relativity was on everyone’s lips,110 the Irish physicist Joseph Larmor had been invited to give the first plenary lecture. Larmor, who had devised the formulas for Lorentz transformation before Hendrik Antoon Lorentz himself,111 referred to both Hilbert and Klein in his lecture “Questions in Physical Interdetermination”: Thus one can concur with Hilbert and Klein that the essential feature of the new analysis, which has merged gravitation in the scheme of space and time, is that the extremal relations that determine extension after Riemann on the basis of distance are now involved in altered form along with the extremal relations of Action that determine dynamical sequence.112

Although Fehr, Greenhill, D.E. Smith and other members of the ICMI attended the conference in Strasbourg, the commission’s final report was not presented. Instead, it was published in the journal L’Enseignement mathématique.113 The proceedings of the congress in Strasbourg had not yet been published when Klein, after the 1920 conference of the German Natural Scientists and Physicians (Gesellschaft deutscher Naturforscher und Ärzte, GDNÄ) in Bad Nau107 See KLEIN 1909–16. 108 Regarding this history, see also GISPERT 2021. 109 In Sweden, a prize question had already been formulated in preparation of this scheduled Congress; see Jahresbericht DMV 24 (1915) Abt. 2, p. 69. 110 Observations of the solar eclipse on May 29, 1919 had verified Einstein’s theoretical prediction (based on his general theory of relativity) that the path of light is altered by gravity. 111 Klein was familiar with Larmor’s book Aether and Matter (1900) and he mentioned Larmor’s results in lectures on the theory of relativity (1916–18); see KLEIN 1927, p. 72. 112 See Joseph Larmor, “Questions in Physical Interdetermination,” Comptes Rendus du Congrès International des Mathématiciens (Strasbourg, 22–30 Septembre 1920), ed. Henri Villat (Toulouse: Libraire de l’Université, 1921), pp. 3–40, at p. 28. 113 See FEHR 1920, and Jahresbericht der DMV 29 (1920) Abt. 2, p. 43.

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heim, arranged for meetings in Göttingen. He had stopped traveling for health reasons, and he wanted to pass the baton to younger researchers without withdrawing completely. Under the motto “the younger generation should steer the course,” he invited Walther Lietzmann and Georg Wolff to discuss matters with him on September 27 and 28, 1920, and he had conversations on September 30th with Friedrich Poske, Georg Hamel, Rudolf Rothe, and Heinrich E. Timerding to test their suitability for certain offices.114 Timerding had been the chairman of the German Committee for Mathematical and Natural-Scientific Instruction (Deutscher Ausschuss für mathematischen und naturwissenschaftlichen Unterricht = DAMNU) since 1912, and he had been supported in this capacity by Lietzmann since 1919. Now, in 1920, Klein managed to have Lietzmann, Rothe, and Wolff appointed to the German ICMI subcommittee. At the same time, Klein proposed that this subcommittee should submit a report to L’Enseignement mathématique. By preserving the German subcommittee, “the continuity of work will be maintained, and it will be ensured that the valuable achievements of the ICMI will not go to waste,” as was stated in the Jahresbericht der DMV.115 One year later, however, Ludwig Bieberbach, who was then the secretary (Schriftführer) of the German Mathematical Society, reported: After the Entente countries had placed their mutual scientific exchange on a new basis, by focusing on academic conferences held among allies, and had thereby limited this interaction “pour le moment,” as Fehr believes, to allied nations and a few neutral countries, the ICMI’s board members (Klein, Greenhill, Fehr, [David Eugene] Smith) were asked for their opinion, and they decided to dissolve the ICMI, which had been founded on a different basis.116

In the following years, the German Mathematical Society nevertheless continued to list the ICMI subcommittee as an existing commission.117 Klein did not live to see the revival, in 1928, of the ICMI’s activities at the International Congress of Mathematicians in Bologna, under the presidency of D.E. Smith. There, German mathematicians were once again allowed to participate,118 whereas they had still been excluded from the 1924 congress in Toronto. Klein, as ever, continued to keep a close eye on international developments, and after the war he renewed his correspondence with scholars abroad; his former doctoral student Virgil Snyder, for instance, reported to him about the attitude of American mathematicians toward the Toronto decision (see Appendix 11).

114 [UBG] Cod. MS. F. Klein 5D, fol. 10v. About Rudolf Rothe, for instance, Klein concluded that his perspective was “too narrow.” 115 See Jahresbericht der DMV 30 (1921) Abt. 2, p. 29. 116 Ibid. 31 (1922) Abt. 2, p. 59. 117 As late as 1927, we read in Jahresbericht der DMV 36 (1927) Abt. 2, p. 1: “Internationale Mathematische Unterrichtskommission, einges.[etzt] 1908: W. Lietzmann, R. Rothe, G. Wolff (seit 1920).” 118 Nationalistic tendencies still remained, however; see SIEGMUND-SCHULTZE 2016a.

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9.3.2 Countering the Restriction of Mathematics and the Natural Sciences When statements pertaining to secondary education threatened to impede intellectual development, Klein weighed in on the conversation: I imagine the goal of all education to be a product of two factors, α ⋅ β , where α refers to the development of ethics, willpower, and – for all I care – also to physical training, while β refers to intellectual development and general ability. It may be that, for some time, our secondary schools failed to give enough consideration to factor α. That said, it recently seems as though the importance of factor β is in danger of being underestimated. And yet the reconstruction of our country largely depends on the presence of enough people who have a sufficient amount of β at their disposal. To train such people in a systematic way seems, to me, to be the real objective of our secondary schools. In order to improve this situation, I have long supported the reform movement in mathematical instruction, which aims to create an immediate relationship between the instructional material and the tasks of practical life […].119

Klein prompted Lietzmann to publish this explanation of his views in the Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht. It was a response to statements made by the Association of Academically Trained Teachers in Germany (Verband akademisch gebildeter Lehrer Deutschlands), which, at a conference held in Jena in May of 1921, had concentrated primarily on ethical education, a trend that had already become apparent during the war. In the spring of 1917, the Social Democratic Party (SPD) had put forth a resolution in parliament calling for a national School Conference to be held after the war in order to discuss matters relating to pedagogy, education legislation, and the organization of schools.120 Mathematical and scientific education did not feature at all in these discussions. As early as November 17, 1917, Klein had therefore sent invitations to a debate, hosted by the Göttingen Association for the Promotion of Applied Physics and Mathematics, “on the lessons of the war with respect to the teaching of mathematics, mechanics, and physics.” Numerous guests of honor (from the Ministry of Culture, DAMNU, the military, etc.) had accepted the invitation to attend. They adopted a resolution to underscore the importance of mathematics and science.121 On the occasion of the twentieth anniversary of the Göttingen Association in 1918, Klein had argued yet again: Moreover, I would like to say that our resolution is by no means intended to oppose the general movement in schools, which are now more strongly focused on developing the students’ willpower and thus on cultivating ethical subjects. We do not, however, want instruction to lose itself again in the indeterminate perception of vaguely imagined generalities; rather, we want instruction to be tied to a clear knowledge of reality that is adjusted to suit each level of education. Nor do want this knowledge to be merely practical but rather, if possible, combined with theoretical insight.122

119 Walther Lietzmann, “F. Klein über die Aufgaben unserer höheren Schulen,” Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht 52 (1921), pp. 267–68. 120 See REICHSSCHULKONFERENZ 1920, pp. 11–12. 121 [UBG] Math. Arch. 5036, esp. fols. 53–54, 135–37. 122 KLEIN 1918, p. 224.

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On December 15, 1918, Klein wrote to Konrad Haenisch, who had meanwhile been made the SPD Minister of Culture in Prussia. Klein thanked him for the congratulatory telegraph on the golden anniversary of his doctorate, three days earlier, and he added: “I would like to be able to provide modest assistance with the forthcoming educational reform.”123 Klein enclosed with this letter the publications of DAMNU and the most recent volumes of the German ICMI treatises on mathematical instruction. He referred to the aforementioned resolution by the Göttingen Association, and he diplomatically emphasized his continuing efforts to promote the interests of the overall educational system. The year 1919 had barely begun when Klein, on January 7th, gave a lecture in a session of the Göttingen Mathematical Society titled “Mathematischer Unterricht an den verschiedenen Schularten” [Mathematical Instruction at the Different Types of Schools]. The goals of this talk were, first, to provide a retrospective overview of his work on the topic and, second, to raise prospective pedagogical questions with an eye toward school reform.124 Prospectively, he formulated theses that accorded with his long-held ideas and also pertained to the issue of a unified school system (Einheitsschulsystem), which was much-discussed at the time: a)

The entire school system should form an ideal whole, so that its individual levels properly fit together […]. b) If they are not to atrophy, all of these things have to be anchored at the university. […] An even more varied training in state-economic disciplines. Cf. philosophy, psychology. New professorships for general pedagogy. But also for the didactics of individual subjects. c) Consider changes in the teaching staff? […] Only a few to teach the highest levels of mathematics, others with a broader foundation. (Make researchers and teachers separate). […]125 d) To do everything to ensure that mathematics (or, more generally, the exact disciplines) will gain even more validity in the schools and in Germany’s public life.

In order to respond to the educational issues that were being discussed throughout Germany, the Philosophical Faculty at the University of Göttingen formed a pedagogical committee on March 24, 1919, and it announced on May 19, 1919: “Messrs. Klein and [Edward] Schröder have been asked to prepare materials for the issues to be addressed at the Reich School Conference and to work in conjunctions with the representatives from Berlin.”126 Klein added Walther Lietzmann to this committee and sent him as a delegate to this School Conference, which took place from June 11 to June 19, 1920 in Berlin under the direction of the Minister of the Interior.

123 [UBG] Cod. MS. F. Klein 2E, fol. 72 (a draft of a letter from Klein to Haenisch, December 15, 1918). 124 Ibid. 22F, fols. 1–3v. 125 Klein had added here: “I wanted to combine everything but, over the years, I was only able to do so by alternating [between research and teaching] (after I had earlier pushed myself past the breaking point [mich kaputt gemacht hatte])” (ibid., fol. 3). 126 [UAG] Phil. Fak. III, vol. 5, fols. 208, 210.

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The subjects of mathematics and the natural sciences, however, played no more than an ancillary role at the Reich School Conference. Not a single report or committee was explicitly devoted to them. In comparison to the Prussian school conferences (1890, 1900), there was a far greater number of participants: representatives from state governments, district and school administrations, and various groups of teachers from all of Germany. With respect to mathematical and scientific subjects, Lietzmann functioned as the representative for the German Committee for Mathematical and Natural-Scientific Instruction (DAMNU), and Friedrich Poske participated as a representative of the (teachers’) Association for the Promotion of Mathematical and Natural-Scientific Instruction (Verein zur Förderung des mathematisch-naturwissenschaftlichen Unterrichts). There were also eight representatives from the German Committee for Technical Education.127 Everyone involved, including Lietzmann, engaged primarily in discussions about the organizational and structural issues of the school system. Lietzmann was the only participant to criticize the planned reduction of educational hours for mathematics and the sciences, and in doing so he followed Klein’s example by citing the cases of France and England, where schools placed a greater emphasis on the exact sciences.128 Ultimately, the idea of creating a unified school system throughout the entire German Empire never came to fruition; in Jena alone, it was decided that elementary school teachers were required to have a university education – one of the aspects discussed at the Berlin conference. At the aforementioned meetings held in September of 1920 in Göttingen (see Section 9.3.1), Klein discussed the new educational reforms taking place in Germany. He asked the participants: “Who is now creating the new curricula? Who is working on the examination regulations? […] We also have to prepare ourselves to create curricula for the new institutions at universities.”129 Furthermore, Klein was familiar with Georg Hamel’s idea of establishing an alliance: a Federation of Mathematical Societies and Associations in Germany (Reichsverband mathematischer Gesellschaften und Vereine) in order to coordinate proposals for educational reforms, and he supported it. Hamel, who had earned his doctorate in Göttingen and had worked as Klein’s assistant in 1901/02,130 had presented this idea in September of 1920 at the aforementioned GDNÄ meeting in Bad Nauheim. Now, Klein encouraged Hamel “simply to take the initiative, without pushing Berlin into the forefront.”131 Adolf Krazer, then a board member of the German Mathematical Society, informed Klein:

127 128 129 130

Regarding this Deutscher Ausschuss für technisches Schulwesen, see Section 8.1.1. See REICHSCHULKONFERENZ 1920, p. 706. [UBG] Cod. MS. F. Klein 5D, fol. 10v. In his dissertation – “Über die Geometrieen, in denen die Geraden die Kürzesten sind” [On the Geometries in which the Straight Lines Are the Shortest] (1901) – Georg Hamel had combined Hilbert’s axiomatic tendencies with Klein’s mechanical orientation. 131 [UBG] Cod. MS. F. Klein 5D, fol. 10v.

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9 The First World War and the Postwar Period I have recently been busy carrying out the resolutions of the Bad Nauheim meeting. Regarding the Federation suggested by [Georg] Hamel, I believe that I will be able to bring the matter to a satisfactory conclusion. Given that the opinion was unanimously expressed in Bad Nauheim that this federation would neither be a mathematical association nor that it would disturb the independent work of existing associations, the “German Federation of TechnicalScientific Associations” can serve as a model. Founded in 1916, the latter […] includes 13 large associations and it pursues similar purposes to those which Hamel had in mind. In this sense, I will prepare a draft for the organization and circulate it soon.132

The new Federation of Mathematical Societies and Associations was formed on January 6–7, 1921 at a special meeting of the German Mathematical Society, which was held at Klein’s home in Göttingen. Hamel was named its chairman. Together, this Federation, the German Committee for Mathematical and NaturalScientific Instruction, and the Association for the Promotion of Mathematical and Natural-Scientific Instruction sent a joint resolution to the Ministry of Culture in Berlin on January 20, 1921. The items in this resolution had already been agreed upon at the 1920 conference in Bad Nauheim, and they followed Klein’s ideas: that mathematics (and sciences) should not be reduced in the school curriculum; that, for mathematical instruction, the basic ideas of the Meran reform should still remain significant, but a stronger emphasis should be placed on applications to technology and business; that teaching appointments for didactics would be welcome, but they should be instituted at all universities and Technische Hochschulen.133

In 1901, when secondary-school curricula were discussed under Klein’s leadership,134 four weekly hours of instruction were devoted to mathematics and two to the natural sciences (physics and chemistry) in the upper classes of Prussian humanistic Gymnasien (whose main task was to provide a thorough education in the classical languages). In the Realgymnasien, five weekly hours were devoted to both mathematics and the natural sciences; and in the Oberrealschulen, five hours were reserved for mathematics and six for the natural sciences. The primary objective of Realgymnasien and Oberrealschulen was to prepare students to study engineering and the natural sciences. The German defeat in the First World War gave rise to an anti-technology sentiment, and thus educational authorities wanted to reduce the number of instructional hours devoted to mathematics and the sciences at Realgymnasien and Oberrealschulen in favor of so-called cultural subjects. In addition to the three types of boys’ schools mentioned above, there were also secondary schools for girls and newly established Deutsche Oberschulen, where mathematics was only taught to a limited extent or on an optional basis. With Klein’s guidance, Lietzmann published an article about the state of the reform in which he encouraged secondary school teachers to become involved, as was the case in France, Italy, and the United States. Lietzmann concluded the 132 [UBG] Cod. MS. F. Klein 4A: 29, fol. 10 (Krazer to Klein, October 11, 1920). 133 K. Körner, “Die 86. Versammlung der GDNÄ in Bad Nauheim vom 19.–25. September 1920,” Zeitschrift für mathematischen und naturwiss. Unterricht 52 (1921), p. 83. 134 [UBG] Cod. MS. F. Klein 31, fol. 1.

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article with these words: “It stands to hope that, in this matter, the creation of the Federation of Mathematical Societies will cause things to change.”135 The federation was officially constituted on September 23, 1922 at the meeting of the Society of German Natural Scientists and Physicians in Leipzig. Despite Klein’s warnings, however, its working committee consisted entirely of mathematicians from Berlin.136 An advisory council was formed to represent the individual societies and associations; in 1922, the representatives from Göttingen were Felix Klein, Richard Courant, Walther Lietzmann, and Emil Wiechert.137 Even though Klein could hardly leave his house by this time, he remained engaged. He recognized that the danger of reduced mathematical instruction at secondary schools had not been averted. When, in March of 1924, a memorandum was presented by the then Prussian Minister of Culture Otto Boelitz, who was a member of the German People’s Party (Deutsche Volkspartei),138 on the “The Reorganization of Prussian Secondary Education,” Klein wrote the following to Theodor Valentiner, who was then the Kurator of the University of Göttingen: “I am especially eager to discuss with you the reforms of secondary education that Boelitz’s Ministry has considered and about which a potentially decisive conference will be held in Berlin tomorrow.”139 In Boelitz’s memorandum, the opinion that mathematical and scientific instruction could be reduced at secondary schools was justified with the authority of Georg Kerschensteiner, who as a reform pedagogue and honorary professor in Munich had taken part in the 1920 School Conference in Berlin. Kerschensteiner, who had once been Klein’s student, had completed his doctorate in 1883 under Alexander Brill’s supervision at the University of Munich. He now let Klein know that he had by no means endorsed such a reduction.140 Equipped with this new information, Klein prompted, in June of 1924, the Mathematical and NaturalScientific Faculty of the University of Göttingen to formulate a resolution in opposition to the key points of Boelitz’s memorandum: its biased emphasis on socalled cultural subjects such as German, religion, history, and geography; its restriction of mathematics and the natural sciences; its plan to eliminate Realgymnasien; and its general opinion that “the economic and technical age lies behind us.”141

135 Walther Lietzmann, “Die Mathematik in der Schulreform,” Jahresbericht der DMV 30 (1921) Abt. 1, pp. 59–68. 136 In 1933, this working committee (led by Georg Hamel and Ludwig Bieberbach) immediately expelled its Jewish members. See MEHRTENS 1989 [1985], and 1987; TOBIES 1993b. 137 For a full list of members on this council, see Jahresbericht der DMV 32 (1923) Abt. 2, p. 14. 138 Boelitz held this office from November 17, 1921 to January 6, 1925. The national liberal German People’s Party, which was created in 1918 under the leadership of Gustav Stresemann, occasionally formed coalitions with the Social Democratic Party. 139 [UAG] Kuratorial-Akten, 4 Vb/216 (Klein to Theodor Valentiner, May 20, 1924). 140 See KLEIN 2016 [31924], p. 300. 141 [UAG] Math.-Nat. Generalia: No. 25; 21. Regarding the planned changes, see Jahresbericht der DMV 33 (1925) Abt. 2, pp. 61–67 (p. 62 contains a commentary by Felix Klein).

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This resolution from Göttingen – along with protests from other associations142 – had an influential effect on new Richtlinien für die Lehrpläne der höheren Schulen Preußens [Guidelines for the Curricula of Prussia’s Secondary Schools], which was edited by the ministerial official Hans Richert and published in 1925.143 The “Richert Reform,” which introduced new organizational measures and a more “civic” orientation to secondary education, ultimately paved the way for differential and integral calculus to become a binding feature of the Prussian mathematics curriculum at the Realanstalten. In the third edition of Klein’s Elementary Mathematics from a Higher Standpoint (vol. 2), Klein and Seyfarth commented in 1925 that these guidelines now corresponded to the revised curriculum proposed in Meran by the Breslau Education Commission (see Section 8.3.4.1).144 In July of 1924, after a ten-year hiatus, a continuing education course for teachers was offered again in Göttingen (with 113 participants), introduced by Carl Metzner, an official at the Prussian Ministry of Culture – who had passed his teaching examinations in Göttingen in 1902 (mathematics, physics, geography).145 In 1925, at the next course of this type after Klein’s death, Metzner gave a commemorative speech in his honor in which he promised to continue to forge ahead with Klein’s vision and, “in Klein’s spirit, to convey to the youth and keep alive the cultural significance of mathematics and its applications.”146 9.4 SUPPORT FOR RESEARCH After Germany’s defeat in the First World War, the slogan “science as a substitute for power” (Wissenschaft als Machtersatz) came to define the development of the scientific policy and scientific organization in the Weimar Republic. Discourses about Germany’s weaknesses in other areas served to mobilize the resources of science, which had itself entered a state of emergency on account of the war and the subsequent period of inflation. Industrialists, government officials, and scientists (each with their own set of goals) were all interested in new ways of organizing research funding. Felix Klein played an active role in these activities, and he directed funding to mathematical projects.

142 Oppositional opinions were collected and published in a book: Das höhere Schulwesen: Stimmen gegen die Neuordnung des preußischen höheren Schulwesens, edited by the Deutscher Verband technisch-wissenschaftlicher Vereine (Berlin: VDI Verlag, 1924). 143 Charles A. Noble, an American mathematician who had studied under Klein and Hilbert (see Section 7.3) and later became a professor at the University of California (Berkeley), visited Germany in 1926 and wrote an analysis of the state of mathematical education there. See his article “The Teaching of Mathematics in German Secondary Schools and the Training of Teachers for These Schools,” The American Mathematical Monthly 34/6 (1927), pp. 286–93. 144 See KLEIN 2016 [31925], p. 302. 145 [BBF]; [UAG] Math.-Nat. Fak 25 (on the continuing education course in 1924). 146 Regarding Metzner’s speech, see “Zum Gedächtnis von Felix Klein,” Deutsches PhilologenBlatt 33 (1925), pp. 492–93.

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9.4.1 The Emergency Association of German Science The Emergency Association of German Science (Notgemeinschaft der Deutschen Wissenschaft) was created in Berlin on October 30, 1920 (since 1929, it has been known as the German Research Foundation – Deutsche Forschungsgemeinschaft). Friedrich Schmidt-Ott, the repeatly mentioned official at the Prussian Ministry of Culture, became the president of this “self-governing umbrella organization,” the purpose of which was to promote and regulate humanistic, scientific, technical, and medical research.147 Its first vice president (by rank) was the mathematician Walther Dyck,148 and its second vice president was the chemist Fritz Haber149. The organizations for which the Emergency Association was responsible included the universities and Technische Hochschulen affiliated with the Association of German Universities (Verband Deutscher Hochschulen, established in 1919), the academies of science, the Kaiser Wilhelm Society, the Society of German Natural Scientists and Physicians (including its affiliated professional societies), and the Federation of Technical-Scientific Associations (est. 1916). Funding came primarily from the state budget and, to a lesser extent, from an (industrial) donors’ organization. A central committee, which was chaired by Adolf von Harnack, processed applications from expert committees and directed them to special committees (committees devoted to publishing, research stipendiums, apparatuses and materials, research travel, etc.). On November 23, 1920, the presidium and the central committee of the Emergency Association resolved to allow, at first provisionally for one year, the academies of science and the Association of German Universities to determine the organization of the expert committees. Schmidt-Ott, who visited Klein in Göttingen at the end of June in 1920,150 informed him officially on January 10, 1921: The composition of the expert committee for mathematics, astronomy, and geodesy was put in the hands of the Society of Sciences [Academy] in Göttingen. The latter elected you to serve as the chairman of this expert committee. In addition to yourself, the committee will consist of Privy Councilor Professor A. Krazer (Karlsruhe), Professor I. Schur (Berlin), Privy Councilor Professor Dr. Louis Krüger (Potsdam), and Professor Dr. Bauschinger (Leipzig).151

Schmidt-Ott expressed his hope that “you are prepared to accept the position offered to you in the overall interest of German science.” Klein replied that he had 147 On the establishment of the Emergency Association and on the term “umbrella organization” (Dachverband), which did not imply a democratic institution but rather a union against “the specter of Bolshevism” and a lobby for promoting the interests of science against the new SPD government, see Jochen KIRCHHOFF 2003 (the quotation here is from p. 77). 148 On Walther Dyck’s activity on the Emergency Association, see HASHAGEN 2003, pp. 619–51. 149 In 1918, Fritz Haber was a Nobel laureate in chemistry for his invention of the catalytic synthesis of ammonia from nitrogen and hydrogen (the industrial procedure is known as the Haber-Bosch process), but he had also developed poisonous gases and masterminded their use as weapons of mass destruction in the First World War. 150 [StA Berlin] Rep. 92 Schmidt-Ott, No. 2 (a letter from Klein dated July 2, 1920). 151 This and the following quotation are from [UBG] Cod. MS. F. Klein 4A, fol. 61–61v.

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not only accepted this role but that he had also already conducted (in Göttingen) the first committee meeting. For future meetings in Berlin, Klein requested that Issai Schur should act as his proxy. On January 5, 1921, Fritz Haber already had written that Klein could send a representative to attend the meetings in Berlin, and he added the following bit of praise: “The weight of your personality remains, and the sum of judgment, experience, and personal connections that you embody in your chairmanship is extremely useful for the matters at hand.”152 A letter by Richard von Mises dated November 6, 1920 indicates that he thought Klein would include him on the expert committee: I returned from Göttingen on Monday evening with good success. I was very happy to find Klein still in very good shape; he is quite the same, with an amazing mental freshness and activity, although it seems that he is paralyzed in his legs. […] Since I have now become the representative of applied mathematics in the Emergency Association of German Science […], there is new work for her [my secretary]. Unfortunately, this “emergency” will also lead to a meeting in Göttingen during the Christmas holidays.153

In addition to mathematics, the expert committee chaired by Klein also represented astronomy and geodesy. The members chosen to serve on it tended to be older and established professors who were less keen to promote their own personal agendas. Carl Runge, a professor of applied mathematics, was similarly chosen to lead the expert committee for physics in order to balance out competing interests. In December of 1920, Richard von Mises, who was more than twenty years younger than Klein and Runge, had submitted an application to the Emergency Association to receive funding to purchase instruments for his institute,154 but his application was rejected because the Emergency Association did not allocate funds for the purpose of a given institute’s teaching capabilities.155 Klein’s correspondence with Adolf Krazer documents their coordinated approach to supporting certain projects. On November 9, 1920, Klein wrote to him: It is still unclear what authority and, especially, what financial means each committee of the Emergency Association has. Nevertheless, I am happy to continue our provisional discussions. […] I have intentionally excluded all individual journals, including Crelle’s Journal and even the Archiv der Mathematik und Physik. Otherwise, we would experience a competition not only among editorial boards but also among publishers, and we would waste the funding at hand or sustain certain publications that ought to die off (under today’s circumstances, the number of journals must be reduced). – An exception is the Jahresbericht [der DMV], if we can succeed in providing it with the character of a generally valid bulletin of current events. In this respect, I am open to different publishers and forms of publication. I certainly have nothing against Teubner in itself. But without corporate profit. […] My suggestions are made 152 Ibid., fol. 52 (Fritz Haber to Klein). 153 A letter from Richard von Mises to his mother dated November 6, 1920. The English translation here is slightly modified from SIEGMUND-SCHULTZE 2020, p. 11. 154 Thanks to funding from the Göttingen Association and the Carl Zeiss Foundation, instruments of the kind von Mises wished for had long been available at the University of Göttingen and the University of Jena; they were used especially for teaching purposes. For von Mises’s efforts to equip his institute in Berlin accordingly, see SIEGMUND-SCHULTZE 2021. 155 See the analysis in TOBIES 1981b.

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with great caution, and with this same caution I approach the reorganization of the German Mathematical Society and the Mathematical Federation.156

Klein and Krazer agreed early on that the ENCYKLOPÄDIE, the edition of Gauß’s works, the review journal Jahrbuch über die Fortschritte der Mathematik, and the Jahresbericht der Deutschen Mathematiker-Vereinigung should receive financial support.157 These projects were deemed to be of primary importance when Expert Committee 5 (for mathematics, astronomy, geodesy), which Klein chaired, met on January 7–8, 1921 in Göttingen. Carl Runge, as the chairman of Expert Committee 6 (for physics, astrophysics, geophysics, and meteorology), and Richard Courant (as a secretary [Schriftführer]) had also been invited to this meeting. The majority of funding for mathematics came from the publications committee. By March 31, 1922, “Klein’s” committee received 792,755 Mark to support publications and 174,800 Mark in support of individual research projects. This placed it in seventh place among the twenty expert committees (behind physics, theology, chemistry, biology, mineralogy, and classical philology).158 Klein thought beyond disciplinary boundaries; together with Otto Blumenthal and in coordination with Dutch scientists, for instance, he helped to organize means for the Emergency Association’s library committee to acquire missing specialist literature (e.g. missing journal volumes from the war period) from abroad. At the peak of hyperinflation in the Weimar Republic, the Emergency Association also provided funding for the edition of Klein’s collected works. In November of 1922, 50,000 Mark were granted to support the second volume and 100,000 Mark were allocated for the third.159 An additional 300,000 Mark of funding were directed to the project in January of 1923.160 By this point, however, Klein had already stepped down from his position as the chair of the expert committee. Klein had withdrawn his candidacy when the committees were to be newly elected in March 1922 for the following four years. His colleagues knew that, to a great extent, the successes were due to his personal engagement and diplomatic skills. Bauschinger urgently requested Klein to maintain his role: “There is no one there who could replace you to any extent.”161 Issai Schur wrote to Klein: In any case, this success is new evidence for how extraordinarily difficult it would be to replace you as the chairman of the expert committee. I still have not given up hope that Mr. Courant will inform me that you have set aside your intention to resign and that you would not object to being reelected.162

156 157 158 159 160

[UBG] Cod. MS. F. Klein 4A, fol. 18 (a draft of Klein’s letter to Krazer, November 9, 1920). Ibid., 4A: 29, fol. 7 (a letter from Krazer to Klein dated November 7, 1920). See Bericht der Notgemeinschaft über ihre Tätigkeit 1921/22 (Berlin, 1922), p. 12. [UBG] Cod. MS. F. Klein 4A, fol. 96 (Schmidt-Ott to Klein, March 20, 1922). Ibid., fols. 315, 323 (letters from Schmidt-Ott to Klein, Nov. 11, 1922 and January 26, 1923). On the funding provided for other mathematical works and projects, see TOBIES 1981b. 161 [UBG] Cod. MS. F. Klein 4A, fol. 214 (Bauschinger to Klein, September 4, 1921). 162 Ibid., fols. 219–219v (Issai Schur to Klein, September 16, 1921).

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On September 11, 1921, Klein had let Schmidt-Ott know that, going ahead, he would have to concentrate all of his remaining energy on editing his collected works.163 He also shared with him the names of the elected members of the new expert committee. They were the same as before: Krüger (geodesy), Bauschinger (astronomy), two members for “pure” mathematics (Krazer and Issai Schur), and additionally Carl Runge for applied mathematics (he was no longer the chairman of the expert committee for physics, where – as Klein wrote – “the names Einstein and Stark are the focus of two enemy camps.”)164 The election of the committee members for “pure” mathematics had already taken place on September 20, 1921, at the annual meeting of the German Mathematical Society in Jena.165 Ludwig Bieberbach, however, managed to replace Issai Schur on the committee.166 Klein continued to remain available as an advisor. On July 19, 1922, SchmidtOtt asked for Klein’s opinion about how to concentrate the distribution of funding more strongly “on research devoted to large problems and on the production of summary reports on broader areas.” In his response, which is dated July 27, 1922, Klein told him that the promotion of “general undertakings” (the ENCYKLOPÄDIE, the edition of Gauß’s works, and the review journal Jahrbuch über die Fortschritte der Mathematik, which was edited by Leon Lichtenstein at the time),167 still seemed to be of the greatest importance, as he had already stressed in his report from January. Klein thought that other new scientific plans would be “hampered by the progressive deterioration of general conditions.”168 Meanwhile, Klein managed all of this from his home. From here, he also saw to it that the Göttingen Association, which he had founded in 1898, was handed over in an orderly manner to a new organization, though he was somewhat disappointed by this transition. 9.4.2 The Gauss-Weber / Helmholtz Society Under the Treaty of Versailles, which was signed on June 28, 1919, the German chemical industry was stripped of all its patents and trademarks. Industrialists in this branch therefore developed a keen interest in reviving innovation and establishing organizations for the promotion of chemistry: the Emil Fischer Society for the Promotion of Chemical Research (est. June 15, 1920), the Adolf Baeyer So163 Ibid., fols. 225, 257 (Klein’s drafts of letters to Schmidt-Ott, Sept. 23, and Nov. 7, 1921). 164 Ibid. 5E, fol. 5 (Klein to Schmidt-Ott, Jan. 20, 1921). The reference here is to the outspoken anti-Semitism of the physicist and Nobel-Prize winner Johannes Stark, who for this reason had been rejected as a possible candidate to succeed Eduard Riecke (d. 1915) in Göttingen. 165 See in detail Jahresbericht der DMV 30 (1921) Abt. 2, p. 104 (TOBIES 2019, p. 474). 166 [UBG] Cod. MS. F. Klein 4A: 254 (Issai Schur to Klein, November 14, 1921). 167 On the history of this review journal that time, see SIEGMUND-SCHULTZE 1993, pp. 40–44. 168 [UBG] Cod. MS. F. Klein 4A, fols. 308–09v (a letter from Schmidt-Ott to Klein dated July 19, 1922; a draft of Klein’s reply dated July 27, 1922). Regarding the further development of support for mathematical projects, see TOBIES 1981b.

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ciety for the Promotion of Chemical Literature (est. June 16, 1920), the Justus Liebig Society for the Promotion of Chemical Education (est. 1920). Klein kept an eye on these developments. He joined the new Society for Technical Physics (1919), offered advice about its statutes,169 and attempted to acquire further financial support for mathematical, physical, and technical research in Göttingen. Henry Theodore von Böttinger, who had been highly active on behalf of the University of Göttingen (see Section 8.1.1), died on June 9, 1920, and Klein announced his passing to the members of the Göttingen Association for the Promotion of Applied Physics and Mathematics on June 21st. The following day, Klein wrote to the Krupp manager Emil Ehrensberger in order to secure funding for the construction of his envisioned Mathematical Institute. He concluded his letter with the words: “Let’s not despair but rather develop new initiatives.”170 Before his death, Böttinger had suggested that Carl Duisberg should replace him on the board of the Göttingen Association. Duisberg had been a general director and board member of the Friedrich Bayer dyestuff factories since 1912. He directed the Liebig Society mentioned above, managed the establishment and the statutes of the Emergency Association’s donors’ organization (est. December 14, 1920), and developed numerous other initiatives, so that, as far as Göttingen was concerned, he saw himself as no more than a “liquidator.”171 The Göttingen Association’s decision to operate on the basis “of the free collaboration of its members” and without any formal statutes now had an unfavorable effect. The industrial members were now less willing to donate. Personnel changes in Göttingen meant that professorships in mathematics and physics remained unoccupied for some time. Duisberg thus sought to incorporate the association into a new and larger society for the promotion of physical and technical research, one that would support not only such research in Göttingen but throughout Germany at large. Duisberg appointed Dr.-Ing. Albert Vögler, the general director of the DeutschLuxemburgische Bergwerks- und Hütten A.G. (a mining company headquartered in Dortmund), to serve as the chairman of this new organization, which was called the Helmholtz Society for the Promotion of Physical-Technical Research (est. October 28, 1920).172 Vögler, who was conservative and nationalistic, had supported an aggressive annexation policy during the First World War, and later he made large donations to the National Socialist German Workers’ Party.173 Klein noted: “All sorts of scruples concerning the new society.” He felt that it lacked a sufficient number of “outstanding scholars” on its first administrative board; he felt that it lacked the ability to organize research activity. Only the interest accrued by the Helmholtz Society was allowed to be donated, whereas the

169 Ibid. 5B, fols. 57, 64–71, 124–28 (Klein’s correspondence with Georg Gehlhoff, the founder of this society, 1919). 170 Ibid. 5C, fols. 42–43v (Klein’s letters, June 21 and 22, 1920). 171 Ibid. 5D, fols. 36–39 (a letter from Duisberg to Klein dated November 13, 1920). 172 Ibid. 4B, fols. 1–3. 173 See KOHL 2002.

