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Table of contents :
Foreword
Preface
Style and Translation
Evolution of the Project and Acknowledgments
Contents
Illustrations
1 Pieri’s Contributions to Foundations
and Philosophy of Mathematics
1.1 Pieri, the Man, the Scholar, the Teacher
1.2 Philosophy of Mathematics and Mathematical Logic
1.3 Foundations of Geometry
2 Pieri’s Philosophy of Deductive Sciences
2.1 Primitive Concepts
2.2 Definitions
2.3 Definitions by Abstraction
2.4 Postulates, or Primitive Propositions
2.5 Proofs
2.6 Abstract Deductive Science
2.7 Logic and Mathematics
2.8 Pieri's Letter to Russell
2.9 Metamathematics
2.10 Semantics and Model Theory
2.11 Nominalism
3 Two Paths to Logical Consequence: Pieri and the Peano School
3.1 Tarski’s Definition of Consequence
3.2 Aristotle’s Counterexample Method
3.3 Independence of the Parallel Postulate
3.4 Logical Consequence in a Model-Theoretic Context: The Peano School
3.4.1 Peano
3.4.2 Pieri
3.4.3 Padoa
4 Pieri’s 1900 Paris Paper
ON GEOMETRY ENVISAGED AS A PURELY LOGICAL SYSTEM
§ I
§ II
§ III
§ IV
§ V
§ VI
§ VII
5 Pieri and Projective Geometry
5.1 Pieri’s Studies, Research, and Teaching
5.2 Evolution of Projective Ideas and Methods
5.3 Synthetic Projective Geometry as an Autonomous Field
5.4 Geometry as a Logical System
5.5 The Transformational Approach
5.6 Multidimensional Projective Geometry
5.7 From Duality to Plurality
6 Pieri’s 1898 Geometry of Position
Memoir
THE PRINCIPLES OF THE GEOMETRY OF POSITION
LIST OF ABBREVIATIONS
§ 1 The Primitive Entities
POSTULATE I
POSTULATE II
POSTULATE III
POSTULATES IV AND V
POSTULATE VI
POSTULATE VII
POSTULATE VIII
POSTULATE IX
POSTULATE X
§ 2 The Alignment Relation and the Projective Line
POSTULATE XI
§ 3 The Visual of a Form and Projective Planes
POSTULATE XII
§ 4 The Plane Quadrangle and the Harmonic Relation
POSTULATE XIII.
POSTULATE XIV
§ 5 The Projective Segment
POSTULATE XV
POSTULATE XVI
POSTULATE XVII
§ 6 Further Properties of Segments
§ 7 Natural Orderings and Senses of a Projective Line
§ 8 The Projective Triangle
§ 9 Segmental Transformations
POSTULATE XVIII
§ 10 Harmonic Correspondences and STAUDT’s Theorem
§ 11 Projective Hyperplanes of the Third Species and Ordinary Space
POSTULATE XIX
§ 12 Projective Hyperplanes of the nth Species and Absolute Projective Space
POSTULATE XIX'
POSTULATE XX'
APPENDIX
7 Transformational Geometry
7.1 Motions and Transformations
7.2 Isometries and Similarities
7.3 Transformations as Tools
7.4 Transformations in Foundational Studies
7.5 Postlude
8 Pieri’s 1900 Point and Motion
Memoir
ON ELEMENTARY GEOMETRY
LIST OF ABBREVIATIONS
§1 Generalities about point and about motion. The relation of collinearity among points. Line, plane, and sphere are introduced.
POSTULATE I
POSTULATES II and III
POSTULATE IV
POSTULATE V
POSTULATE VI
POSTULATE VII
POSTULATE VIII
POSTULATE IX
§2 Rotating a line onto itself. Midpoint of a pair of points.
Rotating a plane onto itself. Orthogonality relation
among three points or between two intersecting lines
POSTULATE X
POSTULATE XI
POSTULATE XII
POSTULATE XIII
POSTULATE XIV
§3 Rotating one plane onto another. Orthogonality of lines and planes. Various properties relating to lines, planes, and spheres.
POSTULATE XV
POSTULATE XVI
§ 4 Points internal or external to a sphere. Segments, rays, half-planes, angles, and so on.
POSTULATE XVII
POSTULATE XVIII
POSTULATE XIX
§5 Relation less than or greater than between two segments or
between two angles. Triangle is introduced. Congruence of triangles and other propositions of the first and third books of Euclid.
§6 Sum of two segments. Other properties of triangles, circles,
and so on. Continuity of a line.
POSTULATE XX
9 Pieri’s Works on Foundations and Philosophy of Mathematics
9.1 Course Materials and a Translation
9.1.1 Higher Geometry Lectures by Riccardo De Paolis
1882–1883
1883–1884
9.1.2 Geometry of Position by G. K. C. von Staudt
Pieri’s Treatment of the Fundamental Theorem
Pieri’s Handwritten Notations
9.1.3 Projective Geometry: Lectures at the Military Academy
9.1.4 Course Records from Catania University Archives
9.1.5 Projective Geometry: Lectures at Parma
Pieri 1910
Pieri 1911c
9.1.6 Descriptive Geometry: Lectures at Parma
9.2 Foundations of Projective Geometry
9.2.1 Principles That Support the Geometry of Position
9.2.2 Postulates for Abstract Projective Geometry of Hyperspaces
9.2.3 Primitive Entities of Abstract Projective Geometry
9.2.4 Intermezzo (1897b)
9.2.5 Principles of the Geometry of Position Composed into a Deductive Logical System
9.2.6 New Method for Developing Projective Geometry Deductively
9.2.7 Principles That Support the Geometry of Lines
9.2.8 Staudt’s Fundamental Theorem and the Principles of Projective Geometry
Introduction (§1)
Proof of the Fundamental Theorem (§2)
Results Not Dependent on Continuity Principles (§3–§6)
Continuity and Archimedean Principles
Pieri’s Archimedean Postulate XVIII' (§1, §7–§9)
Pieri’s Postulate XVIII'' (§9)
Real Projective Geometry of Degree < 2 (§10–§11)
Conclusion
9.2.9 New Principles of Complex Projective Geometry
Background
Axiomatic Framework
Incidence Postulates and Dependence (§1)
Chains (§2–§4)
Coordinates and Cross Ratios (§7)
Further Topics, from §4–§7 and 1906a
Conclusion
9.2.10 On the Staudtian Definition of Homography
9.3 Foundations of Elementary and Inversive Geometry
9.3.1 On Elementary Geometry as a Hypothetical Deductive System: Monograph on Point and Motion
9.3.2 Elementary Geometry Based on the Notions of Point and Sphere
9.3.3 New Principles of the Geometry of Inversions
9.4 Arithmetic, Logic, and Philosophy of Science
9.4.1 Geometry Envisioned as a Purely Logical System
9.4.2 On an Arithmetical Definition of the Irrationals
9.4.3 A Look at the New Logico-Mathematical Direction of the Deductive Sciences
9.4.4 On the Consistency of the Axioms of Arithmetic
9.4.5 On the Axioms of Arithmetic
10 Central Themes and Impact of Pieri’s Work
10.1 Philosophical Themes in Pieri’s Research
10.2 Themes in Foundations of Geometry
Pieri’s Views on Abstract Mathematics
10.2.1 Geometry as an Abstract Science
10.2.2 Geometry from a Synthetic Perspective
10.2.3 Geometry from a Transformational Point of View
10.2.4 Geometries Constructed as Autonomous Disciplines
10.2.5 Continuity and Archimedean Principles
10.2.6 Minimizing the Number of Primitive Notions
10.3 Pedagogical Themes
Projective Geometry
Inversive and Elementary Geometry
10.4 Pieri’s Impact
10.4.1 Philosophy
10.4.2 Foundations of Geometry
Projective Geometry
Inversive Geometry
Elementary Geometry
10.4.3 Pedagogy
10.5 Opportunities for Future Research
Foundations of Geometry
Pedagogy
Logic and Philosophy
Appendix
A.1 Errata and Addenda for Marchisotto and Smith 2007
A.1.1 Errors and Corrections
A.1.2 New Items for Chapter 6: Pieri’s Works
A.2 Two Letters from Louis Couturat
A.3 Russell’s Annotations on Principles of the Geometry of Position
A.4 Pieri’s 1905b letter to Oswald Veblen
Bibliography
Permissions and Credits
Index of Persons
Index of Subjects
Recommend Papers

The Legacy of Mario Pieri in Foundations and Philosophy of Mathematics [1 ed.]
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Elena Anne Corie Marchisotto Francisco Rodríguez-Consuegra James T. Smith

The Legacy of Mario Pieri in Foundations and Philosophy of Mathematics

Elena Anne Corie Marchisotto Francisco Rodríguez-Consuegra • James T. Smith

The Legacy of Mario Pieri in Foundations and Philosophy of Mathematics

Elena Anne Corie Marchisotto Department of Mathematics California State University, Northridge Malibu, CA, USA

Francisco Rodríguez-Consuegra Depto. Lógica y Filosofía de la Ciencia Fac. Filosofía y Ciencias de la Educación Universidad de Valencia Valencia, Spain

James T. Smith Department of Mathematics and Statistics San Francisco State University San Francisco, CA, USA

ISBN 978-0-8176-4822-0 ISBN 978-0-8176-4823-7 (eBook) https://doi.org/10.1007/978-0-8176-4823-7 Mathematics Subject Classification (2020): 01-02, 01A55, 01A65, 01A75, 03-03, 03A05, 51-03, 51F20, 51G05, 97-03, 97G40, 97G50, 97G80 © Springer Science+Business Media, LLC, part of Springer Nature 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. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Science+Business Media, LLC The registered company address is: 1 New York Plaza, New York, NY 10004, U.S.A.

Foreword by David E. Rowe

Mario Pieri (1860–1913) has been one of the less-sung heroes in the history of Italian geometry, in part owing to his premature death, but probably also because of the breadth of his mathematical interests. In the 2007 first volume of this projected trilogy, The Legacy of Mario Pieri in Geometry and Arithmetic, Elena Anne Marchisotto and James T. Smith appended a comprehensive bibliography of his works, both published and unpublished, that attests to this impressive eclecticism. If Pieri is mainly remembered today for his contributions to axiomatic research in the foundations of geometry, it should not be forgotten that he was also a leading authority on enumerative geometry, a field in which he published no fewer than seventeen papers. One of his last works, Méthodes énumeratives (Pieri [1915] 1991), appeared in Jules Molk’s French edition of the Encyklopädie der mathematischen Wissenschaften. That article was a revised translation of H. G. Zeuthen’s 1905 survey of a field in geometry far removed from foundational research. (We need only recall that David Hilbert’s fifteenth Paris problem called for a rigorous proof of the Schubert calculus, which required far more machinery than he and his contemporaries could possibly imagine.) As the authors of this second volume make clear, Mario Pieri was a scholar as well as an original thinker. His mathematics was shaped by various currents of thought, some of which would profoundly influence later developments; several others did not, which hardly means they lack historical interest. Pieri studied at the Scuola Reale Normale Superiore in Pisa, where he came under the influence of Italy’s leading differential geometer, Luigi Bianchi (1856–1928). Much more decisive for his subsequent career, however, was the period that followed in Turin, where Pieri gained his first academic appointments, at the military academy and then also at the university. His association with Giuseppe Peano (1858–1932) led to a lifelong interest in foundations of arithmetic and geometry, strongly guided by symbolic logic. During the 1890s, Peano and his disciples, including Pieri, published their work in the journal Rivista di matematica as well as in the various editions of Peano’s Formulaire mathématique. This connection with the Peano school represents the most familiar current of thought to which Pieri belonged. The present volume, however, reveals several other important influences on his work, beginning with Peano’s colleague in Turin, Corrado Segre (1863–1924). During the 1880s, Segre had been studying the work of G. K. C. von Staudt (1798–1867), a pioneering figure who cultivated a systematic approach to synthetic projective geometry. Following Segre’s suggestion, Pieri prepared an annotated Italian translation of Staudt’s 1847 Geometrie der Lage as well as parts of his 1856–1860 Beiträge zur Geometrie der Lage. This translation appeared in 1889 under the title Geometria di Posizione, and included an essay by Segre about Staudt’s life and work.

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In the years that followed, Pieri elaborated on this theory, which he presented in axiomatic form in The Principles of the Geometry of Position Composed into a Deductive Logical System, (Pieri 1898c, translated in chapter 6). For this study, he employed nineteen sequentially independent axioms, deducing additional theorems at each stage of the set. To appreciate this accomplishment in historical context, Pieri’s rigorous development of synthetic projective geometry was methodologically very similar to Hilbert’s axiomatic treatment of Euclidean geometry in his Grundlagen der Geometrie, which only appeared one year later ([1899] 1971). Leaving aside questions of priority, Mario Pieri, Alessandro Padoa, and other Italians were strong advocates of axiomatization in foundational studies. The present authors thus rightly raise the question why Hilbert’s study overshadowed contemporary work in Italy, in particular that of Pieri. There can be no simple answer to this question, but an aspect that ought to be borne in mind is that Hilbert was following a very different trajectory when he published Grundlagen der Geometrie. Not long before, he had spent several years writing his Zahlbericht ([1897] 1998), in which he recast—some might say even recreated—the theory of algebraic number fields. When he afterward turned to foundations of geometry, Hilbert recognized that one could obtain different types of Euclidean geometries analytically by using appropriate number fields (those that satisfied the theorem of Pythagoras). This point has been consistently overlooked in the secondary literature, which rarely discusses issues that go beyond the first two chapters of Grundlagen der Geometrie. If, however, one simply reads chapter 7, the last chapter in that booklet, it is hard to miss this perspective. Once seen, it becomes clear that Hilbert aimed not just to give a rigorous framework for studying geometries that satisfied the parallel postulate; he also wanted to place these geometries on a firm arithmetical basis by means of segment operations, which could be shown to satisfy the axioms for standard number systems. This goal, in fact, forms the true centerpiece of his whole work, which opened the way for many subsequent investigations and eventually led to the modernization of the foundations of geometry. This interplay between geometry and axiomatically defined number systems would also eventually come to dominate the theory of projective spaces. Mathematicians would later recast the concept of a projective space in a more algebraic guise that starts with the notion of a vector space over an arbitrary field. One then obtains a projective space by viewing its one-dimensional subspaces as points. Purely synthetic approaches did not die out completely, but since Pieri’s work was largely dedicated to completing and modernizing Staudt’s theory, it marks the end of that particular story. As the authors describe, its highpoint was the fundamental theorem of projective geometry in its classical form, which states that a projective mapping of a line is completely determined by its values at three points. (The modern formulations also must take into account the automorphisms of the ground field, but for the real number system there is only one, the identity mapping.) Staudt, Felix Klein, and others stumbled in their attempts to prove this theorem, which is hardly surprising when we realize how long it took to fathom the many subtleties presented by the continuum of real numbers. In 1900, Pieri gained appointment to a chair at the University of Catania, a position he held until he moved to Parma in 1908. On arriving there, he published Elementary

Foreword

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Geometry as a Hypothetical Deductive System: Monograph on Point and Motion (Pieri 1900a, translated in chapter 8—“elementary geometry” here means absolute geometry, those theorems provable without any assumptions regarding parallelism). In this axiomatic study, Pieri refined an earlier one by Peano (1894b) that employed a similar combination of geometric objects together with special transformations. This approach thus occupies an intermediary position between two extremes identified by Hilbert. In his Grundlagen der Geometrie, Hilbert revoked Euclid’s appeal to superposition by making the SAS theorem an axiom. Then, in 1902, he stood this static theory on its head by taking a group of motions as the starting point for foundations of geometry. Unlike Peano and Pieri, Hilbert laid little stress on pursuing a minimalist program. Pieri’s study, by contrast, was aimed at building geometry on a logical basis with only minimal assumptions, though also with an eye toward didactical considerations. As a first model for such a study, Pieri pointed to Moritz Pasch’s 1882b Vorlesungen über neuere Geometrie. Pieri lauded this classic as marking the beginning of a renovated order of ideas regarding the foundations of geometry, a work truly inspired by the intention to make all of geometry share in the clarity and deductive perfection and in the almost crystalline form that we have before our eyes in arithmetic. (See chapter 8, page 251.)

Pasch employed four primitive concepts: point, segment, planar surface, and the relation of congruence between two figures. Peano afterward used three—point, segment, and motion—which Pieri then reduced to point and motion. Pieri further described how Staudt’s projective theory would make it possible to suppress all forms of transformations, but noted that the ensuing complications made the price much too high for the practical demands of teaching elementary geometry. Pieri ended his introduction with some general remarks about his methodological preferences: ... I profess to be greatly indebted to algebraic logic, in which I recognize the most opportune and most valid instrument for this kind of studies: not only for the effectiveness of the symbols in themselves, but so much more by virtue of the intellectual habits that the methods and doctrines of this science are shown capable of teaching and promoting, and also certainly for their suggestive faculty, which often leads to observations and research not otherwise fostered. All of our propositions can be easily translated into symbolic logic; in fact the largest part were conceived and written from the start according to the ideography constructed by PEANO. But in consideration of many for whom the symbolism of mathematical logic is not familiar ... we abandon this form of exposition, after having extracted the greatest benefit possible, and keeping only very few easy abbreviations (gathered and declared here next) for the simple objective of reducing somewhat the bulk of this essay. (Chapter 8, page 254.)

Interestingly enough, in 1908 Pieri returned to the challenge of ridding motion from the axioms of elementary geometry. In Elementary Geometry Based on the Notions of Point and Sphere (Pieri 1908a, translated in volume 1, chapter 3), Pieri showed how one could start with the primitive concepts of point and the three-point relation satisfied by two points that are equidistant from the third. He began, in fact, by citing his earlier remarks in his preface to the Point and Motion monograph, noting that the 1908a work

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represented the realization of the theory he then had in mind. He saw, however, few analogies between these two works, since the earlier one dealt with “the idea of motion, or congruence of figures, as a point transformation, more in its role as a primitive idea, and therefore undefined except through postulates.” In his later theory, “all of the fundamental geometric operations and relations” had to be defined “in terms of point and equidistance from a point,” the primitive concepts. (See the first volume, §3.0, page 161). Indeed, in this theory the analogy was closer to Staudt’s projective geometry, which introduced collineations and correlations by means of the primitive concepts projective point and alignment of projective points. Pieri’s stringent derivation of elementary geometry based on those two primitive concepts did not go unnoticed: one mathematician who studied it was the young Alfred Tarski. These brief comments hopefully give some idea of the kinds of interests that motivated Mario Pieri’s work and the important intellectual impulses that ran through it. The best known of these arose through his close affiliation with the Peano school, but this was only one of the currents that found expression in his writings. Segre’s interest in higherdimensional projective geometry also exerted a strong influence on Pieri, who clearly mixed his progressive ideas together with a profound knowledge of classical projective geometry. The authors of this volume have produced a very worthy sequel to their first volume, to which they refer in many places. They have made extraordinary efforts not only to describe Pieri’s work in considerable detail, but even more importantly to contextualize what he accomplished. Their commentaries abound with insights into related investigations by his predecessors and contemporaries, coupled with a wealth of information about the influence of Pieri’s works on several distinguished readers. Historians of mathematics, especially those with an interest in understanding the mathematical world in which Mario Pieri moved, can be truly thankful to Elena Anne Marchisotto, Francisco Rodríguez-Consuegra, and James T. Smith for their diligent scholarly efforts.

David E. Rowe Johannes Gutenberg University Mainz, Germany

Preface The Italian mathematician Mario Pieri (1860–1913) played a major role in the development of algebraic geometry and foundations of mathematics around the turn of the twentieth century. He was a bridge between the research groups of Corrado Segre and Giuseppe Peano in Turin. In foundations of mathematics, Pieri created axiomatizations of real and complex projective geometry, of inversive geometry, of absolute (neutral) geometry based on the notions of point and motion, of Euclidean geometry based on point and equidistance, and of the arithmetic of natural numbers. His developments of geometry were born of his 1889a critical study of G. K. C. von Staudt’s famous 1847 monograph Geometrie der Lage. Pieri’s research differed greatly from that of others. He broke new ground with novel choices of primitive terms. His incisive presentation of the hypothetical-deductive viewpoint, along with work of others in the Peano school, prepared the scene for deep studies of logical foundations of mathematical theories. This is the subject of the present book. The Italians’ precise formulations, and the practice of David Hilbert and his followers, established the abstract approach that soon became standard in foundations research and in the exposition of higher mathematics. In algebraic geometry, Pieri emphasized birational and enumerative geometry. The thread connecting much of his research is multidimensional projective geometry. That subject was both the stimulus and the proving ground for Pieri’s foundational work. Although the mainstream of algebraic geometry research soon departed from Pieri’s approach, his results continue today to stimulate new developments in enumerative geometry. Pieri’s extensive research in algebraic geometry, which can be based on his axiomatic foundation, is not covered by this book. For a summary, see the box on page xv. Pieri died in 1913. There immediately followed thirty-five years of turmoil and catastrophe in Italy and the rest of the world. During that period the work of the Segre and Peano schools lost much of its prominence, and recognition of Pieri’s contributions disappeared almost entirely from the mathematical literature. This is the second in a series of books to examine Pieri’s life and work. The primary goal of the series is to make Pieri’s research in diverse fields—mathematical logic and philosophy of mathematics, foundations of projective, inversive, and elementary geometry, algebraic and differential geometry, and vector analysis—accessible to today’s scholars and to assess its importance (much yet unrecognized) in historical and modern contexts. The first book, The Legacy of Mario Pieri in Geometry and Arithmetic, by Elena A. Marchisotto and James T. Smith,1 introduced Pieri with a detailed account of his personal 1

Marchisotto and Smith 2007. Throughout the present book, that volume will be referred to as M&S 2007. Appendix 1 of the present book is a list of errata and addenda for M&S 2007.

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and scholarly life. An overview of the totality of Pieri’s results, demonstrating the breadth and depth of his research, was followed by a closer look at his foundational work. The book featured English translations and analyses of two of his major works in foundations: the paper On the Axioms of Arithmetic, which Peano judged superior to his own axiomatization of natural-number arithmetic; and the memoir Elementary Geometry Based on the Notions of Point and Sphere, which Alfred Tarski used as a model for his geometrical theories and as a basis for his development of the geometry of solids.2 A chapter explored Pieri’s influence on the research of his contemporaries as well as modern scholars, and proposed reasons for his relative obscurity. The volume concluded with a listing of Pieri’s published and unpublished works known to the authors in 2007. All of those, including reviews, lecture notes, surviving letters, and collections of works, were identified and classified, and all reviews, letters, and collections described in detail.3 In this second book, The Legacy of Mario Pieri in Foundations and Philosophy of Mathematics, these authors continue focusing on Pieri’s foundational research. For his philosophical work, they are joined by the Spanish philosopher and Russell scholar Francisco Rodríguez-Consuegra, who has contributed two chapters devoted to that aspect of Pieri’s research. The book includes an English translation and analysis of Pieri’s axiomatization, The Principles of the Geometry of Position Composed into a Deductive Logical System, which was judged by Bertrand Russell as “the best work on the present subject” and used by Alfred N. Whitehead as a basis for his own axiomatization.4 Also included is an English translation and analysis of Elementary Geometry as a Hypothetical Deductive System: Monograph on Point and Motion, Pieri’s axiomatization of absolute geometry—Euclidean geometry as taught in his era in elementary courses, except for the theorems dependent on the Euclidean parallel axiom.5 The transformational approaches in these axiomatizations are analyzed in the context of Felix Klein’s Erlanger program and the Riemann–Helmholtz space problem. Other central themes of Pieri’s axiomatizations are identified, and the impact of his implementation of these themes is examined. The book also includes a translation of Pieri’s invited presentation in Paris at the 1900 International Congress of Philosophy, Geometry Envisioned as a Purely Logical System, which demonstrates his commitment to these themes.6 That was one of the first papers that presented the modern axiomatic method in detail to a general audience. Chapters 1 to 4 of this volume set the stage for exploration of Pieri’s axiomatizations of geometry and their impact by exposing his underlying philosophy of mathematics and the goals for his research. Chapter 1 is a brief overview of Pieri’s contributions to the foundations and philosophy of mathematics; it is streamlined to give an idea of the spirit 2

Pieri 1907a, 1908a; Tarski [1927] 1983. See also subsection 9.3.2.

3

M&S 2007, chapter 6. Two letters, a review, and seven sets of lecture notes and outlines have been discovered since then; they are described in subsections 9.1.1, 9.1.4, 9.1.6, and appendix 1 of the present book.

4

Chapter 6 and subsection 9.2.5. Pieri 1898c. Russell 1903, 382. Whitehead [1906] 1971.

5

Chapter 8 and subsection 9.3.1. Pieri 1900a.

6

Chapter 4. Pieri [1900] 1901.

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of the book. Chapters 2 and 3 were constructed by Rodríguez-Consuegra. Chapter 2 provides an overview of Pieri’s philosophy of deductive sciences and its relation to that of other researchers. A concise exposition of these ideas, it is intended to convey the essential details of their context and use. The conception of mathematical theories as hypothetical-deductive systems—a term Pieri himself invented—is explored. Chapter 3 examines Pieri’s approach to logical consequence, which emanated from the concepts of independence and consistency. It is compared to the notion derived from Tarski’s concepts of satisfaction and truth. These two chapters refer frequently to Pieri’s Paris presentation, which is translated in chapter 4. Chapters 5 and 6 are focused on projective geometry, which Pieri embraced in research and teaching. Chapter 5 examines the origins of his interest in the subject and places his results in a historical setting. Specific characteristics of Pieri’s approaches to projective geometry are identified in relation to the work of other scholars. In particular, he clarified the pervasive role of duality in projective geometry by means of his development of the subject as a hypothetical-deductive system. Pieri’s contributions to higherdimensional geometry are featured. His most important work in projective geometry, the memoir The Principles of the Geometry of Position Composed into a Deductive Logical System, is translated in its entirety in chapter 6. Chapters 7 and 8 treat transformational geometry. Chapter 7 explores from a broad perspective the development of transformational methods up to Pieri’s time, extending the discussion in chapter 5 of Pieri’s use of them specifically in projective geometry. It provides a historical context for his innovative axiomatization of absolute geometry, Elementary Geometry as a Hypothetical Deductive System: Monograph on Point and Motion, which is translated in its entirety in chapter 8. Pieri’s own exposition for a general audience, is included in his Paris address, translated in chapter 4. Point and Motion was an initial foray into the construction of foundations for various geometrical theories, based on properties of their characteristic transformation groups. That field soon took a direction different from Pieri’s thrust, but nevertheless continued to follow his methodology. A few of its high points are discussed in the concluding section 7.5 of chapter 7. Chapter 9 provides summaries and analyses of Pieri’s individual published works in this field, commenting on their reception or impact. This includes the publications translated in chapters 4, 6, and 8. For others, enough details of the theories are expounded to provide tastes of their deductive methods. Chapter 9 also summarizes his surviving unpublished classroom materials on projective and descriptive geometry. It is expected that readers will concentrate on the translations and summaries of a few works of particular interest. Chapter 10, the culminating narrative of this book, assesses the impact on the mathematical community of Pieri’s innovative research into the foundations and philosophy of his subject.

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The appendix contains four sections: • Errata and addenda for M&S 2007, • The first publication of two 1906 letters to Pieri from Louis Couturat, • A comprehensive report about Bertrand Russell’s annotations on his personal copy of Pieri’s 1898c Geometry of Position memoir, • The first publication of Pieri’s 1905b letter to Oswald Veblen. The extensive bibliography lists all and only works referred to in the present volume. Each entry indicates where citations occur. The author–date system is employed for citations: for example, Pieri 1900a is a citation for a work published under Pieri’s name in 1900. Sometimes an author is to be inferred from the context, so that a date alone may also serve as a citation of a work. In bibliographic references, sections of works cited are indicated by the symbol §. In references to the present book, the word “section” is spelled out, and introductory material for a chapter n is regarded as its section n.0. Many of Pieri’s papers are reprinted in Pieri’s 1980 collected works. The page numbers of that publication are displayed in addition to the original ones. Since this book emphasizes Pieri’s research in just two areas, many of his works in other areas are not mentioned here. A complete bibliography of his works, published and unpublished, is contained in chapter 6 of M&S 2007, augmented by appendix 1.2 of the present book. This book’s index lists both subjects and persons. The latter entries include personal dates when known. Ninety-six persons most distinguished or most closely associated with Pieri are the subjects of biographical sketches in M&S 2007. Thirteen more are featured in the present book, as indicated by italics in its index of persons. Style and Translation Boxes such as the one on the facing page provide discussion that informs or supplements the main narrative. They may be read out of sequence. Many are biographical. When necessary for clarity or precision, but sparingly, the narrative text in this book employs the concise mathematical terminology and symbolic notation now common in undergraduate courses in higher algebra. The translations in this book are meant to be as faithful as possible to the published French (chapter 4) and original Italian versions (chapters 6 and 8), while maintaining aspects of Pieri’s work that are tied to his expository style. He did not employ today’s ultraconcise language, but nevertheless worded his presentations with impeccable precision. The translations use subjunctives and related auxiliary verbs uncommon in conventional modern English. Readers should interpret some such instances as indications that Pieri may have been shading his meaning differently from what might be conveyed by shorter, more familiar English expressions. Indeed, Pieri often alluded to the difficulties of rendering details in ordinary prose. Annotations for selected narrative are provided by the authors in [bracketed] footnotes to assist today’s readers. In most cases, readers can proceed with the same caution they would use with English mathematical or philosophical prose written in Pieri’s time or a decade or two earlier. But for a definitive interpretation, they should consult the original.

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Classes, Membership, and Guillemets. Pieri followed a distinctive practice, attributed to Giuseppe Peano,* for describing classes and membership. Not employed in mathematics today, it occasionally led to translation problems in chapters 4, 6, and 8. They were connected with Pieri’s widespread use of quotation marks, always of the same type, but for several different purposes. These sentences from Pieri 1898c display the situation: (1) “„ Segno che, posto innanzi ad un ente di data classe, denota l’insieme di tutti gl’individui che si hanno per eguali al medesimo risp.o alla classe. (Introduction, page 5) (2) Il “punto projettivo„, significato anche da “[0]„ è una classe. (§1P1, page 6) (3) “a 0 k„ ... k essendo una classe, denota... “a è un k„ ... . (ibid.) (4) Se a è un punto prj, dicesi “eguale ad a„ ... o “a„ ciascun punto prj il quale appartenga ad ogni figura contenente a. (§1P4, page 7)

Sentence (1) says that for any entity a of a given class, a denotes the set of all individuals that are regarded as equal to a: that is, the singleton set a whose sole member is a.† Pieri used the quotation marks to name the enclosed symbol. In contemporary mathematical practice,‡ followed in this book, prose is generally arranged to make this type of quotation unnecessary. Sentence (2) says that «projective point», signified also by [0], is a class. Here too, the pair of quotation marks that Pieri used to form the name “[0]„ are omitted in translation. But the first pair is different: it is used to name the class corresponding to the enclosed phrase. This use of quotation marks in (2) is also different from their much more common uses in direct quotation of a source phrase or in establishing distance from the usual meaning of the enclosed phrase. For these reasons, the translations in this book use guillemets («þ») for the type of quotation that occurs in the first line of this paragraph. Translated word-for-word, sentence (3) would say “a 0 k” ... k being a class, stands for “a is a k” ... .

The expression “a is a k” is not used now in foundations of mathematics, because, for example, if k were a type of line, it might not be clear whether a should be a line or a point on it. Bertrand Russell considered this situation in detail (1903, 56, 67–68). He regarded the first two instances of k as referring to a class, but regarding the third he claimed that “a is a k” is a propositional function, as here, when, and only when, k is a class-concept.” He criticized Peano’s and Pieri’s convention, claiming, Peano, not I think quite consciously, identifies the class with the class-concept; thus the relation of an individual to its class is, for him, expressed by “is a.”

Occasionally, as in section 6.11, page 214, the translations in this book insert “[member of]” after a phrase such as “is a” when appropriate to forestall this confusion. Sentence (4) is an especially illustrative example. A word-for-word translation would read, If a is a projective point, each projective point that should belong to every figure containing a is called “equal to a” ... or “a”.

That would be wrong: a is a projective point that belongs to every figure containing a but a is not called a. The translation in section 6.0 on page 148 replaces the terminal verb phrase by is called equal to a ... or [a member of] {a}.

These four examples were chosen for simplicity. Most readers would be able to discern Pieri’s intent here even from word-for-word translations. But these conventions sometimes cause perplexity when they come into play later, inside more complicated definitions and proofs. * Peano 1894a, 1897a, and 1897b show evolving conventions. † See the chapter 6 introduction, page 146. In place of notation like a, the translations in this book use the now standard symbolism {a}. Pieri’s wording would support varying interpretations of equality, needed for applications of Peano’s logic, but Pieri did not pursue that possibility. ‡ See Tarski [1936] 1995, 58–60.

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The only intentional modernizations are • punctuation, • bibliographic citations, which have been altered to refer to entries in the bibliography of the present book, • minor changes in personal names, to conform to entries in the index, • occasional changes in mathematical symbols, where Pieri’s are inconsistent with today’s mathematical practice, and • use of a few common English mathematical terms invented more recently than Pieri’s equivalents, some of which have disappeared from use. To conform to contemporary English standards and to enhance readability, punctuation in the original is often omitted from the translation, or altered, or replaced by a word. Distinctive treatment for some of Pieri’s quotation marks is discussed in the box on page xiii. For information about citation conventions, consult the initial paragraphs of the bibliography. For information about naming, consult the initial paragraph of the index of persons. Changes in symbols or terminology are discussed in the chapters where they occur, often with footnoted explanations. Evolution of the Project and Acknowledgments This project to study and examine the life and legacy of Mario Pieri stemmed from Marchisotto’s 1990 New York University doctoral research, and her correspondence with Rodríguez-Consuegra. The latter had become interested in Pieri through his 1988 University of Barcelona doctoral research that led to his 1991 book on Russell. The two published a plan for the project in their joint 1993 paper. Over the intervening years, much research by Marchisotto, and her many conversations, particularly with H. S. M. Coxeter, Steven R. Givant, Ivor Grattan-Guinness, Jeremy Gray, Steven L. Kleiman, Ann Kostant, Gian-Carlo Rota, David E. Rowe, James T. Smith, and Janusz Tarski, led to considerable expansion of the scope of the project. To make this complexity manageable, a series of three books was planned, the first of which, by Marchisotto and Smith, was published in 2007. For this second volume, Rodríguez-Consuegra’s contributions of chapters 2 and 3 bring to fruition the discussion of Pieri’s philosophy of mathematics that was planned for the original project in 1993. Smith’s interest originated during his studies of the foundations of transformational geometry. At his 1970 doctoral thesis defense, Alfred Tarski asked him a single question: where did this field begin? Tarski disagreed with Smith’s naive answer—Hilbert’s Grundlagen der Geometrie—and suggested studying Pieri! Marchisotto rekindled that interest after thirty years’ neglect. Smith’s contribution includes not only his collaboration with Marchisotto and Rodríguez-Consuegra on the translations and mathematical and historical content, but also his assistance with the overall execution of the plan, notably verification of details, editing and formatting, and creating a detailed bibliography and comprehensive index. The last book of the series was intended to survey, in a historical setting, Pieri’s contributions to algebraic and differential geometry and vector analysis: see the box on page xv.

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The authors particularly acknowledge the assistance of Ivor Grattan-Guinness for calling attention to Alessandro Padoa’s 1900 Rome lecture notes, cited in subsection 3.4.3; Victor Pambuccian’s suggestions for recent research to cite in 10.5; and inspiration, assistance, or support by the other mathematicians and historians of mathematics just mentioned, as well as Kenneth Blackwell, Gianfranco Bolzoni, Francesco, Marco, and Vittorio Campetti, Paola Cantù, Maria Grazia Ciampini, Valeria Cinquini, Salvatore Coen, Salvatore Consoli, Helen Cullura Corie, Sofia Gallo De Maio, Angelo Fabbi, Livia Giacardi, Joseph Gubeladze, Haragauri N. Gupta, Jemma Lorenat, Erika Luciano, Joseph A. Marchisotto, Michelle Anne Marchisotto, Andrew McFarland, Marta Menghini, Riccardo Migliari, Philippe Nabonnand, Pier Daniele Napolitani, Raffaello Romagnoli, Emma Sallent Del Colombo, Helen M. Smith, Jerry N. Stinner, Alfred Tarski, and Michael Markus Toepell. We are immensely thankful for the library and archival services provided by the Universities of California, Catania, Parma, Pisa, Southern Illinois, and Turin, the Scuola Reale Normale Superiore, the Biblioteca Statale di Lucca, the Library of Congress, and San Francisco State University.

Differential and Algebraic Geometry; Vector Analysis. Pieri studied for the laureate at the University of Pisa, with the geometers Luigi Bianchi and Riccardo De Paolis. Bianchi evidently supervised Pieri’s 1884 doctoral dissertation, on singularities of Jacobian algebraic varieties, as well as his dissertation for the Scuola Reale Normale Superiore that same year, on topics in differential geometry.* Pieri continued research in those areas at the University of Turin, working with the group established there by Enrico D’Ovidio and Corrado Segre, until he was appointed professor in Catania in 1900.† Pieri’s six doctoral students all wrote dissertations on algebraic geometry in Catania during 1902–1908. During this span of twenty-four years, Pieri produced thirty-two research papers on differential and algebraic geometry.‡ His emphases were birational and enumerative algebraic geometry. Pieri became particularly well known in the latter area, extending the work of Hermann Schubert. Pieri formulas are today an important tool in that discipline. In 1909, Pieri was invited to translate into French and edit the German article on enumerative methods, by the Danish mathematician H. G. Zeuthen, in Felix Klein’s monumental encyclopedia.§ Also near the end of his career, Pieri developed a new interest, vector analysis: he completed two research papers and an instructive appendix for the pioneering book by his friend from Pisa and Turin, Cesare Burali-Forti.2 The present authors originally planned a third volume, to present Pieri’s achievements mentioned in this box. Thus, those were only briefly described in M&S 2007 and hardly at all in the present book. Age is interfering with our plan. We hope others—younger, versed in these areas, facile with the languages, and interested in investigating mathematical and social history—will continue this phase of our project. We will offer advice as appropriate, and, always, our encouragement. * Pieri 1884b, 1884c. † During that period Pieri began his research on logic and foundations of mathematics, the subject of the present book. ‡ See M&S 2007, §1.2.1, §6.1, §6.2. § Pieri [1915] 1991. 2 See M&S 2007, §1.2.1, §6.3. Burali-Forti’s approach to vector analysis was soon eclipsed by the methods used today.

Elena A. Marchisotto James T. Smith 2019

Contents Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi 1 Pieri’s Contributions to Foundations and Philosophy of Mathematics . . . . . . 1.1 Pieri, the Man, the Scholar, the Teacher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Philosophy of Mathematics and Mathematical Logic . . . . . . . . . . . . . . . . . . . 1.3 Foundations of Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 4

2 Pieri's Philosophy of Deductive Sciences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1 Primitive Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Definitions by Abstraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 Postulates, or Primitive Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.5 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.6 Abstract Deductive Science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.7 Logic and Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.8 Pieri’s Letter to Russell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.9 Metamathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.10 Semantics and Model Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.11 Nominalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3 Two Paths to Logical Consequence: Pieri and the Peano School . . . . . . . . 25 3.1 Tarski’s Definition of Consequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Aristotle’s Counterexample Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3 Independence of the Parallel Postulate . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.4 Logical Consequence in a Model-Theoretic Context: The Peano School . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Peano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Pieri . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Padoa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36 36 37 41

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4 Pieri’s 1900 Paris Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 §I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 §II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 §III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 §IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 §V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 §VI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 §VII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46 58 60 62 64 66 68 72

5 Pieri and Projective Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.1 Pieri’s Studies, Research, and Teaching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.2 Evolution of Projective Ideas and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.3 Synthetic Projective Geometry as an Autonomous Field . . . . . . . . . . . . . . . . 95 5.4 Geometry as a Logical System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.5 The Transformational Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.6 Multidimensional Projective Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.7 From Duality to Plurality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6 Pieri’s 1898 Geometry of Position Memoir . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 §1 The Primitive Entities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 §2 The Alignment Relation and the Projective Line . . . . . . . . . . . . . . . . 6.3 §3 The Visual of a Form and Projective Planes . . . . . . . . . . . . . . . . . . . . 6.4 §4 The Plane Quadrangle and the Harmonic Relationship . . . . . . . . . . . 6.5 §5 The Projective Segment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 §6 Further Properties of Segments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 §7 Natural Orderings and Senses of a Projective Line . . . . . . . . . . . . . . 6.8 §8 The Projective Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 §9 Segmental Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10 §10 Harmonic Correspondences and Staudt’s Theorem . . . . . . . . . . . . . . 6.11 §11 Projective Hyperplanes of the Third Species and Ordinary Space . . . 6.12 §12 Projective Hyperplanes of the nth Species and Absolute Projective Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.13 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

136 147 153 156 163 168 175 181 187 194 206 211

7 Transformational Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Motions and Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Isometries and Similarities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Transformations as Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Transformations in Foundational Studies . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Postlude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

223 223 226 229 234 240

216 220

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8 Pieri’s 1900 Point-and-Motion Memoir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 §1 Generalities about Point and Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 §2 Rotating a Line or Plane onto Itself; Midpoint; Orthogonality . . . . . 8.3 §3 Rotating One Plane onto Another; Properties of Lines, etc. . . . . . . . 8.4 §4 Points Internal or External to a Sphere; Segments, Rays, etc. . . . . . 8.5 §5 Relation Less or Greater between Segments or Angles; Triangles . . 8.6 §6 Sum of Two Segments; Continuity of a Line; Other Properties . . . .

245 256 266 274 282 292 300

9 Pieri’s Works on Foundations and Philosophy of Mathematics . . . . . . . . . . . 307 9.1 Course Materials and a Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 9.1.1 9.1.2 9.1.3 9.1.4 9.1.5 9.1.6

Higher Geometry Lectures by Riccardo De Paolis (Pieri 1883–1884) . . . Geometry of Position by G. K. C. von Staudt (translation: Pieri 1889a) . Projective Geometry: Lectures at the Military Academy (Pieri 1891c) . . Course Records from Catania University Archives (Pieri 1901–1908) . . Projective Geometry: Lectures at Parma (Pieri 1910, 1911c) . . . . . . . . . . Descriptive Geometry: Lectures at Parma (Pieri 1912f) . . . . . . . . . . . . .

308 313 324 329 331 336

9.2 Foundations of Projective Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 9.2.1 9.2.2 9.2.3 9.2.4 9.2.5 9.2.6 9.2.7 9.2.8 9.2.9 9.2.10

Principles That Support the Geometry of Position (1895a, 1896a–b) . . . Postulates for Abstract Projective Geometry of Hyperspaces (1896c) . . . Primitive Entities of Abstract Projective Geometry (1897c) . . . . . . . . . . Intermezzo (1897b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Principles of the Geometry of Position Composed into a Deductive Logical System (1898c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . New Method for Developing Projective Geometry Deductively (1898b) . Principles That Support the Geometry of Lines (1901b) . . . . . . . . . . . . . Staudt’s Fundamental Theorem and the Principles of Projective Geometry (1904a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . New Principles of Complex Projective Geometry (1905c, 1906a) . . . . . . . On the Staudtian Definition of Homography (1906f) . . . . . . . . . . . . . . .

342 349 352 355 359 369 376 381 394 412

9.3 Foundations of Elementary and Inversive Geometry . . . . . . . . . . . . . . . 415 9.3.1 9.3.2 9.3.3

On Elementary Geometry as a Hypothetical-Deductive System: Monograph on Point and Motion (1900a) . . . . . . . . . . . . . . . . . . . . . . 416 Elementary Geometry Based on the Notions of Point and Sphere (1908a, 1915) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 New Principles of the Geometry of Inversions (1911d, 1912c) . . . . . . . . 432

9.4 Arithmetic, Logic, and Philosophy of Science . . . . . . . . . . . . . . . . . . . . . 434 9.4.1 9.4.2

Geometry Envisioned as a Purely Logical System ([1900] 1901) . . . . . . 435 On an Arithmetical Definition of the Irrationals (1906e) . . . . . . . . . . . . 437

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9.4.3 9.4.4 9.4.5 10

A Look at the New Logico-Mathematical Direction of the Deductive Sciences (1906d) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 On the Consistency of the Axioms of Arithmetic (1906g) . . . . . . . . . . . . . 455 On the Axioms of Arithmetic (1907a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

Central Themes and Impact of Pieri’s Work . . . . . . . . . . . . . . . . . . . . . . . . 465 10.1

Philosophical Themes in Pieri’s Research . . . . . . . . . . . . . . . . . . . . . . 466

10.2

Themes in Foundations of Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 467

10.2.1 10.2.2 10.2.3 10.2.4 10.2.5 10.2.6

Geometry as an Abstract Science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometry from a Synthetic Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . Geometry from a Transformational Point of View . . . . . . . . . . . . . . . . . Geometries Constructed as Autonomous Disciplines . . . . . . . . . . . . . . . Continuity and Archimedean Principles . . . . . . . . . . . . . . . . . . . . . . . . . Minimizing the Number of Primitive Notions . . . . . . . . . . . . . . . . . . . . .

10.3

Pedagogical Themes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481

10.4

Pieri’s Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

469 471 472 474 476 478

10.4.1 Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 10.4.2 Foundations of Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 10.4.3 Pedagogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 10.5

Opportunities for Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Errata and Addenda for Marchisotto and Smith 2007 . . . . . . . . . . . . . . 2 Two Letters from Louis Couturat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Russell’s Annotations on Principles of the Geometry of Position . . . . . . 4 Pieri’s 1905b Letter to Oswald Veblen . . . . . . . . . . . . . . . . . . . . . . . . . . .

507 507 512 516 519

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521 Permissions and Credits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 Index of Persons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 Index of Subjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590

Illustrations Portraits

section, page

Alfred Tarski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1, Bernard Bolzano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1, Ernst Schröder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.0, Louis Couturat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.0, Mario Pieri around 1900 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.0, Bertrand Russell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.0, Johannes Kepler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2, Girard Desargues and Marin Mersenne . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2, Gaspard Monge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2, Jean-Victor Poncelet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2, Gaston Darboux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3, Sophus Lie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5, Felix Klein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5, Johannes Thomae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5, Joseph Diez Gergonne . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7, Julius Plücker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7, Mario Pieri around 1895 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5, Hermann Wiener . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3, Mario Pieri around 1899, Smoking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.0, Giovanni Vailati . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1, Figures

26 26 52 52 52 52 84 84 88 88 103 116 116 116 130 132 169 233 248 348

section, page

Pieri’s Handwork for His Staudt Translation . . . . . . . . . . . . . . . . . . . . . . . frontispiece Allegory of Dialectics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.0,

6

Euclid’s Parallel Postulate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3, Klein’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3, Padoa’s 1900 Lectures on Algebra and Geometry, title page . . . . . . . . . . . . . . 3.4,

33 35 45

Universal Exposition, Paris 1900 Poster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eiffel Tower and Grounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Italian Pavilion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Looking Northeast over the Seine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Palace of Congresses and of Social Economy . . . . . . . . . . . . . . . . . . . . . . . . Dynamos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . International Congress of Philosophy, Section III Proceedings, Title Page . . Pieri’s 1900 Paris Paper, First Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.0, 4.0, 4.0, 4.0, 4.0, 4.0, 4.0, 4.0,

46 46 46 49 49 49 55 10

Allegory of Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.0, Theorem of Desargues, with Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2,

76 86

xxi

Illustrations

xxii

section, page

Figures

Introducing Homogeneous Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Harmonic Quadruple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pole/Polar Correspondences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two-Column Composition for Dual Statements . . . . . . . . . . . . . . . . . . . . . . .

5.2, 92 5.3, 97 5.3, 97 5.7, 128 5.7, 131

Turin, Piazza Castello, 1934 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . University of Turin, Rectorate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pieri’s Apartment Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pieri’s 1898 “Geometry of Position” Memoir, First Page . . . . . . . . . . . . . . . . . Definition of Projective Segment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Set a, b, c x, of Points Following x in the Natural Order a, b, c . . . . . . .

6.0, 6.0, 6.0, 6.0, 6.5, 6.7,

137 137 137 140 169 180

Plane Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parallel Desargues Condition (4a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concurrent Desargues Condition (4b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pappus–Pascal Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.2, 7.5, 7.5, 7.5,

227 243 243 243

Postcard Depicting Catania Harbor and Mt. Etna . . . . . . . . . . . . . . . . . . . . . Cathedral of St. Agatha . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Across the Cathedral Square: Pieri’s Hotel . . . . . . . . . . . . . . . . . . . . . . . . . . . Pieri’s 1900 “Point and Motion” Memoir, First Page . . . . . . . . . . . . . . . . . . .

8.0, 8.0, 8.0, 8.0,

246 246 246 249

9.1.1, 9.1.1, 9.1.2, 9.1.2, 9.1.2, 9.1.4, 9.1.4, 9.1.5, 9.1.6, 9.2.4, 9.2.4, 9.2.5, 9.2.7, 9.2.7, 9.2.8, 9.2.9, 9.3.2, 9.4.3,

309 310 314 320 321 330 330 333 340 354 354 365 379 379 384 405 430 444

Lectures on Higher Geometry by De Paolis, Title Page . . . . . . . . . . . . . . . . De Paolis Lecture on Generalities about Forms, First Page . . . . . . . . . . . . Pieri’s 1889 Translation of Staudt’s 1847 Geometrie der Lage . . . . . . . . . Figures for Pieri’s Footnote to 1889a, Paragraph 102 . . . . . . . . . . . . . . . . . Figure for Pieri’s Footnote to 1889a, Paragraph 106 . . . . . . . . . . . . . . . . . . Pieri’s Lecture Report, Projective Geometry . . . . . . . . . . . . . . . . . . . . . . . . . Pieri’s Lecture Report, Higher Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . Pieri 1910, Example Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pieri 1912f, Example Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Advertisement for 1893 Premiere of Manon Lescaut in Turin . . . . . . . . . . Opera Review, 2 February 1893 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pieri’s Continuity Postulate XIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trilateral Frames and Join of a Line with a Pencil . . . . . . . . . . . . . . . . . . Reference Quadruple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Archimedean Principle, Projective Version . . . . . . . . . . . . . . . . . . . . . . . . . Pieri 1905c, Figure 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pieri’s Point and Sphere Memoir, 1915 Translation . . . . . . . . . . . . . . . . . . Pieri’s 1906 Address to the University of Catania . . . . . . . . . . . . . . . . . . . .

Pieri’s Definition of “Q Lies Somewhere Between P, R” . . . . . . . . . . . . . . 10.2.2, 480 Russell’s Figures for Pieri 1898c §6.2 and §6.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . First Appended Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Second Appended Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

appendix 3, 517 appendix 3, 518 appendix 3, 519

1 Pieri’s Contributions to Foundations and Philosophy of Mathematics The research of Mario Pieri (1860–1913) was widely known and valued in his lifetime. With the passage of time it largely faded from view, save for the recent attention of a relatively small number of scholars. The years since the 2007 publication of the first book in this series, The Legacy of Mario Pieri in Geometry and Arithmetic, have seen renewed interest in Pieri’s life and work.1 Significant to this is the increased accessibility, on the Internet, of most of Pieri’s publications. More mathematicians, philosophers, and historians are examining their content. The present book contributes to and supports those endeavors. It continues, from both philosophical and mathematical perspectives, the analysis of Pieri’s research in foundations of mathematics begun in the earlier book.2 This first chapter gives a brief overview of Pieri’s life and his contributions to the philosophy and foundations of mathematics that are treated in the chapters that follow. The preface outlines the content of each chapter, including the final one, which summarizes the central themes and impact of Pieri’s research in these areas, within its historical context. 1.1 Pieri, the Man, the Scholar, the Teacher To appreciate this work, it is important to know something about the man.3 Mario Pieri was born into a world of some privilege and much culture. His father, Pellegrino, was an attorney. Pieri’s older brother and mentor, Silvio, became a professor of the comparative linguistics of classical and Romance languages. A younger brother, Ferruccio, was 1

Marchisotto and Smith 2007; throughout the present book, this volume will be referred to as M&S 2007. An example of increased interest is the paper Ingaliso 2010. A notable exception to the neglect has been the widespread use of the Pieri formulas in enumerative algebraic geometry. See the box at the end of the preface, page xv.

2

For extensive biographical information about Pieri and his contemporaries, and a survey of his research in these and other areas, consult the first volume, M&S 2007, chapters 1, 2, and 6. That book also provided English translations and analyses of two of his most important papers in those areas (Pieri 1908a, 1907a).

3

M&S 2007, §1.1, presents a detailed biography of Pieri. See also the introductions to chapters 4, 6, and 8 of the present book.

© Springer Science+Business Media, LLC, part of Springer Nature 2021 E. A. C. Marchisotto et al., The Legacy of Mario Pieri in Foundations and Philosophy of Mathematics, https://doi.org/10.1007/978-0-8176-4823-7_1

1

2

1 Pieri’s Contributions

a poet. They had another brother and four sisters. Mario grew up in a family who valued learning. As a child, he developed a love of mathematics and music. As a student, he excelled, earning honors and recognition by his instructors. As a researcher, he combined the discipline of careful and persistent study with the courage to forge paths untraversed or not fully explored by others. With humility and modesty, and always giving credit where due, he sought to correct imperfections in published works. As a professor, Pieri sought to expose his students to an understanding of topics that embraced different approaches. He strove to instill in them an appreciation that comes from the hard work of learning. Pieri’s love of learning was not lost on his students. His Catania doctoral student Giorgio Aprile thanked him enthusiastically: You began to make me love science. The work that you proposed for me has led me to enjoy the delights of true study.4

The depth and breadth of Pieri’s scholarship, stemming from this lifelong love of learning that embraced mathematics, philosophy, history, and more, was evident in an address that he presented to the university faculty in Catania.5 Pieri referred to sixty-nine scholars from at least eleven countries, citing the contributions of many of them repeatedly. Here again, beyond demonstrating a mastery of his topic, Pieri’s generosity of spirit and skill as a communicator shone brightly. He began the address with analogies between poetry and mathematics. This was intended to accommodate a diverse audience, many of whom were not versed in the topics he planned to discuss, and some of whom may have had related anxieties. Always respectful of his peers, Pieri assumed the role of instructor, endeavoring, despite the complexity of his topics, to make his assertions intelligible for nonmathematicians. Pieri died prematurely at age fifty-two, of throat cancer. Five friends and colleagues published obituaries for him in professional journals.6 All attested to his scholarship, lifelong love of learning, generosity of spirit, and humility. Francesco Campetti, Pieri’s great-grandnephew, revealed My grandmother Beatrice always told me that Pieri used to spend quite all his time studying, but he didn’t like to be considered a genius … He was really modest.7

1.2 Philosophy of Mathematics and Mathematical Logic Mario Pieri’s presentation, On Geometry Envisioned as a Purely Logical System, delivered at the 1900 International Congress of Philosophy in Paris, provides an excellent overview of some of the main points of his philosophy of logic and mathematics. It is translated in full in chapter 4.8 Pieri considered logic as essential for mathematics—indeed, as the 4

Aprile 1910.

5

Pieri 1906d, described in subsection 9.4.3.

6

Castelnuovo 1913, Giambelli 1913, Levi 1913–1914, Peano 1913, Rindi [1913] 1919.

7

Campetti 2005.

8

Pieri [1900] 1901 is further described in subsection 9.4.1.

1.2 Philosophy of Mathematics and Mathematical Logic

3

basis of all deductive sciences. In this presentation, Pieri applied that view to pioneer his abstract approach to deductive theories. He demonstrated the construction of a major part of geometry 9 as a hypothetical-deductive system: a formal system in which primitive concepts, arbitrarily selected, are implicitly defined by postulates, which are neither true nor false but only express conditions that may or may not be verified. In such a system, theorems are derived solely from the postulates, by applying laws of logic with no appeal to intuition.10 Chapter 2 provides an overview of Pieri’s philosophy of deductive sciences and its relation to that of other foundations researchers, notably Bertrand Russell and Gottlob Frege. The conception of mathematical theories as hypothetical-deductive systems is explored. Chapter 3 examines Pieri’s approach to logical consequence, interpreting it in relation to that of Aristotle, Giuseppe Peano, Alfred Tarski, and others. Pieri’s idea of logical consequence, which emanated from the concepts of independence and consistency, is compared with that derived from Tarski’s concepts of satisfaction and truth. The appearance of the formal systems of the Peano school illustrates the connection between logical consequence and the formal point of view, which arose through the development of the concept of independence. These two chapters refer frequently to Pieri’s 1900 Paris presentation. There, Pieri explained why this approach to geometry was such an important goal. Six years later, in his more comprehensive Catania address, A Look at the New LogicoMathematical Direction of the Deductive Sciences,5 Pieri expressed his views on many issues in logic and foundations of mathematics. Positioning Peano’s logic within the evolution of thought about the relationship of logic and mathematics, Pieri championed its use, notably in the study of mathematics as hypothetical-deductive science. That same year, in his paper On an Arithmetical Definition of the Irrationals,11 Pieri proposed a logical device that would facilitate overloading symbols in Peano’s ideography, and used it to address the question, how best to define the real number system in terms of rational numbers. Pieri’s philosophy of deductive sciences and his approach to logical consequence have earned him a place alongside Peano, Russell, Tarski, and others.12 Pieri’s focus on metamathematical issues yielded a paper On the Consistency of the Axioms of Arithmetic.13 There, he constructed a model of the theory N of natural numbers within the theory F of finite sets, thus proving the consistency of N relative to that of F. Pieri’s practice of improving the works of others can be seen in his paper, On the Axioms of Arithmetic.14 He showed that the recursion principle in the axiomatizations of natural-number arithmetic by Peano and Alessandro Padoa could be replaced by a weak version of the minimum principle, that each nonempty set K of natural numbers must have an element that is not the successor of any element of K. 9

See sections 4.6 and 4.7.

10

See chapter 2 and sections 4.3 and 4.5.

11

Pieri 1906e, described in subsection 9.4.2.

12

See chapters 2 and 3, section 10.1, and subsection 10.4.1.

13

Pieri 1906g, discussed in subsection 9.4.4.

14

Pieri 1907a, translated and described in M&S 2007, §4.2, §4.3. See subsection 9.4.5 of the present book.

4

1 Pieri’s Contributions

The overwhelming evidence of Pieri’s implementation of his philosophy of mathematics and application of metamathematical strategies can be found in his constructions of geometrical theories as hypothetical-deductive systems. That is the focus of the next section. 1.3 Foundations of Geometry To appreciate Pieri’s research on foundations of geometry, it is important to trace the path that led him there. It started with research at the Universities of Pisa and Turin on algebraic geometry. A crucial moment occurred when Corrado Segre invited Pieri to translate and annotate G. K. C. von Staudt’s monograph Geometrie der Lage that pioneered the synthetic approach to projective geometry.15 That experience, in addition to the opportunity to collaborate in Giuseppe Peano’s school of logic, foundations, and pedagogy of mathematics, expanded Pieri’s focus. During 1895–1912, he published thirteen papers in foundations of geometry: ten on real and complex projective geometry, two on elementary geometry (absolute and Euclidean), and one on inversive geometry.16 Two of Pieri’s axiomatizations of geometric theories are translated in the present book: • The Principles of the Geometry of Position Composed into a Deductive Logical System,17 • Elementary Geometry as a Hypothetical Deductive System: Monograph on Point and Motion.18 A third was translated in the earlier book in this series: • Elementary Geometry Based on the Notions of Point and Sphere.19 These works provide convincing justification for Peano’s characterization of Pieri’s contributions as “constituting an epoch in the study of the principles of geometry.” 20 The previous and present books, in the sections just cited, analyze what Pieri accomplished. Chapter 10 of the present book gives an overview of their central themes and their impact relative to the time when they appeared, as well as to future mathematical study, and provide further support for Peano’s evaluation of Pieri’s legacy. Pieri embraced projective geometry in research and teaching. Chapter 5 examines the origins of his interest and places his results in a historical setting. Specific characteristics of his approaches to projective geometry are identified in relation to the work of other scholars, notably Staudt and Felix Klein. Pieri clarified the pervasive role of duality in projective geometry by presenting the subject as a hypothetical-deductive system. His 15

See the box at the end of the preface, page xv, and subsection 9.1.2.

16

See sections 9.2 and 9.3, and M&S 2007, §6.4. Several of these papers were published in multiple parts.

17

Pieri 1898c, translated in chapter 6 and discussed in subsection 9.2.5.

18

Pieri 1900a, translated in chapter 8 and discussed in 9.3.1 and M&S 2007, §2.5.1, §3.10.1.

19

Pieri 1908a, translated in M&S 2007, chapter 3, and described there in §2.5.2, §3.10.4. See also subsection 9.3.2 of the present book.

20

Peano [1915] 1973, 171.

1.3 Foundations of Geometry

5

contributions to higher-dimensional projective geometry are also featured. His most important work in projective geometry, the memoir The Principles of the Geometry of Position Composed into a Deductive Logical System, is translated in its entirety in chapter 6.17 Chapter 7 explores from a broad perspective the development of transformational methods up to Pieri’s time. This discussion extends that in chapter 5 on the use of such methods specifically in projective geometry, and provides a historical context for Pieri’s axiomatization of absolute geometry, Elementary Geometry as a Hypothetical Deductive System: Monograph on Point and Motion.18 That innovative paper is translated in its entirety in chapter 8. Pieri’s own exposition for a general audience, the address translated in chapter 4, includes a summary of his Point and Motion paper. That work was an initial foray into the development of a whole field, which constructed and analyzed the foundations for various geometrical theories, based on properties of their characteristic transformation groups. This field soon took a direction different from Pieri’s thrust, but nevertheless continued to follow his methodology. A few of its highpoints are discussed in the concluding section of chapter 7. From the translations in chapters 4, 6, and 8, readers can ascertain the extensive details and extraordinary precision of Pieri’s arguments, as well as absorb his style of presenting them. The summaries in chapter 9 of all of Pieri’s works in foundations and philosophy of mathematics highlight noteworthy features and explain their context and reception. These summaries present the crucial elements of the theories, sufficient to provide insights into their content and methodologies. The translated publications “should enable readers to get the gist of Pieri’s strategies, and the summaries should provide further mathematical, philosophical, and historical context for them. The summaries in chapter 9 complete the picture of a scholar who embraced mathematics with a keen awareness of its history and its future, and made fruitful contributions toward its advancement. The body of all Pieri’s foundational works reveals his steadfast belief in the power of axiomatization and the effectiveness of logic in creating axiomatic theories. He constructed his axiomatizations with rigor and precision to expose the special ways in which they promote understanding of the subject at hand and distinguish it from others. In the foreword for the first book of this series the British historian Ivor GrattanGuinness described Pieri as “a most able contributor to geometry, arithmetic and mathematical analysis, and mathematical logic during his rather short life.” 21 The breadth and depth of the research that Pieri has given us to explore underlines the tragedy of the loss of a scholar who had much more to contribute.

21

M&S 2007, v.

Allegory of Dialectics Laurent de La Hyre (1649)1

2 1

Pieri’s Philosophy of Deductive Sciences

This chapter2 presents and discusses Mario Pieri’s main contributions to the philosophy of deductive, or formal, sciences. These include • a logicist philosophy of mathematics, previous to Bertrand Russell’s and independent from Gottlob Frege’s, • a conception of axiomatic systems clearly described as hypothetical-deductive, • a view of primitive concepts fluctuating between the implicit-definition approach and one using an explicit “second-order” definition, • systematic use of metamathematical ideas, and • a clearly model-theoretic, semantic, conception of axiomatics. The presentation is very concise, allowing rapid exposition of these ideas without depending on inessential details of their context or employment. Those are addressed in later chapters. Chapter 2 proceeds in large part by paraphrasing key passages from Pieri’s works.

1

One of the seven classical liberal arts, dialectics included what is now called logic. According to Rosenberg and Thuillier 1988, 300–301, the woman represents the goddess Minerva, who was often depicted wearing a plumed metal helmet and accompanied by an owl. For the Allegory on Dialectics, Laurent de La Hyre followed in many respects the popular iconological design of Cesare Ripa (1609, 119). Ripa regarded the helmet as signifying intellectual vigor. The book at lower right presumably represents the dialogue Gorgias, in which Plato (1987) attacked the sophistic argumentation of its title character. The incongruous crow at upper left probably represents the Sophist Corax of Syracuse—his Greek name means crow— who was known for developing the reverse-probability technique of argument (see Aristotle 1991, 1402a17, and O’Grady 2008). Ripa suggested that the two feathers on Minerva’s helmet, white and black, should show that arguments of that sort can easily defend a true claim or its opposite, “just as easily as the wind can blow the feathers. [Both] arguments, as formulated by a vigorous intellect would, like the feathers, seem supported by the solidity of the helmet.” The double-pointed darts in Minerva’s right hand, like a probability argument, can strike both ways. The moon atop her helmet referred to Ripa’s citation of the comparison of dialectics and the moon by the philosopher Clitomachus: the moon alternately waxes and wanes (Valeriano [1562] 1602, 474). Minerva’s left fist, according to Ripa, should indicate a conversational gesture to help demonstrate that the ultimate aim of this art is disputation. La Hyre’s painting measures 1.05 ×1.10 m. For further information about him and his connection with projective geometry, see a box in section 5.2, page 85.

2

Preliminary materials related to this chapter were published in Rodríguez-Consuegra 1993 and 1997a.

© Springer Science+Business Media, LLC, part of Springer Nature 2021 E. A. C. Marchisotto et al., The Legacy of Mario Pieri in Foundations and Philosophy of Mathematics, https://doi.org/10.1007/978-0-8176-4823-7_2

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2.1 Primitive Concepts In Pieri’s view, primitive concepts are elementary (simple, noncomposite) undefined notions that supply the raw material for postulates, or primitive propositions. Consequently, they can be assigned any meaning compatible with the postulates; that is, they should satisfy at least those properties established by the postulates. Thus, the primitive concepts can be understood in an arbitrary way within the limits set by the postulates. This could be expressed by saying that they are somehow defined by the postulates in such a way that readers can gradually come to know their exact meaning without having to use intuitive cognition. The primitive concepts are also used for decomposing or analyzing the remaining concepts of a science by means of (nominal) definitions and logic. In any axiomatic system there should be as few primitive concepts as possible, to maintain the advantages obtained by minimizing the number of postulates.3 Pieri felt that, according to the strictly deductive method introduced by Moritz Pasch, the content of the primitive concepts must be as indeterminate as possible, so that they are only used according to the logical relationships expressed by the primitive propositions, or postulates. Then, even if we should assign them an arbitrary interpretation, we are not forced to associate them with any specific image or representation; it is enough to understand the system as a whole.4 However, as we shall see, elsewhere in his writing Pieri favored a more clearly model-theoretic view of the primitive concepts. 2.2 Definitions According to Pieri, definitions should always be nominal, consisting of simple assignments of names to things that are already known or that have been acquired in the course of the deductive treatment. They are expressed by means of the primitive (undefined) concepts already introduced, and the logical notions in use. In contrast, a so-called real definition (definition of a thing or entity in itself) is a system of predicates sufficient to qualify some concepts, with a deductive aim; in this case, he preferred to say that those concepts remained undefined. Thus, primitive concepts can only be defined by postulates.5 In definitions by postulates, where an entity enjoys, with others, certain relationships that are actually verified (satisfied), we can say that it is determined as a solution or root of a certain system of logical “equations,” just as the unknown quantity of a system of mathematical equations would be. If the system can be “solved” when the other entities are already known, we can speak of a nominal definition.

3

Pieri 1895a, 607; 1896c, 11; 1897c, 343; 1898c, 2, 6 (sections 6.0 and 6.1 of the present book); 1900a, 173–174 (section 8.0); [1900] 1901, 378–379 (section 4.3); 1906d, 58. Pieri used the Italian and French equivalents of primitive concept, entity, idea, and notion apparently interchangeably.

4

Pasch 1882b. Pieri 1900a, 178 (section 8.1); [1900] 1901, 373 (section 4.1).

5

Pieri 1898c, 3 (section 6.0); 1900a, 173–174 (section 8.0); [1900] 1901, 378 (section 4.3).

2.3 Definitions by Abstraction

9

In summary, all of Pieri’s definitions can be divided into two types: explicit (nominal), and implicit (real or by postulates). A nominal definition is merely a linguistic abbreviation, which can induce a condensation of ideas and thus establish a conventional identity or an equality by definition. On the other hand, by leaving aside their truth or falsehood, we can consider a set of postulates to be an implicit definition of the primitive concepts occurring in them.6 2.3 Definitions by Abstraction In a somewhat obscure way, Pieri described definitions by abstraction as those where a new concept is created from a relation that manifests some characteristic properties of equality.7 A well-known case, already considered by Gottlob Frege, was the concept of direction in Euclidean geometry: two lines are said to have the same direction if and only if they are parallel. Another example, considered by Giuseppe Peano, is cardinality: two classes have the same cardinal number if and only if there is a one-to-one correspondence between them.8 These definitions by abstraction would not be explicit according to Pieri’s classification as described in section 2.2. Pieri introduced the method now in common use for transforming (implicit) definitions by abstraction into (explicit) nominal definitions, which was then followed by Cesare Burali-Forti and Bertrand Russell.9 Russell’s most obvious application was his well-known definition of the cardinal number of a class as the class of all classes with which it can be put in one-to-one correspondence. The method is described in only two places in Pieri’s writings, where a property based on an equivalence relation R is replaced by membership in an equivalence class consisting of all elements related by R to a particular one. In one of those places, Pieri introduced the property, being a sum of two given segments, as being congruent to a certain segment constructed from them; then he noted that any segment having that property—that is, any member of the congruence class of the constructed segment—can be called a sum of the first two.10 In the other example, Pieri replaced the definition by abstraction of the direction of an ordering of a projective line r —and thus of the sense of r —as the common property 6

Pieri [1900] 1901, 378 (section 4.3 of the present book); 1906d, 68; 1906g, 203; 1908a, 446–447 (M&S 2007, §3.9, Note 1).

7

Pieri 1906d, 48. Giuseppe Peano had introduced the term in 1894a, §38, with a more explicit description: any equivalence relation R defines by abstraction a function  such that for all x, y in the domain of R,  x =  y if and only if x R y. Cesare Burali-Forti (1894a, §5–§7) included definition by abstraction as the last in his list of types of mathematical definition. In 1899, §12, Peano declared that the first type, nominal definition, was best. For further information, see Rodríguez-Consuegra 1991, §3.4.3.

8

Frege [1884] 1953, §64. Peano 1894a, §38–§39.

9

Shortly after Pieri’s death, Peano wrote, “The lamented Prof. Pieri, and later Russell and Burali-Forti, have already asserted that every definition by abstraction can be rendered nominal by setting  x = { y : x R y }” (Peano 1913, 765). Peano was probably referring to Burali-Forti [1900] 1901 and Russell 1903, §109–§111. For further information, see Rodríguez-Consuegra 1991, §3.5.2, §5.2, and Mancosu 2016, §2.2.

10

Pieri 1900a, 216 (section 8.6).

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of all corresponding orderings of r, by a nominal definition of a direction of r as the class of all its orderings that correspond with a given one. It is interesting how Pieri clearly stated that the nominal definition is preferable to the definition by abstraction, describing the latter as indirect.11 2.4 Postulates, or Primitive Propositions Pieri held that primitive propositions, or postulates, express the relations among primitive concepts; they must be used when deducing facts in a science, in the sense that the facts can be nothing but logical combinations of the primitive propositions. Thus, it could be said that the set of postulates defines the primitive concepts. Primitive propositions can be considered a priori judgments from which theorems (derived propositions) are deduced, proved, or demonstrated. This distinction between primitive and derived propositions is in line with the one already established between primitive and defined concepts. Indeed, the aim of defining an object by means of other objects is to deduce a fact from other facts. This explains the convenience of reducing the primitive concepts and primitive propositions to a minimum number, and of accepting the help provided by some postulates, which, as general premises from logic and mathematics, are necessary to any science.12 Postulates can also be regarded as conditional propositions, neither true nor false, which merely express conditions that may or may not be satisfied, depending upon the interpretation of the primitive concepts. For example, a certain equation might be true if its variables should represent real numbers, but false for quaternions.13 The very clear semantical point of view is to be noted. When primitive concepts are assigned physical meanings drawn from direct observation of the external world, axioms or primitive propositions become gratuitous statements: more or less evident judgments that are not demonstrated, but will be used to prove theorems. Their certainty will be merely intuitive or empirical, and they are accepted because not everything can be proved. From a formal perspective, such judgments are called postulates because they are only “conditions, or premises, on which depend the validity or consistency of the whole system.” Their truth or falsehood may be abstracted, and they can be understood as mere conditions which together constitute an implicit definition of the primitive concepts, without affecting the system as a logical edifice. Therefore, since the only criterion in effect is that the postulates must be sufficient to

11

Pieri 1898c, 37 (section 6.7 of the present book); Russell had highlighted that passage in his copy of that paper. See Russell and Pieri 1898 (appendix 3 of the present book), and Rodríguez-Consuegra 1991, 132, 156.

12

Pieri 1897c, 351; 1898c, 3–4 (section 6.0); 1900a, 173–174, 203 (sections 8.0 and 8.4); [1900] 1901, 377–380 (section 4.3); 1906d, 58–60.

13

Pieri [1900] 1901, 388–389 (section 4.5).

2.5 Proofs

11

deduce a set of theorems, the postulates may be chosen arbitrarily. Thus, what in one system appears as a postulate can act as a theorem in another.14 2.5 Proofs From his very first works, Pieri held a semantic conception of proofs, already latent in Peano and all of his school (see chapter 3). Pieri clearly anticipated the Tarskian way, even before Tarski was born. Thus, if P and Q are propositions about entities represented by letters a, b, c, ... , which can be changed as you please (for their only meaning is that assigned in P), then the assertion “from P(a, b, c, ...) one deduces Q(a, b, c, ...)” means the same as whatever a, b, c, ... should be, if P(a, b, c, ...) is true of them, then Q(a, b, c, ...) will also be true; whoever asserts P about these objects cannot deny Q for them.15

A similar definition—not so clear, though—appeared some years later: The statement that a certain proposition P is a consequence of other propositions A, B, ... , or that from A, B, ... one deduces P, signifies exactly that a being equipped with reason who accepts A, B, ... as true cannot ignore the truth of P: in sum, that it is not permitted to affirm A, B, ... and at the same time to deny P.” 16

As we shall see in the next section, for Pieri deductive sciences are hypotheticaldeductive systems. That is relevant to this discussion of proofs, because he defined such a system as A succession of propositions, whose truth depends only on certain postulates or hypotheses peculiar to each discipline, and on logical axioms, in such a way that the whole is developed through the power inherent in just these principles, combined algebraically with each other according to the laws and precepts that inform the logical calculus.17

Pieri’s implicit distinction between postulates and rules of inference should be underlined: that was much less clear in Giuseppe Peano’s writing. Similarly, it should be emphasized that here these rules seem to belong to logic, and are regarded not as mere postulates, but as truths. 2.6 Abstract Deductive Science According to Pieri, when a branch of science takes as its basis a series of concepts and postulates understood in the sense previously explained, it becomes an abstract deductive science, autonomous and independent from any other mathematical or physical doctrine. 14

Pieri 1908a, 447–448 (M&S 2007, 267–268).

15

Pieri 1898c, 9 (section 6.1 of the present book).

16

Pieri 1908a, 447–448 (M&S 2007, 267–268).

17

Pieri 1906d, 59–60.

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This took place, for instance, with projective geometry, which had been regarded as a part of elementary Euclidean geometry.18 This abstract approach to geometry thus results from not appealing to any physical interpretation of the premises, or to any geometric evidence or intuition; this had long been the practice in arithmetic. It is opposed to the physico-geometrical approach, according to which primitive entities and axioms are drawn from direct observation of the external world, by identifying them with ideas acquired through empirical intuition of certain objects and physical facts.19 As Pieri clearly indicated, the abstract approach was first presented by G. K. C. von Staudt and then by Arthur Cayley, Felix Klein, and Riccardo De Paolis, as well as in celebrated writings on non-Euclidean geometry and on multidimensional spaces.20 Indeed, the discovery of non-Euclidean geometries confirmed the ideal coexistence of several geometrical systems, each one free of contradiction, and therefore all true, subjectively speaking, and equally possible. Consequently, geometry became not only a science of real things but also a doctrine of the possible. This destroyed the Kantian notion of space as an a priori form within the mind. The abstract approach is also a speculative approach, since its concepts are mere creations of our mind and its postulates, sheer acts of our will, or a priori subjective truths, or truths by definition (without excluding their primordial rooting in external facts). So, such concepts and postulates are arbitrary, at least as far as they do not aim at a pre-established end, guiding thought. All this, together with the rigor and detailed analysis of the deductive method of mathematical logic, aims at establishing geometry as a hypothetical-deductive system or ideal science, which—given its method and premises—is independent from intuition and limited to the study of a certain order of logical relations.21 Under this conception, geometry is a hypothetical doctrine of all that can be depicted or represented by points and figures, in the same way that arithmetic studies everything that can be numbered or represented with numbers. Each can subsist independent from any particular interpretation of their primitive concepts, and thus become a formal science of a given order of logical relations. So, the formal nature of a discipline depends on the comparison of its objects only as far as their common traits are concerned, through relations that establish similar behavior for those objects. In this regard, geometry is not likely to depend on the spatial interpretation (referring to the extension of bodies) any 18

Pieri 1895a, 607; 1896c, 9–10; 1906d, 43.

19

Pieri 1896c, 10; 1898b, 781. Compare Pasch 1882b, 17. Pieri did not mention David Hilbert’s foundational work on geometry during the 1890s. Like Moritz Pasch, Hilbert also recognized the underlying role of intuition (see Hilbert 2004).

20

See Pieri 1898c, 2 (section 6.0 in the present book). Pieri cited no specific works by these authors, save Staudt 1847.

21

Pieri 1898b, 781; 1898c, 3 (section 6.0); 1900a, 173 (section 8.0); [1900] 1901, 368, 374 (sections 4.0 and 4.1). Pieri sometimes used the term logica algebraica for mathematical logic.

2.7 Logic and Mathematics

13

more than arithmetic depends on the economic interpretation (referring to currency), even though those interpretations may have practical interest.22 It is then obvious that for Pieri there is an identity between geometry and arithmetic with regard to application of the “logical method.” All this leads to a splendid general characterization of hypothetical-deductive systems (see section 2.10). One advantage of accurately distinguishing primitive from derived ideas is the possibility of abstracting from the meaning of their categories, which enables us to avoid any special interpretation and operate symbolically with expressions having variable content, linked with each other by known logical relations. In this sense, only postulates provide primitive concepts with content, and progressively limit their initial indetermination, as the system develops, until they are implicitly defined in such a way that readers do not need to rely on intuitive cognition. On the contrary, they will acquire an exact idea of what the primitive terms should mean and express. Thus, several orders of concrete and particular things are covered by a single general and abstract doctrine, just as an algebraic solution of a quantitative problem always entails several numerically different cases, which, in addition, differ from one another as a result of the quality of the data.23 Indeed, all deductive sciences can be interpreted as hypothetical-deductive systems, in which the truth of theorems has only a logico-formal nature and a hypothetical necessity that depends on the conditions or hypotheses that define their range of validity.24 2.7 Logic and Mathematics According to Pieri, logic is a universal science, which embraces in its basic notions the widest set of objective interpretations. Its laws are unchangeable, as they are constitutive principles of human reason. Logic is the formal science par excellence, since it studies objects in general, real or not, possible or impossible, independently from their existence.25 The more general logical categories—individual, class, membership, inclusion, representation, negation, and a few more—are common and necessary to any human discourse. In geometry, they make possible the combination of the primitive and derived notions.26 Like Gottlob Frege and Bertrand Russell, Pieri conceived of logic as an insurmountable, inescapable universal language 27: this helps to explain Pieri’s conviction that the consistency of logical premises cannot be demonstrated. For Pieri, again in line with Frege and Russell, formal logic, despite its extreme generality, is not just a list of tautologies, simple and sterile. That bias originated in a 22

Pieri [1900] 1901, 374, 376 (section 4.2 of the present book); 1906d, 43, 44, 46.

23

Pieri [1900] 1901, 381 (section 4.3); 1906d, 41; 1908a, 103 (M&S 2007, §3.9, note 3).

24

Pieri 1906d, 57, 58.

25

Pieri [1900] 1901, 370 (section 4.1); 1906d, 41, 44.

26

Pieri 1898b, 782; 1900a, 173 (section 8.0); [1900] 1901, 384 (section 4.4); 1908a, 446 (M&S 2007, §3.9, note 1). See also Rodríguez-Consuegra 1994.

27

See also section 2.9.

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simplistic conception of logic, according to which this general science would be based on the principles of identity, contradiction, and excluded middle, regarded as laws of thought, and on the distinction between analytic and synthetic judgments (the former being constrained to repeat and deploy the content of our concepts, whereas only the latter could increase our knowledge). For Pieri, that conception had already been overcome: deductive reasoning is something far more reaching, and its deductions can associate several principles until it draws a more general set of consequences, including some not entailed by any one of them. Thus, logic is similar to what Leibniz called combinatorics, the art of inventing by reasoning. We can then admit that it is a synthetic method guided by some kind of rational intuition or perception of the logical relationships between principles and consequences, and nothing else.28 A good ideographic algorithm was for Pieri a useful instrument for the discipline of thought, because it excludes ambiguity and other defects of spoken or written ordinary language.29 Pieri admitted using “algebraic logic,” later called symbolic or mathematical logic or logistic, in developing his research, but he used that language in publications only until 1899. Mathematical logic is like a microscope through which the smallest differences between ideas can be noticed, which are usually unperceivable due to the defects of ordinary language. To attain sufficient accuracy, its employment is required. We could say then that mathematical logic, or the microscopic analysis of thought, is some sort of positivism of deductive reason.30 Mathematical logic studies the formal properties of logical relationships and operations, and their ideographic representation using a small number of symbols with precise meaning, which comply with certain predetermined rules, similar to those of algebra. Yet it is not a new logic. It is the perfection of the formal logic of Aristotle and the Scholastics: it is the modern form of deductive logic. Consequently, formal logic forms the propaedeutics—introductory principles—of all truly scientific and critical philosophy.31 From about 1905, Pieri took an unequivocally logicist viewpoint. Of course, his conception of logic, like Frege’s and Russell’s, included what we now call set theory. Pieri attributed to logic the arithmetic principles, established by Richard Dedekind and others, that characterize the primitive ideas of natural number and successor. In particular, the principle of mathematical induction is attributed to logic. The process through which arithmetic has become more and more abstract, to the extent that it has become the ideal deductively organized science of all that is numerable, had caused many, including Dedekind and Giuseppe Peano, to consider it as a legitimate branch of logic.32 The ideas of number and function in arithmetic, and those of figure and transformation in geometry, reproduce special aspects or modifications of two fundamental logical categories: class and representation. These concepts of transformation, function, and 28

Pieri 1906d, 75–76.

29

Pieri 1898c, 4–5 (section 6.0 of the present book); 1906d, 32–33.

30

Pieri [1900] 1901, 382 (section 4.3); 1906d, 31.

31

Pieri 1906d, 26, 80.

32

Pieri 1896c, 9; 1898c, 4 (section 6.0); [1900] 1901, 370 (section 4.1).

2.7 Logic and Mathematics

15

correspondence, which are hardly distinguishable from each other and are included in the more general concept of relation, belong legitimately to logic. That is why Pieri called his 1898 projective-geometry system logico-deductive, and why he referred to geometry as a purely logical system in the title of his 1900 Paris presentation.33 Moreover, as we saw in section 2.6, geometrical theories, regarded as hypothetical-deductive systems, are nothing but formal analyses of an order of logical relations, as is the case with arithmetic. Trying to exclude from the realm of pure logic the generic notion of natural number and the principles of arithmetic, raises great difficulties (which Pieri considered not at all insurmountable, as some seemed to believe). In this regard, we must not believe that the concepts denoted by the expressions one, each, all, any, none, as well as the general idea of set or class and the singular/plural distinction, involve or assume the notion of natural number as something already formed or acquired. Without such expressions discourse would not be possible. They belong to logic before they belong to arithmetic, since they express certain primitive ideas par excellence, such as those represented by the symbols for class and membership, which are at the basis of all reasoning. Nothing can be drawn from nothing, so it is not surprising to find in logical notions some fragments of the general idea of natural number, which is defined using them. But this is not a petitio principii, any more than is definition of the real number system by means of its subsystem of rationals.34 Pieri noted that, following Russell and Henri Fehr,35 we could consider a fusion of logic and mathematics: logic is mathematical due to its form, and mathematics is logical due to its method. Mathematics gets its certainty from its union with logic, while the permanence of logical principles derives, to a great extent, from its being founded on the use of mathematics.36 However, Pieri believed that Russell’s thesis of the identity of logic and pure mathematics was not totally demonstrated, and said that constructing certain entities solely from logical constants is not enough to ensure their existence: no definition can create its own raw material. Therefore, Russell’s demonstration of the existence of the natural numbers did not seem sound enough to him, although it would be enough to add a single existential assumption, recently discussed by Peano: there is at least one infinite class of entities.37

33

Pieri 1898c, 43 (section 6.9 of the present book); 1908a, 448 (M&S 2007, §3.9, note 4); Pieri [1900] 1901, title (section 4.0).

34

Pieri 1906g, 201–202.

35

Russell 1903, I, 9. Couturat 1904, 1037. See subsection 9.4.3, page 449.

36

Pieri 1906d, 39, 42–43.

37

Russell 1903, 497. Pieri 1906d, 69–70. Peano 1906, 141. There has been considerable study of the question, how to construe the notion of natural number without assuming the existence of an infinite set. See, for example, Quine 1963, chapter 4.

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2.8 Pieri's Letter to Russell One of the present authors, Francisco Rodríguez-Consuegra, found in the Russell Archives a letter from Mario Pieri to Bertrand Russell dated 1903. It can be regarded as an indirect testimony of the Peano school’s important influence on Russell. The letter, really a postcard, reads as follows.38 Most distinguished Doctor Catania (Sicily), 22 May 1903 I thank you heartily for the splendid gift of the Principles of Mathematics; and even more for the favorable opinions you have expressed about some of my works. The principal assumption of your book conforms perfectly with my viewpoint. I, too, have always believed that the primitive objects of pure mathematics can all be defined by means of some logical categories (Cl, , (such that), etc. ... ); in short, that the undefinable can be eliminated from all deductive sciences except from Logic; and that the primitive objects of that [science] are not subject to different interpretations; and therefore that they must rightly be called logical constants. I have indeed affirmed all these [opinions], although very timidly, somewhere in my works (as probable truths), and still remember that Professor [Giuseppe] Peano disagreed with me on this opinion. I shall read the book with great interest. To you the most obliged Mario Pieri

This letter, which Russell carefully transcribed but never published, has several important points. First, Pieri emphasized that the nominal definitions that he himself devised as alternatives to definitions by abstraction (see section 2.3) not only introduce concepts in an explicit way, but also eliminate them. This elimination is rather doubtful in Russell’s own work, since his Platonism from the time of his 1903 Principles prevented him from admitting, for instance, that the concept of natural number could be eliminated only by having the well-known explicit definition available in terms of the corresponding class of classes. Second, it is clear that Pieri was completely aware of his own priority over Russell with regard to the logicist idea, that some or all of mathematics is reducible to logic. Third, although Pieri partially adhered to the tradition of implicit definition (from Joseph Gergonne to David Hilbert),39 in his last analysis any arithmetic or geometric notion could be reduced to logical ideas. Thus, Pieri’s work shows a clearly logicist flavor, not only because he explicitly stated, as Peano had already done in 1889, that projective geometry is a logical combination of primitive propositions and a complex construction of primitive ideas, but also because, while admitting specifically geometric ideas, he held in the last analysis that all notions used in such maneuvers could be reduced to purely logical concepts. Given this, Pieri’s letter is a good way to understand better the origins of Russell’s logicism.

38

Pieri 1903a. The present authors supplied the words in [brackets].

39

Gergonne 1818; Hilbert [1899] 1971.

2.9 Metamathematics

17

2.9 Metamathematics In Pieri’s view, to choose the notions that should be taken as primitive in a hypotheticaldeductive system, we can follow this criterion: a field of science is normally characterized by a maximum group of transformations that cannot alter the properties studied by that field. Thus, we should choose primitive concepts invariant under that fundamental transformation group. This idea, originally discussed in Felix Klein’s Erlanger program, was later applied by Alfred Tarski in an attempt to define the notion logical constant.40 In selecting postulates, the canons of the strictly deductive method must be followed. No one of them should be decomposable into separate assertions with less deductive strength. And none should be stronger than necessary for its deductive role. Their number may increase as long as the deductive power of the whole system remains the same. Likewise, the number may decrease if certain premises are replaced by a principle equivalent to their conjunction.41 The set of primitive notions should be irreducible, in such a way that no one of them can be defined explicitly in terms of the others. And the set of postulates should be independent, in such a way that no one of them can be deduced from the others.42 Pieri did not claim that his postulates for the geometry of position or the projective geometry of hyperspace form independent sets, that condition being very close to ideal perfection. However, the appendix to Pieri 1898c focused on demonstrating the sequential independence of most of those postulates by the usual model-theoretic method—that is, the independence of each one from the preceding postulates. That let Pieri hope that he was not too far from this goal. He simply held that his postulates are sufficient, together with the logical axioms, to sustain the whole edifice of this abstract geometry.43 The most important points in Pieri’s 1898c appendix can be summarized as follows. A set of propositions is called independent if no one of them is a consequence of the rest—that is, if for each of those propositions, entities can be found that do not satisfy it, but do satisfy the others. Since the postulates of a deductive science are conditions imposed on the primitive notions, to prove their independence it would suffice to identify a corresponding set of interpretations, or specifications, of the primitive entities, such that for each interpretation all of the postulates except the corresponding one would be satisfied. But Pieri noted that such results had only been obtained on very few occasions, and only for very restricted postulate sets. In that appendix, Pieri proved the sequential independence of most of his postulates I–XIX, using interpretations, or models, from logic, arithmetic, analysis, and geometry. 40

Pieri [1900] 1901, 389 (section 4.5 of the present book). Klein [1872] 1892–1893. Tarski 1986b. M&S 2007, §5.2, shows how Tarski was influenced during the 1920s by Pieri’s work on foundations of geometry. Whether Tarski 1986b itself was inspired by Pieri is not known.

41

Pieri 1898c, 3 (section 6.0).

42

Pieri [1900] 1901, 380 (section 4.3).

43

Pieri 1896c, 10; 1898c, 4 (section 6.0).

18

2 Pieri’s Philosophy of Deductive Sciences

To prove that all postulates of a deductive system are compatible—that is, they do not contradict each other—it suffices to find a single interpretation that satisfies them all. This is equivalent to showing that the negation of each postulate is independent from the other postulates. Pieri provided that interpretation: analytic geometry with four variable homogeneous coordinates. His very clear semantic point of view is notable here. This accords with Pieri’s definition of logical consequence, which in fact is just the contrary of the semantic notion of independence.44 Pieri returned to this subject in a footnote to his 1904a article on foundations of projective geometry.45 In that account, the consistency of two propositions A and B belonging to a deductive system , or the deductive independence of either from the other, is affirmed relative to some realm  of rational or empirical cognition, whose truth is conceded a priori by finding an interpretation of the primitive concepts of , and of those involved in A and B, that satisfies the postulates or basic principles of  and exhibits the properties signified by A and B at the same time, or by exactly one of them for proving independence. Thus, the verification depends on the postulates of , if the realm is purely deductive, or on intuitive empirical principles in any other case. In any of these cases, from the principles of , which Pieri regarded as including the axioms of logic, it would be deduced that A does not exclude B or does not entail B. If  should be a realm in pure deductive logic, ideal perfection would be reached, since consistency or independence of A and B will have been established through pure logic. If systems  and  should coincide, this method would yield the usual type of proof of independence: it would provide two interpretations of the primitive concepts of  one satisfying A and the other satisfying B, but neither satisfying both. If A and B belonged to  as well as , the method would not yield their consistency: that would constitute a petitio principii, since  was already assumed consistent. (For the same reason, it is not possible to establish by this method the consistency of pure logic.) In the other cases, this method would yield a proof of independence or consistency of A and B relative to the system , whose consistency has already been granted.46 The search for a direct and absolute demonstration of the consistency of the axioms of arithmetic within arithmetic itself should thus fail; however, Pieri felt that it might succeed within pure logic. He felt that David Hilbert’s 1900 precept provided an acceptable logico-deductive goal: Hilbert claimed that such a demonstration would require a direct and absolute method that should be sought within pure logic without resorting to any auxiliary system whose mathematical existence, or consistency, could be doubted. A demonstration within pure logic, however, would involve serious difficulties: excluding from that field the notion of natural number and principles of arithmetic. Nevertheless, Pieri was optimistic: he wrote that he could not share his colleague Alessandro Padoa’s 44

Pieri 1898c, 60 (section 6.13 of the present book). See section 2.5 and chapter 3.

45

Pieri included the same text, rearranged, in his 1906g article on the consistency of arithmetic.

46

In 1904a, 331, Pieri required for an independence proof that A, B should belong to : thus, that they should be consistent. In 1906g, 198, he dropped that requirement.

2.9 Metamathematics

19

opinion that no direct demonstration of consistency could be found by known methods of reasoning.47 In his 1906g article48 on the consistency of the axioms of arithmetic, Pieri insisted on using semantic model-theoretic methods, extended somewhat the discussion of logic in the 1904a paper, and quoted Gottlob Frege’s opinion that the only way to establish the consistency of a concept with rigor is to prove that something falls under it. According to Pieri, proving by a direct method that the premises of a system are not contradictory—that no contradiction will ever be found even should the series of deductions be continued indefinitely—raises a serious difficulty: there are infinitely many possible consequences of the axioms. That would require reasoning by recurrence over the infinite set of derived propositions. However, even should the induction principle be accepted among the logical axioms, we would not know how to decide whether the set of derived propositions is numerable, amenable to application of the principle. For Pieri it would thus never be possible to prove deductively the consistency of the whole system of logical premises. It is then a priori evident that no criterion of consistency will lead to sound results as far as premises necessary to the discourse are concerned. Indeed, in order to apply the criterion, reasoning would be required and consequently logical axioms needed as well. According to Hilbert, these difficulties could be overcome by introducing the fundamental principles one at a time, ensuring their consistency step by step. However, to Pieri it did not seem that the consistency criterion that should be followed for the first propositions can be based on them only, nor applied without auxiliary elements, since every demonstration depends on many principles.49 In 1906g, Pieri attempted to establish the consistency of the Dedekind–Peano axioms, which, he held, implicitly define the natural numbers and their arithmetic. Pieri carried out his argument in a realm of pure logic, which he regarded as including set theory. To this end, he interpreted within logic the notions of natural number and successor. The expression n is a natural number should stand for n is the class of all classes equinumerous with a particular finite50 class; and successor of n, for the class of all classes equinumerous with a union p c {q} where p 0 n and q ó p. Pieri noted that this amounts to the nominal definition of natural number and successor.51 Pieri felt that with this contribution, consistency followed from the earlier works of Georg Cantor, Frege, 47

Hilbert [1900] 2000, 414; Padoa 1903; Pieri 1904a, 330–331. They wrote about axioms for real, signedinteger, and natural-number arithmetic, and evidently regarded the corresponding consistency statements as interdeducible.

48

Pieri 1906g was published in French in a leading philosophical journal, and attracted readers different from those of most of his papers.

49

Hilbert [1904] 1970, 131. Pieri 1906g, 200.

50

For Pieri, two classes were equinumerous if there was a bijective mapping between them, and a class was finite if it was not equinumerous with any proper subclass. This definition of finiteness is due to Richard Dedekind ([1888] 1963, 63).

51

Perhaps in error, Pieri used the word somme for successor in the relevant discussion in 1906g, 207, although on page 203 he had written suivant.

20

2 Pieri’s Philosophy of Deductive Sciences

Cesare Burali-Forti, and Bertrand Russell.52 He did not intend to recommend these definitions in place of those already proposed by Russell, but asked whether they might be preferable. He noted that Russell had not employed the notion of finite set at this stage, but had formulated his definition of finite number in a way that incorporated the induction principle. Pieri had based his approach on two logical axioms added to those in Giuseppe Peano’s Formulaire 53 : that there should exist at least one infinite set, and that the union of an infinite class of classes should be infinite. These two axioms thus served somehow in place of the induction principle. Russell’s avoidance of the second one did simplify the logic, but precluded any analysis of that principle.54 Curiously, Pieri did not seem to realize that set theory would not be a suitable basis for his consistency proof, precisely because its own consistency was doubtful in view of the paradoxes.55 Perhaps he tacitly considered that Russell’s work on the paradoxes was enough to overcome them; but that would be strange, since Russell had not yet published anything precise about the theory of types (the appendix to Russell 1903 on types is anything but precise). 2.10 Semantics and Model Theory In his 1898c memoir on projective geometry, explaining the relationship between his alternative postulates XIX and XIXr that specify three or more dimensions,56 Pieri expressed considerations fundamental for understanding his semantic conception of a formal system. He wrote that the interpretation or specification of primitive entities is totally arbitrary: whenever they are assigned special meanings so that all the postulates become true, an interpretation or representation of the system is obtained. For instance, projective point (0) and projective line (1) can be reinterpreted as projective plane (2) and pencil of projective planes, respectively; under that interpretation (2) becomes (0), just by its formal definition. This reciprocity is also applicable to the relationship between the notions of point and line in a plane, and line and pencil of lines. It lends importance to the idea of interpretation: the duality law, already long familiar in projective geometry, 52

Pieri cited no specific work of Cantor. Elsewhere in 1906g, Pieri had cited Frege [1884] 1953, Burali-Forti 1896–1897, and Russell 1903.

53

Peano et al 1895–1908, volume 4.

54

Some of these ideas from Pieri 1906g are found also in 1906d, 61–64.

55

Set theory developed later in a way that was much more precise than the logic that Pieri employed, but would not conveniently accommodate the definition of natural number that he had presented. With the now standard Zermelo–Fraenkel set theory (ZF), each of Pieri’s nonzero natural numbers would be a proper class. ZF permits only very restricted manipulations of proper classes; in particular, they cannot be considered as members of other classes. This problem is usually overcome by defining 0 to be the empty set, 1 = {0}, 2 = {0, 1}, 3 = {0, 1, 2}, and so on, and postulating that these constitute the elements of a set. In 1955 Dana S. Scott discovered a modification of the earlier definition that can be used in ZF: after developing the theory of ordinal numbers and rank, define n is a natural number to mean n is the set of all sets of least rank equinumerous with a particular finite set. The details are presented, for example, in Levy 1979, §II.7, §III.2.

56

Pieri 1898c, §11 (section 6.11 of the present book).

2.10 Semantics and Model Theory

21

can thus be seen as an instance of a more general principle, a plurality law, which belongs to geometry no more than it does to any other deductive science. This conception allowed Pieri to present a “model-theoretic” definition of ordinary projective space. Any interpretation of the primitive concepts (point, and line joining two points) that satisfies postulates I–XIX is a representation of ordinary projective space. Thus, Pieri defined «ordinary projective space» as the class of all possible interpretations that satisfy postulates I–XIX.57 In this way, Pieri clearly overcame a problem with Giuseppe Peano’s approach: Pieri managed to distinguish the implicit-definition approach from the far more modern semantic one. According to the semantic approach, an axiomatic system does not implicitly define its primitive concepts; rather, it explicitly defines a second-order concept. Peano’s obscureness had been noticeable in the 1899–1893 editions of his Formulaire, where he construed number as “that which one obtains by abstraction from” all the interpretations of his well-known postulates for arithmetic. In 1903 Bertrand Russell explicitly objected to Peano’s presentation. He offered as a clarification simply “the class of ” all those interpretations: that is, the method Pieri had introduced in a geometrical context in 1898c. Later, Pieri noted that the method can be used with any axiomatic theory. Thus, Pieri clearly surpassed David Hilbert in this regard, whose writings had openly depended on the implicit-definition approach, and Pieri clearly anticipated Russell.58 Pieri did not regard such a plurality of interpretations as indetermination: although the mind cannot contemplate the full content of the primitive concepts, which is remote and complex, the postulates always let us decide whether or not a given object satisfies their requirements. This led him to the idea that we are to some extent free to specify a variety of interpretations for the primitive concepts of arithmetic and geometry, distinct from each other and even quite obscure. Thus, it happens that arithmetic and geometry embrace all sorts of concepts, numerable and figurable.59 Pieri considered relationships among hypothetical-deductive systems at the end of his 1898b paper on projective geometry, an axiomatic presentation different from his now more standard one, 1898c. The primitive concepts of 1898c were projective point and the projective line joining two such points; those of 1898b were projective line and the relation that holds between two projective lines just when they have some point in common. Pieri noted that both systems have the same content despite their construction from different materials. Pieri explained that two such systems should be considered logically equivalent if the primitive concepts of each one can be defined in terms of those of the 57

Pieri 1898c, 55–56 (section 6.11 of the present book). For the use of guillemets («...»), see a box in the preface, page xiii.

58

Peano et al. 1895–1908, volume 2, number 3, page 30; volume 3, page 44; volume 4, page 34. Peano continued, “or in other words ... the system N0 that has all the properties stated by the [postulates], and only those.” In 1903, §122, Russell objected that those interpretations were not distinguishable from one another, and the system N0 not defined at all. Russell did not mention Pieri, but in fact had read Pieri 1898c carefully (Rodríguez-Consuegra 1991, 156). Pieri 1906d, 49–50. In that address to the Catania faculty, Pieri in fact credited Russell with the method! Hilbert [1899] 1971, chapter 1.

59

Pieri 1900a, 178 (section 8.1).

22

2 Pieri’s Philosophy of Deductive Sciences

other, and (with such definitions), the postulates of each are consequences of those of the other.60 In his 1906d address to the Parma faculty, Pieri discussed the notion of categoricity. He regarded a hypothetical-deductive system as categorical if between any two interpretations of its primitive concepts there should always exist a perfect correspondence, such that “in certain respects the corresponding classes of objects that satisfy the postulates can be regarded as the same.” Pieri’s discussion echoed that of Oswald Veblen, who had introduced the idea in 1904. To a noncategorical system can always be appended new postulates that constrain its realm and open avenues to new possibilities. On the contrary, in a categorical system, any true proposition that can be stated in terms of the primitive concepts is always deducible from the postulates.61 2.11 Nominalism One of Pieri’s very few purely philosophical digressions, in his 1900 Paris lecture,62 stemmed from his formalistic conception of hypothetical-deductive systems, according to which primitive concepts are determined only by the postulates. For Pieri, the old controversy between realists and nominalists must be decided theoretically in favor of the latter.63 However, he would introduce a new phrase, geometry as a purely logical system, to designate the new general and abstract conception according to which geometry is an exclusively deductive and purely rational science, even though from an educational viewpoint it still needs to be regarded as a physics of extension. By 1906, however, Pieri saw nominalism as a conventionalist failure. If in considering the purely formal side of the mathematical disciplines we lose sight of all real or possible material, then we fall prey to the nominalist defect, whose genuine form consists of interpreting such disciplines as complexes of symbols and arbitrary conventions void of content and without objective value. Pieri felt that this fault is very frequently committed by mathematicians in regarding mathematical ideas merely as free creations of the mind.64 In his view, mathematicians were becoming more nominalistic day by day. But theirs was not the coarse, empirical nominalism of Thomas Hobbes.65 Rather, they would reason as follows: 60

Pieri 1898b, 797. See also Pieri [1900] 1901, 380–381 (section 4.3 of the present book).

61

Pieri 1906d, 46. Veblen 1904, 346. This early connection between categoricity and completeness (deducibility of all true statements), already stated by Veblen, was not yet fully developed, but nonetheless interesting. In 1981 John Corcoran raised the question whether the difference between those ideas had yet been fully recognized.

62

Pieri [1900] 1901, §II (translated in section 4.2).

63

For realists even abstract words have objective reference in a third realm of things, beyond the physical and mental realms; Gottlob Frege and the early Bertrand Russell followed Plato about this. For nominalists, all conceptual terms are just names, without objective reference.

64

Pieri 1906d, 30. Pieri cited Cantor [1883] 1996, §8, as an example.

65

For information about Hobbes, consult Peters 1967.

2.11 Nominalism

23

If in the physical or mental universe there exists something that should satisfy the conditions I have imposed on the symbols, then [under this interpretation] the various facts that I have demonstrated will also have to be seen as true.

Or even, “My premises must be taken as consistent unless proven otherwise.” This is clearly different from denying the symbols any real or possible content; rather, we should operate with them without worrying about a particular interpretation. If accused of wasting time, nominalists will rightfully answer that their imaginary worlds are fully attractive to them, and that the satisfaction and honor of human understanding are more than enough to justify their scientific research. Unfortunately, Pieri never developed this interesting shade of philosophy, halfway between what was later called if-then-ism and the unmistakable underlying formalism.66

66

Pieri 1906d, 31–32. The two quotations are quoted there as typical statements. For more information on if-then-ism, consult Putnam 1967.

3 Two Paths to Logical Consequence: Pieri and the Peano School

This chapter1 has two main goals. First, it will explore the “negative” avenue leading from the concepts of independence and consistency to that of logical consequence. This can be seen as opposed to the “positive” route based on Alfred Tarski’s concept of satisfaction and truth, for which only Bernard Bolzano, and sometimes Aristotle, are regarded as precedents. Second, it will display the nexus between logical consequence and the formal point of view, which arose with the development of the concept of independence. That can be seen clearly when studying the appearance of the first truly formal systems within the Peano school. Section 3.1 introduces today’s usual definition of logical consequence, commonly attributed to Tarski. That is compared briefly with Bolzano’s definition, now recognized as a fundamental precedent. Section 3.2 describes and discusses Aristotle’s counterexample method. Although that is sometimes considered a main antecedent to the semantic notion of independence, the present discussion concludes otherwise. Section 3.3 studies the independence of Euclid’s parallel postulate via Nikolai I. Lobachevsky’s alternative postulate and Felix Klein’s model, which established the consistency of nonEuclidean geometry. That can be regarded as a sign that a semantic definition of logical consequence was already possible well before the work of the Peano school. The only thing still lacking was the concept of a formal, or at least abstract, system. Section 3.4 describes the rise of such concepts in the Peano school, through Mario Pieri and Alessandro Padoa. By 1900 Padoa had provided a fully semantic definition of logical consequence.

1

Preliminary materials related to this chapter were published in Rodríguez-Consuegra 1996, 1997b, and 1997c.

© Springer Science+Business Media, LLC, part of Springer Nature 2021 E. A. C. Marchisotto et al., The Legacy of Mario Pieri in Foundations and Philosophy of Mathematics, https://doi.org/10.1007/978-0-8176-4823-7_3

25

Alfred Tarski

Bernard Bolzano

3.1 Tarski’s Definition of Consequence

27

3.1 Tarski’s Definition of Consequence Alfred Tarski presented his canonical definition of logical consequence in 1935: To the fundamental concepts of semantics belongs the concept of the satisfaction of a sentential function by ... a sequence of objects. ... [The] intuitive sense of such turns of phrase as ... the triple of numbers 2, 3, and 5 satisfies the equation “x + y = z,” can ... arouse no doubt. ... One of the concepts which can be defined with the help of the concept of satisfaction is the concept of model. ... Let L now be an arbitrary class of sentences. We replace all extra-logical constants occurring in the sentences of ... L by corresponding variables, that is equal constants by equal variables, different by different; thereby we arrive at a class of sentential functions Lr. An arbitrary sequence of objects which satisfies each sentential function of ... Lr we shall call a model ... of the class of sentences L ... [Now] the concept of following logically can be defined in the following way: the sentence X f o l l o w s l o g i c a l l y from the sentences of the class K if and only if every model of the class K is at the same time a model of the sentence X.2

In that essay, Tarski referred to Rudolf Carnap as his only predecessor. Later, he also recognized Bernard Bolzano. Indeed, Bolzano’s definition, a century older, is very similar: If we assert that certain propositions A, B, C, D, ... M, N, O, ... stand in the relation of compatibility with respect to ideas i, j, ... , then we assert ... no more than that there are certain ideas whose substitution for i, j, ... turns all of those propositions into true ones. ... Let us consider ... the case that among the compatible propositions A, B, C, D, ... M, N, O, ... the following relation obtains: all ideas whose substitution for the variable ideas i, j, ... turns a certain part of these propositions, namely A, B, C, D, ... into truths, also [make] a certain other part of these propositions, namely M, N, O, ... true. This special relation ... between propositions A, B, C, D, ... on the one hand and M, N, O, ... on the other is of special importance, since it puts us in a position to infer the truth of M, N, O, ... , once we have recognized the truth of A, B, C, D, ... . I wish to give the name of deducibility [Ableitbarkeit] to this relation ... . Hence I say that propositions M, N, O, ... are deducible from propositions A, B,C, D, ... with respect to variable parts i, j, ... , if every class of ideas whose substitution for i, j, ... makes all of A, B, C, D, ... true, also makes all of M, N, O, ... true.

Intuitively, it appears that Tarski and Bolzano were both trying to define the same idea.3 Some differences, however, stand out from different perspectives, as follows. First, Tarski’s notion of consequence is defined for linguistic expressions in the framework of formalized languages, whereas Bolzano’s notion of deducibility applies to propositions, which are nonlinguistic . Therefore, in the absence of the concept of a logical calculus, which depends on the formalization of logic, it is not possible even to imagine the equivalence between consequence and derivability for first-order languages, which had already been demonstrated by Kurt Gödel before Tarski published his definition of consequence. 2

Tarski [1935] 2002, 185–186; this quotation was extracted from its translation of Tarski’s German version. The present authors converted its use of augmented letterspacing for emphasis to italics, except where Tarski used both. For information about the relationship of the various versions and subsequent literature, see the introduction to Tarski [1935] 2002.

3

Carnap 1935. Tarski [1935] 1956, 417. Bolzano [1837] 1972, §155.2. During 1913–1923 Kazimierz Ajdukiewicz ([1921] 1966) introduced to Polish scientists a definition of logical consequence that can be regarded as a predecessor of Tarski’s. Ajdukiewicz knew Bolzano’s definition and adopted Pieri’s. As a doctoral student Tarski became very familiar with Ajdukiewicz’s work. See Batóg 1995, §1; Tarski [1930] 1983, 32; and subsection 10.4.1 of the present book.

28

3 Two Paths to Logical Consequence

Second, Bolzano’s condition that the sequence of antecedents and consequents must be compatible does not follow from the modern notion of consequence. Thus, his theory of deducibility is less general and more complicated. Sometime after 1940, Tarski modified his definition of logical consequence. Over the next decade, his new definition gradually became standard: “A sentence ... is said to be a logical consequence of a set A of sentences if it is satisfied in every realization ... in which all sentences of A are satisfied ...” Besides interpretations of the nonlogical constants, Tarski’s notion of realization includes a nonempty domain over which the individual variables should vary.4 This domain was not considered in his original definition. Accordingly, the sentence › x, y [ x = / y], for example, would then have been deemed logically true—a logical consequence of any set A. With the now standard definition, however, that is not so.5 A possible similarity between the two definitions could be drawn from the assumption that establishing a relation of logical consequence requires considering extralogical knowledge, since consequence is relative to a variable set of ideas that are substantive, not merely formal but having real content.6 If this is interpreted as a criticism of Bolzano’s definition because his line between logical and extralogical ideas is unclear, the same criticism would apply to Tarski. Tarski mentioned this point in the same paper; he had become aware of it several years earlier. Here is found another part of the current criticism of Tarski’s definition.7 From the perspective of a philosopher, other interesting differences can also be noted. In particular, for Bolzano two propositions with no common ideas cannot be deduced from one another, whereas for Tarski an analytic (formally true) proposition is a logical consequence of any proposition, even if there is no common idea between them. Paul B. Thompson has interpreted this as a sign that Bolzano’ s philosophical approach is epistemic: concerned with the possibility of systematizing propositions into a science. On the other hand, Tarski’s approach is clearly ontic: concerned with intrinsic characteristics of the propositions themselves. Thus, Bolzano would exclude the possibility that the truth of a conclusion might be independent of the truth or falsehood of the premises, since his 4

Tarski, Mostowski, and Robinson 1953, 8.

5

According to Ignacio Jané (2006), a version of Tarski’s original notion of logical consequence saw common informal use by postulate theorists during the early 1900s, starting with the Peano school and then the American researchers. Paolo Mancosu (2006, §3) showed that Tarski was still using his original definition in 1940. In 1953 Tarski did not emphasize that his new notion was different. That led to confusion in subsequent literature. For example, Jan Berg (1962, §IV.5.B, 116–117) pointed out the differences between Bolzano and Tarski described in the preceding two paragraphs, but then confused Tarski’s original definition with the now standard one. The difference became well known only through Etchemendy 1988. See also Corcoran and Sagüillo 2011.

6

Vega 1984, 174–175.

7

Tarski [1935] 2002, 188. According to Mancosu 2005, §2, Tarski had already discussed the point with Carnap in 1930. W. V. O. Quine’s 1951 pursuit of this question is well known. In fact, he had discussed it with Carnap in 1933 and probably with Tarski as well (Quine 1991, 267). For further information, see Etchemendy 1988, §2.

3.1 Tarski’s Definition of Consequence

29

objective would be to describe the epistemic connection of a set of propositions within a theory.8 How did the semantic definition of consequence arise, following the negative path from the concept of independence? The most relevant consideration here is the very narrow conceptual nexus between logical consequence on the one hand, and the formal nature of theories and the languages used in defining it, on the other.9 Logical consequence cannot be defined unless it is understood that axiomatic theories should be essentially formal—concerned with form, not content. This essential formality is required by the very fact that axiomatic theories are held together by logical consequence. To see this, let K be the class of axioms of a theory T. Then, S is a theorem of T if and only if it is a logical consequence of K. This relationship does not depend on a particular interpretation of K and S. Mathematicians studying an axiomatic theory need not care about the referents of its sentences, even if they use a model as a guide. The key point is that, for any interpretation, if the axioms are true, so are the theorems. Since the claim that S should be a logical consequence of K is independent of any particular interpretation of K and S, it remains applicable to the uninterpreted statements of K c {S}. Thus, the study of an axiomatic theory is a study of uninterpreted statements. It follows that a full semantic definition of logical consequence could not have appeared until the modern concept of an axiomatic formal system was available, which includes at least a clear idea of the formal nature of the nexus between axioms and theorems. Of course, this is just a necessary condition. However, as shown in section 3.4, the mere recognition of such concepts, which took place in the Peano school at the end of the nineteenth century, immediately led to the modern definition of logical consequence. On the other hand, an analysis of the historical precedents of the notion of logical consequence—namely, Aristotle’s counterexample method and the proof of the independence of the parallel postulate via Felix Klein’s model—leads to the conclusion that it was precisely the lack of the concept of an axiomatic formal system that made the definition of logical consequence remain impossible. The next section shows that with Aristotle’s method it is nearly impossible to speak properly about today’s idea of independence. Section 3.3 will show that the independence demonstrated via Klein’s model could not have been generalized, along the negative path, until a semantic definition of logical consequence had been attained.

8

Bolzano [1837] 1972, §155.21; Thompson 1981, §3.

9

The rest of this paragraph follows closely the extremely clear, and more detailed, description in Torretti [1978] 1984, 192–196.

30

3 Two Paths to Logical Consequence

3.2 Aristotle’s Counterexample Method This section describes the argument according to which Aristotle is the earliest known precursor of the modern concept of independence. His writings include various instances of the counterexample method. It is used to rule out certain forms of syllogism10 by providing, for the first, middle, and last terms, interpretations that make the premises true but the conclusion false. Aristotle’s use of this method suggests that he was prepared to prove the independence of the conclusion from the premises in a way very similar to today’s usual method of proving independence by considering certain interpretations. To what extent is that claim justified? The following text by Aristotle is most suitable for illustrating the application of the counterexample method: But if the first term belongs to all the middle, but the middle to none of the last term, there will be no deduction between the extremes; for nothing necessary follows from the terms being so related. For it is possible that the first should belong to all or none of the last, so that neither a particular nor a universal conclusion is necessary. But if there is no necessary consequence, there cannot be a deduction from these premises. Terms of belonging to all: animal, man, horse; terms of belonging to none: animal, man, stone.11

The next displays analyze in detail the syllogistic figure and modes to which Aristotle referred. It identifies the two ostensible conclusions to which he alluded, and identifies the two triads of interpretations that show that the syllogistic forms in question are invalid. The two remaining ostensible conclusions are included for later discussion.12 mode premise 1 premise 2

figure

( A) Each M is a P. (E) No S is an M.

ostensible ( A) Each S is a conclusions (E) No S is a discussed later

1st triad

2nd triad

P = animal M = man S = horse

P = animal M = man S = stone

P. P.

( I ) Some S is a P. (O) Some S is not a P.

Under the two interpretative triads, the differences between the ostensible conclusions become obvious:

10

See the box on page 31 for information about syllogisms.

11

Rose 1968, 37. The displayed passage is a very close paraphrase of Aristotle 1989, A4, 26a2 ff.

12

The ensuing discussion follows Rose 1968, 37 ff., and Lear 1980, chapter 4.

31

3.2 Aristotle’s Counterexample Method

1st triad

2nd triad

Each man is an animal No horse is a man

Each man is an animal No stone is a man

Each horse is an animal ( A) No horse is an animal (E) Some horse is not an animal (O)

No stone is an animal (E) Each stone is an animal ( A) Some stone is an animal ( I )

The following points emerge immediately from this analysis.

A SYLLOGISM is a type of argument from two categorical sentences (the premises) to a third (the conclusion). A categorical sentence must consist of a quantifier followed by a subject term t1 , a copula, possibly negated, and a predicate term t 2 : [ Each or no or some ] t1 [ is a or is not a ] t2 . It is called particular if the existential quantifier some is present; otherwise, it is general. If t1 should be a singular term, designating a unique individual, the quantifier may be omitted. A categorical sentence has one of four modes, depending on the quantifier and the negation: ( A) ( I)

Each t1 is a t2 . Some t1 is a t2 .

(E) (O)

No t1 is a t2 . Some t1 is not a t2 .

For such an argument to be called a syllogism, each two of the three sentences must have exactly one term in common; the middle term is common to the premises. The first and last terms are the predicate and subject of the conclusion. Letters M, P, and S, respectively, are conventionally used to denote them. A syllogism has one of four figures, depending on the arrangement of terms in the premises: Figure: Premises: Conclusion:

1

2

3

4

M P S M —— S P

P M S M —— S P

M P M S —— S P

P M MS —— S P

For further information, consult Prior 1967.

(1) Aristotle’s procedure only shows that for a figure 1 syllogism with AE premises there are two interpretive triads that make the premises true, and which make the ostensible A or E conclusion true, respectively. Neither of those conclusions can be derived legitimately from the premises. (2) It is not at all necessary to speak about interpretative triads in this way, let alone provide them, because simple scratch work with the variables M, P, S suffices to show that the conclusion S P is not legitimate in either the A or the E mode. (3) It is thus strange that Aristotle did not just use triads that make the premises true but those conclusions false: for example, “Each tree is a plant, but no man is a tree; ergo each man is a plant.” It is not easy to regard Aristotle as a precursor of the modern concept of independence.13

13

Łukasiewicz (1957, §20) suggested that Aristotle had used triads that make the premises true but those conclusions false, but apparently misunderstood the structure of Aristotle’s discussion.

32

3 Two Paths to Logical Consequence

But Aristotle was more ambitious. Instead of just rejecting specific modes of the conclusion, he wanted to reject the AE pair of premises altogether, with regard to all ostensible conclusions. Therefore, under his first triad, the conclusion A is true and thus compatible with the premises; that excludes the possibility of conclusions in modes E and O. Under his second triad, the conclusion E is true, and thus compatible with the premises; that excludes the conclusions in modes A and I. Therefore, the truth of premises A and E does not make any ostensible conclusion “necessary,” in Aristotle’s words. What he did show is that the AE pair of premises is syllogistically sterile.14 Is this a real use of the counterexample method? That can be held only indirectly: in the end, Aristotle rejected false conclusions, even if he did so through true ostensible conclusions. This in no way supports a view of Aristotle as the first precursor for the modern concept of independence. More clearly, Aristotle’s work should not be considered a clear and genuine precedent for the method that proves independence through the use of interpretations. 3.3 Independence of the Parallel Postulate This section presents the traditional postulates for Euclidean and hyperbolic plane geometry, and sketches the argument that the consistency of Euclidean geometry implies that of hyperbolic. It then highlights a deficiency in the discussions that usually follow such presentations in the familiar literature. Euclid’s five postulates are well known: 1. To draw a straight line from any point to any point. 2. To produce a finite straight line continuously in a straight line. 3. To describe a circle with any centre and distance. 4. That all right angles are equal to one another. 5. That, if a straight line [k] falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.15

The deductive strength of postulate 5, the parallel postulate, can be more easily understood if it is realized that the first four postulates imply this proposition: For each line g and each point O not on g there is at least one line h through O that does not intersect g.16

14

One should realize, however, that from AE premises the conclusions of modes A and E may be quite true, but merely ostensible—not really deduced from the premises.

15

Euclid [1908] 1956, volume I, 154–155. Euclid’s language is far different from that of the present book. The most visible differences can be alleviated by replacing the phrases to draw, to produce, and to describe by there exists. A thorough interpretation lies beyond the present scope, and is not needed for the analysis in this section. Interested readers may consult Mueller 1981 for that, and compare the modern approaches of Hilbert [1899] 1971 and J. Smith 2000, chapter 3.

16

This result, Euclid [1908] 1956, I.31, depends on the exterior angle inequality, I.16, and other early consequences of postulates 1–4.

33

3.3 Independence of the Parallel Postulate

O k

h l

hr g

Euclid’s Parallel Postulate:

Ê+Ê Ê < 180E

Proof sketch (see the figure above): drop a perpendicular l from O to g, then erect h z l through O; intersection of g and h would contradict the exterior angle inequality. Postulate 5 then implies that any other line hr through O must intersect g, because g and hr must make the interior angles less than two right angles on the one side or the other of l. Thus, Euclid’s postulate implies the following proposition: 5r. For each line g and each point O not on g there is at most one line h through O that does not intersect g.

On the other hand, any line hr through O that makes those interior angles less than two right angles must differ from h; statement 5r would imply that hr must intersect g. That is, proposition 5r implies postulate 5: they are equivalent. It is easier to analyze the deductive role of the simpler proposition 5r.17 Centuries ago the complexity18 of postulate 5 suggested the question whether it could be deduced from the first four postulates. After several others attempted that unsuccessfully, Nikolai I. Lobachevsky published during 1829–1840 some discussions of the consequences of replacing Euclid’s parallel postulate by the following proposition L, the negative of postulate 5r: L. For some line g and point O not on g there is more than one line h through O that does not intersect g.

Lobachevsky showed that L would entail a stronger proposition: for each line g and point O not on g there is more than one line h through O that does not intersect g. Moreover, no line between two such nonintersecting lines h could itself intersect g; thus, there must be infinitely many lines through O that do not intersect g. Lobachevsky used continuity to show the existence of two lines h and hr through O that do not intersect g, such that any line through O must intersect g if it lies outside the vertical angles formed by h and hr. These he called parallel to g; the other nonintersecting lines, he called hyperparallel. From these considerations he developed from postulates 1–4 and L an extensive theory, now called hyperbolic non-Euclidean geometry. That no contradictions appeared in it suggested that none should ever appear there. And should

17

Arguing that hr must intersect g on the appropriate side of line k in figure 1 is left to the reader. The form 5r of the parallel postulate is often named for the Scottish mathematician and geoscientist John Playfair.

18

Not just linguistic complexity: the figures suggested by postulates 1–4 are in some sense limited in extent, but postulate 5 claims the existence of points arbitrarily distant from its given points.

34

3 Two Paths to Logical Consequence

that be the case, there could be no valid derivation of Euclid’s postulate 5 from postulates 1–4, because its conclusion would contradict Lobachevsky’s proposition L.19 During 1868–1883, several ways of completing this argument were discovered, all using the general method for proving independence through interpretations. Within Euclidean geometry, mathematical structures were constructed, each of which satisfies Euclid’s postulates 1–4 and Lobachevsky’s proposition L, but not postulate 5r. This implied that 5r is independent from postulates 1–4, since there is at least one interpretation that makes those true but 5r false. At the same time, under the assumption that the underlying Euclidean geometry is consistent, these models verified the consistency of hyperbolic geometry, since its postulates are all true under those interpretations.20 For the postulates of plane hyperbolic geometry, the simplest model to describe was introduced by Felix Klein in 1871. It is constructed in the interior of a fixed circle K in the Euclidean plane. Notions mentioned in the postulates are interpreted as follows. Corresponding Features Euclidean Geometry point line order of points on a line motion

Klein’s Model point internal to K (not on the boundary) chord of K (without its boundary points) order of points on a chord transformation of the interior of K onto itself that maps chords onto chords

This list suffices, once it is realized that other notions mentioned in the postulates can be defined in terms of these and congruence, and congruence means related by a motion. Obviously, postulate 5r is false under this interpretation: the figure on the facing page shows a chord g and point O in the Klein model, with several chords through O that do not intersect g inside K. Verifying that the model satisfies the remaining postulates is possible but tedious, because it is necessary to check all those that Euclid assumed but did not state explicitly.21 The argument just presented shows that if Euclidean plane geometry is consistent, so is hyperbolic. It can be extended to show that if absolute plane geometry is consistent, 19

Lobachevsky [1829–1830] 1899. Carl Friedrich Gauss and Bolyai János are also credited with the same discovery, at about the same time. See Bonola [1906] 1955 for further information and an English translation of a related 1840 paper by Lobachevsky.

20

For further information, consult Bonola [1906] 1955, Stillwell 1996, Stump 2007, and RodríguezConsuegra 2020, bibliography, 104–107.

21

Klein 1996, §12. Other models may be regarded as somewhat harder to describe but less tedious to reason about. Verification of the remaining postulates is facilitated either by adopting a postulate system designed for that purpose, or by developing and using tools of projective geometry or complex analysis. Readers will find a fairly complete discussion in Aleksandrov et al. 1963, chapter 17, §4.2, §5. The use of motions in the postulate system employed there is reminiscent of Pieri 1900a, translated in chapter 8 of the present book. For alternative presentations, see Bonola [1906] 1955, §84–§91, and Greenberg 1980, chapter 7.

35

3.3 Independence of the Parallel Postulate

K

h

O hr

A

hyperparallels

g

In Klein’s Model h, hr Are Parallel to g so is hyperbolic. (Absolute geometry is the theory derived from all the postulates of Euclidean geometry except for the parallel axiom.) If absolute geometry is consistent, it has a model, and in that model either postulate 5r or its negation, which is equivalent to Lobachevsky’s postulate L, is true. The latter case implies the consistency of hyperbolic geometry directly. The former implies the consistency of Euclidean geometry, hence that of hyperbolic geometry by the earlier argument. The lessons drawn from this episode in the history of mathematics are often limited to a few general comments about relationships among consistency, relative consistency, independence, and logical consequence. It is not usually pointed out that proving the independence of a set S of postulates is the same as proving the negation of logical consequence; and thus, whoever can define the former, in terms of the existence of an interpretation that satisfies all postulates in S but one, should also be able to define logical consequence in terms of the concept of the totality of interpretations. This lack is exemplified by a passage from a respected historical survey by the noted mathematician Morris Kline: The fact that hyperbolic geometry is consistent implies that the Euclidean parallel axiom is independent of the other Euclidean axioms. If this were not the case, that is, if the Euclidean parallel axiom were derivable from the other axioms, it would also be a theorem of hyperbolic geometry for, aside from the parallel axiom, the other axioms of Euclidean geometry are the same as those of hyperbolic geometry. But this theorem would contradict the parallel axiom of hyperbolic geometry and hyperbolic geometry would be inconsistent.22

Thus, the generally accepted interpretation of this episode is limited to underlining the independence of the parallel axiom and the relative consistency of hyperbolic geometry, these indeed being the two basic objectives of this application of the method of interpretations. Of course, some sort of equivalence between independence and negation of logical consequence is present: if the former can be defined in general through the method of interpretations, then so should the latter. But for this to be possible, mathematics had to wait until a certain notion of formal system was available, and an abstract and general understanding of the concepts of interpretation and model. The next section shows that the Peano school met those requirements, more than thirty years before Alfred Tarski’s celebrated definition. 22

Kline 1972, 916. See also Courant and Robbins 1941, chapter 4, §9.

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3.4 Logical Consequence in a Model-Theoretic Context: The Peano School Some members of the Peano school, notably Giuseppe Peano himself, Mario Pieri, and Alessandro Padoa, developed a view of axiom systems as formal systems that can be regarded as a semantic, model-theoretic approach. In that approach, nominal definitions work as schemes that can be seen as independent of any interpretation. Under this conception these mathematicians arrived at what can be regarded as a full semantic conception of logical consequence. They should be acknowledged as having anticipated Alfred Tarski’s definition of that concept in his celebrated paper [1935] 2002. To develop this idea each of their roles will be discussed separately.23 3.4.1 Peano Addressing the question of definition by postulates, Giuseppe Peano suggested that the postulates of an axiomatic theory “determine or, if one wishes, define the primitive concepts, when no direct definition has been given.” Somewhat later, he wrote that they determine a concept N “that is obtained by abstraction from all these systems [that satisfy them] ... .” 24 In the same paragraph, Peano granted that a set of postulates could be satisfied by an infinity of different systems of entities. This vagueness must have highlighted the need to make those claims more precise. The question was, how do his famous postulates for arithmetic, for instance, define the primitive concepts natural number, zero, and successor, or the locution, N is the system that satisfies the postulates? The problem of nominal definitions under this rather model-theoretic viewpoint is that no possible definition—that is, none whose definiens is built up from the primitive concepts—could depend on any interpretation of those ideas.25 The same definition would remain valid under different interpretations that satisfy the postulates. The constructions actually implemented in the definitions would be mere schemes void of any intuitive content until some particular interpretation should be specified. That would be fine from a formal viewpoint. But for Peano the primitive concepts did have intuitive content, which we try to grasp through the postulates. What a nominal definition should actually do would be to point out familiar content in the definiens. For Peano the intuitive or empirical content of the primitive concepts was precisely the ultimate guarantee that when we build up a whole system we are not constructing mere void schemes to be fulfilled later by any system of entities whatsoever, but are actually handling real entities whose ontological status cannot be dissolved into formal schemes.26 23

For further information, see Borga 2005.

24

Peano 1894a, §44; Peano et al. 1895–1908, volume II (1898), §2, 2. The latter idea can also be regarded as a way to transform a definition by postulates into a nominal definition—see subsection 3.4.2. Peano’s junior colleague Cesare Burali-Forti introduced (1894a, §6) the idea of definition by postulates, then called definition of the third species or definition in itself, and used it ([1900] 1901, 295) to describe Peano’s arithmetic and geometric postulate systems. See Borga 1985, §1.7, for further information.

25

A nominal definition allows readers always to replace the definiendum (term being defined) by the definiens (defining phrase).

26

Peano 1894b, 75.

3.4.2 Logical Consequence in a Model-Theoretic Context: Pieri

37

In summary, for Peano the formal aspects of an axiom system seemed to be a clear disadvantage in trying to grasp the ultimate essence of the primitive ideas, which we acquire through a previous process of empirical induction. For him, the formal character of his axiom systems was somehow opposed to the epistemological character he wanted for their primitive ideas and propositions: those were to be entities and facts previously known. The ensuing subsections will show that this problem was later gradually avoided by Pieri and, mainly, by Alessandro Padoa. Even so, Peano was perfectly able to use the method of counterexamples, or interpretations or models, to prove the relative independence of certain axioms. Although he had used the method since at least 1889, only in 1894 did he give it an explicit theoretical account: It is possible to prove the independence of some postulates from others by means of examples. The examples ... are obtained by attributing to the undefined symbols ... completely arbitrary meanings; and if it is found that the fundamental symbols, through this new meaning, should satisfy a set of primitive propositions, but not all of them, it will be inferred that [the remaining ones] are not necessary consequences of those; that is to say, that the second set of propositions express properties ... which were not yet expressed in the first.27

The first implicit equivalence between independence and the negation of logical consequence appeared here. But Peano evidently lacked interest in following that development. This equivalence may also have merely followed from Peano’s usual way of understanding logical consequence from the positive side. As early as 1889 he wrote, The sign means one deduces. ... If propositions a, b contain indeterminate entities x, y, ... , that is, are conditions on these same entities, then a x, y, . . . b means, whatever x, y, ... may be, from proposition a one deduces proposition b.28

However, Peano never seemed interested in clarifying the semantic interpretation that could be involved in this manner of explaining logical implication.29 The ensuing subsections will show how Pieri and Padoa improved on this still obscure position. 3.4.2 Pieri Mario Pieri contributed three new ideas to the developing theory of logical consequence: • his method for transforming definitions by abstraction into nominal definitions; • his conception of axiom systems as mere hypothetical-deductive systems, which, for the first time, made a true model-theoretic approach possible; and

27

Peano 1894b, 62.

28

Peano [1889] 1973, Logicae notationes, II. The displayed translation is by the present authors.

29

Burali-Forti (1894a, 64) explained logical implication in terms of satisfaction and conditions (expressions containing a variable): the expression ax . x. bx should be read, “Whatever should be the x that satisfy the condition ax , they also satisfy the condition bx .”

38

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• his method for transforming definitions by postulates into nominal definitions, which arose from that conception. This subsection will show how Pieri’s view of definitions, nominal and by abstraction, simplified the features that his formulation of axiom systems would have to account for. The essential role of the latter in developing a definition of logical consequence was already discussed in section 3.1. Pieri’s specific contribution is elaborated here, followed by a discussion of his use of these methods to supplant reliance on any concept of definition by postulates. That made commonplace the emphasis on the class of all possible interpretations of a logical theory that is characteristic of today’s mathematical logic. The two published examples of Pieri’s method for transforming definitions by abstraction into nominal definitions were discussed in section 2.3. His method was strictly equivalent to Bertrand Russell’s famous 1903 definition of the cardinal number of a class, not presented implicitly merely by stipulating that two such cardinals are equal just when the classes are related bijectively, but explicitly by regarding the cardinal number of a class C as the class of all classes bijectively related to C.30 With that method Pieri intended to replace obscure abstractions by what he described as “true definitions.” Pieri’s second contribution, even more important, was his formal view of axiom systems. According to Pieri, an axiom system should consist of a set of primitive concepts without an a priori meaning, a set of postulates that are neither true nor false, and a set of theorems that can be derived from them using explicitly stated logical principles. The primitive concepts and postulates convey nothing substantive until an interpretation of the primitive terms is provided. Thus, Pieri presented for the first time a full scheme for the new approach to axiomatics, opposed to the old Euclidean view. No epistemological or ontological justification is needed for the primitive concepts of a such a system; they are merely implicitly defined by the postulates, which do not represent facts, but mere logical relations between the primitive terms. The intuitive empirical approach of Giuseppe Peano and Moritz Pasch was now completely transcended.31 From this viewpoint nominal definitions are no longer intuitive constructions that build up derivative concepts from simple ones, but mere abstract devices, notwithstanding the fact that developing the theory may require keeping in mind a particular interpretation of the primitive ideas. This led Pieri to generalizations of two notions that had previously appeared in geometry: • Joseph Gergonne’s principle of duality, according to which certain demonstrations in projective geometry remain valid for dual forms of theorems, which are obtained merely by interchanging certain primitive concepts

30

Russell 1903, chapter 11. Peano had suggested this explicit definition in 1901 (Peano et al. 1895–1908, vol. 3, §32), but rejected it for a reason that Russell would not accept. See also Rodríguez-Consuegra 1991, §3.5.2, §5.2.

31

Peano followed this formal approach (1889, 4), but nevertheless insisted on a tie to experience (1894b, 75). Pieri broke that tie. For a brief comparison of their viewpoints with that of Pasch, see Smith 2010, §3–§5. For more information on Peano and Pasch, see M. Segre 1994 and Schlimm 2010, respectively.

3.4.2 Logical Consequence in a Model-Theoretic Context: Pieri

39

• Felix Klein’s method of classifying geometries according to their properties that are invariant under certain transformation groups. Pieri thought that the principle of duality is just a particular form of a more general principle: the principle of plurality, according to which the primitive concepts of a science are indeterminate except as specified by the postulates, so that they may be replaced by other sets of undefined terms without affecting the proofs of the theorems. This general principle concerns not only geometry, but all deductive theories.32 Pieri wrote that Klein’s classification method can provide a way to select the concepts that should be taken as primitive. Since a science is usually characterized by a group of transformations that do not change the essential properties under consideration, its primitive concepts should be invariant under that group, but not under any wider group.33 Unfortunately, Pieri did not work out those ideas, nor did he apply them to his particular investigations, but they clearly show the completely modern way in which he saw the more abstract features of axiomatic theories. Because of his resorting to Klein’s idea, Pieri can be regarded as the first mathematician to pioneer its use, later widespread, in describing formal sciences. A general analysis suggests that Pieri’s formal view of primitive concepts and nominal definitions even led him to consider whether some indetermination in the primitive concepts would ultimately and unavoidably cloud the meaning of the primitive propositions of any deductive science. At that point Pieri was evidently not yet able to draw the ultimate implications of his own model-theoretic viewpoint as clearly as Alessandro Padoa would soon do.34 For Pieri the plurality affecting primitive concepts does not amount to complete indetermination: although the mind cannot grasp the complete domain of these ideas, the postulates always determine whether or not any given object belongs to it. Thus, apparently, Pieri did not see the possibility of nonstandard interpretations of his systems, even though he had been in possession of the concept of categoricity since at least 1906.35 Again, unfortunately, Pieri did not apply this concept to any of his axiomatizations. Pieri’s abstract conception of definitions also made a true model-theoretic description of implicit definitions possible. According to Pieri, the postulates somehow “define” not only the primitive concepts of a theory, but also a certain global concept. For instance, Pieri wrote in 1898 that his postulates for projective geometry define a global concept, the class of ordinary projective spaces—the interpretations that satisfy the postulates.36 32

Pieri 1898c, §11, translated in section 6.11 of the present book. See also the concluding paragraphs of section 5.7. Pieri highlighted the important role of plurality in algebraic geometry, where the interpretations of the primitive concepts of projective geometry are fixed “arbitrarily, case by case, so that the same algebraic variety can be identified in turn with the most heterogeneous figures and even with geometric entities beyond the ordinary ...” ([1915] 1991, 327).

33

Pieri [1900] 1901, §5, translated in section 4.5.

34

Padoa’s work is discussed in subsection 3.4.3.

35

Pieri 1906d, 46. See also Veblen 1904, 346.

36

Pieri 1898c, §11.

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3 Two Paths to Logical Consequence

Thus, he not only introduced the approach used later, in 1903, by Bertrand Russell, but showed a mastery of the abstract axiomatic method that clearly can be regarded as superior to that evinced by David Hilbert’s famous [1899] 1971 book. Gottlob Frege severely criticized the latter for its implicit-definition approach, and later for its formulation of postulates in terms of defined—not primitive—concepts.37 The model-theoretic approach also enabled Pieri to apply metatheoretic devices systematically, starting at least with his 1898c axiomatization of projective geometry. In the appendix of that paper,38 he constructed models to demonstrate the sequential independence of most of his postulates, and presented a model to demonstrate their consistency. Pieri completed this work about two years before publication of Hilbert’s book.39 For Pieri the only way to prove consistency was to exhibit a model, as his later publications show; thus, he explicitly supported Frege’s view that the consistency of a concept means essentially the existence of some object falling under it.40 It is curious that Hilbert then believed that mere syntactic consistency ensures mathematical existence, even though he still could prove consistency only by exhibiting a model.41 Pieri’s definition of logical consequence appeared in 1898c in this full model-theoretic context. He defined independence in the usual way: ... propositions P, Q, R, ... will be called “independent of each other” if it should happen that no one [of them] should be a consequence of those remaining: that is, for each one it should prove possible to find some x, y, z, ... that make it false, but that do satisfy the others.42

In the previous paragraph Pieri insisted that this conception of independence was based on a clear semantic conception of the negation of logical consequence in terms of sets of objects or models—that is, in clear semantic terms: Given several conditional propositions P(x, y, z, ...), Q(x, y, z, ...), R(x, y, z, ...), and so on about variable entities x, y, z, ... no doubt can fall on the meaning of the assertions “from P and Q one cannot deduce R,” [and] “R is not a consequence of P and Q,” ... Both of these express no more than this specific proposition: “there exist some x, y, z, ... for which P and Q are true, but R is not true.”

37

Russell 1903, part VI; Hilbert [1899] 1971. Concerning Frege’s criticism, consult Dummett 1991b and the literature cited there. In 1942 Paul Bernays, the editor of Hilbert’s later editions, suggested a simple response—in effect, to consider Hilbert’s postulates from Pieri’s point of view. Steven R. Givant (1999, 50) reported Alfred Tarski’s comparison, during the 1920s, of Pieri’s and Hilbert’s formulations of postulates.

38

Translated in section 6.13 of the present book.

39

Hilbert had lectured on related material during the preceding years. The editors of Hilbert 2004 place the beginning of Hilbert’s metamathematical work on geometry in 1895. The present authors have found no connection between Hilbert’s lectures and Pieri’s geometrical work before 1898.

40

See section 2.9.

41

Frege [1884] 1953, §94–§95. Hilbert [1900] 2000, 414; [1904] 1970, 134. In the latter publication Hilbert began to address the problem of finding some other kind of consistency proof for arithmetic.

42

The four quotations in this paragraph are from Pieri 1898c, §13, §13, §1, and §9, translated in sections 6.13, 6.13, 6.1, and 6.9, respectively.

3.4.3 Logical Consequence in a Model-Theoretic Context: Padoa

41

Again, Pieri based this semantic conception of the negation of logical consequence on a previously introduced positive version of the same concept: ... the statement “from P(a, b, c, ...) one deduces Q(a, b, c, ...)” ... is to be considered the same as “whatever a, b, c, ... should be, if P(a, b, c, ...) is true of them, then Q(a, b, c, ...) will also be true ... .”

Or, in terms even more clear: If P(x, y, z, ...), Q(x, y, z, ...) are propositions about variable objects x, y, z, ... (see §1), the notation “P(x, y, z, ...) x, y Q(x, y, z, ...)” means “x, y, whatever they may be, just by satisfying P(x, y, z, ...) will also have to satisfy Q(x, y, z, ...).” [This] can also be read, “from P follows Q, with respect to x, y, z, ... .”

Pieri did not write here explicitly about models or interpretations, nor, in particular, about “all models” or “every interpretation,” thus making the quantifiers range over every domain. It is possible that he was simply thinking of proofs in the usual truth-functional (conditional) way, as Peano himself had done before. But it seems very likely that he was so familiar with the model-theoretic approach that in writing this way he was expressing something quite similar to the model-theoretic definition of logical consequence.43 At any rate, as shown in the next subsection, his colleague Padoa did reach the full definition only three years later. 3.4.3 Padoa In the first part of the paper he read before the 1900 Paris International Congress of Philosophy, Alessandro Padoa established a general approach to his formal and modeltheoretic conception of axiom systems.44 It included a famous method for proving that a set of undefined concepts of a system is independent with respect to the set of its postulates. Padoa’s method is an extension of Giuseppe Peano’s method for proving independence of the postulates, as follows. To prove the independence of an undefined concept C it suffices to find two interpretations of the whole set of undefined concepts, both of which satisfy all the postulates, and which agree except for their interpretations of C. If C were definable in terms of the other undefined concepts, then the two interpretations of C would have to agree. Padoa’s introduction of that method was only possible thanks to his general view of axiom systems as mere formal, uninterpreted schemes. Thus, to devise this method, Padoa needed to regard deductive theories in a new way. Ideas and facts commonly distinguished earlier as undefined concepts and unproved propositions should now be regarded as mere empirical and psychological aids for constructing the theory. They must remain independent of the theory itself, considered as a formal system. Thus, Padoa interpreted the postulates of a formal theory only as conditions to be imposed on the undefined concepts, and ignored any requirement of 43

Three further examples of Pieri’s model-theoretic reasoning are apparent in his 1901b and 1905c axiomatizations of the geometry of lines and complex projective geometry. See subsection 9.2.7 of the present book, pages 377–378, and footnotes on pages 402 and 409 of 9.2.9.

44

Padoa [1900] 1901.

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simplicity for them. Under this general view, for the first time he described nominal definitions as mere relations between new symbols and those previously introduced, which can establish meanings for the new symbols as soon as an interpretation of the system as a whole is chosen. That same year, Padoa presented both aspects of his completely model-theoretic conception of theories—that is, the already quite formal conception, and the consequent character of nominal definitions—in a much clearer way in a little-known series of lectures at the University of Rome.45 A clearer conception of the relationship between the usual method for proving independence of the postulates, and the notion of logical consequence can also be found there, even a literally “Tarskian” definition of logical consequence. These three points are illustrated by passages in the following paragraphs. Regarding the formal view of axiom systems, Padoa wrote, We shall consider deductive theories in their formal aspect; that is, we shall imagine that, at the beginning of the theory, the symbols that represent the ideas assumed as primitive should be devoid of meaning.46

Concerning nominal definitions, he explained, In deductive theories regarded under the formal aspect, definitions that are called “nominal” by logicians do not single out the meaning of the derivative ideas. Each definition expresses only a relationship between the meaning of the defined idea and the interpretation of the primitive ideas, in a manner that distinguishes the former implicitly once the latter are established.47

Thus, Padoa made quite explicit the tendency to regard nominal definitions as mere abstract relations, which is only implicit in the writing of Peano and Pieri. This was possible thanks to Padoa’s explicit effort to distinguish the logical, the semantic, and the epistemological elements involved in the transition from the old Euclidean axiomatics to the new, formal, modern, framework. In his 1900 Paris paper, Padoa wrote that to prove that a given member of a set of propositions is independent of the others it suffices to exhibit an interpretation of the undefined concepts under which the given proposition is false but all the others true. In 45

Padoa 1900. The cover of the book of notes for Padoa’s lectures is shown on page 45; the inscription means “compliments of the author.” The present authors have not been able to study notes from Padoa’s similar lecture series in 1898 in Brussels and 1899 in Pavia. Consult Luciano 2009 for further information.

46

Padoa 1900, 17. This seems to be the first published use of the expression formal in that context by the Peano school. In the original, the quotation reads, Noi consideremo le teorie deduttive sotto l’aspetto formale; immagineremo cioè che, all’inizio della teoria, I simboli che rappresentano le idee assunte quali primitive, sieno sprovvisti di significato.

47

Padoa 1900, 17. In the original, the quotation reads, Nelle teorie deduttive considerate sotto l’aspetto formale, nemmeno le Df che I logici dicono “nominali” individuano il significato delle )))))))))) idee derivate. Ogni Df esprime soltanto una relazione ))))))) fra il )))))))))))))))))))))) significato della idea definita e )))))))))))))))))))))))))))) l’interpretazione delle idee primitive; per modo che, fissata, questa, quello risulta implicitamente individuato. “Df” is an abbreviation for “Definizione.”

3.4.3 Logical Consequence in a Model-Theoretic Context: Padoa

43

this he followed the tradition of Peano and Pieri. However, Padoa also made explicit the underlying reason for this method, which the others in the Peano school left implicit: if an interpretation is found that satisfies all those propositions but one, “then this proposition is not a logical consequence of the other propositions; that is, it is not possible to deduce the proposition in question from the other ... propositions.” 48 Padoa’s 1900 Rome lectures contain a full definition along these lines of what is today called logical consequence: If a primitive proposition, say , were deducible from the preceding [ones], that would mean that each interpretation of the primitive ideas that should satisfy the primitive propositions preceding proposition  must also satisfy proposition .49

This appears strictly equivalent to Alfred Tarski’s famous definition, according to which a given proposition  is a logical consequence of a set  of propositions if and only if every model that satisfies  also satisfies .50 And only Padoa was able to make it completely explicit that the familiar method for proving independence was based on an underlying concept of logical consequence in this sense.51 That only Padoa was able to do this seems to be due to his complete mastery of the formal, model-theoretic approach. Before him, every member of the school knew about the usual methods of proving consistency and independence. But Padoa, in addition, usually considered axiom systems from the model-theoretic viewpoint, and was especially interested in analyzing the abstract structure of deductive theories as a goal that is interesting in itself. He alone was able to draw the corresponding conclusions. Padoa’s argument relating independence and logical consequence can be presented explicitly and concisely as follows. Denote by  a set of statements and by M an interpretation, and use the abbreviations “sat” and “ind” for satisfies and is independent of. 48

Padoa [1900] 1901, 323. The quotation is from Jean Van Heijenoort’s translation, page 122. In the original, it reads Alors, cette P n’est pas une conséquence logique des autres P; c’est-à-dire: il n’est pas possible de déduire la P considérée des autres P non-démontrées.

49

Padoa 1900,18. This precedes the introduction of the method for proving independence. In the original, the quotation reads Se una Pp, ad es. , fosse deducibili dalle precedenti, ciò significherebbe che ogni interpretazione delle idee primitive, la quale verifichi le Pp precedenti la P , deve pure verificare la P . “P”, “es.”, and “P p” are abbreviations for “Proposizione,” “essemplo,” and “Proposizione primitiva.”

50

It can be objected that Tarski’s definition of logical consequence was completely clear and acceptable only because it was preceded by a full analysis of the concept of truth for formal languages, in terms of satisfaction (Tarski [1936] 1983), whereas the work of Padoa and Pieri work lacks such a foundation. Two questions come immediately to mind. (1) Should these earlier definitions of logical consequence, which seem perfectly canonical, be denied credit just because they were not based on a prior definition of truth? (2) Might not Tarski have already known his celebrated definition, but resisted making it public until his full definition of truth was available?

51

It does not matter that Padoa said nothing about the difference between derivability and logical consequence—that is, the difference between syntactic and semantic deducibility, which was recognized much later.

44

3 Two Paths to Logical Consequence

The usual definition of independence in model-theoretic terms, assuming the consistency of the whole system, is  ind 

]

Replace  ind  by ¬ (

 )

› M ( M sat  &

¬ (

]

¬ (M

sat ) ) .

 ) and › M by

¬ œM ¬

¬ œM ¬( M

sat  &

¬ (M

to obtain

sat ) ) .

Boolean logic then yields this definition of the logical consequence relation  : (  )

] œM ( M sat 

| M sat  ) .

This is precisely what Padoa, and much later Tarski, actually did. Padoa included similar definitions in two later publications. The first, which appeared in 1912, is phrased in terms of conditions, or propositions with at least one variable and thus dependent on an interpretation: Two conditions involving the same variable x being given ... it can happen that every time the first is satisfied by an interpretation of x, the second is also found to be satisfied by the same interpretation of x. In this case, we say that the first condition implies the second.52

The second appeared in 1930 and is phrased in terms of solutions, or interpretations that satisfy a set of propositions: It is customary to say that a condition is a consequence of other conditions when each solution of the latter is a solution of the former. Alternatively, the former is said to be independent of the latter just when at least one solution of the latter is not a solution of the former.53

Thus, once again, the relation between logical consequence and independence is crystal clear. The concluding section 10.5 of this book suggests this question: why did these definitions remain so little known for so many years?

52

Padoa 1912, 44. In the original, the quotation reads Deux conditions par rapport à une même variable x étant données ... , il peut arriver que, toutes les fois que la première est vérifiée par une interprétation de x, la seconde aussi se trouve verifiée par la méme interprétation de x. En ce cas, nous dirons que la première condition implique la seconde.

53

Padoa 1930, 20. In the original, the quotation reads Usa dire che una condizione è conseguenza di altre condizioni quando ogni soluzioni di queste è una soluzione di quella. Altrimenti, si dice che quella è indipendente da queste: cioè, quando almeno una soluzione di queste non è soluzione di quella. Padoa 1930 was published in the encyclopedia Berzolari et al. 1930–1953, which had been planned decades earlier. In May 1910, Luigi Berzolari had invited Pieri to contribute that article; two years later, Pieri took his final medical leave. Padoa may have started drafting the article not long after that. See also Luciano 2012a, 39.

Summary of Lectures on Algebra and Geometry as Deductive Theories Given by Prof. Alessandro Padoa at the Royal University of Rome 1900

Italian Pavilion

Universal Exposition Paris 14 April to 12 November 1900

4 Pieri’s 1900 Paris Paper This chapter contains an English translation of Mario Pieri’s [1900] 1901 paper Geometry Envisioned as a Purely Logical System. It was originally presented at the first International Congress of Philosophy, held in Paris in August 1900 in conjunction with the Universal International Exposition. It was one of the first papers that presented the modern axiomatic method in detail to a general audience. The following introductory paragraphs describe the Exposition, the Congress, and the immediate setting of the presentation of the paper. Several quotations let participants tell their own stories. Some remarks then relate Pieri’s paper to other publications, and describe some conventions used in the translation, which starts with Pieri’s own introduction. The 1900 Paris Exposition was one of a continuing series of fairs organized to display the benefits of industrialization, encourage appreciation of arts and cultures in a worldwide context, and promote nationalism. The first, held in 1851 at the Crystal Palace in London, was followed every four years or so by increasingly grand expositions in major cities of the world. The 1900 Exposition was the fifth one held in Paris. Planning started in 1892. In his 1895 proposal to the French government, Commissaire Général Alfred Picard argued, The Exposition of 1900 ... will give a new impetus to industry and to commerce, ... constitute a vast lobby for studies and education of the public, ... attest once again the material growth of the nation, and, what is still better, add to its glory and to its radiance abroad. ... The Republic will have closed the nineteenth century with dignity and will have attested to its desire to remain in the vanguard of civilization.1

The Exposition took place during 14 April–12 November 1900 in central Paris along the Seine from the Pont d’Iena past the Pont d’Alma to the Pont Alexandre III. The total attendance exceeded that of any previous exposition: more than fifty million, greater than the entire population of France in 1900. Nevertheless, that was only two-thirds of the anticipated figure, and many regarded the exposition as a financial disaster. The unexpectedly low attendance was due in part to the prohibitive cost of tickets. According to historian Richard D. Mandell, “nothing the Exposition might have produced could reach the heights that its enthusiasts had hoped for.” Many French citizens believed that it

1

Mandell 1967, chapter 1. Picard 1902, volume I, 7, 194.

© Springer Science+Business Media, LLC, part of Springer Nature 2021 E. A. C. Marchisotto et al., The Legacy of Mario Pieri in Foundations and Philosophy of Mathematics, https://doi.org/10.1007/978-0-8176-4823-7_4

47

48

4 Pieri’s 1900 Paris Paper

showed the embarrassing superiority of the accomplishments of France’s rivals Germany, Great Britain, and the United States. Mandell concluded, To nineteenth-century intellectuals raised on faith in science, reason, and progress, it seemed that the most complete and expensive demonstration to celebrate science, reason, and progress produced an impression of human uselessness, finiteness, and debility.2

The grandeur of the Exposition was overwhelming: the American scholar Henry Adams, undeterred by the price, “haunted it, aching to absorb knowledge and helpless to find it.” Adams wrote about his reaction to the dynamo, or electrical generator, one of the most familiar of the exhibits: To Adams the dynamo became a symbol of infinity. As he grew accustomed to the great gallery of machines, he began to feel the forty-foot dynamos as a moral force, much as the early Christians felt the Cross.3

One hundred twenty-seven international conferences were held in conjunction with the Exposition, fulfilling its planned role as a lobby for studies and education. Examples include conferences on copyrights, fisheries, postal service, public health, and teaching. Most took place in the pure white Palace of Congresses and of Social Economy on the north bank of the Seine by the Pont d’Alma.4 The International Congress of Philosophy was organized by the editorial committee of the Revue de métaphysique et de morale, a leading philosophical journal founded in 1893. Its editor wrote, The scholars and the philosophers of the entire world are asked to unite here upon the common ground of reason and of free thought. ... The occasion of the Universal Exposition was deemed a favorable one for carrying out the project of uniting these thinkers and of soliciting them to labor together without distinction of nationality or boundary in the common cause of truth and morality which appeals solely to reason ... and to give thus a sort of reality to that ideal society which is called by the beautiful name of humanity.5

The Congress took place during 1–5 August 1900. A report containing brief summaries of the papers presented there was published later that year: it constituted the entire issue 5 of volume 8 (1900) of the Revue. The four-volume proceedings 6 of the Congress, with the full texts of the papers, was published over the next three years. This was the first of a series of international philosophy congresses that recur every five years or so.

2

Mandell 1967, 111–117.

3

Adams [1918] 1931, chapter 25: “The dynamo and the Virgin,” 379–380. See the bottom illustration on the facing page.

4

Mandell 1967, 68. The top illustration on the facing page shows the view toward the northeast over the buildings along the Seine between the Pont d’Alma at left and the Pont Alexandre III (De Olivares et al. 1900, antepenultimate page of last issue, number 93). The middle illustration shows the Palace.

5

Léon 1900, 621.

6

International Congress of Philosophy 1900 and 1900–1903.

1900 Universal Exposition, Paris

Palace of Congresses and of Social Economy

Dynamos in the Gallery of Machines

50

4 Pieri’s 1900 Paris Paper

The French logician Louis Couturat organized Section III of the Congress, devoted to the logic and history of science. A year in advance, he wrote to Pieri, I invite you to the International Congress of Philosophy that will be held in Paris during 2–7 August 1900 (immediately before the Congress of Mathematicians). Giuseppe Peano has volunteered to take part in the Committee on Patronage and I have asked him to invite all his collaborators. ... Questions of the logic of sciences and of the history of sciences have a major place indeed in the program of the Congress, as you will be able to judge from the extract attached here. ... Ernst Schröder (also a member of the Committee) designated you as the most capable of notably treating question IV or V. ... We hope that you would like to let the Congress benefit from your insights about questions that you have especially studied, and that you would honor us with a report ... The attached list is purely suggestive and not limiting. We shall accept memoirs in Italian and publish them in French translation, even if the authors cannot assist in person at the Congress. However, I hope that you will be able to come here, and that I shall also have the honor of making your acquaintance.

The topics in the attached list that Couturat suggested to Pieri were IV. V.

Postulates of geometry, their origin and value. Methods of geometry: analytic geometry, projective geometry, geometric calculus (quaternions).

Two months before the Congress, Couturat wrote to Pieri that he had received the manuscript of Pieri’s paper. Twenty minutes would be allotted at the Congress for an oral summary. If Pieri could not attend, he should send a summary to be read there. In fact, Pieri did not go to Paris. He had been appointed professor at the University of Catania in Sicily, and had just moved there from Turin.7 Couturat was unique in the French mathematical community for his serious interest in logic.8 At the time of the Congress, he was studying unpublished manuscripts of Leibniz. In 1894 Peano and his school had launched the Formulaire de Mathématiques, which embraced Leibniz’s ideas about a universal language to be used to create a dictionary of all knowledge. Couturat’s correspondence with Peano reveals their common vision of implementing Leibniz’s program. Couturat probably invited Pieri to the Congress simply as a member of Peano’s school. But he deferred to Schröder to recommend specific topics for Pieri.9 The suggestion that Pieri speak about postulates of geometry may have been motivated by Schröder’s desire for the Congress to highlight Peano’s symbolic logic, which Pieri used for his axiomatizations. Peano had acknowledged in his book on the geometric calculus that its logical basis had in turn been derived from Schröder’s 1877 pamphlet. Schröder may have also suggested geometric calculus as a topic because Pieri, in his research on algebraic geometry, used the enumerative calculus that Hermann Schubert had developed, modeling techniques from Schröder’s logical calculus.10

7

Couturat [1899–1901] 1997. M&S 2007, 32–34.

8

For information about Couturat see the biographical sketch in M&S 2007, 74; Eisele 1971; Beaufront n.d.; and Luciano and Roero 2005.

9

Grattan-Guinness 1997, §16.7, 670. Couturat [1898] 2005, 7. Peano 1894a, §1.

10

Peano [1888] 2000, xiv. Pieri 1893d, 534. Schröder 1877. Schubert [1879] 1979. For comparison of Schröder’s and Peano’s logical systems, see Jourdain 1914 and Peckhaus 1991.

Introduction

51

Ernst Schröder was born in 1841 in Mannheim, in the Grand Duchy of Baden. His father was a scientist and secondary-school teacher and administrator there; his mother, the daughter of a pastor. The eldest of their four children, Ernst was rather solitary, but very active in individual sports. He never married. Schröder earned the doctorate at the University of Heidelberg in 1862 with a thesis on stellated polygons supervised by Otto Hesse. He then studied physics at Königsberg with Franz Neumann, earned a teaching credential, and completed habilitation in Zürich in 1865. After several teaching positions and military service, Schröder became professor at the Technical University in Karlsruhe in 1876. From childhood he pursued the study of languages, and in later life, of botany. He was regarded as particularly even-tempered and gentle. Schröder’s main research lay in closely related subjects in logic and algebra. Starting with an 1873 book for teachers and students, he aimed to design logic as a calculating discipline, extending work of Charles S. Peirce. Publication of Schröder’s comprehensive three-volume Vorlesungen über die Algebra der Logik began in 1890, but was completed only after his premature death in 1902. It introduced many now standard features of logic and became a standard resource for that subject. (For further information, see Lüroth 1903 and Dipert 1991.)

Section III of the Congress met in five sessions during 2–5 August, with the French mathematician Jules Tannery presiding. It featured twenty-four papers, fifteen presented in person. Those participants included several noted scholars: the mathematics historian Moritz Cantor, logician Ernst Schröder, mathematicians Henri Poincaré and Giuseppe Peano, and two members of Peano’s school: Alessandro Padoa and Giuseppe Vailati. Padoa’s introduction, entitled Logical Introduction to Any Deductive Theory, became famous: it featured his now standard method for proving definitional independence of concepts described by a deductive theory.11 The youthful philosopher Bertrand Russell presented a paper on space and time. Summaries were read for the nine other papers. Two of those, by Pieri and by Cesare Burali-Forti, were presented by Couturat.12 Fifty years later Russell described the impact of the Congress on his intellectual life: The congress was a turning point in my intellectual life, because I there met Peano ... In discussions ... I observed that he was always more precise than anyone else, and that he invariably got the better of any argument ... I decided that this must be owing to his mathematical logic ... By the end of August I had become completely familiar with all the work of his school ... The time was one of intellectual intoxication ... Suddenly, in the space of a few weeks, I discovered what appeared to be definitive answers to the problems which had baffled me for years ... I was introducing a new mathematical technique, by which regions formerly abandoned to the vaguenesses of philosophers were conquered for the precision of exact formulae. Intellectually, the month of September 1900 was the highest point of my life.13

11

Padoa [1900] 1901, Vailati [1900] 1901.

12

Couturat [1900] 1899, International Congress of Philosophy 1900. The presenters came from six countries: Austria, France, Germany, Great Britain, Italy, and Russia.

13

Russell [1900] 1901; [1951] 1967, 232–233. Grattan-Guinness 2000, 290–291, contains an account of Russell’s epiphany based on materials in the Bertrand Russell Archives. In an 8 February 1901 letter to Pieri, Couturat reported that Russell held Pieri’s works “in high esteem and would advise anyone who wants to learn the logical principles of projective geometry to read them” (Couturat [1899–1901] 1997, 45). See also Desmet and Weber 2010, 158.

Ernst Schröder in the 1890s

Mario Pieri around 1900

Louis Couturat around 1900

Bertrand Russell in 1907

Introduction

53

In 1960 the mathematician Hans Freudenthal, deploring much nineteenth-century philosophical writing about geometry, vividly described the scene that Russell encountered in Paris: Somebody ... has defined a philosopher as a man who at midnight in a dark room looks for a black cat that isn’t there. With an eye on those philosophers of the nineties I would like to add: Meanwhile the black cat is sitting in the broad light of the adjoining room. Then the door opens—I mean the door of the Paris Philosophical Congress of 1900. In the field of philosophy of sciences the Italian phalanx was supreme: Peano, Burali-Forti, Padoa, Pieri absolutely dominated the discussion. For Russell, who read a paper that was philosophical in the worst sense, Paris was the road to Damascus.14

Among those attending were Alfred N. Whitehead and Samuel Dickstein. Whitehead was as impressed as Russell by the results achieved by the Peano school. Whitehead based his influential 1906 work The Axioms of Projective Geometry on Pieri’s 1898c Geometry of Position memoir.15 Dickstein was instrumental in keeping mathematics alive in Poland during Russian oppression. He had recently published a Polish translation of Peano’s book on geometric calculus.16 Later, Dickstein would spur publication of Pieri’s work in Polish and help foster the remarkable development of mathematics in Poland after 1915. In turn, Pieri’s work inspired that of Alfred Tarski in Poland and the United States.17 Within a year, Couturat published in French two detailed reports about the papers related to mathematics that had been presented at the Congress, and the American mathematician Edgar O. Lovett published a similar one in English. A summary of Pieri’s paper appeared in each of them.18 As mentioned earlier, Pieri had submitted the full text of his Paris paper to Couturat in May 1900. In a 30 May 1900 letter, Couturat suggested that the length of Pieri’s manuscript might cause a problem, but that evidently never materialized. Early in 1901, Couturat asked Pieri to proofread the French translation of the manuscript. That appeared the following year in the Congress proceedings.19 The content of Pieri’s Paris paper is discussed further in chapter 2 and subsection 9.4.1 of the present book. By the time it was presented at the Congress, Pieri had been investigating the foundations of geometry for at least a decade. He had started by translating into Italian the pioneering projective-geometry treatise Geometrie der Lage by G. K. C. von Staudt, and had continued with a series of four papers of his own work on foundations 14

Freudenthal 1962, 616. Burali-Forti and Pieri were present only in the summaries read by Couturat. According to the New Testament, Paul the Apostle was converted to Christianity while traveling from Jerusalem to Damascus (Acts 9:3–9).

15

Grattan-Guinness 2002, 432. Whitehead [1906] 1971; Pieri 1898c is translated and analyzed in chapter 6 of the present book.

16

Peano [1888] 1897 is a Polish translation of Peano [1888] 2000.

17

Pieri 1915. See M&S 2007, §5.2; and McFarlands and Smith 2014, §9.2.

18

Couturat [1900] 1899, 404–405; International Congress of Philosophy 1900, 593–594; Lovett 1900–1901, 171–172.

19

Couturat [1899–1901] 1997. Pieri [1900] 1901. Pieri [1900] 1901 is the same as the version in Pieri’s collected works on foundations of mathematics (1980, 235–272) except for pagination. Unfortunately, Pieri’s Italian manuscript has disappeared.

54

4 Pieri’s 1900 Paris Paper

of that subject. These were subsumed and refined in Pieri’s 1898c Geometry of Position memoir. He then turned to the foundations of Euclidean geometry, with his 1900a work, Point and Motion.20 In their introductions Pieri described his approach to the axiomatic method in some detail. In his Paris paper, translated in the present chapter, he referred to those papers briefly, and greatly expanded the discussion of his fundamental idea, the hypothetical-deductive system. The present authors hope that their translation of Pieri’s Paris paper will not only reveal Pieri’s understanding of geometry as an abstract deductive theory, but also confirm that he pioneered this approach to deductive theories independently of and simultaneously with the formalistic program that David Hilbert unveiled in his [1899] 1971 Grundlagen der Geometrie. In sections 4.2, 4.6, and 4.7, Pieri cited particular features of Euclid’s Elements without specifying the edition. Similar passages in Pieri 1900a, written at roughly the same time (see the translation in chapter 8 of the present book), indicate that he was referring to the Euclid 1885 textbook edited by Enrico Betti and Francesco Brioschi. Since that is now difficult to access, and the enumerations of its features are not standard, Pieri’s citations have been altered here to refer to Thomas L. Heath’s [1908] 1956 edition of Euclid. A citation of the form I.2 will refer to book I, proposition 2. The translation in this chapter is meant to be as faithful as possible to the published French version, which is the voice of Louis Couturat speaking in 1900 for Mario Pieri. For translation conventions, see the general discussion in the preface, pages xii–xiv. Editorial comments are inserted in square brackets [like these], usually as footnotes, to document changes in mathematical terms, to note or suggest corrections for occasional errors in the original, and to explain a few passages that may seem puzzling.

20

Staudt 1889. Pieri 1895a, 1896a–c, 1898c, and 1900a: the first four of these memoirs are summarized in subsections 9.2.1 and 9.2.2 of the present book, and the last two are translated in chapters 6 and 8.

Proceedings of the 1900 International Congress of Philosophy Section III Title Page

Pieri’s Paris Paper First Page

ON GEOMETRY ENVISAGED AS A PURELY LOGICAL SYSTEM By MARIO PIERI, Professor at the University of Catania

The intellectual movement concerning the guiding principles of mathematics in general and especially in geometry, with regard to both logical analysis of the premises (definitions, axioms, and so on) and to the critique of the methods that inform the fundamental doctrines, has attained today a development so remarkable that students as well as scholars are not always in a position to follow its principal phases, nor to absorb and coordinate its details. On the contrary, reconstructive works are comparatively rather rare, in which one might use each new idea to reform or modify a part of the edifice of the science in conformity with the growing speculative requirements. Thus, on the occasion of this grand intellectual recapitulation of the century, our International Congress of Philosophy, it will not seem inappropriate that I should permit myself to submit to the judgment of learned scholars, even nonmathematical ones, the conclusions of some studies that I have completed during recent years21 with a view to setting elementary geometry on foundations that are accepted, and, I will say, almost required by the general evolution of the sciences.22 The title of the present communication reveals rather clearly my point of view concerning the end to which this evolution is directed and toward which geometry appears to be tending in its intensive development. I maintain with assurance that this science, in its loftiest as well as in its most modest parts, will continue affirming and consolidating itself more and more as the study of a certain type of logical relations; freeing itself little by little from the ties that still bind it, although rather weakly, to intuition, and consequently assuming the form and qualities of a purely deductive and abstract ideal science, such as Arithmetic. I do not deny that such an opinion would find a great many opponents. But, if I do not deceive myself, it agrees so closely with the truth and finds so many confirmations in the present and past circumstances of our scientific thinking (as I shall try to prove), that it is permissible to maintain it openly without shadow of paradox. Whatever one should think about this subject, it is good that the question be posed and argued, because it is, so to speak, in the air that we breathe; and the present reflections could serve, at the very least, to better explain and characterize the trend of which I shall speak.

21

I allude particularly to the following monographs (not very accessible because of the language in which they are written and the nature of the publications in which they are found): Pieri 1898c [translated in chapter 6 of the present book], 1900a [translated in chapter 8], 1898b.

22

Among the numerous authors that one could cite in support of this opinion, I shall mention here particularly Edmond Goblot (1898, introduction) and Édouard Le Roy (1899–1900, [1900] 1901).

58

4 Pieri’s 1900 Paris Paper

§I The movement that I have in mind concerns the ideal coordination of results, and touches only indirectly on their essence. As usually happens for all intensive progress,23 this trend is the fruit of an internal upheaval within established results, in consequence of which these appear from a new and unexpected perspective, in some new combinations and new relationships, from which arise criteria and schemes for classification, and so on; whereas extensive progress, generally speaking, only yields new truths that enlarge the domain of this or that science and open new branches of knowledge without altering its structure and fundamental characteristics. Let us consider arithmetic, for example. It is growing in extent, even in the most recent times, and expanding greatly in the part called number theory (Adrien-Marie LEGENDRE, Carl Friedrich GAUSS, Gustav LEJEUNE DIRICHLET, ...). And in depth, through the fact that, being originally a computational art created in response to the needs of commerce and exchange (of which there is still a trace in some of its old terms), it is becoming little by little the ideal science of everything enumerable, and in our time receives a deductive organization that one can call perfect. The degree of abstraction that it has attained is so remarkable that many (Richard DEDEKIND, Giuseppe PEANO, ...) now consider it with good reason as a part of Logic: that is to say, as a branch of that universal science which embraces in its basic notions the broadest set of objective interpretations. But if the general idea of number underwent such a transformation (as well as those of the unit, of succession, of finite or infinite class, and so on ... , that march along with it and which are the primary material of arithmetic), then it would be a great wonder if nothing similar should have been happening to the general idea of point, which, with some others that are subordinate to it, is the substratum for every geometric concept. Already according to many indications, it is permissible to conclude that, in effect, the notion of point is also undergoing—it, too—a fully comparable evolution, in the sense of continually increasing abstraction. Long ago, through the artifice of coordinates, Cartesian geometry introduced in place of geometric reasoning and constructions the methodical use of analytic operations and transformations, which by routes distant from intuition leads to consideration of geometrical questions in a much more general context and from a point of view more elevated than customary; not to mention so many other effects that led the science of extension to receive in its bosom through the intermediary of coordinates some unusual, if not foreign, interpretations of geometric concepts, and consequently to extend its conclusions beyond the field that was reserved for it by tradition. But a perhaps yet more powerful influence was that which the multiple speculations about the geometries called non-Euclidean yielded: speculations well over a century old—Girolamo SACCHERI ([1733] 1920), Johann LAMBERT ([1766] 1895)—that had no 23

[The two occurrences of italics in this long sentence and the first two in the next paragraph do not appear in the original.]

4.1 Section I

59

effective impact on thinking until more recent times (GAUSS, Nikolai I. LOBACHEVSKY, BOLYAI János, Bernhard RIEMANN, Arthur CAYLEY, Eugenio BELTRAMI). These would confirm and certify the ideal coexistence of several geometric systems, each of which is in itself free from contradictions and absurdities, and which are therefore all true, subjectively speaking, and all equally possible. Already in 1792, GAUSS had conceived the principle of a geometry in which Euclid’s postulate is not valid; and it was in trying in vain to prove this postulate that the limits of experimental geometry were transcended for the first time. From this moment one started to speak of abstract geometric entities, without worrying about their greater or lesser agreement with the reality of physical space. Also, from then on, geometry was no longer just a science of real objects: it started to assume as well the quality of a doctrine of the possible. And the Kantian notion of space as an a priori form of the human spirit was ruined, in order to make room for the contingency of the properties that one attributes to the extension of bodies.24 Then, Julius PLÜCKER, undertaking to consider a variable geometric entity, which might be completely arbitrary, as playing the role of a generating element of a space (that is to say, as a point), conferred directly on this notion of point a value and a sense much more general than usual, without the intermediary of any representation. From then until today, in discussing curves or surfaces, for example, one alludes to certain classes for which one does not specify the elements, provided that they can be determined in a unique way and through a continuous law by means of numerical values attributed to one or two variable parameters. And this general meaning could be developed and extended further and further with the notion of higher spaces (with however many dimensions one wishes): notions that absolutely force the partial abandonment of the ordinary physical concepts concerning the class of points. One could expect another effect of the trends and of the methods that distinguish the geometric calculus of Hermann GRASSMANN and the enumerative geometry of Hermann SCHUBERT: the one and the other characterized by the use of certain algorithms that perform the role of the classical proof process algebraically, although operating in that way directly on the geometric entities. In this manner, one is brought naturally and as though by logical necessity to admit, with Professor Moritz PASCH, that

24

The notion of geometrical truths is singularly clear and distinct in [an 1835 work of ] Franz A. TAURINUS, as it appears, for example, in the following passage: In effect, there is for Geometry an internal and an external truth. The first requires that Geometry form a completely deductive system closed in itself, free from logical contradiction, and with no question about its applicability to the phenomena of the external world. ... However, if Geometry should not be merely an idle result of a productive imagination, but also of practical importance, then one asks whether an external truth should also be attributed to Geometry, a distinction that is no longer purely mathematical. [Translated from Stäckel 1899, 420–421; the italics are Pieri’s.]

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if geometry is to be really deductive, the inference process must in fact be entirely independent of the meaning of the geometric concepts, just as it must be independent of the figures; only those relations between the geometric concepts may come into consideration, which were laid down in the [primitive] propositions and definitions used.25

This amounts to saying, in short, that a primitive entity of any deductive system whatever (as point would be, in geometry) must admit interpretations that are arbitrary within certain limits assigned by the primitive propositions—in such a way that the content of the words or symbols that one employs to describe any primitive subject should be determined uniquely by the primitive propositions that bear on this subject, and moreover that anyone should be free to attach to these words or to these symbols a meaning ad libitum, provided that it is compatible with the general attributes imposed on this entity by the primitive propositions. Such is geometry as a hypothetical doctrine: one could describe it, in short, as the science of everything that is figurable (that is to say, representable by points and figures), in the same manner that arithmetic studies everything that can be interpreted as a number.26 But, if I am not mistaken, once having arrived at this height of ideal representation, nothing prevents us from conceiving the whole of geometry as a purely speculative and abstract system, where the objects are pure creations of our intellect and the postulates, simple acts of our volition (mental choices, subjective a priori truths, truths by definition), without failing to appreciate that they often have their deepest root in some external fact. Beyond that, the ones and the others will be arbitrary, at least insofar as we do not arrange them according to a pre-established purpose that might be supposed to guide our thinking.27

§ II Some find that it is sufficient to refer to the primary origins of a fact to classify it, and they infer immediately from that the essentially objective character of geometry. But 25

PASCH 1882b §12, 98. [Translated from Pasch; the italics are Pieri’s. Pieri omitted the word “used” (benutzten). The word postulates here corresponds to Pieri’s postulats and Pasch’s Sätze. But Pasch considered both Kernsätze and Lehrsätze: postulates and theorems.]