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Göttingen Association had thrived off the funding from entry fees and annual dues (5000 Mark to join, plus a minimum yearly contribution of 500 Mark). Klein dismissed the new society as a “collection box” of capital.174 Klein was also displeased that the new society was named after Helmholtz, who, in 1893, had argued against the creation of university institutes for technical physics. Klein’s suggestion that it should be called the Gauß-Weber Society – in order to underscore the tight bond between mathematics and physics – had been rejected by Duisberg with the comment that Helmholtz’s name was more widely known. The new society established its headquarters in Munich, even though Duisberg had written to Klein on July 20, 1920 that it would be in Göttingen. Mathematics did not appear in the new name, even though Duisberg had spoken of mathematics.175 Education was likewise dropped as a funding sector.176 In the interest of the Göttingen Association, Klein continued to play a role in hiring new physics professors: Max Reich to replace the late Hermann Theodor Simon, Max Born to replace Peter Debye, James Franck to replace Woldemar Voigt. When Born informed Klein on July 15, 1920 that he and Franck were interested in actively maintaining the connection between science and industry, this was clearly an effort to gain Klein’s support.177 This special connection in Göttingen, however, was nearing its end. Born had written: I am very pleased that the Göttingen Faculty wants to recommend Prof. Franck to the Ministry. It is indeed a considerable responsibility that I have taken upon myself by declaring, in such definite terms, that someone is the right person for physics in Göttingen; I believe, however, that I am quite certain about the matter, because Franck is indeed suited to the grand style of education. […] Franck is certainly well aware of the Göttingen Association […] He will certainly be, just as I am, willing to offer you his full assistance to achieve your great goal of harnessing science to stimulate industry and, conversely, interesting industry in promoting science. Your new foundation, the Gauß-Weber Society, is in fact a matter of the utmost importance; it is remarkable that you were able to bring it about despite the difficulties that your poor health has caused you. For I myself have begun to campaign in a similar way. For years, my idea has been to create a “Calculation Institute for Theoretical and Technical Physics.” […] If I can be of any use to the goals of your undertaking, I stand with heart and soul at your disposal.178

174 [UBG] Cod. MS. F. Klein 4B, fols., 5, 7. Duisberg informed Klein on January 22, 1921 that industry would prefer to donate funding for specific purposes. For the Helmholtz Society, 35 million Mark had meanwhile been raised, whereas the industrial donations to the Emergency Society (for “science in general”) were very low. Klein felt that the field of fluid dynamics was in need of more support, and he wanted to secure 1 million Mark of funding from the Helmholtz Society to keep Ludwig Prandtl in Göttingen. Duisberg rejected this proposal. 175 Ibid. 5C; 5D (Duisberg’s letters to Klein; drafts of Klein’s letters to Duisberg and Vögler). 176 Ibid. 5E, fols. 117–18, 131. The mayor of Göttingen wrote to Klein (Febr. 17, 1922) and asked him to apply for funding from the Helmholtz Society to support the Göttingen School of Mechanics (initiated by the Göttingen Association). The Helmholtz Society rejected this proposal on April 27, 1922 with the justification that its sole purpose was to support research. 177 See SCHIRRMACHER 2019, pp. 75–78. 178 [UBG] Cod. MS. F. Klein 5C, fols. 68–69 (Max Born to Klein, July 15, 1920).

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With Duisberg’s approval, Klein was able to ensure that the Leverkusen dyestuffs factory would donate (via the Göttingen Association) funding of 15,000 Mark to pay for the printing costs of the first volume of his collected works: In expressing my especially great joy about the fact that it is possible for us to repay a small portion of our debt of gratitude to you, the founder and guiding spirit [spiritus rector] of the Göttingen Association, I remain, with the wish that things might go better for you in the year that has just begun than they went for all of us last year in our downtrodden German fatherland, admiringly and sincerely yours, C. Duisberg.179

For the transition of the Göttingen Association, Klein ultimately found a twopronged approach, which he explained as follows to the physicist Wilhelm Westphal, who was also an advisor to the Prussian Ministry of Culture: Beginning on October 1, 1921, the Göttingen Association was in fact absorbed by the Helmholtz Society, whereby a small portion of the sums allotted by the industrialists to the Helmholtz Society – it may be 450,000 Mark – was expressly reserved for the institutes created by the Göttingen Association. Prof. Prandtl became a board member of the Helmholtz Society, while I have been named an honorary member. On the other hand, in addition to the funding that the Göttingen Association had accumulated for the construction of the Mathematical Institute, which totals approximately 350,000 Mark, we also have a sum of 700,000 Mark, which the donors’ organization of the Emergency Association expressly made available to the institutes of the Göttingen Association in order to create a special “Foundation for Mathematics and Physics, Particularly Their Applications” for the University Association here. Within the University Association, this foundation is under the control of a particular board of trustees, which includes Duisberg180 from the industrial side and Courant, [Carl] Runge, and myself from the university side. Courant has taken on the managing role. In general, as is the case with Helmholtz Society, only the interest earned from the capital of this foundation is to be used; its future is thus hard-pressed by the progressive devaluation of the Mark. My younger colleagues will thus have to make an effort to create, each according to his own opportunities, a broad basis of support and wide-ranging relations within this new foundation. I myself feel as though I am too old to adapt to these new conditions.181

Klein had already expressed the idea of creating a foundation within Göttingen’s University Association (Universitätsbund) in a letter to Schmidt-Ott on August 3, 1920. From this correspondence, it is also clear that he hoped to preserve the 300,000 Mark that the Ministry of Finance had once promised him for the construction of the Mathematical Institute.182 On July 3, 1922, Klein (as the former co-chair of the Göttingen Association) and Richard Courant (as the director of the new foundation) sent a final protocol of the Göttingen Association as well as the statutes of the “Foundation for Mathematics, Physics, and Their Applications (formerly the Göttingen Association)” to all the members. They informed them that “the goals previously pursued by the 179 Ibid. 8: 533/A and A/1, fols. 91 and 92 (Duisberg to Klein, January 6, 1921). 180 Alongside Dr.-Ing. Carl Still from Recklinghausen, and Dr.-Ing. Max Walter, the director of Norddeutscher Lloyd in Bremen (see NEUBAUR 1907, pp. 612–13). 181 [UBG] Cod. MS. F. Klein 5E, fol. 144 (Klein’s draft of a letter to Westphal, June 8, 1922). 182 Ibid. 5D, fol. 127 (a draft of a letter from Klein to Schmidt-Ott dated August 3, 1920).

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Göttingen Association would be carried forward in light of further developing conditions and, if necessary, in light of the special wishes of future donors.” The foundation’s assets were managed by Göttingen’s University Association.183 On the basis of this support (and with Rockefeller funding provided by the International Education Board), Courant saw to it that the opening of the new Mathematical Institute on Bunsenstraße would be celebrated in 1929.184 9.5 END OF LIFE Heinrich Behnke, who was a student in Göttingen at the time, related a story about how Klein, in the winter of 1918/19, planned ahead for his death: About Klein, [Erich] Hecke later told the following macabre story. One evening, Klein summoned one of his closest collaborators and explained to him that he would die overnight. He was to come back early in the morning, collect this report, then go to the registry office, then to the publishing house, etc. etc. Assigned with these tasks, the man came as wished the next morning at the appointed hour. But Klein was not dead. He was extremely annoyed, however, that fate had not followed his program.185

This not only indicates the great extent to which Klein planned everything programmatically. It also reveals that Klein’s illness was already quite advanced. To Schmidt-Ott, Klein reported in 1921 that he was suffering from “muscular atrophy, which the doctors are unable to stop,” so that he was only able to walk around in his room “with difficulty and with a cane.”186 Klein informed Arnold Sommerfeld about his “extensive muscular rheumatism.”187 As early as December 6, 1919, after learning of the death of Adolf Hurwitz, Klein had written the following to the historian Alfred Stern in Zurich, the son of M.A. Stern: Dear Friend! This card lay addressed but still unwritten on my desk for several days. There was far too much to do. Many thanks for your news about Hurwitz’s death and for the other information! When one thinks about the misery of our conditions, one may become envious of those who have passed away before us. Whether we like it or not, we still have to muddle through. My wife and I myself now number among those whose physical abilities have greatly suffered. It bothers me the most that I can no longer walk properly, but I have also observed that I work much more slowly […] than before. That said, I am still active in general university affairs and scientifically – in the latter case, for instance, with the goal of preparing an edition of my old works. Given the current circumstances, however, it is doubtful whether I will find a publisher for it. Best regards from your old F. Klein188

183 184 185 186 187 188

[UBG] Cod. MS. F. Klein 5D, fols. 2–6. See SIEGMUND-SCHULTZE 2001, pp. 144–56, 277–78. BEHNKE 1978, p. 35. [UBG] Cod. MS. F. Klein 5E, fol. 5v (Klein to Schmidt-Ott, January 20, 1921). [UBG] Cod. MS. Teubner 49 (a postcard from Klein to Sommerfeld, December 12, 1922). [Deutsches Museum] HS 1968-4 (a postcard from Klein to Alfred Stern, December 6, 1919).

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Klein, of course, was able to find a publisher, and we also know that he remained active until his last breath. When his seventy-fifth birthday came around on April 25, 1924, his former students and colleagues insisted on honoring him, even though his health prevented him from celebrating.189 In 1924, the Carl Zeiss Foundation in Jena, which had donated 100,000 Mark to establish an Ernst Abbe Memorial Prize, chose Klein to be its first recipient (for his mathematical works). The committee members who had made this decision were Paul Koebe, Robert Fricke, and Hermann Weyl.190 In the same year, the German Mathematical Society, then chaired by Otto Blumenthal, named Klein its first honorary member. In 1924, too, Richard von Mises published biographical essays about Klein in the Zeitschrift für angewandte Mathematik und Mechanik and in the Deutsche Allgemeine Zeitung. There he described Klein’s multifaceted activity and wrote in awe about his “extraordinary diligence and dutifulness.”191 In 1921, when the German Association of Engineers decided to publish the Zeitschrift für angewandte Mathematik und Mechanik, Klein expressed his “special satisfaction and joy that engineers and mathematicians had found common ground.”192 Von Mises, who, along with Ludwig Prandtl and Hans Reissner, was a board member of the Society for Applied Mathematics and Mechanics (Gesellschaft für angewandte Mathematik und Mechanik), which had been founded in 1922, also worked to ensure that Klein was named an honorary member of this organization.193 Moreover, as the interim vice chancellor (hoc tempore procancellarius) of the University of Berlin, Richard von Mises arranged for Klein to receive a honorary degree (“Doctoris rerum politicarum dignitatem et ornamenta”) in 1924 (see Fig. 46). Representatives of applied mathematics, physics, and technology – fields which Klein had long supported – were able to set aside nationalistic grudges somewhat earlier than the governing body of the International Congress of Mathematicians. Thus, in 1924 and at the initiation of Theodore von Kármán, the first International Congress for Applied Mechanics was held in Delft in the Netherlands.194 The executive committee of the Congress sent the following telegram to Klein: The International Congress for Applied Mechanics, which met from April 22nd to the 28th in Delft, and which includes members from America, Australia, Belgium, Bulgaria, Czechoslovakia, Egypt, England, France, Germany, Holland, Italy, Norway, Poland, Russia, Scotland, Spain, Switzerland, and Turkey, sends on the occasion of your 75th birthday its best wishes in

189 [MPI Archiv] 1078, fol. 4 (a letter from Prandtl to Richard von Mises dated February 21, 1924). I am indebted to Reinhard Siegmund-Schultze for bringing this letter to my attention. 190 See Jahresbericht DMV 30 (1921), p. 74, Math. Ann. 94 (1925), p. 176, and TOBIES 2020b. 191 MISES 1924. 192 [UBG] Cod. MS. F. Klein 5D, fol. 106 (a letter from Klein to Diedrich Meyer, the director of the German Association of Engineers (VDI), written in February of 1921), published in Zeitschrift des VDI 65 (1921), p. 332, reprinted in ZAMM 5 (1925), pp. 358–59; English trans. in SIEGMUND-SCHULTZE 2020a (Appendix B), p. 26. 193 [MPI Archiv] 2378, fol. 60 (a letter from von Mises to Prandtl, December 12, 1923). 194 See BATTIMELLI 2016; KÁRMÁN 1967.

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9 The First World War and the Postwar Period grateful recognition of the inestimable services that you have rendered both to the science of mathematics and to mechanics!195

On his way back from Delft, von Mises delivered to Klein the certificate of his honorary membership in the Society for Applied Mathematics and Mechanics and, as well, the certificate of his honorary doctorate from Berlin. He reported: On my way back from Delft, I spent a few hours in Göttingen to visit Klein. Despite his 75 years and his severe illness, which leaves him entirely immobile, he sits upright at his desk and painstakingly works. Everything that he needs is within reach, and the neighboring room is not occupied by a nurse but rather by his assistant, to whom he dictates letters and manuscripts, etc. There has never been such a phenomenon of will.196

N. Wiener described the situation a few months before Klein’s death. He saw him […] in his great study, a pleasant, high, airy room lined with bookcases and with a large table in the middle covered with an orderly disorder of books and open periodicals. The great man sat in an armchair behind the table, with a rug about his knees. He was bearded, had a fine, chiseled face, and carried about him an aura of the wisdom of the ages.197

Lietzmann, who was there more often, provided further details about the rooms: The pictures on the wall (engravings by Raphael), […] the desk, the round conference table in front of the sofa, the bookcases here as well as in the neighboring room and in the bedroom, the locations where the recent publications were placed, and so on. When Klein was no longer able to leave his bed, a place for it, too, was found in his study.198

On June 23, 1925, David Hilbert composed a brief eulogy after Klein had died the evening before (see Appendix 12). As his last will, Klein had insisted that his funeral should not involve any official pageantry and speeches.199 The funeral took place on Thursday, June 26th at 3:30 at the chapel of Göttingen’s central cemetery,200 where the only speaker besides the pastor was Carl Runge. The Göttingen Mathematical Society and the directors of the Mathematical Institute organized a public memorial celebration, which was held on July 31, 1925 at 11 in the morning in the university auditorium on Wilhelmsplatz. Hilbert signed the invitation, and Richard Courant gave the eulogy.201 Klein’s programmatic planning extended beyond his death. After his death, Klein’s extensive library was acquired by the Hebrew University of Jerusalem, whose construction had begun on July 24, 1918. In December of 1923, Paul Koebe had attempted, with the help of the Carl Zeiss Foundation, to acquire Klein’s library for Jena. Klein, however, had replied that now was not the time to 195 C.B. Biezeno and J.M. Burgers, eds., Proceedings of the First International Congress for Applied Mechanics in Delft, 1924 (Delft: J. Waltman, Jr., 1925), p. xv (original in German). 196 A letter from Richard von Mises to his mother, May 7, 1924. I would like to thank Reinhard Siegmund-Schultze for bringing this source to my attention. 197 Norbert WIENER 2017 [1956], p. 292. 198 Walther LIETZMANN 1960, p. 54. 199 [MPI Archiv] 1078, fol. 33 (Prandtl to Richard von Mises, June 25, 1925). 200 [Archiv TU München] A letter from Carl Runge to W. Dyck dated June 24, 1925. 201 See COURANT 1925.

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sell the library and that he had already made an agreement with his relatives and with Richard Courant a year and a half earlier.202 For his gravestone (see Fig. 45), Klein himself had chosen the Latin words sincere et constanter (honestly and steadily), thus borrowing the motto of the Prussian Order of the Red Eagle.203 On July 25, 1925, Klein’s family returned, as was obligatory, his orders of merit to the Prussian state: the Order of the Red Eagle (2nd Class with oak leaves, bestowed at the 150th anniversary of the Göttingen Society of Sciences), the Order of the Crown (2nd Class), and the stars belonging to both orders. His Bavarian Maximilian Order, which he had received in 1898 (see Section 4.3.1), also had to be surrendered.204 Felix Klein’s son Otto, who then lived in Hanover, took care of affairs for his mother and ensured that she would receive her (one-time) entitled payment of 1,400 Mark from the university’s retirement fund for the widows and orphans of professors. This process dragged on until 1926, and it involved providing detailed information about the family’s financial conditions and taking into account the service time that Klein had accrued outside of Prussia (13 years and 182 days). The value of Anna Klein’s property was estimated to be 45,000 Mark. In the end, Felix Klein’s total annual income had been 13,690 Mark (11,550 Mark plus a cost-of-living allowance of 1,140 M and fringe benefits of 1,000 M). For Anna Klein, an annual widow’s allowance of 6,571.80 Mark was calculated and paid in monthly installments of 547.65 Mark.205 Anna Klein died two years after her husband, on October 18, 1927. On November 27, 1927, her sister Sophie Hegel, who had taken care of her to the end, wrote to Walther Dyck in Munich: She pined greatly for the end. This thought has repeatedly consoled me whenever I feel terribly lonely in the old, familiar rooms where I spent so many lovely years with the two dear people. I have much to thank them for, and my good sister also decided that I should remain in the house and have two rooms on the second floor; in addition, “the children” want to keep their father’s former study for themselves in order to have a place to stay in their parents’ house. That is also a very pleasant thought to me, and thus, in the future as well, the Klein home will always remain open to old friends.206

The rest of the family remained connected. Together with her niece Elisabeth Staiger (Felix Klein’s youngest daughter, then a teacher in Kiel), Sophie Hegel spent the Christmas of 1927 in Magdeburg at the home of her nephew Otto Klein.207 On October 12, 1927, Felix Klein’s daughter Elisabeth and his brother Alfred both attended the unveiling of a commemorative plaque on his birthplace in Düsseldorf. The Göttinger Zeitung reported on March 13, 1928 that the street 202 [UBG] Cod. MS. F. Klein 10: 509B (Koebe to Klein, December 9, 1923, with a draft of Klein’s reply). 203 See Georg Prange, “Felix Klein zum Gedächtnis,” Hannoverscher Kurier (August 2, 1925). 204 For Klein’s honors, see http://hans.sub.uni-goettingen.de/nachlaesse/Klein.pdf (pp. 94–98). 205 [UAG] Kur. 9038, fols. 7, 9–25. 206 [BStBibl] Dyckiania (a letter from Sophie Hegel to Dyck, November 27, 1927). 207 Ibid. (a letter from Elisabeth Staiger to Dyck dated December 28, 1927).

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formerly known as Lindenstraße had been renamed the Felix-Klein-Straße. In Düsseldorf and in Erlangen, where Klein first worked as a professor, streets were also named after him. In Munich, Constantin Carathéodory stated: “It is only by illuminating him from all angles that one can come to understand his significance.”208

Figure 41: Felix Klein, a drawing by Leonard Nelson [Hillebrand].

208 CARATHÉODORY 1925, p. 2.

10 CONCLUDING REMARKS I wish to quote Cantor, who once said: “The essence of science lies in its freedom,” that is, mathematics can deal with anything it desires, as long as it draws only correct conclusions from premises. While I theoretically accept Cantor’s sentence, I add a practical restriction, which seems to me essential, namely that everyone who has freedom also has responsibility. I do not want, therefore, to plead for an absolute arbitrariness in the construction of mathematical ideas, but I do want to recommend to everyone that they keep in mind the whole of science.1

Felix Klein extended Georg Cantor’s maxim, which in its original form pertained only to mathematics,2 to science as a whole. Klein’s sense of responsibility for mathematics and its broad range of application may very well have led to his inclusion, on June 7, 1923, into the Order Pour le Mérite für Wissenschaften und Künste along with Albert Einstein, the writer Gerhart Hauptmann, the sculptor Hugo Lederer, and the painter Max Liebermann.3 The Weimar Constitution of 1919 had banned all Orders, among them the military class of this Prussian Order of merit, the history of which dates back to 1667. The peace class of this Order, however, which had been introduced in 1842 at the suggestion of Alexander von Humboldt, became, since the 1920s, a self-supportive “loose association of eminent scholars and artists,” and it still exists today. Before Klein, the following mathematicians had been made members of the peace class (when it was still a distinction conferred by the Prussian state): Gauss (1842), Jacobi (1842), Cauchy (1849), Poncelet (1863), Weierstrass (1875), Hermite (1878), G.G. Stokes (1879), Carl Neumann (1897), Luigi Cremona (1902), and Ludwig Sylow (1904). Even though outstanding scholars were and continue to be responsible for advancing mathematical knowledge, such people do not act in isolation but rather within a certain framework with given conditions. In their remarks about the history of science in Plücker’s obituary (see Section 8.3.1), Clebsch and Klein made it clear that it is necessary to take these conditions and circumstances into account when evaluating and classifying the achievements of mathematicians. In what follows, I will revisit the thesis formulated in my introduction concerning the continuity of Klein’s activity, and I will take another look at the other questions that guided my research (10.1). In addition, I will also outline the extent to which Klein can be regarded as a pioneering figure (10.2). 1 2 3

KLEIN 2016 [31928], p. 170. See Georg Cantor, “Ueber unendliche, lineare Punktmannichfaltigkeiten, 5,” Math. Ann. 21 (1883), pp. 545–91, at p. 564: “The essence of mathematics lies precisely in its freedom.” This translation is from EWALD 1996, p. 896. See also SIEGMUND-SCHULTZE 2016b, p. 230. See Orden Pour le Mérite für Wissenschaften und Künste, Die Mitglieder des Ordens: Zweiter Band, 1882–1952 (Berlin: Gebr. Mann, 1978), p. 310; and FUHRMANN 1992.

© Springer Nature Switzerland AG 2021 R. Tobies, Felix Klein, Vita Mathematica 20, https://doi.org/10.1007/978-3-030-75785-4_10

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10.1 A SUMMARY OF FINDINGS The sources that I have analyzed confirm the thesis that Klein’s career path was characterized much more by continuity than discontinuity. Such continuity is evident in his “universal” view of mathematics and its applications, his creation of effective organizational conditions for mathematical activity, and his focus on mathematical instruction in schools. The way in which he managed his health problems also remained remarkably consistent. On the Continuity of Klein’s Field of Research Klein recognized early on that it would be insufficient to focus his research on geometry alone and that he would have to expand his interests to include other areas and applications. His wide-ranging perspective, which he acquired from Plücker and Clebsch, was evident from early on, for instance in his application to supplement the mathematics and physics collection of the Erlangen University Library in November of 1872 (see Appendix 2). After meeting the Scottish mathematical physicist Peter Guthrie Tait, moreover, Klein wrote the following to Sophus Lie on September 14, 1873: Of the mathematicians known outside of England, I have so far only seen Tait from Edinburgh. He espouses a research direction that seems to be overpowering in England and threatens to become the only approach in the near future – the direction that appreciates mathematical research only to the extent that it might lead to direct applications. […] Remarkably, Cayley is respected very little by this group. […] When I expressed a different opinion, Tait replied with the question: “What good are the 27 lines of the F3 [cubic surface]?” You will perhaps be even more irritated by such a view than I am. It made an impression on me in the sense that I would like to discuss some example of its applications later on in order to demonstrate to people that we really are much better at mathematics than they are – and this is aside from the fact that I am interested in physics as well as mathematics. Above all, however, I am astonished by this opposition that I am encountering here. In Germany, I am accustomed to a completely different type of opposition, namely that between geometricians and the function-theoretical school, which, whenever possible, attempts to deny that we are mathematicians at all. Here in England, we are regarded as the genuine mathematicians, but for that very reason we are also regarded as being superfluous.4

Klein’s impulse to balance out opposite views and to integrate new research areas into his arsenal of methods led him to function theory, both to the geometric and physical approach taken by Riemann and to the analytical methods used by Weierstrass and his students (see Section 5.5.2). At the same time, Klein delved more and more into applied fields – without setting aside his pure mathematical research. In his inaugural address at the University of Erlangen in 1872 (see Section 3.2), Klein admittedly expressed the 4

[Oslo] A letter from Klein to Lie dated September 14, 1873. Regarding the twenty-seven lines of the cubic surface, see Fig. 11 in Section 2.8.2, and Section 6.3.1.

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opinion that technical applications were outside the purview of academic mathematics, but his mind changed when he encountered engineers and scientists conducting research at the Technische Hochschule in Munich (see Section 4.3.1), where he was employed from 1875 to 1880. This is reflected in his 1880 inaugural address in Leipzig, in which he emphasized the “universal applicability” of mathematics (see 5.1). He declared that the applications of mathematics to technology were urgently needed at a time when such applications were still largely regarded as “dirty” (schmutzige) mathematics by the German mathematical community. As early as 1881 in Leipzig, he had formulated a research program for mechanics and mathematical physics (see 5.5). He was briefly distracted from this program by his intensive discussion with Poincaré (see 5.5.3 to 5.5.5). His 1888 memorandum to the Prussian Ministry of Culture (see Section 6.4.2) and his many other activities, articles, speeches, and lectures (especially during his first trip to the United States in 1893; see Section 7.4.1) evidence his broad, Gauss-oriented program. The year 1895, when Hilbert began his professorship in Göttingen – at Klein’s instigation – did not interrupt or deeply change his program,5 given that he had begun to pursue his all-embracing goals before this time. Klein himself regarded 1893 as a special year in which he increasingly succeeded in implementing his program (see Chapter 7 and p. 546), which had long been in the works, in Göttingen. With his own program (Klein used the term “program” excessively) and with the appointment of Hilbert, Klein strove to develop mathematics and its applications in a universal way. Klein regarded Hilbert as his scientific descendant – trained by Hurwitz and Lindemann. Klein had praised Hilbert as “the rising man” (see Sections 6.3.7.3 and 7.9), who, like himself, “favored unified, holistic approaches,” as David Rowe succinctly put it.6 Klein’s aforementioned interlude with uniformization theory and his exchanges with Poincaré came at a time when he thought that he had already handed over this research domain to Hurwitz (see 5.5). Only in retrospect did Klein (rightly) consider his results in this area the absolute high point of his research. He was especially pleased that his original idea of proof (by continuity) for the uniformization theorems eventually proved to be effective (see 5.5.4.4 and 8.2.1). There were some previous highlights, however, that Klein described as his “peak,” and there were some later ones as well. So far, there has been little discussion of the fact that Klein’s book on the icosahedron culminates in his own proof of one of Kronecker’s theorems.7 Kronecker himself had yet to find a valid proof for it, and Klein rejoiced in November of 1876 (and even later) at the results that he had achieved (see Sections 4.2.1 and 5.5.6). During the period when his research concentrated mainly on applied mathematics, he did not neglect pure ma-

5 6 7

On the presumed change in Klein’s undertakings in 1895, see ROWE 2019c. ROWE 1997. KLEIN 1884. The English translation of this book, which was first published in 1888, has been reprinted as recently as 2019.

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thematics (see Section 8.2.1). He continued, for example, to carry out his work on automorphic functions with Fricke, Hilbert, Minkowski, Koebe, and Brouwer. The emphasis of Klein’s activity did shift over the years, yet this shift was not qualitative but quantitative – also toward the writing of (more) monographs. When he decided to summarize his results in monographs, however, this was not a fallback solution due to health problems – as one repeatedly reads. Rather, Klein was following the French example of publishing books, which he had already underscored in his inaugural address in Leipzig in 1880 (see 5.1). The manuscript of his first monograph, Ueber Riemann’s Theorie der algebraischen Funktionen,8 in which he first described the “Klein bottle,” was completed in October of 1881, before he studied Poincaré’s work in greater depth (see 5.5.1.2). He also wrote his second monograph (on the icosahedron) on his own; and for his later books he sought and found collaborators. It is true that Klein’s students eventually worked out many of his ideas, but this cannot be described as a radical change. Klein passed along his bright ideas to students from early on in his career, even while he was still a Privatdozent.9 The fact of the matter was that, over time, he had more and more students and a growing staff as well. Klein’s way of writing mathematics remained largely consistent. He explained mathematics with a rich vocabulary, and from early on he preferred the French manner of writing at the time (see Section 2.6.3). He wanted his findings to be understood by everyone – not just by specialists in a given area. Thus he later regarded symbolic logic’s “banishment of common language” and Russell and Whitehead’s “dispute over the finite number of words” as “grossly one-sided approaches” that have “lost sight of the overall purpose of mathematics.”10 Nevertheless, Klein would eventually attempt to delve into all new areas of research and even to promote those areas that were further away from his own (such as Edmund Landau’s analytic number theory; see Section 8.2.2). Creating Favorable Conditions for Good Scientific Work Even as a young man, Klein felt responsible for this. He contributed to review journals, including the Bulletin des sciences mathématiques et astronomiques in France and the Jahrbuch über die Fortschritte der Mathematik in Germany (see Sections 2.6.1 and 2.8.3.4). Since his first professorship, Klein cared deeply about the holdings of the mathematical library (see Appendix 2). In order to increase the library’s holdings of international literature, he organized an exchange of publications between the Societas Physico-Medica Erlangensis and British societies (see

8 KLEIN 1882. The English translation appeared in 1893 (see also Fig. 25). 9 On his first doctoral students J. Diekmann and F. Lindemann, see p. 107 and p. 115. 10 [UBG] Cod. MS. F. Klein 22A, fol. 74a (Klein’s notes from 1915). See Bertrand Russell and Alfred North Whitehead, Principia Mathematica, 3 vols. (Cambridge: Cambridge University Press, 1910–13).

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Section 3.3), whereas his later colleagues eschewed any efforts of this sort (see Section 5.7.3).11 Even before Klein became an editor of Mathematische Annalen, he had established an exchange between this journal and Darboux’s Bulletin (see Section 2.4.2). Klein also fostered academic interaction with Eastern Europe (see Section 5.4.2.5). His sense of responsibility for the state of the mathematics library in Göttingen did not end with his retirement, as is evident in a letter from Edmund Landau to the Ministry of Culture (see the introduction to Chapter 9). Focusing on Problems of Mathematical Instruction at Schools Klein did not wait to direct his attention to this issue until the leaders of industry began to call for educational reforms in science and mathematics. As early as his doctoral examination from 1868, he made a statement about changes that ought to be made in the mathematical curriculum of secondary schools (see Section 2.3.4). In a letter to Gaston Darboux in 1872, Klein expressed his opinion about this, and he used his inaugural addresses as a professor (both in 1872 in Bavaria, and 1880 in Saxony) to stress this issue as well. In the spring of 1893, Klein himself visited secondary schools in order to learn which topics might be useful for advanced continuing education courses, which he first began to organize for Prussian teachers of mathematics in 1892 (see Section 7.3). During the later stages of the German Monarchy, Klein became a driving force behind new educational reforms. This had international effects. He was not only elected chairman of the first International Commission on Mathematical Instruction; he also became a model for people who instigated similar developments in other countries. For example, Tsuruichi Hayashi, the Japanese translator of Klein’s book Famous Problems of Elementary Geometry,12 was even referred to in his homeland as the “Felix Klein of Japan” (see 7.3 and 8.3.4). At the beginning of the Weimar Republic, Klein took steps to influence new developments in educational reform (see 9.3.2). With respect to educational policy in Germany, Klein’s activity was consistent regardless of historical upheavals. Klein’s Handling of His Health Problems His illness in the fall of 1882 – often discussed, also by Klein himself, in the context of his correspondence with Poincaré – should not be described as a “mental collapse” or as a case of “lethargy and depression.”13 This would imply that Klein

11 Klein established mathematics as one of the Societas Physico-Medica Erlangensis’s official fields of interest (see the introduction to Chapter 3), and he was the driving force behind the reorganization of the academies of sciences in Leipzig and Göttingen (see 5.7.3 and 6.4.3). 12 KLEIN 1895b. 13 Quotation from PARSHALL/ROWE 1994, p. 186.

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had lost his inner drive to go on working, but he remained mathematically active (see esp. Section 5.5.2.3). The sources reveal that he suffered from overwork, asthma, and digestive problems (see 5.3.1). This pattern of symptoms accompanied Klein both before and after this particular point in time. His way of coping with his health problems was the same whenever they flared up: he would reduce the number of lecture courses that he had to teach while continuing to focus on his research seminars and other projects (see 4.4, 5.3.1, and 8.5.1). In 1912, a spa doctor ultimately diagnosed him as having neurasthenia – chronic exhaustion – on account of “overexertion in his profession” (see Appendix 8). Klein decided to retire, but he remained active as an emeritus professor. Klein regularly expressed self-doubt about whether he could accomplish his broad research program and do justice to the high scientific demands that he made on himself. In 1892, when he complained to Paul Gordan that “I have lost the strength to delve into particular mathematical problems because of all the general issues that I have in mind,” Gordan consoled him: So it is for everyone who compares his meager human achievements with the ideal. Now, your ideal is greater than that of other people, and thus this rift is more gaping. […] You are deeply immersed in a large part of mathematics, more deeply than most people; you see the threads that connect the different domains, and you want to create a unified form of mathematical thinking. That is too difficult for one person. You must be satisfied with the testimony of your peers that you are an extremely useful member of the scholarly world who breathes life into things when the strength of others is already waning.14

A Summary of the Aspects that Guided the Research for the Present Biography Regarding the first of these aspects, which was how Klein was able to become an internationally renowned mathematician, it should be stressed that Klein already showed himself to be a citizen of the world at the young age of twenty-three. He stated then that “mathematics is a thoroughly international science and that the progress of the productive mathematician is considerably hindered if he does not have a universal overview of the findings of others at the same time” (see Appendix 2). In this respect, Klein followed Clebsch’s program of “uniting people and research areas” (see Section 2.4.1) and his approach differed from that of more narrowly oriented mathematicians (see Section 5.5.2.4). Klein created connections between different areas of mathematics (geometry, algebra, function theory, number theory), and he also made connections between mathematics and its neighboring disciplines (applications, philosophy, history, psychology, education). He oriented himself toward international developments, and he found numerous likeminded people in Germany and abroad to follow his visions.

14 [UBG] Cod. MS. F. Klein 9: 469 (a letter from Gordan to Klein dated October 14, 1892). For Klein’s letter to Gordan, which is dated October 8, 1892, see ibid., 12: 508.

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Through Plücker and Clebsch, Klein had been integrated into the international research community at an early stage, and he had maintained and strengthened his contacts through his repeated trips to France, Great Britain, Italy, the United States, and elsewhere (see Sections 2.6, 3.3, 3.4, 6.3.7.1, 7.4, and 7.8). On the basis of his edition of Plücker’s work on line geometry,15 Gaston Darboux had reached out to the twenty-year-old Klein as early as March of 1870, which was even before Klein traveled to Paris for the first time. Beginning in 1880, Darboux sent a string of young French mathematicians to study with Klein. Klein also developed close relations with numerous British mathematicians (Cayley, Sylvester, Greenhill, Forsyth, among others) and also with the algebraic-geometric school in Italy (see 3.3 and 3.4). Francesco Brioschi was instrumental in instigating Klein’s first Italian publication: “Sull’equazione dell’icosaedro nella risoluzione delle equazioni del quinto grado” (1877). In 1877, the Reale Istituto Lombardo di Scienze e Lettere in Milan became the first foreign academy to elect Klein as a member. Young Italian mathematicians had been coming to study under him since 1878; he benefited from their results, which he disseminated in the proceedings of various academies and in Mathematische Annalen (see 3.4 and 4.2.1). In the interest of the latter journal, whose program he defined for decades, Klein deliberately expanded his contacts to Eastern Europe as well (see 2.4.2, 5.4.2.5, and 6.3.7.1). Klein’s lecture courses and research seminars became a point of attraction. Already during Klein’s time in Erlangen, Sophus Lie recommended that Scandinavian mathematicians should not go to Berlin but rather to Klein in Erlangen, where their research would be better supported. Enthusiastic reports by the likes of Irving W. Stringham (USA), Poul Heegaard (Denmark), and Wilhelm Wirtinger (Austria) indicate that Klein went out of his way to create a productive work environment. Word spread beyond Germany’s borders that Klein was able to recognize talent, steer it in the right direction, and publish the results quickly. In this way, he created an international network that extended as far as Japan and India.16 The remarkable scope of this network is clear from the large number of foreign researchers whom Klein was able to recruit as contributors to the ENCYKLOPÄDIE (see the Index of Names at the end of this book) and as participants in the International Commission on Mathematical Instruction, not to mention the number of foreign mathematicians who donated money to entice Max Liebermann to paint Klein’s portrait (see Section 8.5.2 and Appendix 10). Klein’s influential and inspirational effect on people was based on his collaborative way of working, which brings us to the second aspect that guided my research. The prominent Berlin mathematician Leopold Kronecker believed that mathematicians did not need to form a (scientific) school and that collaborative

15 PLÜCKER 1869. 16 In 1910, Klein became an honorary member of the Calcutta Mathematical Society ([UBG] Cod. MS. F. Klein 114: 41).