26

The necessity of subsuming the immense variety of phenomena in the confining framework of a small number of logically manageable intellectual schemes is manifested to an eminent degree in the most advanced parts of Physics, where one has a great interest in being able sometimes to evoke a whole class of natural laws solely with the magic of language: for example, with a certain system of differential equations. And this is good above all if one expects that such a system sooner or later can play the role of a bona fide rational definition of the ideas that it represents.

27

Thus, for example, it will seem appropriate that the geometry of position should be governed entirely by certain fundamental laws (such as the principles of projection and duality), which are thus said to inform and impart its character; and that by means of suitable interpretations (that is to say, by some nominal definitions) one should be able to discern easily in the outcomes the ensemble of facts that compose ordinary elementary geometry and the twin metric geometries.

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more often than not, the origin is nothing more than a moment in relation to continuous evolution. In order to be clear about the present nature and most proximate goals of any institution, one must preferably consider the history of its sequential development, modifications, and adaptations. At the onset of any intellectual formulation of the facts, the eminently subjective role of cognition sometimes finds itself obscured and as though eclipsed by the imbalance between the multiplicity and variety of objects and the inability of our mind to master and discern them. But if the science, in the last analysis, is nothing but a schematic representation of the world, having as goal the dominion of reason over the facts, born and reared for the rational needs of mankind—in short, created for mankind—this subjective character will have to be manifested sooner or later in each branch of our knowledge, beginning with the most developed parts, and to increase more and more with the progress of studies. Just as there was a time when one did not conceive of negative numbers other than as debts, likewise today through a similar lack of abstraction many have the habit of linking to the idea of point the ordinary spatial representation. One does not deny the heuristic importance, nor any less the didactic value, of such a concrete interpretation of geometric entities: but that only amounts to the advantage of any representation that is intuitive and fully conforming to an ordinary abstraction. To maintain that the postulates of geometry, for example, are just rigorous forms of the intuitive concept of physical space (which just impose a certain character of stability, with a rational cachet, on facts of spatial intuition), is to my mind to give excessive weight to an objective representation by making it a condition sine qua non for the very existence of geometry, whereas this could very well continue to subsist without that. Today, geometry can exist independently of any special interpretation of its primitive concepts, just like arithmetic. Observe how currency (unit of account) was for a long time and still remains among the concrete interpretations of the entity number, one of the most familiar and intuitive. Anyone who would today regard this palpable representation, or another similar one, as the basis of the science of numbers would be imitating, it seems to me, those who conceive geometry just as a science of facts inherent in the extension of physical objects. This science of extension, among the diverse aspects of the abstract concept of geometry, will assuredly be the one that maintains the greatest practical interest, but it will not constitute geometry, any more than accounting constitutes arithmetic. From this one can only conclude that the old controversy between realists and nominalists must be decided theoretically in favor of the latter. But I would find it more suitable to adopt a new term to designate the more doctrinal position and the new point of view, much more general and abstract, that one would adopt when one regards geometry as an exclusively deductive and purely rational science. The geometry that is taught today, although it might be closer to this degree of intellectual elaboration than any other field of science (apart from logic and arithmetic), is nonetheless distinguished from them in several respects. Consequences are not always derived there from premises by pure logic: arguments from evidence (or, as one says now, from intuition) hide behind the best

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tailored syllogisms,28 or even are invoked openly. The primitive notions there are more numerous than is necessary, etc. These problems can be avoided in part; but it is clear that, given adolescents’ aptitudes,29 the requirements of school, and good didactic practices, and so on, it will never be permitted in this field to make very heavy use of abstractions, nor to turn the minds of young people away from the phenomena of extension. Thus, in the schools, elementary geometry can perhaps never escape the character of a physics of extension that it has possessed since the most distant antiquity.

§ III A hypothetical-deductive system, as I intend it, must not only distinguish organically the a priori or primitive propositions from the derived or deduced propositions (that is to say, the definitions and postulates from the theorems), but also, and in the same measure, classify the notions that these propositions are about, by discerning among them the basic ideas, primitive or undecomposed, and by keeping them well separated from those that are reproductions or formal derivations (or can be obtained in that way), and which are in effect composed of the primitives, combined with each other and with the categories of logic. The two distinctions are in truth strongly analogous. The second is no less ancient than the other one, and does not seem to have less value, but nevertheless, before our time, mathematicians have not accorded it equal importance in practice.30 In fact, one seeks most often to reduce the postulates and the axioms to a minimum, without trying in general to define all the notions that figure in the deduction from a minimum [number of] fundamental ideas. In this way, the advantage gained on the one hand can 28

As one sees, for example, in EUCLID I.1 and I.22.

29

[Pieri’s phrase was qualités de l’esprit.]

30

“The first concepts, with which a science begins, whatever it should be, must be clear and reduced to the smallest number. Only then can they form a solid and adequate foundation for the edifice of the science” (Lobachevsky [1829–1830] 1898, §1, 2). If by definition one intends a pure and simple imposition of a name for things already known or acquired by the science, the primitive ideas will be the undefined concepts. But one still understands “definition” in a larger sense: this is just as one says, for example, that the primitive concepts are not defined except by the postulates. In effect, the latter attribute to these concepts certain properties that suffice to characterize them with a view to the deductive goals that one proposes for oneself. To avoid all ambiguity, the term nominal definition will be used when one wants to exclude the real definition, or definition of a thing. With regard to definitions: a entity that is defined in itself, that is to say as it is used with others already known or not yet defined in certain given relations and which verifies them, is specified as the root of certain logical equations, just as the unknown x in a system of two simultaneous equations f (x, y) = 0, (x, y) = 0 in x, y. But here as in algebra it will sometimes be possible by appropriately chosen methods to solve the system of logical equations with respect to the entities not yet defined that figure in them. These will then furnish a nominal definition in the proper sense of the word. If, for example, this is a matter of a single entity being given in relation to some others already known, or regarded as such, this resolution can be effected in many cases by known logical algorithms. Thus, it seems more exact and more appropriate to distinguish between “explicit definitions” and “implicit definitions”: a distinction that embraces all the types of definitions. Compare the memoirs Peano [1900] 1901 and Burali-Forti [1900] 1901 in these proceedings.

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quite often be lost on the other, because of the number and quality of the primitive ideas on which the system rests.31 Thus, we have two types of primitive ideas, which must be regarded as given a priori, to remain in the domain of abstractions, but which must find an image more or less exact and in accordance, if not perfect agreement, with every sort of object and phenomenon to which one would apply the system in whole or in part: namely, the primitive concepts, or categories, on the one hand and the primitive propositions, or postulates, on the other. These correspond in some way to the ideas usually suggested by the interrogatives what and how. The acquired ideas will be those that are defined explicitly by means of the primitive ideas, or that one accepts as known (similar to the quantities that one adjoins to a given field 32 in the theory of equations); and the derived propositions will be those that one proves deductively by means of the primitive propositions. Moreover, one does not exclude the incorporation of categories and postulates belonging to other deductive systems, such as logic and arithmetic, that one adopts by way of general premises necessary for any system. As far as possible, it will be appropriate that the primitive ideas be irreducible among themselves in the sense that it not be possible to define any one of them explicitly in terms of the others. And similarly: that the postulates be independent from each other, in such a way that it should not be possible to derive any one of them from the others. But one does not hide the fact that those are conditions that nearly approach ideal perfection, but which are never fulfilled except in very rare cases.33 A precise distinction between the primitive or simple ideas and the derived or composite ideas offers this benefit among others: to be able more often than not to decide the logical equivalence of two systems with a small number of comparisons. In effect it suffices (and it is necessary for equivalence) that it be possible to define the primitive concepts of each system explicitly in terms of those of the other, and moreover that the primitive propositions of the one be consequences of those of the other. But a more considerable advantage is the possibility to disregard the meaning of the categories, so that it is permitted in the discussion to give them any special interpretation in arguing and operating symbolically 31

Thus, although the possibility of developing the whole system of ordinary geometry from only three primitive ideas (see §IV) should be beyond doubt (by virtue of earlier examples), one nevertheless still sees proposed for this same role of “fundamental concepts of geometry” notions such as those of rigid body, part of a body, space, part of a space, occupying a space, time, repose, and motion. (See Killing 1893–1898.)

32

[Pieri’s term was domaine de rationalité.]

33

Given several conditional propositions P(x, y, z, ...), Q(x, y, z, ...), R(x, y, z, ...), and so on, about variable entities x, y, z, ... , one can have no doubt about the validity of the assertions “from P and Q one cannot deduce R,” and “R is not a consequence of P and Q.” These two phrases express nothing other than that this particular proposition, “there exist some x, y, z, ... for which P and Q are true, but R is not true.” Thus P, Q, R, ... will be independent from each other, if for each of them, one can find some x, y, z, ... that do not satisfy it, while the others do satisfy it. Consequently, to establish the absolute mutual independence of a given system of n postulates, it is necessary to produce n interpretations of the primitive concepts, each one of which leaves unsatisfied a single one of these n principles. On the contrary, to prove that these n primitive propositions are all compatible (that is to say, noncontradictory), a single concrete example suffices, which satisfies them all at once (consequently, a single example also suffices to conclude that the contrary proposition, or negation, of each of them is independent of the rest). Compare Alessandro PADOA’s memoir [1901] 1900 [presented at this Congress].

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on expressions of variable content but still interconnected by known relations, and consequently to embrace in a single general and abstract doctrine several types of concrete and particular objects ( just as the algebraic solution of a quantitative problem always embraces several cases, that are numerically different, and moreover that differ from each other in the type of the data). Here one will have to recognize the utility of a good ideographic algorithm, as an instrument appropriate for guiding and disciplining thought, for excluding ambiguities, tacit assumptions, mental restrictions, insinuations, and other defects nearly inseparable from ordinary language, whether spoken or written, and so harmful in speculative research. Thus, great value must be attached to the use of formal procedures that belong to algebraic logic; and this not just for the effectiveness of the symbols in themselves, but also because of the intellectual habits that the methods and doctrines of this science prove capable of engendering and developing, and moreover for their suggestive power, which often leads to observations and investigations where no other paths go. Mathematical logic is like a microscope, appropriate for observing the smallest differences in ideas, differences that the defects of ordinary language more often than not render imperceptible in the absence of any instrument to enlarge them. In my opinion, whoever scorns the advantages of such an instrument, particularly in this area of studies (where errors often result from ambiguities and from misunderstandings of details insignificant in appearance) deliberately rejects the most powerful tool that one has today for supporting and directing our mind in intellectual operations that demand great precision. The researches cited at the beginning of this memoir (I propose to summarize later those that are about elementary geometry) have been thought out with and written first in the pasigraphy constructed by Professor Giuseppe PEANO.34

§ IV Among the studies devoted to organizing the science of figures as a hypothetical-deductive system, the work of Professor Moritz PASCH (see section I), inspired by the intention to establish geometry upon axioms that agree easily and immediately with reality, marks the debut of a new order of ideas concerning the foundation of this science, aiming at what it might lend to the clarity, to the deductive perfection, and in this almost crystalline form that arithmetic offers us. PASCH’s primitive notions are four in number: point, the relation of being situated between two points (that is to say, segment, as introduced in the geometry of Euclid and of Lobachevsky), plane surface or finite portion of a plane, and the relation of congruence between two figures. PASCH’s principles have since been analyzed with the instrument of algebraic logic and reproduced, with notable modifications

34

See especially Peano 1897a.

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65

in structure and form, by Giuseppe PEANO.35 He has reduced the categories to just three, namely point, segment, and motion: plane is defined formally in terms of segment and point, and the author has preferred the concept of motion, as a special transformation from points to points, to that of congruent figures, from which it is hardly distinguished logically—but the former is perhaps more manageable in deductions. In turn, it has been possible recently to reduce these primitive ideas to just two: encompassing in the notions of point and of motion (this being intended as a representation of points by points, and devoid of all mechanical significance) the concepts introduced in elementary geometry, even including segment. By means of these two ideas, with the aid of the more general logical categories of individual, class, membership, inclusion, representation, negation, and some others (familiar and, one can say, necessary for all human discourse), one is in a position to give a nominal definition for each of the other concepts and thus obtain a geometric system where the author is pleased to enjoy easier progress and greater deductive simplicity in comparison with preceding systems. The act of using the simplest motions, such as translations, rotations, symmetries,36 and so on, and their products, more broadly than usual in definitions and arguments, confers on the system as a whole a certain ease of manner that is not devoid of clarity and effectiveness.37 On the other hand, one knows how often Euclid and the authors of later treatises relied on constructions of congruent figures, and thus masked the use of motions, which they did not succeed in excluding; and how the systematic adoption on a grand scale of transformations of space and of their groups distinguishes and governs, one can say, all modern geometry. A suitable way to exclude the concept of motion from the foundations of elementary geometry (by making it, in short, a derived idea, as some would like) might be to define congruence of figures in terms of the more general notion of homography, which could itself serve as a primitive idea,38 or which could be generated, according to the method of G. K. C. von STAUDT, for example, as a product of other notions of projective geometry, which can be reduced to just two.39 And for the rest40 one could follow the analytic procedures of Arthur Cayley and Felix Klein that under the name of projective-metric determinations reproduce in analytic terms all the properties of motions, whether Euclidean or non-Euclidean, considered as representations of points by points. But everyone sees how a reform of such grand consequence, that would lead directly to preceding instruction in ordinary elementary geometry by that of the pure geometry of position, is hardly recommended. On the other hand, I do not find any other method that,

35

In the memoirs Peano 1889 and Peano 1894b.

36

[Pieri used the singular: symétrie.]

37

Some remarkable traces of an attempt in this direction are found already, for example, in LEIBNIZ’s Characteristica geometrica ([1679] 1971). See also M. CANTOR 1880–1898, volume 3, 31–35, and COUTURAT [1901] 1985, chapter 9.

38

For example, in the manner indicated in the last of the memoirs cited at the beginning [Pieri 1898b].

39

As in the first of the works just mentioned [Pieri 1898c, translated in chapter 6 of the present book].

40

[of the notions of elementary Geometry]

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while satisfying the requirements of rigorous deduction, might succeed in proscribing motion from the elements of geometry. If one does not want to erase from the principles of geometry all traces of motion, it remains to reduce its role to smaller proportions. Restricting the primitive sphere of motion is a task feasible and assured of success; in fact, there is no lack of examples of progress already realized in this direction. In the Elementi di geometria by Giuseppe Veronese,41 the relation of congruence, insofar as it plays the role of a primitive idea (that is to say, defined exclusively by the postulates), is not regarded as a transformation from space to itself, nor of an arbitrarily given figure to another, but only as a relation between two segments, that is to say, in the last analysis, as a relation between four points. That persuaded me one day to try to restrict still further the aspect of motion not explicitly defined, by reducing it to a relation between only three points. I no longer have any doubt about the possibility of composing all of elementary geometry with only these two elements: (1) point; (2) a certain relation between three points a, b, c, that can be rendered by the phrases “b and c are equidistant from a” or else “the pair ac is congruent with the pair ab,” and so on. Indeed, on the basis of this relation one can explicitly define the most general motion of a figure given arbitrarily. But the excessive complexity to which the greater part of such a system finally leads (given the numerous logical requirements that one wants to satisfy) sparks the desire, however, for new studies in this direction. However, we cannot conceal from ourselves that the idea of motion is employed as a primitive more than is necessary, when one introduces it as a transformation of the entire class of points onto itself.

§V The obstacle just mentioned has suggested to me some reflections about other conditions that one usually requires or that one could require of the fundamental concepts and postulates of a deductive system. I believe that it would be out of place to expect luminous evidence in the premises, especially since one does not attach to the primitive ideas a meaning concrete and well circumscribed in the field of reality. Indeed, since one deliberately does not seek any sensible images for the abstract notions under discussion, since one talks about the symbols (for a point or a motion, for example) as about the numbers of Arithmetic or about algebraic formulas, speaking of intuitive evidence makes no sense. (This is why the term postulate is preferable to axiom.) In this it will not be forbidden, for example, for one who is treating elementary geometry, to have always before the eyes a real interpretation of the symbols, and to adapt the premises to the images; as long as one does not assume grounds of suitability, dependent on the applications that one has in mind, for a fundamental condition of geometry. How could one take account of intuitive

41

Veronese 1897.

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67

evidence for the postulates that distinguish the so-called non-Euclidean geometries, after having found evident [Euclid’s] axiom XII about parallels, or vice versa? Moreover, this requirement of intuitive clarity would not suggest itself to anyone who might consider that the primitive concepts (except for the categories of logic, necessary for all discourse and consequently impossible to describe with words) can be given, if one wishes, by means of implicit definitions or logical descriptions that play in some way the role of postulates, or else as roots of a certain system of simultaneous logical equations (see footnote 30 in section III). For example, one gives the name point or motion respectively to each specification of classes  and  that enjoy the following properties: (here, the series of premises about point or motion, designated respectively by  and ). This logical device, which is certainly not advisable when it is a matter of presenting several primitive concepts at the same time, is commonly employed, and then with advantage, in cases where it is possible to present one primitive concept separately, independently of the others, by means of statements condensed in a small number of propositions. For example, the concept of curve (open, without nodes, continuous, ...) can be introduced for the needs of analysis situs without effort, and without leaving the domain of the greatest abstraction, in the form of “class of elements, that admits two continuous orders, one inverse to the other, without end points,” where the notions of order, continuity, and so on, might be understood by means of ideas established previously, and can be specified in turn in a similar way.42 Such a description in effect includes a system of postulates: but since these, in the form of definitions, sufficiently manifest their capacity as conditional propositions about the primitive ideas (to wit, their naturally arbitrary character, and so on) it happens that no one asks whether they are evident in themselves or not. The postulates, like all conditional propositions, are neither true nor false: they express only some conditions that can sometimes be verified and sometimes not. Thus, the equation (x + y)2 = x2 + 2 x y + y2 is true if x, y are real numbers, and false if it refers to quaternions (attaching to each hypothesis the familiar meaning of the symbols +, ×, and so on). Be that as it may, provided one is reasoning about things logically determined (it matters little in what way), one runs hardly any risk of failure through too much abstraction or generality of concepts. Fortunately, every fear in this regard is without foundation, as so many examples show. In choosing certain notions that one would take as primitive in preference to others, one could use the following criterion: each field of science is ordinarily characterized by a largest group of transformations that cannot change the properties that this science studies. As far as possible, one agrees that the primitive concepts should also be invariant with respect to this fundamental group, without being so for any larger group. A fault of 42

Enriques 1898, 224–225.

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analysis situs, as it is constituted today, is to presuppose the Euclidean notion of length, which is not at all invariant with respect to the group of continuous transformations. It is just the same in G. K. C. von STAUDT ’s geometry of position, for the notions of semiline, semi-cone, and so on, which are not invariant with respect to homographic transformations. At the same time, I believe it is also preferable to exclude from the primitives of the elements of geometry at least those of curve,43 surface, and solid in general, which are invariant with respect to the transformations of a group much larger than that of elementary geometry (and which should rather belong to analysis situs); and as well, for the same reason, the notions of segment and of line,44 which are invariants for the group of affinities, which includes the principal group, or that of similarities. Some others also, and for various reasons,45 propose to avoid as much as possible in elementary geometry the generic notions of space, curve, surface, and so on, that are in no way necessary to its development. Just as in projective geometry one defines in succession curves and surfaces of the second order, of the third, and so on (and often even plane) without recourse to the more elevated and complex notions of algebraic curves and surfaces (of order n) much less to the concepts of curve and surface in general, likewise in elementary geometry one can introduce line, plane, sphere, circle, segment, and so on, little by little as needed, by means of appropriate constructions founded on some concepts best suited to this science. These criteria are without doubt incomplete and not absolute, but nevertheless a little less vague and indeterminate than the usual reasons of simplicity so often invoked to justify anything one wishes.

§ VI Let us describe briefly how one could construct a system of elementary geometry starting from the notions of point and motion. One will postulate (1) that « point» and « motion» are classes, or general ideas;46 (2) that there exists at least one point; (3) that if p is a point, there always exists some point different from p. Deductive use of these three principles is rather rare. Many believe that (1) and (2) are premises more logical than geometric, and with regard to (3), instead of stating it in the postulates, they prefer to introduce it each time in conditional form in the hypotheses of the few theorems that depend on it. It is very important to understand with certainty the sense of the words “equal or different points.” If p is a point, we will call “equal to p” or “coincident with p” each point that belongs to every figure that 43

[Pieri’s word for curve was ligne.]

44

[Pieri’s phrase was “de segment et de droite.”]

45

See Peano 1894b, 53.

46

[For this translation’s use of guillemets («þ»), and Pieri’s class-membership conventions, see a box in the preface, page xiii. (This footnote will be cited several times in this chapter.)]

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contains p. And by “figure” we will intend any class or multiplicity of points whatsoever. Thus defined, the equality of points is a relative equality, which moreover subsumes absolute equality or identity of concepts, in such a way that two points equal according to the definition just given, that is, with respect to the class of all points, will still be equal with respect to any logical attribute or characteristic whatsoever (considering that each property enjoyed by one of the two points will always be shared by a class of points, all of which have it in common). Notice that the term “figure” is not understood by everyone, even in the sense just defined. If we have to consider later some systems of lines, of planes, of spheres, and so on, not as classes of classes of points, but as simple classes, we will have to distinguish them by another name, instead of simply being able to call them figures. Saying that one figure  “is included or situated” in another [figure]  is equivalent to affirming that each point of  belongs to  as well. If moreover each point of  is a point of , only then will the two figures be called equal to each other or coincident. (The habit of calling two congruent or superposable figures “equal,” without further explanation, must be proscribed, in my opinion, and there is no lack of examples of ambiguity born, so to speak, from suggestions of this term.) By the first of these deductive premises, the name « point » also has the meaning of “class of [all] points” or of “space.” 46 This latter term can thus easily be avoided; and that will not be surprising if one considers that it does not occur in the language of APOLLO47 NIUS and ARCHIMEDES. Next, one can state that (4) Every motion is an injective 48 transformation from points to points; that is to say, assigns to each point a point, and to each pair of distinct points a pair of points that are also distinct. The class of motions thus is included in the general category that supports the nouns function, representation, transformation, and so on. (5) For any motion  there always exists another motion (that will be called the “inverse of ,” or  –1 ) that assigns to each point x a point that the motion  transforms into x. Principle (4) implies that there cannot exist two motions not equal to each other but nevertheless both capable of mapping each such point x to a point that  transforms into x. Thus, given premises (4) and (5), each motion will be a reversible or reciprocal transformation from points to points (a bijective correspondence of space with itself)— an expression that includes both premises, and can thus also take their place. One will recall in this regard that, following the rules of the deductive method, it seems appropriate to avoid as much as possible that a postulate could easily be decomposed into several statements, distinct and of less weight, and that it should have greater strength 47

Peano 1894b, 52.

48

[Pieri’s adjective was isomorphe; his word for bijective was reciproque, or biunivoque.]

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than what is required by its deductive role. In that way the number of postulates can increase noticeably, whereas the total of the arbitrary conditions that they impose on the primitive ideas remains constant (even if it does not diminish). Moreover, this number can decrease merely by substituting for certain premises a statement equivalent to their logical product. But if an increase results from the tendency to resolve statements into elementary or distinct parts, one could not regret it, particularly if one thinks that this is the most appropriate way to gradually eliminate superfluous assumptions.49 (6) Two motions  and  performed successively, the second on the result of the first, have the same effect as a single motion, called their composition or product. It is not asserted that such a product be commutative, but from principles (4) and (6) it follows that a succession of three or more motions is again equivalent to one motion, and that this composite operation will be associative, like ordinary multiplication. Principles (4), (5), (6), and (12) establish50 that the motions constitute what is called in modern parlance “a transitive group of transformations” in the most general sense. But the particular characteristics of this group, which suffice to distinguish it from every other analogous class of transformations from points to points, will be completely specified only by the full set of postulates. Further, from principles (5) and (6) it follows that if  and  are motions, the operations –1 , –1  are motions, each of which leaves all points fixed. A transformation that assigns each individual of a class to itself is usually called an identical transformation; and it is clear that two such transformations applied to the same class are equivalent. From this [follows] the theorem that “if there exists a motion, then every identical transformation of the class « point»46 is likewise a motion.” It seems appropriate to call such a motion improper (but this usage is not recognized elsewhere), and effective or proper those motions for which one can always find at least one point that should be transformed into another, different one. (7) For each pair of distinct points there exists at least one proper motion that leaves both of them fixed. One can visualize this proposition via the perceptible fact that a body can always be moved when one fixes any two of its points. And the motion of rigid bodies will serve as a concrete image, intuitive and fully conforming to the abstract idea of motion, provided that one ignores time and focuses attention exclusively on two states of the moving body (the initial and final positions) in making the distinction between proper and improper motions. From principle (7) one can conclude immediately the existence of a proper motion, in view of the preceding (2) and (3). (8) Given that a, b, c are distinct points, if there exists a proper motion that maps each one to itself, any motion that leaves each of a and b fixed will also leave c fixed. This is a principle with great deductive power, which consequently states a rather restrictive condition on the classes «point» and «motion».46 It permits us to construct the notion of “line” and to recognize some of its most remarkable properties. Indeed, we can call 49

It is already known that one need not regard as impossible even a Geometry without any apparent postulate.

50

[(12) is in §VII, section 4.7.]

4.6 Section VI

71

three points a, b, c collinear if there exists a proper motion that represents each one by itself; and “join of a with b,” or ab, the class of all points x such that the three points a, b, x should be collinear. Then from the premises already established it follows that if a and b are noncoincident points, the join of a with b is not distinguishable from the locus of all points that remain fixed with respect to any motion whatever that maps a to a and b to b; that there exists at least one point outside the line ab; that through a and b there passes no more than one line; and so on. The definition just given for collinearity of points or for line (which stems in substance from LEIBNIZ 51 ) is not appropriate for the geometry of hyperspace, nor is principle (8). To obtain the higher spaces, one must therefore renounce this postulate and that definition, just as one renounces the postulate of parallels and the ordinary definition (and also one omits the axiom that two lines cannot enclose a space) to pass to the geometry of Nikolai I. Lobachevsky or Bernhard Riemann.52 By the plane of three noncollinear points a, b, c one can understand the figure abc consisting of all the points in lines53 that join point a with the various points of bc, or point b with points of ca, or point c with points of ab. This definition (which hardly differs from the one for projective plane attributed to RIEMANN) is stated as a theorem by Moritz PASCH.54 It is equally appropriate for each of the geometries called hyperbolic, parabolic, and elliptic (of the second species); but the last is excluded by principle (8).55 From this definition, combined with other premises including principle (9) it follows immediately that two planes that have three noncollinear points in common [must] coincide; that a line that has two points in a plane lies entirely in it; and so on. (9) If a, b, c are noncollinear points and d, a point of the line bc other than b, the plane abd will lie entirely in the plane abc. Given two points a and b, the term “sphere through b around a,” or “sphere through b with center a,” abbreviated with the notation ba, for example, designates the class of all points to which b is mapped by some motion that leaves a fixed. Each of the general ideas that are introduced little by little is proved to be an invariant with respect to motion: for example, each motion that maps a to ar and b to br transforms the sphere ba into the sphere brar , and so on. And with regard to spheres one can already

51

Leibniz [1679] 1971, §14 (10 August 1670), 145.

52

Whoever would like a geometry of four dimensions, for example, without deviating too far from this one, could begin by substituting for the definitions of collinearity and line some analogous definitions of coplanarity and plane, and for principles (7) and (8) two analogous propositions about three or four points respectively. But here is not the place to explain more fully.

53

[Pieri’s description was “... figure occupée indistinctement par toutes les droites ... .” The present authors inserted the name abc in this sentence.]

54

Pasch 1882b, Lehrsatz 12, 25–26. [For the connection to Riemann see section 5.6, page 121.]

55

[Pieri was probably alluding to the elliptic plane as described in Bonola [1906] 1955, §74–§75, and excluding the spherical plane.]

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4 Pieri’s 1900 Paris Paper

prove, for example, the property stated in EUCLID III.11 and III.12.56 It is unnecessary to say how one can introduce the [concept of] circle by means of a sphere and a plane that contains its center.

§ VII One will see here the statement of the other primitive propositions, with some of the principal definitions that are linked with them and that are necessary for understanding them. In spite of the presence of some abbreviating terms (such as line, plane, sphere, segment, and so on) whose aim is to encapsulate several ideas with some signs occupying little space, the primitive propositions still do not cease to be solely about the categories of point and motion. There is no need here for anything else to develop deductively this part of elementary geometry, which does not depend on Euclid’s parallel axiom.57 (10) If a and b are noncoincident points, there exists a motion that maps a to itself, and which assigns to b as image a certain point different from b but belonging to ab. Such a motion is certainly proper, and maps the line ab to itself in such a way that, except for a, no point of ab corresponds to itself. This same principle is reproduced in another form in the proposition, “if a and b are points distinct from each other, the sphere through b around a and the line joining a with b meet again at a point different from b.” (11) If one assumes that a, b are noncoincident points and that  and  are motions, each of which leaves a fixed but maps b to a point different from b and belonging to ab, it follows that the images of point b under  and  coincide. This amounts to saying that the line ab and sphere ba cannot meet in more than two distinct points. (12) If a and b are distinct points, there exists a motion that transports a to b while mapping some point of ab to itself. This is equivalent to affirming the existence of a sphere that passes through a and through b, and has its center at a point of ab. Two such spheres cannot coexist if they do not have the same center, and this leads to the notion of midpoint or center of a pair of points, whereas the two preceding principles provide those of points symmetric with respect to a given point, and so on. The same facts 56

[Euclid [1908] 1956, volume 2, 24–32. These propositions say that the line joining the centers of tangent coplanar circles passes through their point of contact.]

57

Euclid [1908] 1955, volume 1, 155. [Pieri wrote “l’axiome XII du Ier libre d’EUCLIDE ,” referring tacitly to Euclid 1885.] The memoir Pieri 1900a [translated in chapter 8 of the present book] studies most of the elementary geometric facts that do not involve comparison between sizes of surfaces or solids (without excluding, however, convex plane angles, triangles, the relation of larger or smaller between segments or plane angles, and so on) and which are, moreover, independent of the axiom of parallels just mentioned. Consequently, these properties belong no less to the geometry of Lobachevsky than to that of Euclid, because they live, so to speak, on the ground common to these two geometries ( pangeometry). Nevertheless, one can find there the greatest part of books I and III of Euclid and something of XI: I believe this, in short, is sufficient to ensure that ordinary elementary geometry can be established conveniently on the twenty postulates that I propose, and on the axiom of parallels.

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then permit definition of certain motions, such as translations and point reflections,58 and so on, for example. (13) Given three noncollinear points a, b, c, there must exist a motion that transforms both a and b into themselves, and which maps c to a point different from c but belonging to the plane abc. (Compare postulate 10.) Such a motion necessarily maps the entire plane abc to itself and can be called a reflection of that plane onto itself about a and b as pivots. From this follows the existence of at least one other point common to the spheres ca and c b besides c, and all ambiguity is removed by virtue of the following principle. (14) If a, b, c are noncollinear points, there cannot exist in the plane abc two points, distinct from each other and from c, that would be common to the two spheres ca and cb . From this follow, for example, EUCLID III.9 and III.10,59 and also the theorems that a line cannot meet a sphere in more than two distinct points; that two reflections of a plane abc across the line ab determine equal mappings from the plane abc to itself; that such a motion maps the plane abc to itself in a manner symmetric or involutory; and so on. Perpendicularity is introduced in the form of a relation between only three points by the following definition (thus reducing it to the simplest terms and stripping it of everything superfluous): supposing that a, b, c should be points, a different from b and c, when one says “the pair (a, c) is perpendicular to the pair (a, b),” or when one writes (a, c) z (a, b), one wants only to affirm the existence of a motion that maps a and b to themselves and c to a point different from c but belonging to the line ca. In short, the statement (a, c) z (a, b) expresses the proposition, “points a, b, c are not collinear, and when one reflects the plane abc onto itself about a and b as pivots, c falls back on the line ca.” One now proves propositions 3, 16, and 18 of book III; the permutability of pairs (a, c) and (a, b) with respect to the symbol z; and finally, that the relation (a, c) z (a, b) is a consequence of the fact that the sphere through c around b contains the point symmetric to c with respect to a. Perpendicularity of the lines ac and ab depends on that of the two pairs (a, c) and (a, b); one establishes in turn the existence of a single line perpendicular to another and passing through a point outside it; the nonexistence of more than one line normal to another at a point on it in a plane passing through it; and so on.60 58

[Pieri’s terms for translation and reflection were glissement and rabattement.]

59

[Euclid [1908] 1956, volume 2. Proposition 9 says that if point c lies inside a circle C, points d, e, f are distinct points on it, and segments cd, ce, cf are equal, then c is the center of C; proposition 10, that two distinct circles cannot meet in more than two distinct points.]

60

[In this context in French or English, normal means perpendicular. Euclid [1908] 1956, volume 2. Proposition 3 says that if b, br are distinct points on a circle C whose center c does not lie on bbr, and a is a point on bbr, then a is the midpoint of b, br if and only if (a, c) z (a, b). Proposition 16 has three parts. The first says, in effect, that if a is a point on a circle C with center c, and b is a point distinct from a such that ( a, c) z ( a, b), then ab meets C only at a. The second part says further, in effect, that no other line through a can meet C at just one point. The third part of proposition 16, which (continued on the next page)

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4 Pieri’s 1900 Paris Paper

Postulating further the following two principles, (15) if a, b, c are noncollinear points, there exists a point outside the plane abc; (16) if a, b, c, d are noncoplanar points, there exists a motion that, while mapping a and b to themselves, transforms d into a point of the plane abc; 61 one can then deduce immediately all the elementary propositions (in common treatments) where the notion of segment62 plays no role, nor that of parallel lines. One [can] prove, for example, that there are infinitely many lines perpendicular to a single line at a given point on it, and that these are all situated in a plane, which one can call perpendicular to the line at this point; and then the other customary theorems about perpendicular lines and planes. From that emerges the proposition that two planes with a common point meet along a line. In most of the modern treatments, this latter fact is proved (as well as some others no less fundamental) by means of premises about the sides, where one conceives a line as partitioned by an arbitrary one of its points, or a plane by one of its lines, or space by a plane. But we have made here no appeal whatever to these notions, which are defined with care later in the cited work,63 relying on the last postulates, 17 through 20. To introduce segment, we say that a point x of the line ab is situated between a and b when it lies inside the sphere whose poles are these points. This is equivalent to requiring (by virtue of another definition) that the plane perpendicular to the line at x intersects the sphere at points different from a and from b (the points interior to the sphere being the centers of its chords). Thus, the notion of segment,62 which is most often taken as primitive, is decomposed into the elementary concepts of point and motion, and reconstructed from those alone through a sequence of purely nominal definitions. Finally, here is the place to state the four following postulates. (17) Let a, b, c be distinct collinear points. If a plane perpendicular to their line at a point different from a, b, c meets one of the three spheres constructed on the point-pairs (a, b), (a, c), (b, c) as poles, it must meet another one, as well.

(continued from the preceding page)

involves a horn angle, is irrelevant to the present discussion. Proposition 18 says that if a is a point on a circle C with center c, and b is a point distinct from a such that ab meets C at just one point, then ( a, c) z ( a, b).]

61

As one sees, postulates 13 and 16 could, if desired, be condensed into one. And by showing (through a very slight modification of form) that the motion considered in principle 13 transforms the entire plane abc into itself, one could also eliminate postulate 9, by concealing it under a principle that affirms the possibility of reflecting a plane onto itself.

62

[Pieri’s phrase was notion de droite terminée ou de segment. For Pieri, segments are closed: see definition P28 in §1 of Pieri 1900a (section 8.1 of the present book) and definitions P2 and P6 and theorem P3 of §4 (section 8.4).]

63

[Pieri 1900a, translated in chapter 8 of the present book.]

4.7 Section VII

75

(18) If a, b, c are points, and if c lies between a and b, no point can be situated at the same time between a and c and between b and c. (19) Given noncollinear points a, b, c, whenever a line lying in the plane abc passes between a and b (that is to say, meets the line ab between these two points), it must also pass between points a and c or between points b and c, unless it should contain one of the points a, b, c. This postulate corresponds to Moritz PASCH’s principle IV about plane surfaces.64 In more restricted and cleaner form: in the plane of the points a, b, c, there does not exist any line that meets only one of the three segments ab, ac, bc.65 (20) If a and b are distinct points and k, a nonempty figure situated entirely in the segment ab, there must be a limit superior (or inferior) of k, that is to say, a point x interior to the segment ab or else coincident with b, such that (1) no point of k should be situated between x and b; (2) whatever point y is selected between a and x, there always exist some points of k lying between y and x, or else equal to x.66 To sketch, as briefly as it might be, the manner of constructing the other more elevated parts of the Elements would surpass the bounds of this memoir. On the other hand, we should be permitted to indicate that a construction completely analogous to that with which we are occupied here can also be carried out for the geometry of position, understood (in the manner of G. K. C. von STAUDT) as an autonomous deductive science, and freed of all links to the premises of elementary geometry. A careful study leads, there too, to distinguishing, among all the conceptual material, just two primitive notions. The one (which it does not seem possible to avoid) is that of projective point. The other can be chosen as one of these notions: join of two projective points (or projective line), or homography, which plays the same role in projective geometry as congruence in elementary geometry. With these few elements is reconstructed any projective idea at all, no matter how complex: for example, projective segment and triangle, sense of a projective line, projective hyperspace, absolute projective space (with an infinite number of dimensions), and so on, without deviating from the strictest observance of the rules of the pure deductive method, renewed and reinforced in our era by algebraic logic. May 1900

64

Pasch 1882a, 21.

65

If we should only suppose a, b, c distinct instead of noncollinear, principle 17 would still remain established that way; but certain notable facts that follow solely from premises 17 and 18 would appear linked with premise 19. It is very easy to reduce the preceding premises to a smaller number of suitable logical combinations or syntheses, whereas the contrary operation (which we force ourselves to employ here as much as possible) is not as easy.

66

[Pieri neglected to specify the changes necessary to convert (20) to the correct statement for limit inferior of k , that is to say, a point x interior to the segment ab or else coincident with a, such that (1) no point of k should be situated between x and a; (2) whatever point y is selected between b and x, there always exist some points of k lying between y and x, or else equal to x. It is not clear whether Pieri intended to postulate both forms of (20) or just the first. A similar question concerns postulate XX (P15) in Pieri 1900a, §6, translated in chapter 8 of the present book.]

Allegory of Geometry Laurent de La Hyre (1649) 1

5 1

Pieri and Projective Geometry

Projective geometry can be described as the geometry of the straightedge, in comparison with Euclid’s geometry of straightedge and compass. A projective plane can be thought of as a Euclidean plane extended to include ideal points at which parallel Euclidean lines intersect, and an ideal line containing just those ideal points. From a transformational point of view, projective geometry is the study of the projectively invariant properties of figures, which remain unchanged under one-to-one mappings of the projective plane onto itself that relate collinear points to collinear points and concurrent lines to concurrent lines. In comparison, Euclidean geometry is the study of properties that remain invariant under translation, rotation, reflection, and change of scale. In this chapter 2 we place Pieri’s contributions to projective geometry in historical perspective. In section 5.1 we discuss how he embraced the subject in school and university, in research, and in teaching. In later sections we identify specific characteristics of his approaches to projective geometry and look at other scholars’ particular contributions that would eventually play significant roles in Pieri’s explorations. The discussion will show where his research fit into the historical context, but it is in no way fully representative of all geometers who contributed to the development of the subject, nor of all who influenced Pieri. 1

This painting (oil, on canvas) was commissioned for the Paris house of a wealthy adviser to the French monarchy, Gédéon Tallemant: the largest (1.04 × 2.18 m) of a series representing the seven liberal arts. It contains several allusions to the work of Girard Desargues, a close acquaintance of the artist. Desargues had pioneered applications of classical geometry, particularly to perspective drawing. In her left hand, Geometry holds a compass and try square; a plumb square sits on the ground below. According to the popular Iconologia of Cesare Ripa (1609, 196), the compass signifies the scope of classical geometry, whereas the other more versatile instruments represent her practical concerns. (Ripa furnished detailed guidelines for other allegories in the series, but only these few for Geometry.) In her right hand Geometry holds a paper on which are drawn familiar diagrams related to Euclid’s propositions I. 47, II.9, and III.36 (see [1908] 1956, volumes 1 and 2), all of which Desargues had proposed to generalize. The canvas on the easel depicts freshly cut stones, alluding to his study of that practice. The globe recalls the widespread application of geometry to cartography. The sphinx looks upon examples of ancient Egyptian geometry to the right, but her deterioration reflects their eclipse by later developments in the foreground (Williams 1968). The canvas on the easel actually displays faint white guidelines that an artist might employ in loosely following Desargues’s method of perspective; they are too faint to appear here. Close examination of the painting, in the Legion of Honor in San Francisco, suggests that the description of the lines by Judith V. Field and Martin Kemp (Field 1987, 215–217) is inaccurate. A copy by La Hyre of the central portion of this painting also survives: he omitted the easel and the pyramid.

2

Preliminary forms of parts of this chapter appeared in Marchisotto 2006. For more information, see the articles in Bioesmat-Martagon 2010, particularly Voelke 2010.

© Springer Science+Business Media, LLC, part of Springer Nature 2021 E. A. C. Marchisotto et al., The Legacy of Mario Pieri in Foundations and Philosophy of Mathematics, https://doi.org/10.1007/978-0-8176-4823-7_5

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In section 5.2 we discuss several early contributions that prepared the way for Pieri’s geometrical research, which he conducted in the environment of two different schools of thought: the analytic and the synthetic. Pieri’s research in algebraic and differential geometry followed the analytic approach, which translates properties of geometric figures into relationships among numerical coordinates of their components and uses algebra to derive new properties. Here, however, we emphasize background for Pieri’s researches in synthetic foundations for projective geometry. In particular, we focus on the introduction of ideal points at infinity, the methods of projection and section, the principle of continuity, and the introduction of imaginary (complex projective) points with a goal of illustrating how these ideas and methods impacted Pieri’s research. In the seventeenth century, Girard Desargues promoted the methods of projection and section and introduced ideal points at infinity to extend the Euclidean plane. Johannes Kepler also used projection and section, with his own interpretation of ideal points. Kepler appealed to a continuity principle which, reconfigured in various ways for projective and enumerative geometry, would be controversial for years to come. The work of Desargues and Kepler led by the late eighteenth century to the synthetic descriptive geometry of Gaspard Monge, which would inspire the synthetic projective geometry of Jean-Victor Poncelet and Michel Chasles in the early nineteenth. By the turn of the twentieth century, Pieri would build on these results in his research on the foundations of synthetic projective geometry in the school of Giuseppe Peano and on enumerative algebraic geometry in the school of Corrado Segre, both at the University of Turin. Section 5.3 acknowledges G. K. C. von Staudt, whose 1847 work Geometrie der Lage (Geometry of Position) particularly inspired Pieri to investigate projective geometry; and it identifies others who influenced Pieri in disseminating Staudt’s ideas.3 In 5.4 we focus on Moritz Pasch, Peano, Cesare Burali-Forti, and Alessandro Padoa for their influence on Pieri’s foundational studies, and in 5.5 we pay tribute to Felix Klein and others for Pieri’s transformational approach to the subject. In 5.6 we set Pieri’s contributions to higher-dimensional studies of geometry in their historical context. And in 5.7 we explain a phrase that Pieri himself created to capture the essence of all his axiomatic constructions: mathematics as a hypothetical-deductive system. In particular, we show how he used this idea to clarify the pervasive role of the principle of duality in projective geometry. 5.1 Pieri’s Studies, Research, and Teaching In this section we discuss briefly how Pieri became familiar with projective geometry in school and university. We show his reliance on the subject for his research in algebraic

3

Leibniz used this term—geometria sitū s (geometry of position)—in a letter in 1679; Leonhard Euler used it in the title of his famous 1736 paper on the Königsberg bridge problem (Struik 1969, 183). Lazare Carnot called his 1803 treatise on projective geometry Géométrie de position. Poncelet employed the term projective property ( propriété projectif ) as early as 1822. The exact term géométrie projective was employed in the brief 1859 overview of the field of geometry by Olry Terquem, editor of the journal Nouvelles Annales de Mathématiques. Pieri used the terms geometria di posizione and geometria projettiva interchangeably.

5.1 Pieri’s Studies, Research, and Teaching

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geometry, and how he subsequently developed a specialization in its foundations. We conclude with remarks about its role in his teaching.4 School and University. Pieri may have been introduced to projective geometry during 1879–1880, his final year in secondary school at the Royal Technical Institute in Bologna, by Augusto Righi, his physics teacher.5 Righi had recently published papers on projective geometry, and was beginning a prolific scientific career. By the turn of the century he would become Italy’s leading physicist. At the University of Bologna the next year, Pieri studied projective geometry with Pietro Boschi. On the required examination for analytic and projective geometry, he earned a perfect score and was awarded honors. Pieri’s examiners were Boschi, Matteo Fiorini, and Salvatore Pincherle. During 1881–1882, his first year at the Scuola Reale Normale Superiore in Pisa, Pieri enrolled in a compulsory course with Angiolo Nardi Dei to study projective and descriptive geometry with design; he earned a perfect score on that examination, too.6 Algebraic and differential geometry were the subjects of Pieri’s 1884b and 1884c Pisa doctoral dissertations, and the focus of his early research. Until his untimely death in 1913, he would continue to be a productive researcher in those fields, even as he turned his attention to research in vector analysis and foundations of arithmetic and geometry (projective, Euclidean, and inversive). With respect to teaching, projective geometry became the focus of Pieri’s professional life. Research in Projective Geometry. Pieri became professor of projective and descriptive geometry at the Royal Military Academy in Turin in 1886, and in 1888 was also appointed assistant to the corresponding chair at the nearby University. He participated in two of the three schools of mathematical research that flourished simultaneously there at the turn of the twentieth century: Corrado Segre’s school of algebraic geometry and Giuseppe Peano’s school, which made important contributions to analysis, logic, foundations, linguistics, and teaching.7 Influenced by Segre, Pieri applied to algebraic geometry the ideas and methods of multidimensional projective geometry. Pieri used only techniques of projective geometry, unlike most contemporary researchers, who employed Euclidean techniques as well. Pieri wrote several papers that characterize different types of birational transformations of

4

See M&S 2007, §1.1–§1.2, for information about Pieri’s family life and an overview of his research.

5

Today this school is known as the Istituto di Istruzione Superiore Crescenzi Pacinotti. Pieri’s curriculum there qualified him to study mathematics, physics, and engineering at any Italian university. See M&S 2007, §1.1, §1.3, for information about the Italian schools and for a biographical sketch of Righi.

6

See M&S 2007, §1.1, §1.3, for much more information about the universities at Bologna, Pisa, and Turin, Pieri’s work there, and biographical sketches of the mathematicians just mentioned. At that time, according to Giacardi 2015, the standard physical-science curriculum included a course in geometry with a component on drafting (designo). See also Menghini 2006, 38–43.

7

Giacardi 2001, 139, 146–147, 151. The third school, led by Vito Volterra, concentrated on functional analysis and its applications.

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a projective space.8 Like other mathematicians of the Segre school, Pieri sought to classify algebraic varieties up to birational transformations and to better illuminate birationally invariant properties of varieties.9 Pieri made important contributions to enumerative algebraic geometry that were also set in the context of projective geometry. The object of classical enumerative geometry was to find the number of geometric figures satisfying given geometric conditions, in terms of birationally invariant features of the figures and conditions.10 This area underwent significant development after Hermann Schubert introduced in 1879 a symbolic calculus based on the representation of a geometric condition by an algebraic symbol. Pieri made productive use of the calculus to extend the results of Schubert and others.11 After intensive study and translation of G. K. C. von Staudt’s fundamental 1847 treatment of projective geometry, and inspired by Peano, Pieri began to focus on the foundations of this subject.12 Between 1895 and 1906, Pieri produced thirteen research papers on foundations of projective geometry (see section 9.2). His work built on the results of many scholars, but was notable in the paths he pursued to establish the subject for the first time as an autonomous science. Pieri explored projective geometry as an abstract formal system rather than as a study of space. His goal was to base it exclusively on projective concepts, with no appeal in its construction to ideas drawn from intuition. Pieri contributed to the formative discussion of multidimensional projective geometry. His axiomatizations of projective space and hyperspace reflected his allegiance to Peano’s goals of founding mathematical theories on the smallest number of primitive concepts, employing the axiomatic method and symbolic logic to develop mathematics abstractly. These axiomatizations used Peano’s logical calculus explicitly or implicitly, continually revisited ideas proposed by Staudt, and promoted the goals of Felix Klein’s Erlanger program. Primarily for his work in foundations of projective geometry, Pieri received

8

Let S and Sr be projective spaces, each with a system of homogeneous coordinates. A mapping  : S 6 Sr is called rational, or a Cremona transformation, if its component functions can all be defined by homogeneous polynomials of the same degree. Varieties V f S and V r f Sr are called birationally equivalent if there exist rational mappings  : S 6 Sr and r : Sr 6 S and subvarieties U   V and Ur   V r such that the restrictions of  to V – U and r to V r – Ur are inverses of each other. (U and Ur must contain all points where the component functions of  and r, respectively, are all zero.) By slight abuse of language  and r are then called mutually inverse birational mappings from V to V r and from V r to V.

9

Brigaglia and Ciliberto 1995, 6.

10

Kleiman 1977, 299. A typical problem is to determine the decrease in the number of varieties in a pencil that have a double point, due to the presence of an ordinary k-fold point. Luigi Cremona (1864, 196) had shown that for curves in a plane this reduction is (k – 1)(3k + 1). In 1886b, Pieri proved that for surfaces in three-dimensional space it is (k – 1)2(4k + 2).

11

Schubert [1879] 1989; Kleiman 1976, 450–451. For a brief summary of Pieri’s work in algebraic and differential geometry and vector analysis see M&S 2007, §1.2.1, and Marchisotto 2010, §3, 332–349.

12

Segre in fact suggested that Pieri undertake his 1889a annotated translation of Staudt 1847, and wrote the preface for it (see Segre [1887] 1997).

5.1 Pieri’s Studies, Research, and Teaching

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honorable mention in the competition for the 1904 Lobachevsky Prize of the PhysicoMathematical Society of Kazan.13 The historical and mathematical facts concerning Pieri’s research in projective geometry are remarkable from several points of view: • his choice to explore the subject and make it the focus of his research, after being trained as an algebraic geometer; • his initiation into the field with his intensive study of Staudt’s seminal work, and his continual revisitation of Staudt’s ideas to improve and extend them; • his establishing the subject as an autonomous science; • his foundational approach to the subject, which emerged from his association with the school of Peano at Turin, and yet differed in many respects from the works of its other participants; • his efforts to connect the construction of the subject as an abstract logical system with his metamathematical research and his views on pedagogy; • his transformational approach to the subject, which embraced Klein’s Erlanger program; • his interest in higher-dimensional aspects of the subject. Teaching. Pieri’s teaching career was largely focused on projective geometry.14 He taught projective and descriptive geometry at the Military Academy in Turin and served as assistant to the chair of that subject at the nearby University.15 Pieri published the lectures for his Academy course in projective geometry as the book Pieri 1891c. It was very well received by Segre and others at the University, and probably contributed to Pieri’s appointment there as libero docente that same year.16 In that capacity, Pieri gave courses in projective geometry and fundamentals of geometry. In 1900, Pieri was appointed to the chair of projective and descriptive geometry at the University of Catania in Sicily. Eight years later he joined the faculty at the University of Parma as full professor and director of the School of Projective and Descriptive Geometry with Design. Pieri remained there until he took medical leave in 1912. Pieri was truly committed to teaching. Comments in his research papers reveal that he believed geometry should be taught as an abstract deductive science, with an emphasis on the role of transformations. He believed that such an approach would provide insights into the nature of a geometry, elementary or projective, as an autonomous science. It 13

David Hilbert won the prize, for the second edition of his [1899] 1971 Foundations of Geometry. It was awarded jointly for the quality of the research and of the nomination. Henri Poincaré’s [1902] 1903–1904 nomination of Hilbert was more than twenty pages long; Peano’s 1905 nomination of Pieri was a scant four pages.

14

However, Pieri’s doctoral students pursued topics in algebraic geometry.

15

According to Menghini 2009, instruction at the military academy was considered to be at university level.

16

Analogous to habilitation in northern Europe, the libera docenza was required for appointment as an independent teacher. See M&S 2007, §1.1, for further information about Italian universities at that time, and detailed discussion of Pieri’s employment.

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appears that Pieri was never able to implement fully the pedagogical strategies he proposed. Nonetheless, an examination of his lecture notes for his projective geometry classes demonstrates the impact of his research on his teaching. Pieri’s lectures for the Military Academy in Turin17 were developed after his 1889a translation of Staudt’s seminal 1847 treatment of the subject, but before Pieri’s first axiomatization of projective geometry (1895a, 1896a–b). They reflect the traditional approach, adopted by Peano and others, based on the idea of projective geometry as an extension of Euclidean. Pieri’s extant university lectures, developed after he had done foundational research in the subject, tell a different story. Registries from the University of Catania provide lists of topics of Pieri’s lectures during his tenure there. They give no substantive details on the pedagogy that he employed, although they do suggest that, like any good teacher, Pieri tried different strategies in different years. The Catania registries reveal changes both in course content and in the order in which topics are introduced, as compared to Pieri’s course at the Military Academy. The changes model some but not all aspects of Pieri’s axiomatizations, which sever all ties to Euclidean geometry.18 Pieri’s lecture notes from the University of Parma19 give more details of the influence of his research on his teaching. In these we see how Pieri explicitly alerted students to the more “desirable” view of projective geometry as an autonomous subject, rather than as an extension of Euclidean geometry. Pieri began those notes with a message for students. His military-academy notes contained no such introduction. 5.2 Evolution of Projective Ideas and Methods This section traces the synthetic approach to projective geometry from its genesis in the work of Girard Desargues, through the evolution of its methods in the descriptive geometry of Gaspard Monge and the projective geometry of Jean-Victor Poncelet, to the early research of Mario Pieri in Corrado Segre’s school of algebraic geometry, which also influenced Pieri’s teaching. From its origins in antiquity, projective geometry had been seen as part of Euclidean geometry.20 In his 1639 Rough Draft on conics, Desargues envisioned a projective plane as a Euclidean plane augmented by the introduction of ideal

17

1891c. See section 9.1 of the present book.

18

See section 9.1.

19

1910, 1911c. See section 9.1.

20

The origins of projective geometry can be traced back through the Euclidean context to ancient Greece, in the writings of Euclid, Apollonius, Menelaus, Pappus, and others. Notwithstanding the proliferation of projective ideas in antiquity, interest in them remained virtually dormant until the fifteenth century, when they were revived in response to the need for artistic representations of three-dimensional figures on two-dimensional canvases.

83

5.2 Evolution of Projective Ideas and Methods

points.21 In such a view, the projective plane consists of all ordinary points and lines of the Euclidean plane and, in addition, a set of ideal points lying on one ideal line at infinity, so that exactly one ideal point lies on each ordinary line. In an analogous way, three-dimensional Euclidean space can be extended to projective space by adding an ideal plane at infinity consisting of all the ideal points and lines added to the ordinary planes. As early as 1604, Johannes Kepler had introduced ideal points in his study of conics. He proposed that all conic sections with a common vertex and one common focus have essentially the same shape, related by an analogy principle that “expresses the inner resemblance of contrasted figures ... which are connected by innumerable intermediate forms.” 22 Choosing any point on a circle as the vertex V and regarding its center C as two coincident foci, he noted that by moving the second focus F away from C along the ray VC to infinity, the circle is transformed via ellipses into a parabola; and if F proceeds “beyond” infinity to the opposite “end” of the line VC, the parabola becomes a hyperbola. Kepler noted,

6

:

... geometrical terms ought to be at our service for analogy. I love analogies most of all: they are my most faithful teachers, aware of all the hidden secrets of nature. In geometry in particular they are to be taken up since they restrict the infinity of cases between their respective extremes ... and place the whole essence of any subject vividly before the eyes.23

When Desargues undertook the projective study of conic sections, he made the innovative step of connecting the notion of ideal points at infinity to the idea of perspective. He recognized that the study of conic sections could be unified by the fact that all nondegenerate conics are projections of circles: hyperbolas, parabolas, and ellipses are those which respectively intersect, touch, or do not meet the line at infinity in their plane.24 Mathematicians would embrace this extended environment for projective geometry, recognizing that the properties of certain mathematical forms are more easily comprehended when interpreted in this context.25 However, the use of projective methods in conjunction with various forms of the principle of analogy to produce ideal points would come under severe scrutiny in the years that followed. [The narrative continues after the figure on page 86.]

21

Desargues [1639] 1987, 70–71.

22

Taylor 1881, lviii.

23

Kepler 2000, 109. See also Davis 1975, 678–680; that study differs from Taylor 1881 on Kepler’s interpretation of the degenerate straight-line conic. See also Field and Gray 1987, appendix 4, 184–188, 221–222.

24

Field and Gray 1987, 53. Whether Desargues actually based his study on Kepler’s work is open to speculation: see Field 1997, 183–186.

25

According to H. S. M. Coxeter (1971, 310), “For more than two centuries this procedure was believed to be unique” to projective geometry, until Maxime Bôcher (1914, 194) recognized that “adding a single ideal point to the Euclidean plane (to make an inversive plane) is entirely analogous to what Desargues did, and equally useful.”

Johannes Kepler in 1610

Girard Desargues and Marin Mersenne in 1643: Detail of an 1889 Mural by Théobald Chartran in the Sorbonne, Paris

Girard Desargues was born in 1591 in Lyon to one of the richest families in France, functionaries in service to king and parliament. Few personal details of his life are known, but he was evidently well educated, talented in classical mathematics, and familiar with its Greek sources. His father died around 1613 and was succeeded in the family by his elder brothers; they both died in 1629. Girard practiced engineering and architectural design, but his main occupation was evidently managing the family estate. He frequented the circle of intellectuals centered around Father Marin Mersenne in Paris, including René Descartes, Blaise Pascal, and others. Desargues published limited-circulation pamphlets on perspective drawing and on projective geometry in 1636 and 1639. Written in French rather than academic Latin, these emphasized practical matters but were addressed to learned readers, probably just to the Mersenne circle. They were republished in 1648 in somewhat enlarged form by Abraham Bosse, a well-known engraver who also published guidebooks for artists. That edition contains what is now known as the Desargues theorem about perspective triangles. (See the figure on page 86.) Desargues died in Lyon in 1661. Desargues employed his own peculiar terminology, which detracted from the accessibility of his work and delayed its impact. Not really publicized until 1685, by the mathematician Philippe de La Hire, it still remained obscure two centuries later. La Hire had known Desargues personally through his father, Laurent de La Hyre.* Laurent de La Hyre was born in 1606 in Paris. His father was a government official and sometime artist. Laurent studied the techniques and in 1631 was licensed as a painter. His mother died in 1638 and he married the next year. By then he was caring for his six younger siblings and his wife’s two sisters. Laurent’s father died in 1643, and his own health soon began to deteriorate. Four of the sisters entered a convent in 1649. La Hyre became closely associated with the printmaker Abraham Bosse. Through Bosse he met Girard Desargues, whom he retained to design his new house. Nearby lived the wealthy and flamboyant Gédéon Tallemant, legal counselor to the king.† Tallemant commissioned La Hyre to fashion for his own house a series of large allegorical paintings corresponding to the seven classical liberal arts. Two of those works, Géométrie and Dialectique, completed in 1649 and 1650, are depicted to introduce chapters on geometry and logic in the present book, pages 76 and 6. They are described in detail there. Géométrie features graphical references to work of Desargues. This was a time of turmoil in Paris. The Thirty Years’ War came to an end in 1648, and the participating countries were in economic and political chaos. King Louis XIV was an infant, his mother Anne of Austria was regent, and the Italian cardinal Jules Mazarin was ruling France as chief minister. A rebellion, La Fronde, raged in Paris during 1648–1653 and threatened the regime. La Hyre’s health failed and he died in 1656 after a year’s severe illness. He left a widow and five minor children, including Philippe, who became a noted mathematician.‡ *For further information about Desargues, consult Field 1997, 191–229, and Dhombres and Sakarovitch 1994. † Tallemant was a cousin of Gédéon Tallemant des Réaux, author of many vignettes of notable figures of that

era.

‡ For further information, consult Rosenberg and Thuillier 1988; Jal 1872; Monmarqué 1836, v–xi; and Wine

et al. 1993, 22–26.

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Theorem of Desargues. If ABC and ArBrCr are perspective from O, then the intersections K, L, M of corresponding pairs of their edge lines are collinear.

M Br

g

B A

O

f

Ar K

C

Cr e

er

h

Application. To draw the line through a given point K on a canvas toward the off-canvas intersection L of two given lines e, er, choose a point O and lines f, g, h through O intersecting e, er at A, B, C, Ar, Br, Cr as shown, so that intersection M falls on the canvas; then L will lie on the line KM.

L [continued from page 83]

The methods of Desargues were virtually ignored until the closing decades of the eighteenth century, when Monge revived them to create what he called descriptive geometry (Monge 1798). In courses at l’École Polytechnique, and in the many papers that he published on this subject, Monge fostered the use of projection and section26 to analyze objects in space by examining their projections onto planes. Monge’s drafting plane was divided into four quadrants by two lines intersecting at right angles. Following the architects’ tradition of double projections, he projected a spatial body orthogonally onto adjoining portions of two perpendicular half-planes, which he then folded out into two adjoining quadrants. As appropriate, a projection onto a third half-plane could be folded out onto another quadrant, and the remaining quadrant used to represent relations revealed by these projections.27 In these projections onto planes by collections of parallel lines, Monge made implicit use of Kepler’s principle of analogy to link the spatial and planar objects and manipulate them. In 1822 Monge’s student Poncelet revived the projective geometry of Desargues, generalizing Monge’s methods by projecting along pencils of intersecting lines as had Desargues. Poncelet extended Kepler’s principle of analogy, calling his own version the “principle of continuity”: if in keeping the same given properties, we begin to vary the original figure by ... a continuous motion ... the properties and relations found for the first system remain applicable to successive stages of this system, provided always that we have regard for the particular modifications which may occur as when certain magnitudes vanish or change in direction or sign ...

26

In three dimensions this process can be described as follows: given a point O not on a figure F, the set P of all lines joining O to the points of F is called the projection of F from O. Given a plane  not through O, the set of intersections of  with the lines of P is called the section of P by .

27

See Grattan-Guinness 1997, 369, and Coolidge [1940] 1963, 109–114.

5.2 Evolution of Projective Ideas and Methods

87

This is ... a thing which in our day is generally admitted as a sort of axiom whose truth is manifest, incontestable, and which need not be proved. ... the objects to which it is applied should be ... continuous or subject to some laws that one could regard that way. Some objects can even change position through a series of variations that the system undergoes, others can diverge to infinity ... ; the general relations then survive the modifications, without ceasing to apply to the system.28

With regard to their principles of analogy and continuity Kepler and Poncelet employed the same type of language involving motion that is found in Euclid’s works.29 Poncelet distinguished three types of geometry: ancient pure geometry, analytic geometry or the method of coordinates, and his own modern pure geometry, which admitted the use of infinity and infinitesimals and invariants of varying figures but avoided coordinate algebra or any calculations that are not derived from continuously changing intermediate forms of given figures.30 Following the approach that Desargues had taken, Poncelet precisely introduced the new points, lines, and the plane at infinity to form the space of projective geometry. Analyzing the behavior “at infinity” of various conic sections, he noted that the principle of continuity implies that the points at infinity of a plane can be considered ideally as distributed on a unique line, situated likewise at infinity in this plane.

Further, he claimed, in three dimensions, all the points at infinity ... can be supposed to belong to one and the same plane ... This principle is just an extension of that in [the previous quotation] and can be deduced directly by means 31 of the principle of continuity.

Investigating systems of conic sections, Poncelet encountered some purely geometric problems that seemed to require the use of complex scalars. But he wanted to generate and analyze geometric figures with no appeal to analytic methods. He devised a way of handling some examples of this type by using, instead of coordinates, properties of involutory transformations on certain lines related to the problems. Poncelet could introduce imaginary points into the projective plane in a way analogous to his introduction of ideal points at infinity and subject in the same way to his principle of continuity: [There] was no attempt to provide a literal representation of imaginary objects. Instead, ... one could “view” the imaginary points through constructing the real lines that contained them. ... Poncelet’s new definitions weakened the boundary between real and imaginary objects, and as justification he pointed to the diverse modes of existence already present in geometry with respect to infinity and infinitesimals.32

28

Poncelet 1822, xix–xxvii. The first two quoted sentences were translated by Vera Sanford in Smith [1929] 1959, 315–316.

29

Euclid’s first postulate is to draw a line from one point to another. He defined sphere as the figure comprehended when a semicircle, its diameter remaining fixed, is carried around and restored to its initial position (Euclid [1908] 1956, volume 1, book I, postulate 1; volume 3, book XI, definition 14; see Elkind 2013).

30

Poncelet 1817, 142–143; Lorenat 2015a, 161–162.

31

Poncelet 1822, §1, chapter 2, note 96, 48–49; supplement, note 580, 361.

32

Lorenat 2015a, 158, 182. For more information, see Coolidge [1940] 1963, §5.2, 92–95; Wilson 1992, (ii)–(iv), 152–161.

Gaspard Monge around 1810

Jean-Victor Poncelet in 1848

5.2 Evolution of Projective Ideas and Methods

Jean-Victor Poncelet was born in Metz, in the Lorraine region of France, in 1788. The son of a wealthy lawyer, he was reared until age fifteen by loving foster parents in a nearby town. Poncelet was schooled at a lycée in Metz and during 1807–1810 at l’École Polytechnique in Paris, where he studied with Gaspard Monge, Lazare Carnot, and other notable scientists. Poncelet joined the army and was assigned first to further study at the School for Application of Artillery and Engineering in Metz,* then detailed as an engineering officer to serve in Napoleon’s 1812 Russian campaign. Captured during the disastrous French retreat from Moscow, Poncelet was marched a thousand kilometers in winter from Smolensk to Saratov, Russia, where he was imprisoned for a year. During that time he reviewed the fundamentals of mathematics from memory and drafted a manuscript for his Treatise on Projective Properties of Figures, which pioneered many aspects of projective geometry. Poncelet returned to Metz in 1814 to a promotion, to teaching at the same school, and to supervising manufacture of military equipment at the arsenal there. In 1822 he published his treatise, with a subtitle indicating his main professional emphasis: A Work Useful for Those Who Are Involved with Applications of Descriptive Geometry and Geometric Operations on Land. Poncelet also published research results in the Journal de mathématiques pures et appliquées, edited by Joseph Diez Gergonne. These drew intense criticism from the leading mathematician Augustin-Louis Cauchy. Moreover, Gergonne himself began contributing to foundations of projective geometry, and an unpleasant priority dispute arose between them. In 1825 Poncelet was promoted to professor of mechanics at Metz. Evidently discouraged by controversy in pure geometry, he changed his research direction to applied mechanics—for example, to the design of waterwheels for powering industry. After a brief foray into politics in Metz, he was promoted again, admitted to the mechanics section of the French Academy of Sciences, and in 1835 appointed professor at the Sorbonne in Paris, specifically to study mechanics. During the 1840s, Poncelet achieved international acclaim, married, and was promoted twice more. During the 1848 disturbances he resigned his professorship, was elected to political office as a moderate republican, and promoted to general and commander of l’École Polytechnique. During the uprising that June, he dealt with student activism; afterward, he concentrated on curriculum reform. Throughout these years, Poncelet had been publishing steadily in mechanics. After retiring in 1850 he chaired the mechanics sections of the 1851 and 1855 Universal Expositions in London and Paris, forerunners of the 1900 Paris Exposition featured in the introduction to chapter 4, and he published a major survey of progress in mechanics and engineering. Poncelet closed his career in the 1860s by collecting and publishing or republishing much of his early work in geometry. He died in 1867 after a long illness.† * l’École d’Application de l’Artillerie et du Génie de Metz † For more information see Chatzis 1998, Didion 1869, and Taton 1975.

89

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Immediately after Poncelet announced his principle in 1818, its validity came under fire, particularly from his contemporaries, Augustin-Louis Cauchy and Siméon Poisson.33 Questions persisted for decades. Addressing students in 1891, Segre remarked that introducing imaginary and infinitely distant elements into geometry proved to be most useful before it was justified with perfect logic. And Poncelet, who learned to employ [it] to great advantage ... never succeeded in establishing it rigorously in spite of the many pages which he wrote on the subject.34

In 1904, Gaston Darboux agreed in part: Poncelet ... imagined the famous principle of continuity, which gave rise to such long discussions between him and Cauchy. The principle is an excellent one when properly enunciated, and may be of the greatest service. Poncelet made the mistake of refusing to present it as a simple consequence of Analysis; Cauchy, on the other hand, would not recognise that his own objections, valid no doubt in the case of certain transcendental figures, lost their force in [Poncelet’s] applications ... . Whatever opinion may be held on the subject of such a discussion, it at least shewed, in the clearest manner, that Poncelet’s geometrical system rested on an analytical basis ... .35

More recently, the philosopher Mark Wilson has noted, Whatever its shortcomings in terms of clarity, the “continuity” principle represents the new projective elements as the natural outgrowth of the domain of traditional geometry. This “organic” connection through “continuity” gives one the right, thought Poncelet, to claim that the new extension elements represent the correct extension of Euclidean geometry. The new elements are added to complete the mechanism that makes Euclidean geometry work. Accordingly, the “ideal” elements are “just as real” as the original elements of geometry, even though we can form no visual picture of ... the imaginary elements.36

In separate studies, mathematicians and historians Jean Dieudonné and Julian L. Coolidge reported that Poncelet, using the continuity principle, had “enabled a tremendous expansion of all geometric concepts ... with the systematic introduction of points at infinity and imaginary points”; in promoting these ideas and methods of synthetic projective geometry, Poncelet had “placed the subject in the right light” and inspired “a goodly number of able geometers.” 37 Shortly after Poncelet’s pioneering work, his contemporary August F. Möbius introduced the basic algebraic technique of homogeneous coordinates, which treated all points —ordinary points, ideal points at infinity, and imaginary points—simply as projective points, with no distinction between them.38 Building on the analytic geometry of René Descartes and Pierre de Fermat, analytic geometers used this method to achieve many 33

See Gray 2007, §4.2, 47–50.

34

C. Segre [1891] 1903–1904, 454.

35

Darboux 1904–1905, 104. Darboux also supported his last claim by referring to Poncelet’s notes, written before 1817.

36

Wilson 1992, 161. This statement is supported by the deeper study Nabonnand 2008a, 43–50, which is based largely on Poncelet’s prolix philosophical essay [1818] 1864.

37

Dieudonné [1974] 1985, 8; Coolidge [1940] 1963, 95.

38

Möbius 1827. See the box on page 92.

91

5.2 Evolution of Projective Ideas and Methods

of the results acclaimed in the previous paragraph. Nonetheless, the rise of analytic geometry39 did not signal an end to synthetic geometry. Synthetic geometers continued to follow Poncelet, using the continuity principle to justify treating ordinary points and ideal points at infinity similarly. And, despite the controversies over the principle, the synthetic approach of Poncelet was widely accepted and promoted before the turn of the twentieth century. Regarding the expansion noted in the previous paragraph, Dieudonné claimed with hyperbole: The movement is so successful that, for close to 100 years, “geometry” will mean the geometry of the complex projective plane ... or of complex projective three-space ... .40

In his 1891 summary of tendencies in geometric investigations Segre proclaimed, The heroic age, so to speak, of synthetic geometry, in which the object was not merely to give new results to the science, but in which all, from Poncelet to Jakob Steiner, from Michel Chasles to G. K. C. von Staudt, were engaged in the struggle to prove the usefulness of the geometric method to the analysts who were not willing to recognize it—that age has passed; and nowadays battle is no longer necessary.41

So, battle over, by Pieri’s time projective geometry was being studied in two different ways: analytically and synthetically. Analytic approaches translated properties of geometric figures into equations and used algebraic calculations to derive new properties. Synthetic methods characteristically studied such properties by directly considering geometric figures, including those that had been introduced only analytically. Specifically, For Poncelet the description or painting of the figure through invoking imagined or visualized sensible objects ... was the characteristic feature of a purely geometric approach.

Imaginary or infinite objects could become sensible through projection or the principle of continuity.42 In the spirit of peaceful coexistence of synthetic and analytic methods, Darboux even suggested (see the quotation on page 90) that the continuity principle could be regarded as an analytic technique. In his 1875 textbook on synthetic projective geometry, Hermann Hankel went even further: the principle, seen from the standpoint of the old geometers, resembled a true loosening of the reins and seemed to threaten in the highest degree the present evident nature and solidity of geometry. ...This was much more a gift that pure geometry received from analysis ... . Only the habit of regarding real and imaginary quantities as analytically equivalent leads to that principle, which could never have been discovered without analytic geometry. Thus, pure geometry was in its turn reimbursed for analysis’s long occupying the exclusive interest of mathematicians; indeed it was perhaps an advantage that geometry had left fallow for a time while for the other discipline it bore rich fruit, that could now be so beneficially sown on a fresh field.43 [This paragraph continues on page 93.] 39

Voelke 2010, §3, and Boyer 1956, chapter 9, describe the rise of analytic geometry.

40

Major contributors to the algebraic approach included Lazare Carnot, Julius Plücker, Otto Hesse, Arthur Cayley. Dieudonné [1974] 1985, 8.

41

C. Segre [1891] 1903–1904, 452. Worldwide research productivity in synthetic geometry reached a maximum in 1887, but analytic geometry continued to grow (Cajori 1919, 278–279, based on White 1915, 110–111).

42

Lorenat 2015a, 166.

43

Hankel 1875, 9–10.

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5 Pieri and Projective Geometry

Homogeneous Coordinates. A convenient way ¯ = { t [ x, y, z] : t 0 } P to visualize the extension of a Euclidean plane  ` to include ideal points at infinity is to regard it as y the plane with equation z = 1 in threedimensional space 3 with real Cartesian coordi: x nates x, y, z; let x, y serve also as a Cartesian 7z = 1 P = [ x, y, z] coordinate system in . (See the figure at right.) The lines and planes through the origin can be regarded as the projective points and lines of ¯, y the extended or projective plane. In this repreO ¯ is the set of all sentation a projective point P scalar multiples t [ x, y, z] of any point in 3 7z = 0 different from the origin O; any nonzero memx Q = [ x, y, 0] ber of this set is called a homogeneous coordinate ¯_ ¯. Further, a projective line g¯ is the Q triple for P set of all solutions [ x, y, z] of a homogeneous linear equation a x + b y + c z = 0 for some non¯ lies on ¯g just when a homogeneous coordinate zero triple of scalars,* and the point P ¯ satisfies a homogeneous linear equation for g triple for P ¯. A point P in  with Cartesian coor¯ —the line through the origin and P —with dinates x , y belongs to just one projective point P ¯ with homogeneous coordinate triple [ x, y, 1]. For any nonzero scalar z, the projective point P homogeneous coordinate triple [ x, y, z] also has homogeneous coordinate triple [ x/z, y/z, 1], and hence corresponds to (contains) the point P in  with Cartesian coordinates x'z, y'z . Now consider ¯ with homogeneous coordinate triple [ x, y, 0] is a line in 3 the case z = 0. A projective point Q through the origin and parallel to . It does not correspond to any point in , but is called an ideal point at infinity. In this representation, all ideal points at infinity lie on the x, y plane of 3, which is the projective line with homogeneous linear equation z = 0. The algebra in this discussion can be generalized to explain the use of homogeneous coordinates and homogeneous linear equations in higher dimensions. It also still applies when all scalars are allowed to be complex. Unfortunately, the figure no longer provides visual confirmation in these cases. Complexification and Homogenization (Example). Algebraic solution of the equations of two circles in the Euclidean plane, of the form ( x – a) 2 + ( y – b) 2 = c 2, yields zero (0), one (1), or two (2) intersections. In case (0) the circles are disjoint. In (1) they are tangent and the algebraic solution is a double root. In (2) the solutions are simple roots. If complex scalars are admitted, cases (0) and (2) coalesce: disjoint circles have two distinct simple intersections, perhaps not real. This makes circles seem different from other conics, two of which can have as many as four intersections. To study this situation projectively, homogenize the Cartesian equation of a circle C : a finite projective point with homogeneous coordinate triple [ x, y, z] lies on the projective counterpart C¯ just in case its image, with Cartesian coordinates x'z , y'z , lies on C . Substituting those fractions for x, y in the equation of C and clearing denominators yields the homogeneous equation ( x – az) 2 + ( y – bz) 2 = c 2 z 2 of C¯. Thus, C¯ always passes through the circular points at infinity with homogeneous coordinate triples [1, ± i, 0]. In the complex projective analysis two circles always intersect in four points, counted by multiplicity. When the circles are concentric, the values of the partial derivatives of the corresponding polynomials at the circular points coincide, so those are double intersections. * stands for a row of three items and [ x, y, z], for a column. This permits use of the linear-algebra convention, row times column, for expressing a homogeneous linear equation.

5.2 Evolution of Projective Ideas and Methods

93

[continued from page 91]

Research in both analytic and synthetic geometry thrived. Joseph Diez Gergonne’s journal Annales de mathématiques pures et appliquées served during 1810–1832 as a forum for publications in both analytic and synthetic projective geometry (as well as algebraic geometry). In Italy during 1860–1875, the analytic approach was advocated by the school of Giuseppe Battaglini in Naples, and the synthetic by that of Luigi Cremona in Bologna and Milan. Pieri presented both analytic and synthetic arguments in his lectures to students. His classroom lessons included discussions of the ideal elements of Desargues and how they were used to construct a projective plane from a Euclidean one.44 However, in his research into the foundations of projective geometry, Pieri relied solely on the synthetic approach. In this project, he was pursuing a challenge that Segre presented in 1891 after analyzing the “heroic age” of synthetic geometry: By the study of the great geometers some scholar should become inspired to search and relate how and by whose work have been obtained the most important advances in modern geometry.45

In fact, it is through Pieri’s immersion in Segre’s school of algebraic geometry that we can trace the strong influence of Poncelet and the other great synthetic geometers on Pieri’s thinking, both with respect to the principle of continuity and the method of projection and section. In his research in enumerative algebraic geometry, Pieri, like others in Segre’s school, worked entirely within the framework of projective geometry and used the Schubert calculus to predict the number of geometric figures that satisfy certain conditions. Here are two example problems: 46 • to find the number of lines that intersect four given lines in three-dimensional space; • to find the number of points common to two twisted cubic curves that lie on the same quadric surface. The figures and conditions investigated may stem from elementary geometry, as in the first example, or they may originate in analytic geometry, as in the second. Such problems were solved by reducing them to consideration of special cases, appealing to a version of the continuity principle formulated by Hermann Schubert to count solutions of the corresponding systems of algebraic equations, and examining their algebraic multiplicities. It was customary in algebraic geometry to restrict proofs of theorems to cases where the points or algebraic varieties under consideration are in general position: that

44

See subsections 9.1.3–9.1.5.

45

C. Segre [1891] 1903–1904, 452. Segre used the word giovane, which connotes young scholar.

46

Schubert [1879] 1979, 13, 23; Coolidge [1940] 1963, 180–181.

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is, not satisfying various special conditions, often left unstated.47 Schubert’s basic idea was to specialize the parameters on which the objects under consideration depend, connecting the general case of the given problem by “continuous variation” to another case that has a trivial solution. According to his principle of conservation of number, the number of solutions is invariant as long as it remains finite and solutions are counted with appropriate multiplicities. Thus, Schubert could construct proofs for theorems that did not depend on any general-position condition.48 Schubert and members of Segre’s school followed Poncelet’s synthetic approach to projective geometry. For them, description of figures “through invoking imagined or visualized sensible objects ... was the characteristic feature of a purely geometric approach.” 42 For them, enumerative geometry was more than just the algebraic analysis of solutions of systems of multivariate algebraic equations. Reliance on the principle of conservation of number gave rise to a long controversy. In 1900 this motivated David Hilbert to include the following as the fifteenth of his famous problems that would guide the development of mathematics in the twentieth century: To establish rigorously and with an exact determination of the limits of their validity those geometrical numbers which Schubert especially has determined on the basis of the so-called principle of ... conservation of number, by means of the enumerative calculus developed by 49 him.

Over the following decades many mathematicians, including Pieri, contributed repeatedly toward a solution of this problem using a variety of algebraic tools. While Schubert’s calculus is regarded as basically secure, various questions evidently still remain unsettled.50 Pieri’s extensive use of the process of projection and section in his work on algebraic geometry served as a guiding principle in his axiomatizations of projective geometry.51 In the tradition of Monge and Poncelet, he used projective methods to construct geometric figures and determine their mathematical properties. Emulating other synthetic geometers such as Steiner, who systematically used perspective in his 1832 monograph to generate new geometric figures, Pieri appealed to it to generate figures in space and hyperspace using the concept of the visual).52 In his axiomatizations, Pieri explicitly

47

The notion of generic point—one in general position—was eventually made precise in 1926 by B. L. Van der Waerden (see 1971, 172–173).

48

Schubert [1879] 1979. See also Pieri [1915] 1991, an updated translation of the major 1905 report by H. G. Zeuthen on enumerative methods in algebraic geometry.

49

Hilbert [1900] 2000, 426; Schubert 1879.

50

For details of this controversy and brief discussions of the methods involved in attempting to settle it, see the studies Kleiman 1976 and Xambó-Descamps 1996. The latter emphasizes the work of the Italian mathematician Francesco Severi. Another study, Kleiman and Laksov 1972, very effectively presents the results of the Dutch mathematician B. L. van der Waerden, who employed methods of algebraic topology; and Laksov 1994 considers in detail the 1904 work of Giovanni Giambelli. According to the eminent combinatorialist Richard P. Stanley, “The work of classical geometers such as Schubert, Pieri, and Giambelli on the Schubert calculus was vindicated rigorously by Charles Ehresmann, Van der Waerden, W. V. D. Hodge, and others.” (Stanley 1999, 399)

51

Pieri 1895a, 607. See also subsection 9.2.1 of the present book, page 343.

52

See 9.2.2, page 351.

5.3 Synthetic Projective Geometry as an Autonomous Field

95

acknowledged such seminal results as the Desargues triangle theorem53 and its importance with respect to the fundamental theorem of projective geometry54 attributed to Staudt. Pieri joined a distinguished group of Italian geometers who advanced the synthetic approach to geometry. However, virtually all of them admitted nonprojective concepts in their constructions.55 Pieri stood apart from them. From his very first axiomatization of projective geometry he endeavored to construct the subject as a science in its own right, with no appeal to concepts extraneous to it. In this endeavor he owed his inspiration to Staudt. That is what we discuss next. 5.3 Synthetic Projective Geometry as an Autonomous Field In 1847 G. K. C. von Staudt published a book that presented projective geometry in a way that made no apparent appeal to metric ideas: Geometrie der Lage (Geometry of Position).56 He aimed to influence instruction in geometry as well as its underpinnings: Perhaps this writing will induce some teachers to preface to their lessons in the geometry of measure the essentials of the geometry of position, so that their students should gain at the very beginning that overview without which proper understanding of the individual theorems and their relation to the whole is not really possible.57

In the 1856–1860 three-volume work Beiträge zur (Contributions to) Geometrie der Lage Staudt extended his results, mainly to include and formalize synthetic complex projective geometry. The significance of these books with respect to the foundations of geometry 53

Pieri never failed to stress the importance of the Desargues theorem ([1636] 1987) and the significant role it plays in foundations of projective geometry (for examples, see subsections 9.1.3, 9.1.5, and page 361 in 9.2.5). The theorem states that if two triangles are perspective from a point (that is, the three lines joining pairs of corresponding vertices are concurrent) then they are perspective from a line (the intersections of corresponding pairs of edge lines are collinear). See the figure on page 86. The theorem holds in a model M of the projective incidence axioms if and only if M is isomorphic to the coordinate geometry of the same dimension over a (possibly noncommutative) division ring R; it also entails the existence of certain projective transformations (see Artin 1957, 71).

54

Pieri (1895a, §7, 625) noted that analysis of the premises on which projective geometry is based would not be complete if it did not lead at least to Staudt’s fundamental theorem, which like the Desargues theorem is a cornerstone of that doctrine.

Staudt’s theorem says that the only projective transformation of the points of a line that has three distinct fixed points is the identity. 55

Projective geometry can be defined as the study of properties of figures that remain invariant under the process of projection and section (see footnote 26). Properties of incidence are preserved by this process and so are called projectively invariant. Distance, angle magnitude, linear order, and parallelism are not projectively invariant, and so are not considered purely projective concepts by this definition. Pieri’s colleagues admitted nonprojective concepts such as linear order (Fano 1892 and Enriques 1894, for example) and distance (De Paolis 1880–1881).

56

For this term see footnote 3 in the introduction to this chapter. For a summary of the contents of the chapters of Staudt 1847, see Reich 2005.

57

Staudt 1847, iv.

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5 Pieri and Projective Geometry

was captured by the mathematician Hans Freudenthal: if one should follow the pattern of Hilbert 1896 and divide the history of projective geometry into naive, formal, and critical periods, then it would not be too farfetched to count the naive period up to Staudt, to have him open the formal one, and to start the critical period with Moritz Pasch, or the Italians, or with David Hilbert ... , rather than contributing to the contents of projective geometry, [Staudt] made an attempt at its formal recasting ... .58

Staudt proceeded from an intuitively conceived space containing the finite real points, lines, and planes of Euclidean geometry. He tacitly assumed their familiar order properties, but employed no concept of distance. In addition to these objects, he introduced and specified the behavior of their infinite and imaginary counterparts, making precise the somewhat vague practices of Jean-Victor Poncelet and his followers. Staudt’s “formal recasting” of projective geometry synthetically as an autonomous science focused to a large extent on his reformulation, in purely projective terms, of the concepts of harmonic quadruple and projective mapping, his construction of an algebra of scalars and a coordinate system, and his ingenious steps toward eliminating the metric notion of cross ratio from projective geometry.59 (For examples of metric reasoning, see the box on page 97.) Traditionally, the cross ratio L( A, B; C, D) of collinear points A, B, C, D was defined in terms of distance. It was central to projective geometry because it is invariant under projection and section, even though distance is not. The box on page 97 illustrates this. Just as the concept of distance provides a means to express Euclidean properties of figures, the cross ratio provided a means to express projective properties—for example, a criterion for separation of four points. Traditional expositions of projective geometry had begun with the cross ratio, basing much of the subject on that idea. Staudt wanted to free projective geometry from all metric notions. Thus, he needed to replace the cross ratio with a concept, based only on incidence, that would enjoy the same type of relationship to projective properties. To do this, Staudt devised a purely projective way to describe the concept of harmonic separation by means of a complete quadrangle. Traditionally, harmonic separation was based on the cross ratio: two points A and B on a line g were said to harmonically separate two other points C and D on g if their cross ratio is –1. Staudt adapted the construction of the fourth harmonic D that uses a complete quadrangle as described in the box on page 97. He made that into a definition of the fourth harmonic solely in terms of intersections of points and lines: If on a line three points A, B,C are given, and then a quadrangle is constructed so that a diagonal passes through the second of the given points, whereas two opposite edge lines intersect at each of the other two, then the other diagonal of the quadrangle intersects that line in a fourth point D, which is determined by the three given points and is called their fourth harmonic point.60

58

Hilbert 1896, 124; Freudenthal 1974, 191.

59

For more details on this discussion, see Freudenthal 1974 and Marchisotto 2006, §3–§5.

60

Staudt 1847, 43.

97

5.3 Synthetic Projective Geometry as an Autonomous Field

O

Cross Ratio. In traditional Euclidean geometry, the cross ratio of four points A, B, C, D on a directed line g is defined in terms of signed distances as L( A, B; C, D) =

CA / CB DA / DB

.

g If these points are joined to a point O not on g, the A C B D directed segments correspond to directed angles with vertex O. (See the figure.) Repeated application of the law of sines yields an expression for the cross ratio just in terms of the sines of those angles. Thus, the cross ratio is invariant under the depicted projection and section.* Harmonic Quadruples. In traditional projective geometry, a quadruple A, B, C, D of collinear points was called harmonic if their cross ratio is –1. Interchanging A with B or C with D does not affect this condition. When it holds, A and B are said to separate C and D harmonically, and C and D are called harmonic conjugates with respect to A and B. In fact, D is the only conjugate of C with respect to A and B: if L( A, B; C, Dr) = –1 for any point Dr, then DA / DB = DrA / DrB, and thus D = Dr. This permits the traditional definition of D as the fourth harmonic of A, B, C. Harmonic quadruples occur in almost every study of traditional projective or Euclidean triangle geometry. One such consideration yields an algorithm for constructing the fourth harmonic D, illustrated in the figure at right. Given points A, B, C on the line g, select any point O not on g, then any point P on OA. Construct the intersections Q = OC 1 PB and R = OB 1 QA, and finally D = PR 1 g.

O P Q A

C

R

B

D

g

Points O, P, Q, R and the six lines joining them constitute a figure known as a complete quadrangle. By Ceva’s and Menelaus’s theorems applied to  OAB with center Q and transversal PR,

OP AC BR PA CB RO

=1

OP AD BR PA DB RO

= –1 .

Consequently, the cross ratio ( CA / CB)/(DA / DB) = –1 and D is the fourth harmonic. This shows† that the constructed D is independent of the auxiliary points O and P. The fourth harmonic can always be constructed this way: if D is the fourth harmonic defined by the cross ratio, use this algorithm to construct point Dr. The preceding argument shows that Dr = D. All applications of the fourth harmonic can thus be done by means of complete quadrangles instead of cross ratios. * The symbol L is due to Veblen and Young 1910–1918, volume 1, 160. This result was discussed by Pappus of Alexandria ([c. 340] 1986, volume 1, §196, 262–263; volume 2, 457–458, 560–562). Cross ratios are often called anharmonic ratios. † A proof in the style of this paragraph, but arranged differently, can be found in Carnot 1803, 280–283. That

book introduced the systematic use of signed distances (Eves 1963–1965, volume 1, 64). The proof given here is from Möbius 1827, 239. See also Poncelet 1822, 13, and Steiner 1832, 18.

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5 Pieri and Projective Geometry

Instead of citing metric results as in the box on page 97, Staudt applied the Desargues triangle theorem three times to prove that D is independent of the auxiliary quadrangle.61 With this definition, Staudt could present all necessary applications of harmonic separation without referring to the distance-based notion of cross ratios. Staudt then used harmonic separation to redefine a type of transformation fundamental in projective geometry. Traditional projective geometry included the theorem that any triple of collinear points can be related to any other by a composition of perspectivities, the simplest correspondences between points on lines in a projective plane.62 A perspective image is a cross section of a projection. The top figure in the box on page 97 provides an example. Such compositions, called projectivities, were regarded as the fundamental transformations of projective geometry. They preserve cross ratios because perspectivities do. Are projectivities the only such transformations that preserve cross ratios? In the traditional approach to projective geometry it was easy to answer affirmatively, as follows. Given a transformation  from a line g to a line gr that preserves cross ratios, select any three points A, B, C on g, find a projectivity  that maps their images back to the original points, and let  =  . Then,  preserves cross ratios and fixes A, B, C. For any point D on g, set D* = (D); then, L( A, B; C, D) = L( A, B; C, D*) and therefore D = D*. Thus,  is the identity on g and  = –1, a projectivity.

This result facilitated verification that a given transformation should be a projectivity: merely check that it preserves cross ratios, instead of constructing a sequence of perspectivities. Staudt reformulated the traditional approach by replacing the condition of preserving cross ratios of quadruples of points with that of preserving the harmonic property. A correspondence between points on lines g and h in a projective plane that maps any harmonic quadruple of points on g to another one on h is called harmonic. To verify that a given correspondence is a projectivity, one checks instead that it is harmonic. This permitted Staudt to remove from this part of projective geometry the consideration of cross ratios and consequent dependence on metric techniques. To validate this approach, Staudt needed, in the argument displayed in the previous paragraph, to replace the implication if  preserves cross ratios and fixes three points on a line g, then  is the identity on g with this one: if  is harmonic and fixes three points on a line g, then  is the identity on g. The latter implication is Staudt’s fundamental theorem of projective geometry. The theorem is valid in the real projective plane, but Staudt’s proof was faulty: he relied on 61

The Desargues theorem is depicted in a box on page 86 in section 5.2. In three dimensions it can be established without appeal to metric ideas. Staudt’s uniqueness proof is presented in detail in Coxeter [1949] 1961, §2.5.

62

For a proof, see Coxeter [1949] 1961, §2.7.

5.3 Synthetic Projective Geometry as an Autonomous Field

99

unstated assumptions about the order of points on a line and about its continuity. Mathematicians learned some decades later that some such assumptions cannot be avoided.63 In his 1856–1860 Beiträge volumes, Staudt extended the geometry of position to include imaginary points, which he defined in terms of involutions. To make his fundamental theorem valid in the complex plane, he had to constrain his notion of projectivity to preserve not just harmonic quadruples but also the sense of imaginary points. See the box on page 100. Sometimes in projective geometry, as here, it is necessary to consider the analogy between points on a line and scalars in the real or complex number system. Staudt’s approach to projective geometry first incorporated real Euclidean lines, extended them with ideal points at infinity and imaginary points, then augmented them with new ideal and imaginary lines and planes. If the points on even one line are endowed with the properties of real or complex numbers, then those on any other would enjoy them as well, by virtue of a projectivity relating the lines. The properties of the points on a Euclidean line analogous to the algebra of real numbers, however, stem from metric notions: parallel translations and changes of scale. Can those be based on purely projective ideas instead? Staudt began an affirmative solution in the Beiträge, with his algebra of throws. Staudt’s method was based on the idea that in traditional projective geometry two quadruples A, B, C, D and Ar, Br, Cr, Dr with the same cross ratio are always related by a projectivity.64 Thus, at least part of the traditional role of the cross ratio can be played by equivalence classes T( A, B; C, D) of quadruples related by projectivities. Staudt called these throws.65 In traditional projective geometry it is possible to anticipate some consistent results when some of the arguments to the cross-ratio function coincide or lie at infinity, provided the latter are regarded as infinitely distant from the others and conventions such as 0 A  = 1 =  /  are used cautiously to handle  as an infinite scalar. This heuristic suggests a particularly simple uniform way to represent scalars x by cross ratios and thus by throws. Staudt designated three arbitarily selected points on a projective line g by the symbols , O, I : they play the roles of the point at infinity, with coordinate , and of the points with coordinates 0 and 1. If X were the point on g with coordinate x, then I  / IO  /1 L(, O; I, X ) = = = x. X  / XO /x Staudt thus reasoned that the throw T(, O; I, X ) could be used as the scalar coordinate x of X. 63

Staudt’s theorem is not valid if imaginary points are introduced: in complex projective geometry conjugation is a harmonic transformation of a real line and fixes all real points on it but is not the identity. Assuming the Desargues theorem, Staudt’s holds if and only if the coordinate field R is commutative and has no trivial automorphism. Those conditions follow from assumptions such as the Pappus–Pascal theorem and Archimedes principle. See subsection 10.2.5, page 476; Freudenthal 1974, 193; and Hartshorne [1967] 2009, chapters 5–6.

64

Proof. As noted on page 98, A, B, C and Ar, Br, Cr are related by a projectivity ; if D* = (D), then L( Ar, Br; Cr, D*) = L( A, B; C, D) = L( Ar, Br; Cr, Dr), and thus D* = Dr.

65

Staudt 1856–1860, 15. Some authors use the noun cast instead of throw. Staudt’s word was Wurf. Some use jet in French, and some, simply equivalents of the noun quadruple.

100

5 Pieri and Projective Geometry

Staudt’s Framework for Complex Projective Geometry. As remarked on page 96, G. K. C. von Staudt’s projective geometry included not just the real Euclidean points, lines, and planes but also the analogous ideal elements at infinity and imaginary elements. Mario Pieri’s noted 1898c study, translated in chapter 6 of the present book, provided a rigorous axiomatic foundation for Staudt’s 1847 book, which considered only real projective geometry, without imaginary elements. Therefore, most of this chapter is devoted to real geometry, and this section is devoted to Staudt’s work that underlay Pieri’s achievement. Some notable aspects of Staudt’s theory cannot be explained adequately without mentioning his provisions for handling imaginary elements. They are outlined in this box. A more comprehensive discussion is to be found in subsection 9.2.9. Staudt’s theory relied on Euclidean properties of order and orientation. Hard to describe with complete precision, these were generally assumed tacitly in Staudt’s time. Staudt treated them precisely enough, however, for use in his complicated constructions. Their most salient feature is that a real projective line has two possible senses, or cyclic orders. A second central feature of Staudt’s theory was the notion of elliptic involution: a self-inverse harmonic transformation of the points of a projective line that has no fixed point. Elliptic involutions arise in applications of projective geometry in such a way that it would simplify discussions if they had fixed points. Investigations in complex analytic geometry had shown that these should have complex coordinates satisfying the real equation of the line, and they should occur in conjugate pairs. Previously, points at infinity had been introduced in a consistent way as new names for parallel pencils, and lines and the plane at infinity were introduced to make the projective incidence axioms hold. Staudt described the consistent extension of the notions of order and orientation to these elements. Making others’ earlier informal methods precise, Staudt introduced imaginary points in an analogous way as new names for ordered pairs consisting of an elliptic involution on a real line g and one of the senses of g. Once coordinates are introduced, the two senses yield conjugate points. He introduced imaginary lines and planes in a consistent way as well, to make the projective incidence axioms hold.* In order to make the fundamental theorem valid for imaginary elements, Staudt had to replace his notion of harmonic correspondence with a stronger version: harmonic correspondence that preserves the senses of the imaginary points.† For this version of the notion Staudt provided no distinctive name beyond the adjective projektivisch, which he used for all versions. Julian L. Coolidge used the phrase relating figures of four elements to others having the same relation with regard to sense.‡ Hans Freudenthal reported, ... there were hardly any contemporaries who tried to understand von Staudt’s method ... This subject has never become popular, which is not to be wondered at. ... Everybody would ask: Is this worth the formidable effort? ... It is not true that von Staudt’s [1856–1860] Beiträge could be erased from the course of history of mathematics. ... It would not be until twenty years after von Staudt that a wave of abstractions as bold as von Staudt’s sets in. It strikes one that this step was done in geometry earlier than in algebra. ... [Was it] because there the abstraction could safely be guided by intuition? #

Riding that wave, Pieri took up the challenge. In his 1905c and 1906a papers he provided a rigorous axiomatic foundation for Staudt’s complex projective geometry. These papers are considered in detail in subsection 9.2.9. *Staudt 1856–1860, §7.

† Ibid., §17, n. 226.

‡ Coolidge 1900, §5, 188.

#Freudenthal 1974, 196–198.

5.3 Synthetic Projective Geometry as an Autonomous Field

101

In classical Euclidean geometry, two points O and I determine a coordinate system on their line g : they have coordinates 0 and 1, respectively. Given points on g with coordinates x =/ 0 and y it is easy to use parallel lines to construct the points with coordinates x + y, –x, x A y, and 1/ x; distances are not explicitly involved. Such constructions have counterparts in the projective plane: parallel lines correspond to lines meeting at infinity. These constructions can be adapted to apply to throws corresponding to points on a projective line g. Thus, Staudt built, entirely with projective tools, an algebra of throws that models some aspects of the algebra of numbers. He verified for finite throws what are now called the axioms for the theory of fields, but did not consider the details of order and continuity in that structure.66 Given three points A, B, C with coordinates a, b, c, on a projective line g, the familiar formula for a cross ratio L(A, B; C, X ) can be used to define a linear fractional transformation x ² xr of the coordinates x of points X on g: xr =

( c  a) / ( c  b)

.

( x  a) / ( x  b)

The corresponding transformation  of the points X with coordinates x maps A, B, C to , O, I. Moreover, it is a projectivity.67 Therefore, T(A, B; C, D) = T(, O; I, X ) = x = L(A, B; C, X ) .68 Thus, the notion of throw captures the essence of the cross ratio. Staudt redefined scalars in purely projective terms, as throws, and established a particularly simple connection between cross ratios and throws analogous to the classical definition of cross ratio in terms of distance. By means of the theory of throws Staudt “freed the cross ratio from metric bondage.” 69 According to Freudenthal 1974, “Staudt started something that was essentially new,” the harbinger of a new period. He was “the first to raise the foundational question and to aspire to purity of methods in projective geometry.” His reformulations of familiar concepts such as harmonic quadruple, projectivity, and coordinate arithmetic to gain insight into their fundamental properties was abstract and astonishing.70 Many years elapsed until others dared to consider abstractions as bold as Staudt’s and to take such 66

Staudt’s algebra of throws is valid in both real and complex projective geometry. For the material in this paragraph, see Freudenthal 1974, 198–199, Veblen and Young 1910–1918, volume 1, chapter 6, and Staudt 1856–1860: vol. 1, §1, n. 24–28; §9, n. 142–144; vol. 2, §19–21, n. 256–291; §27, n. 393–396; §29, n. 404–412.

67

To see this, note that x ² xr is a composition of operations of the form x ² x + y, x ² –x, x ² x A y, and x ² 1/ x. Their constructions, mentioned in the previous paragraph, explicitly show that the corresponding point transformations are compositions of perspectivities. 68 This argument can be continued to show that every projectivity  on g corresponds to a linear fractional coordinate transformation. If A, B, C = –1(), –1(O), –1(I), then the composition  , a projectivity, fixes , O, I. By Staudt’s fundamental theorem it is the identity, and thus  = –1, a linear fractional transformation. 69

Henderson and Lasley 1938, 9.

70

Freudenthal 1974, 190–199.

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5 Pieri and Projective Geometry

liberties with definitions. That his proof of the fundamental theorem relied on hidden continuity assumptions did not detract from its influence, but spurred the best minds of the late nineteenth and early twentieth centuries to refine his work, make his arguments rigorous, and ascertain the true range of their validity. Among those mathematicians was Mario Pieri. On the one hand, Pieri was inspired by Staudt’s ingenious methods to rid projective geometry of metric concepts. On the other, like many of his contemporaries, he was acutely aware of the flaws in Staudt’s treatment. Pieri’s engagement with Staudt’s legacy began around 1886 with Corrado Segre’s suggestion that Pieri should translate and analyze Staudt’s 1847 Geometrie der Lage (see subsection 9.1.2).71 Pieri chose axiomatics as a vehicle for making Staudt’s arguments rigorous (see section 5.4). A major influence on Pieri’s work in disseminating and clarifying Staudt’s ideas was Theodor Reye’s monograph on projective and algebraic geometry, which first appeared in 1866. As late as 1922, Heinrich Timerding noted, its historical value for the development of geometry will remain undisputed and it is still unmatched for its didactic effectiveness.

Reye’s ideas appear to have provided a foundation for much of Pieri’s work. Pieri adopted and extended Reye’s arguments, and sought new ways to support them.72 In 1873 Felix Klein reported a gap in the proof of the fundamental theorem of projective geometry that Staudt had presented in 1847.73 During the next decades he and several others provided improved arguments in that direction. These contributions involved both application of continuity assumptions 74 and the introduction of real numbers as scalar coordinates, mentioned earlier. They contributed to Pieri’s results in his 1898c and 1904a papers, translated in chapter 6 and discussed in detail in subsections 9.2.5, 9.2.8, and 10.2.5. Three of them stand out particularly as inspiring Pieri’s research: Thomae 1873, Darboux 1880, and De Paolis 1880–1881.

71

Segre was Pieri’s senior colleague at the University of Turin. For a biographical sketch of Segre, see M&S 2007, 107–108.

72

Timerding 1922, 192. In his 1889a translation of Staudt 1847, Pieri proposed some modifications, crediting Reye (Marchisotto 2006, §7). Reye’s organization of the subject and some specific achievements are cited in Pieri 1897c, §5, 349, and in 1898c, §5, 27 (translated in section 6.5 in the present book, page 173). Pieri, in his own papers on algebraic and inversive geometry, also cited Reye’s work. For a biographical sketch of Reye, see M&S 2007, 99–100.

73

Klein 1873, §5, 139. Staudt 1847, §9, paragraph 106, page 50. Evidently, no one had reported the gap in print until then (Natucci 1952, 13). But Karl Weierstrass had presented an alternate proof in his geometry lectures sometime during 1858–1870. Klein had visited Weierstrass in 1870 and discussed the notion of continuity and the work of Arthur Cayley. See Weierstrass 1903b, Knoblauch 1903, and Voelke 2008, 288.

74

Near the beginning of this period, Richard Dedekind published Continuity and Irrational Numbers ([1872] 1963), which showed how to formulate useful continuity assumptions.

Gaston Darboux around 1884

Jean-Gaston Darboux was born in Nîmes in southern France in 1842, the elder of two children of a haberdasher. His father died when Jean-Gaston was seven. He studied at lycées there and at Montpellier, then entered the École Normale Supérieur in Paris. He completed those studies in 1866 with a dissertation on differential geometry of surfaces, supervised by Michel Chasles and others. Darboux taught in secondary schools until 1872. During that time he founded the Bulletin des sciences mathématiques et astronomiques, which is still published. He taught at the École during 1872–1881, then succeeded Chasles at the Sorbonne in 1880. There he pursued a long career, serving as dean of the science faculty during 1889–1903. His bibliography in Lebon 1913 lists 419 publications. In an 1880 article on projective geometry he provided a major step toward Pieri’s later axiomatization of that subject. Darboux’s four-volume 1887–1896 monograph on differential geometry became a classic. His 1904–1905 survey article on geometrical methods provided background for the present book. Darboux married in 1872; his family included a son and a daughter. In 1884 he was elected to the French Academy of Sciences. During his life he received comparable recognition from fourteen other countries. Darboux died in 1917.* *For further information about Darboux, consult Forsyth 1917 and Lebon 1913.

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5 Pieri and Projective Geometry

In his 1873 booklet on foundations of projective geometry, Johannes Thomae emphasized, as a basic concept, rotation of a pencil of lines through a fixed center. Based on that, he described motion of a point along any other line in the plane of the pencil, giving the line a “left-right” orientation and identifying the interior and exterior of any segment in it. This let him adapt and present more cleanly Staudt’s proof of the fundamental theorem.75 Reye adapted this for the 1877 second edition of his influential text Geometrie der Lage: Staudt’s proof of the fundamental theorem of projective geometry, in view of the justified objections that Klein has raised against it, has been replaced with a proof originating from 76 Thomae.

In 1880 Gaston Darboux provided two arguments important in the evolution of thought about projectivities. First, using real coordinates and analytic methods he showed that any transformation of the points of a line that preserves harmonic quadruples must be linear, and hence continuous and order-preserving. And if it fixes three distinct points, then it must be the identity: Staudt’s fundamental theorem.77 Darboux’s second argument, without referring to coordinates, partially supported the following result: if points A and B are collinear with but do not separate points E and F, then there exist points C and D such that A, B, C, D and C, D, E, F are harmonic quadruples.78

Darboux proceeded to derive Staudt’s theorem from this one, using results of Jakob Lüroth and H. G. Zeuthen published by Klein in 1874. Darboux’s second argument involved notions of order, motion, and continuity that had not yet been organized formally and precisely enough to constitute a complete argument for the fundamental theorem. Darboux’s work was widely appreciated, and Pieri frequently referred to it. Pieri eventually recast the result displayed above as a definition of the phrase “E belongs to the projective segment determined by A, B that contains F;” this in turn enabled him to define segment solely in terms of the incidence of points and lines.79 The efforts of Thomae and Darboux and others to completely justify Staudt’s theorem led to deeper consideration of the role of motion in projective geometry. Continuous motion had been central to Staudt’s presentation. Among the basic notions described on his first page were point and line: “when a point moves, it describes a line.” Motion was possible in two senses, forward and backward. If a moving point should proceed from point a to b to c, but never return to a previous position, then b was said to separate a, c. If it should proceed beyond c to d and then return to its starting position, then a, c were said to separate b, d. Staudt (very informally) called a line straight if “its position cannot change when it is held fast at two points.” After extending these concepts 75

Thomae did not refer to Klein's work, but the two had common interests and were situated rather close to each other, in Halle and Göttingen, respectively. For further information about Thomae, see a box in section 5.5, page 117.

76

Reye 1877–1882, part 1 (1882 edition), xiii. Staudt 1847, 45.

77

Darboux 1880, 55–58. Darboux used nonhomogeneous coordinates.

78

Darboux 1880, 58. The converse theorem—if such points C, D exist, then A, B cannot separate E, F —is elementary.

79

Darboux did not mention Thomae’s work. Pieri 1895a, §7; see subsection 9.2.1.

5.3 Synthetic Projective Geometry as an Autonomous Field

105

by introducing points and a line at infinity, Staudt noted that an extended line is separated by any two of its points into two complementary segments. Thomae and Reye, on the other hand, considered only straight lines. At the beginning, Reye introduced the notion of segment informally as bounded part of a line, without defining bounded. Soon after, referring to motion along a line, he introduced points at infinity and separation of two pairs of points.80 A complete argument for Staudt’s fundamental theorem would emerge from formal organization of the notions of order and motion for points on a line, and of its continuity. Steps in that direction were provided by Riccardo De Paolis soon after Darboux’s work. De Paolis had earned the laureate in Rome in 1875, studying under Luigi Cremona. In 1880 he was appointed professor in Pisa. De Paolis was Pieri’s teacher there during 1883–1884 and continued counseling Pieri after that. The 1880–1881 paper On the Foundations of Projective Geometry by De Paolis has been called the first important Italian contribution to that subject. It built on results in Staudt 1847, Thomae 1873, and Reye 1877–1880, and constituted a major step toward filling the gaps in Staudt’s proof of his fundamental theorem, particularly the issue of continuity, following suggestions of Klein in 1873 and 1874. (De Paolis felt that Thomae’s and Reye’s proofs still suffered from the same defects as Staudt’s.)81 De Paolis’s main concept was the fundamental form of the first kind. It consists of an ordered set and a concept of motion within it, which allows his definition of projective segment: if an element E of such a form starting from an initial position E1 arrives at a final [position] E2 we shall call the part of the form described by E the segment E1 E2 .

To reflect the idea that two distinct points should determine two complementary segments, De Paolis would specify a particular element E of E1 E2 , writing E1 E E2 to identify the segment containing E without referring explicitly to the direction of the motion. Elements E1 , E2 are said to separate E from elements of the segment complementary to E1 E E2 .82 Using these ideas De Paolis derived the theorem of Darboux stated earlier, without using coordinate methods, then applied it to show that a harmonic transformation : X ² Xr between the points on two lines g and gr preserves order: 83

80

Staudt 1847, §1, paragraphs 2–7, pages 1–3; §5, paragraph 61, page 27. Reye 1866–1868, 16. De Paolis 1880–1881, 491–492.

81

Avellone et al. 2002, 366, 369. For a biographical sketch of De Paolis, see M&S 2007, 78. Descendants of Pieri’s family gave the current authors access to Pieri 1883–1884, Geometria superiore dalle lezioni del Pr. Riccardo De Paolis: handwritten notes from the University of Pisa.

82

De Paolis 1880–1881, 489. De Paolis was tacitly excluding complex projective geometry. He used the verb separate (493) but did not explicitly define it.

83

De Paolis 1880–1881, 493.

106

5 Pieri and Projective Geometry

If four points A, B, E, F are situated on g so that A and B do not separate E from F, then there will exist points C and D on g such that A, B, C, D and C, D, E, F are harmonic quadruples. So are Ar, Br, Cr, Dr and Cr, Dr, Er, Fr, because  is harmonic. On gr, therefore, Cr and Dr must separate Ar from Br and Er from Fr, and thus Ar and Br cannot separate Er from Fr.

De Paolis used this result and those of Lüroth and Zeuthen mentioned earlier to achieve his ultimate aim of coordinatizing real projective space on the basis of purely geometric considerations. Instead of using the algebra of throws to play the role of the real number system, he introduced by projective methods a harmonic system to define an injection  from the rational numbers to the points on a line g. Postulating completeness of g and selecting arbitrarily a single point  on g, he extended  to a bijection from the real numbers to the set of points on g different from . The weakness in De Paolis’s approach was that the motion inherent in his main concept, like the continuity ideas proposed earlier by Klein, was still based on Euclidean geometry. Nevertheless, De Paolis’s work “constitutes a leap ahead with regard to the existing approaches, and it had a deep impact on the contemporary Italian school.” 84 Building on the results of these and other scholars, Pieri focused not only on making Staudt’s arguments rigorous, but also on eliminating all dependence on concepts external to projective geometry. Staudt’s endeavor to replace metric concepts had been conducted in a Euclidean environment that allowed for motion and parallel lines. Pieri devoted himself to the following particular tasks: • replacing Staudt’s Euclidean environment with a purely projective one—in all his axiomatizations of projective geometry, Pieri would assume a projective environment a priori, postulating the relationships between its fundamental entities (see subsection 9.2.5 of the present book); • correcting Staudt’s proof of his fundamental theorem (subsections 9.1.2, 9.2.1, 9.2.5, and 9.2.8); and • refining Staudt’s definition of projectivity so that it relied on a minimal set of assumptions (section 5.5, chapter 7, and subsection 9.2.10). Staudt inspired Pieri’s research in projective geometry. In this rich environment of his colleagues’ results, Pieri would be the first to make Staudt’s vision for projective geometry a reality. Moritz Pasch, Giuseppe Peano, and the members of Peano’s school would guide Pieri in the methods he chose for this achievement. In his 1898c paper, translated in chapter 6, Pieri would construct projective geometry as a science in itself, free from all metric ideas.

84

Avellone et al. 2002, 366–367, 369. De Paolis 1880–1881. De Paolis’s harmonic system was essentially the net introduced by August F. Möbius ([1827] 1959) and often termed “net of rationality.” Compare the discussion of Euclidean motion in section 5.2, pages 86–87.

107

5.4 Geometry as a Logical System

5.4 Geometry as a Logical System 85 Pieri started research in projective geometry amid a proliferation of attempts to establish the subject on a rigorous basis. He entered the arena benefiting from the research of those who believed that the construction of geometry as a logical system provided one avenue to pursue. This section highlights the influence of Moritz Pasch, Giuseppe Peano, Cesare Burali-Forti, and Alessandro Padoa on Pieri’s foundational approach and shows how Pieri forged his own path as he built upon their work. Choosing Axiomatics. Largely because of the influence of Peano and his school, Pieri changed his original research direction, algebraic and differential geometry, to include investigations into mathematical logic and axiomatics. Of the seventeen papers he produced on foundations of geometry, thirteen are about projective geometry. These were devoted to G. K. C. von Staudt’s vision of the subject, but inspired in their construction by the work of Pasch and Peano. Pasch had produced the first axiomatic treatment of projective geometry in 1882, and Peano expanded the modern axiomatic view that Pasch initiated.86 Pieri adopted the axiomatic approaches of Pasch and Peano; there was deep crossfertilization between their ideas and his. However, with respect to the idea that geometry describes the physical world, Pieri distanced himself from those scholars. Pieri embraced Pasch’s work, but not his empiricist point of view. Pieri often cited Pasch’s results and freely expressed his debt to Pasch.87 The link between them was affirmed by the philosopher Ernest Nagel, who observed, The writings of Pasch contain one of the first clear expressions of the view that pure geometry is a “hypothetico-deductive”system, whose “axioms” are in effect simply “implicit definitions” of the terms they contain.

Nagel acknowledged Pieri’s introduction of the “neat” term “hypothetical-deductive system” and its general application by the Peano school.88 In opposition to empiricist doctrine, Pieri believed, like Peano, that undefined terms can have meanings divorced from the real world and that logical consequences of the postulates should be developed without

85

Parts of this section appeared in Marchisotto 2011.

86

Pasch 1882b. Peano 1889, 32–33; 1894b, 54–55. For a brief sketch of Pasch’s life see M&S 2007, §1.3. Peano’s life and work and his relationship to Pieri are discussed in detail there in §5.1, §5.3.1.

87

Pasch [1896] 1997; see the discussion in subsection 9.2.1, page 343. See also these translations or discussions of Pieri’s papers: paper location 1898b . . . . . 9.2.6 1898c . . . . . chapter 6

88

paper location 1900a . . . . . . . chapter 8 [1900] 1901 . . chapter 4

paper location 1904a . . . . 9.2.8 1908a . . . . M&S 2007, chapter 3

Nagel 1939, §62, 193–194, 199. Nagel referred to Pieri 1895a, which employed the notion of hypotheticaldeductive system. That term came into use gradually, for example in Pieri 1898b and in the titles of Pieri 1898c and 1900a and the exposition in [1900] 1901, translated in chapters 6, 8, and 4 of the present book.

108

5 Pieri and Projective Geometry

regard to those meanings.89 They both believed that postulates should be simple statements. Pieri explicitly revealed that he crafted his postulates so that they could not easily be broken down into smaller components. He regarded their irreducibility and independence as goals “touching on ideal perfection.” Both Pieri and Peano were concerned with the completeness of their sets of postulates, in the sense that they should provide the raw materials for rigorous proofs of all desired theorems.90 Pieri rejected Peano’s belief that for a work to merit the name “geometry” it is necessary that the postulates express the result of observations of physical figures.91 Pieri did not tie his postulates to any specific interpretation or to application in our physical world. He sought to demonstrate how the form of the deductive science prevails over its substance. Pieri believed that geometry could exist independently from any special interpretation of its primitive concepts: he invited interpretations completely removed from any sense conferred by ordinary intuition of space. That being said, Pieri did believe that despite the freedom to choose assumptions in axiomatizations, ultimately the choice would be inspired or confirmed by real-life applications. Although intuition of our world plays no role in deductions within an abstract logical system, it should be used, notably in the classroom, to motivate or confirm the choice of primitive notions and postulates.92 Pieri also distanced himself from Peano’s view of the relationship between projective geometry and elementary geometry (the ordinary geometry of the plane and space). Peano envisioned real projective geometry as a part of general geometry (Euclidean or non-Euclidean) that could be derived from elementary geometry. Peano made it clear that his treatment of the geometry of position could not contradict the axioms of elementary geometry: indeed, it stemmed from them. Peano started from ordinary geometry and proceeded to the stage where projective entities could be introduced. His postulates established the relationships among the ordinary entities of general geometry; then he introduced “opportune definitions” of projective entities.93 For example, an ideal point at infinity can be defined as a bundle of lines. Pieri, on the other hand, would not consider elementary geometry in his construction of projective geometry. He would assume a projective environment a priori, postulating the relationships between its fundamental entities. A deep cross-fertilization of ideas between Peano and Pieri was reflected in their axiomatic constructions. In particular, both addressed the need to achieve rigor and simplicity for the subject, as well as the instrument to accomplish that: algebraic logic.

89

Julius Plücker had already claimed that “Every geometrical relation is to be viewed as the pictorial representation of an analytic relation which, irrespective of every interpretation, has its independent validity” (1846, 322, quoted in Nagel 1939, 191). Pieri acknowledged that precedent (1906d, 43).

90

Peano 1894b, 55, 61–64. Pieri 1895a, §1, 607; 1896c, §1, 10. See also subsection 9.2.2 of the present book.

91

Peano 1894b, 54, 75.

92

See Pieri 1894b, 39. This last aspect of Pieri’s concept of hypothetical-deductive system has often been neglected: see, for example, Moritz 1928, 417.

93

Peano 1894b, 73, 75.

5.4 Geometry as a Logical System

109

Using Mathematical Logic. Pieri adopted the view, “logic is a mathematical doctrine through its form and mathematics is a logical doctrine through its method.” This permeated not only his research but his teaching. He believed that the fusion of logic with mathematics was “of high importance for the philosophy of mathematics” and had fostered a “reform of great consequence” in studies, such as those of Peano and Bertrand Russell, that systematically promoted it. Pieri proposed that teaching geometry as a logical system was the ideal approach to pedagogy.94 A significant feature of Peano’s program was his investigations into mathematical logic, with the goal of establishing a symbolic notation for formal reasoning.95 Pieri was deeply influenced by Peano and other members of his school, notably Burali-Forti and Padoa, who were ardent champions of Peano’s mathematical logic.96 In his axiomatic expositions of mathematical theories, Pieri utilized Peano’s symbolic language implicitly or explicitly. But again Pieri would forge his own path. His recourse to the logical calculus coincided only in part with Peano’s intention for its use. Like Peano, he recognized its utility for achieving rigor, clarity of exposition, and ease of analysis. But unlike Peano, he also saw it as a basis for constructing mathematical theories as abstract logical systems, in which theorems are derived from postulates by application of the laws of logic, independent of any appeal to intuition. Burali-Forti’s expositions and his further development of Peano’s logic provided important tools for Pieri. In particular, Pieri utilized Burali-Forti’s notation and his ideas on definition and class membership to create axiomatizations of mathematics as abstract logical systems.97 Pieri aligned himself with Padoa with respect to his formal approach to geometry.98 Unlike Peano, they did not tie their postulates to any specific interpretation or to any application in our physical world. They both understood logical consequence of the postulates as truth in all interpretations that satisfy them.99

94

Pieri 1906d, 41, 49. For the first quotation, Pieri cited the report Couturat (1904, 1037), which was quoting the address Sur la fusion progressive de la Logique et des Mathématiques given by Henri Fehr at the 1904 Second International Congress of Philosophy in Geneva. See also Ingaliso 2011, 237, 242.

95

Rodríguez-Consuegra 1991, 92; Peano 1894a, 3.

96

For brief biographical sketches of Burali-Forti and Padoa and an account of Pieri’s relationship to Peano, see M&S 2007, §1.3, §5.1.

97

Pieri cited Burali-Forti’s contributions (1894a, 1894b, 1896–1897, [1900] 1901) in Pieri 1895a, §10; 1896c, §2, §4; 1906g, 204–205; and in 1898c, 1900a, and [1900] 1901, translated in sections 6.0, 8.1, and 4.3 of the present book.

98

Pieri cited Padoa’s work ([1900] 1901, [1900] 1902, 1902]) in Pieri 1896c, §2; 1906g, 203–204; and in [1900] 1901, 1907a, and 1908a—the last three are translated in section 4.3 and M&S 2007, §4.2, §3.0.

99

See Jané 2006, 24–25, and section 3.4 of the present book.

110

5 Pieri and Projective Geometry

Minimizing Assumptions. Pieri was committed to Peano’s goal of reducing the number of undefined terms.100 Pieri’s first two axiomatizations of real projective geometry 101 were based on three primitives: projective point, projective line joining two points, and projective segment determined by three points. To put Pieri’s choice of primitives into perspective, compare it with those of Pasch and Peano. In 1882 Pasch had admitted four undefined terms: point, Euclidean segment between two points, coplanar set of points, and congruence of figures. In 1894b Peano reduced those to only three: point, segment between two points, and motion.102 But the three primitives of Pieri’s first two axiomatizations once again reflect an important departure from Peano. Although both took point as an undefined term, their choices for the other primitives differed.103 Although they both included segment as undefined, Pieri’s was projective, determined by three points, whereas for Peano, like Pasch, a segment was determined by its two extremities. Peano, unlike Pieri, did not choose line as primitive. By assuming as primitive the segment ab —the class of points lying between two distinct points a and b —Peano could define the ray arb as the set of points c such that b 0 ac and then the line joining a and b as the union ab c { a, b } c arb c bra.104 Peano’s reasons for eschewing line may have gone beyond his desire to reduce the number of primitives. Since he envisioned geometry as the science of the space in which we live, Peano, like Pasch, may have regarded segment as a notion more fundamental than line for the part of geometry that establishes a body of postulates based on spatial intuition. He related it to the practices of drafting, smithing, and agricultural surveying: one verifies that a point c lies between a and b when a person located at a should see that 105 the object c covers b.

Pieri rejected the use of Euclidean segment as primitive because he was axiomatizing projective geometry independently of Euclidean geometry. The property of being a Euclidean segment is not invariant under projectivities: Euclidean segments are not objects of projective geometry. Using point, projective segment, and projective line as primitives,

100

For example, Pieri [1900] 1901, §III, 379, translated in section 4.3 of the present book, page 63.

101

Pieri 1895a and 1896a–c.

102

Peano’s 1894b axiomatization was built on Peano 1889, which developed the elementary geometry of incidence and betweenness on the basis of two primitives—point and segment—and seventeen postulates. He acknowledged (1894b, 55) that the first eleven had been inspired by Pasch 1882b. Although previous geometers, particularly Staudt, had employed the idea of motion in formulating the concept of segment (see section 5.3, pages 104–105), Pasch had avoided that by adopting postulates characterizing segments, and Peano followed Pasch (1882b, §1) in that regard. Peano’s 1894b treatment of motion proceeded in other directions.

103

Pieri’s primitives not only differed, they had no connection with experience. Peano had made it clear that for him the primitives should be simple ideas “known to all men” (Peano 1894, 52).

104

Peano 1894b, 60–61. Peano’s segments do not contain their end points.

105

Peano 1894b, 54–55. See also E. H. Moore 1902, 144.

5.5 The Transformational Approach

111

Pieri provided in 1895a an unencumbered route into elementary projective geometry, without depending on any nonprojective concepts.106 Pieri regarded projective segment as undefined only in his first two axiomatizations of projective geometry. By his third construction, 1897c, he was able to eliminate segment from the list of primitive notions by defining projective segment entirely in terms of incidence of points and projective lines. Peano embraced Pieri’s decision, calling this reduction of primitives “truly remarkable.” 107 In 1898c Pieri used the same two undefined terms. In 1898b and 1901b, he constructed projective geometry based on two different pairs of primitive notions: for the former, point and homography, and for the latter, projective line and whether two lines intersect. The efforts of Pieri and others were recognized for advancing a critical synthetic approach to geometry. In a 1904 address in St. Louis, the American mathematician James Pierpont observed, The critical synthetic direction represents a return to the old synthetic methods of Euclid, Nikolai I. Lobachevsky and Bolyai János, with the added feature of a refined and exacting logic. ... Geometric intuition has no place in this order of ideas, which regards geometry as a mere division of pure logic. The efforts of this school have already been crowned with eminent success, and much may be expected from it in the future. Its leaders are Peano, Giuseppe Veronese, Pieri, Padoa, Burali-Forti, and Tullio Levi-Civita in Italy, David Hilbert in Germany, E. H. Moore in America, and Russell in England.108

Pieri was often mentioned, without bibliographic citation, in general statements about contributions to axiomatic research in the nineteenth century. For example, writing about the “logical grounding of geometry upon sets of axioms,” D. M. Y. Sommerville credited Pasch for inaugurating the effort, noting however that “we must go back to von Staudt for the true beginnings.” Then, Sommerville gave credit to continued endeavors in this direction by Hilbert and by “an Italian school represented by Peano and Pieri,” as well as to Oswald Veblen, the “chief representative in America.” Sommerville indicated that such research had led to “severe logical examination of the foundations of mathematics represented by the Principia Mathematica of Russell and Alfred N. Whitehead.” 109 5.5 The Transformational Approach This section describes some developments in the study of geometric transformations that relate particularly to Mario Pieri’s research in foundations of projective geometry. A more comprehensive background in transformational geometry is offered in chapter 7, which 106

Pieri 1895a, §7, 626. In §9, 637, Pieri did prove that a projective line is the union of two projective segments and two points, but his proof depended on his postulates about projective lines.

107

Peano 1905, 93.

108

Pierpont 1904–1905, 158. In the corresponding list in the 1906–1908 republication of Pierpont’s address, Russell was replaced by Pasch.

109

Sommerville 1914, 193. Whitehead and Russell 1910–1913.

112

5 Pieri and Projective Geometry

prepares for the translation of Pieri’s 1900a Point and Motion monograph in chapter 8. Transformations play several roles in foundations of projective geometry. Previous sections discussed transformations of a projective line, or from one line to another, such as projectivities, defined as compositions of sequences of perspectivities or as transformations that preserve the harmonic relationship among quadruples of points.110 The present section, however, is concerned with transformations of an entire projective space onto itself. In his 1872 Erlanger program, Felix Klein proposed that to every group of transformations of a projective space S corresponds its geometry, which studies properties of figures in S that are invariant under those transformations, but not under all those in any larger group. Geometries of two spaces could be considered the same if the groups are isomorphic.111 In that context we may consider a projective space S and define projective geometry as the study of properties invariant under the group G of all homographies, or collineations—transformations of S that preserve the relation of collinearity.112 The role of S could also be played by a space consisting of other entities, such as lines, that inherit structure from an encompassing projective space. Klein also noted that in two dimensions, for example, the space S might be enlarged to consist of the points and the lines in a projective plane, and the group G, to include correlations: transformations that map points to lines and lines to points while preserving incidence relationships; this extension will be discussed in section 5.7, on duality. Klein’s program was significantly inspired by collaboration with his close friend, Sophus Lie. During the next two decades, Lie vigorously developed his allied research area, continuous groups. Whereas finite transformation groups can represent discrete symmetries of geometric objects or algebraic systems, such as icosahedra or roots of polynomials, Lie’s groups can represent the continuously varying symmetries of smooth manifolds or sets of solutions of differential equations. Lie’s work quickly attracted many followers. At first, however, Klein did not broadly publicize his Erlanger program, except perhaps in its role as a common element of his own investigations of various subjects.113 According to mathematics historian Thomas Hawkins, the earliest study of geometry in the spirit of the Erlanger program was the 1885 paper on conics by the very young professor Corrado Segre in Turin. In 1890 Segre’s student Gino Fano translated Klein’s 1872 booklet into Italian.114 Klein had opened that with the observation,

110

Point and Motion is an axiomatization of absolute, or neutral, geometry from the transformational point of view. According to the fundamental theorem of G. K. C. von Staudt, these definitions of projectivity are equivalent (see section 5.3 of the present book, page 98).

111

Klein 1872, §1–§3. See Kunle and Fladt 1974, Birkhoff and Bennett 1988, and Gray 2005 for an overview of the program, and Hawkins 1984, 459–461, for details about isomorphism. For a biographical sketch of Klein see M&S 2007, §1.3.

112

By Staudt’s theorem the collineations in two or three dimensions are the compositions of sequences of central collineations, each of which leaves fixed all lines through a central point and all points on an axial line or plane, respectively. Properties such as connectedness or cardinality that are preserved under larger transformation groups fall outside projective geometry.

113

See Rowe 1989 and Hawkins 1984. For a biographical sketch of Lie, see the box on page 119.

114

Hawkins 1994, 187–190. C. Segre 1885. Klein [1872] 1889–1890.

5.5 The Transformational Approach

113

among the advances of the last fifty years in the field of geometry, the development of projective geometry occupies the first place.

Interest remaining high, Klein increased his emphasis on publicizing the Erlanger program, and repeated that assessment in 1893 in its improved version in the leading journal Mathematische Annalen.115 Research in transformational geometry increased, particularly in Italy. In 1986 the renowned mathematician Jean Dieudonné wrote about Klein: ... nobody else in the 19th century expressed so forcefully, and with such clarity and concision, fundamental concepts such as group actions or isomorphisms of structures, with which we are now familiar. ... I think we are justified in seeing in [it] the heralding of the coming of modern mathematics.116

This approach to geometry was followed by many noted scholars; there is no question that Pieri was influenced by them. Pieri had joined the vibrant research community in Turin as professor at its military academy in 1886, then also as assistant to the chair of projective geometry, and libero docente, at the adjoining university. This group was led by Enrico D’Ovidio; besides Segre and Fano, it included—off and on—the geometers Guido Castelnuovo, Federigo Enriques, and Gino Loria. In the tradition of nineteenthcentury research schools, Pieri used projective geometry to establish a setting for his work in algebraic geometry. In particular, he exploited the relationship between the two subjects with respect to invariance under transformations—that is, projective geometry as the study of properties of figures that are preserved by groups of projective transformations, and algebraic geometry as the study of properties of homogeneous forms that are preserved by groups of linear or birational substitutions. He explicitly interpreted some of his results within the framework of the Erlanger program. Pieri also saw the transformational approach as one that should be pursued in instruction and exposition.117 Pieri found inspiration in Klein’s ideas, and joined others in carrying the momentum forward. Pieri embraced Klein’s vision of geometry as expressed in the Erlanger program, and adopted Klein’s slogan, Pure mathematics progresses in proportion to how known problems are examined in greater depth and detail according to new methods. As we better understand the old problems, new ones arise on their own.

However, Pieri distanced himself from Klein’s view that the postulates of geometry are simply rigorous forms of our intuitive concept of space.118

115

Klein 1872, 3. Klein’s student Mellen W. Haskell published an English translation, Klein [1872] 1892– 1893, in the Bulletin of the New York Mathematical Society. Mathematics historian David E. Rowe (1989, 265–266) has noted that Klein’s renewed emphasis, from Göttingen, coincided with the decline of the leaders of the competing Berlin school, and the rise of geometry in Italy with its waning in Germany.

116

Dieudonné 1986. For an analysis of Klein’s program that compares its historical impact on research in different areas of geometry, see Gray 2005.

117

Pieri 1890b. Pieri 1898b, 782–783. For biographical sketches see M&S 2007, §1.3.

118

Klein [1894] 1898b, xxi (expressed in memory of Bernhard Riemann). Pieri included this slogan as an epigraph in Pieri 1900a, 173, translated in section 8.0. See also Klein [1893] 1894b, 41; Pieri 1906d, 26.

114

5 Pieri and Projective Geometry

The founders of projective geometry developed techniques for investigating problems that can be stated solely in terms of incidence of points, lines, and planes. These involved properties of figures that persist under the geometric transformations of projection and section. They showed how to extend Euclidean geometry to projective geometry by adjoining ideal points, lines, and a plane at infinity. They developed many synthetic methods for reasoning about these objects and their relationships directly, and introduced algebraic devices such as real homogeneous coordinates and linear equations to handle them without resorting further to Euclidean notions of distance or angle measure. Building on work of Staudt and others, Pieri constructed a full axiomatic presentation of real projective geometry completely independent of metric or other Euclidean ideas, and including Staudt’s fundamental theorem about projectivities. Transformations play a major role in Pieri’s development, even beyond their role in that theorem.119 Arthur Cayley and Felix Klein showed how to start from real analytic projective geometry, choose a suitable metric form—a bilinear function of the homogeneous coordinates of a projective point—and devise formulas that behave like distance and angle measures in Euclidean or one of the familiar non-Euclidean geometries, depending on the choice of metric. Some details of a version of this development are given in the box on page 115. Because the collineations described there are precisely those whose restrictions to the Euclidean or non-Euclidean points correspond to motions, it is possible to derive properties of Euclidean or non-Euclidean geometry by first selecting the appropriate metric form and then studying the group generated by these collineations. This possibility would inspire Pieri’s axiomatizations of projective and elementary geometry based on homographies or on motions.120 A project for implementing this method is described in the next paragraphs, in Pieri’s words. Reporting in Giuseppe Peano’s journal Rivista di matematica, Pieri reviewed a particularly elegant presentation of the Cayley–Klein theory by the German mathematician Johannes Thomae soon after its 1894 publication: Conic Sections in a Purely Projective Treatment.121 Instead of starting with a metric form, Thomae defined a conic synthetically. Pieri commented, ... the final chapter ... will appear particularly beautiful to every reader. There, in a few pages ... are clearly set out the basics of the ordinary metric geometry of the plane, considered from the point of view of the geometry of position. ... no other book of an elementary nature is known to me, where this topic might be completely developed.122 [The narrative continues after the box on page 117.]

119

See sections 5.2 and 5.3. Pieri 1898c, translated in chapter 6.

120

Cayley 1859; Klein 1872, 1873; Pieri 1898b, 1900a.

121

Pieri 1894b, Thomae 1894. See also M&S 2007, §6.7.2. For a biographical sketch of Thomae, see the box on page 117.

122

Pieri 1894b, 37–38. Thomae (1894, 24) defined a conic as the set of intersections of lines related by a projective but not perspective correspondence between two pencils with different centers.

Cayley–Klein Method. It is possible to start with real projective geometry and from that recover Euclidean geometry and its metric apparatus.* One way, in plane geometry, is to consider a polynomial that is symmetric, homogeneous, and bilinear in the coefficients  = [a1 , a2 , a3 ]  = [b1 , b2 , b3 ] of homogeneous equations for two lines. The lines can be called orthogonal if this polynomial vanishes. For example, the Euclidean lines with equations on the left below are orthogonal when the polynomial (, ) = a1 b1 + a2 b2 = 0, and this condition is applicable just as well to the corresponding homogeneous equations on the right: a1 x1 + a2 x2 + a3 = 0 b1 x1 + b2 x2 + b3 = 0

a1 x1 + a2 x2 + a3 x3 = 0 b1 x1 + b2 x2 + b3 x3 = 0 .

In this Euclidean case, norm and distance formulas can be phrased in terms of : 2 2 =

 ( , )

distance between ,  = 2  – 2.

Algebraic arguments with coordinates and coefficients reveal familiar Euclidean properties of orthogonality and distance. Many such considerations can be generalized to apply to any symmetric homogeneous bilinear polynomial. Used in this way, it can be called a metric form.† For some appropriate choices of metric form, for example (, ) = a1 b1 + a2 b2 (, ) = a1 b1 + a2 b2 + a3 b3 (, ) = a1 b1 + a2 b2 – a3 b3

(Euclidean} (elliptic) (hyperbolic)

the resulting orthogonality relations behave like the Euclidean relation, or like those introduced in earlier decades for elliptic and hyperbolic non-Euclidean plane geometry, respectively. These theories can be completely developed using the metric forms and associated distance formulas. Arthur Cayley pioneered this idea in 1859; Klein carried it further in 1872 and 1873. Determining geometrical properties from the choice of metric form is called the Cayley–Klein method. • In the Euclidean case, exactly one projective line g is self-orthogonal. It has homogeneous equation x3 = 0, is chosen as the line at infinity, and called the absolute line. Projective points not on g are designated as Euclidean points, and the sets of Euclidean points on projective lines different from g constitute the Euclidean lines. • In the elliptic case, no line is self-orthogonal. Elliptic geometry applies to all projective points and lines. • In the hyperbolic case, the graph g of the quadratic polynomial (  is designated as the absolute conic. Coordinate algebra shows that a line is self-orthogonal if and only if it intersects g at exactly one point. A projective point is said to lie inside g, and called a hyperbolic point, if each line through it intersects g at two distinct points. The sets of hyperbolic points on such lines constitute the hyperbolic lines. This structure—hyperbolic points and lines and orthogonality relation with an associated distance formula—satisfies the properties of plane hyperbolic geometry as developed by Bolyai János in [1831] 1891 and Nikolai I. Lobachevsky in [1840] 1914, respectively. It is called the Klein model. The collineations that correspond to motions ‡ of the Euclidean or hyperbolic plane preserve the absolute line or conic g: they map it to itself. The collineations  that correspond to line reflections can be described very succinctly: they are the only ones that preserve g, are self-inverse, and fix exactly two points on g. These  and their compositions are precisely the collineations whose restrictions to the Euclidean or hyperbolic points correspond to motions. * There are several ways to introduce the subject of this paragraph; the terminology varies between them. This one was chosen for its conciseness, not for historical reasons. See Kunle and Fladt 1974 for a comparable detailed presentation. † The term metric refers here to the distance formula for the Euclidean metric. For other metric forms, that relationship is not so close. ‡ Here, motion means distance-preserving transformation. In the Euclidean case, similarities with scale factor = / 1 also preserve the absolute; in the hyperbolic case, there are no similarities except the motions.

116

5 Pieri and Projective Geometry

Sophus Lie around 1870

Felix Klein around 1870

Johannes Thomae around 1870

5.5 The Transformational Approach

117

Carl Johannes Thomae was born in 1840 in Laucha, a village in the German Duchy of SaxeCoburg-Gotha. His father was a teacher and rector of a middle school; his mother stemmed from a merchant family. Thomae was schooled there and in the larger Prussian town of Naumburg, and at the University of Halle. He earned the doctorate in 1864 at the University of Göttingen, with a dissertation on complex analysis. After military service in the Austro-Prussian War, Thomae moved back to Halle, where he taught, continued research, and soon published a book on complex function theory. He served in a noncombat role in the 1870 Franco-Prussian War that year; his younger brother was killed. Appointed ausserordentlicher professor at Halle in 1872, Thomae undertook algebraic geometry as a second research area and in 1873 published a projective-geometry text that influenced Pieri’s investigation of the fundamental theorem of projective geometry. Thomae attained a full professorship in Freiburg in 1874. That year he married a home-town woman who died only a year later after bearing a son. Thomae moved again in 1879 to the University of Jena in the Grand Duchy of Saxe-Weimar-Eisenach. Mathematics had languished there for decades; Thomae set about restoring it. An avid hiker, devoted gardener, amusing storyteller, music lover, and singer, he became a leader in academic society and administration. He remarried in 1892; a daughter was born the next year. Son and daughter eventually pursued careers in art and music, respectively. Thomae’s introduction to his 1880 textbook on complex analysis had presented a formalistic approach to algebra, and triggered a noted controversy with the Jena mathematician and philosopher, Gottlob Frege. As noted in this section, Thomae’s 1893 book on conics influenced Pieri’s approach to the foundations of elementary geometry based on the concepts of point and motion. Thomae’s student, the geometer Heinrich Liebmann, characterized the book: “Thomae placed here ... more value on aesthetic purity of method and richness of geometric insights, than on completely developed axiomatics.” In his later years, Thomae continued to teach, research in, and write about geometry, and to a lesser extent, analysis. He retired in 1914 and died of heart failure in 1921.* * Liebmann 1921, 138. For further information consult that work and Dathe 1997.

[continued from page 114]

Pieri then related Thomae’s approach to his own envisioned research program in foundations of geometry. Having read Thomae’s final chapter, Pieri considered whether it might provide a way to revise elementary geometry, now constituted almost entirely on the concept of motion. ...The foundations of elementary geometry are still controversial, ... whereas little remains to do regarding the geometry of position. This happens because, among the ... geometric correspondences, physical motion is not the simplest, as we would be led to believe from the clear notions we have of all its properties, rendered more familiar in us by daily experience. But recognizing them, discerning in them the primitive from the derived facts, and constructing from them a logical edifice, are something else. ... Instead of resting on the rather varied and superficial ground of correspondence by motion, this should have its foundation in the deeper and more homogeneous soil of the general collineation. ... If the correspondence by motion ...is defined by means of projective translations and rotations, or other special collineations ... it will be possible to do without many postulates that in the current organization of elementary geometry are, or are held to be, necessary.123

123

Pieri 1894b, 38–39. Pieri’s assessment that “little remains to do” in the foundations of projective geometry was evidently an understatement. After this review, during 1895–1898, he published a series of six papers on this subject, tilling that “deeper and more homogeneous soil.”

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Pieri put these words into action with the major 1900a paper, On Elementary Geometry as a Hypothetical Deductive System: Monograph on Point and Motion, translated in chapter 8. But he did not root that system in a projective base: he developed elementary geometry from postulates about two primitive notions, point and motion. This axiomatization fell squarely in synthetic transformational geometry. In his [1900] 1901 Paris address, translated in chapter 4, Pieri discussed both Point and Motion and his 1894 plan for revising elementary geometry, discussed in the previous paragraph. He reported, But everyone sees how a reform of such grand consequence, that would lead directly to preceding instruction in ordinary elementary geometry by that of the pure geometry of position, is hardly recommended. On the other hand, I do not find any other method which, while satisfying the requirements of rigorous deduction, might succeed in proscribing motion from the elements of Geometry.124

The collaboration between Klein and Lie during 1869–1872 led not only to Klein’s Erlanger program, but to Lie’s theory of continuous transformation groups. Since these groups are based on real or complex coordinate geometry, they can be generalized easily to higher dimensions. Lie’s theory achieved great acclaim and became a standard framework for considering questions about transformations, particularly in higher-dimensional geometry.125 Lie named one such question the Riemann–Helmholtz space problem (Raumproblem): To find properties that are satisfied by the Euclidean system as well as both non-Euclidean systems of motions, and by which these three systems are distinguished from all other possible systems of motions of a number manifold.126

Lie’s results included solution of an analogous problem in projective geometry: ... the general projective group of Rn , and the groups isomorphic with it, are the only finitecontinuous groups of point transformations of Rn under which n + 3, but not fewer, different points have an invariant. From this it follows that for the construction of the projective geometry of an n-dimensional number manifold only one axiom is required, namely this: There is in the manifold under consideration a finite-continuous group of point transformations under which n + 2 points have no invariant.127

Pieri’s work led to axiomatic constructions of these groups, which satisfied the criterion in Lie’s result. Thus, Pieri was able to justify his claim that his theories led to results compatible with the most noted recent studies of the subject: Here we shall consider trying a new road: in sum, to establish the geometry of homographic transformations on the concept of homography, not defined except as through postulates; similar to the geometry of motion, or elementary geometry, which, according to the classical

124

Pieri [1900] 1901, 385. See also the translation in section 4.4, page 65.

125

See Hawkins 1984.

126

Lie 1888–1893, volume 3, part 5, 397.

127

Ibid., 524. For Lie, a finite-continuous transformation group was one whose members depended continuously on a finite number of scalar parameters. Lie remarked further that the problem arose with Riemann, and that Hermann von Helmholtz realized that it had not been settled, but that neither had formulated it properly.

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methods, is made to rest (though not always explicitly) on a priori concepts of transformation by motion.

In a footnote to this passage, Pieri wrote Nor should anyone expect that such an idea could contradict that type of reasoning, maximally analytical, in the principles of geometry that are identified with the distinguished names of Riemann, Helmholtz, Lie. Rather, it is to be believed that this should be at least foreshadowed in [Lie’s 1893 words quoted above]. That the space should be a (continuous) numerical variety is a primary axiom of those classical investigations. ... while such a stipulation does not occur in this paper until the last principle, or XX.128

Marius Sophus Lie was born in 1842 in Nordfjordeid in western Norway (then part of Sweden), the youngest of six children of a Lutheran minister. He was schooled first in the town of Moss, on the fjord south of Oslo, then at a Latin school in that city. In 1859 he entered its university. He studied mathematics, not very seriously, and graduated in 1865. Teaching privately and in his former school, he became interested in higher geometry, particularly through the work of Julius Plücker. Lie published a research paper in 1869 in a leading German journal, and won a travel grant. He spent autumn 1869 in Berlin, where he met the younger Felix Klein, who had been Plücker’s student until the latter’s death the previous year. Sharing this interest, the two traveled the next spring to Paris. There, they were greatly impressed by the work of Camille Jordan on group theory, and its possibilities for application in geometry, particularly with regard to the techniques that Plücker had introduced. Their visit was curtailed somewhat by the Franco-Prussian War, but during the next several years Lie and Klein formulated the basic ideas of Klein’s Erlanger program and Lie’s theory of continuous transformation groups. Back in Oslo, Lie earned the doctorate in 1872 with a dissertation on differential geometry and was immediately appointed professor. He married in 1874; his wife bore three children during the next decade. Lie continued research in differential geometry and developed some now-standard methods for solution of partial differential equations. In 1873 he began to investigate systematically the theory of continuous transformation groups. At the University of Leipzig, Klein had been fostering research in this area. In 1884 he arranged for a new doctorate, Friedrich Engel, to visit Lie and help publish Lie’s results. Two years later, Klein moved on to Göttingen, and Lie succeeded him at Leipzig. With Engel’s help Lie’s three-volume treatise on continuous groups was published during 1888–1893. Lie’s health began to decline, and during 1889–1890 he was hospitalized for a nervous breakdown. He recovered, but his friendship with Klein suffered and that with Engel ended. Lie became melancholy and felt underappreciated, in spite of the activity in Germany and France that applied and extended his work, and of his appointments to the Royal Society of London and to the national academies of France and the United States. Lie returned to Oslo in 1898 and died there in 1899, of pernicious anemia. His most noted discovery was that continuous transformation groups, now called Lie groups, could be better understood by “linearizing” them and studying the corresponding vector fields, which have the structure of what is now called a Lie algebra.* *Freudenthal 1973, Fritsche 1991, 1999. In Lie’s time, Oslo was called Christiania.

128

Pieri 1898b, 781. See subsection 9.2.6.

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5.6 Multidimensional Projective Geometry “The first vague outlines of the idea of higher spaces are blurred in the mists of time,” according to the American mathematician Julian L. Coolidge. The prehistory of multidimensional geometry can be traced to antiquity, first in connection with the geometric interpretation of algebraic powers beyond three, then of functions of three or more variables.129 By 1747, Kant had accepted the possibility of more than three spatial dimensions, connecting it with Newton’s inverse-square law of gravitation. Kant concluded that by virtue of this law, [space has] the property of being three-dimensional; ... that this law is arbitrary, and that God could have chosen another, e.g., the inverse-cube, relation; and, finally, that an extension with different properties and dimensions would also have resulted from a different law. ... If it is possible that there are extensions of different dimensions, then it is also very probable that God has really produced them somewhere.130

In 1827, Möbius considered higher-dimensional geometry from a synthetic point of view, observing that mirror-symmetric three-dimensional figures could be rotated into coincidence if they lay in four-dimensional space. However, no systematic studies of spaces of dimensions higher than three were conducted by geometers until the mid-nineteenth century.131 The phrase geometry of n dimensions evidently first occurred in the title of an 1843 paper by Arthur Cayley: Chapters in the Analytical Geometry of (n) Dimensions. It is about the linear algebra of homogeneous functions of degrees one and two in n variables. No geometric terms occur in the paper until the very end, which presents two supposedly familiar applications with n = 4 to the intersections of cones, quadric surfaces, and planes in three-dimensional space.132 Soon afterward, using the synthetic method of projections, Cayley applied four-dimensional geometry to investigate certain configurations of points and lines, including Pascal’s hexagons. Introducing his method of reasoning Cayley wrote, One can in effect, without recourse to any metaphysical notion with regard to the possibility of space of four dimensions, reason as follows (all of this can be translated easily into purely analytical language): Supposing four dimensions for the space, it will be necessary to consider lines determined by two points, that which we call demi-planes determined by three points, and planes determined by four points; two planes then intersect along a demi-plane, etc.133

Decades later, Cayley began a recapitulation of general principles of higher-dimensional geometry with these words:

129

Coolidge [1940] 1963, chapter VI, 231. See also Manning 1914, introduction.

130

Kant [1746–1749] 2012, 27–28. See also De Bianchi and Wells 2015. Extension is the property of a body by which it occupies a portion of space.

131

Manning 1914, 4; Möbius 1827, §140, 171–172. Coolidge [1940] 1963, chapter 6, 231; Rosenfeld 1988, chapter 4.

132

Rosenfeld 1988, 249, chapter 7. Cayley 1843.

133

Cayley 1846, 218, translated by the present authors. Today Cayley’s demi-planes and planes are called planes and hyperplanes, respectively.

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whenever we are concerned with quantities connected together ... which are ... considered as variable or determinable, then the ... relation between the quantities is frequently rendered more intelligible by regarding them (if only two or three in number) as the coordinates of a point in a plane or in space: for more than three quantities ... this can only be obtained by means of the notion of a space of the proper dimensionality; and to use such 134 representation, we require the geometry of such space.

Hermann Grassmann’s 1844 science of linear extension was a forward step in multidimensional geometry, particularly in its application to physical theories, although that was not recognized at the time of publication. Grassmann applied what is now called vector analysis to the geometry of any number of dimensions. He connected the synthetic-geometry precept of dealing with geometric objects, not numbers, to an analytic-geometry strategy of calculating with these objects, explaining, The tendency of this method of calculation for geometry is to unite the synthetic and analytical method: that is, to transplant the advantages of the one into the soil of the other by placing a simple analytical operation at the side of every construction and vice versa.135

Bernhard Riemann’s seminal 1854 study On the Hypotheses Which Lie at the Bases of Geometry introduced the concept of an n-dimensional manifold constructed by geometric recursion. A zero-dimensional manifold is a point; an (n+1)-dimensional manifold was defined as a collection M of elements that can be distributed in a set of n-dimensional manifolds Mt that correspond to the points t of a curve, so that each element of M belongs to at least one of the Mt and overlapping Mt and Mu manifolds are properly related. Thus, a curve, and in particular a line, is a one-dimensional manifold, and a surface, in particular a plane, is a two-dimensional manifold. Riemann showed how to employ a metric, or distance measure to the points of a manifold. Curvature of a manifold could be defined in such a context, and that suggested that a curved three-dimensional space could lie in four-dimensional space.136 Shortly after his 1872 arrival in Turin, Enrico D’Ovidio began publishing about multidimensional projective geometry as a framework for developing the theories of nonEuclidean geometry, following the tradition recently begun by Felix Klein.137 D’Ovidio’s illustrious student Corrado Segre, inspired as well by Grassmann’s work, recognized the essential identity between the algebra of linear transformations and the projective geometry of higher-dimensional spaces. Segre’s 1884 doctoral thesis was a fundamental starting point for the development of Italian projective n-dimensional geometry, ... devoted to the general study, by geometrical tools, of n-dimensional vector spaces (projective spaces), and in particular of bilinear forms defined on them (quadrics).138

The research school of D’Ovidio and Segre at Turin embraced the identification of analytic and synthetic methods in constructing a geometric theory of projective hyper134

Cayley 1870, 51.

135

Grassmann [1844] 1994. Tobies 1996. Grassmann 1845, 340. See also Beutelspacher 1996.

136

Manning 1914, 7–8; Riemann [1854] 1892, 275.

137

D’Ovidio 1873–1875, 1877; Klein 1871.

138

C. Segre 1884a, 1884b. Brigaglia 1996, 155, 160.

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space, and was inspired by the transformational methods of Klein’s Erlanger program.139 Segre was soon to make inspired contributions and indicate novel directions thanks to which, in the last decade of the [nineteenth] century, the algebraico-geometric school of Italy was to take on its most striking characteristics, which proved to be fundamental for future developments.140

Decades later, the British geometer D. M. Y. Sommerville wrote, The wonderful projective geometry of hyperspace has been almost entirely the product of the gifted Italian school of geometers ... .141

By the end of the nineteenth century, multidimensional geometry was being investigated with a variety of methods. Contributions to the field could be found, for example, in the theories of projective metrics, projective transformations and transformation groups, algebraic curves and surfaces—especially birational concepts—and in differential and enumerative geometry and vector analysis. Any initial reluctance of mathematicians to embrace multidimensional geometry appeared to have dissipated. In 1904 Segre observed, Since the [initial 1891] appearance of the present paper multi-dimensional geometry has spread more and more, so that now (among mathematicians!) its opponents have become rare, who at one time were so common. Not only the geometers, but also the abler analysts, no longer hesitate to make use of hyperspace in their researches.142

Segre proposed a classification of research in multidimensional geometry according to its method of defining point: • by coordinates, as a “system of values of n variables ... a necessity in a large number of investigations” • as a geometric form of ordinary space, such as a curve or surface, that depends on any arbitrary number of parameters, following ideas of Julius Plücker and Arthur Cayley • as “points of the same nature as those ... in ordinary space, ... no longer purely analytic forms or geometric forms ... ,” but abstract, purely logical entities.143 Segre’s school of algebraic geometry became a focal point for Italian studies in the field. The first half of the nineteenth century saw the study of algebraic curves in the complex projective plane. In the latter part of the century, that was generalized to a theory of algebraic surfaces in higher-dimensional spaces.

139

Avellone et al. 2002, 379–380; Boi 1990. For more about the Erlanger program consult section 5.5.

140

B. Segre 1963, 401.

141

Sommerville 1929, vii. Sommerville noted that the English mathematician W. K. Clifford had also made an early contribution.

142

C. Segre [1891] 1903–1904, 459.

143

C. Segre [1891] (1903–1904), 459–463.

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In this rich environment Pieri began his research. In his very first published paper (1886a) he generalized a then well known theorem of Enrico Betti and Julius Weingarten,144 showing that the loci of points, the sum and difference of whose normal distances to two given nonparallel base surfaces are constants, constitute a double system of orthogonal surfaces. Pieri then showed that this theorem is a particular case of a more general one about higher-dimensional spaces with arbitrary arclength element. Pieri’s first project in multidimensional projective geometry was his 1887b paper On the Correspondence Principle in an Arbitrary n-Dimensional Linear Space. This was the first generalization of a principle developed more than a decade earlier by Michel Chasles, George Salmon, and H. G. Zeuthen.145 Its first formal version had been concerned with the relation between points X and X r on a line g that holds when a polynomial F( X, Xr), homogeneous in two pairs of scalar variables, vanishes. If to each point X correspond r points Xr and to each Xr correspond  points X, then F has degrees  and r in X and Xr and the homogeneous polynomial F( X, X ) has degree  + r, hence there are  + r points on g that correspond to themselves.146 Pieri extended this result to apply, for any dimension n, to n polynomials each homogeneous in two (n+1)-tuples of scalar variables; he presented a synthetic proof of the extension using complete induction. In his obituary tribute to Pieri, Giovanni Giambelli noted147 the particular importance of this paper, which would underlie a large portion of Pieri’s later research. In the paper, Pieri adopted the third of the approaches in the previous classification, which Segre would soon describe as “geometrical and entirely intuitive.” Following this, and in the tradition of Grassmann and Plücker, Pieri viewed points of higher-dimensional spaces as abstract entities: We shall represent by S n , S n* the two superposed linear spaces, between the elements of which there is an algebraic correspondence, and by Spq an arbitrary space of dimension p and degree q formed from these elements, which solely for brevity of language we suppose should be [called] points, allowed to be elements arbitrary but of the same nature in both of the corresponding spaces.148

144

Betti 1860 and Weingarten 1863.

145

Chasles 1855, 1864. Salmon 1874, 511. Zeuthen 1874.

146

Chasles 1864, 1175. Roots of the polynomials must be counted with respect to their multiplicity. According to mathematician William Fulton (1998, 310), this is one of the primary tools of classical enumerative geometry.

147

Giambelli 1913, 291.

148

C. Segre [1891] 1903–1904, 463. Pieri 1887b, 197. By arbitrary space Pieri meant algebraic variety; his word for degree was ordine. Select N large enough that N-dimensional projective space SN contains Sn and Sn* and accommodates any desired constructions. The dimension and degree of a variety V f SN can be defined as follows: there will always be a smallest nonnegative integer p such that the intersection of V with a generic (N – p)-dimensional subspace SN – p f SN is finite. This p is independent of N and is called the dimension of V. The number of points in the intersection, counted by multiplicity in V, is independent of the choice of the generic SN – p and is called the degree of V. If p = 0, then Spq consists of q points counted by multiplicity. This explanation, provided by Steven L. Kleiman (2016), is close to that by G. H. Halphen (1873–1874, 34), to which Pieri referred in the paper discussed next. Halphen’s discussion, though, is based on the number l of independent equations that define Spq, called its ordre. Those together with N – l others would define a variety with only finitely many points, so that p # N – l. Caution: Halphen’s ordre differs from Pieri’s ordine.

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The following year, in his 1888 paper On a Theorem of n-Dimensional Geometry, Pieri published the first synthetic proof of a higher-dimensional generalization of Bézout’s theorem: if two algebraic varieties V and W of dimensions p and q and degrees  and  lie in general position in an n-dimensional space and p + q $ n, then their intersection will contain a variety D of degree .148, 149 This had been established analytically by G. H. Halphen,150 and synthetically in dimensions n = 2 and 3 by Chasles and Georges Fouret.151 Pieri used his 1887b generalization of the correspondence principle to support the inductive step in the proof. Beginning with his 1891b paper Coincidence Formulas for n Algebraic Series of Pairs of Points of an n-Dimensional Space, Pieri developed from the correspondence principle a sequence of deep studies that included his famous formulas that extended the methods of Hermann Schubert in enumerative geometry.152 Italian mathematicians had been instrumental in placing the theory of birational transformations centrally in algebraic geometry.153 Pieri published several papers aiming to classify all birational transformations of an algebraic variety, and used projective geometry to explore properties that are invariant under such transformations. He employed these projective methods also in broader contexts, such as counting the number of multiple tangents and normals to algebraic curves and surfaces.154 In the 1890b paper On the Projective Geometry of Forms of the Fourth Species, Pieri employed a correspondence  devised by Francesco Chizzoni in 1888 to relate the points in a four-dimensional projective space S4 to the lines in a three-dimensional subspace . To define , Chizzoni and Pieri first selected skew lines ar and br in S4 not lying entirely in  and distinct planes  and  in  whose intersections with ar and br are points A and B, respectively. With a line p in  they associated its intersections with  and  and the planes  and  joining those with ar and br, respec-

149

Gino Loria’s 1888 review of Pieri 1888 in the Jahrbuch über die Fortschritte der Mathematik incorrectly rendered the $ sign in the statement of Pieri’s result as kleiner instead of nicht kleiner. To derive Bézout’s theorem from Pieri’s result, take p = q = 1 and n = 2, so that the dimension d of D is 1 or 0: accordingly, D is a common component of V and W or else D consists of  intersections, counted by multiplicity. For a detailed discussion of the evolution of Bézout’s theorem, Pieri’s generalization of it, and subsequent related studies, see Marchisotto 2017, especially pages 338–341.

150

Halphen (1873–1874, 40, theorem V) derived Pieri’s conclusion from the hypothesis that V, W have ordres l, n with l + m # n. Pieri’s hypothesis n # p + q implies Halphen’s by footnote 148 with N = n.

151

Chasles 1872; Fouret 1872–1873.

152

Pieri’s results constitute a precursor of modern excess-intersection theory (Fulton 1998, 318). Schubert [1879] 1979.

153

Klein [1926–1927] 1979, part 1, chapter 7, 295–319. Rational and birational transformations are defined in footnote 8 of section 5.1 on page 80.

154

Some of Pieri’s results—for example, his 1889b work on triple tangents—can be regarded as instances of the “classical approach” to studying an n-dimensional variety: mapping it birationally onto a hypersurface in (n + 1)-dimensional projective space so that the structure of the singularities appearing on the hypersurface can be investigated, as well as the singularities of the transformation (Kleiman 1977, 304; Kleiman 2016).

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tively; he defined Pr = ( p) to be the intersection  1 .155 Whereas Chizzoni had used  to apply theorems about S4 to deduce properties of the space of lines in , Pieri did the reverse. Pieri’s first result was an explicit instance of a transfer principle emphasized in Klein’s Erlanger program: The projective geometry of the space of points in four dimensions with a fixed triple of independent lines, and the projective geometry of the ordinary space of lines with a fixed trihedron are the same thing.156

Pieri elaborated this result to apply to the geometry of transformations, not just points: The geometry of the linear transformations of a four-dimensional space of points is reflected in the geometry of lines of an ordinary space, in which is assumed as fundamental group the group of all univocal transformations of lines that do not alter the system of tetrahedral complexes having three fixed singular planes in common.157

Pieri proceeded to derive applications, including this final result: The study of the cubic complexes of lines in ordinary space containing three fixed planes is reduced to that of the cubic varieties, with three fixed lines, in four-dimensional space.

Accordingly, he could establish a general theory and a classification of these cubic complexes within the norms of studies already completed.158 In later years he would pursue this thread further. Pieri constructed his 1890b paper according to customs then prevalent in algebraic geometry, omitting many details and justifications. Neither he nor Chizzoni specified for exactly which lines p the correspondence  could be defined. For example, do the planes  and  always intersect at a single point, and what if p should intersect  at A or  at B, thus preventing consideration of  or  as a plane? 159 For a survey of Pieri’s work in multidimensional algebraic geometry see M&S 2007, §1.2.1.160 Pieri’s six doctoral students all completed their studies with him at Catania during 1902–1905 in algebraic and enumerative geometry. Three of them emphasized multidimensional results:

155

 is injective: p is the line joining the intersections with ,  of the planes joining Pr with ar, br.

156

Klein [1872] 1892–1893, §4. Pieri 1890b, §2, 210–211. The three independent lines are ar, br, and the intersection of  with ; the trihedron consists of , , and the intersection of  with the subspace spanned by ar and br.

157

Pieri 1890b, §4, 212.

158

Pieri 1890b, §13, 217–218. Pieri acknowledged underlying results in Chizzoni 1888. For studies already completed, he referred to C. Segre 1888a and 1889c.

159

Conditions on p equivalent to the existence of a unique corresponding point Pr can be formulated as equations and inequations of polynomials in the Plücker coordinates of p and the preselected varieties , ar, br, , and . The same is true when the roles of p and Pr are reversed. Thus,  is a birational mapping from the space of lines in  to that of points in S4 .

160

Further detailed study of Pieri’s work in algebraic and differential geometry is suggested in the box at the end of the preface of the present book.

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Giuseppe Marletta, Fourth-Order Varieties with a Double Plane in FourDimensional Space (1902) Francesco D’Amico, On the Quartic Variety with Three Simple Planes in Four-Dimensional Space (1904) Niccolò Giampaglia, On the Incidence of Points, Lines, and Planes in n-Dimensional Space (1904). During the mid-1890s, Pieri began focusing on establishment of a logical basis for higher-dimensional projective space, including construction of a coordinate system, from a set of independent postulates. In 1891 Segre had advertised the need for such a work: There has not been established nor discussed, as far as I know, any system of independent postulates that serve to characterize linear space of n dimensions, from which one can deduce the representation of its points by coordinates. It would be advantageous if some young scholar should occupy himself with this question (which does not seem difficult).161

Pieri pursued a synthetic approach, defining this space in the same way as ordinary space but eliminating any postulate that restricts it to three dimensions, and modifying others to reflect that decision. The challenge was then to unify it with the analytic approach. Pieri rose to the occasion. Pieri’s 1896c paper A System of Postulates for the Abstract Projective Geometry of Hyperspaces presented a system of twenty postulates based on three primitive notions —point, projective segment, and line joining two points—sufficient to construct projective geometry abstractly in any given dimension. In his very first sentence Pieri observed that the projective geometry of hyperspace, intended as an autonomous science, remained subject to controversy. Indeed, the literature contained abundant details of a polemic on this issue between Pieri’s two mentors, Corrado Segre and Giuseppe Peano.162 Pieri cited previous work of Federico Amodeo, Giuseppe Veronese, Gino Fano, and Federigo Enriques as evidence of recent progress. Veronese is acknowledged today as having made the first systematic use of the synthetic method in treating higher-dimensional geometry. His axiomatization and Fano’s are especially significant, but those and that of Enriques have been judged as lacking in mathematical rigor.163 Pieri’s axiomatization, The Principles of the Geometry of Position Composed into a Deductive Logical System (1898c)164 was the culmination of his research in foundations of multidimensional real projective geometry. There, Pieri constructed real projective geometry of any dimension synthetically on the basis of two primitive notions—point and line joining two points—and twenty postulates. Previous axiomatizations of projective geometry had been based on at least three primitive notions: usually point, line, and order. In 1898c, Pieri handled the order of points on a line by means of a subtle 161

C. Segre 1891a, 60–61.

162

For the polemic consult Peano 1891b, 1891c, 1892, C. Segre 1891b, and Manara and Spoglianti 1977.

163

Amodeo 1891, Veronese 1891, Fano 1892, Enriques 1894. Peano criticized Veronese 1891 harshly in his 1892 review. See also Giacardi 2001, 155, and Voelke 2008, §7. Pieri himself noted lapses in Veronese’s and Fano’s works (see the last footnote in Pieri 1898c, §3, translated in section 6.3).

164

Pieri 1898c is translated in chapter 6 and described in subsection 9.2.5.

5.7 From Duality to Plurality

127

definition of projective segment based solely on incidence.165 The renowned geometer H. S. M. Coxeter called Pieri’s reliance on only two primitives an “astonishing discovery.”166 Pieri’s contributions to the research in foundations of multidimensional geometry did not end with his 1898c paper. His next axiomatization, New Method for Developing Projective Geometry Deductively (1898b) was a construction of three-dimensional projective geometry on the basis of point and homography. But he noted there that by specifying a desired dimension n and replacing his postulate XVII with one stated in terms of n and the defined notion hyperplane, he could develop n-dimensional projective geometry as in 1898c, §11–§12.167 Again building on 1898c, and on Pieri 1904a as well as on results of G. K. C. von Staudt, Theodor Reye, and Segre, Pieri produced an axiomatization of multidimensional complex projective geometry that is also noteworthy for its historical as well as its mathematical importance. With that 1905c paper, New Principles of Complex Projective Geometry,168 Pieri constructed the first synthetic system exhibiting the intrinsic attributes of a geometry in which points can be represented by complex homogeneous coordinates, without making any appeal to real projective geometry. Prior to this, in n-dimensional projective geometry with n > 2, complex points had been introduced analytically as (n+1)-tuples of complex homogeneous coordinates. These had simply been included with the real points, with no assigned geometric meaning beyond their relationship to real points through formulas. Pieri instead assumed complex points a priori, and revealed their properties through his postulates. From that system he derived real projective geometry as a special case. 5.7 From Duality to Plurality Duality is a defining principle of projective geometry that highlights its synthetic underpinnings. It has a rich history that is not without controversy.169 Noting some of its earlier examples, Jean-Victor Poncelet studied duality with respect to an arbitrary conic section C in his 1817–1818 preliminary paper on projective geometry and in chapter 2 of his 1822 Treatise on Projective Properties of Figures. The case of a circle C is presented as an example in the left-hand figure on the next page. Consider a secant line g whose intersections A and B with C are not diametrically opposite. The tangents to C at these points intersect at a unique point P outside C called the pole of g. Given P, one can determine g as the polar of P, the line joining the points where the two tangents through P touch C . As P approaches C , the tangents approach each other. Thus, if P should lie on C , the tangent line g through P can be regarded as the polar of P, and P as the pole of g. If g should pass through the center of C, the tangents would be parallel; their ideal intersection P at infinity can be regarded as the pole of g, and g as the polar of P. Thus, the projective plane is the appropriate 165

See subsection 9.2.1, page 343; 9.2.3; 9.2.5, page 362; and the first definition in 6.5.

166

Coxeter 1949, 7.

167

Pieri 1898b, 793. Pieri 1898c, translated in 6.11 and 6.12.

168

Pieri 1905c, described in 9.2.9.

169

Lorenat 2015b; Kline 1972, 845ff; Coolidge [1940] 1963, 95; Kötter 1901, § XX, 160–167.

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5 Pieri and Projective Geometry

setting for this study. The secant in the figure can be determined from the pole, and hence the pole/polar correspondence between points outside or on C and the secant or tangent lines is bijective. The figure also illustrates a basic property of this correspondence for the circular case: If a certain point P is situated on a straight line h drawn in the plane of a conic section, its polar will pass through the pole Q of this same straight line.170 The right-hand figure, obtained by enlarging the first one horizontally, illustrates the same property of the analogous pole/polar correspondence for the elliptical case. h g

P Q

h

g

P

Q A

A B

B

With Respect to Circle

C

With Respect to Ellipse

E

Pole/Polar Correspondences This basic result entails that the pole of a line joining two points is the intersection of their polars, and the polar of the intersection of two lines is the line joining their poles. In this way, theorems about points, lines, and incidence can be transformed into new theorems by interchanging the term point with line and the phrase intersection of with line joining. This pole/polar theory was generalized easily to apply to arbitrary real conic sections as long as the poles lay outside or on the curves, and to poles inside if imaginary lines are introduced. It can be generalized easily to three dimensions—the polar of a point with respect to a quadric surface is a plane—but Poncelet did not pursue that direction in the works cited. Instead, he applied the method to obtain new planar results via a new idea: The polar reciprocal of a given curve in the plane of a conic section is at once the locus of the poles of all the tangents to this curve and the envelope of the space occupied by the polars 171 of the points of the same curve.

Poncelet expounded the theory of poles and polars extensively during the 1820s, notably in chapter 2 of his 1822 Treatise. As happens frequently in mathematics, the new method was introduced in a context sufficiently complicated—studying polar reciprocals of curves—that the most fundamental aspects of the method were overshadowed. 170

Poncelet 1817–1818, 211. See Altshiller-Court 1952, §379, 179, for a proof that uses a cross ratio.

171

Poncelet 1817–1818, 212.

129

5.7 From Duality to Plurality

Joseph Gergonne, editor of the journal Annales de mathématiques pures et appliquées, published much of Poncelet’s early work. Gergonne stepped into the development of duality theory with his own 1825–1826 paper Philosophical Considerations on the Elements of the Science of Extension. Realizing that the role of the conic section C —central in Poncelet’s theory—was in fact inessential, Gergonne reformulated the pole/ polar relationship as a fundamental principle, in three dimensions, in terms of the language of projective geometry, not the objects it studies. He would present it at the very beginning of an exposition of the subject: some theories in this science depend essentially on metric relations ... and consequently can only be established with the aid of the principles of calculation. Some others, on the contrary, are completely independent of these same relations, and result solely from the situation that the geometric entities ... are found to enjoy with respect to each other. ... An extremely striking character of this [second] part of geometry ... is that with the exception of some theorems symmetric with themselves ... all these theorems are double: that is, in plane geometry, to each theorem there necessarily always corresponds another that is inferred by simply exchanging the two words points and lines with each other; just as in the geometry of space, it is the words points and planes that can be exchanged with each other to pass from a theorem to its correlative. 172

From a large number of examples...of this sort of duality...we limit ourself to indicating ....

Whereas Poncelet had conceived dualization as a correspondence between poles and polars based on a conic section, Gergonne reformulated it as a correspondence between statements of theorems and definitions and their duals, effected by interchanging certain linguistic expressions.173 Whereas Poncelet could derive all his theorems from accepted first principles of projective geometry, Gergonne gave no adequate justification for the general validity of dualization: no assurance that true statements will always correspond to true dual statements. That would come later. In the previous quotation is apparently the first use of the term duality in projective geometry. (Since then it has also been applied to other mathematical correspondences with analogous properties.) By 1847 G. K. C. von Staudt was using the phrase law of reciprocity or duality: Gesetz der Reziprocität oder Dualität. Gesetz is a grander word than Satz, his word for theorem. The Gesetz is about statements and definitions, whereas Staudt’s Sätze are about geometrical objects. In his 1873 textbook, Luigi Cremona described the principle of duality as interchanging the elements point and plane; it is unclear whether these should be words or geometric objects. Theodor Reye and Moritz Pasch used Staudt’s terminology in their influential 1877 and 1882 books, but described reciprocal concepts as well as statements.174

172

Gergonne 1825–1826, 209–210.

173

See Kötter 1901, 160–167, for unpleasant details of a priority dispute between Poncelet and Gergonne about these matters. H. S. M. Coxeter ([1949] 1961, 75) has claimed that Gergonne’s formulation is more general: it supports certain dualization arguments in plane geometry that cannot be based on any conic. Coxeter’s argument was refuted in Pedoe 1975, 276–277.

174

Loria 1903, 222. Staudt 1847, §6, 30 ff. Cremona 1873, §27–§28, 15–16: “principio di dualità,” “scambio degli elementi punto e piano.” Reye [1866–1868] 1877–1882, Vortrag 3, 21: “Begriffe” and “Ausdrücke.” Pasch 1882b, §12, 93.

Joseph Diez Gergonne was born in 1771 in Nancy, capital of the French province of Lorraine. His father, an architect and painter, died when he was twelve. Joseph was educated at the Catholic Collège de Nancy. In 1792, amid revolutionary chaos, he joined the army as an officer, and fought against Prussia in the Battle of Valmy. That event consolidated the French revolution; the king was executed the next year. Gergonne served in the War of the Pyrenees against Spain, then left the army, became a teacher of mathematics at Nîmes in southern France, and married in 1795. Influenced by Gaspard Monge, he pursued mathematical research, chiefly in logic and geometry. In 1810 Gergonne founded the Annales de mathématiques pures et appliquées, the first journal entirely devoted to mathematics. In 1816 he was named professor at the University of Montpellier near the southern coast. During its three decades the Annales was a leading organ for mathematical research, particularly geometry. Gergonne published about two hundred papers of his own there, and very many from leaders throughout western Europe. His research overlapped some of theirs, which led to disputes, particularly with Jean-Victor Poncelet and Julius Plücker. The former championed the synthetic method in geometry, whereas Gergonne favored the analytic. Gergonne is best known for his studies of syllogistic logic, of the relation of axioms and definitions in mathematics, and his exposition of duality in projective geometry. He became university rector in 1830, and soon terminated publication of the Annales. Colleagues resuscitated it under another name in 1836 and competing journals started in Germany in 1826 and Italy in 1858; all three are still publishing.* Gergonne retired in 1844 and died after a long infirmity in 1859.† * Journal de mathématiques pures et appliquées, Journal für die reine und angewandte Mathematik, Annali di matematica pura ed applicata. † For further information, see Lafon 1860, Struik 1970, and Otero 1997.