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work would hinder progress in the field.17 Klein, in contrast, aspired to reproduce Clebsch’s model and thus to create “a school of geometric production.” As a young researcher, Klein spoke about his “social urge to have an effect on others” (see 2.8.3.4 and 3.1.2). His general way of working was based on his approach to mathematical research, which required cooperation. Perpetually analyzing himself, the twenty-two-year-old Klein wrote to Max Noether: Incidentally, the longer I am engaged in mathematics, the more I feel that I am not very well suited to conduct such fundamental investigations. I like to imagine science as a line that connects two points, and it is at these points where the actual scientific work is going on. The one point leads to expanding development [ausbreitende Entwicklung], the other to fundamental deepening [principielle Vertiefung]. I am among the workers at the first point.18

Klein surveyed and systematized developments in mathematics, but he needed discussion partners in order to dig deeper into the state of research and to add greater depth to his own many ideas: “If only I had someone here with whom I could talk reasonably about things. I understand matters very well when I can question someone about them, but hardly at all when I am staring at the printed material in front of me.” He wrote these words to Max Noether on June 30, 1885.19 Klein tested out as many colleagues and students as possible for their potential as collaborators. Whichever partner he preferred to work with at any given phase of his career – male or female (Emmy Noether) – depended on his specific research area, book project, or other activity at the time in question. From a research perspective, his most important cooperation partners were Sophus Lie, Paul Gordan, and Adolf Hurwitz, but there were many other collaborators, as I have shown throughout these pages. The main features of Klein’s collaborative approach warrant further emphasis and will be summarized below. Klein inspired talented young researchers to achieve new results. He supervised his first doctoral student while he was still a Privatdozent in Göttingen (see Section 2.8.2). During his brief time in Erlangen as a young professor, he supervised six doctoral theses and one Habilitation thesis. The Habilitation candidate, Aurel Voss, described how well Klein conveyed his ideas and how he instinctively knew how to steer his students in the direction that best suited them (see 3.1.2). This proved to be a constant feature of Klein’s career as a teacher, as is clear from the way he guided his later students Luigi Bianchi and Hurwitz (see 4.2.4.2). Klein’s “intellectual hegemony,” as Moritz Epple has called it (see 6.3.2), led not only to his supervision of more than fifty doctoral students,20 two of whom were women (Grace Chisholm and Mary F. Winston). He also succeeded in instilling new self-confidence in gifted but “discouraged” mathematicians who had earned their doctoral degrees elsewhere. Examples include Georg Pick (see Section 5.5.7.2) and Wilhelm Wirtinger, the latter of whom remarked: 17 18 19 20

See Kronecker’s comments in Jahresbericht der DMV 1 (1892) II, p. 24. [UBG] Cod MS. F. Klein 12: 534 (Klein to Max Noether, March 12, 1871). Ibid., 12: 605 (a letter from Klein to Max Noether dated June 30, 1885). The list in KLEIN 1923 (GMA III), Appendix, pp. 11–13, is incomplete.

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Forgive me for taking a few minutes of your time at the beginning of the New Year to thank you yet again for so much scientific inspiration and for much more, namely for restoring my confidence, my pleasure in working, and my self-regard. And if I may also express one wish, it is to promote, among the widest circles, the notion that the sciences exist to inform and support one another and should not be confined within proud, rigid, unapproachable, and often useless systems.21

Klein guided and delegated affairs, and such efforts pertained to his students, his own research and book projects, curriculum planning, academic committees, etc.: “Klein has to manage absolutely everything and everybody,” in the (slightly disapproving) words of Otto Hölder (see Section 6.2.2). Wirtinger, who worked as an editor on the second volume of the ENCYKLOPÄDIE (on analysis), referred to Klein as the “onrushing field marshal” of this project (see Section 7.8). In 1911, Anna Klein wrote to her husband, who was then in a sanatorium, that he did not have to give his blessing to everything that happens in Göttingen (see Section 8.5.1), but Klein remained involved in many respects until 1924. Whenever someone seemed unwilling to cooperate with Klein’s program, such as the Siegmund Günther and Wilhelm Schüler in Bavaria or Hans Lorenz in Göttingen,22 Klein could be quite stern. Only a few people – Otto Hölder, Eduard Study,23 and Max Born, for instance – opposed Klein’s leadership (and usually just temporarily). Hölder ultimately came around to recognize Klein’s professional competence, saw that he was “not vindictive,” and became involved in Klein’s projects. Study benefited from a positive letter of recommendation that Klein wrote in his support to the Prussian Ministry of Culture (see Section 5.4.1), and Study’s donation to fund Max Liebermann’s portrait of Klein (see Appendix 10, Fig. 43) could be interpreted as an acknowledgement of Klein’s assistance (and perhaps even as an apology). Klein’s interactions with most of his colleagues were characterized by his diplomatic behavior. In this way, he differed from Hilbert, who was known to be “impulsive.” A notable example of this was when Klein asked Hilbert to attend the meetings of the Göttingen Academy (important for voting), but Hilbert refused to participate as long as the reactionary Germanist Edward Schröder was a member of the executive board in this organization (in a sense, Hilbert thereby ceded the stage to Schröder).24 Klein had skillfully assembled and managed the interdisciplinary editorial board of the journal Mathematische Annalen, and as late as

21 [UBG] Cod MS. F. Klein 12: 365 (Wirtinger to Klein, December 30, 1889). 22 Lorenz, who was Prandtl’s predecessor in Göttingen, was hired away by the Technische Schule in Danzig (see Section 8.1). 23 On Eduard Study, see in particular HARTWICH 2005. 24 Hilbert wrote to Klein on March 7, 1918: “I will not come to the Society of Sciences as long as no one protests against the fact that a man like Schröder has a seat on the board” (FREI 1985, p. 144). E. Schröder, a prominent mediaevalist, was a member of the Alldeutsche Verband, an association that pursued an aggressively nationalistic, militaristic, anti-Semitic, and anti-feminist agenda. As early as 1909, Schröder had attempted to make it difficult for two of Hilbert’s female Jewish students to obtain doctoral degrees. See KÖNIG/PRAUSS/TOBIES 2014.

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1920 he diversified it further by including Einstein and other theoretical physicists (see Sections 2.4.2 and 9.2.2). Hilbert was unable to keep this editorial team together after Klein’s death. Klein remained diplomatic even when his cooperation partners behaved inappropriately (as in the case of Sophus Lie; see 6.3.6) or adopted a backward-looking attitude toward scientific progress. Klein had formed a sort of symbiosis with Lie and Gordan. For a long time, that is, Klein edited mathematical articles for Lie and Gordan, in the sense that he wrote down their (in part) orally communicated results and shaped their ideas into a systematic form. Max Noether reported that Gordan (like Lie) was “clumsy with the pen” (see 3.5). Later, Klein (together with Adolph Mayer) arranged for Lie to be supported by Friedrich Engel – and Gordan would later receive help from Max and Emmy Noether. Klein had brought Gordan to Erlangen to work alongside him in 1874 (see 3.5) and benefited from his algebraic knowledge, especially his theory of invariants of binary forms.25 However, when Gordan opposed Hilbert’s modern theory of invariants, Klein took Hilbert’s side and encouraged him to break away from the old Clebsch-Gordan approach to invariant theory (see 6.3.7.3). In 1894, when Gordan, who was then the president of the German Mathematical Society, opposed adding Hilbert to the editorial board of Mathematische Annalen, Klein diplomatically waited and later pushed through with his decision. About a meeting that took place in Vienna in 1894, Klein reported the following to Walther Dyck, who had been a member of the journal’s editorial staff since 1888 (see 2.4.2 and 5.4.1): Gordan is a peculiar man. In all seriousness, he had a discussion with Max Noether about the question of whether the volume of the Annalen could be reduced by the a priori exclusion of certain research areas, e.g. number theory! And yet I am convinced that number theory is the very subject in which the greatest progress will be made in the near future! For some time, he has probably believed that this is a sensible idea and that he is powerful enough to trim the Annalen back to the standpoint of Clebsch’s school in the early 1870s! When I then spoke with him in Vienna [in 1894] and described such tendencies as reactionary, he backed down from this position.26

Whenever Klein recognized a new mathematical approach to be correct, he endorsed it even despite the oppositional views of his collaborators, and yet he maintained his friendship with them. Klein and his best student and collaborator Adolf Hurwitz maintained a good relationship (see Sections 4.2.4.2, 5.4.1, and 6.3.3), even though Klein had ranked Hurwitz behind the “impulsive, red-bearded” Hilbert when the latter began to outshine Hurwitz scientifically (see Sections 6.3.7.3, 7.9, and Appendix 6). As an important source of creative support, Hurwitz had contributed substantially to the theoretical foundations of Klein’s monographs on elliptic and automorphic functi-

25 Klein included long appreciations of Gordan’s work in his own collected treatises; see KLEIN 1922 (GMA II), pp. 255–61, 380–84, 426–38. 26 [BStBibl] A letter from Klein to Dyck dated August 12, 1894.

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ons.27 Klein did not, however, want to squander Hurwitz’s creative mind on the grunt work of writing these books (see Section 5.5.7.2). As late as 1918, Klein still trusted and relied on Hurwitz’s judgment, as is clear from the following letter: Thanks to a fund established by my friends, now I really can produce an edition of my old works. I have been busy making preparations for this for some time by introducing the material to the assistant I have recruited, Mr. Alexander Ostrowski from Kiev. We will furnish the edition with more or less detailed explanations. I would like to ask you in advance whether you might allow me to send you proofs of the papers related to your field of work, with the request that you might check the correctness and completeness of my explanatory commentary. Best wishes from my house to yours, K.28

Hurwitz predeceased his doctoral supervisor and thus could not contribute. While it is true that Henri Poincaré did not become one of Klein’s mathematical collaborators (unlike so many Italian mathematicians, for instance) but rather a rival, this was not due to Klein but rather to the nationalistic animosities that had been stirred up by his French colleagues (see 5.5.3.2). This did nothing to prevent Klein from evaluating Poincaré’s achievements as outstanding and from calling him, among other things, “the French genius,” a “singular phenomenon,” and a “modern Cauchy.”29 In coordination with Darboux, Klein proposed that Poincaré should be named the first winner of the János Bolyai Prize in 1905 (see 5.4.2.4). In this context, Klein arranged, in 1905 and 1906, for the Göttingen Mathematical Society to host lectures on a number of Poincaré’s works.30 Poincaré was twice an invited guest in Göttingen, in June of 1895 (see Section 7.2) and in April of 1909 – when Hilbert, in Poincaré’s presence, also gave a speech on the occasion of Klein’s sixtieth birthday.31 In 1910, when the idea arose in France to nominate Poincaré for the Nobel Prize, Klein abstained from voting.32 When Paul Appell and Gaston Darboux asked him, in a letter dated March 13, 1914, to serve on a committee intended to prepare a permanent tribute to Poincaré, Klein agreed immediately.33 27 KLEIN/FRICKE 2017 [1890/92]; FRICKE/KLEIN 2017 [1897/1912]. 28 [UBG] Math. Arch. 77: 272 (a postcard from Klein to Hurwitz dated December 20, 1918). 29 [UBG] Cod. MS. F. Klein 22A, fols. 76a, 77 (Klein’s notes in preparation for his historical lectures, Nov. 23. 1915). See also KLEIN 1979 [1926], pp. 355–61; KLEIN 1927, pp. 68–69. 30 See Jahresbericht der DMV 14 (1905) V, p. 586; and 15 (1906) IV, pp. 153, 274. 31 Henri Poincaré visited Göttingen from April 22–28, 1909 to give a series of lectures. For the titles of these lectures, see Jahresbericht der DMV 18 (1909) Abt. 2, pp. 78–79 (see also Section 8.2.1). Klein wrote somewhat critically to Robert Fricke on May 15, 1909: “The Poincaré days were quite satisfactory on a superficial level. On a deeper level, their returns were lower, because Poincaré limited himself to discussing his latest papers, which only elaborate his older ideas. He refrained from highlighting general points and presented the details to a rather uncomprehending audience. Poincaré is also very reserved in conversation. He listens kindly to what he is told, but he replies very little” (quoted from [UA Braunschweig]). 32 [Paris] 82: Klein to Darboux, January 21, 1910. 33 [UBG] Cod. MS. F. Klein 11: 369A (Appell and Darboux to Klein, March 13, 1914); [Paris] 83: Klein to Darboux, March 25, 1914.

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When David Hilbert (the second winner of the Bolyai Prize, likewise agreed upon by Klein and Darboux), who was thirteen years younger than Klein, failed to become the mathematical collaborator whom Klein had hoped to have a “rejuvenating effect” on own work (see 7.9), this was not Hilbert’s fault. Hilbert stood by Klein to the end and cooperated with him in seminars and on other academic matters. It was due to Klein himself, who had been somewhat slow to accept the turn toward the new conceptual mathematics and its abstract methods (see 8.2.2, 8.3.2). When Hilbert, with Minkowski’s encouragement, began to divert more of his attention to mathematical physics, Klein rejoiced and felt that one of the important goals of his wide-ranging Gaussian program had been achieved (8.2.4). The twenty-three unsolved mathematical problems that Hilbert presented at the second International Congress of Mathematicians in Paris in 1900 would go on to influence mathematics considerably. Klein accepted this without any feelings of envy. David Hilbert became, in Hermann Minkowski’s words, the “director general” of mathematical research.34 A document from early 1916 from Klein’s estate reveals that, when he was working on his overview of nineteenth-century mathematics, he wondered how he might appropriately acknowledge Hilbert’s contributions. Klein revisited Hilbert’s famous talk and wrote: “How shall I commemorate these ‘problems’ in my report? Perhaps as the conclusion (of the century)? As evidence that mathematics is not dead.”35 He outlined the problems, adding notes of his own (in italics below): Interplay between thinking and experience. Strictness is simplicity. Clear axiomatization in each case! 1. Cardinality of the continuum.36 2. The inconsistency of arithmetic. Merely postulated. Mathematically, that which is consistent exists. 3. Volume equality of two tetrahedra. 4. The straight line as the shortest connecting line. Minkowski’s geometry. 5. Lie groups without differentiability. Functional equations in general. 6. Axioms of physics. Probability theory. Mechanics. 7. The irrationality (etc.) of certain numbers. 8. Prime number problems. 9. The most general laws of reciprocity. 10. The solvability of Diophantine equations. 11. Quadratic forms with algebraic numerical coefficients. 12. Kronecker’s theorem on Abelian fields arbitrarily extended. This is what I need first of all! Achieved for root functions. Function-theoretical analogies. None of [Hilbert’s] numerous students has understood this. See Fueter […].37 13. Seventh-degree equations cannot be solved nomographically. 34 35 36 37

See MINKOWSKI 1973, p. 108. [UBG] Cod. MS. F. Klein 22A, fols. 83–83v (Klein’s notes, February 4, 1916). Klein grouped together problems 1 through 6 as “foundations.” The Swiss Rudolf Fueter dealt with this topic in his dissertation – “Der Klassenkörper der quadratischen Körper und die komplexe Multiplikation” [The Class Field of Quadratic Fields and Complex Multiplication] (1903, supervised by Hilbert) – and in later works as well.

10.1 A Summary of Findings 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

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The finiteness of systems of forms. Schubert’s enumerative calculus. The topology of curves and surfaces. Definite forms and squares. Tiling space by fundamental regions. [p. 286 Analytical functions, not other classes of functions].38 The analytic nature of regular variational problems. The general boundary-value problem. Linear differential equations with a prescribed monodromy group. Uniformization of analytic relations. Further development of the calculus of variations. (Only here individual explanations). Mathematics as a unified whole!39

Hilbert maintained the role of “director general” in these topics of research, while Klein succeeded in formulating important unsolved problems in some areas of applied mathematics: in fluid dynamics (before he brought Ludwig Prandtl to Göttingen), the statics of structures, the theory of friction, and the special theory of relativity (see 8.2.4). After the general theory of relativity had been worked out in 1915, Klein made some acknowledged contributions to it, mainly in cooperation with Emmy Noether (see Section 9.2.2). Regarding the third aspect that guided my research – Klein’s political stance – it was relevant to consider his roles as a scientific organizer and education policy-maker. At the same time, it was also necessary to examine his views on general political issues and his political attitude during times of war. The issues of scientific organization and educational policy had been on Klein’s agenda long before Hilbert arrived in Göttingen in 1895. Even after Klein had retired, Hilbert left these matters in his hands, so that Abraham Fraenkel, as I have mentioned on more than one occasion, dubbed Klein the “foreign minister” of German mathematics. Working as just such a “foreign minister,” Klein created an alliance among science, the state, and industry that had previously never existed at German universities. That is, in the interest of mathematics, science, and technology, Klein cooperated not only with government officials but also with industrialists. Friedrich Althoff, the dominant official at the Prussian Ministry of Culture (who outlived several ministers and had the ear of the emperor), turned to Klein as his preferred advisor for hiring decisions in mathematics and for other matters, especially after Klein had turned down an offer to work at Yale University in 1896. Althoff created a science policy based on international understanding, and Klein played a part in this. To support Althoff’s goals, he participated in numerous international projects, including the establishment of the Internationale Wochenschrift für Wissenschaft, Kunst und Technik – a publication whose edito38 The page number here (p. 286) refers to Hilbert’s article in the Göttinger Nachrichten (HILBERT 1900). Here, in his eighteenth problem, Hilbert had cited FRICKE/KLEIN 1897. 39 [UBG] Cod. MS. F. Klein 22A, fols. 83–83v (Klein’s notes, February 4, 1916). For the German original, see TOBIES 2019b, pp. 487–88.

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rial board also included Henri Poincaré and H.A. Lorentz. Klein’s initiative to create a Göttingen Institute for Foreigners (Göttinger Institut für Ausländer), which he accomplished in coordination with Althoff and with funding from the industrial donor Henry Theodore Böttinger (who had been raised in England), can also be seen in this context (see Section 8.4). It was this same Böttinger who, with his fortune earned in the German dye industry, supported Klein’s applications-oriented initiatives in Göttingen and even served as the chairman of the Göttingen Association for the Promotion of Applied Physics and Mathematics (see 7.7 and 8.1.1). What was special about this association was that it functioned without a formal statute and that the industrialists – around fifty of them, from all of Germany – donated money not only to support specific research projects at the University of Göttingen but also mathematical and scientific educational reforms and specific educational institutions. Klein regretted that the Helmholtz Society, which succeeded the Göttingen Association in 1920, abandoned this commitment to educational projects (see 9.4.2). In reference to the Göttingen Association, the theoretical physicist and Nobel laureate Max Born later commended Klein’s “wise counsel,” which “had been of such great advantage” to the University of Göttingen.40 The standard by which Klein measured any undertaking was the extent to which it might serve the interests of mathematics, its applications, and education. Whenever he felt that the position of mathematics was in danger, he would present all conceivable arguments (historical as well as military arguments) in its favor. Before, during, and after the First World War, he assembled and mobilized forces in order improve mathematical and scientific education and to ward off those who sought to do it harm (see Sections 8.3.4, 9.1.2, and 9.3). Lewis Pyenson has described Klein’s initiatives as “a carefully constructed programme to preserve the traditional prerogatives of pure mathematics in a material-industrial age.”41 Of course, Klein worked to promote both pure and applied mathematics. In all of his collaborations, including those with government officials and industrialists, Klein’s activity adhered to the concept of appropriately using every resource for the development of mathematics (its applications and instruction included) (see Section 1.2). As one of Klein’s long-term collaborators on educational projects, Walther Lietzmann had the following impression of him: In the image that he created of someone, Klein understood how to separate the essential from the inessential (that is, the factors unrelated to this person’s ability to do a certain task). He was able to ignore things […] such as vanity, boastfulness, opportunism, and even more so political, national, racial, and religious differences. As Hilbert stated in the eulogy which he gave at the Göttingen Mathematical Society on the day after Klein’s death, the secret to his success lay in his incorruptible objectivity.42

40 BORN 1978, p. 208. 41 PYENSON 1979. 42 LIETZMANN 1925, p. 259. For Hilbert’s eulogy, see Appendix 12.

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Regarding his general political position, Felix Klein has often been referred to as a nationalist. Samuel Patterson, for instance, has called him a cosmopolitan and a nationalist,43 while David E. Rowe has recently described him as “an ardent nationalist.”44 These rather strongly worded categorizations, however, do not stand up to a more nuanced analysis. During his first stay in Berlin (1869/70) and also on many later occasions, Klein criticized nationalistically oriented Germans who made condescending remarks about foreign science and scientists (see Sections 2.5.2 and 5.5.2.4). After studying in Paris with Sophus Lie in the spring and summer of 1870, Klein served for several weeks as a paramedic during the Franco-Prussian War. While doing so, he continued to work on mathematics (see 2.7.1). Even before the peace treaty was signed, moreover, Klein reestablished contact with Gaston Darboux. Whereas Darboux responded positively to Klein’s overtures, other French (Camille Jordan, for instance) and German scientists maintained strong nationalistic positions (see 2.7.1). In 1870, Jordan himself annulled his membership in the Royal Society of Sciences (academy) in Göttingen. During the First World War, Felix Klein in Göttingen and Max Planck in Berlin ensured that the French members of the academies there would not be expelled. Émile Picard left the Göttingen Royal Society (as Jordan once had) of his own volition (see 9.1.1). Picard was instrumental in banning German scholars from the Académie des Sciences in Paris. His main reason for doing so was to punish those who had signed the so-called appeal “To the Civilized World.” The appearance of Klein’s signature on this much-discussed document, however, should not be considered in isolation. As of 1911, Anna Klein’s letters to her husband begin to mention the “unrest in the world” and the “cries for war.”45 Felix Klein had clearly perceived the rise of chauvinism, and in this respect it is remarkable that, as early as 1908, he explicitly opposed “national chauvinism” in an official document and called for something to be done about it (see Section 8.4). In doing so, he was in agreement with the policy promoted by representatives in the Prussian Ministry of Culture. Shortly after the outbreak of the First World War in August of 1914, Klein submitted an article to the Jahresbericht der DMV in which he emphasized the international character of mathematics and mentioned the international projects that were in danger (see Section 9.1.1). As Bernhard vom Brocke has already shown, older faculty members at German universities were unable to evade the “campaign to mobilize morale on the home front.”46 More than three thousand university instructors in Germany (including Hilbert and Klein) felt compelled to demonstrate their loyalty to the state in declarations (see Section 9.1.1). There were several of such declarations, and Cordula Tollmien has analyzed the extent to which Göttingen’s professors participa-

43 44 45 46

See PATTERSON 2016. ROWE 2018a, p. 377. [UBG] Cod. MS. F. Klein 10: 376 (Anna to Felix Klein, September 29, 1911). BROCKE 1985.

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ted in them.47 Felix Klein, who remained a member of the Upper House of the Prussian Parliament until the end of the German Empire in 1918 and was thus more widely known, was the only mathematician asked (via telegram) to lend his signature to the aforementioned appeal “To the Civilized World.” Tollmien’s research makes it clear, however, that he had not seen the text and that he had naively assumed that the appeal was meant to calm the turbulence stirring abroad. Little attention has been paid, until now, to the fact that Klein, in the midst of the war in 1916, emphatically supported a parliamentary memorandum in favor of expanding study-abroad programs. The aim of this initiative was to prevent future international conflicts (see Section 9.1.2). Furthermore, it should be stressed that, in 1917 and 1918, German journals published obituaries of the French mathematician Gaston Darboux, and these were written with Klein’s consent. The obituary published by Aurel Voss in the Jahresbericht der DMV in 1918 was both longer and more detailed than that published by Hilbert the year before (see 9.1.1).48 Despite what some authors have written, the sources do not support the image of Klein as a professor who was predominantly “loyal to the emperor” (and this initially surprised me as well). In several speeches given in the Upper House of the Prussian Parliament, Klein vehemently argued on behalf of improving education for all children (including those in kindergarten, elementary school, vocational school, and schools for girls). He was inspired, among other things, by the Perry movement and by books written by the mathematician and education reformer Benchara Branford in Great Britain. Klein did not belong to any political party. He judged parties according to their results and thus even acknowledged, on occasion, the positive influence of the Social Democrats (see Section 8.3.4.1). He did so as early as 1911, during the imperial era, and the Social Democratic Party was hardly “loyal” to the emperor, even though it finally consented to his wartime loans. It should be noted that, as a result of the approval of these loans, the Independent Social Democratic Party of Germany (Unabhängige Sozialdemokratische Partei Deutschlands = USPD) splintered off from the Social Democratic Party and that this new party was joined by Emmy Noether and Leonard Nelson, whose careers Klein helped to promote.49 After the end of the First World War, Klein immediately offered his support to the new Prussian Ministry of Culture, which was led by Social Democrats (see 9.3.2). As the mathematician Kurt Otto Friedrichs reported, Klein deeply regretted the assassination of the Jewish Foreign Minister Walther Rathenau in 1922, and he feared for the prospects of the young republic.50 Klein also rejected the nationalism that flared up after the war, as is clear from his autobiographical sketch 47 See TOLLMIEN 1993. 48 See Jahresbericht der DMV 27 (1918) Abt. 1, pp. 196–217. In 1918, August Gutzmer, the editor of the Jahresbericht der DMV and one of Klein’s close collaborators, also published a profile of Klein, who served as an honorary chairman of the German Mathematical Society in 1918/19. 49 See TOBIES 2012, pp. 105–20. 50 See ROWE 2018a, pp. 376–77.

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published in 1923: “The few foreigners who worked with us in Göttingen at that time seemed to be a stimulating force in general; then there was no trace of the nationalistic antagonism that is now so prevalent in the public sphere.”51 Together with former students and colleagues from abroad, Klein helped to restore international relations after the war. To this end, he renewed contacts with scholars around the world and he made sure that corresponding members were elected to join the Göttingen Academy. The latter included W.F. Osgood and David George Birkhoff in 1922, Guido Castelnuovo in 1923, and Luigi Bianchi in 1924 (see also Section 9.1.2 and Appendix 11).52 Klein’s tireless work ethic, which he had already internalized as a teenager (see 2.1.1), remained with him until his dying day. Klein made reforms when there was something to reform, and he also offered resistance when he deemed it necessary. He passed on this trait (at least) to his daughter Elisabeth, who, in 1933, was demoted from her position as a school principal and had to transfer to another school, as a teacher, on account of her opposition to the regime.53 10.2 A PIONEER Felix Klein was someone who renewed, reconfigured, and reformed many aspects of mathematics and the organization of mathematical research. Not restricted to Germany, these efforts often had international effects. In what follows, I will once again summarize the extent to which he was a pioneer or even a reformer of the academic world. Let us look first at the extent to which Klein can be regarded as a pioneer from an international perspective. With respect to his mathematical research, it must be emphasized that he was the first to rid non-Euclidean geometry of any “mystical” connotations. He did so by combining it with Arthur Cayley’s metric and von Staudt’s metric-free projective geometry (see 2.5.2; 2.5.3). Moreover, Klein was the first to create a unified system to account for the various approaches to geometry at the time. This he did by means of the group concept in his Erlangen Program, which Richard Courant likened to the “ordering force” of the periodic table of elements.54 It should stressed again that Klein used his intuitive concept of the group (see 2.6.2.1) to classify other areas of mathematics (see 4.2), and that he also identified applications in mechanics and the theory of relativity (see 9.2.2).

51 KLEIN 1923a, p. 15. 52 Klein nominated Birkhoff to become a corresponding member at the suggestion of W.F. Osgood. ([UBG] Cod. MS. F. Klein 11: 152–53, letters from Osgood to Klein, December 31, 1921; February 5, 1922; June 17, 1922; and [AdW Göttingen] Pers. 20: 1088, on Castelnuovo, Pers. 20: 1105, on Bianchi. The latter two nominations are written in Klein’s hand). 53 On Elisabeth Staiger (née Klein), see TOBIES 1993a and 2008a. 54 COURANT 1926, p. 200. See also JI/PAPADOPOULOS 2015 and RATAJ/ZÄHLE 2019.

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Klein’s contemporaries likewise thought of him as a pioneer for his role in bringing prominence to Riemann’s geometry program and physical-geometric style of thinking. Addressing Klein in 1909, Hilbert stated: “Riemann was the name that stood on your flag and under whose sign you have been victorious right down the line – victorious against opponents because of the correctness of your ideas […].”55 In Courant’s words, Klein was “the most passionate and successful apostle of the Riemannian spirit.”56 When Riemann’s works were edited in Paris (Œuvres mathématiques de Riemann, 1898), Hermite had arranged for Klein’s famous talk on Riemann (KLEIN 1894, 22pp.) to be printed after his own five-page Préface (see Section 7.4.4, and Fig. 35). In 1893, Klein became the first (and, until 1950, only) foreign mathematician to be awarded the De Morgan Medal by the London Mathematical Society. As early as 1883, Klein became the first German mathematician to be offered a professorship at the University of Oxford (to succeed Henry Smith), and in the same year he received an offer – likewise as the first German mathematician – for a professorship in the United Stated (to succeed James Joseph Sylvester at Johns Hopkins; see Section 5.8.1). Even though Klein turned down these opportunities – as he later did to a second offer from the United States (in 1896, he was offered a professorship at Yale University) – they can be interpreted as indications of his pioneering role. Klein traveled across the Atlantic in 1893 and again in 1896. In the United States, Klein not only disseminated mathematical results developed in Germany and Europe – with the support of his many American former doctoral students.57 At the Mathematical Congress in Chicago, he also was the first to declare that mathematicians should come together internationally (see Section 7.4.1). This resulted in the International Congress of Mathematicians, which has been held since 1897. With his trips to North America, Klein has also been regarded as a trailblazer for establishing German-American professorial exchange programs.58 Klein’s efforts to reform mathematical education also led to international recognition: at the fourth International Congress of Mathematicians, which took place in Rome in 1908, he was named the first president of the International Commission on Mathematical Instruction (ICMI), even though he was not even present at the conference himself (see Section 8.3.4). It was because of Klein’s strong backing that, for the first time, a foreign mathematician was hired as a full professor at a German university: the Norwegian Sophus Lie at the University of Leipzig (1886), the only Saxon university. The extent to which this was unusual at the time is clear from the protests by many German mathematicians, chief among them the prominent Berlin mathematician Karl Weierstrass, whose (political) influence luckily did not extend as far as

55 56 57 58

Quoted from ROWE 2018a, p. 198 (German original in TOBIES 2019b, p. 514). COURANT 1926, p. 202. See, in particular, PARSHALL/ROWE 1994. See BROCKE 1981.

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Saxony (see Section 5.8.3). In 1880, Klein had received the first professorship at a German university devoted exclusively to geometry (this was in Leipzig).59 In Sophus Lie, Klein was able to find a worthy successor for this position (at that time, it was possible for Klein himself to serve on the committee appointed to hire his own replacement, and his views swayed the committee’s decisions). In 1913, when Klein was (finally) elected as a corresponding member of the Academy of Sciences in Berlin, the letter of nomination stated that he was “one of the few mathematicians still able to survey the whole of mathematics” (see Appendix 9). The scientists who supported his election included not only Friedrich Schottky and Max Planck, with whom Klein was on good terms, but also Hermann Amandus Schwarz and Georg Frobenius, with whom Klein had carried out a number of disputes. From today’s perspective, we can probably call Klein, Hilbert, and Poincaré the last of the great “generalists” working in mathematics. Thanks to Klein, the following research areas were established at a German university for the first time: First: descriptive geometry and applied mathematics. Klein introduced the field of descriptive geometry (sometimes called constructive geometry) – which had originated as part of engineering education – at the University of Erlangen in 1874,60 at the University of Leipzig in 1881 (see 5.3.1), and at the University of Göttingen in 1888 (see 6.2.3). This discipline was closely related to kinematics and graphical statics. In 1898, Klein integrated these areas into a new teaching qualification for applied mathematics within the framework of the Prussian examination regulations for teaching candidates (see 8.1.2). In 1904, he was able to establish, at the University of Göttingen, the first full professorship for applied mathematics at a German University (for Carl Runge); this position included descriptive geometry and was intended to promote the development of numerical, graphical, and instrumental methods and their applications to scientific and technical fields. By arguing for the establishment of a precise form of approximation mathematics (see 8.3.2), Klein ultimately paved the way for the development of modern industrial mathematics (engineering and business mathematics).61 Second, Klein played an instrumental role in institutionalizing the field of insurance science, including actuarial mathematics, at the University of Göttingen in 1895, and this was likewise a first at a German university. This later led to the establishment of a separate Institute for Mathematical Statistics (see Section 7.6). 59 Up to that time, geometry professorships, but only for descriptive geometry, had existed at Technische Hochschulen, based on the model of the École Polytechnique in Paris (see also BARBIN/MENGHINI/VOLKERT 2019). 60 [UBG] Cod. MS. W. Lietzmann 1: 43 (a letter from Klein to Lietzmann, June 24, 1910). 61 On the beginnings of this field, see TOBIES 2012; TOBIES/VOGT 2014; regarding its modern developments, see NEUNZERT/PRÄTZEL-WOLTERS 2015, and FRAUNHOFER ITWM 2019. We can also trace a connection between Klein’s promotion of A.A. Markov’s books and so-called “Markov chain Monte Carlo methods,” which are used today for calculating numerical approximations of multi-dimensional integrals (for instance in Bayesian statistics, computational biology, computational physics, and computational linguistics).

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Third, the creation of several new professorships and institutes for scientifictechnical fields – related to mathematical applications – can also be traced back to Klein’s initiative: technical physics, applied electricity (electrical engineering), and geophysics (see 7.7 and 8.1.2). Similarly, aerodynamics was established as a university subject in 1909. In the same year, incidentally, aerodynamics was also introduced in Paris, where Paul Painlevé, who had studied with Klein in 1886 and 1887, established it as an official area of study (see 6.2.3 and 8.1.3).62 Fourth, not only did Klein work on the history of mathematics and reflect on his approach to this subject; in the words of Gert Schubring, he can rightly be called “a founder of the social history of mathematics” in Germany (see Section 8.3.1). Klein established this area of research at the University of Göttingen by ensuring that the first venia legendi for this subject would be awarded in 1908. This was revealed in my examination of the Habilitation files of the candidate in question, Conrad Heinrich Müller (see Section 8.3.1).63 Fifth, on account of Klein’s initiative, Rudolf Schimmack was awarded, in 1911 and likewise in Göttingen, the first ever venia legendi for the didactics of mathematics (see Section 8.3.4.1). Klein also pointed other mathematicians in this direction. As early as 1910, he had arranged for the first textbook on the didactics of mathematics to be published by B.G. Teubner – just as he influenced Teubner’s publication program in the fields of pure and applied mathematics (see 5.6). Sixth, it is possible to see in Klein’s visions the origins of research fields that would not be established until later on, and these should be summarized here. In letters to the Prussian Ministry of Culture in Berlin written at the beginning of the Weimar Republic, Klein suggested that professorships should be established for the didactics of all exact subjects, for general pedagogy, and for university pedagogy. Most surprisingly, the records revealed that Klein considered it sensible to offer specific mathematics courses tailored to meet the different needs of teaching candidates or prospective researchers (see Section 9.3.2). The idea was based on what he saw to be the “double discontinuity” between mathematical instruction at the secondary-school and university levels, which could also be described as a sort of twofold act of forgetting: when someone begins to study mathematics at a university, he or she is told to forget about school mathematics, and yet when the same person returns to secondary school as a teacher, he or she is told to forget all about university-level mathematics. One of the goals of Klein’s published lecture courses, Elementary Mathematics from a Higher Standpoint, was to resolve this problem of double discontinuity.64

62 This is not mentioned in the excellent biography of Ludwig Prandtl (ECKERT 2019a). 63 This information is lacking in the important standard work on the topic (DAUBEN/SCRIBA 2002, p. 493). 64 Klein raises this issue of “double discontinuity” in the preface to the first volume of this book; see KLEIN 2016 [31925], p. 1.