Joseph Diez Gergonne around 1836

5.7 From Duality to Plurality

131

Gergonne invented a page-composition style for exposition of projective geometry that displayed dual statements and definitions vividly: We could very well restrict ourselves to demonstrating half of our theorems, and to deducing the other half with the aid of the theory of poles. But we prefer to demonstrate both of them directly as much not to depart from first elements ... as to emphasize that there is between the demonstrations of the two paired theorems the same correspondence as that between their enunciations. We take the same care, to render this correspondence more apparent, to present the analogous theorems in two columns ... in such a way that the demonstrations can serve to check each other.175

Two-Column Composition for Dual Statements175 Gergonne’s two-column convention remained in vogue for about seven decades. Staudt, Cremona, and Reye used it, albeit sparingly, in their works cited in the previous paragraph; Pasch avoided it altogether. Mario Pieri maintained it in his 1889a translation of Staudt 1847, but avoided it in his own papers on foundations of projective geometry, starting with 1895c. In 1831 the German mathematician Julius Plücker noted that in plane projective geometry an equation ax + by + cz = 0 can be interpreted either as stating incidence of a point with a line, whose homogeneous coordinates are and [ x, y, z], respectively, or as stating incidence of a line with a point, whose coordinates are [a, b, c] and < x, y, z>. This analogy provided the required justification for inferring from a theorem about incidence of points and lines the validity of the dual theorem, with references to points and lines interchanged. In particular, To each proof that can be carried out by connecting general symbols correspond, if we relate these symbols first to point coordinates and second to line coordinates, two such theorems, that are bound to each other via the principle of reciprocity.

Plücker regarded this formulation of the principle as equivalent to Gergonne’s.176

175

Gergonne 1825–1826, 211–212.

176

Plücker 1828–1831, volume 1, viii, and volume 2, vii–ix.

Julius Plücker in 1856

Julius Plücker was born in 1801 in Elberfeld in the Duchy of Berg.* He had two brothers; their father was a merchant. Julius was educated first in Elberfeld, then from 1816–1819 at a gymnasium in the nearby city, Düsseldorf. He then studied history, mathematics, and physical sciences at several universities. In Paris, Plücker studied geometry and mechanics with Gaspard Monge, Siméon Poisson, and others, wrote a dissertation on geometry, submitted it to the University of Marburg in 1823 to earn the doctorate, and completed research for his habilitation—the right to teach—which he obtained from the University of Bonn in 1825. Plücker taught there and at other universities for several years, and published the notable 1828–1831 two-volume book, Developments in Analytic Geometry, which presented a new analytic approach to geometrical research. A stint at the University of Berlin proved uncongenial, probably because Jakob Steiner, a strong advocate of the synthetic approach, had great influence there. In 1836 Plücker finally accepted a full professorship at Bonn, where he remained for the rest of his life. He married the next year; the couple had one son. Plücker’s 1835 and 1845 books polished and extended the analytic methods introduced earlier, deriving from the equations of lines and planes coordinates that represent them as elements constituting hypersurfaces in higher-dimensional projective spaces. His 1839 book on algebraic curves contained what are now known as the Plücker equations for the degree and class of a plane curve.† Always interested in applications of geometry, perhaps discouraged by the lack of response to his geometrical research, but trained in physics as well, Plücker around 1847 turned almost entirely to research in experimental physics. With Bonn collaborators he achieved seminal results in the theory of cathode rays and in spectroscopy. Around 1865 Plücker returned to geometry and hired an able assistant, Felix Klein. Plücker published in 1868 the first of two volumes that further developed his earlier work. They would underlie later much later progress in algebraic geometry. Unfortunately, Plücker died that year. Klein worked through his notes and completed the second volume in 1869.‡ * Berg was absorbed into Prussia at the 1815 Congress of Vienna. Elberfeld is now part of Wuppertal, Germany. † The degree is the number of its intersections with a generic line; the class, the number of its tangents

through a generic point.

‡ For further information, see Dronke 1871, Ernst 1933, and Burau 1975.

133

5.7 From Duality to Plurality

The duality principle was applied in two and three dimensions from its very inception: see the passage from Gergonne 1825–1826 quoted earlier. Although the quoted passage from Plücker 1828–1831 mentions only two-dimensional space,177 the algebra involved can easily be extended to n dimensions with n > 3: a hyperplane in such a space is a subspace of dimension n – 1, and an equation a 0 x0 + AAA + an x n = 0 can be interpreted either as stating incidence of a point with a hyperplane, whose homogeneous coordinates are and [ x 0 , ... , x n], respectively, or as incidence of a hyperplane with a point, whose coordinates are [a 0 , ... , an] and < x 0 , ... , x n>. During the ensuing decades, the techniques now known as linear algebra took shape and gradually fostered graceful exposition of n-dimensional geometries. In the 1873–1875 work of Enrico D’Ovidio, for example, the notion of n-dimensional duality is present but barely extricable from the context of analytic formulas. But only a few years later, Giuseppe Veronese considered arbitrary n-dimensional spaces Rn , writing two spaces Rn–1 intersect in an Rn–2 , three in an Rn–3 , etc., n in an R0 . Just as n arbitrary points determine a space Rn–1 , also n arbitrary spaces Rn–1 determine a point. Therefore, we call the point and the (n – 1)-dimensional space dual spaces in Rn . While m+ 1 arbitrary points determine a space Rm , also m + 1 arbitrary Rn–1 determine a space Rn–m–1 : 178 Rm and Rn–m–1 are also dual.

Corrado Segre reported, With the memoir of Veronese it can be said that the projective geometry of these spaces has for the first time been organized and developed systematically as a geometrical science and not as a disguised sort of analysis.179

From Veronese’s words, however, it is evident that Gergonne’s alternative approach to duality—interchanging linguistic expressions designating corresponding subspaces with their dual expressions—would be awkward in higher dimensions because of the need to refer to subspaces of various dimensions. Pieri’s work in the 1890s would make this approach clear and graceful. Previous sections of this chapter showed Pieri embarking on the foundational study of projective geometry.180 His research on algebraic geometry routinely involved polar

177

Plücker 1832, 124, referred to three-dimensional duality as well.

178

Veronese 1881, 165. Since the 1930s this correspondence between dual subspaces has been described as an involutory anti-isomorphism of the lattice of subspaces: see Birkhoff and Neumann 1936 and Baer 1952.

179

C. Segre 1917, 252.

180

See also Avellone et al. 2002 for general background.

134

5 Pieri and Projective Geometry

reciprocals of given figures. Moreover, in several published proofs he explicitly relied on the principle of duality.181 In his 1891c elementary lectures Pieri stated the twodimensional principle of duality in Gergonne’s form: From each proposition ... can always be derived another ... by interchange of the word point with the word line ... and of the idea of point lying on a line with that of line passing through a point ... .

Pieri continued with the three-dimensional principle, and with a discussion of how he would be using the principle. He sometimes wrote of interchanging words, as above, and sometimes of interchanging ideas or generating elements.182 Pieri was evidently concerned about the treatment of duality in the literature. For example, he wrote to his senior Turin colleague Segre questioning its presentation by Federigo Enriques in 1894. Segre agreed with Pieri and noted that the popular textbook Sannia 1891 proceeded in much the same way.183 Pieri regarded duality as one of the two most important features of the subject. Starting his first research report on foundations, On the Principles that Govern the Geometry of Position, he wrote, The object of the present study is to present a new series of postulates to serve as a basis for projective geometry ... as a deductive science independent of any other body of mathematical or physical doctrine ... and governed in each of its parts by certain fundamental laws, such as the principle of projection and of duality, which ... inform it and give it character.

Pieri felt strongly about this: publications.184

he repeated these phrases in two subsequent

During the 1890s Pieri introduced the method that made Gergonne’s approach to duality precise and graceful. Pieri presented the undefined terms and postulates of projective geometry as a hypothetical-deductive system. In such a presentation the undefined terms, such as point, line (in that order), and incidence relation, are assigned no meaning except for their relationships as stated by the postulates; plane is defined nominally185 from these notions. For Pieri, a projective geometry was what is today called a model—a system of mathematical objects assigned as interpretations of the undefined terms—that satisfies the postulates. From a three-dimensional projective geometry P Pieri could construct a dual geometry D consisting of its planes, lines (in that order), and their inclusion relation, and prove that D satisfies the same projective postulates as P . That is, D is also a projective geometry, and all theorems derived for P are also valid for D with their dual interpretations. Later, Pieri would extend this approach to apply to an n-dimensional projective geometry P. Pieri could construct a dual geometry D consisting of its hyperplanes, (n – 2)-dimensional subspaces (in that order), and their inclusion relation, and prove that D satisfies the same projective postulates 181

For example, in 1892a, Pieri presented some dual three-dimensional theorems in two-column format, and in 1890a, §2, 367, and 1892c, §5, 138, he relied on the principle for proofs in higher dimensions. He referred his students to Chasles [1837] 1875 as a background source on duality (Pieri 1910, 3–4).

182

Pieri 1891c, §117–§120, 180–186.

183

C. Segre [1898] 1997.

184

Pieri 1895a, 607. Pieri 1898c and [1900] 1901, translated in sections 6.0, page 142, and 4.1, page 60.

185

Concerning nominal see section 2.2.

135

5.7 From Duality to Plurality

as P . Again, all theorems derived for tations.186

P

are also valid for

D

with their dual interpre-

Thus, Pieri reformulated Gergonne’s dualization operation—interchange of linguistic expressions with their duals—as a logical step: consideration of the dual geometry. In his very formal retrospective survey of foundations of mathematics, Pieri elevated duality from its status as a fundamental principle of projective geometry to that of a fundamental principle of mathematics: ... certain canons and laws—for example the principle of complete induction, the law of duality and the theorem of Staudt in projective Geometry, the principles of Hamilton and Hertz in mechanics—should serve to represent, embrace, and summarize an immense number of facts.187

Pieri had developed the method of hypothetical-deductive systems for a much broader context of logical and mathematical studies, and he regarded his reformulated law of duality as simply one example application of it. In his 1898c full exposition of projective geometry he described it grandly: here, the law of geometric duality enters as a form of a much more general principle, not belonging to Geometry more than to [any] other deductive science: [a] principle on which might also be conferred the name law of plurality.188

186

In 1895a, 624, Pieri argued that his first ten postulates supported three-dimensional duality. He did not provide all details, and made no claim about duals of his later postulates, which involved his undefined order relation. In 1896c, he extended this presentation to n dimensions. In 1898c, §11, translated in section 6.11, Pieri included a more complete discussion, for n dimensions in general; he had no need there to discuss order, because he had provided in 1897c a nominal definition for order in terms of point, line, and incidence. Proofs of the dual forms of postulates would be facilitated by using undefined notions and postulates for n-dimensional projective geometry that are self-dual, like the now-familiar postulates for plane projective geometry. (That is, changing symbols for all the primitive notions in each postulate to symbols for the corresponding dual concepts should produce obvious consequences of the original postulates.) See Esser 1973 for a survey of such efforts.

187

Pieri 1906d, 40.

188

Pieri 1898c, §11, 56, translated in 6.11, page 216. Pieri repeated this thought in his 1900a memoir on the axiomatization of elementary geometry, translated in 8.1, page 256.

Pieri in Turin.* Pieri earned the laureate from the Scuola Reale Normale Superiore in Pisa in 1884. He secured his first regular academic position two years later in Turin, at the Royal Military Academy. Turin was a manufacturing center with about 300,000 citizens, the seat of government of the Piedmont region in northern Italy, not far from the foothills of the Alps. He moved there with his mother and youngest sister; his father had died in 1882. They lived at 36 Corso San Maurizio. Military-academy professorships carried less status than those in universities, which required further preparation. In 1888 Pieri obtained a second position, across the street at the University of Turin, as assistant to the chair of projective geometry. He earned the venia legendi there in 1891: the right to teach as libero docente, paid solely by his students’ fees, and the right to apply for a university professorship. This is the title under his name on the 1898c and 1900a papers translated in chapters 6 and 8. Pieri sought long and hard for a university professorship, in an era when academic positions were scarce. During that time he continued research in algebraic and differential geometry, but developed a second specialization, logical foundations of geometry, in which he achieved the pioneering results reported in the 1898c and 1900a papers. Pieri’s quest finally succeeded in 1900: at age thirty-nine he won the competition for a professorship at the University of Catania in Sicily, far to the south. That stage in his life is discussed briefly in the chapter 8 introduction.. The photographs on the facing page depict Pieri’s surroundings in Turin; arrows indicate important features. The top figure, from about 1934, displays the fifteenth-century Castello with its Baroque facade, a major feature of the central city. Across from it are the former Royal Theater and behind that, the Military Academy.† These were destroyed by fire and aerial bombardment in 1936 and 1943 and replaced by the present, much larger, opera house. The very narrow street with the arched entrance, Via Verdi,‡ separated those facilities from the main building of the University. The middle figure is a modern photograph of its interior courtyard. The tower in the right background of the top figure is the Mole Antonielliana, originally intended as a synagogue but housing a museum since its completion in 1889. The dark horizontal streak behind the Mole is the line of trees along the broad Corso San Maurizio. Via Verdi runs about 850 meters from the arches to intersect the Corso about 150 meters from the River Po. Pieri’s apartment building stood on the Corso about 150 meters to the left of that intersection. It has been replaced by the modern structure shown in the lower photograph; the original building was probably similar to the one immediately to the left. *For many more details on Pieri’s personal life see M&S 2007, §1.1. † In M&S 2007, 19, the top figure depicted the wrong building for the Military Academy. ‡ Until 1919, Via Verdi was called Via della Zecca (of the Mint). For a view of the courtyard of the former Military Academy building, see M&S 2007, 19.

© Springer Science+Business Media, LLC, part of Springer Nature 2021 E. A. C. Marchisotto et al., The Legacy of Mario Pieri in Foundations and Philosophy of Mathematics, https://doi.org/10.1007/978-0-8176-4823-7_6

136

» Mole Antonelliana » Corso San Maurizio



Royal Theater,



» Military Academy » University of Turin



» Via Verdi (Via della Zecca)

» Castello

Turin, Piazza Castello, 1934 Photograph by Mario Gabinio

.

University of Turin Rectorate

Pieri’s Apartment Location (2008 Photograph)

36 Corso San Maurizio

6 Pieri’s 1898 Geometry of Position Memoir This chapter contains an English translation of Mario Pieri’s 1898c memoir, The Principles of the Geometry of Position Composed into a Deductive Logical System,1 his most important contribution to the foundations of projective geometry. From antiquity, the incidence theorems of what would become projective geometry had been proved using Euclidean metric concepts such as length and angle measure. It took centuries to establish projective geometry as a science in its own right, rather than as an extension of Euclidean geometry. By the nineteenth century it had become evident that some projective methods are more basic than classical Euclidean ones. This led to a desire to establish projective geometry as a science prior to rather than derivative from Euclidean geometry. Pieri would play pivotal and culminating roles in that undertaking. Pieri’s ultimate goal for projective geometry was its rigorous construction as an autonomous subject. The greatest influence on him was G. K. C. von Staudt, through the 1847 book Geometrie der Lage (Geometry of Position). Two decades later, Theodor Reye extended and elaborated on that work in a book with the same German title. Pieri’s first contribution was his Italian translation of Staudt 1847. Inspired by Staudt’s explicit goal of freeing projective geometry from metric notions, Pieri continued, refining and improving the efforts of both Staudt and Reye. With the axiomatization translated in the present chapter, Pieri rigorously developed projective geometry based on the notions point and line joining two points, and finally achieved Staudt’s goal. From those two primitive notions, Pieri constructed projective geometry with real coordinates in any specified dimension.2

1

Pieri 1898c is the same as the version in Pieri’s collected works on foundations of mathematics (1980, 101–162) except for pagination.

2

Staudt 1847, Reye [1866–1868] 1877–1882, Pieri 1889a, Pieri 1898c.

Introduction

139

The present authors have inspected Bertrand Russell’s personally annotated copy of Pieri’s paper.3 The annotations are described in detail in appendix 3. To emphasize that his presentation did not depend on intuitive considerations, and likely in deference to Staudt’s precedent, Pieri included no illustrations. The present authors have included in sections 6.5 and 6.7 two illustrations to help readers visualize Pieri’s notable definitions of projective segment and order of points on a projective line. The translation in this chapter proceeds according to the conventions outlined in the preface on pages xii–xiv. It is meant to be as faithful as possible to the original, preserving aspects of Pieri’s work that are tied to his expository style. In this paper, Pieri abbreviated many Italian words that he used frequently. In the translation, the corresponding English terms are spelled out, and commas separating items in a list or serving as logical conjunctions are often replaced or supplemented by “and”. Editorial comments, enclosed in square brackets [like these], are inserted, usually as footnotes, to document change in mathematical terms, to note or suggest corrections for occasional mathematical errors in the original, and to explain a few passages that may seem puzzling. Pieri used symbols [0], [1], [2], ... to denote the sets of all points, all lines, all planes, and so on. The present authors feel that the stylistic features just noted are superficial, and invite readers to notice that the structure of Pieri’s presentation of projective geometry is uncannily modern!

3

Russell and Pieri 1898.

Pieri’s 1898 “Geometry of Position” Memoir, First Page

THE PRINCIPLES OF THE

GEOMETRY OF POSITION COMPOSED INTO A DEDUCTIVE LOGICAL SYSTEM

MEMOIR BY

MARIO PIERI LIBERO DOCENTE AT THE UNIVERSITY OF TURIN

Approved at the session of 19 December 1897.

INTRODUCTION ... Foundation of these first elements is a task certainly not less difficult than the further development of the most complicated theorems. That pushes into the depths, this into the heights: and the depths and heights are equally unbounded and dark ... ... surely, however, the interest in these is rather great, because the inherent property of mathematics, and with it geometry, to be one of the few parts of human knowledge that enjoy complete certainty and truth, will become doubtful as soon as the theorems and concepts on which it is based become shaky. August CRELLE, On the Theory of the Plane4

Projective geometry has been regarded for a long time as a simple continuation of elementary geometry; and by most it is still preferred to establish its principles with successive extensions of the concepts that govern elementary geometry, derived from observation of the external world and fully conforming to the idea that they are acquired through experimental induction from certain qualities of physical objects and facts. In this way geometry, also preserving in its methods that deductive character which stems from the most remote antiquity, is always presented as an aspect of the physics of extension, rather than as taking a position alongside analysis among the purely mathematical disciplines.

4

Crelle 1853, 20, 17. [The italics are Crelle’s. They are different in Pieri 1898c.]

142

6 Pieri’s 1898 Geometry of Position Memoir

A different and more modern standard—which develops naturally as far as the principles are concerned but should aim as well at a quite different goal—holds that the geometry of position (and along with it the abstract metric geometries derived from it) should be considered as a purely deductive science independent of any other body of mathematical or physical doctrines, and therefore also of the axioms or hypotheses of elementary geometry and of all means of measurement in space with a movable unit, and so on. This other route, opened by G. K. C. VON STAUDT (in the 1847 Geometry of Position and the 1856–1860 Contributions to the Geometry of Position) and then pursued in further studies by Arthur CAYLEY, Felix KLEIN, Riccardo DE PAOLIS, and others (and in the same way by all the celebrated writings on non-Euclidean geometry and on spaces however extended 5), by now leads smoothly to a geometry of position entirely speculative and abstract whose subjects are pure creations of our mind, and postulates, simple acts of our will (without excluding that they should often have their deepest root in some external fact): the one and the others are thus arbitrary, at least insofar as we do not assign them a pre-established goal that should have to provide guidance for thought. Thus, for example, the intention will be considered advantageous that the geometry of position should be in all ways subject to the rule of certain fundamental laws—such as the principles of projection and of duality—which, so to say, inform and impress their character on it; and that by means of suitable interpretations (say, in the light of some nominal definitions) it should be possible to glimpse easily in its results the complex of facts comprising ordinary elementary geometry and its sister metric geometries. Here is not the place to argue about the merits of the one or the other of these ideal processes: both appear to me coherent and legitimate, and in different ways fit into the work of consolidating the geometric edifice on a new basis. Nevertheless, the present work is to be ascribed to the second direction, speculative and abstract. What should be proposed—with analyses unusually precise and detailed, and by means of studies conforming to the increased speculative requirements concerning rigor and to the improvements brought to the deductive method by algebraic logic—[is] to establish solidly the foundations of projective geometry (which is really to speak of almost all of modern geometry) considered as a hypothetical science, and moreover in method, and again in the premises, indeed entirely independent of intuition. Part of the material that is now assembled here was already worked out successively in several memoirs composed in the years 1895–1897 with these titles: On the Principles that Support the Geometry of Position (Pieri 1895a, 1896a, 1896b), A System of Postulates for the Abstract Projective Geometry of Hyperspaces (Pieri 1896c), and On the Primitive Entities of Abstract Projective Geometry (Pieri 1897c).6 But their content does not just reappear arranged in a totality more coherent and organic: rather, it is recast, in many places augmented, in others changed completely. New, or completely revised, are the treatments of harmonic sets,7 of projective segment, of the senses or directions of a line, of projective hyperplanes; some primitive propositions were eliminated (which was 5

[Pieri’s phrase spazi comunque estesi means multidimensional spaces (Loria 1896, 302).]

6

[Pieri used the abbreviations m1 , m2 , m3 for these publications.]

7

[In this context, Pieri’s term for set was gruppo.]

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proposed and sketched in 1897c), others simplified or reduced to statements of less deductive strength, and so on.8 The primitive or undecomposed concepts, in terms of which all the postulates are stated (and are like the raw material of each proposition) will be reduced here to just two: projective point and the join of two projective points. The possibility of decomposing each other concept of projective geometry, however elevated and complex, into just these two elementary notions, to my knowledge, has not been rigorously proved and not even clearly affirmed until now. Thus, for example, in STAUDT the simple angle-space, a portion of a bundle of half-rays, is employed in principle as a primitive notion;9 and in DE PAOLIS the distance between two points (which, like the half-ray, is not a projective concept).10 In the more recent studies of Giuseppe VERONESE, Gino FANO, and Federigo ENRIQUES,11 there is the idea of projective motion, which, among the various modifications of ordered line, natural order, and of sense or direction of a line, and so on, acts as a primitive notion along with point and line without being decomposed nor defined except through postulates.12 In contrast, projective segment is defined here in terms of point and line; and in terms of segment, also the relation of separation of points, the natural orders and directions of a line, and so on. Thus, the distinction between properties of configuration (or involving the mutual incidence of points, lines, and planes) and properties of connection (or involving the separation of elements from each other) appears superfluous and not corresponding to any real and intrinsic difference, because the former and the latter are equally reduced to predicates in the projective categories mentioned. In the choice of postulates it was attempted to observe the canons of the strict deductive method, taking care, for example, that not one should be easy to decompose into several distinct statements of lesser weight (that is, for many reasons, it should not follow the thoroughly modern example of compound postulates)—nor should it have greater strength than what is needed for its deductive role. Their number is nineteen for ordinary projective geometry, seventeen for the projective geometry of the general space or absolute

8

The fundamental premises thus modified can be found on pages 148–152, 154, 159, 166–167, 172, 174, 198, and 214.

9

Staudt 1847, §1, paragraph 10. [Staudt’s term was einfacher Winkelraum; Pieri’s was semplice angoloide.]

10

De Paolis 1880–1881, 489. [See also section 5.3, page 105.]

11

Veronese 1891; Fano 1892, 125; Enriques 1894, 554. [See also subsection 9.2.3 and the box in subsection 9.2.1, page 344.]

12

Definition, [employed here] in a sense rather more strict than usual, should properly be called nominal definition, consisting of the simple imposition of names to things already noted, or that are presumed acquired through deductive steps—in short, expressible in terms of the relevant primitive notions by means of logical categories, not excluding the processes of abstraction and complete induction (see, for example, Burali-Forti 1894a, chapter IV). On the other hand, real definition, of a notion in itself, is a system of predicates sufficient to describe a subject to the deductive ends sought; but here it is preferred to say in such a case that the notion is not defined.—Moreover, the discourse will strive to eliminate all confusion. All the definitions in this essay are purely nominal.

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projective ambient.13 About the subsequent irreducibility of such premises and the independence of each from the others (conditions that almost amount to ideal perfection) we cannot affirm anything in an absolute way; however, the accurate and patient study that we have completed permits the hope not to be too far from that summit.14 But it can be resolutely affirmed that all these suffice together for the requirements of the geometry of position; and it is inferred from this that through them is provided the means to reach the fundamental theorem on projectivity, or theorem of STAUDT (§10), without any other support. Adequately proved in this way by STAUDT and others,15 this in turn permits representation of the points and lines of any space or fundamental form of a given species or dimension by projective coordinates—and thus, for any projective question that might arise about such a form, reduction to a merely algebraic problem. Consequently, every law or fact pertaining to the domain of the geometry of position can in the end only be a logical combination of our primitive propositions, and a statement, more or less complex, about the primitive notions “projective point” and “join of two projective points.”16 The usefulness of a good ideographic algorithm is generally recognized, as an instrument employed to facilitate and impose discipline on reasoning; and to exclude ambiguities, misunderstandings, mental restrictions, insinuations, and other defects nearly inseparable from common language, whether spoken or written—and so harmful to speculative

13

It should not be inappropriate to observe that the number of postulates can increase and yet the sum of the arbitrary conditions that are imposed by them should remain as before, but not decrease; this is the deductive power or weight of the entire system. Also, a decrease [in their number] can result from the substitution, for certain premises, of a principle equivalent to their conjunction. But if an increase is due to an attempt to decompose the statements into elementary and distinct parts, no criticism is possible: the more so if this is thought to be the more suitable way to remove superfluous requirements gradually.— It is not even impossible to consider a geometry without any apparent postulate; and to clarify this it suffices to reflect on the possibility of introducing, in place of each postulate or group of postulates, a special name created to denote [the class of] all entities that satisfy it: that can be done just with formal definitions. Various names (or distinct identifiers) should be used, for example, to distinguish our fundamental notions “point” and “join,” depending on whether it should be desired that they conform to postulates I–XIX, or just to the first eighteen, or to the first seventeen, and so on: and therefore the definitions of those names should come to take the place of the postulates, when all the propositions have been enunciated in terms of them, and not in terms of the generic notions “point” and “line.”

14

See the appendix [section 6.13] for this.

15

Staudt 1856–1860, §19–§21, §29; Lüroth 1874; Sturm 1875; Fiedler 1875; De Paolis 1880–1881; and so on.

16

It is understood that the pure science of reckoning [Scienza del Calcolo] has no postulates of its own, but adopts a few arithmetic principles about the primitive ideas of number (positive integer), of unity, and of the successor to a number: principles that we—following Richard DEDEKIND and others—ascribe instead to logic, because without their application no deductive science seems possible.

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inquiries.17 Thus, the systematic use of the methods and appropriate symbols of algebraic logic is to be held in high regard: in fact, not to make use of them at all and to disregard their advantages (especially in this type of study) would appear to me almost to disregard on purpose the most valid means by which it should be possible today to be prepared for the analysis of ideas. The present researches were conducted and examined in every part according to the system of symbolic notations proposed by Prof. Giuseppe PEANO18 and already somewhat well known and adopted in Italy. However, through regard for the majority of scholars, including nonmathematicians, and to escape the appearance of imposing on the reader a symbolism new and not yet sufficiently widespread,19 the writer has deemed it appropriate in this memoir (save for a few exceptions) to conform to the customary form of discourse. In consequence there remain embedded in the propositions and demonstrations many details, not really avoidable in themselves, that are difficult to render into ordinary prose, because of the excessive prolixity and roughness that they would carry into it, or else such that in ordinary language they should come to lose, at least in large part, those attributes of exactness and precision that constitute their greatest merit. It will be attempted to temper the aridity of the prose with varied and appropriate reflections. Many of these, especially among those inserted into §1, will be of a certain use in reading all the rest. And to reduce somewhat the size of this essay, a few useful abbreviations will be employed. I list next the principal [ones], with some other information not to be neglected.

17

It is not without purpose to refer here to some passages from the work Lectures on Recent Geometry by Moritz PASCH (1882b, §12, 99–100). Although intended to establish geometry on new intuitive and experimental principles, this has acquired in our days well deserved authority in the criticism of foundations: “We meet again in science a group of statements with whose usage we have become comfortable through age-old custom; and, as in daily life, through use of those statements together with all sorts of relationships between the corresponding concepts we become entangled with our thoughts unless we take special account of them, so even in rigorous science it is not easy to succeed in holding unfamiliar contaminations completely at bay. ... Science derives part of its substance directly from the language of daily life. From this source, modes of speaking and intuitions with which one should not formulate scientific propositions are also introduced into mathematics, and have become there the reason that certain developments appear unclear, and that many discussions have arisen, particularly about geometric objects. What role the individual concepts and relations play in the system, and to what extent they are necessary or dispensable for its entirety, comes to light only through absolutely rigorous exposition. Only when the important components are completely assembled, but the superfluous ones excluded, will one possess the proper foundation for those discussions, that thereby they should not become meaningless.” [For the second edition of this work ([1926] 1976, 91–92), Pasch removed the words after mathematics from the third quoted sentence; in the last quoted sentence he changed the confusing phrases after excluded to does one obtain the basis for general discussions about geometry.]

18

See especially the Mathematical Formulary (Peano et al. 1895–1908, volumes I and II).

19

There was a time when one begged the pardon of the reader for each unusual term or symbol that one desired to introduce into the discourse!

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LIST OF ABBREVIATIONS 20 P, Hp, Th stand for “proposition,” “hypothesis,” “conclusion.” Arm(a, b, c), where a, b, c should be collinear points, denotes the harmonic conjugate of c with respect to a and b: that is, the [fourth] harmonic after a, b, c. (a, b, c), where a, b, c should be collinear points, represents the line segment ending in a and c and containing b. Taken from Hermann SCHUBERT are the abbreviations [0] (projective point); [1] (projective line); [2] (projective plane); [n] (projective hyperplane of the nth species). The following are from PEANO.21 But these will be used merely as abbreviations, without making it necessary for the reader to know anything else about them. 

Placed before an entity of a given class, this sign denotes the set of all individuals that are regarded as equal to that same [entity] with respect to the class. It can be read, “equal to” or “coincident with.” (See §1.) 22

0

This always precedes a common noun, that is to say, a symbol for a class, and follows the name or names of one or more of its individuals. It can be understood and read as “is a ... ,” “are some ... ,” or “belong[s] to ... .”

c

Placed between two common nouns, this denotes the union of the two classes: that is, the set of all individuals that belong indiscriminately to the one or the other. It can be read, “or.” 23

1

Placed between two common nouns, this represents the intersection of the two classes: that is, the set of all individuals common to both. It can be read, “and,” or “together with.” 23

-

The negation sign: this stands for “not.” 24

f

Placed between two classes, this would say that the first (on the left) “is contained in” or “lies in” the other, or that each individual of the one also belongs to the other.

20

[Some of Pieri’s text under this heading has been altered. Some abbreviations and symbols have been changed to fit English terms or modern usage. Pieri used for the subset relation f; he used his negation symbol - in the compounds -0 and -= and also in expressions A - B representing differences of classes A and B. In prose passages these have been changed to ó, =, / and A – B. (Pieri’s original symbols have been retained in quoted ideography.) Pieri also listed here abbreviations Cl, Cp, Df, and Tr for are collinear, are coplanar, definition, and theorem. This translation does not employ them. Pieri introduced a few more abbreviations later in this memoir.]

21

Schubert 1886, 27. [In 1905c, Pieri changed from the notation [n] to Sn.] Peano 1890, 183, 192.

22

[Introduced by Peano to denote singleton sets, the symbol  occurs repeatedly in sections 6.9 and 6.10 in formulas rendered in ideography, which are translated in editorial comments. Formulas such as x and x c y, where x, y are individuals, are translated as {x} and { x, y}. In 6.4 Pieri also used Peano’s symbol  for the unique member of a singleton:   x = x. See Peano 1894a, 38; 1897b, 581–582.]

23

[Pieri used the terms summa logica for union or disjunction, and prodotto logico for intersection or conjunction.]

24

[Pieri continued: “For example, ‘-0’ stands for ‘is not equal to,’ and if k is a common noun and a a proper noun, the expression ‘k- a’ denotes the class of those ‘k not equal to a’ (that is, different or distinct from a).” ]

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=

Between two classes this indicates in the same way that each is contained in the other. Between individuals of a class it stands for the group of symbols 0 .25

The notation P1, P2, ... denotes or refers to propositions 1, 2, ... ; and if not accompanied by  d, c, a  any § [section] citation, it must refer to the current §. And the symbol  a, b, c  P7§2 denotes P7 of §2, in which the names of objects a, b, c should be changed to d, c, a, respectively.26 Also, following the example of many [authors], the symbol  is used in place of the phrase “is equal by definition to.” Thus,   a b c where a, b, c should be noncollinear 27 points, means “ is the plane a b c,” or “denoting by  the plane a b c.” Demonstrations will generally be preceded by the word Proof and terminated by the end of the paragraph or an em dash —.28

§1 The Primitive Entities 29 The most important property of the primitive entities of any hypothetical-deductive system is their capacity for arbitrary interpretation, within certain limits assigned by the primitive propositions (axioms or postulates). In other terms, the ideal content of the words or symbols that denote such a primitive subject is determined solely by the primitive propositions that mention them: and the reader has the option to attach to these words and symbols a significance ad libitum, provided this should be compatible with the generic attributes imposed on the entity by the primitive propositions.30 Thus it is, for the words “projective point” and the symbol [0], which are introduced here to represent the same primitive entity. The arbitrariness granted their content will be diminished gradually by the postulates, which will be specified one by one.

25

[For individuals x, y the equation x = y means that x belongs to the set of entities equal to y.]

26

[Pieri’s cross references of the form “(P1)” have often been rendered as “by P1” or “(see P1).”]

27

[In error, Pieri wrote noncoplanar here.]

28

[Pieri enclosed proofs in square brackets, which have been deleted because this translation uses square brackets to signal editorial insertions. No translated proof will itself contain an em dash, nor will it extend beyond a single paragraph, unless specifically noted.]

29

Compare Pieri 1895a, §2–6; 1896c, §2; 1897c, §1–§2.

30

“Also, if geometry is to be really deductive, the inference process must in fact be entirely independent of the meaning of the geometric concepts, just as it must be independent of the figures; only those relations between the geometric concepts may come into consideration, which were laid down in the [primitive] propositions and definitions used.” (Pasch 1882b §12, 98, or [1926] 1976, 90).—See also the note in the appendix [section 6.13 of the present book].

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POSTULATE I P1. « Projective point », signified 31 also by [0] , is a class. To the term class is attached merely the meaning of set, variety, collection, general idea. Thus, the proposition just stated can be seen as equivalent to this: “the symbol [0] is a common noun.” To the idea of class is related that of individual as common noun is to proper noun. This refers to the relation of membership, or belonging, that subordinates the individual to the class (the range of predication of a simple proposition) by means of the symbol 0. (See the introduction.) Thus, the statements abbreviated by “a 0 k” and “a, b 0 k,” where k is a class, mean “a is a [member of ] k” and “a and b are individuals in k.” But for expressing those relations in geometry certain considerations of style sometimes impose the usages “a lies in k,” “k passes through a, b,” “k contains a, b,” and others similar, to which we may be accustomed. Thanks to P1, the conditional proposition “x is a projective point” or “x 0 [0]” always has a meaning: but it could be unsatisfiable32 inasmuch as it has not been said that «projective point» should be a nonempty class. One knows that logicians give meaning even to void or illusory classes, to which no individual belongs, almost as to zero among numbers. Thus it is appropriate to accept the following: POSTULATE II P2. There is at least one projective point. That is, [0] is not an empty class. P3—Definition. Each class of projective points is called a figure or form.—To say that one figure  is contained, or lies, in another [figure]  (  f ) is as much as to affirm that each point of  belongs also to . Moreover, if each point of  is also a point of , the two figures are called equals of each other, or coincident (  = ). Thus, «projective point» is a figure: indeed, the largest figure, because every other [figure] is contained in it. Furthermore, the “class [0] of all projective points” is also a term synonymous with projective space or ambient. P4—Definition. If a is a projective point, each projective point that should belong to every figure containing a is called equal to a or coincident with a, or [a member of ] {a}. In other terms, the locutions “x is equal to a,” “x coincides with a,” and “x 0 {a},” as well as “x = a,” stand in place of “x is a projective point, and there is no figure that passes through a without passing through x.” 33

31

[For this translation’s use of guillemets («þ») and Pieri’s class-membership conventions, see a box in the preface, page xiii. (This footnote will be cited several times in this chapter.)]

32

[Pieri used the adjective assurda.]

33

Compare Peano 1897b, 581–582, §20. [In error, Pieri referred to page 19.]

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6.1 The Primitive Entities (§1)

Thus, “x does not coincide with a” (x = / a) would mean, “there is a class of points that contains a, but x is not a member of it.”—Thus defined, equality presents the qualities that, as a rule, are required of equality between individuals of any class whatsoever (the attributes of equality). These are principally the following: P5—Theorem. (1) If a is a projective point, then «equal to a», or a, is a class of projective points, to which belongs a: that is to say, “a always coincides with a,” or “a 0 {a}” (the reflexive property). (2) If, moreover, b = a and c = b, it follows that c = a (the transitive property). (3) From b = a stems a = b (the symmetric property). Proof. Indeed, in order that a = / b should have held, there would have had to exist (by P4) a figure  containing b among its points and not passing through a; therefore, the figure of points excluded from  (that is, the class [0] – , the complement of  with respect to [0]) would include a but not b, contrary to the hypothesis b = a.34 In place of “the points a and b do not coincide” one often says “a and b are distinct, or different from one another.” (See the introduction.) And in saying “a, b, c, ... are distinct points” it is meant to exclude, without further ado, that any two of them should coincide. The use of the notions and facts included in the propositions 3–5 just stated will not always be signaled to the reader. However, it should be possible to say that not one of our deductions, or very few, should be exempt from them. But one should take good account of these principles, and of some other observations in the present section, through the train of thought of this and the following sections. POSTULATE III P6. If a is a projective point, there exists at least one projective point not coincident with a. Thus, it is affirmed (under the hypothesis a 0 [0]) that [0] – {a}, the intersection of the classes «projective point» and «not equal to a» should not be empty, or that the compound conditional proposition “x 0 [0] and x ó {a}” 35 is not impossible to satisfy with respect to x. —From P2, P6 together it follows that it is not even impossible to satisfy the conditional proposition “x and y are noncoincident projective points.” That is to say, P7—Theorem. There are at least two distinct projective points. Now, supposing that a, b should be distinct projective points, it is appropriate to denote with the phrase join of a with b, or with the symbol ab, a new primitive entity to which should be ascribed initially the quality of class of projective points, or figure, by means of the following two statements: 34

See Peano et al. 1895–1908, volume 2, 7 (paragraphs 80–85).

35

[For x ó {a} Pieri wrote x 0 -a.]

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POSTULATES IV AND V P8. If a, b are projective points and b does not coincide with a, the join of a with b — also denoted by a b — is a class. Each of its members is a projective point. This is as much as to grant, in short, that under that hypothesis, ab should always denote a class contained in [0]. Next, it is established also that ab f ba. That is to say, POSTULATE VI P9. If a and b are projective points, b being different from a, the join of a with b will be entirely contained in the join of b with a. From this [one gets] the theorem P10–Theorem. Assuming a and b are distinct projective points, each of the two joins ab, ba will be contained in the other; thus, these figures will be equal to each  a, b  other or coincident. Proof. This follows from P9 and  b, a P9, considering P8 and P3. POSTULATE VII P11. If a, b are distinct projective points, a must belong to the join of a with b. Thus, from the hypotheses a, b 0 [0] and b = / a one deduces the conclusion a 0 ab. P12–Theorem. If a, b are as above, each of them will have to lie in the join of a with  a, b  b. Proof. From the hypothesis and  b, a  P11 one deduces b 0 ba, then the conclusion, from P10, 11. Thus it stands proved that the class ab is not empty, under the hypothesis of P11. But it is useful to allow more, that this figure should contain at least one point different from a and from b. That is to say, POSTULATE VIII P13. Supposing a, b are projective points and b differs from a, in the join a b lies at least one projective point not coincident with a nor with b. That is, the class ab – {a, b} is not empty. Use of the existential postulates II, III, VIII is rather restricted: only on rare occasions must they be invoked in the demonstration of a theorem. This happens because the hypotheses of our theorems almost always affirm the existence of all the elements that occur in the demonstrations.

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6.1 The Primitive Entities (§1)

The importance of those existential propositions is more critical than deductive, insofar as they serve most of the time to justify the hypotheses of the various theorems, to remove the doubt that these might be impossible to satisfy, and thus that their consequences might be of no speculative or practical interest. It will not be superfluous to observe, once and for all, that the statement “from P(a, b, c, ...) one deduces Q(a, b, c, ...)” —where P and Q should be propositions about the entities represented by the letters a, b, c, ... , changeable at will, thus having no significance independent of that assigned to them by P —is to be considered the same as “whatever a, b, c, ... should be, if P(a, b, c, ...) is true of them, then Q(a, b, c, ...) will also be true; whoever asserts P about these entities cannot deny Q for them.” Sometimes one deduces [this] just about some of the entities a, b, c, ... , without any regard to the others; but this will be stated expressly where it might not be clear in itself. Recall also the correct usage of the terms “is equal to,” “is equivalent to,” and so on, placed between two propositions, P and Q for example, by means of which it is intended to express the fact that “from P one deduces Q and from Q one deduces P,” or that each of the two propositions is a consequence of the other. This might be effected just by the definition of P or Q (nominal equivalence, ), or else by reason of principles already agreed on (real, apodictic equivalence, ]), as happens most of the time.36, 37 POSTULATE IX P14. Suppose that a and b should be projective points, b not coincident with a. If it happens that a point c should belong to the join of a with b without coinciding with a, it will be necessary that the point b should belong to the join of a with c. P15–Theorem. If a, b, c are projective points, b and c being both different from a, each of the two propositions c 0 ab, b 0 ac will be a consequence of the other.  c, b  Proof. This results from P14 and  b, c  P14. P16–Theorem. Supposing a, b, c are projective points, b different from a, and c different from b, the two propositions c 0 ab, a 0 bc will be equivalent to each other. Proof. From the hypothesis and P10 follows the equivalence of the proposi b, a  tions c 0 ab, c 0 ba; and the second of these, by virtue of  a, b  P15, is equivalent to the proposition a 0 bc.

36

Indeed, equivalence of propositions, like [equality] of classes of points, or figures (see P3), enjoys the reflexive, transitive, and symmetric properties (compare P5). Thus, to the idea of a figure or a proposition one should be able to conjoin in familiar ways a new abstract entity, the content of the figure or the deductive capacity or weight of the proposition, regarding the equivalence of two propositions, for example, as the equality of their deductive capacities.

37

[In this translation, Pieri’s symbol == for real equivalence has been changed to ].]

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POSTULATE X P17. Under the hypotheses of Postulate IX, the join of a with c will be contained in the join of a with b. P18–Theorem. Indeed, the two joins ab and ac will coincide. Proof. From the hypothesis and from P14, 8 it follows that c 0 [0] and b 0 ac. Therefore, by virtue of  c, b   b, c  P17 and P17, each of the figures ab, ac will be contained in the other.   P19–Theorem. From a, b being noncoincident projective points and c a point of ab different from b it follows that the joins ab and bc coincide. Proof. From the  b, a  hypothesis it follows (see P9) that c belongs to ba; thus, invoking  a, b  P18, one deduces ba = bc. And also from the hypothesis (see P10) follows ab = ba, and hence the conclusion, being a question of the equality of classes, for which the transitive property holds.  b, a 

 b, c 

From the preceding demonstration is omitted the citation of  a, b  P8,  a, b  P8; and also not seldom afterward, those propositions are allowed to lapse from the discourse where it is stated only that certain objects under discussion should be classes of points, or figures. P20–Theorem. Given projective points a and b distinct from each other, if a point c should belong to ab without falling on a nor on b, the figures ac and bc will coincide. Proof. From P18, 19 together. The following theorem is not to be regarded as superfluous, given our definition P4 of equality of points. P21–Theorem. If a, b should be noncoincident projective points and c, d points equal to a, b, respectively, each of the two joins ab, cd will be contained in the other. Proof. From the hypothesis insofar as a, b, c are concerned, and from P8, 4, 5, follow c 0 ab and c = / b; and then again, by force of P19, ab = bc. But the hypothesis  b, c, d  about points a, b, c is not invalidated by the substitution  a, b, c  . That is, the hypothesis persists after such a change, because b and c do not coincide. Therefore, the hypothesis also entails the conditions d = / c, bc = cd, and with those, the conclusion. This P21 affirms in substance that the principle of substitution holds for projective points, with respect to the content of an arbitrary join. And from this stems also the faculty to substitute points for equal points (with regard to containing as content,38 and vice versa) even in figures much less simple, but defined by means of points and joins: a faculty that we will sometimes use here without further details. On account of P12, 16, propositions P18, 19 are partially invertible: the following also serve in this guise. 38

[Pieri wrote nei riguardi di contenente a contenuto.]

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P22–Theorem. Assuming that a, b, c should be projective points, and a should not coincide with b or with c, each of the two statements “c belongs to ab” and “figures ab and ac coincide” will be a consequence of the other. P23–Theorem. If a, b, c should be projective points, b different from a, and c different from b, the two propositions c 0 ab and ab = bc will be equivalent to each other. P24–Theorem. From the assumption that a, b, c be projective points distinct from each other follows the equivalence of the statements c 0 ab, ac = bc. Serving to close this chapter will be a fundamental proposition commonly known in one of the following forms: “the join of two distinct points is characterized by an arbitrary pair of its points, provided [they are] noncoincident,” “two joins, that have two points in common, coincide,” “through two points passes no more than one line,” and so on. P25–Theorem. Given that a, b should be noncoincident projective points, and that c, d should be projective points belonging to ab but different from each other, it is necessary that the joins ab and cd coincide. Proof. From the hypothesis and from the attributes P5 of equality it follows that c cannot at once coincide with a and with b. Now if c is different from a, it will result from P18 that ab = ac,  c, d  and consequently that d 0 ac; and from here, in view of  b, c  P19, also ac = cd and finally ab = cd. If on the other hand c is distinct from b, P19 entails ab =  b, c, d  bc, d 0 bc; thus, the same P19, or rather  a, b, c  P19, entails bc = cd and consequently also ab = cd. —This proposition, admitted as a postulate in recent works,39 emerges from relationships somewhat simpler than itself, seeing that, for example, Postulate IX (from which it principally derives) certainly expresses a judgment of the same sort as P25, but about a smaller number of points.

§2 The Alignment Relation and the Projective Line P1—Definition. Assuming a, b, c are projective points, any one of the phrases a, b, c colline, or are collinear, or are aligned should convey the fact or statement expressed in the following terms: “there exist two projective points x, y, distinct from each other, such that a, b, c lie together in the join of x with y.” 40 Thus is defined a certain relation among three projective points, symmetric with respect to them, and represented by the expressions “are aligned” or “colline”; and with that also its contrary relation, or negation, so that, in short, to say that “three points are aligned or are not” is as much as to affirm—or deny, respectively—the 39

Veronese 1891, 214, 244; Amodeo 1891, §1.2; Pieri 1896c, §1, 83, footnote; Pieri 1897b; Fano 1891, §1, 109. [Pieri’s 1896c footnote merely cites Enriques 1894 and other items cited in the present footnote.]

40

[Pieri also defined here his abbreviation Cl for “are collinear.” This translation does not employ it.]

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existence of two distinct projective points whose join passes through all three. A formal convention enters here (as with all nominal definitions), which gives rise to a greater facility of thinking and speaking: an abbreviation in notation, corresponding to a condensation of ideas. P2—Theorem. Given that a, b should be noncoincident projective points, and that c should be a point of ab, it follows that points a, b, c colline. Proof. The hypothesis explicitly grants the existence of distinct points a and b for which, by virtue of P12§1, the condition a, b, c 0 ab is satisfied: from this follows the conclusion, by way of  ax,, by  P1. P3—Theorem. If a, b, c are collinear projective points, and b does not coincide with a, then c is forced to belong to ab. Proof. From the assumption (in addition to the hypothesis) that x, y should be distinct projective points and that a, b, c 0 x y,  x, y, a, b  one deduces, referring to  a, b, c, d  P25§1, that x y = ab, and therefore the conclusion. But the existence of the points x, y introduced in this way is admitted with the hypothesis by virtue of P1: thus, the conclusion is a consequence of the hypothesis. The existence of at least three nonaligned points could be denied at present, because it does not follow from the principles accepted up to here. The following principle provides for this; however, it will not see any use in the deductive framework until §4. POSTULATE XI P4. Assuming that a, b should be distinct projective points, there must exist at least one projective point not belonging to a b. P5—Theorem. Under the hypothesis that a, b, c are projective points and b does not coincide with a, the propositions “a, b, c colline” and “c belongs to ab” will be equivalent to each other; and thus their negations will also be equivalent. Proof. This is the conjunction of P2, 3. P6—Theorem. Three projective points a, b, c are certainly aligned, if two of them coincide. Proof. We suppose, for example, that c should coincide with a. Then if b = / a, it suffices to invoke P2 because a 0 ab, and as a consequence also c 0 ab, by the hypothesis and P11, 5§1. If on the other hand b = a, there will nevertheless exist a point x different from a, by P6§1; therefore, since the proposition x 0 [0] – {a} can be conjoined with the hypothesis, it will happen similarly that  a, x  a, b, c 0 ax, and with this the conclusion, by virtue of  x, y  P1. The other parts of the  b, a   c, b  theorem follow from this with the substitutions  a, b  ,  b, c  because by P1 the relation of alignment is symmetric. P7—Theorem. If a, b, c are noncollinear projective points, it will not be possible that two of them should coincide. Proof. The statement is equivalent to P6, by a most familiar principle of logic.

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P8—Theorem. Nor that one of these should belong to the join of the others. Proof. This reduces to P5, thanks to the preceding P7 and to P15, 16§1. P9—Theorem. Moreover, each [point in] ab will be distinct from c, just as each [one in] ac [will be] from b, and each [one in] bc, from a.31 Proof. [This is a] consequence of P8 and of P4, 2, 8§1. P10–Theorem. Under the same hypothesis as P7, there does not exist any point common to the three joins ab, ac, bc. Proof. The unsatisfiability of the compound conditional proposition “y = a and y 0 bc,” and thus also, a fortiori, of “y = a and y 0 ab 1 ac 1 bc,” follows immediately from the hypothesis and from P9. But not even the existence of a point y satisfying “y = / a and y 0 ab 1 ac 1 bc” will be permitted by the hypothesis, because from this, by virtue of P7, 5 and P24§1, it follows  y  c, y  that ab = / ac, whereas the given condition would imply, in view of  c  -,  b, c  P18§1, the equalities ab = ay, ac = ay, and therefore ab = ac. Now, the disjunction of those two propositions, [which are] unsatisfiable with respect to y, is the (unsatisfiable) proposition y 0 ab 1 ac 1 bc. P11–Theorem. Three projective points a, b, c will certainly be collinear if there should exist a projective point d aligned both with a and b, and with a and c, but not coincident with a. Proof. From the combined assumptions that a, b, c 0 [0], that d 0 [0] – {a}, that a, b, d be collinear, and similarly that a, c, d be collinear, it d d  d, b  follows, whatever d might be, through  b  P11§1 and  b, c  -,  b  P3, that a, b, c 0 ad,  a, d  and consequently, in view of  x, y  P1, that a, b, c should be collinear.—Although different in appearance, the proposition that follows is deductively equivalent to P11. P12–Theorem. If a, b, c are noncollinear points and d is a projective point distinct from a it follows either that a, b, d are not collinear or that a, c, d are not collinear. P13–Theorem. If a, b, c, d are projective points such that a, b, d as well as a, c, d and b, c, d colline, [then] points a, b, c will also be aligned. That is, if among four points there are three alignments, the fourth must [also] hold. Proof. If d = a, the hypothesis explicitly contains the conclusion by virtue of P1 and the familiar attributes of equality among points. If on the contrary d = / a, this reduces to P11. P14–Theorem. If a, b, c are noncollinear projective points and d should be an arbitrary projective point, then a, b, d or a, c, d or b, c, d are noncollinear. Proof. P14 is logically equivalent to P13. Given distinct projective points a and b, the entity ab, or join of a with b, is a determined and unique figure. Moreover, abstracting from specific points gives rise to the generic notion of “join of two distinct points”: that is, of one entity that can be conceived in a collective sense as the set of all possible joins. To this is given the name «projective line.» 31

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P15–Definition. With the name «projective line» - —or with the symbol [1] —is represented the class or variety of all the figures in each of which can be found two distinct projective points of which it is the join. In other terms, the statement “r is a projective line” serves to affirm that there exist two noncoincident projective points x, y, and r is, under another name, their join x y. The entity [1] thus defined is indeed not a class of projective points, but a class of classes of points, the one that gives a precise meaning to the compound proposition “r 0 [1] and x 0 r.” And this class of figures is certainly not empty. That there exist projective lines is derived immediately from P15, referring to P7§1 and P4. Moreover, from the postulates accepted until now it would be possible to infer the existence of at least six distinct points and six distinct lines. Thanks to the definition just stated, it would be possible to simplify somewhat the statements of the preceding propositions. Thus, for example, the roles of P1, 4 are now immediately assumed by these two others: P16–Theorem. If a, b, c should be projective points, the statement “a, b, c colline” will be equivalent to the assertion “there exists a projective line containing a, b, and c.” P17–Theorem. Given a projective line, there exists at least one point that does not belong to it. Moreover, perhaps it will not seem inappropriate in a study of the principles to have given a more distinguished role to the concept of collinearity—that is to say, the rather simple projective relationship that three projective points might enjoy —just as coincidence (§1) is the most simple, in fact the unique, projective relationship that could hold between two points.41

§3 The Visual of a Form and Projective Planes 42 Whenever a form  should be given (a class or variety of projective points), and a projective point p, the joins of p with the individual points of  different from p constitute together a new figure that comprises all projective points collinear with p and

41

[The relationship that holds between any pair of points is also projective, as are inequality and the relationship that holds for no pair.]

42

See Pieri 1895a, §4, §5 [discussed in subsection 9.2.1].

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with some point of  different from p. This figure, generated by means of  and p, can be called the visual (Schein) of the form  from the projective point p and the double symbol p will suffice to designate it.43, 44 P1—Definition. Supposing p is a projective point and  is a figure, the “visual of  from p” or “join of p with ” —denoted by p —is the locus of all projective points x with which it should be possible to associate another point y in , different from p, so that x should belong to py: 45 that is, the class of all those x for which the proposition “y 0  – { p} and x 0 py,” conditional on x, y, is satisfiable with respect to y. P2—Theorem. Whatever may be the form  and the projective point p, each point of  noncoincident with p will have to belong to the visual of  from p; that is,  – { p} f p will always hold. Proof. By the hypothesis, if x is a point of , arbitrary but distinct from p, the conditional proposition “x 0  – { p} and x 0 px” will p, x always be satisfied, thanks to  a, b  P12§1: and from this follows the conclusion, in  x view of  y  P1.—It is not even excluded that  should be an empty class, nor that p should be its only member. P3—Theorem. The visual of a figure  from a projective point p will contain  and p, if there should exist at least one point of  noncoincident with p. Proof. By  p virtue of  x  P1 the assertion p 0 p means only that the compound proposition “y 0  – { p} and p 0 py}” is satisfiable with respect to y: but this is true by p, y the hypothesis, in view of  a, b  P11§1. The rest follows from P2. P4—Theorem. If a, b, p are projective points, and a, b are distinct from each other, the figures pAab and pAba coincide. Proof. From the hypothesis and from P10§1 one infers the equivalence of the two conditional propositions y 0 ab and y 0 ba, and    p, ba  thus also, by P8§1 and  pa,,ab   -,  a,   P1, the equivalence of the two conditions x 0 pAab and x 0 pAba.—In this way one also proves the equality, or coincidence, of the visuals of [two or] more coincident forms taken from the same point, or from coincident points.

43

This is more suitable for the class or sheaf [ fascio] of those joins than for the figure they occupy; but here it is taken in the latter sense, the other never occurring.—The concept of “visual” is contained in the more general one of “external projective product of two figures” that arises from joining in all possible ways a point of the one with a point of the other by means of projective lines (whereas, in the external or progressive product of Hermann GRASSMANN, the joins are Euclidean segments).

44

[The German term Schein was used by G. K. C. von Staudt in the same way (1847, §2). Theodor Reye related it explicitly to vision (1866–1868, 7–8). Pieri translated it as visuale in his 1889a translation of Staudt. To Charlotte A. Scott, however, it seemed untranslatable (1900a, 311). The term visuale had long been used in related ways in the contexts of drafting, which Pieri taught: for example, in Sereni 1846, 7, and in Euclid and Heliodorus 1573, 19. Visual cone is a common drafting term in English.]

45

[In error, Pieri wrote ay here instead of py. Pieri sometimes inserted white space where juxtaposition indicated a visual. Because that is so easy to miss, this translation employs a centered dot in such cases: pA will stand for the visual p .]

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P5—Theorem. If a, b should be distinct projective points and p an arbitrary point of ab, the figure pAab will coincide with ab. Proof. By the hypothesis and P5§1, p will be distinct from one or another of the points a and b. Now, if p = / a, it  p follows, invoking  c  P18§1, that ab = ap. Thus, considering also P8§1, the conditional proposition y 0 ab – { p} will prove to be equivalent to another one, “y 0 [0] and ap = a y,” and this in turn to “y 0 [0] – { p} and ab = py,” in view of  p, y   b, c  P23§1, and this, finally, to “y 0 [0] – { p} and ab = py.” In sum, the condition   x 0 pAab on x, which according to P1 is equivalent to affirming that “y 0 ab – { p} and x 0 p y” is satisfiable with respect to y, is resolved into the condition x 0 ab and into the categorical true proposition “there exists a point y different from p  p and such that ab = p y” (true assuming that a = / p and ab = pa, as  c  P18§1  p and  b  P10§1 require). As a consequence, the equation pAab = ab is established.  b, a  And from here, by way of the substitution  a, b  and with regard to P10§1 and P4, it also follows that the conclusion is a consequence of the hypothesis and of the other supposition p = / b. P6—Theorem. Under the assumption that a, b, c should be noncollinear projective points, the join of a with bc —or the figure aAbc —will be the locus of all points x for which the existence of a point y in the line bc is assured, such that x should belong to ay. Proof. Thanks to P7, 9§2, the hypothesis implies that b and c do not coincide, and that each point of the projective line bc should be different from point a; but with this the propositions y 0 bc – {a} and y 0 bc are equivalent, and   it remains only to invoke  ap,, bc   P1. P7—Theorem. Under the hypothesis of P6, the join of a with bc will contain the points a, b, c and their joins ab, ac, bc. Proof. That bc f aAbc follows from the hypothesis  b, a    and  ap,, bc   P3 because, in view of  a, b  P8, 11§1 and P7§2, point b must belong to bc without coinciding with a. Now, all the points of ab satisfy the condition “b 0 bc and x 0 ab” on x; therefore, these too belong to the figure aAbc, thanks to the preceding theorem. And so on. P8—Theorem. If, given the same hypothesis as P6, d should be a projective point noncoincident with a, these two propositions will be equivalent to each other: “d belongs to the figure aAbc,” and “there exists at least one point common to both of the joins ad and bc —[that is,] the lines ad and bc meet.” 46 Proof. Indeed, from the hypothesis and P7, 9§2 one infers that points b and c do not coincide, and that any [point] belonging to bc will be a projective point distinct  y, d  from a. Considering  b, c  P15§1, it follows that the conditional propositions “y 0 bc and d 0 ay” and “y 0 bc and y 0 ad” concerning y are equivalent; and the conclusion results from this in view of P6. From the numerous interpretations that the two symbols [0] and [1] have admitted up to now, those of Euclidean point and Euclidean line are not excluded, 46

[Meet stands for Pieri’s verb incontrarsi. Intersect may be more commonly used in this context now, but that is reserved here for translating tagliere, which Pieri defined in section 6.4 as meet without coinciding.]

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nor are those of Lobachevskian or Riemannian point and line,47 inasmuch as all that precedes could also be ascribed to elementary geometry. This is not so with all the propositions that come next, since the following postulate,48 from now on, attributes to the primitive entities “projective point” and “join of two projective points” a character that is not enjoyed by Euclidean or Lobachevskian points and lines. POSTULATE XII P9. If a, b, c are noncollinear projective points, and moreover ar is a [point of ] b c different from b and from c, and br a [point of ] a c different from a and from c, then the joins of a with ar and of b with br meet (that is, the class aar 1 bbr is not empty). In fact, the projective lines aar and bbr do not fail to meet if ar should fall on one of the points b, c, or br on one of the points a, c. But these restrictions imposed on ar and br are well suited for a primitive proposition; indeed, without them this might state something superfluous. Thus, when possible, we try to avoid deriving from a postulate facts already resulting from other premises. P10–Theorem. If a, b, c are noncollinear projective points and br is a point of ac, the line joining b with br will lie entirely in the figure joining a with bc. Proof. If, in the hypothesis, br coincides with a or with c, the theorem follows from P7, in view also of P21, 10§1. If, on the contrary, br is distinct both from a and from c, b  the hypothesis, thanks to  b  P15, 18§1 and P8§2, will entail the conditions c 0 abr – {a, br}, abr = ac, b ó abr, and consequently also the fact, according to  b, b   b, c  P5§2, that a, b, br are noncollinear. Now, therefore, denoting by d a point of   br, arbitrary but not coincident with b or with br, one deduces by virtue of  b, d, c   c, a, b  P9 the existence of a point common to the lines ad and bc. Thus, by P8 one   is led to conclude that d 0 aAbc. Moreover, even if d should coincide with b or with br, it should not be less true that d 0 aAbc as needed for the conclusion: see P7 and P3§1. P11–Theorem. If a, b, c are noncollinear projective points, the figures aAbc and bAac coincide. Proof. From the hypothesis and the conditional proposition y 0 bc,  b, c, y  whatever might be y, by means of P9§2,  a, b, b  P10 one deduces that y is a projective point distinct from point a, and ay lies in the join bAac. Thus, by P6 each point of the visual aAbc (that is to say, each x for which the proposition “y 0 bc and x 0 ay” about y is satisfiable) will be a point of the figure bAac. Moreover, as it follows from the hypothesis that the visual aAbc is entirely in the visual bAac, it will follow similarly that the latter is contained in the former,  b, a  because the substitution  a, b  does not change the hypothesis.

47

Riemannian of the first species, of course: as, for example, in Euclidean space, the unlimited lines concurrent at a point, and their pencils.

48

[Commonly regarded as a form of the Pasch axiom.]

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P12–Theorem. As always, a, b, c being noncollinear points, should d be a point of the figure aAbc different from b, the line bd will be contained in this visual. Proof. From the hypothesis it emerges first of all that aAbc = bAac and d 0 bAac – {b},  b, a  thanks to P11. Therefore, because of  a, b  P8 the lines ac and bd will intersect. Moreover, of each point x belonging to bd but different from b it will be equally possible to say that ac and bx meet; that is to say, x 0 bAac as required for the  b, d, x  conclusion, considering that bd = bx by virtue of  a, b, c  P18§1, and one can  b, a, x  equally invoke  a, b, d  P8. That also b 0 aAbc results immediately from P7. P13–Theorem. All the visuals aAbc, aAcb, bAac, bAca, cAab, cAba in the hypothesis of P11 coincide in one and the same figure, also called the “projective plane abc.” Proof. It suffices to invoke P4 three times and P11 two times (also considering P7§2) to deduce from the hypothesis the following equations and consequently the conclusion: aAbc = aAcb, bAac = bAca, cAab = cAba, aAbc = bAac, aAcb = cAab. P14–Definition. The expression “ is a projective plane”—often written as  0 [2] — will say precisely, “there exist three noncollinear projective points x, y, z, and the figure xyz is to be called .” That is, the class of all figures, each of which is the visual of a projective line from a projective point not situated on it, is called «projective plane» and 31 represented by the symbol [2]. In the rest of the current section are demonstrated the principal properties commonly attributed to a plane in projective geometry. P15–Theorem. In the event that a, b, c should be noncollinear projective points and d a point of the line bc different from b, it follows that the projective planes abc and abd coincide.49 Proof. Indeed, [this follows] from P1 if one takes account  b, c, d  of P7§2,  a, b, c  P18§1, by which bc = bd. P16–Theorem. As always, should a, b, c be noncollinear projective points and d a point of the line bc, it will be necessary that the projective plane abc coincide with plane abd or with plane acd. Proof. From the hypothesis, in view of P12§2, it follows that either a, b, d or a, c, d are noncollinear. Thus, in the one case points b and d are distinct, and in the other, points c and d, by P7§2. For that reason, it being  c, b  possible to take advantage of P15 or  b, c  P15, respectively, it will happen in the one case that abc = abd, and in the other, acb = acd. In sum, that is the conclusion, because by P13 figures acb and abc are the same. P17–Theorem. [Supposing that] a, b, c are noncollinear projective points and d is a point of plane abc it follows, provided that points a, b, d are noncollinear, that planes abc and abd will coincide. Proof. Under this hypothesis we have (1) points b and c, as well as a and d, do not coincide, by P7§2; (2) from  y y 0 bc – {b} it follows that abc = aby, based on  d  P15; (3) from y 0 ad – {a} it 49

[The hypothesis of P15 entails that a, b, d be noncollinear, so that Pieri could describe abd as a projective plane.]

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follows that ad = ay, and then successively b ó ay, [and then] a, y, b are not  d, y   y, b  collinear, bad = bay, and abd = aby, by way of  b, c  P18§1,  dc  P8§2,  b, c  P5§2,  b, a, d, y   a, b, c, d  P15, and P13; (4) from y 0 bc 1 ad it follows that y = / a and y = / b by   P9§2, and thus also abc = aby and abd = aby, from which one deduces the conclusion; (5) from y 0 bc and d 0 ay follows y 0 bc 1 ad, based on P9§2 and  y, d   b, c  P14§1, and consequently the conclusion. Moreover, by the hypothesis, in view   x of  d  P6), there always exists a point y such that y 0 bc and d 0 ay; thus, under the hypothesis, the conclusion always holds. P18–Theorem. If a, b, c are noncollinear projective points and d, e points on the plane abc, such that points a, d, e are also noncollinear, it must be that plane abc should  d, e, b  coincide with plane ade. Proof. By virtue of  b, c, d  P12§2, the hypothesis entails that either points a, d, b or points a, e, b should be noncollinear. Now in the first case one obtains abc = abd from P17; from that, by P13, abc = adb and  d, b, e  e 0 adb; then adb = ade, by applying  b, c, d  P17 this time. The other case is similar to the first; in fact it can be reduced to this one by exchanging points d and e, which does not alter the hypothesis, and immediately yields abc = aed; and so on. P19–Theorem. Again, a, b, c being noncollinear projective points, if it happens that three points d, e, f, likewise noncollinear, should belong to the projective plane abc, then  d, e, f , a  the projective planes abc, def will coincide. Proof. By means of  a, b, c, d  P14§2 the hypothesis imposes a trilemma: d, e, a or d, f, a or e, f, a are noncollinear. Now if, for example, points d, e, a are noncollinear, one is forced to conclude abc = ade by P18; that is to say, abc = dea by P13, [and] from this, f 0 dea and conse d, e, a, f  quently dea = def, thanks to  a, b, c, d  P17; thus, the equality abc = def stands  f , e   f, d  proved. And from this, by means of the substitutions  e, f  ,  d, f  it also emerges that the hypothesis, combined with the one or the other of the assumptions that d, f, a be noncollinear or e, f, a be noncollinear, should imply abc = dfe or abc = fed. That is to say, abc = def in each case.—P19 is usually stated in this form, [which is] not so clear: “a projective plane is determined by any noncollinear triple of its points.” P20–Theorem. Supposing that a, b, c should be noncollinear projective points and that d, e should be projective points distinct from each other and belonging to the plane abc, it happens that the join of these—that is, the line de —will lie in the plane abc. Proof. Let another assumption be adjoined to the hypothesis, that a, b, d be noncollinear. It follows then by P17 that abc = abd, and hence that e 0 adb by  d, b, e  P13. For that reason and because of  b, c, d  P12 the line de will be completely contained in the plane adb; that is as much as to affirm the conclusion. In this way the hypothesis, together with one or the other of the two suppositions, that a, c, d be noncollinear or that b, c, d be noncollinear, will lead one similarly to conclude that the line de lies in the plane acb or in the plane cba, such being the effects  c, b  c, a of the two substitutions  b, c  and  a, c  on that partial result. Therefore, the conclusion is true under the hypothesis that a, b, d or a, c, d or b, c, d are not collinear, and thus the theorem holds, by virtue of P14§2.

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P21–Theorem. If, a, b, c being noncollinear projective points as always, there should be given in the projective plane abc four points d, e, f, g so that e should not coincide with d nor g with f, there will have to be at least one projective point common to the two joins de, f g. Proof. If it is assumed, beyond the hypothesis, that point  d, e, f  f should not belong to the line de, then by means of  a, b, c  P5§2 and P19 it will follow that points d, e, f are not collinear and thus that planes abc, def coincide. Moreover, g belongs to fde by P13: that is as much as to affirm the conclusion,  f , d, e, g  in view of  a, b, c, d  P8. If on the other hand f 0 de, then f will be a point  f, g  common to de, f g since f 0 f g, as is noted in  a, b  P11§1. As anyone sees, the last three propositions, on which (it can be said) hinges the entirety of projective geometry, can be derived from rather simple premises like our primitive propositions I–XII; of those, only the last two look beyond the confines of the geometry on a line. Postulate XII is truly a statement not very dissimilar to that expressed by P21, inasmuch as the one, like the other, speaks about the intersection of projective lines. But the deductive capacity50 of the first is incomparably less than that of the other. With regard to deductive power that has been employed up to now, the prescribed principle XII could also be replaced by this other one (which contains it, without being equivalent to it): “Given three noncollinear projective points a, b, c, each projective point belongs to the join of a with bc” —that is, “There do not exist any points except in a certain projective plane”—in force of which the class of points would be seen to coincide with the plane abc ([0] = abc). But in view of the restriction this would entail in the scope of the geometry, we shall instead affirm the contrary proposition by means of postulate XIII, which we introduce later, leaving open meanwhile the question of the existence of projective points distinct from those that exist by virtue of postulates II, III, VIII, and XI. Given the definition (see P1, 6) of the visual of a projective line from a point, and considering postulates I–XI, a principle that proves deductively equivalent to postulate XII (and thus also fit to take its place in the context of the succeeding XIII, XIV, ...) is, for example, the statement expressed by P10, where however the restriction that br be distinct from a and from c is adjoined to the hypothesis (in order not to state anything superfluous). And it can be said that with regard to deductive content, this should be related to P20 as postulate XII [is] to P21.51 50

[Pieri’s word for capacity was capacità. See footnote 36 in section 6.1.]

51

In Fano 1892, 109–110, after some combined assumptions, from which our [postulates] I–XI could be extracted (with the exception of VIII, however, of which no further deductive use is made), and for whose same purposes XII serves, both of (our) P20 and P21 are assumed as postulates. And the example given there to prove the independence of these is not conclusive, nor does the assertion appear justified that the statement contained in P10 should not be sufficient to establish the truth of P20. Federico AMODEO (1892, §1.4; 1896, 24, paragraph 5 [in error, Pieri cited §5]) proposes to the same ends (our) P18, or else the alternative, “two lines ab, ac with a (single) common point a (points a, b, c being noncollinear) determine one and only one plane”— both propositions together exceeding the requirement. Prof. [continued…]

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§4 The Plane Quadrangle and the Harmonic Relation 52 P1—Theorem. If a, b, c are noncollinear projective points, and moreover ar is a [point on] bc different from b and from c, and br is a [point on] ac different from a and from c, neither of the lines aar, bbr will coincide with any of ab, ac, bc, nor will aar, bbr coincide with each other; on the contrary, these will meet at a point that will not lie in any of ab, ac, bc. Proof. That aar = / bc and bbr = / ac follows from the hypothesis and from P11§1 and P8§2. And from the hypothesis and P12§2  b   b, a, a   c, a, b, a   c, a, b  one infers, by way of the substitutions  d  ,  a, b, d ,  a, b, c, d  ,  a, c, d  , that points a, b, br will not be aligned, nor points b, a, ar, nor c, a, ar, nor c, b, br. Consequently, by P5§2 and P10§1, bbr = / ab, aar = / ab, aar = / ac, bbr = / bc. Indeed, from the assumption bbr = aar it would follow that b 0 aar, and thus aar = ab: see P11§1,  a, b  P7§2,  b, c  P18§1. Now, because bbr = / aar there can never be two points common to these lines and distinct from each other, P25§1 forbidding that, while thanks to P9§3 there exists a point common to these, indeed a point different from b (since aar = / ab), which thus cannot fall on ab nor on bc (because bbr = / ab and bbr = /  b, a, b, a  bc), and consequently not even on ac (considering that the substitution  a, b, a, b  does not alter the hypothesis).—This yields, for example, the existence of at least seven projective points (distinct from each other) on each projective plane.—With arguments entirely similar to these one verifies the truth of the following theorem. P2—Theorem. Let a, b, c be noncollinear projective points and d a projective point not on any of the lines ab, ac, bc. [Then] it cannot happen that two of the six lines ab, ac, bc, ad, bd, cd should coincide. P3—Definition. We shall say that two projective lines intersect whenever they should have a point in common without coinciding. Moreover— a, b and c, d being pairs of distinct projective points, and having supposed that ab and cd should meet without coinciding—the expression (ab. cd), which can be read, point of intersection of ab with cd, represents the individual in the class formed from the points common to the lines ab and cd: points that are all equal to each other, given P25§1.—In short, the intersection of the two classes ab and cd is set equal to the class {(ab. cd)}. See P4,5§1. P4—Theorem. Let a, b, c be noncollinear projective points and d be a point of the plane abc not belonging to any of ab, ac, bc. Then the lines ab, cd will intersect, as well […continued]

Giuseppe VERONESE (1891) does not properly introduce any “postulate of the plane,” nor is it possible for me to discover from what other projective principles he might succeed in deducing our P19, 20, 21 in his treatise. A premise that differs only in appearance from postulate XII is that inserted by Riccardo DE PAOLIS in the definition which begins his memoir on projective correspondences (1892, §I.1, 504), where it is affirmed, in substance, that each line that meets the two joins ab and bc without containing any of the (noncollinear) points a, b, c must also intersect the other join ca.

52

See Pieri 1896a, §11, 381–393, and 1896b, §14, 464–470.

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as ac, bd and ad, bc; moreover, none of the intersection points (ac. cd), (ac. bd), (ad. bc) will coincide with any of the points a, b, c, d. Proof. Indeed, under those conditions neither of the points c, d will lie in ab, by P8§2. Therefore, (ab. cd)  c, d  will not be coincident with c or with d: see  a, b  P12, 8§1 and P3§1. By P2 it cannot even happen that cd = ca and cd = cb, as would be required by P18§1 should (ab. cd) be coincident with a or with b. That the lines ab and cd should  c, a, b  then intersect emerges from the hypothesis and P13§3,  a, b, c  P8§3, and P3, 2. P5—Theorem. In the hypothesis of P4, should ar, br, cr be the intersection points (bc. ad), (ac. bd), (ab. cd), respectively, these will all be distinct from each other, and the two joins ab, arbr will intersect at a point different from all of a, b, c, d as will ac, arcr and bc, brcr. Proof. From the hypothesis it follows (1) that aar = / ac, ac = abr by P18§1 and P4, 1, and consequently that br = / ar by P21§1; (2) that the assumption ab = arbr would entail ar 0 ab and therefore ab = aar, contrary to P1; (3) that consequently (by P3 and P20, 21§3) lines ab and arbr intersect; (4) that, whenever (ab. arbr) should be coincident with a, it would be necessary to concede that a 0 arbr, and hence aar = abr (by P4 and P24, 12§1) and therefore aar = ac, negating P1; (5) that c, d ó ab, so that it will not even happen that (ab. arbr) should coincide with c or with d. And thus also [follow] the remaining [parts of the conclusion], through substitutions of letters. The stated propositions 1–5 will serve to preface a study of the complete plane quadrangle, MÖBIUS net,53 and of harmonic forms. The notion “harmonic” can be introduced with the following definition, which is the usual one, divested of everything superfluous. P6—Definition. Should a, b be distinct projective points, and c a projective point different from both of them yet belonging to ab, we shall call “harmonic of c with respect to a and b” —or Arm a, b c —any point of ab that should satisfy the following condition on x: “There exist two distinct projective points u and v not belonging to ab but collinear with c, and the intersection points (au. bv), (av. bu) are collinear with x.” Here, the entity Arm a, b c is regarded as a [sub]class of, or figure in, ab; and it cannot be deduced solely from postulates I–XII that all the points of this class should coincide. P7—Theorem. Under the same hypothesis as P6, each one of the [points in] Arm a, b c will be a point of ab distinct from a and from b. Proof. Let it be supposed, beyond the hypothesis, that u, v 0 [0] – ab, v = / u, [and that] c, u, v be collinear. One deduces from this (with respect to u, v) that v 0 uc, [that] a, b, u are not collinear, [and that] v 0 uab by P5§2, P6§3; that a, c, u are not collinear, nor are b, c, u by P18,19§1, P5§2; and consequently that a, u, v are not collinear, nor are b, u, v, and  b, a, u, v  v ó au, v ó bu by P12, 8§2. Therefore, by virtue of  a, b, c, d  P4, 5, by P10§1, and by P13§3, and having set ar, br to stand for the points (au. bv), (av. bu), one concludes from this that the lines ab and arbr intersect at a point different from a and from 53

[See section 6.6, footnote 69.]

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b, and hence that any point x on ab collinear with ar, br, which, given P3,  b, a, u, v   a, b, c, d  P4, 5, and P5§2, cannot be different from the point (ab. arbr), will be distinct   from a and from b. Thus, whatever x may be, one can deduce x 0 ab – {a, b} from the hypothesis and this proposition, conditional on x: “There exist two projective points u, v distinct from each other and not belonging to ab, but collinear with c, and x is a [point on] ab such that x, (au. bv), (av. bv) are collinear.” From this,  c, u  the theorem follows by P6, in view of P4§2,  a, b  P13§1, and so on, proving that under the hypothesis, the quoted proposition is satisfiable with respect to u, v.54 P8—Theorem. Under the same hypothesis as P6, the class Arm a, b c is not empty. Proof.  b, a, u, v  One proceeds as before to conclude, by referring to  a, b, c, d  P4, 5, that lines au and bv, as well as av and bu, intersect at points ar and br distinct from each other and such that lines ab and arbr meet. Now, according to P12, 18, 25§1 and P5§2, the proposition, conditional on u, v, that (in part) supports such a conclusion, is a consequence of the hypothesis and the conditional proposition “u 0 [0] – ab and v 0 cu – {c, u},” [which is] satisfiable with respect to u, v thanks to postulates XI and VIII. Also affirmed with the hypothesis, therefore, will be the categorical proposition, “there exist projective points u, v, x such that u, v 0 [0] – ab, v = / u, c, u, v are collinear, and x 0 ab;” and this is precisely the conclusion, by virtue of P6. P9—Theorem. From the hypothesis of P6 it also follows that the figures Arm a, b c and  b, a  Arm b, a c coincide. Proof. One uses P6 and  a, b  P6 together with P10§1. P10–Theorem. Keeping the hypothesis of P6, if x is a harmonic of c with respect to a and b, [then] vice versa c will be a harmonic of x with respect to a and b. Proof. If alongside the hypothesis is placed the additional proposition u, v 0 [0] – ab; v = / u; c, u, v are collinear; x 0 ab; and (au. bv), (av. bu), x are collinear then the [following] proposition, where ar, br again denote points (au. bv), (av. bu), will follow easily from considerations similar to those involved in P7, 8: c 0 ab; ar, br 0 [0] – ab; br = / ar; x, ar, br are collinear; and (aar. bbr), (abr. arb), c are collinear. Now, if x 0 Arm a, b c [then] by P6 the first of the stated propositions, conditional on u, v, x, will be satisfiable with respect to u, v, and therefore also the second;  x, a , b  but this, with attention given to P7 and to  c, u, v  P6, is as much as to say that c 0 Arm a, b x. To proceed in the best way to further and more remarkable propositions about harmonic sets, it seems appropriate to accept here a new primitive proposition, which in the context of our deductive system seems to have an importance greater than that inherent in the other existential postulates. 54

The intervention of both of the existential postulates VIII and XI should be noted here.

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POSTULATE XIII. P11. If a, b, c are noncollinear projective points, there exists at least one projective point that does not belong to a b c. That is, the figure [0] – abc is not devoid of points. Among the more remarkable facts that can be derived just from fundamental principles I–XIII should be mentioned the theorem of homologous triangles, also called the theorem of DESARGUES, which results principally from the last facts demonstrated in the preceding section and from axioms XIII and VIII. But here, for the statement as well as for the demonstration of this theorem, the Reader is referred to the classic work of G. K. C. von STAUDT; 55 likewise for the demonstration of the following theorem, which is [to be] restated precisely in [terms of ] homologous triangles.56 P12–Theorem. Again let a, b, c be specified by the hypothesis of P6. Then, all the projective points of the figure Arm a, b c necessarily coincide. And after this is established the use of a new term, or symbol, for denoting the individual of the singleton class Arm a, b c. (Compare P3.) P13–Definition. Having assumed that a, b should be noncoincident projective points, and c, a point of ab different from a and from b, the individual in the class Arm a, b c, comprised of some points coincident with each other (see P12), is described as «harmonic to (or after) a, b, c», written31 Arm(a, b, c). In short, the individual of this class is termed Arm(a, b, c)  ¯ Arm a, b c; 57 that is to say, Arm a, b c  { Arm(a, b, c)}. P14–Theorem. Again given points a, b, c as in the hypothesis of P6, and having assumed that ur, vr should be distinct projective points not belonging to ab, but collinear with c, points (aur. bvr), (avr. bur), Arm(a, b, c) will always also be collinear. Proof. From P6, 8, 12, 13. P15–Theorem. Let r, rr be two noncoincident projective lines; and the first contain distinct points a, b, c and besides [these] a point d; and the second, points ar, br, cr, likewise all distinct, and a fourth point dr. Then, when there should exist a point p collinear at once with points a and ar, [with] b and br, [with] c and cr, [and with] d and dr, the propositions “d is the harmonic after a, b, c” and “dr is the harmonic after ar, br, cr” will be both false or both true. Proof. Indeed, it [is to be] proved that p cannot lie in r or in rr, and that the line rr lies entirely in the plane pr. Then, having taken a point u outside the plane pr (see P11), and on the line uc a point v different from u and from c (see P13§1), and thus determined points 55

Staudt 1847, paragraphs 87, 90. Compare Peano [1888] 2000, 76, §48.9.

56

Staudt 1847, paragraph 93.

57

[The notation ¯ x for the member of a class x that has exactly one member is discussed in the introduction, footnote 22. Many authors call Arm(a, b, c) the fourth harmonic.]

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(au. bv), (av. bu), these will be aligned with d, if d = Arm(a, b, c) (see P14). But now, having called vr the point common to the lines cru and pv lying in the plane crpu, it is found that points (aru. brvr), (au.bv), and p are collinear, and likewise points (arvr. bru), (av. bu), and p.58 Thus also, points (aru. brvr), arvr. bru), and dr turn out to be collinear; therefore, dr = Arm(ar, br, cr).59 —Thus is established a remarkable characteristic of the entity Arm: the property of not being changed by projection. To handle the case where point c should coincide with a or with b, it is useful to introduce the following: P16–Definition. If a, b are projective points and b does not coincide with a, the “harmonic after a, b, a” is point a and the “harmonic after a, b, b” is point b. And for many reasons it seems convenient already to settle the single case in which the harmonic of a point and that point itself coincide. Thus the following will be accepted without further ado: POSTULATE XIV P17. Should a, b be distinct projective points, and c a point of a b different from a and from b, the harmonic after a, b, c will not coincide with c ( Arm(a, b, c) = / c). 60 By virtue of P6, 12, 13, Postulate XIV could be replaced by this alternate proposition, equivalent to it: P18–Theorem. Given the hypothesis of P4, points (ab. cd), (ac. bd), (ad. bc) will not be collinear. Indeed, given the projective character of [the notion of ] harmonic set (see P15) it would not be a difficult task to prove that “if there exist three collinear and distinct points ar, br, cr such that Arm(ar, br, cr) should be different from cr —or four points ar, br, cr, dr as in the hypothesis of P4, such that the intersection points (arbr. crdr), (arcr. brdr), (ardr. brcr) should not be collinear—then P17, and therefore P18, must be universally true. For the same reason, in place of postulate XIV, and to play its role, could be taken the particular proposition that results when to the same P17 is prefaced the clause, “there exist three points a, b, c such that ...”—or to P18 the clause “it is not unsatisfiable with respect to a, b, c, d that ... .” 61—It will 58

[In error, in the previous sentence Pieri wrote pc and p for uc and u, respectively; and in this sentence cu and cpu for cru and crpu.]

59

See Staudt 1847, paragraph 94.

60

That this proposition should not follow from the usual premises about the projective categories [0] and [1], and therefore not even from postulates I–XIII, was noticed opportunely by Gino FANO in 1892, 114.

61

See Fano 1892, 114–115.

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not be without purpose to engage the Reader in this observation, because it also applies perfectly to the four postulates XV–XVIII, each of which is concerned with a projective segment specified by means of three collinear points a, b, c, but expresses nothing more in principle than a harmonic property of certain special configurations of points and joins (thus, like each other postulate, burdening the primitive entities, on which a new restriction is thereby imposed). We should therefore have been able to substitute for each one of these an existential postulate, stating in this way the existence of at least three projective points connected by a certain harmonic relation. But on the other hand it can be doubted whether this would make a real simplification in the deductive aspects—it is preferred to leave to the Reader’s judgment the choice of this other form of statement, that requires only the apposition of the clause “there exist r, a, b, c such that ...” in front of each of the primitive propositions XIV–XVIII, and more frequent references to P15. P19–Theorem. Under the hypothesis of P17, calling d the harmonic after a, b, c, it is necessary that the harmonic after c, d, a should coincide with b. Proof. See Staudt 1847, paragraph 96, for the demonstration.—In this is stated the permutability of two harmonic pairs a, b and c, d, provided a be different from b and c from d.

§5 The Projective Segment 62 P1—Definition. Under the premise that a, b, c are points on a projective line r, distinct from each other, the projective segment abc, represented by the symbol (abc), is none other than the locus of a projective point x with which can be associated a point y on r, different from a and from c, such that x should be the harmonic after y, Arm(a, c, y), and b. —In other words (see P6,10,13,17§4), (abc) will be the set of all the points, each of which is a harmonic of b with respect to two points that are harmonic to each other with respect to a and c, but not coincident with them.63 P2—Theorem. As above, let r be a projective line, and a, b, c three points on it, distinct from each other. [Then] the figures (abc), (cba) necessarily coincide. Proof. From c, a P1 and  a, c  P1, by virtue of P9,13§4, and so on. P3—Theorem. Under the same hypothesis as P2, b is always in the segment (abc). Proof. From the hypothesis and from P16§4 can be inferred the existence of a point y on r noncoincident with a or with c but such that b should be the harmonic after y, Arm(a, c, y), and b: indeed, it suffices to take b for y.

62

Compare Pieri 1895a, §8–§9, and 1897c, §4–§6.

63

[See the figure on the facing page, constructed by the present authors. Pieri did not employ any figure.]

Mario Pieri around 1895

a

y

c

x

z

b

For fixed a, b, c, if y, z are harmonic with respect to a, c and b, x harmonic with respect to y, z, then x ranges over the projective segment abc determined by a, c that contains b.

Definition of Projective Segment

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P4—Theorem. But on the contrary, neither of the points a and c belongs to the projective segment (abc). Proof. For a to be a point of the figure (abc), it would be necessary that the condition “y 0 r – { a, c}, yr  Arm(a, c, y), and a = Arm( y, yr, b)” be satisfiable. But this proposition is unsatisfiable with respect to y because, if it were conjoined with the hypothesis, it would yield yr = / y, c = Arm( y, yr, a), and b = Arm( y, yr, a) by P17,19,10,13,16§4, whatever y should be, and consequently b = c by P12§4, which is contrary to the hypothesis. It follows as well that c ó (cba) because interchanging a with c does not affect the hypothesis, and indeed that c ó (abc) by P2.—Points a and c can be called, appropriately, the ends of the projective segment (abc).64 P5—Theorem. Under the same hypothesis about r, a, b, c, and whenever d should be a point of the projective segment (abc), b will have to belong to the projective segment (adc). Proof. The hypothesis of P2 together with “y 0 r – {a, c}, yr Arm(a, c, y), and d = Arm( y, yr, b)” entails that b should be the harmonic after y, yr, and d, by P17,10,12,13,16§4. Now, to suppose that d 0 (abc) is as much as to affirm that the quoted proposition should be unsatisfiable with respect to y by P1, and moreover that d should be a point of r noncoincident with a or with  d, b  b by P4. Thus, the theorem reduces to  b, d  P1. P6—Theorem. Let r, a, b, c be as above, and d represent a point of r noncoincident with a or with c. Then these two propositions will be equivalent: “d belongs to  d, b  (abc)” and “b belongs to (adc)”. Proof. This is P5 with its converse  b, d  P5. P7—Theorem. Again assuming the hypothesis of P2, if it happens that a point d should belong to the segment (abc) without falling on b, it will be necessary that point c should belong to segment (bad). Proof. From the hypothesis of P2 strengthened by “y 0 r – {a, c}, yr  Arm(a, c, y), and d = Arm( y, yr, b), d = / b” it follows by means of P17,16§4 that y = / yr, y = / b, and yr = / b. And from that, by means of P19,16§4, yr = Arm(b, d, y), c = Arm( y, yr, a), and y = / d. Thus, the assumption that d 0 (abc) – { b} together with the hypothesis of P2 is equivalent to denying, according to P4 and P1, that d should coincide with b or with a, and that the proposition “y 0 r – { b, d} and c = Arm( y, Arm(b, d, y), a)” could prove to be unsatisfiable with respect to y. But this is precisely the conclusion, by virtue of  b, a, d, c   a, b, c, d  P1.   P8—Theorem. If on the contrary, d should be a point of r not in the segment (abc) nor coincident with a, [then] point c cannot lie in segment (bad). Proof. By  b, a, d, c  the hypothesis, d = / b (see P3). Thus, in view of  a, b, c, d  P7, the assumption c 0 (bad) leads to the conclusion that d 0 (abc), contrary to the hypothesis. P9—Theorem. The hypothesis of P2 standing firm, if d should be a point in r, arbitrary but different from a and from b, each of the two propositions “d belongs 64

[Pieri often stipulated, “let x 0 (abc),” tacitly assuming that a, b, c should be distinct; and tacitly used the consequences x = / a, c of P4.]