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Regarding the organization of mathematical activity, we have seen that Klein was likewise pioneering in many respects, some of which seem to be taken for granted today. By the time that Klein proposed, in 1893 in Chicago, the aforementioned idea of forming “international unions” of mathematicians, he had already been involved in the creation of numerous discussion groups and societies in Germany, and he had also reorganized a number of existing committees there. Noteworthy in this regard is that he had realized Clebsch’s plan to organize the first conference of mathematicians from all over Germany in 1873 (see Section 2.8.3.4). Klein served three terms as the chairman of the German Mathematical Society (founded in 1890), and he was ultimately named the first honorary member of this organization (see 6.4.4). The Göttingen Mathematical Society, which Klein founded in 1892, served as a model at other universities for the creation of similar colloquia intended to promote the discussion of the latest mathematical results (see 7.2). Throughout his career, whenever Klein arrived as a newly appointed professor at a university or Technische Hochschule, he focused his attention immediately on its institutional framework. In Erlangen, he and the physicist Eugen Lommel created a Mathematical Seminar to train teaching candidates, and he established this university’s first associate professorship – alongside the full professorship – in the field of mathematics (see Section 3.5). At the Technische Hochschule in Munich, he founded its first Mathematical Institute, which included a collection of models (see Section 4.1.1).65 While in Leipzig, he likewise created the first Mathematical Seminar, which he was later able to expand into a Mathematical Institute with a model collection, reading room, and a reference library for students (which was even open on Sunday; see Section 5.2). As a professor in Göttingen, where a seminar and model collection already existed, he was able to add the university’s first reading room for students of mathematics. Of course, collections of mathematical models had already existed before Klein. What was new, however, was Klein’s theoretical approach to the construction of models (see Sections 2.4.3, 2.7.2, and 2.8.2), the mathematical exercises that he devised with them, and the way in which he organized their production and widespread distribution (4.3.3). Klein also recognized the importance of mathematical instruments, and in 1874, as a professor at the University of Erlangen, he became the first German professor of mathematics to acquire a (Frenchmade) mechanical calculating machine (Thomas’s Arithmomètre; Fig. 16). Klein also made use of other instruments and technical achievements, a case being that he was the first German mathematician to use (following the example of intellectuals in France and Italy) novel lithographic processes to reproduce mathematical exercises and to publish autograph copies of his lectures (see Section 4.1.2). Klein’s efforts to coordinate the mathematical curriculum with his colleagues led to lasting innovations that were adopted by other institutions: to the creation of new mathematics courses for engineering students in 1877 (see 4.1.2) and to the 65 Regarding the structural difference between seminars and institutes, see SCHUBRING 2000b.

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first explicit degree programs recommended for teaching candidates in Saxony in 1882 (see Section 5.3.1) and at Göttingen’s university (Prussia) in 1890 (see Section 6.2.3, p. 336). Klein established interdisciplinary research seminars, in which he significantly involved the participation of his younger colleagues.66 Even continuing education courses for secondary school teachers of mathematics, which still exist today, can be traced back to Klein. He first created such courses in Göttingen in 1892. As late as 1925, Carl Metzner, an official at the Ministry of Culture who had studied under Klein, vowed to continue these courses “in Klein’s spirit” (see Section 9.3.2). Felix Klein was the first and, for a long time, only professor of mathematics in Germany to have his own state-funded assistant (internationally, too, this position would not become well-established until later on). At the time, the main justification for this position was his need for someone to manage the model collection and the reading room. Klein first acquired an assistant in 1877 while working at the Technische Hochschule in Munich; he was assigned one in 1881 (beginning with his third semester) at the University of Leipzig, and again in 1892 (beginning with his twelfth semester) at the University of Göttingen (see Section 7.1). Hilbert and Minkowski did not obtain a (shared) assistant until 1904. In order to finance mathematical, scientific, and technical research and instruction at the University of Göttingen, Klein pursued an approach that had not existed before in Germany – with the exception of the Carl Zeiss Foundation in Jena, which was established by Ernst Abbe in 1889.67 The Göttingen Association for the Promotion of Applied Physics and Mathematics, which Klein established in 1898/1900, raised a considerable amount of funding from industry (see Section 8.1.1). In 1920, when new sources of funding for mathematical projects were made available by the Emergency Association of German Science (known as the German Research Foundation since 1929), it was the emeritus Klein who became the first chairman of this organization’s expert committee for mathematics, astronomy, and geodesy (see Section 9.4.1). Klein played a pioneering role in ensuring that women obtained the right to study mathematics at Prussian universities. As of 1893, he made it possible for foreign women to attend lectures and seminars (see Section 7.5), though the Prussian government would not enact a law permitting the official enrolment of women until 1908. On December 14, 1907, when the University of Göttingen elected Klein to serve as its representative in the Upper House of the Prussian Parliament (he was the first professor of mathematics to serve in this capacity), one of the main focuses of his parliamentary speeches concerned the expansion and improvement of women’s (and girls’) education (see Section 8.3.4.1). By carrying out

66 On the seminars that Klein conducted with Hilbert and Minkowski, see Section 5.5.4.4; on his interdisciplinary seminars on applied mathematics, see Section 8.2.4. Regarding the few cases in which Klein’s colleagues were unwilling to participate in his seminars (most notably Otto Hölder and Hans Lorenz), see Sections 6.2.2 and 8.1.2. 67 Regarding Klein’s appreciation of this model in Jena, see Sections 8.1 and 9.3.1.

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his (unofficial) duties as the so-called “foreign minister”68 of German mathematics, Klein worked together with Albert Einstein to ensure that Emmy Noether would become the first female mathematician to be awarded the Habilitation qualification (see Section 9.2.2). He thus paved the way for a ministerial decree issued on February 21, 1920, which stated: “Belonging to the female sex may not be regarded as an impediment to receiving the Habilitation.”69 In accordance with his universal way of thinking (see 8.3.2), Klein became the first and only mathematician to join the Association of German Engineers (in 1895; see 7.7). In January of 1914, he became the only mathematician to join the German Society for the Study of Russia, which had been founded the year before (see 9.1.2), and in 1913 he also cosigned the founding appeal to establish a Society for Positivist Philosophy (see 8.3.2). A philosophical attitude of this sort, which rejects metaphysical thinking and bases knowledge on “positive” (that is, empirical and verifiable) findings, was introduced to Klein early on by Darwin’s theory of evolution and its German proponent Ernst Haeckel (see 2.3.2 and 4.3.3). Overall, it is safe to say that Klein’s trans-disciplinary and international perspective enabled him, in the nineteenth century, to foresee developments and initiatives that would help to secure the position of mathematics in the twentieth century.70 In this respect, we can rightly underscore the words of the mathematician Helmut Neunzert, who stated that Felix Klein was a reformer of the academic world not only in Germany but, in part, internationally as well. In conclusion, it remains necessary to emphasize Klein’s generally open-minded attitude toward new scientific ideas, among them not only Einstein’s theory of relativity but also Arthur Schoenflies’s theory of crystallography based on group theory (see Section 6.3.7.2) or Ludwig Boltzmann’s statistical mechanics. To understand why he was so successful and why he was able to bring so many people into his sphere of influence, we have to acknowledge that a large part of this was due to his ability to put his past achievements behind him, eschew the stubbornness of old age, and stay true to his youthful approach. Klein had studied Kronecker’s methods, for instance, but he later criticized them for being one-sided (see Sections 5.4.2.4 and 6.5.1.1), and they ultimately failed to gain acceptance. Writing in retrospect about Weierstrass, who had suffered greatly from Kronecker’s polemics, Klein thus wrote: Now we would almost like to say that he shouldn’t have taken it so hard; it is only an instance of the fact that all earthly things are subject to the eternal law of motion. The survivor must come to terms with his fate, that for the younger generation it is other ideas that come to the fore. None of us can hinder the world from moving away from and beyond us. We cannot even wish to as, when we were young, we likewise pushed aside the ruling opinions.71

68 69 70 71

FRAENKEL 2016 [1967], p. 138. Quoted from TOBIES 1991b, p. 160. In this regard, see also DASTON 2017. KLEIN 1979 [1926], p. 266-67.

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Figure 42: Felix Klein’s diploma for his honory doctorate from the Jagiellonian University of Krakow (Uniwersytet Jagiellonski w Krakowie), 1900. ([UBG] Cod. MS. F. Klein 113: 5)

APPENDIX: A SELECTION OF DOCUMENTS1 1) A letter from Felix Klein to Heinrich von Mühler, the Prussian Minister of Religious, Educational, and Medical Affairs (Minister of Culture).2 Your Excellency, Düsseldorf – December 19, 1870 In a request dated March 7th of this year, I took the liberty of asking for diplomatic recommendations to travel to France and England for the purpose of undertaking a scientific trip. At the same time, I had offered to submit reports on the conditions of mathematics in these countries upon my return. On March 26th, I was fortunate enough to receive a reply from your Excellency (U 7737) to the effect that the diplomatic recommendations in question had been granted to me, and that your Excellency would be pleased to receive reports on the present state of French and English mathematics. Under the prevailing conditions, unfortunately, the trip could not be undertaken in the way that I intended. My stay in Paris, where I had arrived on April 19th, was suddenly interrupted by the declaration of war on July 16th. I rushed home (Düsseldorf) and, because I was deemed unfit for military service at the moment by the relevant authorities, I joined an association for voluntary medical care, which had meanwhile been established in Bonn. As a member of this association, I spent the period from August 16th to October 2nd, when I was discharged to return home on account of my poor health, in the theater of war. Having only recently recovered, I did not want to make the trip to England because of the time that I had lost. Rather, I have already applied to habilitate in Göttingen and become a Privatdozent of mathematics there, and I intend to move there at the New Year. Given that it is not possible for me to send you reports on French and English mathematics in the way that I intended, I would at least like to enclose a copy of a short report on French mathematics, which I composed together with one of my student friends (Dr. Lie from Christiania), to demonstrate that I had worked along these lines during my stay in Paris. We had prepared this report for the mathematical [student] union at the University of Berlin and had sent it to this organization on July 7th. At the same time, allow me to enclose an article, “Sur une certaine famille de courbes et de surfaces,” which my friend Lie and I coauthored. We presented this work to the Académie des Sciences in two sessions, on the 6th and 13th of June, and the Académie published it in its Comptes Rendus. By choosing this publica1 2

The original German documents are published in TOBIES 2019, pp. 495–524. [Stabi] Sammlung Darmstaedter. See also Section 2.6.3.

© Springer Nature Switzerland AG 2021 R. Tobies, Felix Klein, Vita Mathematica 20, https://doi.org/10.1007/978-3-030-75785-4

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tion venue, we hoped to gain deeper insight into the conditions there and to become personally acquainted with a large number of French mathematicians, and we succeeded in doing so. Finally, allow me to add that we obtained further results – “Ueber die Haupttangenten-Curven der Kummer’schen Fläche vierten Grades mit 16 Knotenpunkten” [On the Main Tangent Curves of the Fourth-Degree Kummer Surface with 16 Nodal Points] – and we recently informed Professor Kummer privately about them. At his request, we submitted this work to the Academy of Sciences in Berlin, which will publish it in its monthly reports with the date of December 15th. By expressing my deepest thanks to your Excellency for your friendly approval of my initial request and by asking for more of your kindness in the future, I remain your Excellency’s most respectfully devoted Dr. Felix Klein. 2) An application submitted by Felix Klein to the Academic Senate of the University of Erlangen for funding to improve the collection of the University Library’s mathematical section (November 15, 1872).3 Royal Academic Senate! For the purposes of a mathematician, a small library may be sufficient, but it must be entirely at his disposal, for he must constantly refer to it in the interests of his research and teaching. The mathematical section of the University Library here, however, is unfortunately not in a state that meets even the most modest requirements. Allow me to begin by briefly explaining its main lacunae to the Royal Senate. The so-called mathematical section of the University Library consists of approximately 1,200 volumes. A great majority of these, however, is utterly worthless for today’s university purposes because they pertain to engineering, architecture, etc. The smaller minority of genuinely mathematical and related works was not collected according to a uniform principle; rather, chance has played an ever-shifting role in these acquisitions, so that, besides the several works that are worthy of attention, there are also almost unbelievable gaps. Of the works by older authors, for example, the writings of Galileo and Newton are available almost in their entirety, but the library has only the last three volumes of the new complete edition of Kepler’s works, and it lacks the most important items in its collection of works by Huygens, Euler, and Lagrange. Regarding the collection of mathematical journals, the German ones (to the extent that they should be considered) are all available, but the foreign journals are entirely lacking. This is all the more regrettable because mathematics is a thoroughly international science, and the progress of a productive mathematician is considerably hindered without him having a universal overview of the findings of others at the same time. In light of the burden that the ongoing acquisition of an 3

[UA Erlangen] Ph. Th. I Pos. 20 V, No. 8. On the context of this application, see Section 3.3.

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additional journal represents for the library’s budget, however, I believe that I should limit my requests. My only proposal is that it should subscribe to a French journal which contains up-to-date reports on recent publications: the Bulletin des sciences mathématiques et astronomiques, edited by Darboux. Some time ago, a number of astronomical journals had also been acquired. With the sole exception of the Annalen published by the observatory in Munich, all of them break off at different times without any apparent reason. For example, complete holdings of the Berliner [Astronomisches] Jahrbuch exist from its beginning in 1776 to 1861. I suggest that the missing volumes should be purchased and that the subscription to this Jahrbuch should be renewed. As far as recent books are concerned, geometry is relatively the best-represented of the mathematical disciplines, given that a preference for geometry has always been cultivated in Erlangen. Yet it is far from the level of completeness that I would hope for it to achieve over time; in particular, the collection lacks certain handbooks that seem to be suited for providing an introduction to the special study of geometry. Other branches of mathematics are in part almost entirely unrepresented, and these are hardly unimportant. On mechanics, for example, there is nothing aside from Poisson’s excellent book; likewise, the most important new works on differential and integral calculus are also lacking; there is nothing to be found on mathematical physics, unless there happens to be something useful in the physics section. In these disciplines, it is necessary to create adequate conditions by filling in the discernible gaps, so that the most necessary items are available – failing which constructive instruction is not conceivable at all. I therefore take the liberty of requesting the Royal Academic Senate to apply to the highest authority [the Bavarian Ministry of Culture] for a sum of 350 Gulden to be allotted from the University’s surplus funds for the purpose of completing the mathematical section of the University Library on the basis of an enclosed cost estimate, the individual items of which are justified in the explanations above. Respectfully and with devotion to the Royal Academic Senate, Felix Klein Professor of Mathematics4

4

Dean Eugen Lommel, a professor of physics known today for the Lommel function and the Lommel differential equation, forwarded Klein’s application with an expert opinion to the Bavarian Ministry of Culture, which granted his request for 350 Gulden ([UA Erlangen] Ph. Th, I. Pos. 20 V. No. 8). – Klein’s abbreviated book titles list (see below) is translated into English; “G” designates a book by a German author or a translation into German; for example: L. Cremona, Einleitung in eine geometrische Theorie der ebenen Curven (Greifswald: Koch, 1865); Poinsot’s Elemente der Statik, als Lehrbuch für den öffentlichen Unterricht und zum Selbststudium (Berlin: Rücker, 1835); Duhamel, Lehrbuch der reinen Mechanik (Braunschweig: Vieweg, 1853).

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Kepler’s Euler.

Collected Works. Vols.1–5 ................................... 25.Introductio in analysin ............................................ 5.Calculus differentialis ............................................. 7.20 Mechanica ............................................................... 6.Methodus inveniendi lineas curvas ......................... 4.Mécanique analytique ........................................... 15.Horologium oscillatorium ....................................... 3.-

Lagrange. Huygens. Updating the Journal Collection. Darboux. Bulletin. Two volumes ........................................... 11.Astronomical Yearbook. Ten volumes G .............. 20.Geometry. Grassmann. Extension Theory 1, 2 G ........................................... 3.15 Plücker. Analyt. geometr. Developments G ........................... 1.10 Algebraic Curves G .................................................. 1.25 Hesse. Lectures. Space, Planes G ......................................... 4.20 Salmon. Geometry of Planes, of Space G .............................. 9.14 Reye. Geometry of Position G ........................................... 3.Cremona. Plane Curves G ........................................................ 2.20 Durège. Curves of the Third Order G .................................... 2.80 Sturm. Surfaces of the Third Order G ................................. 2.20 Lamé. Coordonnées curvilignes ......................................... 2.Mechanics. Jacobi. Lectures on Dynamics G ......................................... 6.Schell. Theory of Motion G ................................................. 4.20 Poinsot. Statics G................................................................... 2.10 Jullien. Mécanique rationnelle ............................................. 4.20 Duhamel. Mechanics G ............................................................ 2.20 Differential and Integral Calculus. Function Theory. Serret.* Differential-Integral Calculus ................................. 7.Bertrand. ** Differential Calculus ............................................. 12.20 Integral Calculus ..................................................... 9.Casorati. *** Function Theory ....................................................... 3.15 Durège. Elliptic Functions G ................................................. 3.Function Theory G................................................... 1.18 Koenigsberger. Elliptic Functions G ................................................. 1.10 Heine. Spherical Functions G.............................................. 2.Lommel. Bessel Functions G .................................................. 1.Neumann. Bessel Functions G ................................................... -.20 Baltzer. Elements G .............................................................. 3.15 Mathematical Physics. Beer. Optics G ................................................................... 2.Elasticity, Capillarity G ........................................... 1.10 Clebsch. Elasticity G .............................................................. 2.20 Lamé. **** Heat ......................................................................... 2.Elasticity ................................................................. 2.Total 205.20 205 Taler, 20 Silbergroschen = 359 Gulden, 55 Kreuzer * **

Klein

J.-A. Serret, Cours de calcul différentiel et intégral, 2 vols. (Paris: Gauthier-Villars, 1868); revised German ed. by Axel Harnack, vol. 1 (Leipzig: B.G. Teubner, 11884). J. Bertrand, Traité de calcul différentiel et de calcul intégral (Paris: G.-Villars, 1864–70).

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F. Casorati, Teorica delle funzioni di variabili complesse, vol. 1 (Pavia: Fratelli Fusi, 1868). G. Lamé, Leçons sur la théorie analytique de la chaleur (Paris: Mallet-Bachelier, 1861); Leçons sur la théorie mathematique et l’elasticité des corps solides (Paris: Bachelier, 1852).

3) Nomination of Dr. Felix Klein, full professor of mathematics at the Technische Hochschule in Munich, to be made an extraordinary member of the mathematicalphysical class of the Royal Bavarian Academy of Sciences, June 7, 1879.5 Since the death of our full member Dr. Otto Hesse, Dr. Felix Klein has been employed as a full professor of mathematics at the Technische Hochschule here (after previously holding the corresponding professorship at the University of Erlangen for a few years). A student of Plücker in Bonn and of Clebsch in Göttingen, he can be described as one of the most productive and ingenious representatives of that younger mathematical school in Germany which has received its direction especially from Clebsch and, like him, has chosen a certain border area between geometry and algebra as its main area of activity. – The enclosed list of publications to date, which is unusually long for such a young author, admittedly contains a few repetitions (in that the author has had some of his articles reprinted essentially unchanged in different places) and several variations on the same theme (first, for instance, a provisional announcement of an idea, which is then presented in more detail and later provided with further explanations). Moreover, some of the more significant works owe their origin to impulses provided by the publications of others (e.g., by Schwarz in Göttingen concerning the connection of the so-called hypergeometric series with the icosahedron equation and, in general, with a new way to solve equations of the fifth degree), but even so there is enough left over to cast the author’s mathematical talent, ingenuity, and astuteness in a favorable light, to secure lasting recognition for his work, and to justify further expectations for the future. Furthermore, he has rendered valuable services as one of the editors of Mathematische Annalen, the primary publication venue of the younger school mentioned above, which counts Klein as one of its most outstanding representatives. In his editorial role, he has also performed a valuable service by occasioning the publication of instructive [mathematical] models, especially pertaining to Plücker’s works. Because a vacancy has arisen on account of our associate member Dr. J. Volhard’s departure to Erlangen, we believe that no scholar here other than Klein who does not already belong to the Academy and whose work is as relevant to our 5

[AdW München] 18791 (minutes of the election sessions). The Royal Bavarian Academy of Sciences had existed since 1759, and Klein was made a member of it on June 25, 1879. The election process involved two stages, one in the mathematical-physical class (in which Klein received 15 of 15 votes), and the other in a general meeting (where 29 of 34 votes were cast in Klein’s favor). When Klein moved to Leipzig, his extraordinary membership was converted into a corresponding membership. Regarding Klein’s election, see also Paul Gordan’s assessment, which is quoted in Section 4.4.

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[mathematical-physical] class could justifiably be given the seat that has become vacant. We therefore propose to the class and eventually the entire academy that Professor Dr. Felix Klein be elected as an extraordinary member of the mathematical-physical class of the Royal Bavarian Academy of Sciences. (We have also sought and received the consent of our absent colleague Dr. C. von Bauernfeind.) Munich – June 7, 1879 Dr. Ludwig Seidel Dr. Gustav Bauer 4) A report by the Philosophical Faculty at the University of Göttingen concerning its decision to propose Felix Klein as the successor to Moritz Abraham Stern, along with separate opinions by the professors Ernst Schering and Hermann Amandus Schwarz (January 1885). 4.1) A report on the Faculty’s hiring proposals for the (third) full professorship for mathematics, to be sent to the Royal Prussian Minister of Culture (Dr. Gustav von Gossler).6 Your Excellency, Göttingen – January 18, 1885 We are honored to submit our most respectful request for the appointment of a full professor of mathematics to fill the chair that has been vacated by the retirement of Professor Stern. The departure of this man, who devoted a long and successful career to our university and whose thorough lectures contributed in no small measure to the flourishing study of mathematics here, has left a conspicuous gap in our teaching staff, and it seems urgently necessary to fill this gap as soon as possible. Out of the glorious past of our university, which counts a number of Germany’s most prominent mathematicians as its own, arises our duty to maintain and promote, with all the means at our disposal, the flourishing of mathematical studies at the university and its importance to the progress of mathematical research. The large range of mathematical disciplines, however, has long made it impossible for any one individual to master and till the field as a whole, and thus the subject of mathematics is represented at all of our universities by multiple teaching positions, and at the larger universities by several full professorships. If our university has been the seedbed in which a relatively large number of today’s representatives of mathematics received their education, it has achieved this success in large part because its larger number of full professorships in mathematics has made it possible to represent the various branches of this highly ramified science and thus to offer students a wide-ranging education. In this regard, we would like to stress that, up until the death of Hofrath [Privy Councilor] Ulrich, there were 6

[UAG] Phil. Fak. 170a, No. 41ss–41tt. Regarding the context of this report, see Section 5.8.2.

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four full professorships for the subject of mathematics here, whereas presently, after the departure of Professor Stern, there are only two scholars working as full professors of mathematics in our midst. In their teaching and research, the remaining two professors have preferred to focus on the theory of analytic functions, mechanics, number theory, curved surfaces, and curves of double curvature. It is therefore our conviction that the most effective way to complement their activities would be to appoint a representative of the geometric-algebraic approach to work alongside them. As such, our first recommendation is Dr. Felix Klein, a full professor of geometry in Leipzig who was born in 1849. Klein began his mathematical studies in Bonn and, having obtained his doctorate there, spent several semesters in Göttingen and Berlin completing them. He finished his Habilitation in Göttingen in 1871, and just a year later he was appointed a full professor in Erlangen. In 1875, he accepted a professorship at the Technische Hochschule in Munich, and he moved to Leipzig in 1880. Even in his earliest works, which proceeded in the direction initiated by Plücker and Cayley, it was possible to recognize the outstanding talent of their author. They were characterized as much by the wealth of their geometric intuition as they were by the breadth of their geometric perspective, and they created the expectation that their author would not limit himself in the further development of his scientific career to the narrow field of purely geometric investigations but would rather also, with the support of the tools of geometry, turn to other problems of mathematics. In fact, his more general examination of the various methods of geometric research led him at first to the theory of transformation groups. This formed the center of his further scientific activity, and since then he has earned a generally respected name in the scientific world through his numerous and extensive studies, which bear witness to the versatility of his intellect. Klein is an excellent teacher whose personality fills his students with sincere admiration and who knows how to inspire his students to conduct mathematical research and to encourage them to carry out independent scientific investigations. He is also very productive as an editor of the Leipzig-based journal Mathematische Annalen, and he is known to be a tireless worker in all the endeavors that he has touched. We would therefore be especially pleased if we were to succeed in bringing this excellent scholar to our university, and we have good reason to believe that he himself is likely inclined to exchange Leipzig for Göttingen for personal reasons. Should we not succeed, however, in gaining Mr. Klein for our university, we are honored to dutifully suggest to your Excellency the following gentlemen secundo loco et pari passu [on equal footing in second place]: Dr. Aurel Voß, born in 1845, currently a professor in Dresden, who was appointed to the Technische Hochschule in Munich at Easter this year; and Dr. Alfred Enneper, born 1830, an associate professor at our university. In this respect, we are aware that the direction of Professor Enneper’s scientific activity is less in line with the focal points mentioned at the beginning of this report. Only if we fail to fill the vacant professorship with a geometrician of Pro-

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fessor Klein’s caliber do we believe that it might be appropriate to take into account the years of loyal service that Professor Enneper has rendered to science and to our university. […] Should unforeseen obstacles stand in the way of appointing Mr. Klein, Mr. Voß, or Mr. Enneper, we obediently ask your Excellency to grant the faculty the opportunity to offer new proposals. The Philosophical Faculty. The Dean. W.[ilhelm] Müller (signed) 4.2) A separate opinion (Separatvotum) on the Faculty’s report, by Ernst Schering, a full professor of mathematics at the University of Göttingen.7 Your Excellency, Göttingen – January 22, 1885 The humble signee is taking the liberty of submitting, in the most respectful manner, his opinion, in so far as it deviates in essential points from the proposal made by the majority of the members of the Philosophical Faculty. The content of the latter hiring recommendation is based on an overestimation of projective geometry, which, in my opinion and in the opinion of other competent experts, leads to a false impression of the benefits that appointing Professor Felix Klein to Göttingen might bring to our university. In contrast to this proposed appointment, the appointment of our Associate Professor Alfred Enneper or Professor Georg Hettner in Berlin as a full professor in Göttingen would be of considerably greater benefit to the study and further training of the strict methods that were introduced by the great Göttingen mathematicians Gauss and Dirichlet and were also used with brilliant success by Riemann. Furthermore, such an appointment would also be of greater benefit to the education of the mathematics teachers to be employed at secondary schools, for whom certainty and clarity of thought are the most important matters. The promotion of […] Associate Professor Enneper would give well-deserved recognition to the merits of his now twenty-six successful years of academic teaching and to his valuable scholarly achievements, and it would ensure that a greater number of scholarly disciplines formerly taught by Professor Stern would still be well-represented by a full professor. In addition, this would create the possibility of appointing a new scholar to the vacant associate professorship [i.e., Enneper’s], who could also […] provide a necessary supplement to the courses offered here for the education of secondary school teachers. As a Privatdozent in Göttingen and as an associate professor in Berlin, Mr. Hettner has developed his academic activity with unusual success, not only in terms of the number of students who attend his courses but also in his ability to teach them with strictly correct and clear thinking. Regarding his research, he has

7

[UAG] Kur. 5956, fols. 9–10v.

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made significant contributions with his investigations of determinants composed of hyperelliptic integrals. That Mr. Hettner has not published anything during his appointment at the University of Berlin is essentially due to the fact that the conditions imposed upon him there require him to take on […] a heavy teaching load. His intimate familiarity with the highest branches of various mathematical fields guarantees, however, that he will go on to produce extremely valuable scholarly achievements. His research areas overlap perfectly with the disciplines represented by Professor Stern. Schering (signed) 4.3) A separate opinion (Separatvotum) on the Faculty’s report, by H.A. Schwarz, a full professor of mathematics at the University of Göttingen.8 Your Excellency, Göttingen – January 25, 1885 Please allow me to present my views, which differ in some points from the report submitted by the majority of my colleagues and which agree in essential respects with the opinion of my immediate colleague, Prof. Schering, regarding the candidates to be considered for the appointment of a third full professor of mathematics to our faculty. One of the merits of Prof. Stern, who has now retired, was his ability to meet, with tireless dedication and lasting success as long as his energies allowed, the needs not only of advanced students of mathematics but also of beginners. It is to his great credit that the students of mathematics at our university have always had the opportunity in recent decades to become acquainted with the unavoidable foundations of almost all higher mathematical disciplines (algebra, differential and integral calculus, and elementary mechanics) through the lectures of an excellent teacher and through the seminar exercises that he conducted. It is in great part due to this merit of Prof. Stern, which cannot be appreciated highly enough by his immediate colleagues, that the study of mathematics at our university could so greatly flourish. If other scholars were able to devote their mathematical lectures to the cultivation of special disciplines and were able to find a larger number of well-prepared students to attend them, this was undoubtedly only made possible by the fact that the most urgent needs of mathematical instruction were already taken care of in the best possible way by Prof. Stern. This aspect of providing sufficient care for the most urgent needs of mathematical education will also remain, in the future, of the utmost importance for the training of capable secondary school teachers. I am convinced that it will only be possible to maintain the flourishing of mathematical studies at our university if, in selecting the scientist to succeed Prof. Stern, we insist on the requirement that he must not only possess indisputably 8

[UAG] Kur. 5956, fols. 11–14.

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outstanding teaching abilities but must also offer the guarantee that, together with the other representatives of mathematics on our faculty, he will adequately tend to the needs of beginners. The subject of algebra is represented by neither of the two current full professors of mathematics here [Schwarz, Schering]; this subject, which is extremely important for the training of future secondary school teacher, is likewise not one of the disciplines represented by the current associate professor of mathematics [Enneper]. Professor Schering and I agree that, in the interest of the fullest possible representation of mathematical disciplines at our university, it is highly desirable that Professor Stern’s successor should be able to take over the representation of algebra in its entirety – without, of course, having his teaching activity restricted in any other way. Likewise in agreement with Prof. Schering, I humbly believe that Dr. Georg Hettner, born in 1854 and currently an associate professor at the University of Berlin, can be described to your Excellency as just such a scholar. Prof. Hettner conducted his algebraic studies under the direction of the most prominent researcher in the field of algebra among all living mathematicians, Mr. Kronecker in Berlin. Moreover, Prof. Hettner is also one of the most talented students of Mr. Kummer and Mr. Weierstrass in Berlin. Regarding the praise that Prof. Hettner has earned, I agree with all of the points made by Prof. Schering in his separate opinion, both with respect to the value of his scientific work and in terms of his eminent talent as a teacher. With respect to the proposals that the majority of the Faculty submitted to your Excellency concerning the appointment of Prof. Stern’s successor, I cannot oppose the proposals in favor of Professors Felix Klein and Aurel Voß. If the appointment of Prof. Klein is successful, an outstanding teacher and an important scholar will be gained for our university and for our Prussian fatherland. The same applies to Prof. Voß. In light of the criteria presented above, however, I regret very much that I cannot accept the suggestion to promote Prof. Enneper to full professor as Prof. Stern’s successor. Your Excellency’s most obedient H.A. Schwarz (signed)

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5) On the scientific polemic between Felix Klein and Lazarus Fuchs. An excerpt of a letter (in draft form) from Felix Klein to Wilhelm Förster (a professor of astronomy at the University of Berlin), January 15, 1892.9 […] Here are just a few more remarks about my relations with Fuchs. The enclosed package contains, above all, the autograph edition of a lecture that I gave in the summer of 1891.10 There, on pp. 66–89, you will find a presentation on the historical development of the study that, ten years ago, brought about the polemic between Fuchs and myself. Although this presentation was of course originally meant for my students’ ears, it will hopefully be intelligible to someone reading it at some remove. My intention in this presentation, in which I provide a precise account of my earlier ideas, is to leave an impression of utmost candor. You will also gather from the booklet that I am once again working energetically on these very issues and that I am in the process of preparing a final presentation on the entire area of research. [[From the same text, incidentally, you will also learn how I consider the task of the geometrician, which is to understand the whole field of mathematics – and, I might add, its applications – from the perspective of geometric intuition.]]11 I should further point out that there have also been two indirect conflicts between Fuchs and myself in recent years […]: 1) Fuchs publishes a theory that turns out to be just plain wrong. 2) A younger mathematician [Hurwitz] notices this, addresses Fuchs himself about it, and discovers that the latter is not as understanding as he had hoped. 3) He [Hurwitz] sends me his view on the matter12 for publication in the Göttinger Nachrichten or Mathematische Annalen, which I, finding them correct, in turn published after smoothing out the manner of his expression when necessary. As it was with Hurwitz in 1887,13 so it is now with the Russian mathematicians Nekrasov and Anisimov. To speak only of the latter: The false developments that Fuchs had originally published in Crelle’s Journal 75 were replaced by him by other erroneous statements, after Anisimov had pointed out his errors, in Crelle’s Journal 106, though now in a form with which A. felt personally dissatisfied. Nekrasov provided the

9 10 11 12

13

[UBG] Cod. MS. F. Klein 1C: 2. On the context of this letter, see Section 6.5.1.1; regarding this polemic, see also Sections 5.5.5 and 5.8.2. The lecture in question was on the theory of linear differential equations. Klein later deleted the sentence printed here in double brackets. In response to Lazarus Fuchs’s article “Über diejenigen algebraischen Gebilde, welche eine Involution zulassen” (published in the Sitzungsberichte of the Berlin Academy, July 1886), Hurwitz had written to Klein: “I recently found that all of these curves can be represented by equations f(s2, z) = 0, from which it immediately follows that they are by no means exhausted by the hyperelliptic curves, as has recently been claimed […].” ([UBG] Cod. MS. F. Klein 9: 1034, letter, Dec. 28, 1886). For further details, see TOBIES 2019, pp. 505–506.

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correct theory in Annalen 38,14 but he made a mistake in one of his secondary points, namely when he wanted to uncover the inner reason for the mistake made by Fuchs in vol. 106. Fuchs noticed this and reacted in vol. 108 with an abrasive reply and held firm, without any reservation, about the theory that he had presented in vol. 106! This was followed by passionate letters from Nekrasov and Anisimov to myself, which I just now, during the Christmas holiday, put into a subdued form and will soon publish in Mathematische Annalen.15 Of course, every incident of this sort only serves to renew and increase the blind hatred16 that Fuchs has directed toward me. He has gone so far as to disregard the basic laws of decency. My student, Dr. Fricke, who has been living in Berlin in recent years (and whom I hold in exceptionally high esteem both personally and as a mathematician), has told me a few stories about this. A year ago, he sent Fuchs the first volume of my book on elliptic modular functions, which Fricke edited (and which contains no polemics whatsoever), with a request for a personal letter of reference: he received no reply at all. It was precisely at that time when Fricke approached Kronecker (whose research was closely aligned with his own) with the question of how Kr[onecker] would assess his application to the Berlin faculty for approval to submit his Habilitation. Kr[onecker] flatly replied that he was in no position to support an application about which the faculty would yet have to make a decision. If you take all of this together, then the question of how the relationship between F[uchs] and myself would develop if we were to work together at the same university is self-evident. I would certainly not behave provocatively but would rather avoid all external conflicts as much as possible. However, I would not abandon my research plan any more than I would cease publishing what I think is right. No one can demand that I should deny my entire past for the sake of receiving a professorship in Berlin. (And even the mere fact that I might be summoned to Berlin would be a grave insult to Fuchs.) Personally, this perspective doesn’t really frighten me very much – my concerns, which I expressed in a letter to A[lthoff], are of a completely different nature – but I do not know how the matter looks from a more general point of view and whether you would want to take responsibility for having helped to bring about such unpleasant conditions. […]

14 See P.A. Nekrassoff [Nekrasov], “Ueber den Fuchs’schen Grenzkreis,” Math. Ann. 38 (1891), pp. 82–90. This article concludes with the following words: “Thus, the cases in which Fuchs’s theorems do not apply should by no means be called exceptional cases.” Hurwitz informed Klein: “Fuchs, [Meyer] Hamburger, and their colleagues are reportedly extremely agitated about Nekrasov’s publication in the Annalen […].” ([UBG] Cod. MS. F. Klein 9: 1090, a letter dated May 1891). 15 See W.A. Anissimoff [V.A. Anisimov], “Ueber den Fuchs’schen Grenzkreis,” Math. Ann. 40 (1892), pp. 145–48. – Anisimov had graduated in Moscow and completed his doctoral thesis, The Fuchsian Boundary Circle and its Applications, at the University of Warsaw in 1892. 16 Later, in this drafted letter, Klein replaced the term “blind hatred” with “feeling of antipathy.”