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to (abc),” “c belongs to (bad)” will be a consequence of the other. Proof. This  b, a, d, c  is P7, conjoined with the converse  a, b, c, d  P7. P10–Theorem. Having assumed the hypothesis of P2, if a point d will lie in the segment (abc) without falling on b, [then] as a consequence point a will lie in segment (bcd). Proof. The hypothesis entails the relation d 0 (cba) – {b} by P2, from c, a which, via  a, c  P7, the conclusion results immediately. P11–Theorem. If, besides the hypothesis of P2 it is assumed that d should be a point of r noncoincident with b or with c, the assertions “d lies in (abc)” and “a lies c, a in (bcd)” are equivalent. Proof. This [follows] from  a, c  P9 and P2. Thanks to our P1, «projective segment» enjoys ipso facto the property projective, as intended in P15§4.31 Thus: P12–Theorem. Being given two distinct projective lines r, rr, to which should belong points a, b, c, d and ar, br, cr, dr respectively, such that a, b, c as well as ar, br, cr should be distinct from each other, and should there exist a projective point p aligned with a, ar, [with] b, br, [with] c, cr, [and with] d, dr, then each of the two propositions “d belongs to the segment (abc)” and “dr belongs to the segment (arbrcr)” will be a consequence of the other. To indicate that a point d, while lying on the projective line r, should not belong to the segment (abc), nor coincide with a or with c, one can adopt the expression, conforming to the use of many, “points b and d are separated by the points (or by means of the points) a and c.” Thus, the locution “points b and d are not separated by points a and c” will stand in place of the alternative, “point d belongs to segment (abc), but does not fall on either of its ends a and c.” —Propositions 2, 6, 9, 11 show that the relation of separation between points a and c on the one hand and points b and d on the other is not disturbed by interchanging points a and c with each other, or points b and d, or the pairs (a, c) and (b, d). And P12 also bears witness to the projectivity of this relation.—As for the most familiar proposition that two pairs of points, all four distinct, on a projective line do or do not permit the existence of a third pair harmonic to both, according to whether they do not or do separate each other, this is no more than a simple modification of our definition P1 of “segment.” This appears informed by the same criterion that inspires the celebrated definition by G. K. C. von Staudt of projective correspondence between simple forms: a definition, as everyone knows, based on the notion of harmonic set.65 65

Staudt 1847, paragraph 108.—“Projective segment” is not to be confused with “segment identified by its ends,” which appears, for example, in the metric Geometry of Euclid and of Lobachevsky, or which was chosen for the role of primitive notion by Moritz PASCH (1882b) and Giuseppe PEANO (1889, 1894b), principally because of the deep intuitiveness that comes to it from the common physical significance of the term “point.” To them I am indebted for many precepts, although the proposition that the primitive notions and axioms should have invariant qualities under projection and duality did not figure at all in their designs. The prevalence here accorded the notion of projective segment over that of separation of points stemmed from the desire to reduce as much as possible each geometric concept to that of “class of projective points,” [which is] deemed simpler and more manageable than any other.

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POSTULATE XV P13. Let a, b, c be distinct projective points on a line r, and suppose that d should be a point of r not belonging to the segment ( a b c) nor coincident with a or with c. It is necessary that this point d should belong to the segment ( b c a). —The restriction that d be different from c is made here only in order not to affirm anything superfluous in a primitive proposi b, c, a  tion. However, this can be removed immediately with attention to  a, b, c  P3. Thus P13 could be reworded in this form: P14–Theorem. Given the hypothesis of P8, point d will have to lie in (bca). P15–Theorem. Under the same hypothesis of P8, point d will lie in segment (acb).  b, c, a  Proof. As verified by  a, b, c  P2 and P14. P16–Theorem. If r, a, b, c are specified as in the hypothesis of P2, [then] any point different from c that should lie in r without belonging to the segment (abc) will c, a have to belong to segment (bac). Proof. This is from  a, c  P14 and P2. P17–Theorem. Under the hypothesis of P13, point c will lie in segment (abd). Proof. It follows from the hypothesis, by virtue of P16, that d 0 (bac). Therefore, the  b, a  conclusion ensues by  a, b  P7. P18–Theorem. And point a, in segment (cbd). Proof. By P2 the substitution does not affect the hypothesis of P17.

 c, a   a, c   

P19–Theorem. And point b, in segment (acd). Proof. Since c 0 (abd) by P17, one  d, c  deduces the conclusion from the hypothesis, citing  c, d  P5. P20–Theorem. Having again assumed the hypothesis of P13, point d will be common to segments (bca) and (cab). Proof. This is the logical product of P13 and P16  b, a  because (bac) = (cab) follows from  a, b  P2.—With the following principle we accept as true the converse of this P20, thus: POSTULATE XVI P21. Let a, b, c be distinct projective points on a line r. If a point d should belong to both of the segments ( b c a) and ( c a b), then it cannot belong to the segment ( a b c). These last propositions 20 and 21 say in sum, “should it happen that the pairs a, c and b, d separate each other, then the pairs a, b and c, d cannot separate each other, nor can a, d and b, c, and conversely.” Thus, “four distinct points on a line can always be distributed, in fact in only one way, into two pairs that separate each other.” P21 can also be phrased symmetrically as follows:

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P22–Theorem. There cannot exist any point common to all three of the segments (abc), (bca), and (cab) determined by collinear and distinct projective points a, b, c. It can now be proved in a fully rigorous manner, resting on the projectivity of

«segment» and without departing from the more common treatments,66 that the harmonic after a, b, c is separated from c by means of a and b. That is to say,

P23–Theorem. The harmonic after three collinear and distinct projective points a, b, c does not belong to the segment (acb), nor coincide with a or with b. Proof. Set d  Arm(a, b, c) and, as in the preceding section, let uvarbr be a structural quadrangle67 of the points a, b, c, d: thus, u, v 0 [0] – ab, u = / v, ar  (au.bv),68 br  (av.bu), cr  (uv.arbr) (see P3,1,2,5,6,13,14§4). The point d will be distinct not only from a and from b (by P7,13§4), but just as well from c (by P17§4). And moreover, cr [will be] distinct from ar and br, and these from each other. Now, from the assumption d ó (cba) one would deduce d ó (crbrar) by projection from u (see P12) and from here by projection from v, that d ó (cab), which cannot be,  c, a   a, c  P15 forbidding it. Therefore, it is necessary to concede that d 0 (cba). But in   the same way one concludes that d 0 (bac). Thus, the condition d ó (acb) follows, by P21. P24–Theorem. Given points a, b, c, distinct from each other, on a projective line r, there will always exist a projective point on r not belonging to the segment (abc), and not coincident with a or with c. Proof. Such, in fact, is the fourth harmonic  c, b  after a, c, b. See P7,8,13§4 and  b, c  P23.—Thus has been established the existence of two pairs of points (all four collinear and distinct) which do not permit a third pair harmonic to both, by P1. For that reason the hypotheses of P13,17–20 can never be impugned. Indeed, the existence of at least three projective points in any segment is assured, that is, P25–Theorem. Let a, b, c be points, all collinear and different from each other. Then, Arm(a, b, c) as well as Arm(b, c, a) will lie in the segment (abc), distinct from each other and from b. Proof. From the hypothesis (see P23) we have Arm(a, b, c) 0 r – (acb) – {a, b} and Arm(b, c, a) 0 r – (bac) – {b, c}; consequently, referring to  c, b   b, a   b, c  P2,15 and  a, b  P16,15, also Arm(a, b, c) 0 (abc) – (bca) and Arm(b, c, a) 0     (abc) 1 (bca). Based on the postulates assumed so far, it does not seem possible to affirm, among other things, the existence of infinitely many points on a projective line, nor the fact that these are a closed form, re-entrant in itself. The following provides for that:

66

See, for example, Reye 1877, part 1, 32ff.

67

[Pieri had defined quadrangolo costruttivo, used as quadrangolo costruttore is used here, in his 1891c textbook, §43, 67. See also Amodeo 1891, 524.]

68

[In error, Pieri wrote (au.bu).]

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POSTULATE XVII P26. Given that a, b, c should be collinear and distinct projective points, and that d should be a point other than b in the segment ( a b c), then if e should be a point of the segment ( a d c), one concludes that point e lies in the segment ( a b c). This is a relation among five points a, b, c, d, e: from that [stems] its greater force in comparison with postulates XIV, XV, and XVI, which are relations among four. It can be transformed in several ways, for example thus: “If points a and c (on a projective line) should not be separated by points b and d, nor by points d and e, they will not even be separated by points b and e.” Further, taking account of P21,22, one converts P26 into P27–Theorem. There does not exist any point common to all three segments (adc), (bca), (cab), where a, b, c should be collinear and distinct points, and d a point of the segment (abc). P28–Theorem. Given collinear and distinct projective points a, b, c, and having selected at liberty a point d in the segment (abc), it will always happen that segments (abc) and (adc) coincide. Proof. In truth, any point e whatever of the segment (adc) will lie in (abc), even when d = b, as evidenced by P26 with P1. (In case d = b [the latter] would yield (adc) = (abc), in view also of P13§4 and P21§1.) Vice versa, each point f of segment (abc) will lie in (adc) because d 0 (abc) ]  d, b, f  b 0 (adc) by P6, after which one returns to  b, d, e  P26, and so on. P29–Theorem. Having assumed the hypothesis of P2, if one will consider a point d in r not belonging to segment (abc), but different from a, the segment (abc) will always be contained in the segment (abd). Proof. Of the two cases d = c and d= / c, it suffices to examine the second. Now, if alongside the hypothesis one should assume the conditional proposition x 0 (abc), from which results (abc) = (axc) x and d ó (axc) by P28, it will happen that x 0 (acd), given P4 and  b  P19. But  d, c  since c 0 (abd) by P17, (abd) and (acd) also coincide, thanks to  c, d  P28 and P3, and thus it also happens that x 0 (abd), which was to be shown. P30–Theorem. If besides the hypothesis of P2 one assumes that d and e should be points of the segment (abc), it will be necessary that segments (adc) and (aec) coincide. Proof. They are both equal to segment (abc), by P28. P31–Theorem. And if points d and e should be distinct, c will lie in segment (dae). Proof. Since e 0 (adc) by P28, d should not coincide with a or with c, by P4.  d, e  For this, refer to  b, d  P7. P32–Theorem. Given the hypothesis of P13, there cannot exist a point common to the segments (abc) and (adc). Proof. From “x 0 (abc) 1 (adc)” would follow (abc) = (axc), (adc) = (axc) by P28, hence also (abc) = (adc), which is contrary

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to the hypothesis because d 0 (adc) – (abc) by P3. Therefore, the quoted proposition is unsatisfiable with respect to x.

§6 Further Properties of Segments In the present section and the next are unfolded various further important consequences of the preceding postulates (especially the last ones). Like the majority of those properties, these appear for the most part wrapped up in the vague notions of closed, or re-entrant line. The material of sections 6.1–6.6, with postulate XVIII on the continuity of segments in 6.9, is sufficient preparation for the study of homographic transformations, and permits us, as we shall see in 6.10, to establish G. K. C. von STAUDT’s fundamental theorem without any concern for the senses or directions of a line (to which, among other things, section 6.7 is dedicated). P1—Theorem. Under the premise that a, b, c should be points of a projective line r, if a point d lies in segment (abc) without falling on b, [then] the figure (abc) – { b, d} is not empty. And, if a point e should be given in the class (abc) – { b, d}, the figure (abc) – { b, d, e} is not empty. And so on. Proof. One of the segments (bca) and (cab) must exclude point d, by P21§5. Let us assume, for example, that d ó (bca): that is to say, by P20§5, that d 0 (cab) 1 (abc). Now,  c, d  by means of  b, c  P24§5, we know that the figure r – (acd) – { a, d} contains some  b, c, a  points, necessarily different from b, because, according to  a, b, c  P18§5, b 0 (acd).  c, d  Thus, the same can be said about the figure (adc) – { d} according to  b, c  P15§5, and about figure (abc) – { d} by P28§5. Then let e be a point in the nonempty class (abc) – (acd) – {b, d}. By P24§5, the point f = Arm(a, e, c) will belong to the figure r – (ace) – {a, e}, and hence will lie in segment (aec) by P4,15§5: that is, [in] (abc) by P28§5. It cannot fall on d, because d 0 (ace) by P4,17§5; nor on b, because b 0 (acd), while segment (acd) is contained in segment (ace) by P29§5. The same holds for the other case, where d ó (cab), just by exchanging the letters a and c with each other. The method of demonstration just described69 can be continued ad libitum, considering the harmonic g after a, c, f, then the harmonic h after a, c, g, and so on, because they all will prove different from each other and from the preceding [points]. Thus, within a given segment the presence of as many points as are desired can be established. And by a method similar to this is also demonstrated the infinity of the harmonic series a0 , a1, a2 , ... a k –1 , a k , a k +1 , ... constructed from three collinear and distinct points a0 , a1 , a2 in such a way that a 3  Arm(a0 , a 2 , a1), ... , ak +1  Arm(a 0 , a k , a k –1 ). 69

[This is the Möbius net construction mentioned in section 6.4, footnote 53.]

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P2—Theorem. Under the premise that a, b, c are points, distinct from each other, on a projective line r, and having selected points d, e arbitrarily in segment (abc), if it happens that d belongs to the segment (ace), it will not be possible that e should belong to segment (acd). Proof. By P4§5 it emerges from the hypothesis  c, e  that d, e 0 r – {a, c} and e = / d. Consequently, as in  b, c  P6§5 and P31§5, [it follows] that c 0 (ade) 1 (dae), and finally that e 0 (dac) 1 (adc), in confor d, e, c   d, a, e, c  d mity with  b, c, d  -,  a, b, c, d  P9§5. But this last condition, by virtue of  b  P2§5 and  c, d, e   b, c, d  P21§5, is equivalent to e ó (acd), which is the conclusion.   P3—Theorem. Let a, b, c, d, e be points on a projective line r with a, b, c different from each other, while points d and e, not coincident with a, are excluded from segment (abc). Then it will not be possible that at the same time e should lie in segment (abd) and d in segment (abe). Proof. Supposing that d and e should both differ from a, the hypothesis yields d, e 0 (acb) by P15§5; thus, it suffices  c, b  then to refer to  b, c  P2. Moreover, the theorem is true by P4§5 if just one of the points d, e should be distinct from a. P4—Theorem. If r, a, b, c are as above, and it is assumed that d, e should be points of segment (abc), and moreover that e should lie in segment (acd), it will be necessary that segment (ace) be included in segment (acd). Proof. Indeed, recalling P4§5 and P2, one deduces that d, e 0 r – {a, c} [and] d ó (ace): and from that also  c, e  the conclusion by virtue of  b, c  P29§5. P5—Theorem. Always given r, a, b, c as above, if d, e should be points of segment (abc), distinct from each other, then either point d will lie in segment (ace), or point e in segment (acd). Proof. From the hypothesis and from P4,28§5 it follows that d, e 0 r – {a, c} and e 0 (adc). And consequently, having taken note of d   P21§5, one of these two cases [occurs]: either e ó (dca) or e ó (cad). But in the  b first case the condition d 0 (ace) holds; in the second, the relation e 0 (acd), as  d, c, a, e   c, a, d  prescribed by  a, b, c, d  P18§5,  a, b, c  P16§5. P6—Theorem. On the assumption that a, b, c should be distinct projective points on a line r, and that d should be a [point of] (abc) different from b, it follows that all the points of r not lying in segment (bad) are in segment (abc). Proof. Having assumed the hypothesis, d certainly falls in r without coinciding with a or with c. With that occurs, moreover, the relation d 0 (acb), or else the alter b native b 0 (acd), thanks to  e  P5 and P4,3§5. But in the one case, having  d, b, c, d   c, b  c, d 0 (adb) by  b, c  P6§5, one can refer to  b, c, d, e  P4 and with that conclude that (abd) f (abc). Therefore, each point of the figure r – (bcd) – {d}, as a member of  b, a, d  figure (abd), will lie in (abc): see  a, b, c  P16§5. The other case follows from the  d, b  first via the substitution  b, d  , having taken notice of P6§5; and precisely, in this way one finds that the figure r – (dab) – {b} lies in the segment (adc). But this suffices for the theorem, given P2,28,3§5. P7—Theorem. Given r, a, b, c as above and having selected in segment (abc) two points d and e, arbitrary but not coincident, all points of r that are excluded from

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segment (dae) will lie in segment (abc). Proof. This is an extension of the  b, d  preceding P6, into which it turns via the exchange  d, e  , assured by (abc) = (adc) and e 0 (adc), and so on. P8—Theorem. Let r, a, b, c be as above; moreover, let d, f be points in segment (abc) such that f should belong to (acd); and let e denote a point of r not coincident with d or with f, such that segment (def ) should be contained in (abc). Then point e will lie in segment (acd). Proof. On the one hand, the hypothesis implies d, f 0 r – {a, c}, f = / d, and (acd) = (df a) by virtue of P4,2§5 and  c, d, f   b, c, d  P28§5. On the other, a ó (de f ) and hence e 0 (dfa), if one refers to    d, e, f , a  P4§5,  a, b, c, d  P19§5. P9—Theorem. Having assumed r, a, b, c as above, let d, e, f be points of the segment (abc) such that e should lie in segment (acd), and f in segment (ace). It follows that segment (de f ) will be contained in segment (abc). Proof. In the first place, one has d, e, f 0 r – {a, c} [and] f 0 (acd) by P4§5 and P4, and consequently e = / d, f= / d, f = / e, [and] (a f e) = (ace) by P4,28§5. Now d ó (ace) by P2, then d ó (a f e) and consequently d 0 ( f ea) [and] a ó( f ed), this because of  f ,e  b, c  P13,11§5. From that results ( f ea) = ( f da) [and] ( f ed) f ( f ea), as prescribed    f , e, d, a   f , e, a  by  a, b, c  P28§5,  a, b, c, d  P29§5, and thus (def ) f (adf ) by P2§5. But because f 0 (acd), one also has d ó (acf ) by P2, or else c ó (adf ) by P6§5; thus, (adf ) will be contained in turn in (adc) on account of P29§5. By P28§5 this is as much as to say in (abc), which was to be shown. P10–Theorem. Under the same hypothesis as P9, point e cannot belong to segment (daf ), nor coincide with d or with f. Proof. [This follows] because here, as previously, the relations d, e, f 0 r – {a}, d = / e, e = / f, and d ó (af e) subsist; with these, also e ó ( fad) by P8§5, and so on. P11–Theorem. As always, let a, b, c be distinct points of a projective line r, and let us assume that d, f should be points of segment (abc) such that f should belong to segment (acd); then let e be a point of r external to segment (daf ), but different from d and from f. I say that point e should lie in segment (acd), and f in segment (ace). Proof. By P4,28,6,31§5, points a, c, d, f satisfy the conditions d, f 0 r – {a, c}, f = / d, (afd) = (acd), d 0 (caf ), and c 0 (daf ); and consequently (caf ) = (cdf ) and (daf ) = (dcf ). About point e one deduces in view of  d, a, f , e   a, b, c, d  P13§5 that e 0 (af d) and moreover that e 0 (acd). But from this, as was   mentioned, we should also have e = / a by P4§5, e ó (dc f ), hence e 0 (cdf ) – {a} by P16§5, and consequently e 0 (caf ) – {a}. Thus again, the relation f 0 (ace) holds. P12–Theorem. Given r, a, b, c as above, if d, e should be points external to segment (abc), the one and the other different from each of the points a and c, it will be necessary that point e lie in segment (adc); for that reason segments (adc) and (aec) will coincide. Proof. From the hypothesis we should have d = / b and a = / b by way of P3,15,17§5; [moreover,] c, d, e 0 (acb), c 0 (abd), and c 0 (abe). Then

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6 Pieri’s 1898 Geometry of Position Memoir  d, c 

(acd) = (abd), conforming with  c, d  P28§5. Thus, should we assume that  c, b, c  e 0 (acd), it will happen that e 0 (abd), then e ó (dac) because of  b, c, f  P10, and finally e 0 (adc) according to P16§5. If on the other hand e ó (acd), it will be  d, e  equally true that e 0 (adc) by P15§5. The rest is stated in  b, d  P28§5. P13–Theorem. If on a projective line r are given three distinct points a, b, c [and] further, a fourth point d, not internal to segment (abc) nor coincident with a or with c, [then] every other point e of the line r is forced to belong to one of the segments (abc) [or] (adc) if it does not coincide with a or with c: that is, r = (abc) c (adc) c {a, c}. Proof. Let it be noted that the assumption e 0 r – (abc) – {a, c} (in addition to the hypothesis) carries with it the condition e 0 (adc) by virtue of P12. Then the conditional proposition e 0 r will imply immediately that e 0 (abc) c (adc) c {a, c}: that the second should be a consequence of the first was already stated in P1§5.—This proposition and the last one of the preceding section assert, in substance, that “Two distinct projective points a and c divide a projective line into two parts (abc) and (adc), which do not have any point in common, but taken together with their common ends a and c reproduce the line.” P14–Theorem. Let r, a, b, c be as above, and in r let there be points ar, br, cr as well, distinct from each other and such that the segment (arbrcr) should be contained in segment (abc) without either one of points ar and cr falling on a or on c; moreover, let cr belong to the segment (acar). Then if points d, e are selected at will in segment (arbrcr), each of the two propositions “e should lie in (acd)” and “e should lie in (arcrd)” will be a consequence of the other. Proof. Appealing to P4§5, P12, and P7,21,29§5 one obtains successively, a, c ó (arbrcr), a 0 (arc cr), cr 0 (cara), cr ó (ara c), and (ara c) f (ara cr). From there, about points d, e one deduces by P4,28,6,11,9,29§5) [that] d, e 0 (abc) – {ar, cr, a, c}, (ard cr) = (arbrcr), a ó (ard cr), d ó (ara cr), ar ó (dcra), cr ó (dara), and (dcra) f (dcrar). But from these and the first [list of results], with the aid of P2,6§5 and P12, one deduces that d ó (arac) and c, cr ó (aard), [and that] (acd) = (acrd) and (acd) f (arcrd). Thus, the [one] proposition e 0 (acd) cannot stand without the other, e 0 (arcrd). Finally, from the proposition e 0 (arcrd) it follows that d ó (arcre), thanks to P2. Then, because exchanging points d, c is allowed in the results just mentioned, thanks to the symmetry of the hypothesis with respect to these [points], so that d 0 (ace) cannot stand without d 0 (arcre), it follows as well that d ó (ace), and consequently that e 0 (acd) by P2.

x

b

Case a1: x 0 (abc) & x ó (acb)

a

= a

x

c

b

c

6

b

Case a2: x 0 (abc) & x 0 (acb)

x

a

= a

b

c

x

c

6

The Set a, b, c x of Points Following x in the Natural Order a, b, c: Shown Here as ))))) )))))

b

Case b: x ó (abc) c {a, c}

a

= a

b

c

c

x

6

x

b

Case c: x = a

x=a = a c

b

x

b

c

6

Case d: x = c

x=a = a c

b

x=c

6

Each circle is cut at point a and unwound onto the line, so that point a becomes its “point at .”

6.7 Natural Orderings and Senses of a Projective Line (§7)

181

§7 Natural Orderings and Senses of a Projective Line P1—Definition. Under the premise that a, b, c are distinct points of a projective line r, the phrase «following x in the natural order a, b, c », abbreviated31 by a, b, c x, will denote70 ( a) the segment (acx), if x is a [point in] (abc); ( b ) the figure r – (acx) – {a, x}, if x is a point of r outside segment (abc) but different from a and from c; ( c ) the figure r – {a}, if x is a point not different from a; ( d) the figure r – (abc) – {a, c}, if x is a point not different from c. By means of this proposition, under its hypothesis about r, a, b, c, a certain function or transformation symbol a, b, c is fully defined that, placed before a point of the projective line r, produces an entire class of points on r. Moreover, the group of symbols a, b, c r draws precise meaning from it (see P13§6). P2—Definition. Under the same hypothesis as P1, the “natural ordering of r in the sequence a, b, c” —or, more briefly, “natural order a, b, c” or “a, b, c r” —denotes that relation (from [points on] r to subclasses of r) that associates to each point x of the line the points represented by a, b, c x. All the properties of segments in the two preceding sections could be phrased readily as inherent facts about the natural order a, b, c. Indeed, this is the way that it is usually done: [by] assuming some properties of the natural order, or projective motion, in the sense a, b, c in order to deduce the most important results about segments, separation of points, etc.71 Thus, with P1, propositions 2, 4, 5 of §6 say in sum nothing more than this: “Let a, b, c be three distinct points on the projective line r, and d, e be points [selected] at will from segment (abc). Then (1) it cannot happen that d should follow e and at the same time e follow d in the natural order a, b, c. [And] (2) if point e should follow d in the natural order a, b, c, each point of r that should follow e in this same order will likewise follow d. [Moreover] (3) if d, e are distinct from each other, one of these two cases will occur: e will follow d in the natural order a, b, c or d will follow e. These are precisely the properties that are usually investigated as order principles, appropriate for ordering the elements of a class. And soon it can be seen, in P3–6, that all three are indeed subsumed under the idea signified by the words “following x in the natural order a, b, c,” for an arbitrary point x on the line. 70

[See the figure on the facing page, constructed by the present authors. Pieri did not employ any figure.]

71

In addition to [the works of] the cited authors, compare Vailati 1892. In contrast, Vailati 1895a and 1895b study postulates to be assumed when the notion of “separation” should be given without definition. But there is probably no account anywhere of our definition of these concepts, based on simpler projectivegeometric notions: point and line.

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6 Pieri’s 1898 Geometry of Position Memoir

P3—Theorem. Having assumed that a, b, c should be three distinct points of a projective line r, and also x, y points of r, it is contradictory to suppose that y should follow x in the natural order a, b, c and at the same time x follow y. Proof. Select a point d in the figure r – (abc) – {a, c}, as is always possible by P24§5. According to P13§6 the hypothesis x, y 0 r will be equivalent to the conjunction of two propositions taken together: “x 0 (abc) or x 0 (adc) or x = a or x = c” and “y 0 (abc) or y 0 (adc) or y = a or y = c.” As such it reduces ipso facto to the disjunction, or polylemma, “x, y 0 (abc)” or “ x 0 (abc), y 0 (adc)” or “x 0 (adc), y 0 (abc)” or “x, y 0 (adc)” or “x 0 (abc), y 0 {a, c}” or “x 0 {a, c}, y 0 (abc)” or “x 0 (adc), y 0 {a, c}” or “x 0 {a, c}, y 0 (adc)” or “x, y = a” or “x = a, y = c” or “x = c, y = a” or “x, y = c.” Now, if it should happen that “x, y 0 (abc),” x, y [the conclusion] reduces to  d, e  P2§6. Under the alternative assumption “x 0 (abc), y 0 (adc),” and consequently y 0 r – (axc) – {a, c} by P28,32,4§5, it happens that x 0 (ac y), by P19§5: this situation suffices to prevent x from belonging  y, x  to a, b, c y (see  x, y  P1b). The case where “x 0 (adc), y 0 (abc)” is only distinguished from the preceding one by the interchange of letters x, y. Nor does the particular hypothesis “x, y 0 (adc)” permit the consistency of the conditions y 0 a, b, c x and x 0 a, b, c y, because assuming y 0 r – (ac x) – {a, x} and x 0 r – (ac y) – {a, y} at once (see P1b) would quickly lead one to conclude by P4,17§5  d, x, y  that x 0 (ac y) and y 0 (ac x), contrary to the statement of  b, d, e  P2§6. In each one of the other eight cases, a glance at definition P1a–d suffices to verify the desired result. P4—Theorem. Let r, a, b, c be as above and moreover, x, y [be] noncoincident points of r. Then, one of two conditions [will be satisfied]: either y will follow x in the natural order a, b, c, or x will follow y. Proof. Also to be considered here are the same cases as in the preceding theorem, except for just two, to wit “x, y = a” and “x, y = c,” which are excluded directly by the condition y = / x. The first hypothesis x, y “x, y 0 (abc)” was already contemplated in  d, e  P5§6. If instead, “x 0 (abc), y 0 (adc),” it follows as in the preceding demonstration that y 0 r – (a xc) – {a}; and after this, also y 0 (acx) by P15§5; that is to say y 0 a, b, c x, by P1a. So the conclusion is true for this case, as well as for the one in which “x 0 (adc), y 0 (abc)” is assumed. Under the fourth hypothesis, that is to say “x, y 0 (adc),”  d, x, y  the conditions y 0 (acx) and x 0 (ac y) cannot be valid at the same time,  b, d, e  P2§6 prohibiting that. For this reason, seeing that x, y = / a, it must happen that y 0 r – (ac x) – {a, x}, or else x 0 r – (ac y) – {a, y}, which is what is desired, by P1b. When “x 0 (abc), y 0 { a, c}” it follows by P3,4§5 that x 0 r – {a} and y = a, or else y 0 (acx); thus, x 0 a, b, c y in the one case and y 0 a, b, c x in the other, on y account of  x  P1c and P1a. For the case “x 0 {a, c}, y 0 (abc)” refer to the one preceding, exchanging x with y. And [under the hypothesis] “x 0 (adc),  y y 0 {a, c},” in the one case x 0 a, b, c y will result from  x  P1c; in the other, by P13§6  y x 0 r – (abc) – {a, c}, and here also x 0 a, b, c y, in conformity with  x  P1d. When it is assumed that “x 0 {a, c}, y 0 (adc),” nothing happens differently, except for  y, x  the exchange  x, y  . Finally, if either “x = a, y = c” or “x = c, y = a,” nonetheless y 0 a, b, c x or x 0 a, b, c y would be [true], by P1c.

6.7 Natural Orderings and Senses of a Projective Line (§7)

183

P5—Theorem. Given r, a, b, c as above, if x, y are points of r and y follows x in the natural order a, b, c, all points that follow y in this order will also follow x: that is, the figure a, b, c y will be contained in the figure a, b, c x. Proof.72 Under the assumption that d should, as above, be a point of the segment r – (abc) – {a, c}, four cases will be possible with respect to x: x 0 (abc), x 0 (adc), x = a, [and] x = c (see P13§6). To the particular assumption x 0 (abc) corresponds a, b, c x = (ac x) by P1a, then also y 0 (ac x) by the hypothesis; and here we can distinguish [three subcases] depending on whether y 0 (abc) or y 0 (adc) or y = c. If y 0 (abc), and thus x and y [are] distinct from a as from c, it will happen that a, b, c y = (ac y); and because then x ó (ac y) by P2§6, it follows immediately, referring to P29§5, that (ac y) f (ac x), in conformity with the conclusion. If y 0 (adc), and consequently y 0 r – (abc) – {a, c} [and] a, b, c y = r – (acy) – {a, y} by P13§6 and P1b, it follows first that (abc) f (aby) [and] b 0 (ac y) by P29,19§5, whence (aby) = (ac y) by P28§5. For that reason (abc) f (ac y), and in the same way (adc) f (acx). Thus, r – (acy) f r – (abc); by P13§6 and P3§5, that is to say r – (acy) – {a, y} f (acx), which was to be shown. Finally, still under the hypothesis x 0 (abc), if one assumes y = c it will follow as before that (adc) f (acx): and this is precisely what is desired, because now a, b, c y = (adc) by P1d, P13§6. Continuing, we assume x 0 (adc), and thus also a, b, c x = r – (acx) – {a, x} and y 0 r – (acx) – {a, x} by P1b. Now from x 0 r – (abc) – {a, c} it follows, just as it did before for y, that (abc) f (abx), [which] thus entails y 0 r – (abc) – {a, c} and consequently a, b, c y = r – (acy) – {a, y} by P1b. But from y 0 r – (acx) – {a} it also follows that (acx) f (acy) by P29§5, and then that r – (acy) – {a, y} f r – (acx) – {a, x}, a result that, whatever may have been said, precisely states the conclusion. Under the third particular assumption x = a, the conclusion emerges immediately from P1c. Finally, that is when it is assumed that x = c, we will have to consider that a, b, c x = r – (abc) – {a, c} by P1d. But from y 0 r – (abc) – {a, c} it follows in the usual way that (abc) f (acy) and consequently that r – (acy) – {a, y} f r – (abc) – {a, c}; then here again the condition a, b, c y f a, b, c x is satisfied. P6—Definition. The hypothesis of P1 about r, a, b, c standing firm, and x being an arbitrary point of r, we designate by the notation –a, b, c x —which can be rendered as «preceding x in the natural order a, b, c » —the class31 of all those points y of r that satisfy the relation “x follows y in the sequence a, b, c.” In other words, –a, b, c r is the relation inverse to a, b, c r. It is easily seen that P7—Theorem. Supposing r, a, b, c are as above, no point precedes a in the natural order a, b, c, but a precedes each other point of r. P8—Theorem. And for any point x of r, arbitrary but different from a, the figures –a, b, c x and {a} c a, c, b x coincide. This is as much as to say, “provided that x should not coincide with a, the [points] preceding x in the natural order 72

[P5 and its proof fill three paragraphs.]

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6 Pieri’s 1898 Geometry of Position Memoir

a, b, c, except for the single point a, are those following x in the natural order a, c, b.” —For this reason a, b, c r and its inverse relation do not have exactly the same nature (as would appear in common usage of the words “preceding” and “following”): however, [the inverse] is hardly to be distinguished from the order in the sequence a, c, b. —And perhaps it will not be superfluous to note that two points, one of which follows or precedes the other, are necessarily distinct by P1,6. P9—Definition. Given a projective line r, the common noun «natural ordering of r » —or r —stands31 for the class of all possible natural orderings conforming with definition P2, each one of which is determined by three distinct points a, b, c selected arbitrarily on r. The definitions of concordant or discordant natural orders and of the senses or directions of a projective line arise as follows, with many relations that derive from them. Proof of these is left in large part to the reader, brevitatis causa.73 One cannot ignore the qualities that such notions inherit from the concept of motion: for example, a great intuitiveness in the form that invests them, and as well, a conceptional content remarkable for the many facts that are gathered and, so to speak, condensed in them. But these gifts, eminently synthetic, precisely because they are so precious on the one hand, are on the other less easily applied to support those aims that analysis sets for itself. P10–Theorem. If a, b, c, just as ar, br, cr, are distinct projective points on the same line r, they will satisfy at least one of the six following conditions, no two of which can be valid at the same time: (1) ar 0 a, b, c br, while br 0 a, b, c cr and ar 0 a, b, c cr (2) br 0 a, b, c ar,

"

br 0 a, b, c cr

"

ar 0 a, b, c cr

(3) br 0 a, b, c ar,

"

br 0 a, b, c cr

"

cr 0 a, b, c ar

(4) br 0 a, b, c ar,

"

cr 0 a, b, c br

"

cr 0 a, b, c ar

(5) ar 0 a, b, c br,

"

cr 0 a, b, c br

"

cr 0 a, b, c ar

(6) ar 0 a, b, c br,

"

cr 0 a, b, c br

"

ar 0 a, b, c cr.

Proof. After each one of the pairs (ar, br), (br, cr), (cr, ar) is subjected to P4, these three propositions will be true at once: “ar 0 a, b, c br or br 0 a, b, c ar,” “br 0 a, b, c cr or cr 0 a, b, c br,” and “cr 0 a, b, c ar or ar 0 a, b, c cr.” Moreover, the horns of each dilemma will exclude each other, by virtue of P3. Now, conjoining these three disjunctions gives rise to eight terms. Two of those, “ar 0 a, b, c br, while br 0 a, b, c cr and cr 0 a, b, c ar,” and “br 0 a, b, c ar,

"

cr 0 a, b, c br

"

ar 0 a, b, c cr,”

disappear thanks to P3,5. The six remaining are those that comprise the conclusion.

73

Though, as it happened to be noted already in section 6.6, these do not recur at all in the sequel.

6.7 Natural Orderings and Senses of a Projective Line (§7)

185

These six cases of the manner in which points ar, br, cr are arranged with respect to the natural order a, b, c can be distributed into two very significant natural sets, because, as will be clarified in the course of this section, cases (2), (4), (6) share many properties, revealing themselves as similar, and the like occurs with (1), (3), (5). The [odd-numbered cases] are distinguished from the others by many characteristics. Thus, it is useful to introduce already the following: P11–Definition. Under the hypothesis of P10, we shall say that the order in the sequence ar, br, cr is concordant with the order in the sequence a, b, c —or further, that ar, br, cr r 0 Ccrd a, b, c r —when any one of the three conditions (2), (4), (6) is satisfied. If instead, points ar, br, cr satisfy (1) or (3) or (5), the ordering in the sequence ar, br, cr will be called nonconcordant, or discordant, with the order in the sequence a, b, c; this is then written ar, br, cr r ó Ccrd a, b, c r. The relation defined here is reflexive, transitive, and symmetric: that is, P12–Theorem. Every natural ordering on r (see P9) is concordant with itself. P13–Theorem. Under the hypothesis of P10 these two statements are equivalent: ar, br, cr r 0 Ccrd a, b, c r and a, b, c r 0 Ccrd ar, br, cr r. P14–Theorem. Moreover, if aO, bO, cO are also distinct points on r and [the statements] ar, br, cr r 0 Ccrd a, b, c r and aO, bO, cO r 0 Ccrd ar, br, cr r are verified at the same time, one infers that aO, bO, cO r 0 Ccrd ar, br, cr r. The next four theorems will be proved more easily. P15–Theorem. If in the hypothesis of P10 the orders ar, br, cr r and a, b, c r are discordant, [then] on the contrary, the orders ar, br, cr r and a, c, b r are concordant, and vice versa. P16–Theorem. Always letting a, b, c be distinct projective points on a line r, the orders a, b, c r, b, c, a r, c, a, b r will all three be concordant with each other, and thus also the orders a, c, b r, c, b, a r, b, a, c r. But the [first three] are discordant with the [latter three]. P17–Theorem. If r, a, b, c are as above, and a point cr different from a and from b given in r, the orderings a, b, c r and a, b, cr r will be concordant or discordant, depending on whether point cr lies or does not lie in the segment (acb). P18–Theorem. Under the hypothesis of P10, if the orders a, b, c r and ar, br, cr r are concordant, while points a and ar coincide, then a, b, c r = ar, br, cr r will be true. See P1,2. The following three theorems, with the preceding one, give singular evidence of the relationship that obtains between different but concordant natural orders, and major importance to the correspondences established with definition P11.

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6 Pieri’s 1898 Geometry of Position Memoir

P19–Theorem. If, under the same hypothesis as P10, the orders a, b, c r and ar, br, cr r are concordant with each other, and if a point x should coincide with ar or should follow ar in the order a, b, c r, [then] the figure a, b, c x will be entirely contained in the figure ar, br, cr x. P20–Theorem. And if, on the contrary, x precedes ar in the natural order a, b, c, the figure –a, b, c x will be contained in the figure –ar, br, cr x. P21–Theorem. Again, given the hypothesis of P10, but having supposed that the orders a, b, c r and ar, br, cr r should be discordant with each other, if the point x lies in a, b, c ar, the figure a, b, c x will lie entirely in –ar, br, cr x. If instead, x lies in –ar, br, cr ar or [coincides with] ar, the figure –a, b, c x [will lie entirely] in ar, br, cr x. Proof. With attention to P8,15 this follows from P20,19 if in these the points b and c are permuted. One way to introduce the direction of a given order a, b, c r, and then the senses of the line r, is suggested by P12–14, and consists in substituting everywhere the locution “orders of the same direction” for that of “concordant orders,” so that the discourse will be conducted obliquely about a new abstract entity, direction, substratum of many concordant natural orders. (That is conceived by thinking of all the orders concordant, for example, with a, b, c r, but abstracting from the special attributes of each one, and from anything that could distinguish them individually.) But this sort of indirect definition (definition by abstraction), although rigorous, is not preferable to a true and appropriate nominal definition such as, for example, this alternative: P22–Definition.74 Given distinct points a, b, c on a projective line r, the class of all natural orderings of r concordant with a, b, c r is called the sense (or direction) a, b, c, designated by S(a, b, c). Thus, S(a, b, c)   r 1 Ccrd a, b, c r. See P9,11. P23–Theorem. By P15, therefore, the direction a, c, b is the class of all natural orderings nonconcordant (that is, discordant) with a, b, c r: S(a, c, b) =  r – Ccrd a, b, c r. P24–Theorem. And consequently  r = S(a, b, c) c S(a, c, b): thus, each natural ordering on r belongs to the sense a, b, c or to the sense a, c, b. P25–Theorem. Moreover, under the hypothesis of P10, the statement “ar, br, cr r is concordant with a, b, c r” reduces to the equality S(ar, br, cr) = S(a, b, c), and vice versa: from this it follows that S(ar, br, cr) should equal one or the other of the senses S(a, b, c), S(a, c, b). Therefore, each projective line admits two distinct directions and no more: [two] senses, an epithet well suited for contrary and opposite [qualities] (see P8). 74

[This is one of the first occurrences of conversion of a definition by abstraction into a nominal definition using equivalence classes. Pieri introduced that technique. See section 2.3.]

187

6.8 The Projective Triangle (§8)

§8 The Projective Triangle 75 It should be noted that among the advantages that the projective segment manifests as a fundamental geometric idea is also this: in passing to forms of higher dimension, the projective segment, determined by its ends and a point, leads intuitively to the projective triangle and to the projective tetrahedron, determined by their vertices and a point, and thus by a way not different from that by which one precedes from the line to the plane and to the higher spaces. Consideration of these figures—projective triangle and projective tetrahedron—proves very useful in many questions of projective geometry, even elementary ones, because the simpler facts inherent in the connection of the plane and in the visual of a plane hinge upon them. G. K. C. von STAUDT introduced and used them in 1847 (see for example paragraphs 172–175 and 188–191), placing in view some properties derived from certain principles about conic surfaces of even order and of odd order. But I do not know whether I might have treated this subject more expansively and with greater simplicity of premises than others. In this section the triangle is studied precisely as a portion of the plane suited to being projected onto another of the same nature, showing how its more intuitive and more often invoked properties might follow immediately from postulates I–XVII. P1—Definition. Let a, b, c be three noncollinear projective points, d a fourth point [taken] at pleasure in the projective plane abc but not lying in any of the lines bc, ca, ab, and let ar  (bc.ad), br  (ca.bd). Then, by projective triangle abcd, signified by the notation (abcd), is meant the set of all points common to the visuals of the two projective segments (barc) and (cbra) taken from points a and b, respectively—thus, the figure a(barc) 1 b(cbra). See definitions P3§4, P1§3, P1§5.76 P2—Theorem. And this figure (abcd) is thus nothing other than the locus of a projective point x, noncoincident with a or with b, for which the lines ax, bx should meet the lines bc, ca, respectively, within the projective segments (barc) and (cbra): in sum, such that lines bc and ax should have an intersection point (bc.ax) belonging to segment (barc), and ca, bx an intersection point (ca.bx) belonging to segment (cbra). Proof. P1 and P2 are equivalent principally because of P1§3 and P14§1. For a clear understanding of this section, pay particular attention to the first five propositions of section 6.4. P3—Theorem. Under the same hypothesis as P1, each point of the projective triangle abcd will lie in the projective plane abc, but not on any of the lines bc, ca, ab. Thus (abcd) f abc – bc – ca – ab. Proof. From x 0 (abcd) follows the existence of y1 and y2 such that y1 0 bc – {b, c}, y2 0 ca – {c, a}, and x = (ay1 . by2 ): see P1; 75

See Pieri 1896a, §11, §12.

76

Compare Staudt 1847, paragraph 175.

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6 Pieri’s 1898 Geometry of Position Memoir

P7,8§2; P2–4§4; P1,4§5. From that, also x 0 abc – bc – ca – ab on account of P6§3 and P1§4. P4—Theorem. And point d will always lie in the projective triangle abcd. Proof. From P2; P12§1; P2,4§4; P3§5. P5—Theorem. In fact, the whole segment (adar), as well as segment (bdbr), will be contained in (abcd). Proof. If x 0 (a dar), it will happen first of all that x 0 abc – {a, b, c}, then (thanks to P12§5 but having omitted every other citation) the lines ca and bx will intersect at a point in segment (abrc), and bc, ax at a point in segment (barc). P6—Theorem. Having assumed again the hypothesis of P1, and moreover denoted by cr the intersection point (ab. cd), if then e should be a point different from a and from b for which the lines ae and be meet segment (barc) and segment (cbra), respectively, it will be necessary that e differ from c and the line ce will have to intersect the segment (acrb). Proof. First of all, bc, ae intersect at a point different from each of a, b, c as do ca, be (see P12§1; P7,8§2; P2–4§4; P1,4§5) in such a way that a(bc.ae) = ae, b(ca.be) = ae, b(ca. be) = be, and b(ca. bd) = bd  (bc.ae),(ca.bd)  (see P22§1). Then,, because of  a , b  P1§4, ae and bd also intersect, so that (ae. bd) 0 abc – bc – ca – ab. Let f  (ae. bd), so that f ó {a, c}, af = ae, (bc.ae) = / f, f and (ab. cf ) ó {a, b}, according to  d  P4§4.77 Now by projection from point a (see P12§5), having seen that (bc.ae) 0 (barc) and having referred to P12§1 and P2,6§2, one deduces that f 0 (bdbr), and from here in the same way, or rather by projection from c, that (ab.cf ) 0 (bcra). From that [follows] the equality (b(ab.cf )a) =  (bc.ae),(ca.be)  (acrb) by P28,2§5. On the other hand, referring first to  a , b  P1§4 then to e   P2–4§4, one deduces that ae will differ from ca, ab; that ae, be intersect at a d point of the plane abc but not on any of bc, ca, ab; that this point coincides with e but differs from c; and finally that ab, ce intersect. Thus, projecting ac onto ae from b, then ae onto ab from c, one comes to conclude that e 0 ((bc.ae) f a), (ab.ce) 0 (b(ab.cf )a), and (ab.ce) 0 (acrb), which was to be shown. Through the property just demonstrated, which is among the principal [ones] about a triangle, propositions 1 and 2 take more regular and symmetric form in the following P7,9. P7—Theorem. Let a, b, c, d be as above in the hypothesis of P1 and denote by ar, br, cr the projections of point d on bc, ca, ab from points a, b, c. Then, triangle (abcd) will be the class of points (different from each vertex) whose images, projected from points a, b, c (vertices), fall within the segments (edges) (barc), (cbra), (acrb), respectively.78 Proof. P6 shows that if two of the images of a point e distinct from each of a, b, c fall in the segments just mentioned, so does the third: thus, P7 is equivalent to P2. 77

[For the last of these conditions, Pieri wrote (ab.cf ) 0 [0] - a - b.]

78

[In error, Pieri wrote (abrc) for (acrb). Vertex and edge are defined formally in P10.]

6.8 The Projective Triangle (§8)

189

P8—Theorem. Under the same hypothesis as P7 the triangles (abcd), (bcad), (cabd), (cbad), (bacd), and (acbd) will all coincide as just one. Proof. This follows from P7, with attention to P2,4§4; P10,12,25§1; P2§5. P9—Theorem. Moreover, these equalities will be satisfied: (abcd) = b(cbra) 1 c(acrb) = c(acrb) 1 a(barc) = a(barc) 1 b(cbra) = a(barc) 1 b(cbra) 1 c(acrb). Proof. These arise from P1, permuting letters a, b, c cyclically with reference to P8. P10–Definition. The hypothesis of P7 about projective points a, b, c, d, ar, br, cr standing firm, the edges of triangle (abcd) are the projective segments (barc), (cbra), (acrb) and the vertices, points a, b, c. The boundary of the triangle is the figure (barc) c (cbra) c (acrb) c {a, b, c}: the union of the edges and the [set of] vertices. The projective segment ab – (acrb) – {a, b} will be [called] the complement of the edge (acrb) (see P13§6). In sum, P6, P7, P9 say that each point where the rays79 meet that project two points, selected arbitrarily on two edges, from the vertices opposite them, is always in a ray projecting from the third vertex a point on the third edge, and belongs to the triangle. The points of the boundary remain excluded from the triangle, according to P3. That is to say, P11–Theorem. No point on the boundary of a projective triangle belongs to the triangle. P12–Theorem. Under the hypothesis of P7, if a point e lies in the projective triangle abcd, figures (abcd) and (abce) will coincide. Proof. According to P3,2, e 0 abc – bc – ca – ab, (bc.ae) 0 (barc), and (ca.be) 0 (cbra). From these facts stem equations (barc) = (b (bc.ae) c) and (cbra) = (c (ca.be) a), according to P28§5 and considering also P3,4§4; P12§1; P7§2. These immediately validate the conclusion, e thanks to  d  P2.—In other words, two projective triangles coincide if they have [their] vertices and one point in common. P13–Theorem. If besides the hypothesis of P7 we suppose that cO should be a point of the line ba,80 the proposition “cO is outside the projective segment (acrb) and does not coincide with a or with b” will be equivalent to this alternative: “lines bc and ca are intersected by dcO at points that should lie within segments (barc) and (cbra).” Proof. By P7§2 and P7,21§3 it can be granted that cO is necessarily different from d and that dcO should intersect each one of bc and ca. The proposition cO ó (acrb) c {a, b} will be equivalent to cO 0 (cbra) 1 (bacr) by P13,20§5; P4§4. And this [is equivalent to] “(ca.dcO) 0 (cbra) and (bc.dcO) 0 (barc)” because the points (ca.dcO), c, br, a are images of the points cO, cr, b, a from the “eye” d, and the same happens for (bc. dcO), b, ar, c with respect to cO, b, a, cr: see P12§5; P12§1; P2,6§2.

79

[Pieri’s informal term here was raggi.]

80

[In error, Pieri wrote bd instead of ba.]

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Considering that triangle (abcd) does not change under permutation of its vertices (see P8), nor if (see P12) in place of point d another point of the same triangle is selected arbitrarily, P13 can be stated in this alternate form: P14–Theorem. If a line containing a point internal to a triangle meets the complement of a side, it will have to intersect each one of the other two sides, and vice versa. See P10. P15–Theorem. Under the same hypothesis as P7 each line lying in the plane abc, provided that it pass through a point internal to the triangle (abcd) without containing any of its vertices, will intersect two sides of the same triangle and the complement of the third. Proof. Let aO, bO, cO be the traces81 of that line on bc, ca, ab (see P21§3, etc.). If cO 0 (acrb), there results the dilemma “aO ó (barc) or bO ó (cbra),” it being impossible that all three of the points ar, br, cr should lie on the boundary, by P14. Thus, one or another of the points cO, aO, bO will lie on the complement of a side. And then refer to P14. P16–Theorem. Assuming the hypothesis of P7, take on the line ab a point cO external to the segment (acrb) but different from a and from b, and let aO, bO be the points where bc, ca are intersected by dcO. Then the segment (aOdbO) will be wholly internal to triangle (abcd), and segment (aOcObO) wholly external. Proof. It is easy to see that cO is a [point on] abc different from d, that dcO is a line in plane abc that cannot coincide with ab nor with ca, that aO is different from b, c, d, cO and bO different from a, c, d, cO, aO: see P13; P12,25§1; P7,8§2; P20,21§3; P4§4; P4§5. Further, with a projection from c, one verifies that cO ó (bOdaO) by P2,6§2; P12§5. And consequently, (bOdaO) f (bOdcO) by P29§5. Now let x be a point of the segment (aOdbO), and thus different from a and from c. By P12§5 the lines bc and ax intersect at a point of (barc), and ab, cx at a point of (acrb). Therefore, it is necessary that x belong to the figure (abcd), according to P2,8. Assuming y 0 (aOcObO) on the other hand, ab and cy will intersect at a point of (bcOa): that is to say, at a point external to (acrb) by P2,32§5. Therefore, y will be external to (abcd) by P7. In the presence of P15, and then because by P12 any [point] whatever of (abcd) can take the place of point d, P16 says in substance that any line of the plane, that should contain a point internal to a triangle without passing through a vertex, is divided by the points where it intersects the boundary into two parts, the one internal to the triangle and the other external. That the same situation should occur if that line should pass through a vertex is already partially affirmed by P5; the rest [is left] to the reader. P17–Theorem. Assume that a, b, c are noncollinear points and d [is] a point of the plane abc but not on any of bc, ca, ab, and denote by ar, br, cr the intersection points (bc.ad), (ca.bd), (ab.cd). [Then] the line that joins two points aO and bO taken 81

[Pieri wrote traccie, evidently here meaning intersections.]

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6.8 The Projective Triangle (§8)

at will in segments (barc) and (cbra) will have to intersect the join ab at a point external to segment (acrb) and different from its ends a and b. Proof. (Having omitted the customary citations, etc.) the lines aaO and bbO intersect at a point e different from c and not belonging to ab, by P1§4; and ab, ce at a point cO of (acrb) by P6; therefore, (acrb) = (acOb). But the point (ab.aObO) coincides with Arm(a, b, cO), according to P5,12,14§4. See P23§5. P17 states that a line that joins two points taken at pleasure on two edges of a triangle will not meet the third edge. And this is the theorem that in the works of STAUDT governs the separation of harmonic pairs.82 The same property can be inverted in this way: if a line meets the complement of one edge and either one of the other two, it will also have to intersect the remaining [edge]. Thus: P18–Theorem. Given a, b, c, d, ar, br, cr anew as above, if  is a point of the projective segment ab – (acrb) – {a, b}, and aO a point of segment (barc), [then] the lines ca and aO intersect at a point of (cbra). Proof. Having set bO  (ca.aO) (see P5§4, etc.), then e  (aaO. bbO) and cO  (ab.ce) as above, it follows that (bc.ae) = aO, (ca.be) = bO, and (ab.aObO) = . Therefore,  0 ab – (acOb) – {a, b}  e, a , b , c  and cr, cO 0 ab – (a b) – {a, b} according to  d, a, b, c  P17 and P3,6§5. And consequently, cO 0 (acrb) and bO 0 (cbra), by P13§6, P6. From P18 follows immediately this alternative: each line that joins two points selected from the complements of two of the three edges [of a projective triangle] also intersects the complement of the third. Thus: P19–Theorem. The hypothesis of P17 standing firm, and having selected points , a arbitrarily in segments ab – (acrb) – {a, b} and bc – (barc) – {b, c}, the lines ca and a will intersect on the segment ca – (cbra) – {c, a}. And from P17–P19 together it follows that a line lying in the plane of a triangle but not containing any vertex either does not meet the boundary or intersects it at two points. That is to say, P20–Theorem. Under the hypothesis of P17, if , b, and  should be the traces on bc, ca, ab of an arbitrary line lying in the plane abc but not passing through any of the points a, b, c, this proposition will have to be true: [first] [second] [third] [fourth]

 ó (barc),  0 (barc),  0 (barc),  ó (barc),

 ó (cbra),  0 (cbra), ó (cbra),  0 (cbra),

 ó (acrb), or  ó (acrb), or  0 (acrb), or  0 (acrb).

Likewise, the following is notable:

82

Staudt 1847, paragraph 93, resulting there from certain principles that can well be attributed to Analysis situs (paragraphs 15–18 and 20).

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P21–Theorem. Resting again on the hypothesis of P17, the line joining two points aO and bO chosen within the segments (barc) and (cbra) always intersects the line cd in the interior of the triangle (abcd). See P5. Proof. Observe first (having omitted the customary citations) that aObO is a line distinct from each of bc, ca, ab and that (caOb) = (carb). The lines cd, aObO will intersect [at a point] different from a; and ab, aObO [at a point] outside (acrb), etc. Thus, having set   (ab.aObO),   (cd.aObO), and recalled P19,12§5, etc., one deduces that cr 0 (ab ) and  0 (bOaO); then, by projection from a, that (bc.a) 0 (carb).83 For the same reason [and by] P9, point  belongs to the triangle. Thus, it has been proved that the line joining two points taken arbitrarily on two edges penetrates within a triangle; and immediately from this it follows that the segments into which the line is divided by those two points are the one external and the other internal to the triangle, thanks to P16,12. And more generally, it can also be concluded that any line in the plane abc penetrates into the triangle (abcd) if it should intersect one of its edges, because this [line], by virtue of P20, will have to intersect another edge as well or pass through a vertex; then one refers to P21 or to P5. P22–Theorem. Having again assumed the hypothesis of P17, let e be a point of abc not situated on any of bc, ca, ab; and aO, bO, cO, its projections (bc.ae), (ca.be), (ab.ce) on these lines. Then, if aO and bO are external to the segments (barc) and (cbra), point cO will necessarily fall in segment (acrb). Proof. In the presence of  e, a, b, c  e  d  P2,4§4,  d, a, b, c  P17, etc., and having set   (ab.aObO), these relations will be     satisfied: aO ó {b, c}, bO ó {c, a}, cO ó {a, b}, and  ó (acOb) c {a, b}. But  ó  b, c, a, a, b  (acrb) thanks to  a, b, c, ,   P19; then cr, cO ó (a b) and cO 0 (acrb), as from P6§5  , b, c, c  and  b, c, d, e  P13§6. From the definition P1 of projective triangle and from P6,8,22 it follows that if a point of the plane abc not lying in any of bc, ca, ab should be projected from points a, b, c on these lines, all three images will fall on the boundary of triangle (abcd), or else a single one of these will fall on the boundary, indeed depending on whether that point is internal or external to the triangle. [This] will be maintained by the following: P23–Theorem. Given the same hypothesis as P22, there are no other possible cases than the following: (1) (2) (3) (4)

aO 0 (barc), aO 0 (barc), aO ó (barc), aO ó (barc),

bO 0 (cbra), bO ó (cbra), bO 0 (cbra), bO ó (cbra),

cO 0 (acrb) cO ó (acrb) cO ó (acrb) cO ó (acrb) .

P24–Theorem. Again, given three noncollinear points a, b, c let d, e, f, g be points of the plane abc, none lying in any of bc, ca, ab. Further, should e, f, g not belong to the 83

[In error, Pieri wrote (bc.ad) here instead of (bc.a).]

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6.8 The Projective Triangle (§8)

projective triangle (abcd), but the points (bc.ae), (ca.bf ), (ab.cg) fall on the segments (barc), (cbra), (acrb), and holding firm the previous meaning of ar, br, cr, then any point whatever of the plane abc that does not lie on any of bc, ca, ab will always lie in one of the four triangles (abcd), (abce), (abcf ), (abcg), nor can it lie in more than one. Proof. Besides the notations already used in P22 set a“  (bc.af ), b“  (ca.bf ), c“  (ab.cf ), aiv  (bc.ag), biv  (ca.bg), and civ  (ab.cg), while by P28§5 we will have (barc) = (baOc), (cbra) = (cb“a), (acrb) = (ac iv b). And further, bO, b iv ó (cbra); cO, c“ ó (acrb); a“, aiv ó (barc) by P6–8. Now, if p is any point whatever of the figure abc – bc – ca – ab, it will have to satisfy the quadrilemma contemplated in P23, putting p, (bc.ap), (ca.bp), (ab.cp) there in place of e, aO, bO, cO. But as far as it is claimed, and having regard to P13§6, those possible cases will be equivalent to these alternatives: (1) (2) (3) (4)

(bc.ap) 0 (barc) , (bc.ap) 0 (baOc) , (bc.ap) 0 (ba“c), (bc.ap) 0 (baivc) ,

(ca.bp) 0 (cbra) , (ca.bp) 0 (cbOa) , (ca.bp) 0 (cb“a), (ca.bp) 0 (cbiva) ,

(ab.cp) 0 (acrb) (ab.cp) 0 (acOb) (ab.cp) 0 (ac“b) (ab.cp) 0 (acivb) .

These, by virtue of P7, resolve immediately into (1) p 0 (abcd), (2) p 0 (abce), (3) p 0 (abcf ), (4) p 0 (abcg). Nor can it happen that two of these triangles should have a common point, because, were that so, for example, for (abcd) and (abce), or for (abce) and (abcf ), one would arrive at (abcd) = (abce) then e 0 (abcd) in the one case by P12, P4; and e 0 (abcf ) then bO 0 (cb“a), that is to say, bO 0 (cbra), in the other. Therefore, the lines ab, bc, ca divide the plane abc into triangles (abcd), (abce), (abcf ), (abcg) in such a way that, with these lines removed, the plane is the union of four triangles, any two of which have no common point. And it will not be superfluous to demonstrate this: P25–Theorem. Under the same hypothesis as P24, a line r of the plane abc, arbitrary but nevertheless not passing through any of the points a, b, c, always penetrates into three of the four triangles (abcd), (abce), (abcf ), (abcg): or rather, into exactly three. Proof. Indeed, having selected a point p on r that does not belong to any of bc, ca, ab, as will always be legitimate in view of, for example, P23§5, etc., this [point] will lie in just one of the four triangles by P24: for example, in (abcd). Then by P15, r will have to intersect the boundary of (abcd) twice: at points u and v, for example, situated on the sides (barc) and (cbra), respectively. Then by P17, point w, the intersection of r with ab, will lie outside segment (acrb), no less than points cO and c“ (see the preceding demonstration); that is to say, by P13§6, it will lie in segment (acOb) or (ac“b), which do not differ from each other. But as seen before, neither do segments (barc) and (baOc) differ from each other, nor segments (cbra) and (cb“a). Therefore, r also intersects the boundary of each of the triangles (abce), (abcf ) twice; or else penetrates into them, by P21. Finally, because none of the points u, v, w belongs to the boundary (baivc) c (cbiva) c (acivb) c {a, b, c} of the remaining triangle since (acrb) = (acivb), while by P32§5 segments (barc) and (baivc), as well as (cbra) and (cbiva), have

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6 Pieri’s 1898 Geometry of Position Memoir

no common points, it follows from P15 that the line r should be wholly external to the triangle (abcg).

§9 Segmental Transformations 84 As in algebra, which rests entirely on the ideas of number and function, the concepts of figure and transformation are in the same way fundamental to geometry, because these just as those serve only to reproduce special aspects or variations of two fundamental logical categories: class and representation. A representation  always presupposes two classes r, rr of which one, rr for example, reflects the other by means of  so that each individual of r should have as an image some individual of rr. Then,  will be [called] a transformation from r to rr or a representation85 of r in rr. A representation of r in rr will be called univocal 86 if individuals equal to each other in r are always represented by individuals equal to each other in rr. [It is called] injective if, besides being univocal, it cannot associate individuals equal to each other in rr with individuals not equal to each other in r (that is, if distinct individuals in r should always have distinct images in rr). A univocal transformation of r in rr is called bijective if its inverse is in the same way a univocal representation of rr in r.87 A correspondence made this way between the classes r and rr will always be injective in both directions; but an injective representation need not be bijective. If  is a univocal representation of r in rr, the symbol a, where a should be any individual of r whatever, denotes the transform of a: that is, the individual of rr that is the image of a by means of . Etc.88 In this section are noted some properties of certain injective representations from one projective line into another, which map segments into segments in the following way: P1—Definition. If r, rr are two projective lines, the name «segmental representation from r into rr» —which31 will sometimes be abbreviated by the sign Tr, rr —stands collectively for the set of all representations  of r in rr which satisfy the following conditions: (1)  should be an injective representation of r in rr; (2) if a, b, c are projective points on r, distinct but otherwise arbitrary, and d any point whatever of segment (abc), the point d should always lie in the projective segment (a, b, c). 84

See Pieri 1896b, §13.

85

[In this translation, transformation is used when the representation is understood to be bijective.]

86

[that is, a function]

87

[Pieri’s word for injective was isomorfa. Following Peano 1894a, §26, Pieri also defined Simile as its synonym, capitalized to distinguish it from the lowercase simile, which for Peano meant bijective. Pieri’s word for image was immagine: it stood for an element of the range of , never for a set of such elements. Pieri’s words for bijective and inverse were reciproca and inversa.]

88

For broader and more precise accounts see for example Peano 1894a, §19–§27, and Peano et al. 1895–1908, volume II, §1.

6.9 Segmental Transformations (§9)

195

P2—Theorem. If r, rr are projective lines,  is a segmental function from r into rr, and a, b, c are points of r distinct from each other, [then] each point d that, while lying in r, should not belong to segment (abc) nor coincide with a or with c, is mapped to a point d not belonging to the segment (a b c), nor coincident with either of the points a, c. Proof. From the hypothesis follows d 0 (bca) 1 (cab) by P20§5, so that d 0 (b c a) 1 (c a b) by P1, and thus the conclusion by P21§5. P3—Theorem. If r, rr, rO are projective lines,  is a [member of ] Tr, rr , and  a [member of ] Trr, rO , [then]   will be a [member of ] Tr, rO . That is, the composition of two segmental functions is a segmental function.89 This section is a preparation for the theorem of G. K. C. von STAUDT in section 10 on harmonic correspondences; however, the subject should in itself not be without interest. First of all it is necessary to stipulate this: P4—Definition. Let r be a projective line,  a segmental function from r into itself, a, b, c three distinct points on r, and ar, br, cr their transforms by ; and in addition assume that segment (arbrcr) should lie with both of its ends in segment (abc), and point cr in segment (acar). Then, we shall designate by h the set of all points x of (abc) satisfying these conditions: (1) that the image xr of x should always lie in (acx); (2) that if x1 is a point of (abc) such that x should lie in (ac x1), its transform x1r should always belong to (ac x1). From now on, we shall represent by k the set of all [members of ] (abc) that are not [in] h. Then, this is demonstrated: P5—Theorem. Under the hypothesis of P4 just stated, neither of the figures h, k is empty.90

89

[Pieri’s word for composition was prodotto.]

90

In a subject where it is so easy to err by disregarding even the smallest details, the reader should not be displeased by an exposition somewhat more precise and a format more rigid and more meticulous than usual. If P(x, y, z, ...), Q(x, y, z, ...) are propositions about variable objects x, y, z, ... (see section 1), the notation “P(x, y, z, ...) x, y Q(x, y, z, ...)” means “x, y, whatever they may be, just by satisfying P(x, y, z, ...) will also have to satisfy Q(x, y, z, ...).” [This] can also be read, “from P follows Q, with respect to x, y, z, ... .” Absent any subscript to the sign ‘ ’, the deduction is understood to extend to all the letters x, y, z, ... . The punctuation marks ‘.’, ‘:’, ‘ˆ’, ‘::’, etc., serve in place of parentheses to collect and distinguish the various parts of the discussion according to their importance. A point ‘.’ conjoins two propositions. Two points ‘ :’ link with each other a set of propositions already separated by points, in just this way: so that the principal division is that indicated by a larger number of points. See Peano 1894a, §10.

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Proof.91 ()

Hp . P1 . P1,4§5 : : ar, br, cr 0 r . br -= ar . cr -= br . cr -= ar . ar, cr 0 (abc) - a - c [By the hypothesis, P1, and P1,4§5, ar, br, cr are distinct points on r, and ar, cr each belongs to the segment (abc) and differs from a, c.]

( ) Hp . x 0 r - (acar) - a . xr   x . () . P14§5 : x: x 0 (cara) . P2,28§5 : x: x 0 (abc) . P1 : x: xr 0 (arbrcr) . P28§5 : x: (arxrcr) (abc) .  a, x, c   d, e, f  P8§6 : x: xr 0 (acar) . x, xr 0 (abc)   [Assume the hypothesis. According to () points a, c, ar must be distinct, as well as ar, br, cr. Consider any point x 0 r – (acar) – {a} and set xr   x. Then a a  c, a  r – (acar) – {a} f (cara) by  b, c  P14§5, hence x 0 (cara). By  b  P2§5 and  d  P28§5, (cara) = (aarc) = (abc), hence x 0 (abc). Since  is segmental, P1 entails xr 0 (arbrcr). By the hypothesis, (arbrcr) f (abc), hence xr 0 (abc). By  aa,,bb,, cc,,dx  P28§5, a, x, c (arxrcr) = (arbrcr) f (abc). Finally, by  d, e, f  P8§6, xr 0 (acar) and x, xr 0 (a b c).] ( )

Hp . x 0 r - (acar) - a - ar . () . P16§5 : x: x 0 (caar) . P9§5 : x . ar 0 (ac x) [Assume the hypothesis. According to () points a, c, ar must be distinct. Consider  c, a  any point x 0 r – (acar) – {a, ar}.92 By  b, c  P16§5, x 0 (caar). Then, ar 0 (ac x) by  c, a, a, x   a, b, c, d  P9§5.]  

()

Hp . x 0 r - (acar) - a - ar . xr   x . () . ( ) . ( ) .  x, a   d, e  P4§6 : x. xr 0 (ac x)   [Assume the hypothesis. According to () points a, c, ar must be distinct. Consider any point x 0 r – (acar) – {a, ar} and set xr   x.93 Then, x 0 (abc) by ( ) and  x, a  ar 0 (ac x) by ( ); hence, xr 0 (ac x) by  d, e  P4§6.]

91

[Pieri divided this proof and some later ones into lemmas written almost entirely in symbols. This translation preserves their labels: Greek letters in an order different from today’s. Each lemma serves at once as its statement and as a sketch of its proof. Proofs of later theorems sometimes refer to earlier lemmas. Pieri’s ideography was closely related to that described in Peano 1894a. The use of points in lieu of parentheses did not entirely dispel ambiguity, particularly with regard to instantiation and generalization of variables. To impart this work’s flavor, most of Pieri’s lemmas are stated here in the original symbolic form but with additional white space. The symbols Hp, Th, -, c, and are Pieri’s abbreviations for hypothesis and conclusion of the current theorem, and his versions of the negation sign ¬, union or inclusive disjunction sign w, and implication or inclusion sign | or f. After each lemma stated in symbolic form is inserted [in brackets] its expression in ordinary language. This practice is continued in P8§10.]

92

[Apparently in error, Pieri omitted the condition x ó a —that is, x = / a —which is required for the application of P9§5 in this proof.]

93

[Pieri neglected to define xr in ().]

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6.9 Segmental Transformations (§9)

()

Hp() . x1 0 (abc) . x 0 (ac x1) . x1r  x1 . ( ) . ( ) .  x , x P1,4§5 .  d1, e  P4§6 : x, x . x1 0 r - a - c . ar 0 (ac x1) .  a, x1  () .  d, e  P2§6 : x, x : x1 0 r - (acar) - a - ar .  x1   x  () : x, x . x1r 0 (ac x1)   1

1

1

[Assume the hypothesis of (), which includes that of ( ) and ( ). Consider any point x1 0 (abc), so that x1 0 r by P1§5 and x1 = / a, c by P4§5, and set x1r   x1.  x , x Consider any point x 0 (ac x1). Two of the hypotheses of  d1, e  P4§6 are now satisfied, and ( ) implies the remaining one, x 0 (abc). Its conclusion follows: (ac x) f (ac x1). Now, ( ) yields ar 0 (ac x), hence ar 0 (ac x1). According to () points a, c, ar must be distinct. Since the hypotheses ar, x1 0 (abc) and ar 0 (ac x1)  a, x  of  d, e1  P2§6 are satisfied, so is its conclusion x1 ó (acar). Now, since the hypox theses of  x1  () are satisfied, so is its conclusion x1r 0 (ac x1).] Now from the hypothesis and from ( ), (), () it follows,94 by virtue of P4, that r – (acar) – { a, ar} f h: thus, there certainly exist some points that lie in h. ()

c, a, c, a, y

Hp . y 0 r - (cacr) - c - cr . yr   y .  a, c, a, c, x  ( )() . () . P2,9§5 : y : y, yr 0 (cba) . yr 0 (cay) .  c, a, y, y   a, c, d, e  P2§6 . P1,4§5 : y: y -0 (cayr) . yr 0 r - c - a .    c, a, y, y   a, b, c, d  P9§5 : y : yr -0 (acy) .   P4 : y . y 0 (abc) - h [Assume the hypothesis. According to () points a, c, cr must be distinct. Consider c, a, c, a, y any point y 0 r – (cacr) – { c, cr}, so that the hypotheses of  a, c, a, c, x  ( ) and  c, a, c, a, y   a, c, a, c, x  () are satisfied, and set yr   y. Those results and P2,9§5 imply   c, a, y, y y, yr 0 (cba) and yr 0 (cay). In turn, P4§5 and  a, c, d, e  P2§6 yield y 0 r – {c, a} and c, a, y, y y ó (cayr), which together entail (cayr) = (acy) by  a, b, c, d  P9§5. By P4§5, yr ó (acy), so condition (1) of the definition of h in P4 fails. Thus, y 0 (cba) – h; hence y 0 (abc) – h by P2§5.] From this and the hypothesis it follows via P4 that r – (cacr) – {c, cr} f k. The existence of points that belong to k is thus by P24§5 also without doubt, which was to be proved. [This concludes the demonstration of P5.] Next, this will be proved:

P6—Theorem. Under the hypothesis of P4, a point x of h and a point y of k, even though selected arbitrarily, satisfy the condition y 0 (acx). Proof. ()

Hp . x1, x2 0 (abc) . x1 0 (ac x2) . x 0 (ac x1) . P4§6 : x , x , x . x 0 (ac x 2 ) 1

2

[Assume the hypothesis. Then for all points x1, x2 , x, conditions x1, x2 0 (abc),  x2 , x1  x1 0 (ac x2), and x 0 (ac x1) together imply x 0 (ac x 2), by  d, e  P4§6.] 94

[In error, Pieri cited () instead of () here.]

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6 Pieri’s 1898 Geometry of Position Memoir

( ) Hp . x 0 h :.: x :.: x1 0 (abc) . x 0 (ac x1) . x1r   x1 . P4 :: x :: x1r 0 (ac x1) ˆ x 2 0 (abc) . x1 0 (ac x 2) . x 2r   x2 . () .  xx2  P4 : x : x 2r 0 (ac x 2) .  x1 , x2   x, x  P4 :: x . x1 0 h 1   1

1

2

1

[Assume the hypothesis, let x be any point in h, and x1 be any point in (abc) such that x 0 (ac x1). By condition (2) of the definition of h in P4, x1r 0 (ac x1), where x1r   x1 . Moreover, if x2 should be any point in (abc) such that x1 0 (ac x2 ), then x 0 (ac x2 ) by (); applying condition (2) again with x2 in place of x1 yields x 2r 0 (ac x 2 ), where x 2r   x2 . Now, both conditions of the definition are satisfied with x1, x2 in place of x, x1; therefore, x1 0 h.] ( )

Hp . x 0 h ˆ x ˆ y 0 (abc) . x 0 (ac y) .  y  x  ( ) : y . y 0 h ˆ x ˆ y 0 k . P4 : y . x -0 (ac y)  1 [Assume the hypothesis and let y be any point in k, so that y 0 (abc) and y ó h by the definition of k. If x is any point in h, then x ó (ac y), because y x 0 (ac y) would imply y 0 h by  x1  ( ).] Hp ˆ ˆ x 0 h . y 0 k . ( ) . P4 : x, y : x, y 0 (abc) . x -= y . x -0 (ac y) : P5§6 ˆ . Th] [Assume the hypothesis and let x, y be any points in h, k, respectively. Definition y, x P4 entails x, y 0 (abc) and x = / y, ( ) implies x ó (ac y), and  d, e  P5§6 yields the desired result, y 0 (acx). This concludes the proof of P6.] At this point the need becomes apparent to invoke a new principle that does not follow from the preceding ones: this is the law of continuity of a segment. It is presented here in the following guise, resembling a noted principle or postulate of Richard DEDEKIND. POSTULATE XVIII

P7. Let a, b, c be three points, distinct from each other, on a projective line r. Suppose the projective segment ( a b c) will be divided into parts h and k so that each should contain at least one point, and so that, designating by x an arbitrary point of h and by y an arbitrary point of k , point y should always lie in the segment ( a c x). Then in segment ( a b c) there will have to exist at least one point z such that (1) each point u of ( a b c), provided z should belong to the segment ( a c u), should always be a point of h ; and (2) each point v of ( a b c) should be a point of k whenever it belongs to the segment ( a c z). And two such points z can never be distinct; thus:

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6.9 Segmental Transformations (§9)

P8—Theorem. Under the same hypothesis as P7 about a, b, c, r, h, k, there can never coexist in segment (abc) two points z, zr distinct from each other and satisfying the following two conditions: (1) from the proposition u 0 (abc), together with one or the other of the two propositions z 0 (acu), zr 0 (acu), it follows (in both cases and for any u whatever) that u 0 h; (2) from v 0 (abc), with one or the other of the two [propositions] v 0 (acz), v 0 (aczr), it follows (in both cases and for any v whatever) that v 0 k. Proof. Define these [propositions about arbitrary points z, zr]: ()

E(z, zr)  { z, zr 0 (abc) . zr -= z } [E(z, zr) should stand for “z, zr 0 (abc) and zr = / z”. ]

( ) F(z, zr)  { u 0 (abc) : z 0 (acu) .c. zr 0 (acu ) ˆ v 0 (abc) : v 0 (ac z ) .c. v 0 (aczr) ˆ

. u 0 h :: v. v 0 k }

u

[F(z, zr) should stand for “for any point u 0 (abc), if z 0 (acu) or zr 0 (acu), then u 0 h; and for any point v 0 (abc), if v 0 (acz) or v 0 (aczr), then v 0 k.”] Then: ( )

 z, z 

Hp . E(z, zr) . () . P1,4§5 .  d, e  P7,5§6 ˆ z, zr ˆ z, zr 0 r - a . r - (zazr) (abc) : zr 0 (acz) .c. z 0 (aczr) [Assume the hypothesis and that z, zr satisfy the condition E(z, zr) defined in (),  z, z  so that z, zr = / a, c by P4§5. Then,  d, e  P7§6 implies r – (zazr) f (abc) and  z, z    d, e  P5§6 implies zr 0 (acz) or z 0 (aczr).]  

()

 z, a, z 

Hp . E(z, zr) . () . ( ) .  a, b, c  P24§5 : z, zr . la propos.e “x 0 r - (zazr) - z - zr„ non è assurda in x [Assume the hypothesis and that z, zr satisfy the condition E(z, zr) defined in (), so  z, a, z  that z, zr = / a by ( ). By  a, b, c  P24§5, r – (zazr) – {z, zr} is nonempty.]

()

Hp . E(z, zr) . F(z, zr) . zr 0 (acz) ˆ z, zr ˆ z, x, z x 0 r - (zazr) - z - zr . () . ( ) .  d, e, f  P11§6 : x : x x x 0 (abc) . x 0 (acz) . zr 0 (acx) . P4§5 .  v   u  ( ) : x : x -0 a c c . x 0 k 1 h : x : x 0 r - a - c . x 0 (ac x) : () ˆ z, zr . esiste un x tale, che “x 0 r - a - c„ e “x 0 (ac x)„ [Assume the hypothesis and that z, zr satisfy conditions E(z, zr) and F(z, zr) defined in () and ( ). Then, by () there exists a point x 0 r – (zazr) – {z, zr} and z , x, z  by ( ), x 0 (abc). Further, suppose zr 0 (acz). Then, by  d, e, f  P11§6, x 0 (acz) and zr 0 (acx). These two conditions, with x 0 (abc), fulfill the hypotheses of the two propositions that constitute definition ( ); hence, x 0 h and x 0 k. By the definition of k in the hypothesis of P7, x 0 (acx). Also, x = / a, c by P4§5.]

200

()

6 Pieri’s 1898 Geometry of Position Memoir

Hp . () . P4§5 : . la propos.e “E(z, zr) . F(z, zr) . zr 0 (acz)„ è assurda rispetto a z e zr [By the hypothesis, (), and P4§5, the proposition zr 0 (acz)” does not hold for any z, zr.]

()

“E(z, zr),

F(z, zr),

and



Hp .  zz ,,zz  ) : . la propos.e “E(zr, z) . F(zr, z) . z 0 (aczr)„ è assurda rispetto a zr e z [By the hypothesis and  z, z  (), the proposition “E(zr, z), F(zr, z), and z 0 (aczr)” does not hold for any zr, z.] z ,z

( )

Hp . ( ) . () . () . () . ( ) : . la propos.e “E(z, zr) . F(z, zr)„ è assurda rispetto a z e zr [By the hypothesis, ( ), (), (), (), and ( ), the proposition “E(z, zr) and F(z, zr)” does not hold for any z, zr.95 ] Hp . () . () . ( ) : . Th [The hypothesis, ( ), (), and ( ) imply the desired result. This concludes the proof of P8.] Having made this digression one observes directly, concerning P4–P6, that the figures h and k defined in P4 satisfy the conditions required by the hypothesis of Postulate XVIII. The following holds, by P4–P7:

P9—Theorem. Given the hypothesis of P4, there will certainly exist a point z in the segment (abc) satisfying the following conditions: (1) whatever u may be, provided u 0 (abc) and z 0 (acu), it will happen that u 0 h; (2) whatever v may be, provided v 0 (abc) and v 0 (acz), it will happen that v 0 k. Finally, it will be proved that a point z such as the one whose existence is affirmed by P9 is necessarily fixed 96 by the mapping  given in P4, while no other point u of (abc) such that z should belong to the segment (acu) can be fixed. With this, in conclusion, we shall come to the consequence that interests us most; however, it will help to break the task into several propositions. P10–Theorem. Having assumed the hypothesis of P4 and selected a point z in the segment (abc) noncoincident with its transform z  zr, if the condition “u is a point of (abc) and z is a point in (acu)” always implies u 0 h, one will be able to conclude immediately that z does not belong to the segment (aczr). Proof.

95

[because ( ) yields zr 0 (acz) or z 0 (aczr).]

96

[Pieri’s term was unito. He also provided the alternative tautologous, which is used with other meanings in logic, rhetoric, and algebraic geometry.]

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6.9 Segmental Transformations (§9)

()

Hp . z 0 (abc) . zr  z . z 0 (aczr) . P1 . P1,4§5 : x ˆ z, x, z  z, z  x 0 r - (zazr) - z -zr .  d, e  P7§6 .  d, e, f  P11§6 : x : x 0 (abc) . x 0 (aczr) . z 0 (ac x) [Assume the hypothesis, let z 0 (abc), and set zr  z. By P1, P1§5, and P4§5,  z, z  zr 0 (abc). Assume also z 0 (aczr). By  d, e  P7§6, any point x 0 r – (zazr) – {z, zr} lies z, x, z in (abc). Moreover, x 0 (aczr) and z 0 (ac x) by P2§5 and  d, e, f  P11§6.]

( ) Hp . () . P1 . P1,4§5 : x ˆ x 0 r - (zazr) - z - zr .  z, x  xr  x .  d, e  P2§6 : x : x 0 r - (acz) - a -z . P2 : x : xr 0 r - (arcrzr) - zr .  z, x  ()P5 .  d, e  P14§6 : x : xr 0 r - (aczr) - zr .  z, x   d, e  P5§6 : x : zr 0 (ac xr) .    z, x   d, e  P4§6 : x : x 0 (ac xr) .    x, x   d, e  P2§6 : x : xr -0 (ac x) .   P4 : x . x -0 h [Assume the hypothesis of P10, consider a point x 0 r – (zazr), arbitrary but different from z, zr, and set xr  x. By () and P4§5, x = / a, c and  z, x  z 0 (acx), hence x ó (acz) by  d, e  P2§6. By P2, xr ó (arcrzr) and xr = / zr. By  z, x   z, x  P5() and  d, e  P14§6, xr ó (aczr). By  d, e  P5§6, zr 0 (ac xr); hence, (aczr) f (acxr)  x, x   z, x  by  d, e  P4§6. But () implies x 0 (aczr); hence, x 0 (acxr); then,  d, e  P2§6 yields xr ó (acx), and that implies x ó h by condition (1) of the definition of h in P4.] ( )

Hp() ˆ u 0 (abc) . z 0 (acu) : u . u 0 h :: z ˆ x 0 r - (zazr) - z - zr . () : x . x 0 h [Assume z, zr are points satisfying the hypothesis of (), and conditions u 0 (abc) and z 0 (acu) always imply u 0 h. Then by (), every point x 0 r – (zazr) – {z, zr} belongs to h.]

()

Hp . z 0 (abc) . zr  z : z ˆ x 0 r - (zazr) - z - zr . x 0 h . ( ) :

x

. z -0 (aczr)

[Assume z is a point satisfying the hypothesis. If some point x 0 r – (zazr) – {z, zr} belongs to h, then by ( ), z must fail to satisfy the hypothesis of (). That is, z ó (aczr).] ()

Hp() ˆ u 0 (abc) . z 0 (acu) : u . u 0 h :: z ˆ x 0 r - (zazr) - z - zr . x -0 h . ( ) : x . z -0 (aczr) [Assume that z, zr are points satisfying the hypothesis of (), and that conditions u 0 (abc) and z 0 (acu) together always imply u 0 h. If some point x 0 r – (zazr) – {z, zr} does not belong to h, then by ( ), z and zr must fail to satisfy the hypothesis of (). That is, z ó (aczr).]

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6 Pieri’s 1898 Geometry of Position Memoir

After conjoining () on the left with “z -= zr ˆ u 0 (abc) . z 0 (acu) : u . u 0 h” and () with “z -= zr” (proposition[s] conditional on z), and combining these deductions member by member,97 one obtains ()

Hp() . z -= zr ˆ u 0 (abc) . z 0 (acu) : u . u 0 h :: z ˆ x 0 r - (zazr) - z - zr . () . () : x . z -0 (aczr) [Assume that z, zr are distinct points satisfying the hypothesis of (), and that conditions u 0 (abc) and z 0 (acu) together always imply u 0 h. If there is a point x 0 r – (zazr) – {z, zr}, then z ó (aczr) by () and ().] On the other hand,

()

Hp() . z -= zr . P1 . P1,4,24§5 : z . la proposizione “x 0 r - (zazr) - z - zr„ non è assurda in x [Assume that z, zr are distinct points satisfying the hypothesis of (). Then, by P1 and P1,4,24§5, there is a point x 0 r – (zazr) – { z, zr}.] Therefore, one can eliminate x from ():

( )

Hp() . z -= zr ˆ u 0 (abc) . z 0 (acu) : () . () :: z . z -0 (aczr)

u

. u0h ˆ

[Assume that z, zr are distinct points satisfying the hypothesis of (), and that conditions u 0 (abc) and z 0 (acu) together always imply u 0 h. Then, by () and (), z ó (aczr).] Hp . () . () : . Th] [The hypothesis, (), and () imply the desired result. This concludes the proof of P10.] P11–Theorem. Assume the hypothesis of P4 and select in segment (abc) a point z noncoincident with its transform z  zr. If, as before, the conditional proposition “u 0 (abc) and z 0 (acu)” should always imply u 0 h, and similarly the condition “v 0 (abc) and v 0 (acz )” should entail v 0 k, one will be able to conclude immediately that zr does not belong to the segment (acz). Proof. ()

97

Hp . z 0 (abc) . zr  z . zr 0 (acz) . P1 : z ˆ y, z, z y 0 r - (zraz) - zr - z .  x, z, z  ()P10 . y: y 0 (abc) . y 0 (acz) . zr 0 (acy)

[Pieri’s word for combining was sommando. It seems unnecessary to conjoin the condition z = / zr, since that requirement is already included in the hypotheses of (), ().]