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6) Letters concerning the potential successor to H.A. Schwarz’s full professorship at the University of Göttingen.17 6.1) An excerpt of a letter from Klein to Adolf Hurwitz, February 28, 1892.18 […] Althoff spent three days here and brought the new appointments in Berlin to a conclusion. Besides Frobenius, Schwarz will be appointed full professor [in Berlin] on April 1st. – I myself, if I may say so here, am quite pleased with the way things have turned out. For I feel as though I was treated rather honorably, and I have also gained some freedom of movement. But to remain on point: now it will be necessary to fill Schwarz’s position here in Göttingen, and this should happen in the near future. I also know exactly what suggestions I would like to make to the Faculty (although you must keep in mind, of course, that I am not the Faculty; I even expressly want to reserve the freedom to modify my current ideas over the course of the forthcoming negotiations): You will roughly guess that I intend to propose you and Hilbert as the only two people who would be able to help me safeguard the status of our scientific reputation in relation to Berlin […]. And now the great difficulty, which required a great deal of deliberation before I decided to write to you about it myself. It goes without saying that I will name you first and Hilbert second. There are, however, a number of concerns about appointing you, and the question is to what extent I should express these concerns and perhaps even admit outright that Hilbert’s presence here might ultimately meet our needs to a greater extent than yours. The first matter is that of your unstable health, the importance of which I do not want to overstate, but I cannot ignore it entirely. Second, there is the far more subtle reason that you are much closer to me than Hilbert, not only personally but also in the way that you think mathematically, so that your activity here would perhaps lend Göttingen mathematics an excessively one-sided character. The third issue – I have to bring it up, as repugnant as the matter is to me and as much as I know how justifiably sensitive you are about it – is the Jewish question. It is not that your appointment would pose any difficulties; these I could overcome. However, we already have Schoenflies, to whom I am always interested in offering a permanent position here (a salaried associate professorship). And I will never be able to accomplish this with the faculty or with the minister if you and Schoenflies are employed side by side! But I must come to an end. For me, the decision between you and Hilbert would be difficult enough if I merely had to weigh the reasons for or against either of you objectively. Now, however, there is also the subjective difficulty, which is that I would like least of all to offend you in the present situation; rather, I would like to do everything I can to be helpful to you. Please write me a line of 17 For the context of these developments, see Section 6.5.1.2. 18 [UBG] Math. Arch. 77: 228.

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reassurance immediately, if possible; but in any case, please speak your mind against me just as unreservedly as I have done here. The faculty meeting is scheduled to take place on Thursday, and I will certainly have your answer by then!19 Whatever the outcome, promise me that our personal relationship will not suffer from it. With best wishes I remain Yours, Felix Klein 6.2) An excerpt of a letter (draft) from Felix Klein to Friedrich Althoff, March 7, 1892.20 Esteemed Senior Privy Councilor! Please allow me to submit a report to you today, even before the faculty’s new hiring proposals are ready, concerning Dr. Schoenflies [a Privatdozent at the University of Göttingen]. You touched upon the question of anti-Semitism. The impression I have from all sides is that no one would take any offence at all with one Jew, but that the appointment of two Jews simultaneously would be considered inadmissible.21 So if Hurwitz were to come here (as I still advocate), then I would have to sacrifice Schoenflies. I would deeply regret that, because I have come to appreciate Schoenflies’s unique talent more and more, not only from a personal perspective but also in the interest of our university. In essence, Schoenflies is a highly gifted man; he has a penetrating mind and, what is more, outstanding teaching abilities in the popular sense. I have repeatedly praised his geometric talent. If he lacks anything, it is consistent energy: he must be forced to do things from the outside, but then he works just as quickly as he does surely. (I have no doubt that you will receive different opinions about Sch[oenflies] from other sources. Yet in contrast to what you may hear, I would like to refer to the fact that, from Christmas onwards, I attended the two-hour seminar conducted by Schoenflies and Burkhardt no less than eight times; they were truly scientifically lively, and any doubt about the qualifications of the seminar leader, who repeatedly gave lectures himself, is out of the question for me.) As I have already noted, however, the decisive factor for me is the consideration of the intended appointment [i.e. Schwarz’s successor]. […] 19 Klein was unable at the time to implement Hilbert’s appointment (Hilbert was still a Privatdozent then), and so he fought on behalf of appointing Hurwitz. 20 [UBG] Cod. MS. F. Klein 1C: 2, fol. 22. See Section 6.5.1.2. 21 Klein was aware of the widespread anti-Semitism at the time, but he made his own judgements on professional grounds. On December 13, 1887, for example, Moritz Pasch, who came from a Jewish family, had written to Klein about Klein’s recommendation that Max Noether and Adolf Hurwitz would be fitting candidates for a professorship at the University of Gießen: “Of course, I was not allowed to include any Jewish colleagues on the list, otherwise scholars such as [Max] Noether and Hurwitz would not have been left out of it” (quoted from SCHLIMM 2013, p. 198).

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In this respect, I have no doubt that our main goal in the present situation should be to appoint someone who complements my interests and can thus lead to the creation of a more rigorous school. Prof. Schwarz was excellent in this respect; Prof. Lindemann will not be equally so. […]22 After he had enjoyed the encouragement of a particularly gifted teacher of synthetic geometry at school, Prof. Hurwitz originally studied with me, then also in Berlin with Kronecker and especially with Weierstrass. Prepared in this way, he brings together all the premises for the epitome of modern research, which aims to combine function theory (as a central discipline) with number theory and geometry. And to these premises, Hurwitz also adds a brilliant level of productivity, which never falters despite the obstacles that his somewhat delicate health puts in his way. The number of his publications, and the number of different subjects that Hurwitz has written about, is very high: In my ranking (from my subjective point of view), his best works are those in which he tackles the problems of geometric function theory which I originally raised, and takes them much further than I ever could have done. This work represents the complement that I have been looking for, and it does so in tangible reality and the most satisfying completeness. […] 7) Felix Klein on the draft of Ludwig Bieberbach’s dissertation, which was supervised by the Privatdozent Paul Koebe at the University of Göttingen.23 Dear Mr. Bieberbach! Göttingen – May 15, 1909 I have just read through your dissertation, and I have great reservations about Part I. What is right is not new, and what is new is wrong. The proof: Ad. 1. That the canonical cut systems are by no means determined by the periods of Abelian integrals is not only pointed out but also explained in Math. Ann. XXI, pp. 184–85.24 – Fricke refers to this in Vol. 1 of the Automorphen [Funktionen], p. 324, where he proves that two operations are sufficient to produce all canonical cuts.25 Ad. 2. The theorem on the generation of all binary period transformations in the hyperelliptic case by the monodromy of branching points is only correct for p = 2; for p = 3, there are already 36 separate families [Scharen]. I had H.D. Thompson treat this matter in his dissertation (American Journal XV, available in the reading room […]).26 22 What follows here are some laudatory words about Lindemann, because Lindemann had complained about Klein’s preference to appoint the younger Hurwitz and Hilbert. 23 [Deutsches Museum] Nachlass Ludwig Bieberbach. I am indebted to Reinhard SiegmundSchultze for bringing this letter to my attention. – Regarding the context, see Section 8.5.3. 24 Klein’s reference here is to his own article: “Neue Beiträge zur Riemann’schen Functionentheorie,” Math. Ann. 21 (1883), pp. 141–218. 25 See, for the English translation, FRICKE/KLEIN 2017 [1897], pp. 263–64. 26 See H.D. Thompson, “Hyperelliptische Schnittsysteme und Zusammenordnung der algebraischen und transcendenten Thetacharacteristiken,” Amer. J. of Math. 15 (1893), pp. 91–123.

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But it is also false to reduce the question of any given Riemann surface to two-sheeted surfaces. The source of the error is the fact that Clebsch-Lüroth did not understand the “sheet” of a R[iemann] surface as a piece of the surface that covers the […] plane exactly once, but rather only as any simply connected piece. Now, what should happen in this situation? I am unfortunately extremely busy these days in light of the negotiations that are about to begin again in the Upper House [of Parliament]. Nevertheless, let me ask you to come to the mathematical collection room on Monday at noon. By the way, would you also please show this letter to Dr. Koebe?27 Yours sincerely, F. Klein 8) Dr. Klaus, a neurologist at the Sanatorium for Neurology and Internal Medicine in Hahnenklee: two reports on the state of Felix Klein’s health.28 Report from March 9, 1912 The full professor of mathematics, Privy Councilor Klein from Göttingen, has been in my sanatorium since December of last year. Owing to years of overexertion in his profession, and due to the fact that he has never allowed himself a necessary period of rest and relaxation, he suffers from a state of extreme exhaustion and irritability of his nervous system. There is no trace of an organic disease. This is a matter of genuine neurasthenia, i.e., a temporary and curable weakness of a healthy nervous system. The duration of the healing process will of course be quite long, given the extent and intensity of the damage that has been done. Even though the patient set aside his courses for the winter semester of 1911/12 and came here for a cure, and even though there has been a slight but unmistakable improvement in the state of his nervous exhaustion, it is nevertheless from a medical point of view entirely impossible that Mr. Privy Councilor Klein should be allowed to lecture again in the summer semester of 1912. If he were to return to work prematurely, he would undoubtedly, as a consequence, relapse to his previous state of weakness and thereby prolong the difficult healing of his suffering. I therefore humbly request, for the urgent reasons mentioned above, that my patient be granted further leave until the winter semester of 1912/13. He will continue to remain up here in my sanatorium. Dr. Klaus – Neurologist (signed) 27 Bieberbach had attended Klein’s seminar from 1906/07 to 1908 (which Klein directed together with Hilbert and Minkowski) and he transcribed (with Max Caspar) the lectures that Klein gave on automorphic functions from May 1 to July 31, 1907 (see [Protocols] vol. 26). As a full professor, Klein had to evaluate Bieberbach’s dissertation – “Zur Theorie der automorphen Funktionen” [On the Theory of Automorphic Functions] – which had been supervised by the Privatdozent Paul Koebe. For a brief biography of Bieberbach, including his grades and the subjects of his doctoral examination, see TOBIES 2006, pp. 58. – For Bieberbach’s support of Nazism, see in particular MEHRTENS 1987; 1990; 2004. 28 [UBG] Cod. MS. F. Klein 2D, fols. 65, 66. See Section 8.5.1.

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Report from October 10, 1912 At the end of September of this year, Privy Councilor Professor Dr. Klein, Göttingen, left my sanatorium, where he had been staying since the beginning of the year, and returned to his university town. In the peace and quiet that prevails up here, he ultimately felt so rejuvenated that he regained the hope of teaching his lecture course in the winter semester of 1912/13. I have now examined Mr. Klein in detail since his departure from the sanatorium, and I unfortunately found that his nervous system is still very easily exhausted and that he had suffered from especially severe bouts of intestinal neurosis. As a result, the patient became very ill again, though only temporarily. Given the rapid change of environment and the cessation of his cure, such relapses are not alarming and are not a sign of a worsening prognosis. However, it is necessary to take them into serious consideration. In order to prevent Privy Councilor Klein from falling victim to greater exhaustion again, I have to prohibit him, from a medical point of view and in the interest of his health and full recovery, from giving lectures in the winter semester of 1912/13. Dr. Klaus 9) Nomination of Felix Klein to be made a corresponding member of the Royal Prussian Academy of Sciences in Berlin, February 27, 1913.29 The undersigned have the honor of proposing to the Academy the election of Professor Felix Klein as a corresponding member in the field of mathematics. Klein was born on April 25, 1849 in Düsseldorf. He studied in Bonn, Göttingen, and Berlin; received his doctorate in Bonn in 1868; became a Privatdozent in Göttingen in 1871, a full professor in Erlangen in 1872, and from there he went to the Technische Hochschule in Munich in 1875. In 1880, he accepted an appointment at the University of Leipzig, and he has been teaching at the University of Göttingen since 1887.30 Klein, one of the few mathematicians still able to survey the whole of mathematics, was originally a geometrician. Familiar with the ideas of the ingenious Plücker, he began his career with work pertaining to the theory of line complexes. He set up a simple form for the equation of a second-degree surface in line coordinates; determined, together with Sophus Lie, the main tangent curves of Kummer’s surface; and provided, for the line complexes, an analogue of Dupin’s theorem of curvature. His own investigations and his knowledge of the ideas of Plücker, Staudt, and Sophus Lie led him to consider the questions treated by his predecessors from a common point of view, and in his inaugural address in Erlangen he united all the geometric ideas that were new at the time into a whole by pre29 [BBAW] Bestand PAW (1812–1945), II-III-135, fols. 70r–71v (Friedrich Schottky’s handwriting). 30 The year 1887 is incorrect; Klein began working in Göttingen in April of 1886.

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senting thoughts rather than formulas.31 For a geometrician who thought in such general terms, Riemann’s theory of Abelian functions must have had a strong attraction. The interest that Klein always showed for Riemann is particularly evident in his book on elliptic modular functions. This book – two imposing volumes32 – contains lectures given by Klein and includes many of his previously published articles. One fine chapter in it contains Klein’s presentation of Riemann’s theory. Essentially, the book is concerned with a single, but highly important, function: the modular function. Its function-theoretical character had first been recognized by Gauss, who never published anything about it himself; however, a page from his estate with a few characteristic drawings by Gauss was discovered and published in time to establish Gauss’s priority on this point. Modular functions were already associated with highly interesting algebraic research by Gauss and Jacobi, and Klein treated these ideas with new means and in a new form. Klein, however, regarded this entire work as just a preliminary stage for conducting a comprehensive investigation of those functions which were discovered around that time (in some cases much earlier); Poincaré called these Fuchsian functions and Kleinian functions, while Klein referred to them as automorphic functions. If, as Klein did, we take the Gauss-Jacobi modular functions as our starting point, we have the first example of an automorphic function, and this immediately reveals some of the depth and difficulties of the theory that Klein formulated in competition with Poincaré. If the issue at hand were to define, in a direct way, the simplest and essentially most important functions newly developed by German mathematicians at the time, then one would have to proceed differently. Besides Poincaré, however, it was Felix Klein who did the most work on automorphic functions. These investigations also fill two imposing volumes.33 Klein also made his other Göttingen lectures, which contain the most important results of his work, accessible to wider circles by having them reproduced in autograph copies. The latter include lectures on the icosahedron, on non-Euclidean geometry, on the application of differential and integral calculus to geometry (a revision of principles), on the hypergeometric function, and on Riemann surfaces. It is to be hoped that all of these will be printed as books.34 Then there will exist a large multivolume textbook on analysis, full of unique geometric methods and with a geometric inclination, for Germany – and this is very desirable. Felix Klein’s tireless activity has not been restricted to his own scientific research. It is to his credit that the difficult task of publishing Gauss’s Nachlass has now been achieved almost completely and in an exemplary manner, and that the 31 What is meant here is not Klein’s inaugural address [Antrittsrede] in Erlangen (see Section 3.2) but rather his booklet Erlangen Program (KLEIN 1872). 32 See KLEIN/FRICKE 2017 (1890/92). 33 See FRICKE/KLEIN 2017 (1897/1912). 34 Klein’s lectures on the icosahedron – Vorlesungen über das Ikosaeder (Leipzig: B.G. Teubner, 1884) – already appeared as a book in its first edition (English trans., 1888; repr. 2019). The other lectures mentioned here were first reproduced in handwritten (autograph) copies. Later, Klein himself prepared some of them to be published as books; see Section 9.2.3.

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great work of the mathematical ENCYKLOPÄDIE was begun and vigorously continues. He was indefatigably active at congresses to promote the union of mathematicians and to identify important goals. Almost all academies and mathematical societies count him among their members, and several universities have awarded him an honorary doctorate. In recent years, his zeal has mainly been devoted to elevating the level of mathematical education at universities and secondary schools, and he has been just as tireless in these efforts as he was as a mathematician in the decades before. H.A. Schwarz, [Georg] Frobenius, [Friedrich] Schottky, [Max] Planck35 10) Speeches given on May 25, 1913 upon the presentation of Max Liebermann’s portrait to Felix Klein.36 10.1) A welcome speech by the physicist Eduard Riecke. Dear friend! Old age is generally a somewhat dubious asset; today, however, I am grateful for it, for it gives me the beautiful duty and heartfelt pleasure of addressing a few words to you on behalf of our colleagues. I still vividly remember Clebsch’s course that we attended, where I saw you for the first time and where I, without having exchanged a word with you, told myself that there was something special behind those eyes and that forehead. When I got closer to you over the course of the semester, I recognized that my eyes had not deceived me. During the following semesters, the events of the war took you away from Göttingen, and it was not until the summer of 1871 that we reunited as Privatdozenten in Göttingen. In addition to the geometric problems that occupied you at that time, you had a lively interest in physics; you also gave a lecture course on the theory of light, accompanied by experiments. The Fresnel mirror that was constructed at the time is still in the Institute of Physics today, though not entirely in its original state.37 When Stern resigned from his professorship in 1885,38 the distant hope of winning you back to Göttingen appeared. The endeavor was successful, despite 35 Klein’s election as a corresponding member of the Berlin Academy took place at the same sessions as Hilbert’s. In the mathematical-physical class, both were elected unanimously on May 29, 1913. In the election that took place during the Academy’s general assembly on June 10, 1913, Klein received 40 out of 44 votes, while Hilbert was elected unanimously. 36 [UBG] Cod. MS. F. Klein 107. Regarding the context, see Section 8.5.2. 37 In 1816, the French physicist Augustin-Jean Fresnel had first described his double-mirror experiment, which was used to demonstrate interference phenomena. 38 Moritz Stern had already applied to be released from his duties on April 19, 1884, and his request was approved on October 1, 1884 (see Section 5.8.2).

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the great difficulties involved. Althoff, who was sympathetic to the plan itself, feared that it would be rejected. I accompanied him on the way back from the observatory into town [Göttingen], and it was then that his resolution was made. At the corner where the Schildweg branches off from the yard of the barracks, he asked me: “Can you promise that Klein will accept?” “Yes,” I replied, and thereupon he agreed to initiate your appointment. I was well aware that you, as our new colleague, would not be guided by the saying “quieta non movere,”39 but that is precisely why I considered your appointment so important and why it seemed to me to be a matter of life and death for our university. At that time, there was a danger that it would sink to the level of a purely provincial university, so a new impulse from the outside was urgently needed. That we were not mistaken in this assumption is clear from the history of the last decades: the reorganization of mathematical instruction, the reorganization of the Society of Sciences [i.e., the Academy], the Göttingen Association – to name only those things that pertain especially to Göttingen. In all of this organizational work, we have always admired three things above all: the high standpoint and the broad perspective with which you embraced all the issues under consideration; the way in which you knew how to recognize the relationship between the most varied things in order to direct all of them toward one and the same goal; finally, your perfect sense of justice toward all matters and people, and the idealism and selflessness with which you put your work at the service of the general public. In the end, everything depends on the point of view from which one looks at things; the higher this is, the more personal opinions disappear, and the more clearly objective interests appear. Thus, the way that you have been working for the good of our university, for the good of science, and for the good of our whole culture has unintentionally had another effect. Surely the highest values of man do not lie in the intellectual but in the ethical sphere. Many of your friends and colleagues may have experienced that after chatting with you, they felt like a hiker in the pure mountain air high above the daily hustle and bustle of people, which disappears in an unsubstantial glimmer far below. Everything petty and limited has vanished and fallen away, and the soul is filled with great and pure feeling. For some of your students, this effect may have even been of deeper significance than the immediate effects of your teaching, and so an especially warm thanks for this as well. I am not merely standing here, however, as a representative of your Göttingen colleagues but rather on behalf of a large number of friends, admirers, and students from all parts of the world, and on their behalf I would now like to read what they have to say to you today, what they would like to thank you for, and what their wishes are for you.40

39 Quieta non movere = “Do not move settled things,” i.e., “Don’t rock the boat.” 40 The message that Riecke then read out contains nothing new. What is interesting, however, are the seventy-one names of the people who signed an appeal for donations for the portrait at the beginning of 1912: A. Ackermann-Teubner, L. Bianchi, O. Blumenthal, M. Bôcher, H. v.  

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10.2) Felix Klein’s acceptance speech. My dear colleagues and friends! For the extraordinary honor that you have bestowed on me with your words and with this painting, I would like to express my deepest thanks to you and to those who commissioned it. I regard this picture as a symbol for the hope that the construction of the independent Mathematical Institute, as we have planned to build it in the immediate vicinity of the Institute of Physics, will in fact come about.41 May it there find its place as a symbol of my efforts to bring the mathematical sciences into a lively relationship with their neighboring disciplines to form a large and versatile whole, which goes beyond the achievements of any individual and certainly beyond my own modest contributions. Your address, however, is a dear reminder to me of the many friends and colleagues with whom I have been able to establish a relationship along my way. Cooperation with like-minded people has always been my real element of life. I also hope, on this basis, to be able to encourage the continuation of the summarizing works that I have been managing for years despite the inhibitions that the state of my health imposes on me. In the meantime, new life is blossoming around me, enriching me and pushing me ever onward and upward, and I accompany its increasing importance with ever more participation. I could wish for nothing better than to have the time in which I worked appear later, in retrospective contemplation, as a time that prepared the way for a new ascent. I thank you again and ask you to convey these thanks in a suitable form to everyone involved.   Böttinger, A. v. Brill, H. Burkhardt, C. Carathéodory, G. Darboux, R. Dedekind, W. v. Dyck, E. Ehlers, F. Enriques, H. Fehr, L. Fejér, R. Fricke, R. Fueter, P. Gordan, G. Greenhill, G.B. Guccia, A. Gutzmer, J. Hadamard, O. Henrici, D. Hilbert, E.W. Hobson, A. Höfler, O. Hölder, A. Hurwitz, G. Kerschensteiner, P. Koebe, J. König, A. Krazer, E. Landau, E. Lange, W. Lietzmann, C. v. Linde, F. Lindemann, F. Mertens, F. Meyer, O. v. Miller, G. Mittag-Leffler, J. Molk, E.H. Moore, C.H. Müller, M. Noether, W. Osgood, E. Picard, H. Poincaré, E. Riecke, K. Rohn, C. Runge, R. Schimmack, H. Schotten, F. Schur, A. Sommerfeld, P. Stäckel, V.A. Steklov [W. Steckloff], O. Taaks, A. Thaer, P. Treutlein, A.V. Vasilev [A. Wassiliew ], G. Veronese, W. Voigt, V. Volterra, K. Von der Mühll, A. Voss [Voß], E. Waelsch, H. Weber, A. Wiman, W. Wirtinger, H.G. Zeuthen. (Not all of those who signed this appeal were themselves among the donors, see Figure 43). The appeal attracted 331 donors. They came from Austria-Hungary (J. König, G. Rados, G. Zemplén ...), Australia, Belgium, Canada (J.C. Fields), Denmark (P. Heegaard...), France (P. Appell, E. Borel, B. Boutroux ...), Germany, Great Britain (A. Berry, G. Darwin, A.E.H. Love), Greece (C. Stephanos), India, Italy (G. Castelnuovo, G. Loria, E. d’Ovidio, E. Pascal, C. Segre, ...), Japan (R. Fujisawa, T. Yoshiye), the Netherlands (L.E.J. Brouwer, ...), Portugal (F.G. Teixeira). They were Polish mathematicians (S. Dickstein, K. Zorawski), came from Russia (including Nadezhda N. Gernet, A.N. Krylov [Kriloff], A.A. Markov [Markoff], D.M. Sintsov [Sinzov], ... ), from Sweden, Switzerland (including E. Fiedler, C. Jaccottet, F. Rudio, A. Weiler), the USA (F.N. Cole, F. Franklin, M.W. Haskell, D.E. Smith, V. Snyder, H.W. Tyler, E.B. Van Vleck, F.S. Woods, ...). 41 As mentioned, the institute (Bunsenstraße 3–5) was not completed until 1929; see 9.4.2.

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Figure 43: A list of donors who sponsored Max Liebermann’s painting of Klein’s portrait in 1912 [Hillebrand].

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Figure 43: A list of donors who sponsored Max Liebermann’s painting of Klein’s portrait in 1912 [Hillebrand].

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Figure 43: A list of donors who sponsored Max Liebermann’s painting of Klein’s portrait in 1912 [Hillebrand].

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Figure 43: A list of donors who sponsored Max Liebermann’s painting of Klein’s portrait in 1912 [Hillebrand].

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11) Virgil Snyder from Ithaca (New York) to Felix Klein, a letter, dated July 4, 1924, concerning the International Congress of Mathematicians in Toronto, Canada from 11 August to 16 August 1924.42 Dear Mr. Privy Councilor, My sincerest thanks for your kind letter, which arrived here just as I had to go to the hospital for an operation. I would now like to return your greetings and, at the same time, explain my position regarding the upcoming Congress in Canada. It was during my stay in Rome two years ago that I first learned that the next Congress would be held in the United States, but I heard nothing then about the participation or non-participation of various nations. As far as I know, the matter was never discussed by the American Mathematical Society; it certainly, in any case, never came to a vote. Professor R.C. Archibald,43 who was in Rome at the same time, and I made plans to invite at least some of the mathematicians who would participate in the Congress to give lectures, after its conclusion, at our universities, perhaps in the form of shorter courses that could be taught in a few weeks. In this way, we hoped that the students and faculty at our universities could get to know these men and that we could ease the financial burden of the long journey for the mathematicians themselves. It will interest you to hear that Castelnuovo, Enriques, Levi-Civita, Segre, and Severi,44 to whom I communicated this plan, found it extremely acceptable, and each of them wished that such invitations would also be sent to representatives from Central Europe. I never learned why the financial support that was initially promised to us was then withdrawn, but this made it necessary to cancel the American invitations. As soon as the invitation from Canada arrived and was accepted, I presented our plan (Archibald’s and mine) to the chairman of the Committee of the Cana-

42 [UBG] Cod. MS. F. Klein 11: 1040A (the original letter, written in German, is published in TOBIES 2019, pp. 521–23). Snyder had spent four semesters (1892/93 to 1894) attending Klein’s lecture courses, and he twice lectured in Klein’s seminar. On the basis of Snyder’s presentation on sphere geometry ([Protocols] vol. 11, pp. 265–73), Klein steered him toward his dissertation: “Ueber die linearen Komplexe der Lie’schen Kugelgeometrie” [On the Linear Complexes of Lie’s Sphere Geometry] (1895). As of 1910, Snyder was a full professor at Cornell University (Ithaca, New York). Following in Klein’s footsteps, he concentrated his research primarily on the field of algebraic geometry. See also PARSHALL/ROWE 1994. 43 The Canadian-born Raymond Clare Archibald (as of 1908 at Brown University, Rhode Island, USA) had completed his doctorate in 1900 with Theodor Reye in Straßburg. 44 About these Italian mathematicians, Klein wrote: “The general algebraic problem of birational transformation of surfaces was then developed further [after Clebsch and M. Noether], especially by the young Italian school, to which belonged Segre, Veronese, Enriques, Castelnuovo, and Severi” (KLEIN 1979 [1926], p. 295).

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dian Congress, Professor J.C. Fields45. He approved it and asked me to develop it further, etc. There was no talk about restricting the invitations. I then sent inquiries to the universities of Chicago, Harvard, and Cornell. From Chicago I received the news that participation in the Congress would be restricted. I was asked to abandon my plan because it would only serve to embarrass the Canadian committee. When I asked for further clarification, I received a long letter from Professor Dickson.46 He wrote that the whole world was aware of the situation – why rustle everything up again? But at the meeting of the Mathematical Society in New York soon thereafter, nobody knew anything about it. I spoke with about forty participants. After a long discussion with Mr. Fields, however, I realized that I could not carry out my plan and I gave it up. Yet now I have an opportunity to implement something that is very dear to my heart. Various American organizations (the Mathematical Society, the Physical Society, the National Academy, etc.) were asked to appoint representatives to form the American group of the International Mathematical Union. Of these representatives – seventeen in all – three have now been elected as delegates to Toronto. The elected delegates are Coble, Richardson,47 and Snyder. We were asked to submit the following proposal: “Resolved that the International Mathematical Union request the International Research Council to consider whether the time is ripe for removing the restrictions on membership in the International Union, now imposed by the rules of that Council.”48 Should this matter not be considered, we have decided to withdraw completely from the [Mathematical] Union.49 In the hope that you are back in good health, and with best regards to you, I am Your devoted Virgil Snyder

45 The aforementioned Fields Medal donor, see Section 8.2.2. In the winter semester 1894/95, John Charles Fields had attended Felix Klein’s lectures on number theory ([UBG] Cod. MS. F. Klein 7E, p.175v) and participated in his seminar without giving a lecture. 46 Primarily an expert in algebra and number theory, Leonard Eugene Dickson had studied at the University of Chicago under H. Maschke, O. Bolza, and E.H. Moore. Dickson had also spent time studying under Sophus Lie in Leipzig and C. Jordan in Paris. He was aware of the antiGerman attitude that then prevailed among French mathematicians. See also Section 9.1.1. 47 Regarding A.B. Coble, see also Section 6.3.1. R.G.D. Richardson had studied in Göttingen, before earning his doctorate from Yale (1906). He became a professor at Brown University. 48 This quotation appears in English in the letter. 49 The proposal made by the American delegates in Toronto in 1924 was supported by participants from Denmark, Great Britain, Italy, the Netherlands, Norway, and Sweden. Snyder, the president of the American Mathematical Society in 1927/28, also served as a delegate at the 1928 Congress in Bologna, where German mathematicians were once again allowed to participate, though not without heavy resistance from nationalistically minded mathematicians (see SIEGMUND-SCHULTZE 2016a).

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Figure 44: The certification of Felix Klein’s election as a foreign associate of the National Academy of Sciences of the United States of America, April 21, 1898 ([UBG] Cod. MS. F. Klein 114: 22).

 

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12) David Hilbert’s eulogy for Felix Klein, delivered at the session of the Göttingen Mathematical Society held on June 23, 1925, one day after Klein’s death.50 Ladies and Gentlemen. Allow me to say a few words before today’s agenda begins. Our dear teacher, colleague, and friend Felix Klein passed away gently in his sleep last night. His end was peaceful and painless; it did not come as a surprise but was long foreseen. Now that it is here, however, the event has touched us all deeply and has shocked us all in the most painful way. For until that moment, Felix Klein was still with us; we could visit him, listen to his advice, and see how vivaciously he participated in matters of all sorts. Now this is over. A great spirit, a strong will, and a noble character has been taken from us. – This is not the place to pay tribute to Klein; no such tribute could be made in just a few words. For his activity and work were so varied and so enormous that it is impossible to focus on any single aspect. It is even impossible to decide whether he was most effective as a teacher, as a researcher, or as a personality. As a teacher, we commemorate here above all his brilliant presentations and lectures. But if we want to identify his great feature, we would have to describe how, in contrast to the prevailing trend toward abstraction and formal aspects, he always emphasized what was intuitive [das Anschauliche] and applicable, thus expressing and underscoring the multifaceted nature of mathematics. And with this tendency he was successful, despite strong countercurrents. And the scientific sign in which he conquered was the name Riemann,51 which he wrote onto his banner. As far as Klein the researcher is concerned, there is hardly a single mathematical field that has not been cultivated by him. Especially geometry and, in particular, geometric function theory. It was precisely the most profound theorems about uniformization that he first foresaw; he also provided the bases for the proofs, and today the whole structure stands strong, elaborated by his students.52 He used the remaining energy of his final years to give us an especially precious gift: the three volumes of his collected works, a prime example of how to edit the works of a scholar.53 Yet even if Klein’s activities for the world and for science may be the main issue, for us there is still the essential question of what he created for Göttingen: a new golden era [Blütezeit], and for this he not only laid the foundations but also issued the guidelines for how to perpetuate it into the future. Everything that you see here is the product of his personality: the reading room, the model collection, the […] institutes, the appointments, the goodwill of the ministry, the important figures from industry whom he won over [as Göttingen’s patrons]. We owe this to

50 51 52 53

[UBG] Cod MS. Hilbert 575: No. 3. An allusion to the Latin expression “In hoc signo vinces” [In this sign thou shalt conquer]. See Section 5.5.4. KLEIN 1921/1922/1923.

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his personality, through which he always and everywhere had success on his side. But what was it about his personality that led to such success? The secret to his success lay in his incorruptible objectivity. Grand goals, never petty or personal ancillary goals. Thus did Klein also bequeath his spirit to us, so that we might continue to work in his spirit. Let us continue as long as this spirit does not fade away.

Figure 45: Felix and Anna Klein’s gravestone in Göttingen’s old city cemetery (photograph courtesy of Dr. W. Mahler).

BIBLIOGRAPHY [AdW Göttingen] Archiv der Akademie der Wissenschaften zu Göttingen. [AdW Leipzig] Archiv der Sächsischen Akademie der Wissenschaften zu Leipzig (Record F. Klein: 1 fol. personal data). [AdW München] Archiv der Bayerischen Akademie der Wissenschaften München. [AdW Wien] Archiv der Österreichischen Akademie der Wissenschaften in Wien (Vienna). [Archive St. Petersburg] Archive of the Russian Academy of Science, St. Petersburg (Election proposals in 1895; Estate of A.A. Markov: Fond 173, Opis 1, No. 40). [Archiv TU München] Directorium der Kgl. Polytechnischen Schule München (Dr. Felix Klein, Kgl. o.ö. Professor, Registratur II 5; X2d). [BBAW] Archiv der Berlin-Brandenburgischen Akademie der Wissenschaften (Nachlässe [NL] H.v.Helmholtz; W. Ostwald, H.A. Schwarz [Schwarz to Weierstraß, No. 1254; Weierstraß to Schwarz, No. 1175]); Bestand PAW 1812-1945). [BBF] Archiv der Bibliothek für bildungsgeschichtliche Forschung Berlin: Personalbögen [Personnel files of Prussian secondary school teachers]. [BHSt] Bayerisches Hauptstaatsarchiv München, Akten des Kgl. Staatsministeriums des Innern für Kirchen- und Schulangelegenheiten. [BStBibl] Bayerische Staatsbibliothek München, Handschriftenabteilung, Dyckiania. [Blaschke] Estate of Wilhelm Blaschke. Mathematical Society in Hamburg (MHG), Klein’s letters to Blaschke (made available by Prof. Dr. Alexander Kreuzer, Jahrverwalter of the MHG). [Canada] Canada Institute for Scientific and Technological Information, National Research Council, Ottawa. K.A. Ø S 2. [Debye] Kuhn, T.S.; Uhlenbeck, G., Interview with Peter Debye dated May 3, 1962. Rockefeller Institute, New York City. [Deutsches Museum] Archiv des Deutschen Museums München; Dept. Sondersammlungen. [Gymnasium Düsseldorf] Zeugnis der Reife Felix Kleins (2.8.1865); schriftliche Abiturarbeiten. [Hecke] Estate of Erich Hecke, Mathematical Institute, University of Hamburg. [Hillebrand] Private Estate of the Barbara and Meinolf Hillebrand family in Scheeßel. [Innsbruck] Estate Otto Stolz (Klein’s letters to Otto Stolz, transcribed by Christa Binder). [Lindemann] Private Estate of Irmgard Verholzer née Balder (Ferdinand Lindemann’s granddaughter): Ferdinand Lindemann’s Memoirs, 1911. [Math. Institute Leipzig] Klein, unedited lectures (kept by Frau Ina Letzel). [MPI Archiv] Max-Planck-Archiv Berlin-Dahlem, Repertorium Prandtl, III. Abt., Repositur 61. [Oslo] Nasjonalbiblioteket, Estate of Sophus Lie (I. Klein’s letters to Lie; II. Klein: “Ueber Lie’s und meine Arbeiten aus den Jahren 1870–72” dated Nov. 1, 1892 (published in ROWE 1992a, 588–604); III. Unpublishcd records by Friedrich Engel, prepared for Lie’s Collected papers). [Paris] Bibliothèque de l’Institut de France, Faculté des Sciences de Paris, MS 2719 (Lettres de Klein à Gaston Darboux). [Paris-ÉP] École polytechnique. Archives. Art. VI §2 Sect. a2 (Klein’s letters to Jordan). [Pisa] Scuola Normale Superiore Pisa, Biblioteca, Fondo Enrico Betti. [Protocols] vols. 1-29. Protocols of Klein’s Mathematical Seminars. [UBG], online: http://www.uni-math.gwdg.de/aufzeichnungen/klein-scans/klein/ [Roma] Rom, Istituto Matematico “G. Castelnuovo,” Università di Roma (Felix Klein’s letters to Luigi Cremona). [StA Berlin] Staatsarchiv Berlin, Preußischer Kulturbesitz, Abt. Merseburg. [StA Dresden] Sächsisches Staatsarchiv Dresden, Ministerium für Volksbildung 10210; 10281. © Springer Nature Switzerland AG 2021 R. Tobies, Felix Klein, Vita Mathematica 20, https://doi.org/10.1007/978-3-030-75785-4

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[StA Potsdam] Staatsarchiv Potsdam, Reichsministerium des Innern (RMdI). [StB Berlin] Staatsbibliothek Preußischer Kulturbesitz, Berlin, Handschriftenabteilung. [UA Bonn] Archiv der Universität Bonn. [UA Braunschweig] Archiv der Technischen Universität Braunschweig (Estate Robert Fricke: Felix Klein’s letters to Fricke). [UA Erlangen] Archiv der Universität Erlangen. [UA Frankfurt] Universitätsarchiv Frankfurt am Main (Estate of Max von Laue). [UA Freiburg] Archiv der Universität Freiburg i.Br. (DMV 1, Korrespondenzbuch. Vorbereitung und Gründung der DMV, 1889–1895). [UAG] Archiv der Universität Göttingen. [UB Erlangen] Handschriftenabteilung der Universitätsbibliothek Erlangen (Records of the Physicalisch-medicinische Societät). [UB Frankfurt] Universitätsbibliothek Frankfurt am Main, J.C. Senckenberg, Na 42 (Estate of Wilhelm Lorey). [UBG] Niedersächsische Staats- und Universitätsbibliothek Göttingen, Handschriftenabteilung. (Cod. MS. F. Klein: http://hans.sub.uni-goettingen.de/nachlaesse/Klein.pdf; Cod. MS. Hilbert; Cod. MS. W. Lietzmann; Cod. MS. Teubner 49; Cod. MS. Math.-Arch. 49: Math. Gesellschaft Göttingen; Cod. MS. Math.-Arch. 50-52: Göttinger Vereinigung; Cod. MS. Math.Arch. 76-79: Briefnachlass Adolf Hurwitz). [UA Leipzig] Archiv der Universität Leipzig [UB Leipzig] Universitätsbibliothek Leipzig, Handschriftenabteilung. ABELE, Andrea; NEUNZERT, Helmut; TOBIES, R. (2004). Traumjob Mathematik. Basel: Birkhäuser. ACKERMANN, Gerhard; WEISS, Jürgen (2016). Alfred Ackermann-Teubner (1857–1941): Erfolgreicher Verleger in Leipzig. Leipzig: EAGLE. ADB. Allgemeine Deutsche Biographie (1875–1912). Leipzig: Duncker & Humblot. ADELMANN, Clemens; GERBRACHT, Eberhard H.-A. (2009). “Letters from William Burnside to Robert Fricke: Automorphic Functions, and the Emergence of the Burnside Problem.” Archive for History of Exact Sciences 63, pp. 33–50. AMTLICHER BERICHT (1877) der 50. Versammlung Deutscher Naturforscher und Ärzte in München, vom 17. bis 22. September 1877. Ed. Felix Klein et al. München: F. Straub. AMTLICHES VERZEICHNIS (1869) des Personals und der Studirenden auf der Kgl. Friedrich-Wilhelm-Universität zu Berlin, 1869/70. Berlin: Gustav Schade. ALLMENDINGER, Henrike (2019). “Examples of Klein’s Practice Elementary Mathematics from a Higher Standpoint: Volume I.” In WEIGAND et al., eds., pp. 203–213. ARCHIBALD, Tom (2016). “Counterexamples in Weierstraß’s Work.” In KÖNIG/SPREKEL, eds., pp. 269–85. ASH, Mitchell (2002). “Wissenschaft und Politik als Ressourcen füreinander.” In Wissenschaften und Wissenschaftspolitik – Bestandaufnahmen zu Formationen, Brüchen und Kontinuitäten im Deutschland des 20. Jahrhunderts. Ed. R. vom Bruch. Stuttgart: Steiner, pp. 32–51. — (2016). “Reflexionen zum Ressourcenansatz.” In Wissenschaftspolitik, Forschungspraxis und Ressourcenmobilisierung im NS-Herrschaftssystem. Ed. S. Flachowsky/R. Hachtmann/F. Schmaltz. Göttingen: Wallstein, pp. 535–54. ATZEMA, Eisso J. (1993). The Structure of Systems of Lines in 19th Century Geometrical Optics: Malus’ Theorem and the Description of the Infinitely Thin Pencil. Doctoral Thesis. RijksUniversiteit te Utrecht. AUBIN, David; GOLDSTEIN, Catherine, eds. (2014). The War of Guns and Mathematics: Mathematical Practices and Communities in France and Its Western Allies around World War I. (History of Mathematics, 42). American Mathematical Society. BAIR, J.; BŁASZCZYK, P.; HEINIG, P.; KATZ, M.; SCHÄFERMEYER, J.P.; SHERRY, D. (2017). “Klein vs Mehrtens: Restoring the Reputation of a Great Modern.” Mat. Stud. 48 (2), pp. 189–219.

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Figure 46: Felix Klein’s diploma for his honory doctorate (doctoris rerum politicarum dignitatem et ornamenta) from the University of Berlin, April 25, 1924. ([UBG] Cod. MS. F. Klein 113: 10)

INDEX OF NAMES In the index, “DMV” denotes a member of the German Mathematical Society, “Enc” denotes a contributor to the ENCYKLOPÄDIE, and “Enc(f)” indicates a contributor to the French edition of the ENCYKLOPÄDIE (Encyclopédie des sciences mathématiques pures et appliquées). Names appearing in figures are included in the index only if they appear elsewhere in the book or if they belong to particularly important mathematicians. Abbe, Ernst 1840–1905, physicist, math., entrepreneur, DMV: 35, 438, 565, 590. Abel, Niels Henrik 1802–1929, Norweg. math.: vii, 25, 47, 63, 67-69, 80, 98, 108, 153, 178, 180, 216, 228, 261, 264, 272, 292, 298, 309, 322, 339, 344, 366, 392, 405, 457, 580, 607, 610 Abraham, Max 1875–1922, theor. physicist, DMV, Enc: 44, 394, 465, 542 Ackermann-Teubner, Alfred Gustav Benedictus 1857–1941, publisher, DMV: 215, 300, 301, 303, 304, 305, 306, 307, 429, 612, 614, 624 Afanasyeva (md. Ehrenfest), Tatyana A. 1876–1964, Russian math., Enc: 433 Ahrens, Wilhelm 1872–1927, math., DMV, Enc: 428 Airy, George Biddell 1801–1892, British math., astronom: 470 Aleksandrov, Pavel S. 1896–1982, Russian math., DMV: 533, 538, 649 Alekseyevsky, Vladimir P. 1858–1916, Ukrainian math.: 393 Althoff, Friedrich 1839–1907, Prussian official: 7, 83, 86, 233-34, 242, 31819, 331-32, 335-36, 354-55, 359-60, 363-64, 366, 373-75, 377-81, 384, 387, 401, 407-08, 410-14, 419, 421-23, 434, 438, 440-46, 451, 488, 491, 500-01, 510-12, 519, 546, 581, 605-06, 612, 626, 633, 642, 649 Amelung, Julius *1864, stud., teacher: 201 Ameseder, Adolf 1858–1891, Austrian math.: 231, 248 Amsler-Laffon, Jakob 1823–1912, Swiss math., eng.: 146

Anisimov, Vasily A. 1860–1907, Russian math.: 603, 604 Appell, Paul 1855–1930, French math., Enc(f): 268, 352, 495, 579, 613-14 Archenhold, Friedrich Simon 1861–1939, astr., DMV: 369 Archibald, Raymond Clare 1875–1955, Canadian-American math., hist., DMV: 347, 618 Argelander, Friedrich Wilhelm 1799–1875, astronom: 30, 31 Aronhold, Siegfried 1819–1884, math.: 48, 53, 119 Artin, Emil 1898–1962, Austrian math., DMV: 536 Ascoli, Giulio 1843–1896, Ital. math.: 153 Assmann, Richard 1845–1918, meteorologist: 451-52 Asthöwer, Fritz 1835–1913, eng.: 422 Augustus, né Gaius Octavius 63 BC–14 AD, Roman emperor: 396 Baade, Walter 1893–1960, astronom, DMV: 535, 536 Bach, Carl von 1847–1931, eng.: 304, 614 Bacharach, Isaak 1854–1942, math., DMV: 204 Bäcklund, Albert Victor 1845–1922, Swedish math., DMV: 136 Baeyer, Adolf (as of 1885 Ritter von) 1835– 1917, chem.: 379, 560 Ball, Robert Stawell 1840–1913, Irish math., astromomer: 150, 330, 353 Baltzer, Richard 1818–1887, math.: 102, 311-12, 596 Barkhausen, Heinrich 1881–1956, physicist: 466, 614

© Springer Nature Switzerland AG 2021 R. Tobies, Felix Klein, Vita Mathematica 20, https://doi.org/10.1007/978-3-030-75785-4

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Index of Names

Barnum, Charlotte Cynthia 1860–1934, American math.: 413 Bartling, Friedrich Gottlieb 1798–1875, botanist: 89 Battaglini, Giuseppe 1826–1894, Italian math.: 40, 41, 44, 76, 87, 153-56 Bauer, Gustav 1820–1906, math., DMV: 124, 172-73, 193-94, 209, 241, 598 Bauer, Max 1844–1917, mineral.: 115 Bauernfeind, Karl Maximilian (Max) von 1818–1894, geodesist: 171, 173, 206, 598 Baumgart, Oswald, math.: 230 Bauschinger, Johann 1834–1893, math., construction eng.: 172-73, 201 Bauschinger, Julius 1860–1934, astronom, DMV, Enc: 201, 557, 559-60 Bayer, Friedrich 1851–1920, entrepreneur: 561 Bechmann (as of 1891 Ritter von), August 1834–1907, legal scholar: 144 Becker, Carl Heinrich 1876–1933, oriental., politician: 531-34 Beer, August 1825–1863, math., physicist, chemist: 596 Beetz (as of 1876 von), Wilhelm 1822– 1886, physicist: 172, 173 Behnke, Heinrich 1898–1979, math., DMV: 64, 546, 564, 625, 631 Behrendsen, Otto 1850–1921, Gymn.-Prof.: 392-93, 398, 614, 636 Behrens, Wilhelm 1885–1917, math., PhD with Klein, DMV: 491, 517, 614 Beke, Emanuel (Manó) 1862–1946, Hungarian math., DMV: 393, 493, 527 Beltrami, Eugenio 1835–1900, Italian math.: 54, 87, 102-03, 154-56, 247, 256, 348, 382, 634 Beman, Wooster Woodruff 1850–1922, American math., DMV: 399, 635 Beneke, Carl Gustav 1800–1864, pastor: 481 Beneke, Friedrich Eduard 1798–1854, philosopher: 481-82, 485-86 Benfey, Bruno 1891–1962, pastor: 170 Benfey, Theodor 1809–1881, orient.: 362 Bernays, Paul 1888–1977, math., DMV: 536 Berneker, Erich 1874–1937, Slavicist: 532 Bernstein, Felix 1878–1956, math., DMV: 168, 302, 394, 420, 480, 487, 490-92, 516, 534, 614, 649

Berry, Arthur 1862–1929, British math., DMV: 334, 613-14 Bertheau, Ernst 1812–1888, orient.: 90 Bertini, Eugenio 1846–1933, Italian math.: 153, 157 Bertrand, Joseph 1822–1900, French math.: 84, 312, 351, 572, 596 Berzolari, Luigi 1863–1949, Italian math., Enc: 431 Bessel-Hagen, Erich 1898–1946, math., DMV: 281, 536-37 Betti, Enrico 1823–1892, Ital. math.: 154, 156-157, 199, 623 Beumer, Wilhelm 1848–1929, teacher, politician: 422 Bezold, Wilhelm von 1837–1907, physicist, meteorol.: 202-03 Bianchi, Luigi 1856–1928, Italian math.: 195, 198, 200-01, 210, 214, 359, 576, 585, 612, 614 Bieberbach, Ludwig 1886–1982, math., DMV, Enc: 428, 457, 491, 505, 522, 550, 555, 560, 607-08, 614, 639 Biedermann, Paul *1862, math., PhD with Klein: 67, 226, 230, 293, 311, 614 Biot, Jean-Baptiste 1774–1862, French math.: 37 Bischof, Johann N. 1827–1893, math.: 17273, 175, 212 Bischof, Karl Gustav 1792–1870, geochemist: 32 Bismarck, Otto von 1815–1898, politician: 28, 84, 329, 499, 511 Blake, Edwin Mortimer 1868–1955, American math.: 404 Blaschke, Wilhelm 1885–1962, Austrian math., DMV: 131, 387, 433, 523, 54748, 623, 636 Blumenbach, Johann Friedrich 1752–1840, zoologist, anthropol.: 45 Blumenthal, Otto 1876–1944, math., DMV: v, 57-58, 394, 436, 453-54, 461, 54344, 559, 565, 612, 614, 626, 644 Bobek, Karl 1855–1899, Austrian math., DMV: 230, 248 Bôcher, Maxime 1867–1918, American math., PhD with Klein, DMV, Enc: 100, 302, 334, 344, 345-46, 372, 393, 405, 455, 612, 614, 625 Bödiker, Tonio 1843–1907, Prussian official: 439

Index of Names Boehm, Karl 1873–1958, math., DMV: 237, 426 Boelitz, Otto 1876–1951, pedag., politician: 555 Börnstein, Richard 1852–1913, physicist: 97 Böttger, Adolf, math., teacher: 230, 232 Böttinger (as of 1907 von), Henry Theodore 1848–1920, industrialist: 422-24, 43941, 443-44, 449, 451-53, 500, 502, 512, 561, 582, 613-14, 637 Bohlmann, Georg 1869–1928, math., actuary, DMV, Enc: 393, 419-20, 447, 614, 636 Bokowski, Adalbert 1899–1948, math., DMV: 536 Boltzmann, Ludwig 1844–1906, Austrian physicist, DMV, Enc: vii, 119, 306, 364-65, 379, 426, 450, 482, 489, 512, 591, 632, 637 Bolyai, János (Johann) 1802–1860, Hungarian math.: 70, 72-73, 101-02, 126, 250, 579-80, 642 Bolza, Oskar 1857–1942, math., DMV: 168, 337, 365, 402, 404, 415, 614 Bolzano, Bernhard 1781–1848, Bohemian math., philosopher, theologian: 139 Boole, George 1815–1864, Engl. math.: 56 Borchardt, Carl Wilhelm 1817–1880, math.: 63-64, 247, 267, 282, 284, 337-38, 365, 411 Borel, Émile 1871–1956, French math., Enc, Enc(f): 302, 430, 452, 613-14 Born, Max 1882–1970, physicist, DMV, Enc: viii, 301, 357, 394, 439, 473, 544, 562, 577, 582, 614, 626 Bortkiewicz, Ladislaus von 1868–1931, Polish-Russ. statistician, Enc: 419 Bosse, Robert 1832–1901, lawyer, politician: 400, 438 Bosworth (md. Focke), Anne Lucy 1868– 1907, American math.: 393 Boulanger, Auguste 1866–1923, French math., Enc(f): 333 Boussinesq, Valentin Joseph 1842–1929, French math., physicist: 430, 467 Boutroux, Pierre 1880–1922, French math., Enc(f): 613-14 Boyd, James H. 1862–1946, American math.: 334 Branford, Benchara 1867–1944, Scottish math., : 302, 584

657

Brater (md. Sapper), Agnes 1852–1929, writer: 163 Brater, Karl Ludwig Theodor 1819–1869, publicist: 163 Brater (neé Pfaff), Pauline 1827–1907: 163 Brauer, Richard 1901–1977, math., DMV: 289 Braun, Wilhelm *1852, Math., PhD with Klein: 133, 136 Braune, Christian 1831–1892, anatomist: 310, 313 Bravais, Auguste 1811–1863, French physicist, crystallographer: 356 Brendel, Martin 1862–1939, astronom, DMV: 420, 447, 465, 478, 614 Bretschneider, Wilhelm 1847–1931, math., teacher, prof., PhD with Klein, DMV: 108, 133-35, 614 Brill (as of 1897 von), Alexander 1842– 1935, math., DMV: 47-50, 53, 87, 11920, 161, 173-77, 191, 193, 195, 202-04, 206-07, 209-10, 212, 221, 227, 249, 264, 270, 372, 386, 451, 555, 613-14 Brill, Ludwig, publisher: 108, 175, 206 Brioschi, Francesco 1824–1897, Italian math.: 153-54, 157, 180-84, 186, 189, 209, 288, 575 Brouwer, Luitzen 1881–1966, Dutch math., DMV: 57- 58, 279-80, 457, 522-23, 572, 613-14, 652 Bruhns, Karl Christian 1830–1881, astronomer: 222 Brunel, Georges 1856–1900, French math., Enc: 10, 230, 244-45, 268, 270 Bruns, Heinrich 1848–1919, math., astr., DMV: 67, 114, 222, 229-30, 308, 371, 614, 648 Bryan, George Hartley 1864–1928 British applied math., Enc: 432 Buchheim, Arthur 1859–1888, British math.: 150, 230, 244 Budde, Emil A. 1842–1921, physicist: 42, 614 Büttner, Friedrich 1859–1915, math., teacher 230, 268 Burkhardt, Heinrich 1861–1914, math., DMV, Enc: 172, 263, 331-32, 338-41, 360, 366, 369-70, 384, 392, 394, 40203, 417, 429-30, 463, 606, 613-14, 638 Burnside, William 1852–1927, Engl. math.: 283, 624

658

Index of Names

Busch, Wilhelm 1832–1908, poet, illustrator: 163 Campbell, George Ashley 1870–1954, American math.: 345, 399, 466 Cantor, Georg 1845–1918, math., DMV: 54, 64, 130, 192, 239, 361, 367-72, 374, 454, 483, 491, 511, 525, 538, 569, 627, 631, 642 Cantor, Moritz 1829–1920, math. historian, DMV: 208, 303-04, 306, 479, 495 Capelli, Alfredo 1855–1910, Italian math.: 402 Carathéodory, Constantin 1873–1950, Greek math., DMV: 1, 4, 57, 123, 327, 390, 465, 480, 522, 523, 524, 528, 532, 536, 537, 568, 613-14, 626 Cartan, Élie Joseph 1869–1951, French math., Enc(f): 131, 352, 646-47 Casorati, Felice 1835–1890, Italian math.: 154, 596 Cassirer, Ernst 1874–1945, philosopher: 131, 244, 625 Castelnuovo, Guido 1865–1952, Italian math., Enc: 431, 585, 613-14, 618, 623 Cauchy, Augustin-Louis 1789–1857, French math.: 44, 217, 223, 295, 410, 412, 569, 579 Cauer (née Schelle), Wilhelmine (Minna) 1841–1922, educator: 506 Cayley, Arthur 1821–1895, British math.: 4, 37, 47-48, 52, 54, 66, 70-73, 87, 90, 101-03, 147-52, 190, 192, 199, 204, 211, 214, 242, 290, 314-15, 334, 348, 353, 416, 473, 570, 575, 585, 599 Chasles, Michel 1793–1880, French math.: 50, 68, 74-75, 79, 84, 105, 108, 130, 192, 197, 241 Chebyshev, Pafnuty L. 1821–1894, Russian math.: 177, 250, 355, 436, 485-87 Chisholm (md. Young), Grace 1868–1944, Engl. math., PhD with Klein: xix, 302, 354, 393-94, 399, 413-16, 429, 522, 529-30, 538, 547, 576, 631, 640, 653 Christoffel, Elwin Bruno 1829–1900, math.: 120, 264, 434, 494 Cicero, Marcus Tullius 106–43 BC, Roman statesman, philos.: 23 Clausius, Rudolf 1822–1888, physicist: 147 Clebsch, Alfred 1833–1872, math.: 2-3, 6, 9, 10, 12, 17, 32, 37, 39-40, 45-55, 5759, 61-65, 67-69, 74-75, 77, 88-93, 9598, 100-02, 106-11, 113, 116, 118-20,

124-26, 128-29, 132-34, 136-37, 13940, 153, 155-56, 158-60, 162, 173, 185, 191-92, 198, 209, 212, 216-17, 221, 223, 249, 260, 264-65, 299, 330, 334, 338, 358, 361, 379, 404, 477-79, 482, 510, 526, 546, 569-70, 574-76, 578, 589, 596-97, 608, 611, 618, 627, 638, 642, 647, 650, 652 Clifford, William Kingdon 1845–1879, Engl. math.: 4, 48, 103, 148-49, 152, 192 Coble, Arthur Byron 1878–1966, American math.: 339, 619 Cohn-Vossen, Stephan 1902–1936, math., DMV: 237, 257, 539, 632 Cole, Frank Nelson 1861–1926, American math., PhD with Klein: 231, 246, 290, 337, 613-14 Collatz, Lothar 1910–1990, math., DMV: 448-50, 627 Columbus, Christopher ca. 1451–1506, Italian explorer: 401 Comba, Paul G. 1926–2017, Italian-American amateur astronomer: 27 Copley, Godfrey Sir 1653–1709, English landowner: 38, 151, 354, 518 Cornelius, Hans 1863–1947, philos.: 231 Courant, Richard 1888–1972, math., DMV: vii, 117, 421, 480, 523-25, 536-37, 539, 547, 555, 559, 563, 566-67, 585-86, 614, 627, 643 Cram (md. Klein), Myrthel, American daughter-in-law of F. Klein: x, 166 Crelle, August Leopold 1780–1855, math.: 63, 628 Cremona, Luigi 1830–1903, Italian math., DMV: 7, 38, 52, 87, 108, 154-58, 190, 206, 216, 246, 339, 348, 382, 408, 468, 495, 569, 595-96, 623, 627, 633, 639 Culmann, Karl 1821–1881, German-Swiss eng.: 73, 468 Czapski, Siegfried 1861–1907, physicist: 114 Czermak, Johann Nepomuk 1828–1873, physiologist: 219 Czuber, Emmanuel, 1851–1925, Austrian math., DMV, Enc: 495 Dalwigk, Friedrich von 1864–1943, math., DMV: 393 Darboux, Gaston 1842–1917, French math.: 6, 14, 44, 74-76, 80, 83-85, 87, 96, 99, 100, 102, 107-08, 120, 123, 126-27,

Index of Names 132, 138, 161, 181, 192, 195-96, 209, 213, 244-45, 250, 254, 268, 270, 292, 311-12, 322, 333, 397, 411, 428, 503, 512, 525, 529, 573, 575, 579-80, 58384, 595-96, 613-14, 623, 627, 630, 643 Darwin, Charles Robert 1809–1882, British biologist: 36, 205, 591, 633 Darwin, George Howard 1845–1912, British astronomer, Enc: 353, 613-14 Debye, Peter 1884–1966, Dutch physicist, DMV, Enc: 327, 480, 562, 623, 645 Decoster, Paul 1886–1939, Belgian philosopher: 491 Dedekind, Richard 1831–1916, math., DMV: 51, 94, 108, 184-85, 232, 249, 264, 282, 284, 341-42, 344, 359, 366, 405, 613-14, 643, 645 Dehn, Max 1878–1952, math., DMV, Enc: 95, 338, 395, 614, 627, 640 Des Coudres, Theodor 1862–1926, physicist: 229, 424, 439, 449, 636 Despeyroux, Théodore 1815–1883, French math., physicist: 397 Deussen, Gustav Adolf Hugo *15.10.1837, religion teacher: 23 Dickson, Leonard Eugene 1874–1954, American math.: 619 Dickstein, Samuel 1851–1939, Polish math., math.hist., DMV: 130, 613-14 Diekmann, Joseph 1848–1905, math., PhD with Klein: 107, 113 Diels, Hermann 1848–1922, classical scholar: 444, 513 Diestel, Friedrich 1863–1925, math., librarian, DMV: 392, 394, 614 Diesterweg, Adolph 1790–1866, educator: 531 Dingeldey, Friedrich 1859–1939, math., PhD with Klein, DMV, Enc: 230, 232, 262, 309, 614 Dini, Ulisse 1845–1918, Italian math.: 157, 199 Dirichlet [Lejeune Dirichlet], Peter Gustav 1805–1859, math.: vii, 9, 32, 45, 62, 69, 92, 254, 258, 261-63, 271, 276, 288, 328, 359-60, 399, 479, 491, 523, 545, 600, 640 Domsch, Paul 1860–1918, math., PhD with Klein, DMV: 100, 230-31, 264-65, 310 Donadt, Alfred *1857, math., teacher: 219 Dove, Heinrich Wilhelm 1803–1879, physicist, meteorologist: 69, 88

659

Dove, Richard Wilhelm 1833–1907, Canon Law scholar: 88, 499 Drenckhahn, Friedrich 1894–1977, math., pedag., DMV: 505, 536 Dressler, Heinrich, math. teacher, author: 230, 232, 262, 614 Du Bois-Reymond, Emil Heinrich 1818– 1896, physiologist: 62, 65, 496, 628 Du Bois-Reymond, Paul 1831–1889, math.: 65, 220, 237, 322, 330, 489 Duhamel, Jean-Marie Constant 1797–1872, French math., physicist: 595-96 Dühring, Eugen 1833–1921, philos.: 116 Duisberg, Carl 1861–1935, chemist, industrialist: 498, 561-63, 614, 637 Dupin, Charles 1784–1873, French math., eng.: 80, 99, 609 Durège, Heinrich 1821–1893, math.: 296, 596 Dyck (as of 1901 Ritter von), Walther 1856– 1934, math., PhD with Klein, DMV, Enc: 7, 56-57, 87, 151, 177, 193-95, 200, 204, 208, 210-11, 219, 223, 22527, 230, 233-34, 260-61, 266, 270, 285, 296, 299, 303-05, 308, 311, 339-40, 367, 369-72, 375, 381, 393, 401-02, 410, 426, 429, 430, 433, 447-48, 475, 493, 503, 512-13, 516, 519-21, 543, 557, 566-67, 578, 613-14, 628, 632 Ebert, Hermann 1861–1913, physicist: 169, 614 Ehlers, Ernst 1835–1925, zoologist: 135, 150, 613 Ehlers, J., student: 399 Ehrenfest, Paul 1880–1933, Austrian physicist, Enc: 426, 433 Ehrenfeuchter, Friedrich 1814–1878, theologian, University prof.: 90 Ehrensberger, Emil 1858–1940, chemist, industry manager: 440, 561, 614 Einstein, Albert 1879–1955, physicist, DMV: ix, 44, 57, 168, 235, 394, 396, 472, 490, 513, 525, 535, 538-544, 549, 560, 569, 578, 591, 628, 632, 643, 650 Eisenstein, Gottlob 1823–1852, math.: 199 Elster, Ludwig 1856–1935, economist, senior officer: 518 Eneström, Gustaf 1852–1923, Swedish math., hist., DMV: 232, 304-05, 474, 479 Engel, Friedrich 1861–1941, math., DMV: 68, 77, 81, 104, 128, 231, 233, 239,

660

Index of Names

240, 287, 303, 313, 320, 349, 478, 578, 614, 623, 638 Engels, Hubert 1854–1945, hydraulic eng.: 467 Enneper, Alfred 1830–1885, math.: 18, 93, 95, 120, 238, 317-18, 331, 599-600, 602 Enriques, Federigo 1871–1946, Italian math., math.hist., Enc: 103, 302, 352, 399, 431, 492, 613, 618 Epsteen, Saul *10.8.1878, American math., DMV: 394 Epstein, Paul 1871–1939, math., DMV: 222, 614 Errera, Alfred 1886–1960, Belg. math.: 491 Escherich, Gustav von 1849–1935, Austrian math., DMV: 426-27 Euclid, ca. 300 BC, Greek math.: 44, 70-72, 101, 103, 105, 116, 126, 143, 149, 183, 351, 479, 485, 491, 496, 629 Euler, Leonhard 1707–1783, Swiss math.: 9, 114, 160, 240, 471, 479, 594, 596, 626 Faltings, Gerd *28.7.1954, math., DMV: 460 Fano, Gino 1871–1952, Italian math., DMV, Enc: 10, 41, 130-31, 154, 393, 399, 447, 648 Fanta (Fanla), Ernst 1878–1936, Austrian math., DMV: 394 Faraday, Michael 1791–1867, British physicist: 410 Fedorov, Evgraf S. 1853–1919, Russian math., mineralogist: 6, 356 Fehr, Henri 1870–1954, Swiss math., DMV: 493-94, 507, 517, 549-50, 613-14, 629 Fellmann, Emil 1927–2012, Swiss science historian: 13 Fermat Pierre de 1607–1665, French polymath., lawyer: 381 Fick, Richard 1867–1944, librarian: 380 Fiedler, Ernst 1861–1954, Swiss math., PhD with Klein, DMV: 231, 248, 293, 614 Fiedler, Wilhelm 1832–1912, German-Swiss math., DMV: 41, 71, 102, 248, 349, 652 Fields, John Charles 1863–1932, Canadian math., DMV: 460, 613-14, 619 Fine, Henry Buchard 1858–1928, Americ. math., PhD with Klein: 231, 246, 407

Finsterwalder, Sebastian 1862–1951, math, DMV, Enc: 49, 174, 176, 368, 451, 629 Fischer, Emil 1852–1919, chemist: 560 Fischer, Gottlob, math., Klein’s first assistent: 177 Fischer, Otto 1861–1916, math., physiologist, PhD with Klein, DMV, Enc: 226, 289, 311, 313, 614 Fleck, Ludwik 1896–1961, Polish and Israeli physician, biologist: 8 Fleischer, Hermann, math., PhD Göttingen, DMV: 394 Flender (née Klein), Aline Leonore 1847– 1914, F. Klein’s sister: x, 21 Flender, Hermann August 1839–1882, manufact., F. Klein’s broth.-in-law: x, 21 Föppl, August 1854–1924, mech., DMV, Enc: 222-23, 368, 390, 449, 468, 614 Föppl, Ludwig 1887–1976, math., mech., DMV: 223, 390, 614 Förster, Wilhelm 1832–1921, astronomer: 369, 376, 528, 603 Ford, Lester Randolph 1886–1967, American math.: 255, 629 Forsyth, Andrew Russell 1858–1942, British math.: 103, 258, 353-54, 413, 575 Fourier, Jean Baptiste Joseph 1768–1830, French math.: 48, 94, 146, 195, 237, 241, 293, 407 Fraenkel, Abraham A. 1891–1965, Germanborn Israeli math., DMV: 12, 478, 525, 581, 591, 629 Frahm, Wilhelm 1849–1875, math.: 191 Franck, James 1882–1964, phycisist: 562 Franklin, Fabian 1853–1939, American math., DMV: 290, 334, 360, 404, 41213, 613-14 Frege, Gottlob 1848–1925, logician, DMV: 56 Fresnel, Augustin Jean 1788–1827, French physicist, eng.: 206, 611 Freundlich (Finley-Freundlich), Erwin 1885–1964, math. astron., DMV: 491 Freytag (md. Loeschcke), Thekla 1887– 1932, teacher: 418, 506, 650 Fricke (née Flender), Leonore 1873–1912, Klein’s niece: x, 21 Fricke, Robert 1861–1930, math., PhD with Klein, DMV, Enc: ix, x, 4, 11, 13, 21, 169, 184, 189, 196, 200, 226, 231-32, 258, 272-73, 279-81, 290-91, 293-95,

Index of Names 297-98, 341-44, 367, 375-77, 384, 388, 392, 394, 397, 402, 417, 426, 437, 45457, 465, 544-46, 565, 572, 579, 581, 604, 607, 610, 613-14, 624, 630, 636 Friedrich, Georg *1860, math. PhD with Klein: 231, 293 Friedrich Wilhelm III 1770–1840, King of Prussia: 28 Friedrichs, Kurt Otto 1901–1983, math., DMV: 584 Friesendorff, Theophil 1871–1913, Russian math., eng., DMV: 301 Frobenius, Georg 1849–1917, math., DMV: 66, 108, 357, 372, 374-378, 427-28, 457, 461, 523, 587, 605, 611, 632 Fröbel, Friedrich 1782–1852, pedag.: 531 Fuchs, Lazarus 1833–1902, math., DMV: 4, 6, 64, 97, 181, 247, 253, 267, 274, 282, 284-85, 318, 342-44, 355, 365, 373-76, 436, 454-57, 603 Fueter, Rudolf 1880–1950, Swiss math., DMV: 580, 613, 615 Fujisawa, Rikitarō 1861–1933, Japanese math.: 494, 613, 615, 630 Fujiwara, Matsusaburō 1881–1946, Japanese math., historian, DMV: 399-400 Furtwängler, Philipp 1869–1940, math., PhD with Klein, DMV, Enc: 389, 417, 458-60, 615 Galilei, Galileo 1564–1642, polymath., astronomer, physicist: 594, 648 Gallenkamp, Wilhelm 1820–1890, teacher, author: 496 Galois, Évariste 1811–1832, French math.: 6, 25, 60, 74, 77, 79, 100, 181, 186, 188, 196, 238, 395, 427, 626, 631 Gauss [Gauß], Carl Friedrich 1777–1855, math., astr.: 1, 9, 30, 45, 72-73, 90, 92, 94-95, 101, 114, 126, 129, 151, 156-57, 183-85, 263, 288, 295, 301, 366-67, 372, 402, 437, 439, 449, 458, 478-79, 485-86, 491, 525, 559-60, 562, 569, 571, 600, 610, 642 Gauthier-Villars, Albert 1861–1918, French publisher, DMV: 76-77, 192, 409, 429, 464, 596, 627, 634, 643 Gay-Lussac, Joseph Louis 1778–1850, French chemist, physicist: 203 Gegenbauer, Leopold 1849–1903, Austrian math., DMV: 295-96, 419 Gehlhoff, Georg 1882–1931, physicist: 561 Gehring, Franz 1838–1884, math.: 30-31

661

Geibel, Emanuel 1815–1884, poet: 362 Geiringer (md. Pollazcek, md. von Mises), Hilda 1893–1973, Austrian-American math., DMV: 417 Geiser, Carl Friedrich 1843–1934, Swiss math.: 87 Geißler, Heinrich 1814–1879, glass blower, mechanic: 34, 36, 628 Gentry, Ruth 1862–1917, American math.: 412 Gerbaldi, Francesco 1858–1934, Italian math., DMV: 154, 231, 247, 285 Gerber, Carl von 1823–1891, Saxon Minister of Culture: 211, 216, 218, 315, 319 Gerber, Heinrich 1832–1912, eng.: 471 Gernet, Nadezhda N. 1877–1943, Russ. math., DMV: 372, 396, 613, 615 Gibbs, Josiah Willard 1839–1903, American physicist: 406-07, 637 Gibbs, Oliver Wolcott 1822–1908, American chemist: 408, 620 Gierster, Josef 1854–1893, math., PhD with Klein, DMV: 177, 188, 195-96, 204, 228, 270, 285, 295, 311 Gilman, Daniel Coit 1831–1908, American educator, academic: 315, 407 Goeb, Margarethe 1892–1962, teacher: 522 Göpel, Adolph 1812–1847, math.: 265 Görres, Joseph 1776–1848, philosopher, publisher: 22 Goethe, Johann Wolfgang von 1749–1832, poet: 23, 30-31, 135, 220 Götting, Eduard 1860–1926, teacher, DMV: 392-93, 398, 615, 636 Gontschareff, A., Russian fellow student of Klein: 29 Gordan, Paul 1837–1912, math., DMV: 47, 48, 53-55, 57, 120, 127, 133-34, 137, 144, 158-60, 178, 182-83, 187-89, 19192, 195, 198, 208-12, 214, 223, 234, 242, 248-49, 264-65, 267, 278, 288, 299, 315, 334, 357-58, 369-72, 377, 379, 381, 398, 404, 574, 576, 578, 597, 613, 615, 641 Goßler, Gustav von 1838–1902, Prussian Minister of Culture: 236, 598 Graefe, Walther *26.8.1892, teacher: 390, 535 Graf, Ulrich 1908–1954, math., DMV: 505 Grassmann, Hermann Günther 1809–1877, math.: 2, 37, 90, 104-05, 107, 120, 126, 129, 217, 240-42, 301, 303, 312-