203

6.9 Segmental Transformations (§9)

[Assume the hypothesis of P4, let z 0 (abc), set zr  z, and suppose zr 0 (acz). y, z, z Then, for every point y 0 r – (zraz) – {zr, z},  x, z, z  P10() implies y 0 (abc), y 0 (acz), and zr 0 (acy).] ( ) Hp() . () . P1 . P1,4§5 : z ˆ y 0 r - (zraz) - zr - z . yr  y : y . yr 0 (arcrzr) .  z, y  ()P5 .  d, e  P14§6 : y : yr 0 (aczr) .  y, z   d, e  P4§6 : y . yr 0 (acy)   ( ) Hp . z 0 (abc) : z ˆ  x, y  y1 0 (abc) .  d, e1  P5§6 : y . y1 = z .c. y1 0 (acz) .c. z 0 (acy1) 1

[Assume the hypothesis of P4 and let z 0 (abc). Then, for any point  z, y  y1 0 (abc),  d, e1  P5§6 implies that y1 = z or y1 0 (acz) or z 0 (acy1).] ()

Hp . zr  z . zr 0 (acz) : y1 = z . y1r  y1 . P1 :

ˆ y . y1r 0 (ac y1) z

1

[If z, y1 are points with transforms zr  z and y1r  y1 , and if zr 0 (acz) and y1 = z, then y1r 0 (ac y1).] 98 ()

Hp() . P1 . P1,4§5 : z :: y 0 r - (zraz) - zr - z . y ˆ y1 0 (abc) . y 0 (acy1) . y1 0 (acz) . y1r  y1 :  z, y  ()P5 .  d, e1  P14§6 : y : y1r 0 (aczr) .  y, z  () .  d, e  P4§6 : y : y1r 0 (acy) .  y1   P4§6 : y . y1r 0 (acy1) d 

: y1r 0 (arcrzr) .

y1

1

1

1

[Assume that a point z satisfies the hypothesis of (), so that zr 0 (arbrcr); hence, zr 0 (abc). Let y be any point in r – (zraz) – {zr, z}. Then, for any point y1 in both (abc) and (acz), P1 implies that the transform y1r  y1 belongs to (arbrcr) and to (arcrzr); these facts entail y1r 0 (abc) and y1r 0 (aczr) by the hypothesis  z, y  y, z, z and  d, e1  P14§6, respectively. By (), zr 0 (acy); with  x, z, z  P4§6 that yields  y , y (aczr) f (acy); hence, y1r 0 (acy). Finally, if also y 0 (acy1), then  d1, e  P4§6 yields 99 (acy) f (acy1 ); hence, y1r 0 (acy1).] ()

Hp ˆ u 0 (abc) . z 0 (acu) : u . u 0 h :: y1 0 (abc) . z 0 (acy1) : y : y1 0 h . y1r  y1 . P4 : y . y1r 0 (acy1)

z

ˆ

1

1

[Assume the hypothesis, let z be any point, and suppose that conditions u 0 (abc) and z 0 (acu) together always imply u 0 h. If y1 is any point of (abc) such that z 0 (acy1), then y1 0 h and by the definition of h in P4 the transform y1r  y1 belongs to (acy1).] 98

[This lemma’s utility is unclear.]

99

[Pieri failed to mention the substitution of y for e in his second invocation of P4§6.]

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6 Pieri’s 1898 Geometry of Position Memoir

From the series of propositions ( ), (), (), () is assembled ()

Hp() ˆ u 0 (abc) . z 0 (acu) : u . u 0 h ˆ ( ) . () . () . () :: z :: y 0 r - (zraz) - zr - z . y ˆ y1 0 (abc) . y 0 (acy1) . y1r  y1 : y . y1r 0 (acy1) 1

[Assume that z is a point satisfying the hypothesis of () and that conditions u 0 (abc) and z 0 (acu) together always imply u 0 h. Let y be any point in r – (zraz) – {zr, z}. If y1 is any point of (abc) such that y 0 (acy1), then the transform y1r  y1 belongs to (acy1), by ( ), (), (), and ().] Consequently, ()

Hp() ˆ u 0 (abc) . z 0 (acu) : u . u 0 h ˆ () . ( ) . () . P4 :: z : y 0 r - (zraz) - zr - z . y . y 0 h [Assume that z is a point satisfying the hypothesis of () and that conditions u 0 (abc) and z 0 (acu) together always imply u 0 h. Then, every point y 0 r – (zraz) – {zr, z} belongs to h , by (), ( ), (), and P4.]

( ) Hp() ˆ v 0 (abc) . v 0 (acz) : v . v 0 k :: y 0 r - (zraz) - zr - z . () : y : y 0 k . P4 : y . y -0 h

z

ˆ

[Assume that z is a point satisfying the hypothesis of () and that conditions v 0 (abc) and v 0 (acz) together always imply v 0 k. Then, every point y 0 r – (zraz) – {zr, z} belongs to k , by (); by the definition of k in P4, y ó h.] ()

Hp . z 0 (abc) . zr  z ˆ u 0 (abc) . z 0 (acu) : u . u 0 h :: y 0 r - (zraz) - zr - z . y -0 h . () : y . zr -0 (acz)

z

ˆ

[Assume th e hypothesis, let z 0 (abc), set zr  z, and suppose that conditions u 0 (abc) and z 0 (acu) together always imply u 0 h. Then, no point y 0 r – (zraz) – {zr, z} belongs to h ; hence, zr ó (acz) by ( ).] ( ) Hp . z 0 (abc) . zr  z ˆ v 0 (abc) . v 0 (acz) : v . v 0 k :: y 0 r - (zraz) - zr - z . y 0 h . ( ) : y . zr -0 (acz)

z

ˆ

[Assume the hypothesis, let z 0 (abc), set zr  z, and suppose that conditions v 0 (abc) and v 0 (acz) together always imply v 0 k. Then, every point y 0 r – (zraz) – {zr, z} belongs to h ; hence, zr ó (acz) by ( ).] Finally, conjoining the hypotheses of (), ( ) with the propositions “z -= zr ˆ v 0 (abc) . v 0 (a cz) : v . v 0 k” and “z -= zr ˆ u 0 (abc) . z 0 (acu) : u. u 0 h”

205

6.9 Segmental Transformations (§9)

(conditional on z), respectively, then combining100 member by member [one obtains] ()

Hp . z 0 (abc) . zr  z . z -= zr ˆ u 0 (abc) . z 0 (acu) : u . u 0 h ˆ v 0 (abc) . v 0 (acz) : v . v 0 k :: z ˆ y 0 r - (zraz) - zr - z . () . ( ) : y . zr -0 (acz) [Assume the hypothesis of P4, let z 0 (abc), set zr  z, suppose that conditions u 0 (abc) and z 0 (acu) together always imply u 0 h, and suppose that conditions v 0 (abc) and v 0 (acz) together always imply v 0 k. Then, if there is a point y 0 r – (zraz) – {zr, z}, it will follow that zr ó (acz) by () and ( ).] The desired result follows from this by elimination of y : Hp . () . P1 . P1,4,24§5 : . Th] [The hypothesis, (), P1, and P1,4,24§5 imply the desired conclusion. The proof of P11 is complete.]

P12–Theorem. Again under the hypothesis of P4, it is necessary that there should exist within the projective segment (arbrcr) a point z coinciding with its transform z and, moreover, such that each point u of (abc) should be distinct from its transform u whenever z should always belong to the segment (acu). Proof. ()

Hp . z 0 (abc) . zr  z ˆ u 0 (abc) . z 0 (acu) : u . u 0 h ˆ v 0 (abc) . v 0 (acz) : P10,11 :: z ˆ z -0 (aczr) . zr -0 (acz) :c: z -0 (aczr) . z = zr :c: zr - (acz) . z = zr :c: z = zr ˆ  z, z  P1 .  d, e  P5§6 :: z : z = zr . P1 :: z : z 0 (arbrcr) . z = zr

v

. v0k ˆ

[Assume the hypothesis, let z be a point in (abc), and set zr  z, so that zr 0 (arbrcr); hence, zr 0 (abc). Suppose that conditions u 0 (abc) and z 0 (acu) together always imply u 0 h, and that conditions v 0 (abc) and v 0 (acz) together always imply v 0 k. Then, P10 and P11 imply that one of the following four cases holds: z ó (aczr) and zr ó (acz), or z ó (aczr) and z = zr, or zr ó (acz) and z = zr,  z, z  or z = zr. By  d, e  P5§6 it follows that z = zr, and hence also z 0 (arbrcr).] The first of the three successive deductions with respect to z proceed[ed] by conjoining P10 and P11 after transferring the clause z = / zr from the hypothesis to the conclusion in each of them, conforming to the rule ((P & Q) | R) | ( P | (R w ¬ Q)). On the other hand one has ( ) Hp ˆ u 0 (abc) . z 0 (acu) : 100

[Pieri’s word was sommando.]

u

. u 0 h ˆ P4 :: z ˆ

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6 Pieri’s 1898 Geometry of Position Memoir

u 0 (abc) . z 0 (acu) . ur  u : P1,4§5 : u . ur -= u

u

. ur 0 (acu) .

[Assume the hypothesis and let z be any point. Suppose that conditions u 0 (abc) and z 0 (acu) together always imply u 0 h. By P4, those conditions imply that the transform ur  u belongs to the interval (acu); hence, ur = / u by P4§5.] Now, conjoin () and ( ): ( )

Hp . z 0 (abc) . zr  z . () . ( ) ˆ u 0 (abc) . z 0 (acu) : u . u 0 h ˆ v 0 (abc) . v 0 (acz) : :: z :: z 0 (arbrcr) . z = zr ˆ u 0 (abc) . z 0 (acu) . ur  u : u . u -= ur

v

. v0k

[Assume the hypothesis, let z be a point in (abc), and set zr  z. Suppose that conditions u 0 (abc) and z 0 (acu) together always imply u 0 h, and that conditions v 0 (abc) and v 0 (acz) together always imply v 0 k. By (), z 0 (arbrcr) and z = zr. By ( ), conditions u 0 (abc) and z 0 (acu) imply that u is different from its transform ur  u.] Hp . ( ) . P9 : . Th] [The hypothesis, ( ), and P9 imply the desired conclusion.] With this is to be considered another [result]: P13–Theorem. Let r be a projective line,  be a segmental function from r to itself, a, b, c be three distinct points on r and ar, br, cr their transforms by , and further, suppose segment (arbrcr) should lie with both of its ends in segment (abc) in such a way that ar should fall in (accr). Then it is necessary that there should exist a point z in segment (arbrcr) coincident with its transform  z, and moreover such that each point u of (abc) should be distinct from its transform u whenever z should belong to the segment (acu). Proof. This could be demonstrated similarly to P12; but in what follows it is not necessary to use it.101

§ 10 Harmonic Correspondences and S TAUDT ’s Theorem 102 The notion of the harmonic after three given points (section 6.4) leads directly to a remarkable class of segmental transformations from a projective line to itself. Indeed, 101

Compare Enriques 1894, §10. The bijective transformations studied there under the name “corrispondenze ordinate” [ordered correspondences] can be reduced to segmental functions, but not to the most general among them.

102

Compare Pieri 1896b, §14.

6.10 Harmonic Correspondences and Staudt’s Theorem (§10)

207

the symbol Arm(a, b, x), where a and b should be projective points given at will, but distinct from each other, is defined by P13,16§4 for any point x whatever on the line r  ab, and represents a point varying with x on this line: according to section 6.9, that sign thus denotes a transformation from r to r, if x should denote a variable point on ab. P1—Theorem. If a and b should be distinct points of a projective line r, then Arm(a, b, r) will be bijective: a segmental transformation from r to itself.103 Proof. Bijective by P13,12,10,16§4; segmental by virtue of P13,6§4, P12§5, P1,3§9, and so on. P2—Theorem. And if x is an arbitrary point of r, then Arm(a, b, Arm(a, b, x)) always coincides with x. Thus, the transformation Arm(a, b, r) is not distinct from its —— inverse Arm(a, b, r). Proof. This [follows] from P12,10§4, and so on.—Two univocal representations ,  of the same class r in [a class] rr are said to be equal to each other if the images x, x of the same individual x in r should always be equal to each other in rr. The correspondence Arm(a, b, r) described here can be constructed by the following well-known [procedure]. Having selected a point u not on r and a point ar on the line bu different from b and from u, let a variable point x of r be projected onto the line aar from the center u, and [let] its image be called v. Next, let point v be projected from b to [the image] br on line au, and point br from ar to the [image] xr on r. [Then] xr will be the transform of x (and vice versa).— A remarkable property is that two corresponding points can never separate two others; that is to say, P3—Theorem. In the hypothesis of P1, if x and y should be points distinct from each other and from each of the points a and b, and if one sets xr  Arm(a, b, x) and yr  Arm(a, b, y), then point yr will lie in the segment (x y xr). Proof. Indeed, points a, b, x, y, xr, yr must all be distinct by P10,13§4, and xr ó (axb) by P23§5; [these conditions] entail b ó (x a xr) by P9§5, and thus the disjunction y 0 (x a xr) or y 0 (x bxr) by P13§6. Now, by P16§4 and P2 the segmental transformation Arm(a, b, r) maps104 the points a, b, x, xr, y to points a, b, xr, x, yr, respectively. Thus, from y 0 (x axr) follows yr 0 (xra x) by P1§9, and consequently yr 0 (x y xr) by P2,28§5; and the same [result] is implied by the supposition y 0 (x bxr). P4—Theorem. Given on a projective line r distinct points a, b, c, and a fourth point d that should lie in segment (acb) without falling on c, the existence of a point x can be established that satisfies this proposition: “x is a point of r and Arm(c, d, Arm(a, b, x)) coincides with x.” Or, in common language: “If two pairs (a, b) and (c, d) of distinct points are not separated, there exists at least one pair 103

[Pieri denoted by Arm(a, b, r) the transformation that assigns to each point x 0 r the point Arm(a, b, x).]

104

[Pieri’s word was cangia.]

208

6 Pieri’s 1898 Geometry of Position Memoir

of points that separates them both harmonically.” Proof. This is already stated in P1§5, as has been remarked elsewhere.105 P5—Definition. Let r, rr be two projective lines. A univocal representation  of r in rr is then called harmonic when (1)  is a bijection from r to rr and (2) however points a, b, c be selected on r, as long as a and b should not coincide, to the harmonic after a, b, c always corresponds the harmonic after a, b, c in rr: thus, points  Arm(a, b, c) and Arm(a, b, c) coincide. In sum, “harmonic transformation from r to rr” is synonymous with “bijection from r to rr that always associates four harmonic points to four harmonic points.” 106 P6—Theorem. The inverse of any harmonic transformation is a harmonic transformation. P7—Theorem. Every harmonic transformation is also segmental. Proof. Indeed, [given] three distinct points a, b, c selected at will on r, and in the segment (abc) a point d not coincident with b, one can always affirm the existence of at least two points x, y, harmonically conjugate with respect to both pairs (a, c) and (b, d), by P4. Now, were it the case that d ó (a b c), there could not exist two points harmonic with respect to both a, c and to b, d, by  xa,, yb, , xc,, yd  P3.107 But by P5 two such points certainly do exist: the images of x, y. Thus, one is forced to conclude that d 0 (a b c). See P1§9. P8—Theorem. Let r, rr be two projective lines and , a harmonic transformation from r to rr. If there exist three distinct points in r, each of which should coincide with its image under , then each other point of r will coincide with its image. Thus, any harmonic transformation that admits three distinct fixed points is an identity. This [is] the theorem of G. K. C. von STAUDT. Proof.108 Let e, f, g be points of r, distinct from each other and such that the points e, f, g should coincide with e, f, g, respectively. Having selected at will a point a on r, and denoted by ar its image under  (thus setting ar  a), let us suppose that points ar and a should not coincide: we shall show that from this follows an absurdity. Points e, f, g, a, ar will be distinct from each other, and the last two not separated by the others, by P7. It will be permissible to suppose that a ó (e g f ), and consequently ar ó (e g f ) and (ea f ) = (earf ), by P12§6. 105

[See the paragraph after P12§5.]

106

In both parts (1) and (2) there is something superfluous: something that, given our premises I–XVIII, is a consequence of the others. But here is adopted the celebrated definition of STAUDT without making it the object of analysis.

107

[Pieri omitted the substitution symbol from the P3 citation.]

108

[Pieri signaled the beginning of this proof by the abbreviation Dm, instead of merely by a square bracket. Dm is presumably his abbreviation for Dimostrazione; he did not include it in the list of such abbreviations in section 6.0, page 146. This proof extends to the end of 6.10. Its final paragraph has been split into three parts, rearranged slightly to enhance clarity, and some necessary justifications have been inserted.]

209

6.10 Harmonic Correspondences and Staudt’s Theorem (§10)

Under these assumptions, let c and cr be the harmonics of a and ar with respect to e, f, so that cr = c by P5. Points c and cr will be different from each   other and from the preceding [points]; moreover, P23§5 and  ae,,bf,,xa,, ya,,xc,, cy P3 yield c ó (eaf ), cr ó (earf ), cr 0 (aarc), [and e ó (af c) by P9§5]. Either ar 0 (aec) or  f , e, a  ar 0 (afc), by  b, d, e  P13§6. It will suffice to consider just one of these hypotheses, for example the first, since interchanging the letters e, f has no effect at all on the hypotheses nor on the results [of the following argument]. Now, from ar 0 (aec), cr 0 (aarc), and P28§5 it follows that (aec) = (aarc), cr 0 (aec), and consequently c 0 (eacr) by P7§5. From this, because c, cr ó (eaf ), it follows by P3§6 that cr ó (eac), and then cr0 (ace) by P14§5. Similarly, from ar 0 (aec) and P7,2§5 it follows that a 0 (ecar). But [by P5§5] on the other hand, a, ar ó (ecf ); and we infer from this [and P3§6] that ar ó (eca) and then [by P14§5] that a ar 0 (cae): that is to say, e 0 (acar) [by P7§5]. From all this and  d  P9§6 follows  a  (arecr) c { ar, cr} f (aec). Moreover, [ (ace) f (acar) ] by  d  P4§6; [therefore] cr 0 (acar) and consequently c 0 (arcra) by P6,2§5. The hypotheses of P12§9 about the transformation  and points a, e, c, ar, er, cr are thus verified, and therefore this condition on [a point] z will be satisfiable: z 0 (arecr) . z = z ˆ u 0 (aec) . z 0 (acu) :

u

. u -= u.

[ z 0 (arecr) , z = z, and for each u, if u 0 (aec) and z 0 (acu), then u = / u.] As a consequence, there will exist in segment (arecr) a fixed point z such that, if u 0 r – (acz) – { a, z}, then the point u surely is not fixed: for then, u 0 (cza) [by  c, z, u   b, c, d  P13§5; moreover, z = z 0 (aec) and P2,28§5 yield u 0(cza) = (azc) f (aec),    c, z, u  and thus] u 0 (aec), and finally z 0 (acu) by  b, c, d  P17§5.109 Let it be observed that the relations c ó (eaf ), cr ó (earf ), and (arecr) f (aec) imply (afc) c a c c f (arf cr) by P9§5 and P13§6. Thus, the same P12§9 can be invoked as well with the transformation – inverse to  (see P6) and the points ar, f r, cr, a, f, c. Therefore, there will exist in the segment (af c) a fixed point y such that, if u 0 r –(arcry) – { ar, y}, then the point u is not fixed. And since, from  a, f , c, a, f , c, y, u  (afc) f (arf cr), c 0 (arcra), and y 0 (afc) it follows, according to  a, b, c, a, b, c, d, e  P14§6, that u ó (acy) implies u ó (arcry), a fortiori no point of the figure r – (acy) – { ar, y} can be fixed. This fact and the similar one already observed about z, where it should be noted that, by the hypothesis, neither of the points a, ar should be fixed, permit the following assertion: u 0 r - (acz) - z .c. u 0 r - (acy) - y :

u

. u -= u.

[for each u, if u 0 r – (acz) – {z} or u 0 r – (acy) – { y} then u = / u .] It is not claimed that z should have to be different from e, or y from f . But the [fixed] points y, z are certainly distinct from a, c, ar, cr [because those are not fixed]; nor can it be that y, z coincide with each other, or z with f, or y with e, by 109

[Pieri also claimed that either of the alternatives ar 0 (aec) or ar 0 (afc) could be deduced from the other by interchanging e, f. The translators supplied the substitution symbols that Pieri omitted from the P3, P13§6, P9§6, and P17§5 citations.]

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P32§5. Now, from z 0 (aec) follows z 0 r – (af c), and on the other hand we have y 0 (af c). Therefore, z 0 r – (ayc): that is to say c 0 r – ( yaz), and consequently c 0 (azy) and (azy) = (ac y), as well as c 0 (ayz) and (ayz) = (acz). But by P21,4§5 it is known that u 0 ( yaz) .

u

: u 0 r - (az y) -  y .c. u 0 r - (z ya) - z ;

[for each u, if u 0 ( yaz) then u 0 r – (azy) – { y} or u 0 r – (z ya) – { z}; ] therefore, u 0 ( yaz) .

u

. u -= u.

[for each u, if u 0 ( yaz) then u = / u.] Since e = e, the paragraph before last yields e = z or else e 0 (acz). In the  e, z  former case, ( yaz) = ( yae). In the latter, z ó (ace) by  d, e  P2§6; moreover, y ó (aec) by P32§5 because y 0 (af c); and P13,28,2§5 yield y 0 (eca) and (ace) = (e ya) and hence z ó (eya). Continuing for this case, e ó ( yaz) by P11§5, and hence ( yaz) f ( yae) by P29§5. Thus, this inclusion holds in any case. Similarly, f = y or else f 0 (ac y) because f = f. In the former case, (eay) = (eaf ). In the latter, y ó (acf ) by P7; moreover, c 0 (aef ) by P16§5 because c ó (eaf ), and P28§2 yield (aef ) = (acf ) and y ó (fea). Continuing for this case, f ó (eay) by P11§5, and hence (ea y) f (eaf ) by P29§5. Thus, this inclusion holds in any case, too. [The two preceding paragraphs and P2§5] entail that segment ( yaz) is always included in segment (eaf ): thus, from the supposition a ó (egf ), or rather g ó (eaf ) [by P5§5], introduced at the beginning, it follows that g ó ( yaz). [Moreover, g = / y: otherwise, g ó (eaf ) and  ea,,ab,, fc,, dg  P17§5 would imply f 0 (eag) =   (eay) f (eaf ), contradicting P4§5. Also, g ó (eaf ) and P2§5 imply g ó (eay) = ( yae), and hence e 0 ( yag) by  ay,, ab,, ce,, dg  P17§5. It follows that g = / z: otherwise,   e 0 ( yaz) f ( yae), again contradicting P4§5.] Consequently, Arm( y, z, g) 0 ( yaz) by P23§5. This is precisely the absurdity toward which [this proof ] was directed: that, although it is apparent that no point of the segment ( yaz) could be fixed, nevertheless it is deduced here that the harmonic of g with respect to y and z, which by P5 and P12,13§4 is certainly fixed, lies in this segment.

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§ 11 Projective Hyperplanes of the Third Species and Ordinary Space 110 P1—Definition. If a, b, c, d are projective points, then the phrase a, b, c, d are coplanar is understood to affirm the existence of three noncollinear points x, y, z such that the points a, b, c, d belong to their plane xyz.111 Thus (referring to P14§3), the proposition “a, b, c, d are coplanar” proves to be equivalent to this alternative: there exists a projective plane in which lie the points a, b, c, d. From this definition and from the principles developed in sections 1–3 (that is, from postulates I–XII) stem the following P2–P6, which through many analogies resemble certain other propositions about the incidence relationship (section 6.2). P2—Theorem. If a, b, c, d are projective points and a, b, c are not collinear, each of the two propositions “d belongs to the plane abc” and “a, b, c, d are coplanar” is a consequence of the other. Compare P2,3,5§2. P3—Theorem. If a, b, c, d should be noncoplanar points, they will also be three by three noncollinear. Compare P6,7§2. P4—Theorem. And each one [will be] external to the plane that joins the remaining three. Compare P8§2. P5—Theorem. Let a, b, c, d be noncoplanar points and e be a point noncollinear with points a and b. Then points a, b, c, e cannot be coplanar on the one hand and points a, b, d, e on the other. Compare P11,12§2. P6—Theorem. Four points a, b, c, d are certainly coplanar if there should exist a point e for which a, b, c, e should be coplanar, as well as [the quadruples] a, b, d, e and a, c, d, e and b, c, d, e. Compare P13,14§2. Given noncoplanar points a, b, c, d a new figure is also determined by them: the visual of the plane bcd from the point a (see P3, P7§2, P1§3). Thus (compare P6§3): P7—Theorem. If a, b, c, d should be noncoplanar projective points, the figure aAbcd — the join of a with bcd —is none other than the locus of a projective point x for which this condition is satisfied: there exists a point y on the projective plane bcd such that x belongs to ay. Proof. The hypothesis, thanks to P3,2 and P6§3,

110

[The title phrase is defined on the next page. Pieri used the word specie with the general meaning, type.]

111

[Pieri also defined here the verb compianare, meaning to be coplanar, and the abbreviation Cp for are coplanar, which this translation does not employ.]

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implies that b, c, d should not be collinear, and that each [point in] bcd should be  distinct from a: for this it suffices to cite  a,pbcd ,   P1§3. For an arbitrary figure aAbcd constructed as just described (to which can be attributed from now on the term three-dimensional linear space, or projective hyperplane of the third species, as in P16) are derived, independently of any new principles, properties entirely similar to those already observed in section 6.3 with regard to the projective plane, and which, just as those, rely principally on postulate XII. P8—Theorem. If a, b, c, d are noncoplanar points, each of the planes abc, abd, acd, bcd will be contained in aAbcd. Compare P7§3. Proof. That bcd f aAbcd is  confirmed by  a,pbcd ,   P3§3, seeing that by P3,4 and P7§3, b belongs to bcd without coinciding with a. And because bc f bcd, one will be able to assign to each point x of the plane abc a point y in bcd such that x should belong to ay, by P6§3. Thus, abc f aAbcd, and so on. P9—Theorem. Under the same hypothesis as P8, in order that a point e different from a should fall in aAbcd, it is required, as a necessary and sufficient condition, that the plane bcd and the line ae should meet. Compare P8§3. P10–Theorem. And if, under the same hypothesis of P8, e should be a point of aAbcd, but different from b, the line be will be entirely contained in this figure. Compare P10,12§3. Proof. Select a point ar on bcd in such a way that e 0 aar (see P7);   and a point cr in cd in such a way that ar 0 bcr, conforming to  ba,,cb,,dc  P6§3 and P3. Then, bcr f bcd, and consequently a, b, cr are not collinear (see P4, P9,5§2), abcr f aAbcd by P1§3, and moreover e 0 abcr by P6§3. [The result] reduces to  e, c   d, c  P12§3.   P11–Theorem. As before, a, b, c, d being noncoplanar projective points, all four of the figures aAbcd, bAacd, cAabd, dAabc coincide. Compare P11,13§3. Proof. For an arbitrary point e of aAbcd [and a point cr selected as before], abcr f aAbcd, e 0 (abcr), and then bacr f bAacd, and moreover e 0 (bacr) by P11§3. Therefore, e 0 bAacd. And so on.112 P12–Theorem. Given a, b, c, d as above, and having selected in figure abcd two points e, f at pleasure but not coincident, their join ef will fall entirely in abcd. Compare P20§3. Proof. Let er, f r be two points in bcd for which e 0 aer and f 0 af r (see P7). If er = f r and consequently ef = aer by P4, P25§1, [the result], by P8, reduces to P10. But if er = / f r, the conditions erf r f bcd, a, er, f r being noncollinear, aerf r f abcd, and ef f aerf r all hold by P20§3, P5§2, P1§3, and P7,20§3; therefore, ef f abcd.

112

[P11 enabled Pieri to omit the white spaces, represented here by dots, when referring to figures defined in P7: for example, instead of a bcd he could write simply abcd.]

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P13–Theorem. Having assumed that a, b, c, d should be noncoplanar projective points and e, f, g noncollinear points of the hyperplane abcd, the plane efg will lie entirely in abcd. Proof. In the plane efg select a point x at pleasure, and in the line fg (see P6§3) a point y such that x 0 ey. Now, fg f aAbcd by P12 and P7§2 and consequently y 0 abcd. Therefore, by P9§2 one can refer again to P12 and conclude that ey f abcd and hence x 0 abcd. P14–Theorem. Given a, b, c, d as above, if e, f, g, h are noncoplanar points that belong to abcd, figures abcd and efgh coincide. That is as much as to say, each hyperplane of the third species is determined by four of its points, chosen at pleasure but not coplanar. Compare P19§3. Proof. Indeed, from x 0 efgh, whatever x might be, follows the existence of a point y such that y 0 fgh and x 0 ey, by P7. Now this y, and consequently also ey and x, will lie in abcd by P13,4,12; therefore, efgh f abcd. For the reverse inclusion, observe first that it is permissible to suppose that all three quadruples e, b, c, d and f, e, c, d and g, f, e, d are at once noncoplanar, because at least one of the points e, f, g, h, for example e, falls outside plane bcd by P1,3; moreover, for the same reasons (considering also P2 and P7§3) at least one of the remaining f, g, h, for example f, will be external to plane ecd, and one of the remaining g, h, for example g, external to plane efd. Additionally, let it be noticed that from p 0 q A rst it follows that from [a condition] p 0 qrst follows q 0 prst, provided that q, r, s, t as well as p, r, s, t should be noncoplanar projective points: in fact, from “z 0 rst and p 0 qz” follows “z 0 rst and q 0pz” in view of P3,4 and P16,10§1. From all of the preceding stem the following conditions in sequence, having regard also for P7§3 and P11,8:  0 ebcd, abcd f ebcd, f 0 ebcd, f 0 becd, b 0 fecd, becd f fecd, abcd f fecd, g 0 fecd, g 0 cfed, c 0 gfed, cfed f gfed, abcd f gfed, h 0 gfed, h 0 dgfe, d 0 hgfe, dgfe f hgfe, abcd f hgfe. Thus stands proved the equation abcd = efgh. P15–Theorem. The hypothesis of P8 standing firm, if e, f, g should be noncollinear points and h, i noncoincident points, and if the former as well as the latter should belong to abcd, then the plane efg will have to share at least one point in common with the line hi. Proof. If at least one of the points h, i should lie on the plane efg, the result will be [seen to be] true immediately. If on the contrary each one of the points h, i will be external to the plane efg, certainly h, e, f, g will not be coplanar, by P2,1, and thus abcd = hefg by P14 and consequently i 0 hefg. But by P7, this serves to affirm the existence of a point y such that y 0 efg and i 0 hy; that is to say, a point y common to figures efg and hi, by P14§1 and P4.—From this follows easily yet another P16–Theorem. If a, b, c, d, e, f, g are as above and h, i, l are noncollinear points of abcd, there will always exist a projective line common to the projective planes efg and hil. P17–Definition. The projective hyperplanes of the third species constitute the class, denoted by [3], of all figures satisfying the following condition on : there exist four noncoplanar projective points a, b, c, d, and  coincides with abcd. In sum:

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each [member of] [3] is none other than the visual of a projective plane from a projective point external to it. The converses of P15,16 are worth noting—that is, P18–Theorem. [Any] two projective lines that should not meet, or a projective plane and projective line that should have one but not more than one point in common, or two projective planes that meet along a line without coinciding, should always lie in a single hyperplane of the third species. Propositions 1–18 of this section are independent of postulates XIII–XVIII: but once XIII is admitted, one will not be able to deny the proposition that there should exist at least one [member of ] [3]. Still, one would not be able to assert or deny on the basis of XIII, nor any other principle among those accepted until now, the existence of several figures such as abcd understood not to be coincident with each other, a, b, c, d being noncoplanar projective points. Resolving this kind of question [is up] to our will, still undecided: whether or not the class of projective points is thus exhausted by a single hyperplane of the third species. We shall entertain both hypotheses. At this point it can be said that the first, as that which brings the concept of “class of projective points” closer to that of “physical space,” is more appropriate to the scope of ordinary projective geometry, whereas the other one prepares the ground for a much more general geometry. (See section 6.12.) POSTULATE XIX P19. If a, b, c, d are noncoplanar projective points, then for each projective point e not situated in any of the planes abc, abd, acd, bcd there will have to be at least one point common to the figures ae and bcd. P20–Theorem. Should b, c, d be noncollinear points and a, e noncoincident points, the plane bcd and the line ae will meet. Compare P15. Proof. If at least one of the points a and e will lie in bcd, the result is established immediately. If a, e ó bcd, but e lies in at least one of the planes abc, abd, acd, the join ae will meet bcd on one of the lines bc, cd, bd thanks to P8§3 and P3. The remaining [case] is just the hypothesis considered in P19. In sum, it is proved that “a projective plane and a projective line always have at least one point in common.” But the reader must not lose sight of the difference between this P20 and P15—independent of postulate XIX—which can be rephrased like this: “a line and a plane lying in the same hyperplane of the third species always have at least one common point.” Postulate XIX has the effect of rendering superfluous the restriction in italics; and this happens, as will be seen, in very many cases.— The following theorem brings into view the role of postulate XIX in constraining the projective ambient (or class of projective points) within the confines of one hyperplane of the third species.

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P21–Theorem. If a, b, c, d are noncoplanar projective points, then every projective point will have to lie in abcd. Thus, projective space coincides with abcd: [0] = abcd. Proof. That a 0 abcd is already known. But for each other projective point e with e= / a the existence of a point common to the figures ae, bcd can be asserted, by P20: that is, the point y that satisfies the condition “y 0 bcd and e 0 ay” (see P7). P22–Theorem. Given two projective planes at pleasure, there is always a projective line that lies in both. Proof. Let b, c, d be noncollinear points in a projective plane , and a, e, f, points in a plane r. Under the assumption  = / r (the case where  = r is omitted), at least one of the points a, e, f, for example a, will be external to . Therefore, the lines ae, af (see P7§2) will meet  (by P19§3, P20) at points er, f r necessarily distinct from each other and from a (by P25§1, P10§2). Now, the line erf r will be common to planes aef and bcd (by P20§3). Proposition P16, “two projective planes lying in a single hyperplane of the third species always have a projective line in common,” is true even if abstraction should be made from postulate XIX,113 but its content is then quite different from that of P22. As already revealed by P22, this last postulate also has the effect of establishing a perfect duality between “projective point” and “projective plane,” according to which each proposition that might be a consequence of the primitive propositions I–XIX finds a complement in another, that stems from it by interchanging the entities [0] and [2] and at the same time “join of two distinct points” with “pencil of planes containing the line common to two distinct planes.” The reason behind this fact is quite simple. An arbitrary figure is a system of projective points and of lines joining them, so that, through each particular interpretation that might be accorded the primitive entities “projective point” and “join of two distinct projective points,” all figures will acquire special significance through their definitions, and content having the same level of specific determination that is attributed to that interpretation (see section 6.1). The interpretations or specifications of the primitive entities are entirely arbitrary—on condition that it is agreed to take into account, in each single case, of only the propositions that stem from the premises verified in it. And each time a special significance should be attributed to each of the entities [0] and [1], in which postulates I–XIX should be satisfied without exception, one obtains an interpretation, or representation, of all of the ordinary projective geometry. It is exactly thus in the example under discussion, where the entities [0] and [1] are replaced by the entities «projective plane» and 31 «pencil of projective planes», and after that the entity [2], by virtue solely of its formal definition, acquires the significance of «projective point», while «pencil of planes»

113

[That is, even if postulate XIX should not be assumed.]

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is converted in the same way into «projective line». Such reciprocity or symmetry, which is generally not encountered elsewhere,114 confers on that interpretation a special prominence. On that account, the law of geometric duality reappears as a much more general principle, not belonging to geometry more than to other deductive science: a principle on which might also be conferred the name law of plurality. P23–Definition. Each particular variation or interpretation of the primitive entities “projective point” and “join of two distinct projective points” that should comply with all of postulates I–XIX, is a representation of the ordinary projective ambient (or space) or, more briefly, an ordinary space. «Ordinary space» is the class of all possible interpretations under the dominion of these postulates. From this and from P21,17,7 it follows that each hyperplane of the third species is an ordinary space, and vice versa. But, even if only postulates I–XVIII should have been verified for the entities [0] and [1], it would nevertheless be possible to interpret an arbitrarily given hyperplane of the third species as ordinary space: it suffices for this to alter the generic content of [0] and [1] to “projective point and projective line lying in that same hyperplane.” See P15.

§ 12 Projective Hyperplanes of the n th Species and Absolute Projective Space 115 As we have mentioned, it remains now to examine briefly the hypothesis opposite to postulate XIX (P19§11), and therefore this must be entirely suppressed in the present section.116 Its negation will be accepted in the following form (compare P11§4): POSTULATE XIXr P1. If a, b, c, d are noncoplanar projective points, there exists at least one projective point not situated in a b c d. That is, outside a given hyperplane of the third species lies at least one projective point. Thus the visual of a [member of ] [3] from a point external to it will be a figure, whose nonemptiness117 is impossible to deny at present, although it has indeed been possible to consider and study it independently of postulates XIX and XIXr. 114

Equal reciprocity is manifested by the duality between point and line—that is, the interpretation of the entities “point” and “line” in a plane as “line” and “pencil of lines belonging to a plane”—and the duality between line and plane, or duality in a bundle. In a given hyperplane of the third species there is a duality between “point” and “plane” in the propositions derived from postulates I–XVIII that does not depend on postulate XIX: but this is not to be confounded with the other one.

115

See Pieri 1896c, §1–§4.

116

However, all the propositions before P19§11 can be invoked.

117

Pieri’s word was realità.

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P2—Definition. The class of all figures, each of which is the visual of some hyperplane of the third species from some projective point that does not belong to it, is called a “projective hyperplane of the fourth species,” and represented by [the symbol] [4]. Therefore, the proposition “ is a projective hyperplane of the fourth species”—  0 [4] —summarizes this other one: “there exist five projective points a, b, c, d, e not lying in the same hyperplane of the third species, but such that  = abcde.” With respect to this figure one observes properties very similar to P8–18§11: because of this analogy, it will be permissible to skip over them.118 P3—Theorem. Given an arbitrary line and plane, or a hyperplane of the third species and a line that meets it, or two planes having a common point, or a hyperplane of the third species and a plane containing a line in it, or two hyperplanes of the third species containing the same plane, there will always exist a hyperplane of the fourth species that contains them. Proof. The result depends on [postulate] XIXr inasmuch as [the case where] that arbitrary line and plane should meet is not excluded. And so on. P4—Theorem. Vice versa, two hyperplanes of the third species, a hyperplane of the third species and a plane, a hyperplane of the third species and a line, [or] two arbitrary planes have in common at least one plane, line, or point, respectively, as long as they lie in the same hyperplane of the fourth species. And if in the end we should accept the following: POSTULATE XXr P5. Let a, b, c, d, e be projective points not lying in the same hyperplane of the third species, and f [be] a projective point not lying in any of the figures a b c d, a b c e, a b d e, a c d e, b c d e. [Then] the hyperplane b c d e will always meet the line a f. (Compare P19§11.) it will have as a consequence P6—Theorem. Under the same hypothesis as P5, the figure abcde will coincide with projective space: abcde = [0]. Thus, the class of [all] projective points will be a hyperplane of the fourth species, and vice versa. (Compare P21§11.) Now, from the preceding P4 one can remove the restriction “as long as they lie in the same hyperplane of the fourth species.” Moreover, the entities “hyperplane of the third species” and “pencil of hyperplanes (of the third species) containing a specific plane” will be interpretable as “projective point” and “projective line,” thus establishing a perfect correspondence or reciprocity between the entities [0], [1], [2], [3] on the one hand and the entities [3], [2], [1], [0] on the other. With 118

See for example Veronese 1881.

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postulates I–XVIII, XIXr and XXr we shall have, in short, the projective geometry of four-dimensional space.119 [There is] no impediment to proceeding in this manner as far as one wishes. For five-dimensional projective geometry it would suffice, for example, to remove postulate XXr and assume instead two new principles XXO and XXIO, similar to XIXr and XXr. And thus, by means of a finite number of postulates, precisely n + 16, is given the possibility of defining projective space as hyperplane of the nth species (n being a whole number larger than 2): in that are always included [definitions of ] hyperplanes of the classes [3], [4], [5], ... [n – 1]. And so on. But one can also continue without end this sequence of successive extensions of the class [0]: in this way there results a class containing hyperplanes of arbitrarily large species. One arrives at this by grace of the logical principle of complete induction (or rather, deduction), assembling in a single proposition the definitions of all the entities [2], [3], [4], ... , and thus immediately all the existential postulates XI, XIII, XIXr, XXO, ... . Alongside the definition of the entity [1] (see P15§2) place this one: P7—Definition. Assuming that n should be a whole number larger than one, [n] or «projective hyperplane of the nth species» refers 31 to the class all figures such that for any one of them, let it be x for example, another figure y of the class [n – 1] and a point a outside of y can be found so that x should coincide with the visual of y from a. —This and P15§2 constitute together an inductive definition,120 which completely determines the meaning of the symbol [n], where n is a positive integer given at pleasure. And it is a strictly nominal definition, because it provides in succession the precise meaning of the terms [1], [2], [3], [4], ... up to a number n as large as desired. By this means [one can state] the following: POSTULATE XIr P8. If n is a positive whole number and  is a [member of] [ n], there exists at least one projective point that does not belong to . [This] collects all the existential postulates XI, XIII, XIXr, XXO, assembling them under a single formula,121 and can be used in place of them. Thus, it will be affirmed immediately that each class [n] is nonempty; thus: 119

Here, as always, the term “space” is reserved to denote the largest figure, the class of all projective points (see §1). The [notion of] four-dimensional space presupposes postulate XXr: not just a hyperplane of the fourth species.

120

[Definition] of the second species, according to the classification of Professor Cesare BURALI-FORTI (1894a, 100–103, 126–128).

121

This happens not differently, for example, in the arithmetical axiom “If a is a number, the successor of a is a number,” which can dissolve into infinitely many propositions: “If 1 is a number, the successor of 1 (thus 2) is a number; if 2 is a number ... .”

6.12 Projective Hyperplanes of the nth Species and Absolute Projective Space (§12)

219

P9—Theorem. Provided that n should be a positive whole number, there will always exist some [member of] [n].” Proof. Suppose n > 1 and that there should exist at least one [member of ] [n – 1]. Then the hypothesis y 0 [n – 1] is satisfiable, and thus by P8 also the proposition a 0 [0] – y; therefore, by P7 the condition x 0 [n] will be satisfied by x  ay. On the other hand, it is true by P6§1 and P15§2 that a projective line is nonempty. Thus, the desired result, true for n = 1, will have to hold for arbitrary n, thanks to the principle of complete induction.122 The class [0], for which postulates I–X (§1), XIr (§12), XII (§3), XIV (§4), XV– XVII (§5) and XVIII (§9) are conceded (and on the contrary postulates XI (§2), XIII (§4), XIX (§11), XIXr, XXr, ... (§12) are denied) can be called the absolute projective ambient or general space. The geometry developed from these principles is absolute projective geometry.123

122

Vice versa, from this result follows postulate XIr by P7: thus, it could be substituted for that.

123

The first to speak of space unlimited even in the number of dimensions seems to have been Professor Giuseppe VERONESE (1882, 211). Nor is there lack of interpretations of the undefined entities that satisfy all these postulates. For example, denote by f (n) an arbitrary function of a positive integral variable [with] a single real value, finite and not always zero, and by f1(n) the class of all functions such as af(n), where a represents a parameter capable of assuming any real value, finite and nonzero. Now, the class of all possible such f1(n) will be an interpretation of the aforementioned class [0], and thus an absolute projective space. In this way a projective point can be conceived as a class of 41 sequences af (1), af (2), af (3), ... , af (n – 1), af (n), af (n + 1), ... , ad infinitum where f is understood to have a specific form: the numbers af (1), af (2), af (3), ... (determined only up to a common factor a) can be called successive coordinates of the point f1 . Two projective points 1 and 1 will be distinct from each other (see §1), if the functions (n) and (n) are linearly independent. [In that case] an entire class of projective points a( + ) is determined by them (  and  [being] real parameters, finite and not simultaneously zero), which can be regarded as the join of 1 with 1 . A point 1 will lie outside this join, if the functions (n), (n), (n) should be linearly independent: and then the visual of [the class of points] a( + ) from the point 1, that is to say the plane of points 1 , 1 , 1 , will be seen to be the class (of classes) of functions a( +  + ), where , ,  should be real parameters, finite and not allowed to be zero all at once. An individual (projective point) of this class is given, for example, by the ratios of two of the numbers , ,  to the remaining one, assumed to be nonzero. And so on. Here, postulate XIr resolves itself into the fact that given n + 1 linearly independent functions f (n), there always exists another one linearly independent from them. And it could also be proved that, given two distinct points 1 and 1 and a point 1  a(r + r) different from both, the projective segment (1 1 1 ) (see section 6.5) is to be regarded as consisting of all and only those points of the line  + , for which the ratio /  takes finite positive or negative values respectively, according to whether r/ r is positive or negative. An even simpler example of a general space is that furnished by the complex of all entire algebraic equations with real coefficients (in a single unknown). And so on. Compare Pincherle 1897, chapter I. [In his §15, Salvatore Pincherle described the linear space of all polynomials f (n). Pieri was evidently referring to the space of all equations f (n) = 0.]

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6 Pieri’s 1898 Geometry of Position Memoir

A P P E N D I X

On the Independence of Postulates I–XIX Given several conditional propositions P(x, y, z, ...), Q(x, y, z, ...), R(x, y, z, ...), and so on, about variable entities x, y, z, ... no doubt can fall on the meaning of the assertions “from P and Q one cannot deduce R,” [and] “R is not a consequence of P and Q,” because this emerges immediately from the common logical significance of the phrases “one can deduce” and “is a consequence of” (see section 6.1, page 151, and 6.9, page 195, footnote). Both of these express no more than this specific proposition: “there exist some x, y, z, ... for which P and Q are true, but R is not true.” With this understanding, propositions P, Q, R, ... will be called “independent of each other” if it should happen that no one [of them] should be a consequence of those remaining: that is, for each one it should prove possible to find some x, y, z, ... that make it false, but that do satisfy the others. Now, the postulates of a deductive science are precisely the conditions imposed on its undefined entities. Therefore, to demonstrate the absolute independence of our postulates I–XIX, for example, it would suffice to present nineteen interpretations, or specifications of the entities “projective point” and “join of two projective points,” for each of which there remains unsatisfied just one of these principles. (Even more, this is the method that is presented more directly by the flow of logic—as it is apparently also the only one that might be known with similar purposes.) Results of this type have until now only been achieved in a very few cases, and only for rather more restricted groups of primitive propositions. Here I restrict myself to proving the ordinal independence of almost all of postulates I–XIX: that is, the independence of each of them from those preceding. That postulate II should not be a consequence of postulate I has been shown already in section 6.1. Now, if the primitive entity [0] is specified as a singleton (that is, a class of individuals all equal to each other), postulates I and II will be true, but III [will be] false: thus, it does not follow from those.—If, instead, [0] should be the class of positive whole numbers, and ab, for positive whole numbers a and b, should have its common arithmetic meaning, postulate IV will be false, with I, II, and III being true; but if ab should denote, for example, the class of rational numbers falling between a and b, the first four postulates will be true but postulate V false.—Assigning to [0] an interpretation at will but in accordance with postulates I, II, and III, if one interprets ab (where a, b 0 [0]) as a symbol assumed to represent the class {a} (see section 1), then postulates I–V will be satisfied, but not VI.—Then, if the class [0] should be specified as «Euclidean point», and 31 the class ab as the figure “Euclidean segment terminated by a and b but not containing the ends” (the class of points that lie between a and b), the postulates from I to VI will be satisfied, [with] VII remaining unfulfilled. If along with the interpretation [0]  «Euclidean point» one sets ab  {a, b}, postulates I–VII will be satisfied and VIII false; but if ab should denote the segment ab with its ends a and

6.13 Appendix

221

b included, the postulates from I to VIII will be true, but not all the way to IX. Similarly, if ab will stand instead for “the class formed by points a and b, their midpoint, and the reflections of each of a and b across the other,” the first postulates I–XI will prove satisfied but X unsatisfied.—If [0] should denote the class of whole numbers, and ab likewise, postulate XI will be the first not to be satisfied.—If [0]  «Euclidean point» and ab  “unbounded line passing through a and b,” this interpretation confirms the first eleven postulates, but is in disaccord with XII.—If [0]  «Euclidean line passing through a fixed point» and ab  “the pencil of lines in which a and b should lie” (the hypotheses a, b 0 [0] and a = / b understood) postulates I–XII are true, [but] XIII false. The possibility of satisfying at once the first thirteen postulates in such a way that XIV remains unfulfilled is proved by specifying the class of projective points as the figure (considered in Fano 1892, 114) of fifteen points, Euclidean for example, that arise from the four vertices a, b, c, d of a tetrahedron, an internal point e, the projections ar, br, cr, dr of those vertices from this point on the faces opposite them, and the projections of each triple of vertices, for example of a, b, c made on the edges bc, ca from point dr: these [last projections] will number precisely six on all the edges together. These fifteen points can be arranged in thirty-five triples in such a way that each point belongs to seven different triples, but two points always lie in just one: for example, if f, g, h denote the projections that fall on edges ab, bc, ca, then triples af b, bgc, cha, adrg, bdrh, cdrf, fgh will be found on the face abc. Now, giving the name “join” to the sets thus constructed from those fifteen points, postulates I–XIII are recognized as true and XIV as false, inasmuch as, for example, Arm(a, b, f ) coincides with f. Moreover, postulate XV proves not to be a consequence of I–XIV. Indeed, if postulates I–XV are assumed, given three collinear and distinct points a, b, c, it will be necessary that the point d  Arm(a, b, c) should lie in one or the other of the projective segments (abc), (bca). But in the [first] case, by P1§5 there certainly exists a pair x and y of points harmonic with respect both to a, c and to b, d, and points a, b, c, d, x, y are all distinct from each other: and the same happens if d 0 (bca). On the other hand (and this was also observed in Fano 1892, 116), one can construct a projective space with forty points and one hundred thirty joins, each one consisting of a set of just four distinct points harmonic with respect to each other in all possible permutations, in such a way that the complex of properties collected under postulates I–XIV are satisfied, and all this without ever finding in any line these six points a, b, c, d, x, y that XV would require. The independence of postulate XVI from its predecessors I–XV still remains uncertain. But the futility of each effort to deduce it from these, without proving the opposite, gives me a certain presumption in favor of its independence. And such a presumption is greater with regard to postulate XVII, which, given the preceding I–XVI, could be reduced easily enough to this form: “Let a, b, c, ar, br, cr be six points of a projective plane, all distinct from each other; provided that a, b, c as well as ar, br, cr should be aligned, the common points of the pairs of lines abr, arb and acr, arc and bcr, brc will also have to be collinear.” 124 Now 124

[This is the familiar Pappus–Pascal theorem: if the vertices of a hexagon lie alternately on two coplanar lines, then the intersections of the three pairs of its opposite edge lines are collinear. See the discussions in subsections 9.2.5, page 363, and 9.2.6, page 373.]

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6 Pieri’s 1898 Geometry of Position Memoir

this proposition (as far as I should know) has never been derived in the geometry of position except by deducing it from the fundamental theorem of G. K. C. von STAUDT: that is to say, by invoking as well the principle of continuity of a line, which here is postulate XVIII. A simpler demonstration was even tried from scratch by H. G. ZEUTHEN (1897), but without fruit. This postulate XVIII is certainly independent from its predecessors, in view of the wellknown possibility of a projective geometry with only rational points, where XVIII would be false, with the others, I–XVII, being valid. Finally, if by [0] is meant that class (of classes) whose generic individual is the class of all the complexes arising from (ax1 , ax2 , ax3 , ax4 , ax5) by varying the parameter a through real numerical values, finite and different from zero, after having given at pleasure real numerical values to the x’s, finite and not all zero; and if, given points x  (ax1 , ax2 , ... , ax5) and y  (bx1 , bx2 , ... , bx5) not equal to each other ( y = / x) one should set x y  ( c( x1 +  y1), c( x +  y2), ... , c( x5 +  y5)),125 where  and  should be real variables that are required to remain always finite and cannot be zero at the same time, and c [should be] a parameter like a and b; then one will obtain a representation of the projective geometry of four dimensions (and properly a system of five homogeneous coordinates), which contradicts postulate XIX (satisfying its negation XIXr, but fully conforming to the others, I–XVIII). To prove that all the premises of a deductive system should be consistent—that is, not contradictory with each other—it suffices [to give] a concrete example that should satisfy them all. Now, the complex of four variable homogeneous coordinates [described] as before satisfies all the postulates I–XIX: and really suffices, because the opposite proposition, or negation, of each of these may be shown to be independent of the others. And so on. Turin, October 1897.

125

[Pieri mistakenly wrote  x1 for  y1 here.]

7 Transformational Geometry Section 5.5 presented background information for Pieri’s use of transformational methods in his pioneering axiomatization of projective geometry, which is translated in chapter 6. Pieri went far beyond those methods in his axiomatization of elementary geometry based on point and motion, translated in chapter 8. Describing that work, he remarked, the systematic adoption on a grand scale of transformations of space and of their groups distinguishes and governs, one can say, all modern geometry.1

This chapter 7 presents additional material on the development of transformational methods up to Pieri’s time, to provide background for that study. Yet more material on Pieri’s use of those methods is to be found in subsection 9.3.1, which briefly summarizes Pieri’s Point and Motion paper, and in 9.2.6, which summarizes his 1898b axiomatization of projective geometry based on point and homography. The present chapter ends with a brief account of some other works in a similar vein during the next century, which stemmed from some of the same roots as Point and Motion but were not directly influenced by it. 7.1 Motions and Transformations Transformational geometry is an outgrowth of kinematics,2 the part of mechanics concerned with rigid motion, or movement of a rigid body, but not with its mass, nor with linear or rotational forces that alter its movement. Many studies of transformational geometry, especially in its earlier years, have used the term rigid body to describe a material object that cannot be forced out of shape. Movement of a geometric object can be specified, for example, by a Euclidean construction, by operation of a mechanical drawing device as proposed by René Descartes in 1637,3 or by transforming an analytic formula as directed by Leonhard Euler in 1776. According to the American mathematician Henry B. Newson, such a transformation is an operation which interchanges among themselves the elements of a space but leaves the space, considered as the aggregate of all its elements, unchanged as a whole. ... From the synthetic point of view, the phenomena of a transformation appeal directly to the eye or to the 1

Pieri [1900] 1901, §IV, translated in section 4.4 of the present book, page 65.

2

That name, as cinématique, was introduced by André-Marie Ampère (1834, 48). The term kinetics is often used for the extension of kinematics to include considerations of mass and force. For a comprehensive survey of late-nineteenth-century familiarity with kinematics see Schell 1870.

3

Descartes [1637] 1954, 50–51; Dennis 1977. Euler [1776] 1911– , 12.

© Springer Science+Business Media, LLC, part of Springer Nature 2021 E. A. C. Marchisotto et al., The Legacy of Mario Pieri in Foundations and Philosophy of Mathematics, https://doi.org/10.1007/978-0-8176-4823-7_7

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space intuitions. ... from the analytic point of view the operation is seen through the medium of a ... substitution on the requisite number of variables.4

In both of these views, differing attitudes emerged toward the definition and role of motion in geometry. The French mathematician Michel Chasles, a major nineteenth-century figure both in the history of geometry and in the development of algebraic geometry, contrasted the analytic and synthetic employment of geometric transformations. In a deep study of his historical work, the contemporary historian Karine Chemla has noted that representing the motion of geometric figures by algebraic substitution in formulas describing them allows different versions of a geometric phenomenon to be treated similarly. It suggests relationships between them and fosters generality. Chasles himself mused, In reflecting on the procedures of algebra, and investigating the cause of the immense advantages that it affords in geometry, does one not perceive that [geometry] owes a part of these advantages to the facility of the transformations that one can impose on the expressions that one first introduces there—transformations whose secret and mechanism constitute the true science and the constant object of the investigations of the analyst? Was it not natural to try likewise to introduce some analogous transformations in pure geometry, bearing directly on 5 the proposed figures and their properties?

Chasles invited his readers to consider a purely geometrical counterpart of the algebraic transformation of formulas: transmutation of figures directly, not via algebraic analysis. He credited his predecessor Gaspard Monge with this idea. Whereas earlier geometers had used transformations only to relate figures of the same type, Monge related figures of different types—for example, figures in two and three dimensions, and could transform a property of one into a wholly different property of the other—which might be easier to analyze. According to Chasles, this step also widened both the class of objects considered and the class of transforms used. Moreover, Chasles noted that transformation of formulas extends to the transformation of theorems, a tool that helps identify basic principles, and reveals the reasons for and connections between truths.6 It has been suggested that classical Greek mathematicians, reacting to the Eleatics’ famous discoveries of paradoxes involving motion, tried to eliminate that concept from geometry. As noted earlier,7 Euclid referred to motion only implicitly. His most notable references are to the method of superposition: placing one figure upon another to investigate their congruence. He routinely used some other words, such the Greek word for draw, that seem to suggest motion: drawing from one point toward another. Detailed study of the progress of a moving object over time was pursued in more advanced parts of mechanics, usually not regarded as part of geometry. But the border4

Newson 1902, 580–581. This is an excellent overview of transformational geometry in Pieri’s time.

5

Chasles [1837] 1875, §8, 196. This paragraph is based on Chemla 2016, particularly pages 62–64, 85–86. See section 5.6 of the present book for a discussion of the relationship between Pieri’s algebraic-geometry research and that of Chasles.

6

Chasles 128, 195, 261; Chemla 65, 86.

7

Szabó 1978, §3.28. See section 5.2, page 87.

7.1 Motions and Transformations

225

line between geometry and kinematics was vague. Many geometrical studies followed Euclid’s lead, minimizing the role of motion. In others, the progress of an abstract moving object played a vital role. Johannes Kepler and Jean-Victor Poncelet formulated continuity principles in that way.7 Those ideas underlay fundamental techniques in the new subject of projective geometry, which was studied both synthetically and analytically. Synthetic projective geometry used intuition as a guide and logic as the instrument by which its results are obtained. Analytic projective geometry employed the methods and machinery of algebra and analysis to demonstrate geometrical theorems. Euclid’s synthetic approach to motion involved only the initial and final positions of a moving figure, ignoring whatever path it might trace from the one to the other. Likewise, Euler, using complicated spherical trigonometry in 1776 to derive equations for the coordinates of the terminal position of a point that has undergone a rigid motion, explicitly noted that he was considering only the initial and final positions, not the path followed.8 A modern version of those equations is presented in the next section. Gradually, the conventional description of motion as a transformation became prevalent: only the initial and final positions were considered, whether or not coordinates were involved. And this approach was used in many areas of mathematics, not just kinematics. Section 5.2 showed that the transition was not sudden. For Jean-Victor Poncelet’s 1822 study the progress of a moving point was essential, and G. K. C. von Staudt followed that track in his flawed proof of the fundamental theorem of projective geometry in 1848. Two decades later, the French mathematician Guillaume-Jules Hoüel wrote, This geometric motion, which care must be taken not to confuse with motion in time, the subject of kinematics, cannot depend on any science but pure geometry. It is advantageous to introduce this idea of geometric motion as early and explicitly as possible. One gains from it a great measure of clarity and precision of language, and one finds oneself better prepared to introduce later into motion new notions of time and velocity.9

Felix Klein’s epoch-making 1872 comparative study of geometrical research,10 his Erlanger program, made no reference to the progress over time of a moving point. After Klein’s work, it became convenient to call transformational geometry that part of kinematics pursued in this way, as well as studies of analogous transformations in other geometrical contexts. However, it still required decades for Staudt’s fundamental theorem to be fully understood. Even in 1894, Giuseppe Peano felt it necessary to preface his own work on foundations of geometry with a comment very similar to Hoüel’s just cited words.11

8

Euler considered only motions with a fixed point. That is not a major restriction since any motion  can be regarded as a motion with a fixed point O followed by the translation that moves O to (O); the translation has familiar equations.

9

Hoüel 1867, 59–60. There was a typographical error in the 1867 edition, corrected and clarified in the 1883 second edition, page 70, from which this passage was translated.

10

Klein 1872, featured in section 5.5.

11

Peano 1894b, 75. See the introduction to the following chapter 8, page 247.

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7.2 Isometries and Similarities An isometry is a geometric transformation that preserves distances: for any points P and Q with images Pr and Qr, segments PrQr and PQ have the same length. Rigid motions, described informally in the previous section, are certainly isometries in threedimensional Euclidean geometry. They also preserve orientation: after a rigid motion, a right hand remains a right hand. Isometries that preserve orientation are called direct. Rigid motion can be defined more formally in three-dimensional Euclidean geometry as direct isometry. Some isometries preserve distances but reverse orientation. These are called indirect isometries. For example, when one stands at attention, the reflection  across the plane  perpendicular to the horizontal line segment between the shoulders at its midpoint changes the orientation of the right hand into that of the left. Applying it again changes that back to the original:  is a self-inverse isometry but not a rigid motion. In plane Euclidean geometry there are two types of direct isometries: translations V with a given vector V and rotations O,  through a given angle  about a given center O. The identity is both, with vector and angle zero. The indirect isometries are produced by composing a direct isometry with the reflection g across a line g. The only additional isometries produced this way are the glide reflections g V where V*g. Reflections play another fundamental role in plane Euclidean geometry, as basic components of all isometries, shown by the figure on the facing page. The translation V with vector V is the composition h k , where h and k are parallel lines and the perpendicular vector from k to h is ½ V. The rotation O,  is the composition h k where h and k are lines forming angle ½  from k to h. In particular, when h and k are perpendicular, h k = k h and this is the half-turn O about O. Thus, the direct isometries are compositions of two reflections, and the indirect isometries, of three. Also, the glide reflection g V is equal to g h k = g O where g z h, k and g 1 k = O. Thus, all plane isometries are compositions of two involutions—self-inverse transformations different from the identity. Isometries belong to a larger class of geometric transformations, the similarities. Euclidean geometry itself is often thought of as the study of geometric properties that are invariant under all similarities. A similarity with scale ratio r > 0 is a geometric transformation that multiplies all distances by the fixed number r: for any points P and Q with images Pr and Qr, the length of segment PrQr is r times that of segment PQ. If r = 1, the similarity is an isometry. A similarity with an arbitrary ratio r > 0 is produced by composing an isometry with the expansion or contraction by ratio r about a point O, which leaves O fixed but maps each other point P to the point on the ray from O toward P whose distance from O is r times that of P. Similarities that preserve orientation are called direct. Two figures are called directly or indirectly similar if they are related by a similarity of the appropriate type.

227

7.2 Isometries and Similarities

P

Pr= g P

P

Pr= k P

g

½V

k

Reflection g across line g

PO= h kP

PO= g k P

h

Translation with vector V: V = h k

g

Pr=kP PO= g V P= O V P

O

½

O

k

P P

Rotation through angle : O,  = h k

V

g

Pr= V P

Glide reflection g V = g O

Plane Isometries12 Euclid’s theory of congruence was based on one of his common notions: things that can be made to coincide are equal (in this case, congruent). Two figures could be called congruent if one could be moved to coincide with the other: that is, if they have the same shape and size. Using the technique of superposition, Euclid showed that two triangles are congruent just when their vertices can be labeled so that corresponding edges have equal length—that is, when they are related by an isometry. They are called directly or indirectly congruent depending on the type of the isometry. Using the theory of area, Euclid showed that the three pairs of corresponding angles are equal if and only if the pairs of corresponding edges have proportional lengths.13 Thus, two figures are similar if and only if they have the same shape, but not necessarily the same size. A similarity

12

As shown by the figures F, reflections reverse orientation; compositions of two reflections preserve it.

13

Euclid I.8, I.26, VI.4.

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7 Transformational Geometry

can be considered as preserving shape but not necessarily size: the size of the transformed figure is that of the original, multiplied by the scale ratio. Developed gradually over the centuries since Euclid, the real number system now plays a central role in the theories of congruence and similarity, through their use of lengths. It also provides the coordinates used in analytic geometry. Once a Cartesian coordinate system is established, the equations relating the coordinates of the image Xr of a point X under a similarity  with ratio r have the form Xr = rMX + B, where M is an orthogonal matrix and B, a vector:  m11 m12   x1   b1   r   m11 x1  m12 x2   b1   x1    +  = .  x  = Xr = rMX + B = r A   2 m21 m22   x2   b2   r   m21 x1  m22 x2   b2 

An orthogonal matrix M is one whose transpose M T is its inverse:

1 0   m11 m12   m11 m 21  T 0 1  = I = MM =  m m   m m  . 12 22  22    21   Since transposition does not affect the determinant of a matrix, 1 = det I = det MM T = det M det M T = (det M)2, and therefore det M = ± 1. The direct isometries have determinant 1 and the indirect isometries, –1.14 The theory of plane isometries received little attention throughout the centuries, probably because the problems under consideration to which it might apply were simple enough that they did not require it. The situation with three dimensions was different: problems of practical significance involving motions required solutions that were not obvious. In a 1763 book the Italian aristocrat Giulio Mozzi considered this subject at length as a branch of mechanics. According to his very first proposition, LEMMA I. If a sphere is moved, while its center remains fixed, I say that at each instant of the motion it will have to be rotated around a fixed axis, which will be one of its diameters.

He supported this with a synthetic argument, then stated some further results, including COROLLARY III. Therefore an arbitrary body that moves can have at each instant of the motion just two movements, one a rotation about the center of gravity, and the other progressing in a straight line common to all its parts. COROLLARY IV. Thus again it can be deduced that these two movements reduce themselves to two others, one of which will be rectilinear and common to all parts of the body and parallel to the axis of rotation, which passes through the center of gravity, and the other only rotational, which will have an axis of rotation parallel to the axis mentioned.15

14

This is a handy way to define “direct,” independent of any intuitive notion of orientation. For a focused exposition of the use of vector and matrix algebra in transformational geometry, see Smith 2000, chapters 6, 7, and appendix.

15

Mozzi 1763, 1–3, after the Introduzione. Corollary IV is proved in the next section, page 231.

7.3 Transformations as Tools

229

In 1776 Leonhard Euler derived equations for the coordinates of the terminal position of a point that undergoes an arbitrary rigid motion . Euler’s work involved much trigonometry, and was equivalent to finding an eigenvector of a 3 × 3 orthogonal matrix, which would determine the axis. But vector and matrix algebra was not then available to express this so neatly. In 1830, French mathematicians Michel Chasles and Jean Hachette published a clear exposition of these results.16 The arbitrary three-dimensional direct isometry or rigid motion  =  described in the previous paragraph as the composition   of a rotation  about an axis g with a translation  whose vector is parallel to g, is now called a screw.17 A screw with angle zero is a translation; a screw with vector zero is a rotation; a rotation with angle 180E is a half-turn. Analogously with plane geometry, the indirect isometries are produced by composing a screw with the reflection  across a plane . In this connection arises a practical problem in kinematics: given the axes, angles, and vectors of two screws, to find the axis, angle, and vector of the screw that results from their composition. The French and English mathematicians G. H. Halphen and William Burnside provided methods for that in 1882 and 1890. Their methods were elementary and not trigonometric, but suited to that problem only. Burnside pointed out that his did not depend on the Euclidean parallel axiom, and hence would be appropriate for mechanics based on hyperbolic geometry.18 The first major step toward a coherent theory of isometries, providing tools applicable to the analysis and solution of many problems, was made by the German mathematician Hermann Wiener in a series of papers published during 1890–1894. The next section describes some of those and uses them as an example to analyze the problem considered in the previous paragraph. 7.3 Transformations as Tools In a series of research and expository papers published during the years 1890 to 1894, Hermann Wiener presented tools that use geometrical transformations to analyze and solve many kinds of problems.19 His techniques belonged to both kinematics and foundations of geometry. Wiener distinguished half-turns about axes, or line reflections, as the most basic transformations, but also employed indirect isometries in essential ways. The boxes below and on the facing page provide a glimpse of this theory. The first introduces tools for using the involutory isometries (self-inverse but different from the identity—the reflections across points, lines, and planes) to investigate properties of more 16

Euler 1776; Chasles 1830, 323–324; Hachette 1831.

17

French, German, and Italian equivalents of the term screw are mouvement hélicoïdal, Schraubung, and moto elicoidale.

18

Halphen 1882, 298–299; Burnside 1890.

19

This section is based largely on Schönbeck 1986.

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complicated isometries and of geometric figures. These examples are presented for three dimensions because Wiener did so, and Pieri used similar three-dimensional tools (see section 7.4). In the next box, Wiener’s technique (with newer notation) is used • to prove Giulio Mozzi’s corollary IV, quoted near the end of the previous section—the composition of a rotation with any translation is a screw whose axis is parallel to that of the rotation; and • to analyze the problem considered there—to describe the composition of two screws. This exemplifies a style of argument that Wiener introduced, which became common during the next few decades. The tools described and applied in these boxes were limited to those needed to justify steps in the derivations. Readers may want to investigate a few more tools of this sort —for example, to analyze the compositions g h = h g and g  =  g of two commuting half-turns g and h or of a commuting half-turn g and plane reflection  , as well as analogous results for point reflections—and to formulate the corresponding but slightly simpler two-dimensional results.

Reflections and Half-turns in Three Dimensions For any point O, lines g, h, k, and planes , , , consider these isometries: O —reflection across O, its center; g —half-turn about g, its axis;  —reflection across  , its mirror.

These three disjoint classes constitute all involutory isometries. They correspond one-to-one with their centers, axes, and mirrors, which constitute the sets of their fixed points.

The following properties justify steps in the next box.  (2) (3)

  =  

]

 =  or  z ; in the latter case   =   1  .

 '2  |   is the rotation about 1  through twice the angle from  to .

 *  |   is the translation by twice the perpendicular vector from  to .

(4)

g f , ,  or g z , ,  | (  lies midway between , 

(5)

 [ g ] =  g –1 and  [  ] =   –1 for any isometry .*

]

  () =  ) .

* To see why (5) is true, note that the compositions in (5) are involutory, and compare fixed points.

7.3 Transformations as Tools

231

Composition of a Translation and a Rotation Let  be a translation and  be a rotation about an axis g. Decompose  into component translations:  =  , where  and  have vectors parallel and perpendicular, respectively, to g. Let ,  be the planes through g such that  is perpendicular to the vector of  and  =  . Let  be the plane parallel to  such that  =   . If  * , then  =  and  is the identity, so  = , and hence   is a screw. Otherwise,  1  is a line h * g and   is a rotation  about h, so   =   =    =      =    =  , a screw.* Composition of Two Screws Lemma 1. The composition of two half-turns is a screw. Proof. Consider the half-turns b and c about two lines b and c. There is a line a perpendicular to both b and c; consider the planes  = b w a and  = c w a and let  and  be the planes perpendicular to a through b and c, respectively. Then,  z , so   =   and b c = ( )( ) =  ( )  =  ( )  = ( )( ), which is the composition of a rotation about a and a translation with vector parallel to a: a screw with axis a. Lemma 2. Given a screw  with axis g, and a line h z g, there are lines k and l such that  = h k =  l h . Proof. Let  be the plane through g for which () is the plane  = g w h. Let  be the plane perpendicular to g through h, so that    = h . Let  be the plane perpendicular to g such that (. Let r be the plane perpendicular to g through the midpoint of the intersections of g with  and , so that  r( ) = . Let r be the plane through g for which  r( ) = : it bisects a dihedral angle formed by  and . Then,  = ( r)(  r) =  ( r  ) r =  (   r) r = ( )  r r) = h k , where k = r1 r. Finally, h k = (h k)(h–1 h) = (h k h–1) h =  l h , where l = h(k). Proposition 3. The composition   of two screws is a screw. Proof. There is a line h perpendicular to the axes of the screws. By lemma 2 there are lines k and l such that  =  l h and  = h k . Thus,   = ( l h )(h k ) =  l (h h ) k =  l k . By Lemma 1, that is a screw.† *This is Giulio Mozzi’s corollary IV, quoted in section 7.2, page 228. The proof follows Schell 1870, 75–76. † This discussion was based on the work of Hermann Wiener (1890–1894, parts II–III, 16–23). Wiener analyzed

the derivation of proposition 3 further to determine from the axes, vector lengths, and angles of the screws  and  the analogous data for their composition  .

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From those tools, Wiener derived another, the following three-reflections principle: The composition of the reflections across three lines is involutory if and only if the lines have a common point O or a common perpendicular g; and then the composition is the reflection across another line through O or perpendicular to g, respectively.20

Friedrich Schur devised a three-dimensional analog for his 1899 study of the fundamental theorem of projective geometry: The composition of the reflections across three planes is involutory if and only if the planes all contain or are all perpendicular to the same line g; and then the composition is the reflection across another plane that contains or is perpendicular to g, respectively.21

The three-reflections principle is valid in hyperbolic geometry as well. It unifies the consideration of concurrent and parallel pencils of lines and of planes. According to the historian Jürgen Schönbeck, Wiener was the first to emphasize the transformational tools described here.22 Wiener published little besides the papers already mentioned.23 His techniques evidently became familiar in Germany during the 1890s. The German geometer Eduard Study devoted considerable attention to them in 1891 and, as just noted, Schur applied the three-reflections principle explicitly in 1899. These mathematicians cited their source. Others rarely acknowledged Wiener, but his methods were used here and there, unheralded, in the literature of that time about foundations of geometry. As the biographical sketch in the following box shows, Wiener was active in the organization of German mathematics and in his university’s administration. Perhaps his influence spread through those contacts. Schönbeck noted, From the beginning, Wiener considered not only rigid motions, as was usual in kinematics in his time, but also ... indirect isometries, even though those have “no immediate meaning in mechanics.” For Wiener the lines of space no longer are just the objects of motions, as for his predecessors, but beyond that and above all the carriers of these motions. Thus already by Wiener is anticipated the view that plays a role in mathematics education and which underlies the general idea of an operator: that is, objects are identified with the operations they carry out. Just this concept of Wiener makes possible that transition, from geometric relationships of place to the group-theoretic relations, that is characteristic of reflection geometry in the twentieth century.24

20

Wiener 1894, 75. The German name for the principle, Dreispiegelungssatz, has since come into use.

21

Schur 1899, 404.

22

Schönbeck 1986, 811.

23

The research reports Wiener 1890–1893 were published and reprinted in Leipzig, somewhat obscurely. Wiener 1891, 1894 were published by the German Mathematical Society and have been cited occasionally regarding Wiener’s influential suggestions for an abstract framework for foundations of geometry and investigations into configuration theorems (Schliessungssätze) in elementary and projective geometry.

24

Schönbeck 1986, 811–812. See also Wiener 1890, 83.

Hermann Wiener in 1894

Hermann Ludwig Gustav Wiener was born in 1857 in Karlsruhe, in the Grand Duchy of Baden. His father Christian Wiener, professor of descriptive geometry at the technical institute there, was especially noted for devising and constructing geometric models and using them in instruction. Hermann’s younger brother Otto became a physicist, noted for work on the wave theory of light. Hermann began university studies at Karlsruhe, but completed his doctorate at Munich in 1880 with a dissertation on plane curves. His training was influenced significantly by Felix Klein, a close collaborator at Munich with Wiener’s cousin Alexander Brill. In 1880 Klein moved to Leipzig, and Wiener followed. After a year, though, Wiener returned to Karlsruhe as assistant to his father. There he prepared his Habilitationsschrift, also on geometry. He submitted that to the university at Halle, earned the venia legendi, and took a position there in 1885 as Privatdozent. At Halle, Wiener produced a series of eight articles* that clarified the relationships of several fundamental geometric theorems and techniques, particularly the Desargues and Pappus–Pascal theorems, the fundamental theorem of projective geometry, and the role of involutory transformations in Euclidean and projective geometry. Wiener laid the foundations for a calculus of reflections and half-turns for use in Euclidean geometry and kinematics, which he regarded as an algebra of geometric objects, in contrast to that of the numbers used in coordinate geometry. Wiener also pioneered the abstract axiomatic approach to geometry, emphasizing its usefulness, but never built a complete theory. In fact, David Hilbert’s famous remark that geometry could be regarded as a theory of tables, chairs, and beer steins just as well as one of points, lines, and planes stemmed from Hilbert’s excitement after hearing a presentation by Wiener in 1890. That took place in Halle at the first meeting of the German Mathematical Society, which had just been founded, largely through the efforts of Wiener’s senior colleague, Georg Cantor. Wiener was one of the original members. In 1894 Wiener became professor of mathematics at the technical university in Darmstadt, where he remained until he retired in 1927. Darmstadt lay outside the mainstream, and Wiener never continued his major research work. In 1911 he revised and marketed a set of geometric models originally developed by his father; they came into widespread instructional use in secondary and technical schools. Wiener served his university well: he was twice elected dean of his faculty. Wiener died in 1939.† *Wiener 1890–1894, 1891, 1894.

†Adapted from Smith 2000, 314. For further infor-

mation about Wiener, consult Schönbeck 1986.

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7.4 Transformations in Foundational Studies During the mid-nineteenth century several scholars suggested that transformations should play a central role in the foundations of geometry. For example, in 1851 the Prussian philosopher Friedrich Ueberweg proposed to base elementary geometry on four postulates that stipulated the free mobility of a rigid body: As the senses witness, a material rigid body can: I) if it is unrestrained, move anywhere that another rigid body is not already situated; II) held fixed at a single place, it can no longer move anywhere without restriction, but is however not deprived of all motion; III) held fixed as well at yet another place, it can make all the motions possible by (II) at no further place, but could still be moved; IV) but should a third place of the body be restrained that could still be moved by (III), all its motion will then be altogether impossible.25

Ueberweg’s postulates I–IV would be echoed closely four decades later, in postulates V–VIII of Giuseppe Peano.26 Ueberweg’s countryman and contemporary, the mathematician Bernhard Riemann, speculated on this subject, but not specifically about motions. He posed a straightforward question: Thus arises the problem, to uncover the simplest facts from which the metric relationships of space may be determined.27

Assuming that “space” is a differentiable manifold, Riemann sought to identify properties that would characterize the formula that represents the distance between two of its points. His approach established a major research area in differential geometry. During the years 1866–1870, yet another Prussian scholar, physicist Hermann von Helmholtz, reformulated this question. According to Chilean philosopher Roberto Torretti, Helmholtz’s question asks, essentially, for the foundations of geometry, regarded as a science of physical space. He says that in his investigation of the problem he followed a path not too distant from the one pursued by Riemann...28

Helmholtz provided an informal system of axioms combining proposals of Ueberweg and Riemann. Like Riemann, he postulated that the points of space constitute a threedimensional differentiable manifold. In much the same way as Ueberweg, he stipulated the existence and free mobility of rigid bodies, with a more complete description of the motion possible when two points are fixed. Helmholtz claimed to have solved Riemann’s problem, concluding that only three types of underlying manifold are possible, for each of which the curvature  is constant:

25

Ueberweg 1851, 25.

26

Peano 1894b, 80. See page 238 in the present section.

27

Riemann [1854] 1892, 273.

28

Torretti [1978] 1984, 156. This paragraph and the preceding two are based on Torretti [1978] 1984, particularly §3.1, §4.1.2. Helmholtz did not mention Ueberweg.

7.4 Transformations in Foundational Studies

235

• Euclidean space ( = 0), • the hyperbolic space of Bolyai János and Nikolai I. Lobachevsky ( < 0), • the three-dimensional surface of a four-dimensional sphere ( > 0). Felix Klein’s pioneering 1872 Erlanger program, described in section 5.5, “for many years provided the framework for geometric studies and gave them special impetus.” 29 In Klein’s view any mathematical structure could be studied through the group of automorphisms that preserve it. The generality of this vision offered a level of abstraction that suggested a way to understand how a geometry is determined by the group of transformations that act on its space: the geometry is studied and taught by analyzing and describing the group of transformations that preserve it. While Klein emphasized the group-theoretic treatment of geometry, he was concerned not with the foundations of geometry (in his work neither the geometries nor their groups are characterized axiomatically) but with certain special geometries and groups of historical importance.30

Nevertheless, Klein’s influence pervaded the foundations research recounted in this section. In 1888, the noted Turin algebraic geometer Corrado Segre suggested that his new student Gino Fano translate Klein’s 1872 study into Italian. That brought Fano into correspondence with Klein in Göttingen. Fano’s translation appeared the next year.31 Helmholtz’s ideas spurred later research by the Norwegian mathematician Sophus Lie, published in 1890. Lie applied Klein’s approach to Helmholtz’s question, reformulating it as the Riemann–Helmholtz space problem: To find properties that are satisfied by the Euclidean system as well as both non-Euclidean systems of motions, and by which these three systems are distinguished from all other possible 32 systems of motions of a number manifold.

By using Klein’s framework Lie made Helmholtz’ work susceptible to precise mathematical analysis. But Torretti explained, Lie’s approach to the problem is wholly foreign to the philosophy of physics.33

Lie’s theories gave rise to an enormous literature with applications both in geometry and in analysis. Because of its emphasis on the symmetries of the underlying point manifold rather than on details of its construction, this research followed a track different from the foundational studies of Moritz Pasch, Peano, Pieri, David Hilbert, and their followers. As reported in section 5.5 (page 119), Pieri mentioned Lie’s work to provide a context that illustrated where his own research fit into the growing literature about foundations of geometry. 29

Kunle and Fladt 1974, 462.

30

Freudenthal and Steiner 1974, 533.

31

Klein 1872, [1872] 1889–1890.

32

Lie 1888–1893, volume 3, part 5, 397.

33

Torretti [1978] 1984, 171.

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Hermann Wiener’s 1890–1893 work, described in the previous section, constituted a toolkit for solving geometric problems and organizing arguments. It was not a framework for building a foundation, either. Nevertheless, some of his contributions caught the eye of mathematicians who were investigating foundational questions. For example, Wiener systematically studied groups, which he called bireflectional,34 each element of which should be the composition of two involutions. These include the groups of rigid motions, of isometries, and of projectivities, in two and three dimensions. Studying their common properties pointed the way toward incorporating them at ground level in foundations for various types of geometry. Wiener also suggested clarifications of the relationships between congruence axioms, the Desargues and Pappus–Pascal theorems, and the fundamental theorem of projective geometry. At a September 1891 meeting of the German Society of Natural Scientists and Physicians in Halle, Wiener reported on that subject. During the next decade Hilbert and Friedrich Schur supplied proofs, and those results entered the standard repertoire of foundations of geometry.35 The box on the next page shows how Wiener’s paper spurred interest and led to Hilbert’s reformulation of another aspect of foundations of geometry, the role of geometric postulates. The account in that box shows that Hilbert paid attention to Wiener’s approach to axiomatic foundations throughout the 1890s. In 1899, to prove the Pappus–Pascal theorem, Schur used Wiener’s method of reasoning with reflections in an essential way. Those techniques made no appearance in Hilbert’s famous [1899] 1971 Foundations of Geometry. But in his 1903 paper, New Foundations of Bolyai–Lobachevskian Geometry, Hilbert constructed from geometric postulates a system of scalars, his noted end-calculus, using Wiener’s methods without attribution. Hilbert’s status as a leader of mathematical research suggests that Wiener’s methods gradually became common practice in Germany during the 1890s. In 1891, Segre visited Klein and other mathematicians in Germany. That year, Segre published a plea for the establishment of a system of postulates for multidimensional projective geometry, which was needed to secure some of the procedures of algebraic geometry.36 Fano again took heed, and after completing his doctoral research, turned to the foundations of geometry. In 1892 he published such a system, tersely describing the setting of his postulates with a single sentence: As the basis of our study we take any variety of entities of whatever nature, entities that we shall, call points, for brevity, but for sure independently of their own nature.37

That sentence by Fano reminds one of Wiener’s and Hilbert’s words quoted in the box on the facing page. Fano himself spent the academic year 1893–1894 in Göttingen, then returned to a position in Rome. Perhaps Segre and Fano provided a route for the dissemination to Italy of Wiener’s approaches to axiomatics and transformational geometry. 34

Wiener 1891, 661. Wiener’s term was zweispiegelig.

35

Wiener 1891, 47; Hilbert [1899] 1971, chapters 5–6; Schur 1899. See the figures in the next section.

36

See the biographical sketches of Fano and Segre in M&S 2007, 80–81, 107–108. C. Segre 1891a, 60–61.

37

Fano 1892, 108–109. By varietà Fano presumably meant set.

7.4 Transformations in Foundational Studies

237

Hermann Wiener and David Hilbert. At a meeting of the German Society of Natural Scientists and Physicians in Halle in September 1891, Hermann Wiener suggested an approach to the foundations of geometry more abstract than usual. His brief published report began thus: One can demand of the proof of a mathematical theorem that it use only those assumptions on which the theorem actually depends. The weakest conceivable postulates are the existence of certain objects and of certain operations by which these objects are combined with each other. If it is possible to arrange these objects and operations among themselves so that theorems result without adding postulates, one obtains in these theorems an area of science grounded in itself. ... For geometry such a return to the simplest objects (elements) and operations is of significance, because in reverse one can construct again from these an abstract science, independent of the axioms of geometry, whose theorems however go step by step in parallel with the theorems of geometry. Plane projective geometry provides an example of this. Let the objects be points and lines; the operations, join and intersection ... or, freed from geometrical garb, let elements of two sorts be postulated, and two operations, about which one assumes that the composition of any two elements of the same sort should yield an element of the other sort.*

Wiener’s live presentation may have been more animated. His main subject was the relationship between the Desargues and Pappus–Pascal theorems and the fundamental theorem of projective geometry. David Hilbert, who had recently been teaching that subject in Königsberg, attended. Hilbert’s former student Otto Blumenthal reported in 1922, Hilbert has told me that this presentation had given him such a stimulus to occupy himself with the axioms of geometry, that he had returned to it immediately on the return journey in the train: really a proof that the preoccupation with axiomatic ways of thinking was already present very early with him.†

Blumenthal expanded on this in 1935: In a Berlin waiting room he discussed with two geometers ... about the axiomatics of geometry and expressed the view that “One must always be able to say tables, chairs, beer steins in place of points, lines, planes.” ‡

The mathematics historian Michael-Markus Toepell recounted that Hilbert paid attention to Wiener’s abstract approach and pursued the relationship of those theorems throughout the 1890s.§ Hilbert began his seminal 1899 book Foundations of Geometry with this famous dryer version of his railroadstation quip, We consider three different systems of things: we call the things of the first system points ... ; the things of the second system we call lines ... ; the things of the third system we call planes ...

and stipulated that the exact relationships of these things are to be completely described by the postulates listed next.2 * Wiener 1891, 45–46. By axioms Wiener evidently meant fundamental intuitive principles. 2 Hilbert [1899] 1971, 2. † Blumenthal 1922, 68. ‡ Blumenthal 1935, 402–403. § Toepell 1986a.

Giuseppe Peano published one paper, On the Foundations of Geometry (1894b), that considered geometric transformations from a foundational point of view. His postulates described properties of rigid motions, or direct isometries. Indirect isometries played no role at all. Peano’s use of motions was close enough to Wiener’s that each would have instantly recognized what the other was doing. But Peano was mainly interested in identifying basic properties for use as axioms, whereas Wiener fostered their use in solving problems in kinematics. Peano did not adopt the abstract view of axiomatics that Wiener espoused and Fano displayed. Instead, Peano wrote,

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We shall consider as geometric terms all terms that appear in a geometry book, and which do not belong to general logic. And the first work to be done is thus to distinguish those terms, or those ideas that they represent, as primitive, that are not defined, from the derived ideas, which are defined. ... Whoever begins the study of geometry must already possess these primitive ideas; it is not at all necessary to know the derived ideas, which will be defined one by one as the study progresses.38

Peano’s primitive ideas were point, betweenness, and motion, the second being a ternary relation on the set of points, and the third, a function from points to points. First, he presented postulates for points and betweenness in three dimensions in the spirit of Pasch 1882b,39 and listed or derived from them many standard theorems. Then, as an alternative to Pasch’s treatment of congruence, Peano presented some postulates about motions. According to the first five of these, the motions should form a transitive group of betweenness-preserving transformations of the set of points. Peano’s final postulate asserted that two motions will coincide if they should agree on what would now be called a flag consisting of a point O, the ray *OP from O toward a point P = / O, and the halfplane * g Q from the line g = OP toward a point Q not on g: that is, Given noncollinear points O, P, Q and motions  and , if (O) = (O), [*OP]  [*OP] and [*gQ][*gQ], then  = .40

The remaining two postulates insure that given two flags, there is a motion  that carries the one onto the other.41 By that final postulate,  is unique. Peano used flags to characterize the translation  that moves a point O to a point P= / O, the rotation that carries the ray *OP from O toward P onto the ray *OQ from O toward a point Q noncollinear with O and P, and the half-turn g about the line g = OP. In each case he specified the images of O, of *OP, and of one of the sides of g in a plane through g. Peano used properties of betweenness to show that g is involutory and leaves fixed each point on g. He concluded his paper by sketching proofs that all translations and rotations are compositions of two half-turns. Peano acknowledged that his work was incomplete: ... with new combinations of these motions it is possible to obtain new properties; there is here a vast field of study and research. ... This treatment has not yet assumed that simplicity which is necessary to be introduced in the elements. Further studies can simplify the individual parts ... I shall be happy about my work if it will contribute to rendering more exact the definitions and demonstrations of elementary geometry, and to better analyzing the concepts on which this science is based.42

38

Peano 1894b, 51–52. Peano never mentioned Wiener.