662

Index of Names

13, 407, 478, 483, 596, 631, 642, 644, 646, 650 Green, George 1793–1841, British math., physicist: 343, 479, 494, 517 Greenhill, Alfred George Sir 1847–1927, British math., DMV: 343, 354, 430, 479, 493-95, 549-50, 575, 613, 631 Griess, Jean French math.: 399 Groth, Paul von 1843–1927, mineralogist: vii, 379 Grüning, Martin 1869–1932, eng.: 470 Gruson, Johann Philipp 1768–1857, teacher, math.: 164 Guccia, Giovanni Battista 1855–1914, Italian math., DMV: 613, 615 Günther, Siegmund 1848–1923, math., geographer, math. historian, DMV: 13234, 172, 577 Guilleaume (Freiherr von), Theodor 1861– 1933, entrepreneur: 440 Gundelfinger, Sigmund 1846–1910, math., DMV: 191, 208 Gutzmer, August 1860–1924, math., DMV: 6, 120, 304, 306, 447, 453, 493-94, 498-99, 506, 532, 584, 613, 615, 631, 649 Haber, Fritz 1868–1934, chemist: 557-58 Hadamard, Jacques 1865–1963, French math.: 613, 615 Haeckel, Ernst 1834–1919, zool.: 36, 205, 240, 528, 591, 629, 643, 653 Haenisch, Konrad 1876–1925, journalist, politician: 526, 552 Haeseler, Gottlieb Graf v. 1836–1919, officer: 500, 507 Hagemann, Eberhard 1880–1958, lawyer, F. Klein’s son-in-law: x, 166, 170 Hahn, Hans 1879–1934, Austrian math., DMV: 467, 615, 646 Hall, G. Stanley 1846–1924, psychol.: 354 Halphen, Georges Henri 1844–1889, French math.: 79, 241-42, 249, 267, 269 Hamburger, Meyer 1838–1903, math., DMV: 604 Hamel, Georg 1877–1854, math., DMV: 394, 471, 487, 550, 553, 555, 615 Hamilton, William Rowan 1805–1865, Irish math.: 104-05, 114, 168, 198, 243, 371, 407 Hammerschmidt (md. Klein), Maria Catharina 1787–1871, paternal grandmother of F. Klein: x, 17

Hankel, Hermann 1839–1873, math.: 124 Hankel, Wilhelm Gottlieb 1814–1899, physicist: 321 Hanstein, Johannes von 1822–1880, botanist: 31-32 Hardcastle, Frances 1866–1941, Engl. math.: 257-59, 354, 417, 634 Harkness, James 1864–1923, Engl., Amer., Canad. math., Enc: 184, 258, 417, 630 Harnack (as of 1914 von), Adolf 1851– 1930, theol.: 163, 444, 501, 507, 557 Harnack, Axel 1851–1888, math., PhD with Klein: 135-36, 163, 179, 191-92, 212, 216, 232, 237, 311, 321, 501, 596 Hartnack, Eduard 1826–1891, optometrist: 36 Haskell, Mellen Woodman 1863–1948, American math., PhD with Klein, DMV: 127, 130, 285, 334, 339-41, 372, 613, 615, 634 Hauck, Guido 1845–1905, math., DMV: 304, 500, 502 Haussner, Robert 1863–1948, math., DMV: 479, 615 Hayashi, Tsuruichi 1873–1935, Japanese math., hist., DMV: 399, 573 Hecke, Erich 1887–1947, math., DMV, Enc: 184, 508, 515, 523, 537, 544, 615, 623 Hedrick, E. Raymond 1876–1943, American math., DMV: 399 Heegaard, Poul 1871–1948, Danish math., DMV, Enc: 81, 95, 338, 393, 395, 399, 494, 575, 613, 615, 627-28, 638 Heffter, Lothar 1862–1962, math., DMV: 229, 369-70, 402, 473 Hegel, Friedrich Wilhelm Karl (Ritter von) 1813–1901, hist., Klein’s fath.-in-law: x, 35, 116, 161, 163-65, 167, 637 Hegel, Georg Wilhelm Friedrich 1770– 1831, philosopher: x, 164, 481 Hegel, Georg, 1856–1933, Bavarian colonel, Klein’s broth.-in-law: x, 166 Hegel (md. Lommel), Louise Friederike Caroline 1853–1924, Klein’s sist.-inlaw: x, 166 Hegel, Maria 1855–1929, Klein’s sist.-inlaw: x, 166-67 Hegel (née Tucher von Simmelsdorf), Maria Helene Susanne 1791–1855: x, 164 Hegel, Sophie Louise 1861–1940, Klein’s sist.-in-law: x, 166-67, 286, 354, 567

Index of Names Hegel (née Tucher von Simmelsdorf), Susanne 1826–1878, F. Klein’s motherin-law: x, 165 Hegel, Wilhelm Sigmund 1863–1945, gov. councilor, F. Klein’s broth.-in-law: x, 166-67 Heimsoeth, Friedrich 1818–1877, class. philologist: 35 Heine, Eduard 1821–1881, math.: 119, 596 Heine, Heinrich 1797–1856, poet, writer: ix, 22, 210, 517 Heinemann, Käthe *8.5.1889, math., botan., pedag.: 535, 537 Hellinger, Ernst 1883–1950, math., DMV, Enc: 346, 390-91, 615 Helmert, Robert 1843–1917, geodesist, DMV, Enc: 304, 615, 648 Helmholtz, Hermann von 1821–1894, physicist, physician: 64, 94, 96, 112, 119, 157, 323, 347, 348, 350-52, 374, 376, 387, 401, 405, 410, 421, 440, 466, 468, 484, 525, 560-63, 582, 625, 637 Henneberg, Lebrecht 1850–1933, math., DMV, Enc: 207, 468, 469-70, 615 Henrici, Olaus 1840–1918, math., DMV 48, 146, 185, 192, 613, 615 Hensel, Kurt 1861–1941, math., DMV, Enc: 288-89, 615, 637 Herglotz, Gustav 1881–1953, math., DMV, Enc: 346, 466-67, 523, 615 Hermite, Charles 1822–1901, French math.: viii, 6, 56, 131, 180, 183-84, 186, 192, 265, 267-68, 270, 282, 288, 292, 342, 346, 352, 358-59, 392, 402, 408, 40910, 416, 455, 458-59, 461-64, 509, 569, 586, 631, 643 Herrmann, Oskar *1859, teacher, PhD with Klein: 228, 230 Herrmann, Theodor, math.: 230 Herschel, John 1792–1871, Engl. astr.: 258 Hertz, Heinrich 1857–1894, physicist: 371 Herz, Norbert 1858–1927, Austrian astronomer, DMV: 402 Heß, Wilhelm 1858–1937, math.: 195, 615 Hesse, Otto 1811–1874, math.: 47-49, 139, 172, 176, 178, 596-97, 634 Hettner, Georg 1854–1914, math., DMV: 256, 317-18, 600-02 Heun, Karl 1859–1929, math., DMV, Enc: 473, 486, 615 Hilb, Emil 1882–1929, math., DMV, Enc: 278, 346, 381, 457, 615

663

Hilbert, David 1862–1943, math., DMV, Enc: viii, ix, 1, 6, 8-11, 39, 48, 50, 57, 65, 87-88, 104, 159, 168, 184, 192, 197, 199, 223, 228-29, 231, 235-37, 241-43, 246, 250, 257-58, 263, 269, 279-81, 286, 303-04, 307, 311, 321, 324, 327, 331, 333, 346, 349-53, 35760, 366, 368-73, 375, 377-78, 383, 385, 390-96, 399, 402, 404-05, 417, 421, 425, 434-36, 439, 441-42, 447, 454-61, 465-67, 473, 481, 483-85, 488-90, 497, 504, 510-11, 518-19, 522-23, 528-29, 533, 535, 537-41, 543-47, 549, 553, 556, 566, 571-72, 577-84, 586, 587, 590, 605-08, 611, 613, 615, 621, 624, 629-30, 632, 640-45, 647, 650 Hildebrand, Rudolf, math.: 231, 615 Hillebrand, Meinolf Rudolf *3.3.1937, F. Klein’s great-grandson: v, x, 13, 17-21, 166-67, 414, 520, 568, 614-17, 623 Hillebrandt, Alfred 1853–1927, philologist: 444, 499-500, 531-32 Hinneberg, Paul 1862–1934, hist.: 475, 513 Hirst, Thomas Archer 1830–1892, British math.: 48 Hirzebruch, Friedrich 1927–2012, math., DMV: 222, 629 Hjelmslev (Petersen), Johannes 1873–1950, Danish math.: 433 Hobson, Ernest William 1856–1933, British math., Enc: 613, 615 Hoeck, Karl Friedrich Christian 1794–1877, hist., philol., librarian: 89-90 Höckner, Georg 1860–1938, math., actuary: 230 Höfler, Alois 1853–1922, Austrian math.didact., philos., DMV: 306, 482, 503, 613, 615, 632 Hölder, Otto 1859–1937, math., DMV, Enc: 57, 60, 71, 220-21, 231-34, 238, 246, 266-67, 301, 328-36, 361-62, 375, 427, 535, 577, 590, 613, 615, 632 Höpfner, Ernst 1836–1915, pedag., Prussian official, curator: 389, 414, 419, 439 Hoetzsch, Otto 1876–1946, historian: 533 Hofmann, August Wilhelm 1818–1893, chemist: 36 Holst, Elling Bolt 1849–1915, Norwegian math.: 70, 137, 191, 324, 633 Holzmüller, Gustav 1844–1914, math., DMV: 462, 497

664

Index of Names

Hoppe, Heinrich 1857–1899, math., teacher, DMV: 230-31, 262 Hoppe, Reinhold 1816–1900, math., DMV: 120, 208, 369-70 Hoüel, Jules 1823–1886, French math.: 7576, 102, 630, 632 Humboldt, Alexander von 1769–1859, natural scientist: 569 Humboldt, Wilhelm von 1767–1835, academic: 22 Hurwitz, Adolf 1859–1919, math., DMV: 10, 13, 51, 56, 117, 177-78, 183, 192, 195-99, 210-11, 214, 223, 225, 227-30, 233-36, 238, 240, 242, 248, 252, 255, 260-62, 268, 270-72, 274-75, 284-86, 291-99, 308-09, 311-12, 314, 316, 32021, 323, 326-28, 332-33, 339-41, 346, 348-49, 353, 355, 357, 359-62, 365, 369, 371, 373, 375-79, 381, 384-85, 398, 402, 408, 412, 417, 429-30, 43435, 457, 459, 461, 484, 511, 564, 571, 576, 578-79, 603-07, 613, 615, 624, 632, 641, 644, 652 Hurwitz, Julius 1857–1919, math., DMV: 641 Husserl, Edmund 1859–1938, philos.: 489 Huygens, Christiaan 1629–1695, Dutch math., physic., astr.: 594, 596 Ihlenburg, Wilhelm *1884, math., PhD with Klein: 456 Intze, Otto 1843–1904, eng.: 422 Jaccottet, Charles 1872–1938, Swiss math., PhD with Klein, DMV: 333, 346, 393, 399, 613, 615 Jacobi, Carl Gustav Jacob 1804–1851, math.: vii, 9, 37, 47-48, 63, 69, 180, 183, 235, 264-65, 272, 288, 294-95, 299, 341, 569, 596, 610 Jacobs, Konrad 1928–2015, math., DMV: 6 Jaensch, Erich R. 1883–1940, psychol.: 491 Jahnke, Eugen 1861–1921, math.: 305, 615 Jerrard, George 1804–1863, British math.: 180 Johnson Ada M. *1870, British math.: 354 Jordan, Camille 1838–1922, French math.: 4, 6, 52, 54, 60, 74, 77-79, 85, 87-88, 90, 100, 104, 130, 140, 153, 180-81, 189, 192, 198, 240, 245, 267, 288, 338, 526, 583, 619, 626 Joubert, Charles 1825–1906, French math.: 180

Jürgens, Enno 1849–1907, math., DMV: 369-70 Jullien, Michel Marie 1827–1911, French Jesuit, scholar: 596 Jung, Giuseppe 1845–1924, Italian math., Enc: 157, 190, 208 Kamerlingh Onnes, Heike 1853–1926, Dutch physicist, Enc: 431 Kant, Immanuel 1724–1804, philosopher: 116, 489 Kantor, Seligmann 1857–1903, Austrian math.: 230, 248 Karagiannides [Carajianides], Athanasios 1868–?, Greek math., DMV: 335 Kármán, Theodore von 1881–1963, Hungarian-American math., eng., DMV, Enc: 449, 468, 544, 565, 633 Kasner, Edward 1878–1955, American math.: 131, 246 Kasten, H. math. teacher in Bremen, DMV: 369-70 Katz, David 1884–1953, psychol.: 492, 615 Kayser (née Schleicher), Eleonore 1793– 1875, F. Klein’s mat. grandmother: 20 Kayser, Christian Gottfried 1791–1849, wool merchant, F. Klein’s mat. grandfather: 20 Keesom, Willem Hendrik 1876–1956, Dutch physicist, Enc: 431 Kekulé, August 1829–1896, chemist: 36 Kępiński, Stanisław 1867–1908, Polish math., DMV: 283, 335 Kepler, Johannes 1571–1630, math., astronomer: 14, 594, 596 Kerry, Benno 1858–1889, Austrian philosopher: 489 Kerschensteiner, Georg 1854–1932, math., DMV: 555, 613, 615 Ketteler, Eduard 1836–1900, physicist: 31, 35 Kiepert, Ludwig 1846–1934, math., DMV: 16, 61, 66-68, 116, 119, 161, 184, 186, 199-200, 208, 212, 224, 233, 236, 369, 419, 615, 633 Kiesel, Karl 1812–1903, school director: 22 Kirchberger, Paul 1878–1945, math., teacher, author: 21, 436, 634 Kirchhoff, Arthur 1871–1921, writer: 415 Kirchhoff, Gustav Robert 1824–1887, physicist: 374, 466, 489 Kirdorf, Adolph 1845–1923, mining industrialist: 27, 422

Index of Names Klaus, Dr., neurologist: 515, 517, 608-09 Klein, Alfred 1854–1929, lawyer, F. Klein’s brother: x, 17-22, 166-67, 422 Klein (md. Flender), Aline 1847–1914, F. Klein’s sister: x, 21 Klein (née Hegel), Anna Maria Caroline 1851–1927, F. Klein’s wife: viii, x, 161-63, 166-70, 176, 214, 286, 324, 326, 330, 435-36, 449, 501, 517, 51921, 567, 577, 583, 622 Klein, Carl 1842–1907, mineralogist: 317 Klein (md. Staiger), Elisabeth Marie Aline 1888–1968, teacher, F. Klein’s daughter: viii, x, 166-68, 213, 326, 418, 535, 537, 541, 567, 585, 649-50 Klein, Eugenie 1861–1910, F. Klein’s sister: x, 22, 167 Klein, Johann Peter Friedrich, 1777–1858, smith, F. Klein’s grandfather: x, 17 Klein (md. Süchting), Luise (Louise) 1879– 1961, F. Klein’s daughter: x, xix, 16667, 169, 414 Klein, Otto Karl 1876–1963, eng., F. Klein’s son: x, 166-69, 567 Klein, Peter Caspar 1809–1889, Prussian official, F. Klein’s father: x, 18 Klein, Sophie Elise (née Kayser) 1819– 1890, F. Klein’s mother: x, 19-20 Klein (md. Hagemann), Sophie Eugenie 1885–1965, F. Klein’s daughter: x, 21, 166, 167, 169 Kleine, Friedrich Peter *1731, farmer, F. Klein’s great-grandfather: 17 Kleine (née Schürfeld), Catharina Margarethe, F.Klein’s great-grandmother: 17 Klemm, F. math. teacher in Bremen, DMV: 369-70 Klingenfeld, Friedrich August 1817–1880, math.: 172, 211 Klinkerfues, Ernst Friedrich Wilhelm 1827– 1884, astronomer: 92-93, 95, 120 Klitzkowski, Felix, b. in Danzig [Gdansk], math.: 333 Kluckhohn, August von 1832–1893, historian: 212 Kneser, Helmuth 1898–1973, math., DMV: 536 Kneser, Martin 1928–2004, math., DMV: 533 Knoblauch, Johannes 1855–1915, math., DMV: 89, 229

665

Koebe, Paul 1882–1945, math., DMV: 6, 269, 273, 279-81, 307, 456-57, 514-15, 522-23, 525, 546, 565-67, 572, 607-08, 613, 615, 651 König, Julius 1849–1914, Hungarian math., DMV: 102-03, 120, 249-50, 615 Koenigsberger, Leo 1837–1921, math., DMV: 9, 47, 48, 53, 64, 260, 275, 295, 351, 367, 369, 379, 410, 412, 426, 489, 596, 637 Kötter, Fritz 1857–1912, math., mech., DMV: 463 Kohlrausch, Wilhelm 1855–1936, physicist: 168 Kollert, Julius 1856–1937, physicist, DMV: 230, 262 Kopp, Lajos 1860–1928, Hungarian math.: 130 Koppel, Leopold 1854–1933, banker: 513 Korkin, Aleksandr N. 1837–1908, Russian math.: 250 Korteweg, Diederik J. 1848–1941, Dutch math.: 432 Kortum, Carl Arnold 1745–1824, physician, poet: 163 Kottler, Friedrich 1886–1965, AustrianAmerican physicist: 541 Kovalevskaya (née Korvin-Krukovskaya), Sofya V. 1850–1891, Russian math.: 56, 97-98, 411-12, 463 Kowalewski, Gerhard 1875–1950, math., DMV: 243, 433, 637 Kraepelin, Karl 1848–1915, biologist: 46, 498, 648 Krause, Martin 1851–1920, math., DMV: 402 Krauß (as of 1905 Ritter von), Georg 1826– 1906, entrepreneur: 203, 424 Krazer, Adolf 1858–1926, math., DMV, Enc: 231, 233-34, 237-38, 264-65, 396, 426, 457, 536, 553-54, 557-60, 613, 615 (here Krager=Krazer) Kregel von Sternbach, Karl Friedrich 1717– 1789, philanthropist: 239 Krell, Otto 1866–1938, eng., industr.: 451 Krieg v. Hochfelden, Franz 1857–1919 stud. math.: 231 Kronecker, Leopold 1823–1891, math., DMV: 3, 35, 54, 56, 63-64, 68, 180, 182, 184, 188, 196-97, 229, 235, 23738, 240, 249-50, 260-61, 264, 266, 28889, 292-93, 296, 322, 328, 342, 359,

666

Index of Names

368, 371, 373-74, 376-77, 405, 454, 457, 483, 571, 575-76, 580, 591, 602, 604, 607, 626, 637, 642, 645 Krüger, Louis 1857–1923, math., surveyor, DMV: 557, 560, 615 Krull, Wolfgang 1899–1971, math., DMV, Enc: 240, 536, 637 Krupp von Bohlen und Hallbach, Gustav 1870–1950, entrepreneur: 170, 422, 440, 442, 444, 561, 615 Krylov [Kriloff], Alexej, N. 1863–1945, Russ. naval eng., math., Enc: 480, 613, 615 Kuhn, Thomas S. 1922–1996, American physicist, philosopher: 8, 629 Kummer, Ernst Eduard 1810–1893, math.: 1, 35, 59-63, 65-67, 71, 77-81, 84-88, 98-100, 150, 194, 206-07, 221, 227, 229, 233, 250, 298, 309, 340, 366, 374, 405, 594, 602, 609 Kundt, August 1839–1894, physicist: 376 Kutta, Wilhelm 1867–1944, math., DMV: 448, 450, 453 Ladd-Franklin, Christine 1847–1930, American math., psychologist: 412-13 Lagarde, Paul Anton de 1827–1891, orientalist: 318, 365-66, 638 Lagrange, Joseph-Louis 1736–1813, French math.: 9, 114, 160, 179, 457, 480, 594, 596 Laisant, Charles-Ange 1841–1920, French math.: 76, 305 Lamb, Horace 1849–1934, British math., physicist, Enc: 302, 432 Lamé, Gabriel 1875-1870, French math., physicist: 253-54, 344, 346, 416, 462, 463, 596 Lampe, Emil 1840–1918, math., DMV: 76, 119-20, 305, 369-71, 500, 513 Lanchester, Frederick W. 1868–1946, British polymath., eng.: 452 Landau, Edmund 1877–1938, math., DMV: viii, 327, 441-42, 456, 460-61, 518, 522-23, 526, 532, 545, 572-73, 613, 615, 647 Landolt, Hans Heinrich 1831–1910, Swiss chem.: 31-32, 97 Lange, Ernst *1858, math., Saxon official, PhD with Klein: 222, 228, 230, 232, 308, 615 Lange, Helene 1848–1930, pedag.: 506

Larmor, Joseph 1857–1942, Irish physicist, math.: 431-32, 549 Laski, Gerda 1893–1928, Austrian physicist: 536 Laue, Max von 1879–1960, theor. physicist, DMV, Enc: 357, 541, 615, 624 Laugel, Léonce 1859–1936, French math., trans., DMV: 350, 409, 458-59, 462, 464, 634-35, 643 Launhardt, Wilhelm 1832–1918, eng.: 36263, 366 Lederer, Hugo 1871–1940, sculptor: 569 Legendre, Adrien-Marie 1752–1833, French math.: 30, 252, 272 Leibniz, Gottfried Wilhelm 1646–1716, polymath.: 310, 530-31 Lemoine, Émile 1840–1912, French math.: 402 Lenard, Philipp 1862–1947, physicist: 542, 544 Lerch, Matyás 1860–1922, Czech math., DMV: 402 Levi-Civita, Tullio 1873–1941, Italian math., DMV: 200, 463, 618 Lexis, Wilhelm 1837–1914, statistician, economist: 401, 414, 419-20, 439, 454, 500, 502, 615, 634, 635 Lichtenberg, Georg Christoph 1742–1799, physicist, satirist: 45 Lichtenstein, Leon 1878–1933, PolishGerman math., DMV, Enc: 57, 254, 263, 281, 560 Lichtwark, Alfred 1852–1914, art historian: 519, 642 Lie, Sophus 1842–1899, Norwegian math., DMV: viii, 1, 5-6, 14, 16, 19, 26, 41, 56-57, 59, 61, 65-71, 73-86, 88-91, 9596, 98-102, 105, 107-08, 110, 117, 120, 123, 125-31, 133, 136-38, 143, 147-50, 153, 156, 158-61, 181, 183, 206-09, 223, 225, 239-40, 243, 253, 267-68, 277, 285, 287, 303, 312, 314-15, 32024, 328, 333, 335, 347, 349, 350-52, 374, 389, 393, 404, 411, 427, 435, 447, 463, 484, 491, 517, 519, 570, 575-76, 578, 583, 586, 609, 619, 623, 627-28, 630, 633, 638, 641, 643, 644, 648, 653 Liebermann, Max 1847–1935, painter: ix, 243, 494, 519-21, 528, 531, 569, 575, 577, 611, 614-17, 642 Liebig (as of 1845 Freiherr von), Justus 1803–1873, chemist: 561

Index of Names Liebisch, Theodor 1852–1922, mineral., Enc: 317, 356, 366, 385, 388 Liebmann, Heinrich 1874–1935, math., DMV, Enc: 302, 339, 463, 615, 638 Lietzmann, Walther 1880–1959, math., didactics, DMV: 396, 487, 494, 503, 505, 515-517, 550-55, 566, 582, 587, 613, 615, 624, 629, 632, 638 Linde (as of 1897 Ritter von), Carl 1842– 1934, eng., inventor, entrepreneur: 172, 201-03, 423-24, 443, 516, 613, 615 Lindemann, Ferdinand 1852–1939, math., PhD with F. Klein, DMV: 5, 13, 20, 40, 48, 63, 87-88, 94-97, 113, 115, 117, 120, 133-35, 137, 150, 159, 162-63, 171, 191-92, 198, 208-09, 216, 229, 236, 264-65, 311, 321-23, 333, 348-49, 357-58, 369, 373-74, 376, 381, 388, 393, 398, 401, 430, 434, 571-72, 607, 613, 616, 623, 627, 653 Liouville, Joseph 1809–1882, French math.: 74, 85, 102, 639 Lipschitz, Rudolf 1832–1903, math., DMV: 30-33, 41-42, 63-64, 113, 120, 217, 351, 371, 645 Lissajous, Jules Antoine 1822–1880, French physicist.: 136 Listing, Johann Benedikt 1808–1882, math., physicist: 92-95, 120, 235 Lobachevsky, Nikolai I. 1792–1856, Russian math: 70-73, 101, 126, 251, 324, 350, 408 Loewy, Alfred 1873–1935, math., DMV: 393, 616 Lommel (as of 1892 von), Eugen 1837– 1899, physicist, DMV: x, 124, 160, 166, 533, 589, 595-96 Lommel, Herman 1885–1968, philologist, Indo-Europeanist: 532 Lorentz, Hendrik Antoon 1853–1928, Dutch physicist, Enc: 150, 304, 395, 431, 472-73, 509, 549, 582 Lorenz, Hans 1865–1940, techn. physicist, DMV: 439, 447, 449, 513, 577, 590 Lorey, Wilhelm 1873–1955, math., DMV: 30, 48, 51, 66, 89, 94, 96, 107, 114, 133, 145, 176-77, 219, 232, 275, 313, 399, 418, 420, 481, 505, 510, 527-28, 616, 624, 639, 646

667

Loria, Gino 1862–1954, Italian math., hist., DMV, Enc: 42, 135, 399, 431, 613, 616, 635 Lotze, Rudolf Hermann 1817–1881, philosopher: 89, 116 Love, Edward Hough 1863–1940, British math., Enc: 302, 429, 432, 469, 470, 616 Ludwig, Carl 1816–1895, physiologist: 2, 310, 312-13 Ludwig II, Otto Friedrich Wilhelm von Wittelsbach 1845–1886, King of Bavaria: 124, 171-72 Lüders, Otto 1844–1912, class. philol.: 35 Lüroth, Jacob 1844–1910, math., DMV: 40, 47-48, 50, 53, 87, 100, 120, 173, 208, 212, 313, 369, 375, 608 Luther, Robert Karl Theodor 1822–1900, astronomer: 27 Luzin, Nikolai N. 1883–1950, Russ./Soviet math.: 533 Mach, Ernst 1838–1916, Austrian physicist, philosopher: 296, 490 MacKinnon (md Fitch), Annie Louise 1868– 1940, Canadian-American math.: 406 Maddison, Ada Isabel, 1869–1950, British math.: 417, 483-84, 635 Madelung, Erwin 1881–1972, physicist: 466 Magnus, Gustav 1802–1870, physicist: 62 Maltby, Margaret Eliza 1860–1944, American physicist: 413 Mangoldt, Hans von 1854–1925, math., DMV: 311 Mansion, Paul 1844–1919, Belgian math., DMV: 127-28, 495, 634 Mariotte, Edme ca. 1620–1684, French physicist: 203 Markov [Markoff], Andrey A. 1856–1922, Russian math.: 6, 250-51, 301, 345-46, 355, 487, 587, 613, 616, 623, 651 Marotte, Francisque 1873–1945, French math., teacher: 502 Maschke, Heinrich 1853–1908, math., DMV: 337-38, 365-66, 402, 404, 41213, 619, 640 Massenbach, Leo Freiherr von 1797–1880, lawyer, Prussian official: 19 Maxwell, James Clerk 1831–1879, Scottish physicist: 150, 169, 364, 395, 410, 463, 468-69

668

Index of Names

Mayer, Adolph 1839–1908, math., DMV: 6, 47, 53-57, 75, 91, 100, 118, 120-21, 123, 157, 161, 181-83, 193-95, 208-09, 211, 213-15, 219, 222, 225-27, 239-40, 250, 267, 303, 308, 311-12, 321-22, 324, 333, 335, 349, 355, 369-70, 427, 479, 483, 538, 578, 651 Mazurkiewicz, Stefan 1888–1945, Polish math.: 421 McClintock, Emory 1840–1916, American actuary: 407, 418 Mehmke, Rudolf 1857–1944, math., DMV, Enc: 146, 302-04, 616 Meier, Ernst von 1832–1911, lawyer, curator: 380, 383, 412-13 Mendeleev, Dmitri I. 1834–1907, Russian chemist: 363 Merkel, Friedrich 1845–1919, anatomist: 115, 117, 479 Mertens, Franz [Franciszek] 1840–1927, Polish Austrian math.: 613, 616 Merz, John Theodore 1840–1922, German British chemist, historian, industrialist: 480, 484, 640 Metzler, G.F., listener with Klein: 399 Metzner, Carl 1876–1939, teacher, Prussian official: 556, 590 Meyer, Diedrich, building officer, VDI director: 565, 616 Meyer, Eugen 1868–1930, techn. physicist, DMV: 424, 616 Meyer, Franz 1856–1934, math., DMV, Enc: 4, 12, 108, 193-94, 204, 243, 299, 305, 369-70, 372, 392, 395, 426, 430, 433-34, 616 Meyer, Georg, math. teacher in Bremen, DMV: 369-70 Meyerstein, Moritz 1808–1882, mech.: 120 Michelson, Albert Abraham 1852–1931, American physicist: 395 Mie, Gustav 1868–1957, physicist, DMV: 307 Mill, John Stuart 1806–1873, British philosopher, economist: 36 Miller (ab 1875 von), Oskar 1855–1934, civil eng.: 516, 613 Minding, Ferdinand 1805–1885, GermanRussian math.: 136 Minkowski, Hermann 1864–1909, math., DMV, Enc: viii, 57, 236, 279, 286, 327, 331, 346, 358, 369-70, 377, 391,

402, 434, 436, 441, 454, 456, 458-60, 466, 472-73, 511, 572, 580, 590, 640 Minkowski, Rudolph 1895–1976, GermanAmerican astronomer: 536 Minnigerode, Bernhard 1837–1896, math., mineralogist: 52, 93, 95, 120 Mises, Richard von 1883–1953, math., DMV, Enc: 7, 266, 304, 428, 449, 471, 524, 534, 558, 565-66, 640, 647, 654 Mittag-Leffler, Gösta 1846–1927, Swedish math., DMV: viii, 56, 130, 212, 250, 272, 277, 295, 305, 315, 328, 342, 495, 519, 613, 616, 640, 643, 648 Młodziejewski, Bolesław 1858–1928, Polish-Russian math.: 334 Möbius, August Ferdinand 1790–1868, math.: 37, 104, 114, 140, 208, 215, 217, 237, 248, 257, 301, 310-312, 478, 629, 653 Mohrmann, Hans 1881–1941, math., DMV, Enc: 433, 537 Molien, Theodor 1861–1941, GermanBaltic/Soviet math., DMV: 231, 251, 293, 311 Molk, Jules 1857–1914, French math., DMV, Enc(f): 429, 613, 616, 628, 630 Mollier, Richard 1863–1935, applied physicist, eng.: 424 Mommsen, Theodor 1817–1903, historian: 366, 642 Monge, Gaspard 1746–1818, French math.: 37, 44, 224, 625 Moore, Eliakim Hastings 1862–1932, American math., DMV: 313, 399, 402-04, 413, 613, 616, 619, 640, 641 Morera, Giacinto 1856–1909, Italian math.: 156, 231, 247-48, 293, 311 Morgan, Augustus de 1806–1871, British math: 151-52, 354, 586 Morley, Edward Williams 1838–1923, American chemist: 417 Morrice, George Gavin 1859–1936, British scientist, transl.: 290 Mügge, Otto 1858–1932, mineralogist, Enc: 356, 522, 616 Mühler, Heinrich von 1813–1874, Prussian Minister of Culture: 39, 82, 593 Müller, Conrad Heinrich 1878–1953, math., math.hist., PhD with Klein, DMV, Enc: 390, 394, 429, 457, 470, 479, 515, 537, 588, 613, 616

Index of Names Müller, Felix 1843–1928, math., teacher, DMV: 118, 120 Müller, Georg Elias 1850–1934, psychol.: 258, 317, 484-85, 490, 492, 616 Müller, Hans math., PhD Göttingen (1903), DMV: 394 Müller, Reinhold 1857–1939, math., DMV: 370 Müller, Wilhelm 1812–1890, Germanist: 600 Müller-Breslau, Heinrich 1851–1925, structural eng.: 304, 468, 634 Nachtweh, Alwin 1868–1939, eng.: 168-69, 616 Naumann, Otto 1852–1925, Prussian official: 391, 526, 528, 530, 541 Neesen, Friedrich 1849–1923, math., physicist: 34, 96-97, 108-09, 113-15, 120, 166-67, 208, 345 Nekrasov, Pavel A. 1853–1924, Russian math.: 355, 603-04 Nelson, Leonard 1882–1927, philos.: xx, 481, 489-91, 568, 584, 616 Nernst, Walther 1864–1941, physical chemist: 385, 413, 422-23, 502, 516, 522, 528, 637, 640 Netto, Eugen 1848–1919, math., DMV, Enc: 66, 67, 120, 402 Neugebauer, Otto 1899–1990, AustrianAmerican math.hist., DMV: 480, 537, 539, 642 Neuhäuser, Joseph 1823–1900, philosopher: 30-31 Neumann, Carl 1832–1925, math., DMV: 46-48, 53-55, 63, 108, 140, 215, 22123, 227, 229, 253-56, 262, 306, 311, 321, 323, 569, 596, 652 Neumann, Franz 1798–1895, physicist: 47 Newcomb, Simon 1835–1909, CanadianAmerican math., astronomer: 246, 407 Newton, Isaac 1642–1727, British math., physicist: 31, 491, 594 Nielsen, Jakob 1890–1959, Danish math., DMV: 536 Nimsch, Paul *1860, math., PhD with Klein: 230, 293 Noble, Charles Albert 1867–1962, Amer. math.: 399, 556 Nöggerath, Johann Jakob 1788–1877, mineralogist, geologist: 31-32 Noether, Emmy 1882–1935, math., DMV: ix, 50, 160, 178, 359, 417-18, 536, 539-

669

42, 576, 578, 581, 584, 591, 637, 641, 644-45, 650, 651 Noether, Fritz 1884–1941, math., DMV: 50, 462, 616, 637 Noether, Max 1844–1921, math., DMV: 10, 13, 47, 49, 50, 53, 57-59, 64-65, 68-71, 77, 79, 84, 103, 107, 113, 118-20, 124, 147-48, 155, 158-62, 178, 182, 188-89, 204, 208, 236, 248-49, 278, 285, 296, 323, 334, 341, 351, 354, 357, 366, 372, 402, 406, 417, 462, 481, 576, 578, 606, 613, 616, 618, 643 Nohl, Herman 1879–1960, philosopher, pedagogue: 534 Ocagne, Maurice d’ 1862–1938, French math., Enc(f): 302, 402, 645 Ohrtmann, Carl 1839–1885, math., teacher: 118, 120 Olbricht, Richard 1859–1912, math., school director, PhD with Klein: 230, 616 Oliver, James Edward 1829–1895, American math.: 404, 406 Osgood, William Fogg 1864–1943, American math., DMV, Enc: 249, 280, 334, 340-41, 405-06, 585, 613, 616, 625 Ostrowski, Alexander M. 1893–1986, Ukrainian-Swiss math., DMV: 442, 478, 524, 536, 545-46, 579, 641 Ostwald, Wilhelm 1853–1932, physical chem.: 2, 426, 492, 513, 528, 546, 637 Ovidio, Enrico d’ 1842–1933, Italian math.: 153-54, 247, 613, 616 Padé, Henri Eugène 1863–1953, French math.: 130, 334 Padova, Ernesto 1845–1896, Italian math.: 157 Painlevé, Paul 1863–1933, French math., Enc: 333, 452, 471-72, 588 Paladini, Bernardo 1863–?, Italian math.: 402 Papperitz, Erwin 1857–1938, math., PhD with Klein, DMV, Enc: 231, 280, 36970, 616 Parseval, August von 1861–1942, airship designer: 453 Pascal, Blaise 1623–1662, French math.: 139 Pascal, Ernesto 1865–1940, Italian math.: 156, 334, 341, 366, 613, 616, 634 Pasch, Moritz 1843–1930, math., DMV: 100, 120, 482, 491, 606, 645

670

Index of Names

Pasquier, Ernest 1849–1926, Belgian math.: 120, 127-28 Pasteur, Louis 1822–1895, French microbiologist: 36 Pauli, Wolfgang Ernst 1900–1958, Austrian physicist, Enc: 536, 541, 632 Peano, Giuseppe 1858–1932, Italian math.: 483, 491, 647 Peipers, Johann Philipp David 1838–1912, philosopher: 116-17 Perry, John 1850–1920, Irish eng., math.: 169, 302, 502, 507, 509, 584, 641 Pervushin, Ivan M. 1827–1900, Russian math.: 402 Pestalozzi, Johann Heinrich 1746–1827, Swiss pedagogue: 531 Petermann, August Heinrich 1822–1878, cartographer: 36 Petzoldt, Joseph 1862–1929, philos.: 490 Pfaff, Friedrich 1825–1886, geologist, mineralogist: 145 Pfaff, Hans Ulrich Vitalis 1824–1872, math.: 124, 145, 163 Pfeiffer, Friedrich 1883–1961, math., DMV: 470-71, 616 Pfitzer, Ernst 1846–1906, botan.: 32 Picard, Charles Émile 1856–1941, French math.: 131, 253, 268, 292, 333, 340, 342-43, 352, 357, 410, 456, 459, 495, 512, 529, 531, 583, 613, 616 Pick, Georg 1859–1942, Austrian math., DMV: 10, 248, 293-97, 321, 330, 341, 460, 576 Pickering, Edward Charles 1846–1919, American astronomer: 406 Pieri, Mario 1860–1913, Italian math. Enc(f): 130, 639 Pietzker, Friedrich 1844–1916, teacher, DMV: 400, 503, 616 Pincherle, Salvatore 1853–1936, Italian math., DMV, Enc: 402, 635 Planck, Gottlieb 1824–1910, lawyer: 327 Planck, Max 1858–1947, physicist, DMV: 14, 177-78, 204, 327, 415, 440, 457, 489, 513, 528-30, 541, 583, 587, 632 Plato ca. 428–348 BC, Greek philos.: vii, 35-36, 117, 143 Plücker, Albert, son of Julius P.: 39 Plücker (née Altstätter), Antonie, Julius P.’s wife: 40, 97, 111, 477 Plücker, Julius 1801–1868, math., physicist: vii, 2, 9-10, 17, 22, 28-42, 46-48, 52-

53, 58-62, 65-69, 71, 80, 88-89, 92-93, 96, 111, 113-14, 126, 128-29, 134, 13839, 151, 154-55, 206, 208, 216, 233, 300, 301, 309, 330, 345, 477-78, 510, 536, 569-70, 575, 596-97, 599, 609, 627-29, 642, 653 Pockels, Agnes 1862–1935, physico-chemist: 345 Pockels, Friedrich 1864–1913, math., physicist, DMV, Enc: 302, 344-45, 455, 616 Poincaré, Henri 1854–1912, French math.: viii, 4-6, 102, 130, 189, 215, 245, 250, 252-53, 258, 267-85, 295, 340, 342, 352, 357, 381, 387, 394-95, 403, 410, 456, 458, 467, 485, 495, 512-13, 519, 541, 571-73, 579, 582, 587, 610, 613, 631, 640, 642-43, 646, 651, 652 Poinsot, Louis 1777–1859, French math.: 177, 595-96 Poisson, Siméon Denis 1781–1840, French math., physicist: 595 Pokrovsky, Petr M. 1857–1901, Russian math., DMV: 355 Poncelet, Jean-Victor 1788–1867, French math., eng., physicist: 1, 37-38, 68, 70, 127, 569 Pontani, Bernhard *27.10.1845, teacher: 29 Poske, Friedrich 1852–1925, pedag.: 494, 550, 553, 616 Prandtl, Ludwig 1875–1953, eng., mech., DMV, Enc: viii, 4, 7, 307, 327, 391, 449-53, 465-66, 468, 470-71, 522, 524, 562-63, 565-66, 577, 581, 588, 616, 623, 628, 637 Prange, Georg 1885–1941, math., DMV, Enc: 114 Pringsheim, Alfred 1850–1941, math., DMV, Enc: 204, 509-10, 616 Pringsheim, Nathanael 1823–1894, botanist: 36 Prym, Friedrich 1841–1915, math., DMV: 65, 237, 256, 264, 457 Puiseux, Victor 1820–1883, French math.: 108 Pulfrich, Carl 1858–1927, physicist: 35 Pupin, Mihajlo Idvorski 1854–1935, Serbian-American physicist: 467-68 Rabinowitsch-Kempner, Lydia 1871–1935, bacteriol.: 505, 506 Radicke, Gustav 1810–1883, physicist: 32

Index of Names Rados (Raussnitz), Gusztáv 1862–1942, Hungarian math., DMV: 103, 231, 248-50, 616 Ranke (as of 1865 von), Leopold 1795– 1886, historian: 116, 164-65, 427-28, 475, 477 Raphael 1483–1520, Italian painter: 566 Rathenau, Walther 1867–1922, industrialist, liberal politician: 584 Rausenberger, Otto 1852–1931, math., teacher: 274-75 Rayleigh (Lord), Strutt, John William 1842– 1919, British physicist.: 151-52, 345, 431 Reeß, Maximilian 1845–1901, botanist: 135 Reger, Maximilian (Max) 1873–1916, composer: 167 Reich, Max 1874–1941, physicist: 562 Reichardt, Hans 1908–1991, math., DMV: 73, 548, 642 Reichardt, Willibald A. 1864–1924, math., PhD with Klein: 231, 311, 337, 616 Rein, Wilhelm 1847–1929, pedag.: 492 Reinke, Johannes 1849–1931, botanist: 501 Reissner [Reißner], Hans 1874–1967, eng., math., physicist, DMV, Enc: 468, 565, 616 Repsold, Johann A. 1838–1919, instrument maker: 363, 366 Réthy, Mór (Moritz) 1846–1925, Hungarian math., DMV: 120 Reye, Karl Theodor 1838–1919, math., DMV: 69, 73, 78, 87, 120, 208, 215, 240, 371, 596, 618 Reynolds, Osborne 1842–1912, British physicist: 354 Ricci-Curbastro, Gregorio 1853–1925, Italian math., DMV: 199-200 Richardson, Roland George Dwight 1878– 1949, Canadian-American math., DMV: 346-47, 619 Richelot, Friedrich Julius 1808–1875, math.: 47, 119, 223, 379 Richert, Hans 1869–1940, teacher, education politician: 556 Richter, Otto math. student: 231 Riecke, Eduard 1845–1915, physicist, DMV: 96, 110-17, 120, 161, 235, 305, 316-20, 325, 332, 336, 363, 366, 400, 420, 423-24, 434, 439, 450, 473-74, 519, 522, 546, 560, 611-12, 616, 636, 643

671

Riedler, Alois 1850–1936, Austrian eng., professor in Germany: 401, 424, 443 Riemann, Bernhard 1826–1866, math.: vii, 2, 4-6, 9, 45, 48-49, 51, 53, 58, 65, 68, 72-73, 92, 95, 98, 101, 103, 126, 12829, 135-36, 139-43, 149, 153-54, 15657, 178-81, 184-86, 194, 203, 217, 22223, 228, 237, 249, 252-53, 255-60, 26265, 268, 270-74, 276-78, 280, 282-85, 288, 294-95, 338, 340, 342-44, 352, 354, 357, 360, 366, 379, 389, 393, 406, 409-10, 417, 452, 457, 461, 464, 478, 479, 547, 549, 570, 572, 586, 600, 60708, 610, 621, 631, 634, 638, 640, 64344, 646, 648, 651, 653 Rieppel (as of 1906 von), Anton 1852–1926, eng.: 439-40, 444-45, 469, 471, 616 Ritter, August 1826–1908, astrophysicist, prof. of mechanics, DMV: 370 Ritter, Ernst 1867–1895, math., PhD with Klein, DMV: 331, 387-88, 393, 406, 413, 435 Rockefeller, John D. 1839–1937, American entrepreneur: 564, 647 Rodenberg, Carl Friedrich 1851–1933, math., DMV: 97, 108, 309, 370 Rohn, Karl 1855–1920, math., PhD with Klein, DMV, Enc: 193-94, 204, 20607, 220-23, 226, 236, 298, 308-09, 311, 433, 613, 616 Rohns, Christian Friedrich Andreas 1787– 1853, architect: 169, 327 Rohr, Moritz von 1868–1940, math., inventor: 114, 650 Rosanes, Jacob 1842–1922, math., DMV: 374, 375, 616 Rosemann, Walther 1899–1971, math.: 257, 547, 636 Rosenbach, Friedrich Julius 1842–1923, physician, surgeon: 384 Rosenhain, Johann Georg 1816–1887, math.: 236 Rosenthal, Arthur 1887–1959, math., DMV, Enc: 381, 421, 430, 616 Rost, Georg 1870–1958, math., DMV: 457 Rothe, Rudolf 1873–1942, math., DMV: 550 Routh, Edward John 1831–1907, British math.: 302, 431 Roux, Karl 1826–1894, painter: 210 Rudio, Ferdinand 1856–1929, GermanSwiss math.: 317, 479, 613, 616

672

Index of Names

Rüdenberg, Reinhold 1883–1961, eng.: 466, 616 Ruer, Wilhelm 1848–1932, judicial councilor, poet, Klein’s classmate: 27, 616 Runge, Carl 1856–1927, math., DMV, Enc: viii, 25, 62, 89, 168, 177-78, 229, 261, 303, 327, 391, 421, 445, 448-49, 45253, 465, 469-70, 473, 476, 489, 506, 512, 517, 522-24, 528, 532, 537, 539, 558-60, 566, 587, 613, 616, 632, 643 Runge, Iris Anna 1888–1966, math., chem., physicist: 418, 448, 466, 506-07, 517, 523, 537, 546, 650 Russell, Bertrand 1872–1970, British philosopher, polymath.: 353, 572 Sachs, Eva Henriette 1882–1936, classical scholar, teacher: 35 Sachs, Julius 1832–1897, botanist: 36 Sagorski, Ernst 1847–1929, Klein’s fellow student, teacher: 29, 35, 44 Salmon, George 1819–1904, Irish math.: 40-41, 47, 71, 87, 102, 248, 596, 645 Sanden, Horst von 1883–1965, math., DMV: 453, 616 Sartorius Freiherr von Waltershausen, Wolfgang 1809–1876, geol.: 90 Sauppe, Hermann 1809–1893, classical philologist: 317, 364-66 Scheffers, Georg 1866–1945, math., DMV: 447 Scheibner, Wilhelm 1826–1908, math., DMV: 206, 215-16, 222, 229-30, 233, 237, 239, 308, 311-12, 321 Schell, Wilhelm 1826–1904, math., mech., DMV: 596 Schellbach, Karl Heinrich 1805–1892, math., pedagogue: 63, 65 Schering, Ernst Christian Julius 1833–1897, math., astr., DMV: xvii, 11, 92, 93, 95, 98, 120, 235, 236, 317, 318, 325, 335, 363, 366, 367, 377, 378, 380, 384, 392, 434, 478, 598, 600, 601, 602 Schering (née Malmstén), Maria Heliodora 1848–1920: 367 Schiller, Friedrich 1759–1805, poet, playwright: 25, 454 Schilling, Carl 1857–1933, math., DMV: 370, 616 Schilling, Friedrich (Fritz) 1868–1950, math., DMV: 302, 386-87, 391, 39394, 440, 447, 449, 616, 636, 645 Schilling, Martin publisher: 175

Schimmack, Rudolf 1881–1912, math., didactics, DMV: 302, 390, 446, 50405, 515, 517, 588, 613, 616, 635, 645 Schläfli, Ludwig 1814–1895, Swiss math.: 140-41, 257, 309, 633 Schlegel, Victor 1843–1905, math., teacher, DMV: 313, 402 Schlesinger, Ludwig 1864–1933, Hungarian-German math., DMV: 344, 478, 616 Schlömilch, Oscar 1823–1901, math., DMV: 303 Schmeidler, Werner 1890–1969, math., DMV: 536 Schmidt, Carl, theologian: 155 Schmidt, Erhard 1876–1959, math., DMV: 394 Schmidt (as of 1920 Schmidt-Ott), Friedrich 1860–1956, science politician: 303, 401-02, 413, 444, 450, 453, 530, 533, 557, 559, 560, 563-64, 645 Schmitz, Wilhelm 1846–1900, manager in the company Krupp: 444 Schmoller (since 1908 von), Gustav 1838– 1917, economics scholar: 501 Schneider, Jakob 1818–1898, F. Klein’s math. teacher: 25 Schoenflies (Schönflies), Arthur 1853–1928, math., DMV, Enc: 4, 10, 96, 103-04, 154, 296, 329-32, 335-36, 356, 359-60, 380, 383-84, 387, 392, 402-03, 422, 447-78, 502, 538, 591, 605-06, 617, 633, 640, 642, 646 Scholze, Peter *1987, math., DMV: 460 Schotten, Heinrich 1856–1939, teacher, DMV: 494, 613, 617 Schottky, Friedrich 1851–1935, math., DMV: 234, 256, 265, 270, 283, 375, 377, 392, 428, 457, 587, 609, 611 Schouten, Jan Arnoldus 1883–1971, Dutch math., DMV: 131, 302, 646 Schröder, Edward 1858–1942, Germanist, mediaevalist: 577 Schröder, Ernst 1841–1902, math., DMV: 120, 369-70 Schröder, Johannes 1865–1937, math., teacher, PhD with Klein, DMV: 339, 418 Schrödter, Emil 1855–1928, eng.: 422 Schroeter, Heinrich Eduard 1829–1892, math., DMV: 64, 87, 373, 375-76

Index of Names Schubert, Hermann Cäsar Hannibal 1848– 1911, math., DMV, Enc: 50, 67, 100, 119-20, 241, 368-71, 581 Schüler, Wilhelm, math.: 172, 176-77, 577 Schütz, L., a student in Klein’s courses: 399 Schur, Friedrich 1856–1932, math., DMV: 219, 223, 233-34, 236, 251, 308, 311, 323, 613, 617 Schur, Issai 1875–1941, math., DMV: 523, 557, 558, 559, 560 Schur, Wilhelm 1846–1901, astronomer, DMV: 325, 335, 363, 434 Schwalbe, Bernhard 1841–1901, math., teacher, DMV: 500 Schwarz, Hermann Amandus 1843–1921, math., DMV: 11, 13, 65, 69, 98, 120, 181, 185, 207, 212, 233, 235-36, 238, 256, 261-62, 266, 269, 276, 280, 28283, 295-96, 317, 318-23, 325-26, 32832, 334-36, 342-43, 356, 358, 360, 363, 365, 372, 374-78, 381, 383-84, 387, 393, 412, 428, 456-57, 499, 522, 587, 597-98, 601-02, 605-07, 611, 623, 640 Schwarzschild, Karl 1873–1916, astronomer, DMV, Enc: 394, 420, 453, 461, 465, 467, 617, 636 Scott, Charlotte Angas 1858–1931, BritishAmerican math., DMV: 258, 416-17, 639 Seeger, Johannes, physicist: 44 Seeliger (Ritter von), Hugo 1849–1924, astronomer, DMV: 222, 304 Segre, Corrado 1863–1924, Italian math., DMV, Enc: 6, 41-42, 130, 154, 431, 483, 613, 617-18, 626, 644, 647 Seidel (as of 1882 Ritter von), Ludwig 1821–1896, math., DMV: 173, 193-94, 206, 209, 241, 373, 379, 381, 598 Selenka, Emil 1842–1902, zoologist: 135 Selling, Eduard 1834–1920, math., DMV: 392 Serret, Joseph Alfred 1819–1885, French math.: 232, 596 Severi, Francesco 1879–1961, Italian math: 49, 618, 640 Seyfarth, Friedrich 1891–1960, math., teacher, DMV: 538, 548, 556, 636 Shafarevich, Igor R. 1923–2017, Russian math.: 47, 50, 647 Sibley, Hiram 1807–1888, American entrepreneur: 406

673

Siedentopf, Henry 1872–1940, physicist: 399, 466 Siemens, Werner von 1816–1892, eng., entrepreneur: 438-39, 442, 451 Simon, Hermann Theodor 1870–1918, physicist: 306, 447, 449, 466, 518, 562, 617 Simon, Max 1844–1918, math., hist., DMV: 67, 120, 509, 617 Simony, Oscar 1852–1915, Austrian math.: 208 Sintsov (Sinzow), Dimitrii M. 1867–1946, Russian math. DMV: 130, 251, 372, 613, 617 Sitter, Willem de 1872–1934, Dutch astr.: 542-43, 643 Slaby, Adolf 1849–1913, eng.: 424, 443, 451, 499-502, 635 Slodowy, Peter 1948–2002, math., DMV: 5, 179, 290, 634, 648 Smend, Rudolf 1851–1913, theologian: 434 Smith, David Eugene 1860–1944, American math., hist., pedag., DMV: 396, 399, 493-94, 517, 549-50, 613, 617, 635 Smith, Henry John Stephen 1826–1883, British math.: 148-52, 183, 192, 196, 199, 244, 314, 586 Smith, William Robertson 1846–1894, Scottish orientalist, Old Testament scholar: 96, 105, 123, 147, 183, 353 Snyder, Virgil 1869–1950, American math., PhD with Klein, DMV: 372, 399, 406, 417, 550, 613, 617-19 Sohncke, Leonhard 1842–1897, physicist: 356 Sommerfeld, Arnold 1868–1951, math., physicist, DMV, Enc: 4, 7, 11, 50, 253, 256, 303-04, 331, 345, 388-89, 391, 393, 395, 426, 431-32, 435, 453, 455, 459, 462-63, 472, 539, 544, 564, 613, 617, 628, 636, 648 Sommerfeld (née Höpfner), Johanna 1874– 1955: 389 Sonin, Nikolay Y. 1849–1915, Russian math., DMV: 251, 372, 487 Speiser, Andreas 1885–1970, Swiss math., DMV: 459 Spiegel-Borlinghausen, Adolph von 1792– 1852, officer, Prussian official: 18 Spiess, Otto 1878–1966, Swiss math., hist.: 537

674

Index of Names

Spiro, Eugen (Eugene) 1874–1972, GermanAmerican painter: ix Spitzer, Simon 1826–1887, Austrian math.: 208 Springer, Anton 1825–1891, art hist.: 30-31 Springer, Julius 1880–1968, publisher, DMV: 57, 301, 307, 543, 635 Stäckel, Paul 1862–1919, math., hist., DMV, Enc: 302, 304-05, 349, 427, 443, 471, 476-78, 494, 499, 613, 617 Stähelin, Helene 1891–1970, Swiss math.: 535, 537 Stahl, Hermann von 1843–1909, math., DMV: 457 Staiger, Robert 1882–1914, musicologist, F. Klein’s son-in-law: x, 166-68, 527 Stark, Johannes 1874–1957, exp.physicist: 474, 542, 560 Starke, Dorothea 1902–1943, math. 471 Staude, Otto 1857–1928, math., PhD with Klein, DMV, Enc: 71, 100, 221, 228, 230, 233, 236-37, 264-65, 311-12, 428, 505, 617, 632 Staudt, Karl Georg Christian von 1789– 1867, math.: 70, 73, 79, 97, 103, 12324, 126, 138, 141, 215, 585, 609, 640 Steckel, Fritz 1884–1915, teacher: 491 Steindorff, Ernst 1839–1895, hist.: 116-17 Steiner, Jakob 1796–1863, math.: 37, 47 Steinitz, Ernst 1871–1928, math., DMV, Enc: 42 Steklov (Steckloff), Vladimir A. 1863–1926, Russian math.: 613, 617 Stéphanos, Cyparissos 1857–1917, Greek math., DMV: 130, 495, 613, 617, 642 Stern, Alfred 1846–1936, hist.: 116-17, 564 Stern, Antonie 1892–after 1967, math., DMV: 536 Stern, Moritz Abraham 1807–1894, math., DMV: 51-52, 92-95, 107, 116, 120-21, 127, 132, 235, 317, 329, 364, 564, 598, 601-02, 646 Still, Carl 1868–1951, eng., entrepreneur, DMV: 563 Stöhr, Friedrich, student: 230 Stokes, George Gabriel 1819–1903, Irish math., physicist: 353, 431, 569 Stolz, Otto 1842–1905, Austrian math., DMV: 6, 14, 61, 66-67, 69-71, 100, 107, 116, 119-20, 123, 128, 138-39, 161, 183, 208, 210, 213, 218, 221, 29596, 623, 625

Stresemann, Gustav 1878–1929, politician: 555 Stringham, Irving W. 1847–1909, American math.: 10, 230, 245, 575 Struik, Dirk 1894–2000, Dutch-American math., hist., DMV: 537, 539 Struve, Ludwig von 1858–1920, GermanBaltic math., astronomer: 231 Studt, Konrad von 1838–1921, Prussian Minister of Culture: 499-501 Study, Eduard 1862–1930, math., DMV, Enc: 50, 87, 131, 224, 233, 240, 24244, 246, 323, 330, 339, 351, 357, 369, 402, 404, 542, 577, 617, 627, 631, 637 Stumpf, Carl 1848–1936, philosopher: 11516, 485-86, 489, 648 Sturm, Rudolf 1841–1919, math., DMV: 42, 120, 208, 369-70, 375, 596 Süchting, Friedrich (Fritz) Wilhelm 1874– 1969, eng., F. Klein’s son-in-law: 16667, 169, 516, 641 Sylow, Ludwig 1832–1918, Norweg. math.: 240, 569 Sylvester, James Joseph 1814–1897, British math.: 37, 47-48, 87, 148, 151-52, 245, 246, 314-316, 360, 395, 401, 407, 409, 412, 575, 586, 641 Tägert, Friedrich 1863–1950, math. teacher: 398, 635 Tait, Peter Guthrie 1831–1901, Scottish math., physicist: 94, 96, 105, 147-48, 256, 570 Takagi, Teiji 1875–1960, Japanese math., DMV: 488 Tannery, Jules 1848–1910, French math.: 75-76, 302, 474 Taussky-Todd, Olga 1906–1995, Austrian, later Czech-American math., DMV: 417, 460 Taylor, Brook 1685–1731, Engl. math.: 487 Tedone, Orazio 1870–1922, Italian math., DMV, Enc: 470 Teixeira, Francesco Gomes 1851–1933, Portuguese math., hist.: 495, 613, 617 Tellkampf, Adolph 1798–1869, math., pedag.: 503, 629 Terquem, Orley 1782–1862, French math.: 304 Thaer, Albrecht 1855–1921, math., pedag., DMV: 494, 613, 617 Thomae, Johannes 1840–1921, math., DMV: 124

Index of Names Thomas de Colmar, Charles Xavier 1785– 1870, French inventor: 145-46 Thomas, Sidney Gilchrist 1850–1885, Engl. inventor: 422 Thompson, Henry Dallas, American math., PhD with Klein: 334, 339, 407, 607 Thomson, Joseph John 1856–1940, British physicist: 431, 448-49 Thomson, William (Lord Kelvin) 1824– 1907, British physicist: 94, 96, 146-47, 205, 256 Tietjen, Friedrich 1834–1895, astron.: 376 Tietze, Heinrich 1880–1964, Austrian math., DMV, Enc: 433, 617 Tikhomandritsky, Matvey A. 1844–1921, Russian math.: 250 Tilly, Joseph Marie de 1837–1906, Belgian math.: 348 Timerding, Heinrich Emil 1873–1945, math., DMV, Enc: 4, 476-77, 550 Timoshenko, Stephen P. 1878–1972, Ukrain.-Amer. mechan.: 466, 471, 649 Timpe, Aloys 1882–1959, math., PhD with Klein, DMV, Enc: 302, 390, 470, 480, 617 (Trimpe=Timpe) Toeplitz [Töplitz], Otto 1881–1940, math., DMV, Enc: 14, 346, 490, 510, 617, 630 Tollens, Bernhard 1841–1918, chemist: 115, 117 Treitschke, Heinrich von 1834–1896, hist.: 117 Treutlein, Peter 1845–1912, math., pedag., DMV: 494, 514, 613, 617, 638 Troschel, Franz Hermann 1810–1882, zoologist: 31-32 Trueblood, Mary Esther 1872–1939, American math.: 487 Tyler, Harry Walter 1863–1938, American math., DMV: 249, 334, 404, 613, 617 Uffrecht, Bernhard 1885–1959, math., pedagogue: 491 Ulrich, Georg Karl Justus 1798–1879, math.: 92-94, 120 Uppenkamp, August 1824–1909, teacher: 23 Urysohn, Pavel S. 1898–1924, Russian math.: 533, 538 Valentiner, Herman 1850–1913, Danish math.: 189, 247 Valentiner, Theodor 1869–1952, lawyer, univ. curator: 555

675

Van Vleck, Edward Burr 1863–1943, American math., PhD with Klein, DMV: 334, 346, 372, 404, 406, 613, 617 Van Vleck, John Monroe 1833–1912, American astronomer: 407 Varićak, Vladimir 1865–1942, Serbian math., DMV: 473 Vasilev [Wassiliew], Alexander V. 1853– 1929, Russ. math., DMV: 251, 350, 355, 372, 408, 495, 511, 533, 613, 617, 635 Vermeil, Hermann 1889–1959, math., DMV: 107, 535-36 Veronese, Giuseppe 1854–1917, Italian math., DMV: 154, 230, 246-47, 351, 613, 617-18 Vietoris, Leopold 1891–2002, Austrian math., DMV, Enc: 433 Virchow, Rudolf 1821–1902, physician: 205, 653 Vögler, Albert 1877–1945, entrepreneur: 561-62, 637 Voellmy, Erwin 1886–1951, Swiss math., teacher, DMV: 537 Voigt, Woldemar 1850–1919, theor. physicist, DMV: 317, 325, 335, 345, 363, 434, 439, 461, 466, 522, 562, 613, 617 Volhard, Jakob 1834–1910, chemist: 597 Von der Mühll (VonderMühll), Karl 1841– 1912, Swiss math., DMV: 47, 53-55, 57, 120, 215, 219, 222, 227, 613, 617 Voss [Voß], Aurel 1845–1931, math., DMV, Enc: 4, 41, 51-52, 54, 59-60, 94-95, 97, 100, 108, 119-20, 124, 132-33, 136, 138, 146, 158, 160, 163, 258, 301, 31719, 321-23, 339, 365, 372, 384, 419, 453, 476-77, 492, 529, 546, 576, 584, 599-600, 602, 613, 617, 652 Vries, Gustav de 1866–1934, Dutch math., DMV: 432 Waelsch, Emil 1863–1927, Czech math., DMV: 231, 248-49, 613, 617 Waerden, Bartel Leendert van der 1903– 1996, Dutch math., DMV: 241, 359 Wagner, Ernst Leberecht 1829–1888, physician: 258 Wahrendorff, Ferdinand 1826–1898, physician: 324 Waitz, Georg 1813–1886, hist.: 89, 116-17, 164, 546 Walker, Gilbert 1868–1958, British math., physicist, meteorologist: 431-32

676

Index of Names

Wallach, Otto 1847–1931, chemist: 115, 439, 522, 617, 625 Walter, Max 1857–1935, eng.: 563, 617 Waltershausen, Wolfgang Sartorius von 1809–1876, geol.: 90 Walther, Alwin 1898–1967, math.: 487 Wangerin, Albert 1844–1933, math. DMV, Enc: 419 Warnstedt, Adolf von 1813–1897, lawyer, univ. curator: 318 Weber, Carl Maria von 1786–1826, composer: 35 Weber, Eduard Ritter von 1870–1934, math., DMV, Enc: 393 Weber, Ernst Heinrich 1795–1878, physicist, physician: 308 Weber, Heinrich 1842–1913, math., DMV, Enc: 11, 47, 51, 57, 249, 282, 304, 366, 369-70, 372, 374, 376, 378-80, 383-84, 392-93, 396, 402, 426, 434, 436, 476, 509, 613, 617, 643, 645 Weber, Moritz 1871–1951, eng., DMV: 390-91 Weber, Wilhelm Eduard 1804–1891, physicist: 89, 92-94, 97, 111-12, 120, 235, 317, 326-27, 365, 439 Wedekind, Ludwig 1843–1908, math., PhD with Klein: 129, 133-37, 139, 147, 191, 401 Weichold, Guido *1857, math., PhD with Klein: 230, 261-62, 264-65 Weierstraß, Karl 1815–1897, math., DMV: 9, 13, 35, 41, 48, 50-51, 56, 61, 63-69, 71-72, 97, 119, 139, 142, 151, 183, 192, 197-200, 204, 207, 212, 217, 222, 229, 233, 235, 237-38, 249, 256, 26061, 263-66, 268, 291-92, 294-95, 29899, 318-19, 321-23, 328, 329-30, 33940, 342, 354, 357, 365, 368, 371-74, 376-77, 410-12, 457, 483, 486, 509-10, 569-70, 586, 591, 602, 607, 623-26, 632-33, 637, 644-45, 647, 652 Weiler, Adolf 1851–1916, Swiss math., PhD with Klein, DMV: 42, 97, 108, 110, 132-34, 137 Weingarten, Julius 1836–1910, math., DMV: 100, 200 Weinreich, Hermann 1884–1932, math., pedagogue, DMV: 302, 496 Weiß, Wilhelm 1859–1904, Austrian math., DMV: 40, 54, 177, 231, 248-49

Weitzenböck, Roland 1885–1955, Austrian math., DMV, Enc: 433 Welcker, Friedrich Gottlieb 1784–1868, class. philologist, archeologist: 35 Wellhausen, Julius 1844–1918, biblical scholar, orientalist: 147, 380 Wellmann, H. math. teacher, DMV: 369-70 Wende, Erich 1884–1966, lawyer, administrator: 526 Wenker, Albert †1871, F.Klein’s school friend: 26, 60-61, 85-86, 89 Wernicke, Alexander 1857–1915, math., mech., pedagogue, DMV: 496, 617 Westphal, Wilhelm 1882–1978, physicist: 563 Weyl, Hermann 1885–1955, math., DMV: 2, 131, 142, 243, 480, 491, 517, 544, 565, 617, 645-46, 648 Weyr, Eduard 1852–1903, Czech Austrian math.: 402 Weyr, Emil 1848–1894, Czech Austrian math., DMV: 76, 248, 340 White, Henry Seely 1861–1943, American math., PhD with Klein, DMV: 334, 340-41, 372, 377, 402 Whitehead, Alfred North 1861–1947, British math., philosopher: 244, 572, 639 Wiechert, Emil 1861–1928, geophysicist, DMV, Enc: 440, 447-52, 466, 555, 617, 636 Wiedemann, Eilhard 1852–1928, physicist: 222 Wieghardt, Karl 1874–1924, math., PhD with Klein, DMV, Enc: 391, 469, 617 Wiener, Christian 1826–1896, math., DMV: 48-49, 118, 239 Wiener, Hermann 1857–1939, math., DMV: 231, 233-34, 239, 369-70, 617 Wiener, Norbert 1894–1964, American math., DMV: 437, 566 Wigger, Julius 1871–1934, teacher: 399 Wilamowitz-Moellendorff, Ulrich von 1848–1931, class. philologist: 35, 366, 437, 528, 626 Wilbrandt, Adolf von 1837–1911, writer: 362 Wiles, Andrew *1953, British math.: 381 Wilhelm I 1797–1888, German Emperor and Prussian King: 84, 320 Wilhelm II 1859–1941, German Emperor and Prussian King: 363, 401, 416, 437, 443-44, 451-52, 487, 497, 499

Index of Names Williams, Ella Cornelia, American math. teacher: 412 Wiltheiss, Eduard 1855–1900, math., DMV: 370 Wiman, Anders 1865–1959, Swedish math., DMV, Enc: 99, 189, 247, 288, 338, 613, 617 (Wimmer=Wiman) Windau, Willi 1889–1928, math., DMV: 536 Winkelmann, Max 1879–1946, math., PhD with Klein, DMV: 418, 463, 471, 548, 617 Winston (md. Newson), Mary Frances 1869–1959, American math., PhD with Klein: 346, 404, 413, 415-16, 462, 547, 576, 632 Wirtinger, Wilhelm 1864-1945, Austrian math., DMV, Enc: 4, 131, 141, 184, 280, 292, 340, 341, 366, 371, 417, 429, 457, 482, 575, 576, 577, 613, 617, 628, 630, 643, 653 Wirtz, Karl 1861–1928, prof. of electrical engineering: 231, 617 Witting, Alexander 1861–1946, math., PhD with Klein, DMV: 231, 338, 617, 653 Wöhler, Friedrich 1800–1882, chemist: 45 Wolff, Karl Georg 1886–1977, math., teacher, DMV: 550, 640 Wolfskehl, Paul Friedrich 1856–1906, physician, math., DMV: 381, 395, 538 Woods, Frederick Shenstone 1864–1950, American math., PhD with Klein: 334, 613, 617 Wright, Orville 1871–1948, American aviation pioneer: 450, 452, 640 Wright, Wilbur 1867–1912, American aviation pioneer: 450, 452, 640 Wüllner, Adolf 1835–1908, physicist: 29, 648 Wulff, George V. 1863–1925, Russian crystallographer: 356 Wundt, Wilhelm 1832–1920, psychol., philos.: 219, 229, 310, 492, 528 Wußing, Hans 1927–2011, math.hist.: ix, 5, 78, 126, 653 Yoshiye (Yoshie), Takuji (Takuzi) 1874– 1947, Japanese math.: 394, 487-88, 494, 613, 617 Young, George Paxton 1818–1889, BritishCanadian theol., logician: 25 Young (née Chisholm), Grace, see Chisholm

677

Young, William Henry 1863–1942, British math., DMV: 302, 429, 522, 538, 631, 640, 653 Zacharias, Max 1873–1962, math., DMV, Enc: 433 Zarncke, Friedrich 1825–1891, philologist: 30, 220 Zeiss, Carl 1816–1888, scientific instrument maker: 10, 35, 114, 438, 442, 447, 471, 558, 565-66, 590, 649 Zemplén, Győző 1879–1916, Hungarian physicist, Enc: 468, 613, 617 Zeppelin, Ferdinand Graf von 1838–1917, general, inventor: 451, 617 Zermelo, Ernst 1871–1953, math., DMV, Enc: 307, 394, 456, 490, 514 Zeuthen, Hieronymus Georg 1839–1920, Danish math., Enc: 47, 50, 54, 68, 120, 242, 309, 395, 476-77, 479, 613, 617 Zhukovsky, Nikolay Y. 1847–1921, Russian math., mech.: 372, 450, 453, 463, 466 Zindler, Konrad 1866–1934, Austrian math., DMV, Enc: 41-42 Ziwet, Alexander 1853–1928, Polish-German-American eng., math., DMV: 404, 495, 634 Zoepffel, Richard 1843–1891, church historian: 116 Żorawski, Kazimierz 1866–1953, Polish math., DMV: 335, 613, 617 Zorn, Philipp 1850–1928, prof. of canon and constitutional law: 500 Zühlke, Paul 1877–1957, math., pedagogue, DMV: 505, 617