39

Peano 1894b, 55–76.

40

Peano 1894b, 80. This notation for rays and half-planes is Pieri’s: see section 8.4, P27, P30. For any point set x, the notation [x] stands for the image { (P) : P 0 x}.

41

Peano’s final four postulates closely echo those proposed four decades earlier by Ueberweg. See page 234. Peano did not mention Ueberweg, either.

42

Peano 1894b, 90.

7.4 Transformations in Foundational Studies

239

Mario Pieri built on that 1894b work of Giuseppe Peano, his colleague and mentor. In an overview of the consideration of isometries in foundational studies before Pieri’s, the Italian mathematician Maino Pedrazzi reported, If to Giuseppe Peano one owes perhaps the first attempt to construct plane geometry on isometries ... using axial symmetry, even if not completely, as the basic transformation, it is only with Mario Pieri that one has a complete hypothetical-deductive systematization of plane and solid elementary Geometry, in which all the properties of isometries are developed and utilized.43

Like Peano, Pieri never referred to the work of Hermann Wiener described earlier. Pieri’s identification of fundamental ideas and the organization of some of his arguments is reminiscent of Wiener’s. Pieri did acknowledge that his idea of basing elementary geometry on the concept of motion was inspired by Johannes Thomae’s 1894 book on conic sections.44 Pieri presented his results in detail in his 1900a memoir on Point and Motion and summarized them in his [1900] 1901 Paris address.45 Pieri’s most significant advance beyond Peano 1894b was the discovery that collinearity and betweenness can be defined in terms of point and motion. Thus, Pieri removed betweenness from Peano’s list of undefined notions. Pieri adapted Peano’s postulates in several ways. For example, Peano’s initial postulates for motions described them as constituting a group of betweenness-preserving transformations of the set of all points, and his postulate V stated its transitivity: for any distinct points P, Q there is a motion  such that (P) = Q.46 After defining collinearity and betweenness in terms of point and motion, Pieri could skip the betweenness requirement. But in another regard, Pieri’s transitivity postulate was stronger than Peano’s: in addition to the condition (P) = Q, it required that (R) = R for some point R collinear with P, Q.47 Pieri’s second major advance over Peano’s work was his detailed treatment of continuity. Peano had made no claim to completeness, and in 1894b had ignored this aspect of foundations of geometry. Pieri’s final postulate stipulated the continuity of lines using Richard Dedekind’s technique. This step was not new to Pieri’s Point and Motion memoir: he had employed the same technique in his 1898c Geometry of Position.48 Whereas Peano had ignored indirect isometries in 1894b, Pieri defined point and plane reflections formally in 1900a and used them alongside motions in his arguments. Another difference is that Peano was writing preliminary guidelines and suggested some very broad postulates, particularly about the transitivity of the group of motions, which were phrased in terms of flags, a defined notion. Pieri postulated only those special cases that 43

Pedrazzi 1978, 204.

44

Pieri 1894b. See section 5.5 of the present book, page 117.

45

Translated in chapters 8 and 4, respectively.

46

Peano 1894b, 78–80.

47

Postulate XII, P6§2: see section 8.2, page 267;  could be the half-turn about any line perpendicular to PQ through the midpoint of P and Q. Pieri’s definition of collinearity of three points is remarkably simple: there should exist a motion different from the identity that fixes all three.

48

Dedekind [1872] 1963. See sections 8.6, Postulate XX, and 6.9, Postulate XVIII.

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he found necessary for his proofs, and presented absolutely all details. As a result, his treatment was less concise than Peano’s. Pieri’s practices had a major impact decades later in the geometric and logical research of Alfred Tarski.49 Pieri had applied some aspects of the reasoning underlying Point and Motion slightly earlier in his 1898b paper, New Method for Developing Projective Geometry Deductively. This work was not so closely related to Peano’s 1894b memoir; rather, as its title indicated, it provided an alternative to Pieri’s own 1898c axiomatization of projective geometry. Pieri addressed the Riemann–Helmholtz space problem50 in the projective context: to find properties satisfied by the groups of all projective transformations in various dimensions that distinguish them from all other transformation groups. Pieri presented a system of postulates based on the primitive ideas point and homography (point transformation that maps lines to lines).51 Pieri postulated first a class [0] of points and a group  of point transformations. Using just those notions he constructed point sets that he called lines. Other postulates entailed that [0] and  form a projective geometry in the sense of Pieri 1898c, whose homographies are precisely the transformations in . A final postulate required that these lines be continuous in the sense of Dedekind,48 and hence isomorphic to the lines of real projective geometry. Pieri’s 1898b work thus constituted a solution of the Riemann–Helmholtz space problem.52 7.5 Postlude Hermann Wiener’s results about involutory transformations were incorporated into major works on foundations of geometry for the next century, fulfilling Giuseppe Peano’s prediction of the opening of a vast field of research.53 Peano’s and Pieri’s work provided logical and organizational techniques for this effort, and helped establish the study of isometries as fruitful for foundations research. But most later works progressed in other directions. Some of those developments are recounted here briefly. Many of Wiener’s transformational-geometry tools are valid in non-Euclidean geometry as well as in their original Euclidean context. David Hilbert exploited this feature in his 1903 paper, New Foundations of Bolyai–Lobachevskian Geometry. In particular, Hilbert used Wiener’s three-reflections principle to construct from geometric postulates a system of scalars, the noted end-calculus.54 Soon after, the Berlin mathematician Gerhard

49

M&S 2007, §6.3.2, 367–369; McFarlands and Smith 2014, §9.3–§9.4, 184–194.

50

See page 235 and section 5.5, page 118.

51

Michel Chasles had popularized the term homography in the extended title and in part 2, section 1, of [1837] 1875. Today, a homography is usually called a collineation. G. K. C. von Staudt considered this concept less formally (1847, paragraphs 121, 122).

52

See subsection 9.2.6 for a more detailed description of Pieri 1898b.

53

For example, Reye 1886–1892, volume 2 (1892), cited Wiener six times. Peano is quoted in the previous section, page 238.

54

Hilbert 1903, 144–147.

7.5 Postlude

241

Hessenberg applied reflection techniques to set up a foundation for spherical geometry.55 In 1907 the Danish mathematician Johannes Hjelmslev expanded the previous work of Hilbert and Hessenberg to show how plane geometry can be constructed through exclusive use of plane axioms, without considerations of continuity, completely independent of the question of parallels.

Hjelmslev’s axioms involved motions in an essential way, and his proofs employed Wiener’s reflection techniques very heavily. None of these three mathematicians acknowledged Wiener’s contribution to this field.56 Hjelmslev’s paper marked a shift and a trend in research on foundations of geometry. First, he expanded the use of the term motion (Bewegung) to include both direct and indirect isometries: Each relationship, by which to each line and each point in it are associated a unique line and point in it so that the segments corresponding in this way are congruent, should be called a motion.57

Second, Hjelmslev saw no need to describe in detail elementary logical and geometric notions—for instance relationship and reflection—with which earlier research papers, such as Wiener’s, had been preoccupied: such notions had attained commonplace status. This may explain why none of the three mathematicians just mentioned cited Wiener’s work. After a long hiatus, Hjelmslev’s development of Euclidean and non-Euclidean geometries based on the reflection concept was continued and greatly expanded during the 1930s by the German mathematicians Friedrich Bachmann, Gerhard Thomsen, and others. They rekindled investigation of the Riemann–Helmholtz space problem in the context of synthetic geometry. Identifying properties of involutions that generate the isometry groups of various geometries under consideration, they formulated corresponding postulates that could be applied to a set S of generators of an arbitrary group G . From a group G satisfying these postulates, they could build a structure of points and lines satisfying familiar geometric postulates, as Pieri had built geometric structures decades earlier from points and transformations.58 Further postulates about S could entail that G be isomorphic to the isometry group of that structure. After another long delay, this project reached a high plateau with Bachmann’s book Aufbau der Geometrie aus dem Spiegelungsbegriff. Many researchers have extended its scope to include higher dimensions, other geometries, and other transformation groups.59

55

Hessenberg 1905.

56

Hjelmslev 1907, 449, 454. Hjelmslev cited Schur 1899, which on page 402 cited Wiener 1891.

57

Hjelmslev 1907, 474.

58

Pieri 1898b (see subsection 9.2.6 of the present book) and his 1900a Point and Motion memoir (translated in chapter 8).

59

Bachmann [1957] 1973. Readers may also consult the expository work Bachmann et al. 1974.

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7 Transformational Geometry

Thomsen’s work60 had two emphases. First, like Hjelmslev and Bachmann, Thomsen proposed using reflections in the axiomatic basis of a geometric theory. Second, he enhanced the reflection calculus that Wiener had initiated. Wiener had shown that many interesting plane figures consisting of geometrically related points and lines can be described by equations relating their reflections. For example, points P, Q, R, G constitute a triangle and its centroid if and only if the motion G P G Q G R is the identity. Wiener and Thomsen would encode that condition simply as GPGQGR.61 Their reflection calculus manipulated such compositions directly to prove geometric theorems. Thomsen extended it to three dimensions and admitted logical operations. The theory of reflections has many applications to the study of symmetries, particularly those of crystals, ornaments, and polytopes. Of the works cited in this section, only Wiener mentioned that.62 Moreover, a large repertoire of methods in interactive computer graphics is based on properties of transformation groups in projective and Euclidean geometry that those authors considered, as well as on the techniques of descriptive geometry that they and their colleagues taught. Those applications lie beyond the scope of the present study. Five of the eight authors of works cited previously mentioned the possible application of transformational methods in the teaching of geometry.63 As an outgrowth of such promotion, some of these techniques have, for better or worse, entered the standard secondary-school and undergraduate geometry curricula.64 For example, see the 1966 secondary-school text Transformational Geometry by the Swiss mathematician Max Jeger. A broader treatment of this trend is also beyond the scope of the present book. This section, and its chapter, concludes with a sketch of a remarkable application of geometric transformations, very different from those just discussed: building a field of scalars that can serve as coordinates. It depends on the relationship of Desargues’ theorem (see the figures on the facing page) to certain projective or affine transformations, already studied in the 1890s. This methodology first appeared explicitly in the 1919 paper Segment Calculus and Group Theory by Wilhelm Schwan, who earned the doctorate in 1923 under supervision of the German geometer Max Dehn. Dehn adapted and incorporated a summary of Schwan’s ideas in an appendix to the widely circulated 1926 edition of Moritz Pasch’s famous 1882b book Vorlesungen über neuere Geometrie.65 Schwan himself presented them extensively but less formally in his 1929 book, Elementary Geometry, aimed at a somewhat broader audience but not widely disseminated. In its foreword, Dehn wrote,

60

Thomsen 1933a; see also its partial translation 1933b.

61

Wiener 1893, 559. Note: G P , for example, is the translation whose vector is twice that from P to G.

62

Wiener 1890–1893, 246.

63

Wiener 1890–1893, 14; Peano 1894b, 90; Pieri 1900a, preface, translated in section 8.0, page 252; Thomsen 1933a, viii. The book that contains Bachmann et al. 1974 was aimed at German secondaryschool teachers.

64

See the informative and provocative report Freudenthal 1971 and the comprehensive survey Usiskin 1974.

65

Dehn 1926, 218–229.

243

7.5 Postlude

... the entire work is permeated by the modern geometric ideas with which Felix Klein, especially, has fertilized geometric research, as well as those men filled with the geometric spirit who extended his work—I name only Hermann Wiener and Eduard Study.66

In turn, an adaptation of Dehn’s version was published in brief in 1940 by the Austrian mathematician Emil Artin, who had been a pioneer in the development and application of abstract algebra during the 1920s; its complete, polished form is chapter 2 of Artin’s 1957 book Geometric Algebra.67 Artin’s construction is based not just on the elements of a group T of transformations of the set of points of a geometry, but also on a ring of algebraic transformations of the elements of that group itself: a tour de force of transformational geometry. It is sketched in the box on the following page. All assertions there have straightforward proofs, except those expressly termed laborious.

P

Pr

f

Pr

P Q

Qr

g

f

Q

O

Qr

g

R Rr R

h

h

Rr

Parallel Desargues Condition (4a)

Concurrent Desargues Condition (4b)

f * g * h & P, Pr on f & Q, Qr on g & R, Rr on h & PQ * PrQr & QR * QrRr | RP * RrPr

O on f, g, h & P, Pr on f & Q, Qr on g & R, Rr on h & PQ * PrQr & QR * QrRr | RP * RrPr

T

g

Pappus–Pascal Condition

R

O= / P, Q, R, S, T, U & g = / h & O, P, R, T on g & O, Q, S, U on h & QR * TU & RS * UP | PQ * ST

P O Q

U S

h

66

Schwan 1929, v. For more information about Schwan, consult Kirsch 1981.

67

Artin (1940, 20) acknowledged Dehn’s role, but not Schwan’s. Artin presented this material in a 1955 course at New York University.

244

7 Transformational Geometry

Emil Artin’s Geometric Construction of Scalar Coordinates. Artin’s theory* was cast within the framework of an affine plane: a structure consisting of a set P of points, a family G of point sets called lines, and an incidence relation between them, written on or through, satisfying these axioms: (1) (2) (3)

each two distinct points P, Q lie on a unique line PQ; for each line g, each point P is on a unique line disjoint from g; there are three points not on any common line.†

A set of all lines through a given point O is called a concurrent pencil. Lines that are equal or disjoint are called parallel, abbreviated *. This is an equivalence relation on G ; its equivalence classes are called parallel pencils. Artin considered dilatations: nonconstant functions P ² Pr from P to P such that P= / Q & Pr on g & g * PQ | Qr on g.

The dilatations form a group D of transformations of P . A dilatation different from the identity  can have at most one fixpoint. The identity and dilatations with no fixpoint are called translations. They form a normal subgroup T of D . A trace of a dilatation  is any line containing a point P and also its image Pr. The traces consist of all lines if  = , or the concurrent pencil of lines through a single fixpoint, or a parallel pencil if  has no fixpoint. Artin considered the set K of tracepreserving endomorphisms of the translation group T : operators  :  ²   such that, for every ,  0 T , ( B )  =   B   and ,   have the same traces. Clearly, the constant and identity endomorphisms 0 :  ²  and 1 :  ²  belong to K and K is closed under endomorphism composition:   B  = ( )  for all  0 T . An addition operation ,  ²  +  is defined on K by setting  +  =   . With some labor, Artin showed that the translation group T will be Abelian if it is transitive, and that its transitivity is equivalent to the parallel Desargues’ condition (4a) for distinct parallel lines f, g, h, which he added to the list of axioms: (4a)

f * g * h & P, Pr on f & Q, Qr on g & R, Rr on h & PQ * PrQr & QR * QrRr | RP * RrPr

(See the top left figure on the preceding page.) Under this assumption, inversion—  ²  –1 —is a trace-preserving endomorphism of T , called simply –1, and each  0 K has an additive inverse  B (–1). With some labor involving conjugation in T , Artin showed that each  0 K different from 0 has an inverse with respect to endomorphism composition. Thus, the set K with the operations addition and composition is a division ring with additive and multiplicative identities 0 and 1. (Its associativity follows from that of function composition; its distributivity, directly from the definitions.) Elements of K can be called scalars. For an affine plane consisting of pairs of coordinates selected from a division ring R, this scalar ring K is isomorphic with R. Further, Artin showed that another transitivity condition—that for any distinct collinear points O, P, Q there should exist a dilatation that fixes O and maps P to Q —is equivalent to the concurrent Desargues condition (4b): (4b)

like (4a) except that the lines f, g, h should be concurrent.

(See the top right figure.) With both (4a) and (4b) as additional axioms, Artin arbitrarily selected a point as origin and translations ,  with nonparallel traces, and determined that for each point P there are unique scalar coordinates ,  0 K such that the translation    maps the origin to P. Moreover, the points on any line can be characterized in the usual way by a scalar equation that is linear in their coordinates. Finally, Artin proved the famous theorem explored by David Hilbert half a century earlier: ‡ the Pappus–Pascal axiom (see the bottom figure) is equivalent to the commutativity of K. * Artin 1957, chapter 2, §1–4, §6, §7. † An affine plane results from a projective plane by removing one line and all the points on it. ‡ Hilbert [1899] 1971, chapter 6.

8 Pieri’s 1900 Point and Motion Memoir

This chapter contains an English translation of Mario Pieri’s 1900a memoir, On Elementary Geometry as a Hypothetical Deductive System: Monograph on Point and on Motion.1 By elementary geometry, Pieri meant Euclidean geometry as taught then in elementary courses, except for the theorems dependent on the Euclidean parallel axiom. That body of theorems is known as absolute or neutral geometry. In this memoir Pieri demonstrated that this theory can be presented as a hypothetical-deductive system,2 with no reliance on experience or intuition. Pieri defined its concepts precisely in terms of a selection of undefined primitive notions much simpler than that found in earlier presentations: just point and (direct) motion. He derived its theorems in explicit deductive steps from a set of unproved postulates precisely phrased in terms of these notions, and independent of the parallel axiom. In earlier foundational studies, the concept of motion had not been analyzed definitively. The classical Greek philosophers, Aristotle in particular, seemed to agree that motion of physical bodies, which implicitly refers to the passage of time, should be banned from geometry (except for applications to astronomy).3 Greek mathematicians did not all adhere to such a strict dichotomy; whether Euclid explicitly considered motion in this sense as a fundamental aspect of the foundation in the Elements is arguable. Most of Euclid’s references were to the method of superposition, which seems to involve implicitly some notion of motion. The entire geometric structure of the Elements is based on theorems derived by that method. Mathematicians continued in that vein for centuries, until the use of analytic methods in geometry suggested consideration of motions as 1

Pieri 1900a is the same as the version in Pieri’s collected works on foundations of mathematics (1980, 183–234) except for pagination. The collected works volume lists its date as 1898–1899.

2

The idea of a hypothetical-deductive system is discussed in sections 2.6, 4.3, and 5.4 of the present book.

3

See Wilhelm Killing’s treatise (1893–1898, volume 2, pages 3–4). This view was disputed by Thomas L. Heath (Euclid [1908] 1956, volume 1, 226). Killing did list a few almost explicit references to physical motion in the Elements. See also sections 5.2 and 7.1 of the present book, pages 87 and 224.

© Springer Science+Business Media, LLC, part of Springer Nature 2021 E. A. C. Marchisotto et al., The Legacy of Mario Pieri in Foundations and Philosophy of Mathematics, https://doi.org/10.1007/978-0-8176-4823-7_8

245

Postcard Depicting Catania Harbor and Mt. Etna, Late 1800s. Far left: railroad viaduct and dome of the Cathedral of Saint Agatha, also seen below

Across the Cathedral Square: Pieri’s Hotel, L’Albergo Centrale,

2008

Introduction

247

instantaneous transformations from initial to final coordinates of an object or figure. In Felix Klein’s revolutionary and unifying 1872 Erlanger program all motions are described this way; there is no mention of physical motion. Nevertheless, even late in that century, as discussion of motion in foundations of geometry came into focus, Giuseppe Peano felt it necessary to interpret motion as a transformation, explicitly and in detail: In the physical world we see rigid bodies moving with the elapse of time. We can consider the succession of infinitely many positions that a body assumes, or we can limit this to examining the relations between two positions of the figure, initial and final, without dwelling on the intermediate positions assumed by the body while passing from the one to the other. Only this particular treatment is developed in Euclid, and can be considered as a geometrical treatment. The first approach, more general and much more complicated, plays a role in kinematics, and can be excluded from geometry. ...

Peano then showed how to define various elementary notions in terms of point, segment, and motion and how to derive some basic theorems of geometry from postulates about those concepts. Using the same method, but with one less primitive notion, Pieri greatly expanded the scope of Peano’s results.4 Pieri’s subsidiary goals, some implicit, involved the presentation of his system. He would increase the level of precision and rigor found in Euclid’s Elements and more recent expositions of elementary geometry. He tacitly adopted the fusionist strategy of Riccardo De Paolis: developing plane and solid geometry simultaneously by uniform methods.5 Rather than banishing motion entirely from the foundations of geometry, as some had suggested, Pieri made motion central to his entire presentation. By insuring that his definitions and theorems could be expressed succinctly in Peano’s logical language, he fashioned a remarkably uncluttered geometrical system. This, he expressly hoped, would make it useful for those who would concentrate on improving instruction in the schools.6 After his paper was accepted but before it was published, Pieri finally obtained appointment as a university professor, at the University of Catania, in Sicily. He arrived there from Turin by rail in early February 1900.7 A port and industrial city with about 150,000 inhabitants, Catania was only half the size of Turin, and its university far less prestigious. It was spectacularly located, just south of Mt. Etna. Pieri took lodging at the Central Hotel, across from the main church of the city, the Cathedral of St. Agatha, and near the University. (See the illustrations on the facing page.) Pieri’s choice of accommodation was influenced by a charming letter from Francesco Chizzoni, the professor whom he replaced.8 Pieri’s first weeks in Catania were preoccupied with professional work, preparing lectures on projective and descriptive geometry, and overseeing publication of the memoir translated here. 4

Peano 1894b, 75. See page 252 in the present section.

5

Fusionism was controversial in Pieri's time. See De Paolis 1884, preface; the discussion in M&S 2007, §2.5; and the more extensive account, Ulivi 1977.

6

See pages 252–254.

7

For a brief account of Pieri’s life in Turin, see the chapter 6 introduction.

8

Chizzoni [1900] 1997, translated in M&S 2007, §1.1; many more biographical details are found there. The hotel still exists; it was under reorganization in 2008 but has since reopened.

248

8 Pieri’s 1900 Point and Motion Memoir

The translation is meant to be as faithful as possible to the original, preserving aspects of Pieri’s work that are tied to his expository style. For translation conventions, see the discussion in the preface, pages xii–xiv. Editorial comments, enclosed in square brackets [like these], are inserted, usually as footnotes, to document changes in mathematical terms, to note or suggest corrections for occasional mathematical errors in the original, and to explain a few passages.

Mario Pieri Around 1899

Pieri’s 1900 “Point and Motion” Memoir, First Page

ON

ELEMENTARY GEOMETRY AS A HYPOTHETICAL DEDUCTIVE SYSTEM

MONOGRAPH ON POINT AND ON MOTION BY

MARIO PIERI LIBERO DOCENTE AT THE UNIVERSITY OF TURIN

Approved at the session of 14 May 1899.

P R E FA C E “The first concepts with which a science, whichever it might be, begins must be clear and reduced to the smallest number. Only then can they constitute a solid and sufficient foundation for the structure of teaching.” Nikolai I. LOBACHEVSKY, On the First Principles of Geometry.9 “Pure mathematics progresses in proportion to how known problems are examined in greater depth and detail according to new methods. As we better understand the old problems, new ones arise on their own.” Felix KLEIN, Speech Delivered in Vienna, 27 September 1894.10

By a hypothetical-deductive system we mean any purely deductive theory—or science of reasoning—which not only should distinguish organically the a priori or primitive judgments from those derived or deduced—in short, the axioms and postulates from the theorems—but also in this way and to the same measure should display the various ideas which these judgments are about, thus identifying the parent ideas, primitive or indecomposable, and keeping them well distinguished from those that are reproductions of or formal derivations from them or can be regarded as such, and that after all turn out 9

[Lobachevsky [1829–1830] 1898, 2, translated by the present authors from Friedrich Engel’s German translation. Pieri omitted the clause set off by commas.]

10

[Klein [1894] 1898, xxi, translated by the present authors from Léonce Laugel’s French translation.]

Introduction

251

actually to be composed using the former combined with themselves and with the categories of logic. The two distinctions are in truth very similar; and the second is not less ancient than the other, nor does it seem that it should be accorded a value much different. But despite this, almost no recognition of an equal importance has been given it by mathematicians before our times.11 In truth, for the most part it has been attempted to reduce to a smaller number the axioms and postulates, without conducting any study whatever, generally speaking, to define with the least possible number of fundamental ideas the entities that occur in the deductive treatment; and in this way the advantage that was acquired on one side could very often be lost on the other, given the number and quality of primitive ideas to which it was desired that the system be committed.12 Among the studies aimed at composing the science of figures as a hypotheticaldeductive system should be mentioned here first of all PASCH’s 1882b work Vorlesungen über neuere Geometrie, which marks the beginning of a renovated order of ideas regarding the foundations of geometry, a work truly inspired by the intention to make all of geometry share in the clarity and deductive perfection and in the almost crystalline form that we have before our eyes in arithmetic. The primitive concepts of PASCH are four: point, the relation of lying between two points (that is to say, segment, which is featured in the geometry of Euclid and of Lobachevsky), flat surface or portion of a plane, and the relation of congruence between two figures. This system, as a logical edifice, reappears in every part of what will be the subject of our present essay.13 The principles of PASCH were later analyzed by PEANO with the instrument of algebraic logic; then reproduced in large part by him, with considerable modifications of substance and of form, in the memoirs The Principles of Geometry Logically Exposed (1889) and 11

If by definition one should mean a pure and simple imposition of names to things already known or attributed to the system, the primitive ideas will be the undefined concepts. But “to define” is understood by many in a broader sense: for that reason we shall say that the primitive ideas are “not defined except through postulates.” The latter indeed attribute to them such properties sufficient to characterize them in accordance with the deductive ends that one wishes to achieve. And in order to remove all misunderstanding, the term “definition in a strict sense” or “nominal definition” will be used when it is desired to exclude the “real” definition or definition “di cosa” [“of a thing”—see Marchisotto 2011, 169]. The primitive ideas are perhaps comparable to the raw materials of industry; as are the primitive propositions to simple machines.

12

Thus, although by previous examples [there is] without any doubt the possibility to develop all of the ordinary geometric material from just three relatively very simple fundamental notions such as, for example, point, segment, and motion (Moritz PASCH, Giuseppe PEANO, and so on), we have not long ago also seen proposed for this same service as Grundbegriffe der Geometrie [basic concepts of geometry] no fewer than the notions of rigid body, parts of a body, space, parts of a space, to occupy a space, time, rest, movement (Wilhelm KILLING, in 1898, volume 2, 227). One is made here to think of the “expressions ... which, carried into mathematics through unhealthy influences, represent a step backward that opposes great resistance to its progress. I mention only concepts like space and dimension.” (Pasch 1894, 12.)

13

Because such primitive ideas are to be defined (nominally) in terms of point and transformation by motion, the only geometric categories recognized by our system, and because it is possible to deduce here from its primitive propositions all the axioms of PASCH. But, even leaving out every principle of continuity, it still remains for me to see if, vice versa, those primitive propositions should be a consequence of these axioms.

252

8 Pieri’s 1900 Point and Motion Memoir

On the Foundations of Geometry (1894b). There, the primitive ideas are reduced to just three, which are point, segment, and motion: plane (as well as line) finding itself defined formally in terms of segment and point, and the author having preferred the concept of motion as a special transformation of points to points over that of congruent figures, which is hardly distinguishable from it, logically speaking; but the first is perhaps more manageable for deductive purposes. Also, the system of Professor PEANO can be recovered from ours: I mean, the primitive entities and axioms of that one from the primitive entities and postulates of this one. The system that is now offered for public judgment admits just two primitive ideas: point and motion, this last intended as representation of points by points, and far from any mechanical significance whatever. Beyond that, the author is pleased here to recognize on the whole an easier path and a greater degree of material simplicity in comparison with previous systems; to create from it later the hope of some reforms of the elements of geometry for the school, according to the principles expounded here, or others little different. And truly, if I do not deceive myself, the content of this essay shows, as of now, such a degree of deductive simplicity that educational systems can certainly also take advantage of it. Thus, it could hasten the solution of the “problem of teaching geometry” that is now in the thoughts of all, or at least of those, loving and conscientiously teaching elementary geometry, who recognize in it only too well the many deductive imperfections, and the immense difficulties that are encountered in wanting to surmount or remove them.14 Thus, the idea of contributing in some manner to the educational growth of geometry led me to abandon one approach, which would perhaps be better in the speculative order. I allude to having chosen motion as a fundamental or primitive concept. A convenient method for excluding from the principles of geometry any notion of motion (making it, in short, a derived and composite idea) would be that of defining congruence of figures by means of the more general notion of homography, which itself could function as a primitive entity,15 or else be generated, in the manner of G. K. C. von STAUDT, for example, as a product of other projective-geometric notions, which in truth can be reduced to just two,16 and working, besides, to adhere to the analytical-geometric procedures of Arthur CAYLEY and Felix KLEIN, which as a species of metric-projective determinations reproduce 14

A reform of elementary geometry, with the intent to found it as a purely deductive science and without departing too much from Euclid’s Elements, seems now, from many indications, effectively ripe; and perhaps will be for us a stimulus for further research, more detailed and intrinsic. For a long time it was disputed whether or not elementary geometry should have to be regarded as a hypothetical science, of reason alone. (See, for example, Masci 1885, §III–§IV.) To me it seems that it should always be as we construct it; and that if it is not as yet instituted with the quality of a purely deductive doctrine (that is, of science of the possible rather than of the real), it will nonetheless be able to take such a form sooner or later; and that nothing in it should prevent this evolution, toward which indeed the sciences that are progressing seem properly embarked. But geometry must be recognized as closer than any other to this objective. In that regard see the introduction to Goblot 1898. Nor are there any indications to doubt that even the design of the Caracteristica Geometrica of LEIBNIZ ([1679] 1971) more or less embraces the idea of a hypothetical-deductive system as we intend it. Compare M. Cantor 1880–1898, volume 3, 31–35.

15

For example, in the way that is announced in my note Pieri 1898b.

16

As in the memoir Pieri 1898c [translated in chapter 6].

Introduction

253

in analytic clothes all of the properties of motions, whether Euclidean or non-Euclidean, considered as representations of points by points.17 But everyone sees how little is recommended a reform of such great consequence, which would certainly involve undertaking the teaching of the pure geometry of position before that of ordinary elementary geometry. On the other hand, I do not find any other method that, satisfying good deductive requirements, might serve to proscribe motion from the foundations of elementary geometry. Not wishing to erase from the principles every trace of motion, it remains to reduce its role to a lesser proportion. Restriction of the scope in which motion is taken as primitive is a feasible task with a very promising outcome; in fact, examples of developments already realized in such a direction are not lacking. In Giuseppe VERONESE’s 1897 Elements of Geometry the relation of congruence, insofar as it should have the role of primitive entity (that is, not defined except through postulates) does not act as a transformation of space into itself, nor of any given figure into another, but only like a relation between two segments; in fact, in the final analysis, like a relation among four points.18 This account persuaded me to attempt to restrict the undefined part of motion much more, reducing it to the confines of a relation among just three points. And now there remains for me no doubt about the possibility to compose the whole of elementary geometry with just these two primitive materials: “point” and a certain relation among three points a, b, c that can be interpreted by the phrase “c is as distant from a as b,” or rather, “c belongs to the sphere described by b with center a,” “the pair (a, c) is congruent to the pair (a, b),” and so on, and represented directly, if you please, by means of a symbol like c 0 ba . But the excessive complication in which the greater part of such a system remains shrouded (given the many demands of a logical-deductive character to which we accede) nevertheless leaves the desire if not the need for new studies and further research.19 Chiefly for this [reason], and also in order to try something of benefit in the school, I am induced to present a system where the entity “motion” or “congruence of figures” will come to be used more in the capacity of a primitive.

17

Cayley 1859; Klein 1871, 1873, and others. On the method of reducing the ideas of CAYLEY to exclusively geometric form, see also Thomae 1894, 160–172.

18

Nevertheless, I cannot understand [congruence] in the way the distinguished author may intend, freed from all ties to the concept of motion (Veronese 1897, VII–VIII). If I am not mistaken, the relation signified by the words “segment a is equal to segment b,” which according to the author is introduced by means of postulate II (ibid., 10–11)—but that, like any primitive idea, proves to be determined with respect to the system by virtue of the totality of all those postulates that consider it (although indirectly)—and the other one, “segment a is congruent to segment b” in the sense that, for example, the definitions and axioms of this work (in concert with axiom XII of book I of Euclid) confer on it, are two notions that turn out to be one—that is, are only different in name, since every property enjoyed by one of them is manifested by and found true for the other, and vice versa, without exception or restriction of any kind. For the same reason, the idea contained in the expression “a is converted to b by a motion” is not logically distinguished from those. To me, therefore, it does not seem that the primitive idea of motion might be totally removed from the system of Prof. VERONESE.

19

[That research was completed and published in Pieri 1908a, translated in M&S 2007, chapter 3.]

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The present memoir embraces the major part of the elementary properties that do not establish comparisons between magnitudes of surfaces or solids (without excluding, however, the plane convex angle, the triangle, the relation of less to greater between segments or plane angles, and so on) and that are, for the most part, independent from Axiom XII of the first book of Euclid.20 Those, therefore, do not belong to Lobachevskian geometry to a lesser degree than to Euclidean, resting, that is to say, on ground common to these two geometries ( pangeometry). Here will be recovered, nonetheless, the greatest part of the first and third books of Euclid and several other things: in short, what I deem sufficient to the aim of certifying that elementary geometry can be established comfortably on the twenty postulates of this essay, and on the aforesaid parallel axiom. To the few who will peradventure find here something to praise, I profess to be greatly indebted to algebraic logic, in which I recognize the most opportune and most valid instrument for this kind of studies: not only for the effectiveness of the symbols in themselves, but so much more by virtue of the intellectual habits that the methods and doctrines of this science are shown capable of teaching and promoting, and also certainly for their suggestive faculty, which often leads to observations and research not otherwise fostered. All of our propositions can be easily translated into symbolic logic; in fact, the largest part were conceived and written from the start according to the ideography constructed by PEANO.21 But in consideration of many for whom the symbolism of mathematical logic is not familiar (of which, to tell the truth, only a very small part would be required) we abandon this form of exposition, after having extracted the greatest benefit possible, and keeping only very few easy abbreviations (gathered and declared here next) for the simple objective of reducing somewhat the bulk of this essay.

LIST OF ABBREVIATIONS 22 “Π” denotes «point»; “M”, «motion» .23 With “ab” and “abc” —where a, b, c are points —are represented the “line joining a and b” and the “plane joining a, b, and c.” With “ba” is represented the “sphere with center a that passes through b,” and with

20

[“Axiom XII” refers to the parallel postulate. Pieri’s footnote to proposition P27 in section 8.1 (page 265) indicates that his Euclid citations referred to the Euclid 1885 textbook edited by Enrico Betti and Francesco Brioschi. Since that is now difficult to access, and the enumerations of its features are not standard, the remaining Euclid citations in this chapter have been adapted to refer to Euclid [1908] 1956, edited by Thomas L. Heath. For example, “EUCLID I.2” will refer to book I, proposition 2, in Heath’s edition. For the parallel postulate, see Euclid [1908] 1956, volume 1, 202, postulate 5.]

21

See especially Peano et al. 1895–1908, volume 2, §1.

22

[Some of Pieri’s text under this heading has been altered. Some symbols have been changed to fit English terms or modern usage. Pieri also listed here abbreviations Cl, Df, Dm, Hp, and Pst, for are collinear, definition, demonstration, hypothesis, and postulate; this translation does not employ them.]

23

[For this translation’s use of guillemets («þ») and Pieri’s class-membership conventions, see a box in the preface, page xiii. (This footnote will be cited several times in this chapter.)]

Introduction

255

“Sfr(a, b)” the “sphere with the points a and b as poles.” With “ a/b” and “a*b” respectively, the “reflection24 of a across b” and the “midpoint of a and b.” With “*ab*”, the “line segment terminating in a and b, the ends included.” With “*ab”, the “half-line composed of *ab* and the extension of *ab* across b.” To designate the “convex plane angle that has for edges the half-lines |ab and |ac,” one writes “âbc”; while “|abc|” denotes the “triangle having for edges the segments |ab|, |ac|, and |bc|.” —The sign “z” interposed between two lines, or between a line and a plane, or between two pairs of points, denotes the mutual orthogonality of these figures, and so on. But all these characteristics peculiar to geometry are defined individually in the text. The notation P1, P2, ... , and so on, refers to propositions 1, 2, ... ; and, if not accompa d, c, a  nied by a § citation, refers to the current § [section]. The symbol  a, b, c  P7§2 recalls proposition 7 of §2, in which the names of the entities a, b, c are respectively changed to d, c, a. Following the example of some, the sign “” will be adopted in place of “is equivalent by definition to ...” Thus, r  ab, where a, b are noncoincident points, signifies “r is the line joining a with b” or else “calling ‘r’ the line ab ... .” “0”

always precedes a common noun—that is to say, the sign of a class—and follows the name or names of one or more individuals in it. It can mean and be read: “is a ... ,” “are some ... ,” “belongs to ... .”

“f”

between two classes will say that the first (to the left) “is contained” in the second: in other words, that each individual of the first belongs also to the other.

“=” is the sign of logical equivalence. Between two classes, it will indicate that each is contained in the other. Between two points, it means that these coincide with each other. And so on.25 “1” placed between two common nouns indicates here the intersection26 of the two classes: that is to say, the totality of individuals common to both. Demonstrations will generally be preceded by the word Proof and terminated by the end of the paragraph or an em dash —; 27 but will not be presented with anything more than very brief hints sufficient, nevertheless, to lead from the hypothesis to the thesis. We have not always provided a detailed study to obtain the shortest or the simplest demonstrations; but this is something that will come later.

24

[Pieri’s term was simmetrico.]

25

[Pieri used for the subset relation f. He used his negation symbol - in the compounds -0, -=, and ~z; these have been changed to ó, =, / and z ª .]

26

[Pieri’s term was prodotto logico (logical product).]

27

[Pieri enclosed proofs in square brackets, which have been deleted because this translation uses square brackets to signal editorial insertions. No translated proof will itself contain an em dash, nor will it extend beyond a single paragraph, unless specifically noted.]

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§1 Generalities about point and about motion. The relation of collinearity among points. Line, plane, and sphere are introduced. The system of geometry that is to take shape in this essay proceeds and unfolds with two concepts side by side, not taken from any other deductive science and about which one pretends to know nothing at the beginning: these are point and motion. All the other notions to which we must appeal belong to pure logic (and as such, find themselves distinguished and classified in absolutely deductive form in modern studies on algebraic logic), or else derive their origin from just these two primitive ideas, combined among themselves and with the categories of logic by means of nominal definitions. For the goals of the purely deductive method, it is useful to preserve the greatest indetermination possible for the content of the primitive ideas, which must never appear nor be used other than by dint of the logical relations expressed in the postulates or primitive propositions. For that reason, let us not be obliged to associate with these two terms point and motion any concrete or even specific image—the system as a whole, which is generically affirmed about them in the primitive propositions just mentioned, sufficing for the understanding of everything. It remains nevertheless in the faculty of the reader to attach to these words an arbitrary interpretation, provided that it should not be contradictory to our premises.28 POSTULATE I P1. « Point » and « motion » — Π and M — are general ideas, or classes. That is to say, “the grammatical terms point and motion are common nouns.” 23 POSTULATES II and III P2. There exists at least one point. And if p is a point, there will exist yet another point different from p.

28

In this multiplicity and variety of possible interpretations (and I might almost say mutability of significance) of the primitive ideas, one notices a law of plurality, a clear example of which is provided by the duality that is encountered in projective geometry. But it is not to be considered as indeterminacy. However distant the limits on the content of the primitive ideas should be, and however many things they should embrace that the mind is not used to contemplating at one time, they nevertheless do not cease to exist; and the primitive propositions permit deciding, case by case, whether or not, for example, a given object should belong to the scope of point or of motion. It is certainly true that for the primitive ideas of arithmetic and of geometry, such as whole number, point, and so on, we should be to a certain degree free to impose several determinations of meaning, distinct and often even distant from each other: thus it happens that arithmetic embraces everything numerable, and geometry everything figurable, and so on.

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The deductive use of these three principles is very slight. Many consider I and II as premises more logical than geometric; and, with respect to III, they do not prefer stating it among the postulates rather than introducing it instance by instance in conditional form among the hypotheses of the few theorems that depend upon it. Of much importance is the understanding of the meaning of the terms equal or different points. If p is a point, we call “equal to p” or “coincident with p” each point that should belong to every figure containing p. And by figure we mean any class or variety of points. The equivalence of points thus defined is a relative equivalence, which however involves absolute equivalence, or identity, of concepts, in the sense that two points equal according to the preceding definition, namely with respect to the class of points, will be equal also with respect to any attribute or logical label whatever. This is for the reason that every quality belonging to one of the two points is mirrored by a class of points, all of which have it in common.29 And to this equivalence are due, without any doubt, the reflexive, transitive, and converse or symmetric properties.30 Instead of “the points a and b do not coincide,” one will often say “a and b are different or distinct from each other.” And in saying that “the points a, b, c, ... are distinct,” one will certainly exclude that two of them coincide. It is observed that not even the term figure is understood by all in the sense declared just now; and that if we have to consider some systems of lines, of planes, or of spheres, and so on, not as classes of classes of points, but as simple classes, then it will be necessary to distinguish them with another name, rather than simply calling them figures. To say that one figure  should be contained, or should lie in , another one—  f  —is as much as to affirm that each point of  should also belong to . The two figures will be declared equal to each other, or coincident—  =  —just when in addition each point of  will be a point of . From P1, it follows that «point» is the same as the class of points,23 so that Π will be a figure, in fact the maximum figure, because every other one is contained in it. The term space can easily be avoided here, inasmuch as it will be synonymous with «point» : that is to say, with Π. And so on.31

29

As Professor Cesare BURALI-FORTI points out.

30

For more details compare, for example, Pieri 1898c, §1 [translated in section 6.1].

31

See Peano 1894b, 52.

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POSTULATE IV P3. Every motion is an injective representation 32 of points by points. That is to say, a motion assigns to each point a point, and to every pair of distinct points a pair of distinct points. Therefore, each motion is a representation of the class «point» upon itself, and acts on each point (operates on all of space). A motion is therefore an individual of the category that goes under the names function, representation, and so on.33 A representation, let it be  for example, always associates two classes k and kr, of which one, kr, is the image of the other—that is, corresponds to it according to a rule given at pleasure in such a way that each individual of k should refer to a certain individual of kr. It is understood that from an individual given arbitrarily in a class k, or from several individuals in it, equal to each other (as would be several coincident points in the class Π), our attention should be carried by  onto a single individual of kr, or onto individuals all equal to each other (univocality). But the faculty to invert or permute the two classes (reciprocity) is not presupposed. A representation of k on kr is called injective or similar (ähnlich) if it will not be capable of associating individuals in kr equal to each other to individuals in k not equal to each other: that is to say, if unequal individuals in k always have unequal images in kr. Such, therefore, is a motion as a representation of Π in Π, and so on, by P3. If  should then be a representation of points by points—for example, a motion—and a an arbitrary point, the “image of a according to ” or, as is more often said, the transform of a by , will also be a point, to be called a. Similarly, u being any figure, the “image of u by ” will also be a figure, which we can  p, q, r,  designate with u.34 And thus from the notation   a, b, c,  , where a, b, ... , p, q, ... should be points or figures and  a motion, one will be able to discern immediately that the motion  should represent a by p, b by q, ... , and so on; this will sometimes be signified by the phrase “p, q, ... are the positions of a, b, ... after .” POSTULATE V P4. Whatever be the motion , there also exists a motion —called the “inverse of ” or simply –1 —that associates with each point x a point that is transformed into x by . It will be observed that by virtue of P3, two distinct points cannot coexist that, the one just like the other, should be transformed by  into the same point x: hence, the impossibility of two motions not equal to each other but, the one just like the other, capable of representing an arbitrary point x by a point converted into 32

[Pieri’s term was trasformazione isomorfa. This translation uses the word transformation only for a representation understood to be bijective.]

33

[Pieri included trasformazione in this list, but the corresponding English term is now used differently.]

34

For more extended comparisons, see Peano 1894a, §19–§27, and Peano et al. 1895–1908, volume II, §1.

8.1 Generalities about Point and Motion (§1)

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x by . (Two representations  and , transforming the same class k into another one, kr, are called equal—  =  —if the images  x and  x of the same individual x of k should always be equal in kr.) Therefore, given P3,4 each motion will be an invertible representation—that is, conversive or reciprocal—of points by points (a bijective correspondence of space with itself): this includes both postulates IV and V intrinsically, and could serve in their place. POSTULATE VI P5. Two motions  and  performed successively, the one on that which results from the other, produce the same effect as one motion: the resultant or product of these.—In short, the operation or representation composed from two consecutive motions is equivalent (equal) to one motion. If  be the first operation to perform and  the second, that third motion equivalent to their composition will be indicated by  . It is not said that the motion   should be equivalent to ; for that reason, the one expression is not to be confused with the other. But it results from principles IV and VI that the composition of three or more motions is equivalent to one motion, and that this composite operation will be associative, like ordinary multiplication. With principles IV, V, and VI, as well as XII, it will be established that the motions constitute what these days is called a transitive group of transformations in the most generic sense. But the properties peculiar to this group and sufficient to distinguish it from every other similar class of transformations from points to points will be completely confirmed by means of the totality of the postulates. From postulates V and VI, it follows that if  and  are motions, the operations –1 and –1  are motions, each of which holds fixed every point, so that in short, ( –1) x = x and (–1 ) x = x, whatever should be the point x (whence –1 = –1 ).35 As a rule, any representation whatever that associates each individual with itself is called an identity transformation; and it is clear that two transformations so constructed with respect to the same class are equivalent to each other. From this stems the theorem, “if there exists a motion, any identity transformation of the class «point» is likewise a motion.” 23 Here it appears convenient to identify such a motion as improper, which custom does not otherwise recognize. P6—Definition. Any motion whatever, different from the identity transformation, will be described as effective or proper. That is, «proper motion»  “class of motions for each one of which, let it be  for example, at least one point x can always be found that should be transformed into another one, x, not coincident with x.”

35

One is not averse to using, as a rule in a sense more abstract, words and forms of speaking that awaken images of totally material things, provided that no doubt at all should fall on the logical meaning, which alone is in question.

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POSTULATE VII P7. For each pair of distinct points, there exists at least one proper motion that holds fixed the one as well as the other. That is, by P6, “if a, b are points and a = / b, there will have to be a motion  and a point x such that a = a, b = b, and  x = / x. This proposition can be seen in the perceptible fact that a body is still capable of being moved even though two of its points are fixed. And the movement of rigid bodies is always of a concrete image, intuitive and fully conforming to the entity M, provided that it be abstracted from time, and the observation taken exclusively over two states of the motion—initial and final position—showing the distinction between proper and improper motion. After principle VII one can verify immediately the existence of a proper motion, in view of the preceding II and III. POSTULATE VIII P8. Having assumed that a, b, c should be distinct points, if there should exist a proper motion that should represent each one by itself, then every motion that should leave a and b individually fixed will also hold c fixed. This is a principle of great deductive capacity; and thus signifies as such a very restrictive condition on the class Π. It is now given to us to produce and develop through this the notion of «line» and to recognize some of its more notable properties.23 P9—Definition. If a, b, c are points, the statement “there exists a proper motion that represents by itself each one of a, b, c” is expressed in one of these ways: a, b, c are aligned, or are collinear.36 P10–Theorem. Three points a, b, c are certainly collinear if any two of them, or all three, coincide. Proof. If b = c but a = / b, it is sufficient to refer to postulate VII. And if a = b = c, we will again be with P7, seeing that, by virtue of principle III, one can invoke the existence of a point br different from a. —Therefore, if three points a, b, c do not align, it will be impossible that two of them should coincide. P11–Theorem. These statements are equivalent: “The points a, b, c do not align” and “no motion except the identity transformation is capable of keeping each point a, b, c fixed.” Proof. The present theorem merges with P9, thanks to 36

[Pieri also provided a third alternative whose English cognate collimate is not appropriate here, as well as the abbreviation Cl for are collinear, which this translation does not use.]

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P6.—Therefore, if three noncollinear points should be required to remain fixed, no point at all will be moved.37 It emerges from this that our definition of collinear points cannot suit the geometry of hyperspace. Not even principle VIII. But we consider only elementary geometry, seeking as much as possible to establish the principles in a manner more suitable to deductive simplicity. To produce these higher spaces, it will be necessary, therefore, to renounce that postulate and that definition,38 precisely as one must renounce the postulate of parallels and the ordinary definition of these, or else the principle that two lines cannot enclose space, in order to pass to Lobachevskian or to Riemannian geometry. P12–Definition. Provided that a and b be distinct points, the join of a with b, which is referred to by the sign ab, will be the class of all those points for each of which, let it be x for example, the existence of a motion can be verified that holds a, b, x individually fixed: thus, the class of the points aligned with a and b. P13–Theorem. Under the hypothesis of P12, the points a and b will belong to the join of a with b, and the figures ab and ba will coincide. Proof. The first part comes from P10; the second is true by the symmetry of P12 with respect to a and b. We do not stop here to signal all the facts that derive immediately from the symmetry property of the relation of alignment; as similarly, from the mutual dependence between this and the figure that is discussed in P12. Of this sort is, for example, the following: P14–Theorem. If a, b, c are points, a different from b, each one of the propositions “a, b, and c align” and “c belongs to ab” will be a consequence of the other. And if, in addition, a should be different from c, these statements will also be equivalent: c 0 ab, b 0 ac, and so on. If then we should also appeal to principle VIII, we will obtain propositions of greater weight, like the three that follow: P15–Theorem. Once again, a, b being noncoincident points, the join of a with b does not distinguish itself from the geometric locus of all the points that remain fixed (or each one of which returns to itself) by any motion that represents a as well as b by itself.39 Proof. That each point y of ab (see P12) should be transformed into  y itself by any motion  such that a = a and b = b is certainly stated by  c  P8. 37

[Pieri continued this sentence with the phrase by an effective motion.]

38

See the observations in P22§3, page 279.

39

LEIBNIZ also seems to appeal to the facts considered in P12 and P15 for the definition of line: “Should any body, of which two points should be immovable and fixed, but the body itself nevertheless be moved, then all fixed points of the body will fall on a line that passes through the two fixed points” (Leibniz [1679] 1971, 147).

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That then, vice versa, the statement “from z 0 Π,  0 M, a = a, and b = b it follows, whatever be , that z = z” should also necessarily bring along another one, z 0 ab, is a fact that follows from postulate VII, because the proper motion admitted there as existent will also have to hold z fixed (P12 still present, as is natural). P16–Theorem. There exist three noncollinear points a, b, c. And whatever be the points a and b, provided they be noncoincident, there will exist at least one point outside of the join of a with b. Proof. Given principles II, III, and VII, there will certainly have to exist a pair of distinct points a and b and a proper motion  for which a = a and b = b; thus also, by P6, a point c such that c = / c. Now, a proper motion that should hold each point a, b, c fixed cannot be given, since if it existed, we would, on account of principle VIII, have to infer from it c = c. Thus, it is (see P9), that the points a, b, c should not be aligned. And so on. P17–Theorem. Having assumed that a and b are points, a different from b, if it happens that c and d should be points of the join of a with b, provided they not coincide, the figures ab and cd will coincide. Proof. By the hypothesis we have, by virtue of P8 and P12, that each motion that should represent a by itself as well as b will also hold fixed the points c and d. Thus, by P12, the assumption x 0 ab will imply the existence of a proper motion that transforms to itself each of the points c, d, x. Therefore, from the proposition x 0 ab it follows that x 0 cd and, in particular, that points a and b are in cd; and moreover in the same  c, d, a, b, y  way, by means of the substitution  a, b, c, d, x  , it must happen that from y 0 cd is deduced y 0 ab. This, in short, is as much as to say that ab = cd follows from the hypothesis. P18–Definition. The generic name of line is given to the join of two distinct points, whatever they be. That is, «line» is the class23 of all possible joins. Saying that “r is a line” is as much as to affirm that “there exist two noncoincident points a and b, and r is, by another name, their join ab. The preceding theorem is not different from that which is commonly expressed by saying the following: “through two given points there does not pass more than one line,” “a line is determined by any two of its points,” and so on. All figures and geometric relations that are addressed in this essay are invariants and covariants with respect to motion—that is, are converted by the effect of a motion into others of the same generic name: points into points, lines into lines, spheres into spheres, segments into segments, perpendicular lines into perpendicular lines, triangles into triangles, and so on. Thus, for example: P19–Theorem. If a, b be points given at pleasure, only distinct, and  should be an arbitrary motion, the figure ab transformed by  will become the join of points a and b, so that (ab) = ( a)( b). Proof. Let x be a point in ab and set

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263

ar  a, br  b, and xr  x. For certain, ar and br will be noncoincident points, by P3. I say that xr 0 arbr. Introduce by P7 a proper motion  for which ar = ar and br = br. By P4 and P5 the composite operation –1 , product of the motions , , and –1, will also be a motion such that, further, ( –1 )a = ( –1 )ar = –1ar = a and similarly ( –1 )b = b, from which it is necessary that ( –1 ) x = x by P15. Therefore, we have ( –1 ) xr = x. That is to say (multiplying on the left by )  xr =  x; therefore, xr = xr. Consequently, there will exist a proper motion  that holds all three points ar, br, xr fixed at once: this is as much as to affirm the proposition xr 0 arbr. Vice versa, the assumption yr 0 arbr brings with it –1 yr 0 ab, being able everywhere to substitute ar, br, a, b, –1, yr for a, b, ar, br, , x for the reason that –1 ar = a, –1 br = b. P20–Definition. Given three noncollinear points a, b, c, the figure occupied by all the lines that join the point a with the various points of bc, or the point b with the points of ca, or the point c with those of ab will be called the plane abc, or simply abc. Again: under the same hypothesis, a point x is said to belong to the plane abc whenever it should coincide with one of a, b, c or else the join of a with x should meet bc, or bx and ac should meet, or cx and ab. Observe that the hypothesis that a, b, c be noncollinear will insure that the points a, b, c should be distinct by P10, and that no one of them should belong to the join of the other two, by P14. This definition of a plane (not much different from that of projective plane, which is attributed to Bernhard RIEMANN) is found as a theorem in Moritz PASCH’s Vorlesungen über neuere Geometrie.40 It suits equally each one of the three systems of geometry that are distinguished by the names hyperbolic or Lobachevskian, parabolic or Euclidean, and elliptical of the second species or Kleinian. But the last of these is already excluded by principle VIII, whereas all the propositions of this study agree perfectly with the others. P21–Theorem. Whenever a, b, c should be noncollinear points, the lines ab, ac, and bc will lie on the plane abc; and this will coincide with each of the planes acb, bac, bca, cab, and cba. P22–Theorem. And given a motion , the figure into which the plane abc is transformed will again be a plane: in fact, the plane of points a, b, and c, certainly noncollinear. In short, (abc) = ( a)( b)( c). Compare P19. But from premises I–VIII it does not seem that any proposition about planes should ensue, similar to what one has for lines in P17. For that the following is proposed:

40

[See a paragraph about Riemann in section 5.6, page 121; and Pasch 1882b, 25–26.]

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POSTULATE IX P23. If a, b, c should be noncollinear points and d a point of the line b c, distinct however from b, the plane a b d will be totally contained in the plane a b c. Consequently, (since it follows from the hypothesis by virtue of P10,13,14,17 that c belongs to bd and that a, b, d do not align), the plane abc will also be contained in the plane abd, for which reason the planes abc and abd will coincide. P24–Theorem. Provided that a, b, c be noncollinear points and d a point of the figure abc but not of ab, the two planes abc and abd will coincide. Proof. If d lies in ac or in bc, we will be able to reduce this to postulate IX as before, having also seen P21. If not, the lines ad and bc, or else bd and ca, or else cd and ab, will meet at a point different from the preceding a, b, c, d (see P20 and so on). Now from the fact that a point e, for example, should be common to bc and ad, it follows from P23 that abc = abe, then in the same way, abe = abd. And so, from the assumption f 0 ca 1 bd one concludes, always by virtue of the same principle, that abc = abf and abf = abd, and so on: thus, in each case, abc = abd. P25–Theorem. If a, b, c should be noncollinear points, then d, e points on the plane abc but not aligned with a, it will be inevitable that the plane ade should coincide with the plane abc. Proof. By the hypothesis, b = / a and d = / e by P10, nor can it happen that a, b, d and a, b, c be collinear at one time in the presence of P13, P14, and P17. Thus, if we suppose that a, b, d be noncollinear, then by P21,24, abc = abd and abd = adb; consequently, e 0 abd and adb = ade by P24, whence abc = ade. The same, if it is supposed that a, b, e be noncollinear. P26–Theorem. Again, a, b, c being noncollinear points, if it happens that three points d, e, f belonging to the plane abc are likewise not aligned, then it will be necessary that the planes abc and def coincide. In short, two planes that should have three nonaligned points in common coincide. Proof. The threefold disjunction “a, d, e or a, e, f or a, d, f are noncollinear” is invoked, since its negation would as much as assert the existence of an alignment among the points d, e, f, which is contrary to the hypothesis. But if, for example, a, d, e should be noncollinear, it is necessary to conclude abc = ade, by P25. That is to say, abc = dea (see P21); hence, f 0 dea and consequently dea = def by P24; therefore, the coincidence of abc  f ,e  f ,d  with def stands proved. From here, by means of the substitutions  e, f  ,  d, f  , it arises as well that the hypothesis, combined with the one or the other of the assumptions that a, d, f or a, e, f be noncollinear, requires that abc be equal to dfe or else abc equal to fed; that is, abc = def in each case. P27–Theorem. Given that a, b, c are collinear points and that d, e are noncoincident points of the plane abc, it happens that the join of these—that is, the line de —lies

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totally in the plane abc (EUCLID XI.1).41 Proof. Points d, e, a, points d, e, b, and points d, e, c not possibly being collinear at one time (because that would entail alignment of a, b, c), let a, d, e be noncollinear, for example. Then, the planes abc and ade will coincide by P25; for that reason, de, contained in the plane ade by P21, will thus lie in the plane abc. And so on. Here, the generic entity «plane», or the class of planes,23 will be spoken of, implying (it is somewhat superfluous to say) a definition similar to that proposed not long ago with respect to the entity «line». P28–Definition. Having assumed that a, b should be points, the name sphere through b about a, or sphere through b with center a, or symbolically, b a denotes the class of points for each one of which there exists a motion that should carry it onto b, holding a fixed. In other words, to say that c should belong to the sphere through b about a is as much as to affirm the existence of one such motion , for which a = a and c = b. If the points a and b coincide, the sphere b with center a is restricted to one point: or rather, it is the locus of points that coincide with a, by principles IV and VII. P29–Theorem. If a and b are points, then b belongs to the sphere ba ; moreover, if they are distinct points, then a does not belong to it. Proof. By principles IV and VII. P30–Theorem. Whenever a, b, c should be points and c should belong to the sphere ba , the spheres ba and ca will coincide (EUCLID III.5,6). Proof. In fact, if x 0 ba there will be two motions ,  for which a = a = a, c = b, x = b, and therefore, by P28, there will exist a motion –1 (see P4,5) that carries x onto c holding a fixed; therefore, x 0 ca by P28. Likewise, if y 0 ca there will be a motion  that represents y by c, a by a, so that the transformation –1 will change y into b, a into a, so that y 0 ba . P31–Theorem. From the assumption “a and b are distinct points,  is a motion” follows (ba) = (b)a. Proof. Let x be any point of ba and suppose ar = a, br = b, and xr = x. There exists a motion  for which a = a and  x = b by P28, and a motion r =  –1 that will not change ar, because rar = ( –1)(a) = ( )a= a = ar, but will carry xr onto br for the reason that rxr = ( –1)(x) = ( ) x = b = br. Therefore, xr 0 brar , and so on. From the last two—P30,31— emerges yet another: P32–Theorem. Each motion that should leave fixed a given point a transforms into itself each sphere that should have a as its center. 41

In the citations of EUCLID, I refer to the edition of Professors Enrico BETTI and Francesco BRIOSCHI for the Italian schools: Euclid 1885 (15th printing). [Owing to the limited accessibility of Euclid 1885, all such citations in this translation have been adapted to refer instead to Euclid [1908] 1956.]

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P33–Theorem. If, a, b being distinct points and c an arbitrary point, the spheres ca and cb should not have a common point different from c, this point c will have to belong to the join of a with b (EUCLID III.11,12). Proof. Each motion that should represent a as well as b by itself will have to transform ca into itself no less than cb by P32, and thus also the point c, which by the hypothesis is the only common point; see P15.

§2 Rotating a line onto itself. Midpoint of a pair of points. Rotating a plane onto itself. Orthogonality relation among three points or between two intersecting lines. POSTULATE X P1. If a and b are noncoincident points, there exists a motion that represents a by itself, and gives as the image of b a point different from b but belonging to ab.42 By P6,17§1 such a motion is certainly proper and represents the line ab by itself so that, except for a, no other point of ab corresponds to itself (see P15§1). And thanks to P28§1, the same principle is reproduced as such (although in another guise) in the proposition, “Whenever a and b should be points distinct from one another, the sphere through b about a and the join of a with b will also meet at some point different from b. (See P13,29§1.) POSTULATE XI P2. From the assumption that a, b be distinct points, and that ,  be motions for each of which a should remain fixed and b move to a point different from b but belonging to a b, it will follow that the images of b under  and  coincide. Or rather, this is equivalent: “if a and b are distinct points, the join of a with b and the sphere through b with center a will surely not be able to meet in more than two distinct points. P3—Definition. If a and b are distinct points, the expression reflection43 of b across a, symbolized by b/ a , is supposed to signify the point that lies at once on the line ab and on the sphere ba but is different from b (see P1,2).44 If, on the contrary, the points a and b coincide, the above-mentioned phrase is to denote the point 42

[The Euclidean motions that satisfy P2 are the half-turns about the lines perpendicular to ab at a.]

43

[Pieri’s term was simmetrico.]

44

Here as below (when there are no lurking ambiguities) one will speak of several coincident points as of one single point, conforming to common usage. For example, it would be better to say, “that point which, together with all its equals, constitutes the class of [points in] ab  ba different from b.”

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b, so that a/ b  a. It is observed that by P4,30§1, from br = b/ a follows b = br/ a, whatever these points a and b may be, and that if b = / a, the points b and b/ a are always distinct from each other, by P29§1. P4—Theorem. If as before, a and b are distinct points and  should be a motion that represents a by itself and b by a point br of ab, it follows that in its turn br will be transformed into b by that motion. Proof. If br = b, there is nothing to say. If br = / b, let bO = br. Then, bO will be a [point on] ab different from br since (ab) = abr = ab by P17,19§1, and because on ab no point except for a is fixed, by P15§1. Therefore, with regard to the motions  and –1 (see P4§1), postulate XI can be cited, according to which the points br and –1br, namely bO and b, will coincide. Since br  b and bO  br = ( b) = 2 b, and because the same happens for each point of ab, it can be said that the square of the operation  must transform into itself any point whatever of ab; and that by the motion , the line is transformed involutorily into itself. P5—Theorem. If a, b, c are collinear points, b different from c, and there exists a motion  permuting b with c, but not altering a, so that c should be the reflection of b across a, then any motion whatever that should transform each of the points b and c into the other will have to hold a fixed. Proof. From the assumptions  0 M, b = c, and c = b, it follows that the motion  can alter neither b nor c. Therefore, by P14,15§1, it will not alter a, which lies in bc. Thus, a = a: that is to say, a = a. POSTULATE XII P6. If a and b are distinct points, there exists a motion that carries a onto b and represents some point of ab by itself. That is to say, by P28§1, that given a pair of distinct points a and b one can always find in the join of a with b a point such that the sphere through b around it also contains a, in such a way that a and b turn out to be reflections of each other across it, by P3,4. Two points so constructed in the join of a with b cannot coexist unless they coincide (see P15§1 and P5). From here it is convenient to give a name for that unique point covariant with the two given points a and b. P7—Definition. If a, b are noncoincident points, the point in their join that is distinguished from every other point in it because the sphere through b about it should also contain a, is called the midpoint of a, b or center of the pair a, b and can be denoted by a*b. But if the points a and b coincide, that name is to denote the one or the other, so that a*a  a. Observe the equivalence of these statements: “c is the midpoint of a, b,” “c is the midpoint of b, a,” “b is the reflection of a across c,” “a is the reflection of b across c,” and so on. And, once and for all, if the points

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a and b are distinct from each other, their midpoint a*b will also be distinct from each of them. P8—Theorem. If among the points a, b, c intercedes the relation c = a*b, or else another one, c = a/b , the same is also seen among the positions a, b, c of these points after an arbitrary motion . Proof. Let ar= a, br= b, cr= c, and, for  b, c  example, let c = a*b. By P6,7 there will certainly exist a motion   a, c  , and con–1 sequently a motion  (see P4,5§1) that will transform the point ar into br leaving cr fixed, so that cr in its turn will be the midpoint of ar and br. And so on. P9—Theorem. Given the noncoincident points a and b, there will always be a motion that transforms their join ab into itself, in such a manner that a should go onto b and no point of the line stays fixed. Proof. Having denoted by c the point a/b, certainly different from a and from b by P3, there will exist two  c, a, b   c, b  motions   a, c, b  and   b, c  by P3,4,6; and the product of the two will be  a new motion   ab,,bc,, bc  (see P5,19§1). Nor can any point x of ab be fixed with ab   respect to , because if one wants x = x, one must concede that the sphere xb should pass at once through a, b, and c by P28,29§1, contrary to P2. To this kind of motion is suited the name translation transforming the line45 onto itself, as is the name rotation transforming the line onto itself to all those discussed in the preceding P1–7. P10–Theorem. Again given two distinct points a and b, if it happens that a motion should represent a by b, and b by some point of ab different from b, it will always result from this that b = a/b and –1 a = b/a . Proof. Having set –1 a = a1 (that is, a1 = a), b = br, and denoted by  a certain motion reflecting these points a and b, the one onto the other (see P4,6),  will be a certain motion that transforms a1 into b without altering a, for which reason a1 = b/a by P3 and so on. In the same manner (  designating a motion that exchanges the points b and br, the one with the other), there will exist a motion  that carries a into br without changing b, so that br = a/b . POSTULATE XIII P11. Given three noncollinear points a, b, c there must exist a motion for which a as well as b should be fixed and c transformed into a point different from c but belonging to the plane a b c. A motion so constructed, let it be  for example, is certainly effective or proper (see P6§1), and brings the entire plane abc onto itself by P24§1, seeing that (abc) = ( a)( b)( c) by P22§1 and that a = a, b = b, and c 0 abc – ab by the hypo-

45

[Pieri’s words were scorrimenti della retta.]

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thesis (see P15§1). Therefore, we shall sometimes say, by that motion the plane abc “is overturned or rotates onto itself about the points a and b as hinges.” 46 P12–Theorem. Provided that a, b, c should be noncollinear points, there exists in the plane abc some point different from c and yet common to the spheres ca and cb . Proof. A motion such as that whose existence is affirmed by principle XIII will have to transform each of the spheres ca and cb into itself by P32§1, and hence c into some point common to both (see P29§1). P13–Definition. A circle is the class of points that lie on a sphere and at the same time in a plane that should contain its center. That center will also be the center of the circle. When no ambiguity should be incurred, the notations ca and cb will also denote circles. Thus, in the above-mentioned P12, the circles ca and cb (in the plane abc) meet at least at one point different from c. POSTULATE XIV P14. Whenever a, b, c should be noncollinear points and d, e points of the plane a b c common to both spheres c a and c b but different from c, it will be necessary that these points d and e coincide. Thus, if the centers a and b of the two spheres are distinct points and the two spheres should have a point in common, for example c, not on the join of a with b, the circles ca and cb , lying in the plane abc by P13, surely meet in a point cr different from c by P12, and certainly do not have any other common point different from c and from cr (EUCLID III.9,10). P15–Theorem. If for three noncollinear points a, b, c it happens that c should belong to the sphere ba , this does not meet the line bc in any point different from b and c. In short, a line cannot meet a sphere in more than two distinct points. Proof. Indeed, if according to principle XIII the plane abc is rotated onto itself about b and c as hinges, and ar should emerge from this as the new position of a, then the sphere ba is transformed into bar by P31§1, while each point of the line bc stays fixed. Therefore, each point common to bc and ba , lying at once on the spheres ba and bar and on the plane aarb, will have to coincide with b or with c by postulate XIV. This discourse would require that a, ar, b be noncollinear. But even if it should be supposed that b 0 aar, it will remain beyond doubt that c ó aar, since a, b, c are noncollinear by the hypothesis (see P14,17§1); and so on. P16–Theorem. Given at pleasure two distinct points a, b and a point c in their join ab, the spheres ca and cb do not meet except at c. Compare P33§1. Proof. The supposition that d be a point common to ca and c b but distinct from c, therefore not belonging to ab (because otherwise, the points a and b would have to coincide with point c*d according to P5,7) implies, contrary to P14, the existence of three 46

[  is the half-turn about ab.]

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distinct points in the plane abd, common to both of the circles ca and cb : that is to say, the points c and d plus a third point dr, image of the point d through a rotation of the plane abd onto itself about a and b as hinges (see P11). P17–Theorem. Given noncollinear points a, b, c and an arbitrary point d of the plane abc, if two proper motions  and , the one just as the other, will transform the plane abc into itself, holding fixed each of the points a and b, the one just as the other, it will be inevitable that the points d and d should coincide, and that the point ( d) should be indistinguishable from d. That is, two rotations of the plane abc onto itself about the same points a and b as hinges (see P11) are always equivalent with regard to the plane abc (are equal representations of the plane abc); and by a motion so constructed, the plane abc is transformed involutorily47 onto itself. Compare P4. Proof. From P14,16 it follows that d = d, having seen that the points d, d, and d, beyond being in abc, must also be found at once on the spheres d a and d b by P29,32§1; and that the point d, if it should not belong to ab, will certainly be different from the other two, d and d (see P11,14§1). Nor will the point d, when it should itself be subjected to , cease to belong to the spheres da and db and to the plane abc: thus, either it coincides with d and stays fixed (when d should belong to ab), or it is returned to d, because in the plane abc no point different from d and from d can lie at once in the circles d a and d b , and so on. P18–Theorem. Under the same hypothesis, the midpoint of d and d always lies in ab. Proof. Indeed, if dr  d and consequently dr = d by P17, that midpoint d*d', in keeping with P8, will be transformed into itself by , by P5–7. Now, the points that  does not change are all in ab, since  is a proper motion (see P11,12,6§1 and so on). P19–Definition. Having supposed that a, b, c are points, a different from b and from c, when one says “the pair (a, c) is perpendicular to the pair (a, b)” or writes (a, c) z (a, b), one will merely affirm the existence of a motion for which a as much as b should be fixed, and c transformed into a point different from c but belonging to the line ca. Therefore, in the presence of P14,15§1 and P11,18, the proposition (a, c) z (a, b) will be equivalent to the statement, “the points a, b, c are not collinear and, rotating the plane abc onto itself about a and b as hinges, c falls back onto the line ca,” and also to the statement, “there exists a proper motion that leaves the points a and b fixed, bringing the line ac back onto itself. Here, orthogonality is introduced in the form of a relation among three given points and nothing more, and thus restored to its primitive terms and divested of all that is superfluous (with respect to our system).48 Therefore, it is in the nature of algebraic logic. 47

[Pieri’s word was involontariamente, probably a misspelling.]

48

[Pieri’s parenthetical phrase was di fronte al nostro sistema.]

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P20–Theorem. From supposing that a, b, c should be noncollinear points and (a, c) z (a, b), it follows that the reflection of c across a will lie on the sphere cb (EUCLID III.3). Proof. Rotation of the plane abc onto itself about the points a and b as hinges represents c by its reflection with respect to a (see P3,19) without taking it away from the sphere around b (see P32§1). P21–Theorem. Vice versa, if a, b, c are noncollinear points and c/a 0 cb , then (a, c) z (a, b) (EUCLID III.3). Proof. Indeed, the three points c, c/a, and c (  denoting the cited rotation) will lie at once on the spheres c a and c b and on the plane abc, for which reason c/a will coincide with c by P14. P22–Theorem. If again a, b, c be noncollinear points and the pair (a, c) perpendicular to (a, b), it cannot happen that the sphere through a with center b and the join of a with c meet each other except at a (EUCLID III.16). Proof. If there existed a point d common to a b and ac but different from the point a, and therefore such that a, b, d do not align, then d/a would also belong to the sphere a b , thanks d to  c  P20. But this would coincide with d b by P30§1, so that the distinct points d, a, and d/a would be in the line da and in the sphere, contrary to P15. P23–Theorem. Given noncollinear points a, b, c and a point d on the join of a with b, in order that a point cr lying in the plane abc be common to the spheres c a and c b , and thus common to the two circles c a and c b in this plane, it will have to lie on the sphere c d as well. Proof. The case cr = c is omitted. If cr = / c, the  a, b, abc  image of c by a proper motion   a, b, abc  , the motion existing by virtue of P11, will itself also be common to the spheres c a and c b and to the plane abc, so that c = cr by P14. But d is fixed by ; consequently, cr 0 c d by P28§1. P24–Theorem. From the hypothesis of P21, namely that a, b, c are noncollinear points and c/a 0 c b , it follows that (a, b) z (a, c). Proof. Let cr  c/a, br  b/a, and  be a rotation of the plane acb onto itself about a and c as hinges. I show that b = br. Indeed, having also denoted by bO and m the points b and b*bO, it happens on the one hand that the point m belongs to ac by P18, and on the other, that cr 0 c b , cr = cr, and (c b ) = c bO , and therefore that cr 0 c bO . But, from  b, m  cr 0 c b 1 c bO and m 0 bbO it follows that cr 0 c m , according to  a, d  P23 and having seen that bO, b, c are not collinear, since the supposition c 0 bbO would imply cr = c  b  in keeping with  a  P16. Therefore, m = c*cr by P7; thus, m = a by P15§1 and P5; and therefore bO = b/a = br by P8, and so on. P25–Theorem. Whenever a, b, c be points, a different from b and from c, the statements (a, c) z (a, b) and (a, b) z (a, c) are equivalent. That is, the relation signified with z, between two pairs of points (a, b) and (a, c), will be symmetric49 or commutative. Proof. In this way, from P19,20,24.

49

[Pieri’s word was invertibile.]

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P26–Theorem. Again given noncollinear points a, b, c and then a point br in the join of a with b and a point cr in the join of a with c, each one distinct from a, these propositions will also be equivalent: (a, c) z (a, b) and (a, cr) z (a, br). Proof. Let the reader see. So far, the notion of perpendicularity has been concerned with two pairs of points, consisting of three distinct points and no more. But there is no conflict in also adopting orthogonality between two intersecting lines, as is customary. P27–Definition. Two lines r, s that should have a point in common, let it be a for example, will be called perpendicular, normal, or orthogonal to each other (r z s) if there exists a point b of r and a point c of s such that (a, c) should be perpendicular to (a, b), or (a, b) perpendicular to (a, c). See P19,25,26. By P19 and so on, two lines r and s so arranged (the one perpendicular to the other) are certainly distinct from each other; and, however a point br be chosen in one and a point cr in the other, as long as both are different from the common point a, the pair (a, cr) will always be perpendicular to the pair (a, br) just as this will be to the former, by P25,26 and so on. P28–Theorem. If r is a line and p an external point, a point x can always be found on r so that the relation p x z r should hold (EUCLID I.12). That is, through a given point not on the line r there always passes a line s perpendicular to it. Proof. In r can certainly be found two distinct points a and b, by P18§1: thus, such that a, b, p are not collinear. If  should denote a rotation of the plane abc onto itself about a and b (see P11), the midpoint of p and p will lie on the join of a with b by P18, so that, having let x be such a midpoint, the point p will lie on the line px. Certainly x does not coincide at once with a and with b; let it be different from b, for example. Then, ( x, p) z ( x, b) by P19, and consequently xp z r by P27. P29–Theorem. But in r there cannot coexist two distinct points x and y such that p x and py should both be perpendicular to r. Proof. Because, if they coexisted, the position of p after performing the rotation of the plane xyp about x, y would be common to the lines px, py by P17,19,27; and so on. P30–Theorem. Given noncollinear points a, b, c for which the sphere a b and the line ac should not meet except at a, it is necessary that this line ac be perpendicular to the join of a with b. The same will occur if the sphere a c should not meet ab at any point different from a (EUCLID III.18). See P22. Proof. If ac is not perpendicular to ab, on the latter there will be a point x not equal to a, for which (x, a) z (x, c) by P27,28, and so on, so that another point of ab, that is to say, the  x, c, a  point a/x , will lie on the sphere a c , as  a, b, c  P20 prescribes, without falling on a, which contradicts the hypothesis. And so on.—This and P22 taken together produce the theorem, “In order for a line and a sphere passing through the same point to touch each other at this point—that is, not meet each other elsewhere—it is necessary and sufficient that the line be perpendicular to the join of the center of the sphere with that point.”

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P31–Theorem. Whenever the following propositions should be true at once— “a, b, c are noncollinear points,” “d is a point in the plane abc that does not belong to ab nor coincide with c,” and “(a, c) z (a, b) and (a, d) z (a, b)” —it will likewise be true that the points a, c, d are collinear. Here we have in short the property that is usually enunciated, “Through a given point on a given line, it will not be possible to erect, in the same plane with the line, two different perpendiculars.” Proof. Having set br  b/a , we infer from it by P20,25 that br 0 b c 1 b d , and by P16 that c, d, b are not collinear. Therefore, if we will rotate the plane cdb onto itself about the points c and d as hinges, b will necessarily fall on br by P14; and the midpoint of the pair b and br, namely the point a (see P7), will have to lie in the line cd by P18. Therefore, a 0 cd, which was to be proved. P32–Theorem. Having assumed that a, b, c are noncollinear points, if a motion  is given in such a way that a should coincide with a, and b belong to ab without coinciding with b, and c belong to ac without falling on c, this motion will transform each point of the plane abc into its reflection across a. Proof. By virtue of P4 the hypothesis also implies ( b) = b and ( c) = c, so that the motion 2 will hold the points a, b, c individually fixed; therefore, it will fix any other point, by P11§1. Now, a point x of abc cannot be fixed by  if it is different from the point a, because the assumption “x 0 abc, x = / a, and  x = x” is equivalent to claiming that a, x, b be noncollinear, and thus (a, b) z (a, x) and (a, c) z (a, x) by P19, contrary to the hypothesis that a, b, c be noncollinear (see P31). Therefore, x = / x and (  x) = x. Consequently (see P8 and so on), the midpoint of x and  x is fixed by ; therefore, it coincides with a. P33–Theorem. Whenever the points a, b, c should not be aligned with each other, and (a, c) should be perpendicular to (a, b), there will be a motion that associates with each point of the plane abc its reflection acrossc a.50 Proof. Indeed (see P11 and a, b, / a a, b/ a, c so on), the product   of the two rotations   a, b, c  and   a, b, c  will hold a fixed, transforming b and c into the points b/a and c/a , so that nothing remains but to refer to P32. P34–Theorem. From the hypothesis “a, b, c are points, a different from b and from c, and (a, c) z (a, b) (that is, ac z ab), and  should be an arbitrary motion,” it follows that the pairs of points ( a, c) and ( a, b), just like the lines ( a)( c) and ( a)( b), are also perpendicular. That is, no motion can alter the relation of orthogonality of two lines. Proof. Let the reader see.

50

[Reflection across a is not itself a motion but an indirect isometry.]

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§3 Rotating one plane onto another. Orthogonality of lines and planes. Various properties relating to lines, planes, and spheres. POSTULATE XV P1. If a, b, c are noncollinear points, there exists at least one point outside the plane a b c . P2—Definition. Of four points, a, b, c, d it is said that they are or are not coplanar,51 according as there exist or do not exist three noncollinear points x, y, z whose plane xyz should contain a, b, c, d. Therefore, if the points e, f, g are not collinear, each one of the two propositions “h is a point of the plane e fg” and “the points e, f, g, h are coplanar” will be a consequence of the other. P3—Theorem. If a, b, c, d are noncoplanar points, they will also be three-by-three noncollinear, all four distinct, and each one external to the plane that joins the remaining three. Compare P10,14§1. POSTULATE XVI P4. Points a, b, c, d being noncoplanar, there must exist a motion which, transforming a as well as b into itself, should represent d by a point of the plane a b c. —A motion of this kind will result in transforming the plane abd entirely onto the plane abc, by P22,24§1; hence, one says that abd is rotated onto abc, turning about a and b, and so on.52 P5—Theorem. Given the noncollinear points a, b, c there will be a motion that transforms a into b; b into a point of ab not coincident with a; and c, again to a point of the plane abc. Thus, a plane can glide on itself, so that one of its lines chosen arbitrarily should likewise glide on itself. See P9§2. Proof. There will certainly exist a motion that should represent a by b, and b by some point of ab not coincident with a, by P9§2; let cr be the image of c. If cr 0 abc, nothing else is needed. If, on the contrary, the points a, b, c, cr are not coplanar, it will be sufficient to follow that motion by a rotation of the plane abcr onto the plane abc, conforming to P4 and turning around a and b as hinges. 51

[Pieri provided the abbreviation Cp for are coplanar; this translation does not employ it.]

52

As everyone sees, the two postulates XIII and XVI could, if desired, blend into just one. And, with a slight modification of the wording of principle XIII making it appear that the motion of which one speaks should transform the entire plane abc into itself, one would thus also be able to dispense with postulate IX, concealing it under a principle that affirms the possibility of rotating a plane onto itself. By similar rearrangements (either by suppression of some principles or through substitution of others, and so on), the present system would not be repudiated at all, especially if it had to respond to didactic requirements.

8.3 Rotating One Plane onto Another; Properties of Lines, etc. (§3)

275

P6—Theorem. Again, given the noncollinear points a, b, c there always exists in the plane abc some point cr for which the pair (a, cr) should be normal to (a, b). Or under another guise: given a plane, in this plane a line, and in this line a point, such as abc, ab, and a, there can always be erected, in that plane and through this point, a line perpendicular to the given one (EUCLID, I.11). That two different perpendiculars to ab cannot be constructed as above, is already stated in P31§2. Proof. Let x, for example, be the foot of the perpendicular dropped from c to ab (see P28§2). If x does not coincide with a, it is sufficient to rotate the plane axb onto itself so that the point x should move to a and the line ab return to itself, as is always possible by virtue of P5, and then refer to P34§2. P7—Definition. Two points b and c are called equidistant from a third point a, or this one equally distant from those, whenever the points b and c both lie on one sphere described around a as its center. See P28§1. P8—Theorem. The locus of a point equidistant from two given points a and b, distinct from each other and moreover lying in a given plane that should pass through them, is a line; in fact it is the line normal to ab through the midpoint of a and b. Proof. In this way, from P20,21,25–27,31§2; P6,7; and so on. P9—Theorem. If a, b, c are points aligned and distinct, and d should be an arbitrary point, each point common to two of the spheres d a , d b , d c will lie in all three. Proof. If d 0 ab, we refer to P16§2. If d ó ab, let x, for example, be a point common to the spheres d a and d b but different from d (which lies, as we know, in all three). Now, if x 0 abd we fall back on P23§2. If x ó abd on the contrary, we shall rotate the plane abx onto the plane abd, holding the points a and b as hinges (see P4); for this reason, each one of the two spheres x a and x b , which do not differ from d a and db , is transformed into itself; and the point x falls on d or else onto that point dr different from d but belonging, like d, to both of the circles d a and d b , by P13,14§2. If it does not fall on d, it will be sufficient to follow that by a rotation of the plane abdr onto itself about a and b, in order to say that a motion that should carry x onto d without affecting a or b is always possible. Therefore, x 0 d c , because such a motion also leaves c fixed, by P15,28§1. And so on. P10–Theorem. Points a, b, c being noncollinear, whenever d should be a point of the plane abc, the spheres d a , d b , and dc do not meet except at d. Proof. It can be supposed, for example, that d should belong to the join of a with x, where x is a point of bc different from a and from b (see P20,28§1), so that d b 1 dc f dx by P9, and therefore also d a 1 d x 1 d c f d a 1 dx . But the figure d a 1 d x consists only of d, by P16§2; therefore, so also does the other one, d a 1 d b 1 d c . P11–Theorem. But, having taken any point e whatsoever, each point common to the spheres e a , e b , and e c lies also on the sphere e d . Proof. If d falls on one of the joins bc, ca, and ab, for example on ab, apart from the case where it should coincide with a or with b, in order to conclude that e a 1 e b 1 e c f e d it is sufficient  d, e  to cite  c, d  P9. If not, it will be sufficient to recall that the lines ad and bc, or else

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bd and ca, or cd and ab necessarily meet (see P20§1), so that given the same P9, if dr 0 ad 1 bc, for example, it will be true that e a 1 e dr f e d and e b 1 e c f e dr at the same time, and therefore, that e a 1 eb 1 ec f e d . P12–Theorem. Let a, b, c noncollinear points, d be a point of the plane abc not coincident, however, with a, and then p, a point outside the plane. If it happens that each one of ab and ac should be perpendicular to the join of a with p, this will also be perpendicular to ad (EUCLID XI.4). Or rather, if with the two distinct lines ab and ac that meet at a, a third line pa should be constructed, normal to both at this point, this line pa also turns out to be perpendicular to any line that should lie in their plane and pass through their common point. See P27§2. Proof. The point p/a of the sphere p a (see P3§2) having thus to lie on the sphere p b as well as on the sphere p c by P20§2, and considering that by the hypothesis, (a, p) z (a, b) and (a, p) z (a, c) (see P27§2), also lies on the sphere p d by P11, so that (a, p) z (a, b) is established by P21§2. P13–Definition. Under the same hypothesis as P12, the line pa, perpendicular to all the lines in the plane abc that pass through the point a, is to be called orthogonal, normal, or perpendicular to the plane abc — ap z abc —and this in turn, normal or perpendicular to the line pa — abc z ap. In brief, a line and a plane are defined as perpendicular to each other whenever the line meets the plane orthogonally to two lines in it that are concurrent at the common point and therefore is perpendicular to every line drawn through that point in the plane, by P12. P14–Theorem. If a, b, c, d are noncoplanar points and the lines ab and ac perpendicular to ad, and moreover b and c should be equidistant from a, they will also be equidistant from d, and vice versa. See P2,7. Proof. Let acd be rotated onto abd, pivoting on a and d (see P4): 53 the new position cr of c will be in the join of a with b because (a, cr) and (a, c) are perpendicular to (a, d), by P26,27,34§2; and on the other hand, in the plane abd there do not exist perpendiculars to ad through the same point a that should not be coincident, by P31§2 and so on. Therefore, considering that c belongs to the sphere ba by the hypothesis, cr will have to fall onto b or else onto the point b/a by P2,3§2; thus, in each case cr, and consequently also c, will have to lie on the sphere bd by P20§2. The rest [is left] to the reader. P15–Theorem. Again, a, b, c, d being noncoplanar points, if it will happen that the lines ab and ac should be perpendicular to each other, as well as ab and ad, and ac and cd, then the lines bc and cd will also be perpendicular, so that the line cd will be perpendicular to the plane abc. See P13. This is the “theorem of the three perpendiculars.” 54 Proof. Let dr be the point d/c. Because (c, d) z (c, a) by P26,27§2, the point dr is on the sphere d a by P20§2; and since the lines ac and adr are both perpendicular to ab and thus, consequently, to the line adr thanks 53

[In error, Pieri wrote b instead of d here.]

54

Proved as in Legendre 1802, book V, proposition VI, 141. [Pieri cited proposition IV but not the edition.]

8.3 Rotating One Plane onto Another; Properties of Lines, etc. (§3)

277

 d, d, b 

to P12, one can invoke  b, c, d  P14, whence the point dr lies on the sphere d b as well; this is as much as to conclude cd z cb by P21§2 and so on, which was to be shown.— From here easily [follows] this theorem: P16–Theorem. To a plane and from an external point, a perpendicular line can always be dropped (EUCLID XI.11). To the extent that the substance of the Elements is developed from our principles, it will seem right that the language also go hand in hand, conforming to the methods and the traditional forms by now sanctioned by perennial usage over many centuries. In order that the greatest clarity and efficacy of these methods (due precisely to their broad use, and to their being as though born spontaneously from the more perceptible qualities of familiar objects, instead of descending from on high) should not be corrupted by any doubt and ambiguity about their ideal content, this in the final analysis must always be expressible pasigraphically by the signs Π and M. An expression such as, for example, “draw a line r perpendicular to the given plane from the given point p,” will generally mean “p being an arbitrary point and  an arbitrary plane, there exists a line normal to  and passing through p; let r be this line.”—And, as is done many times, saying “rotate the plane abc about the points a and b onto the plane ... ,” will often be as much as to affirm, “there exists a motion that, keeping the points a and b invariant, transforms the plane abc into the plane ... ;” or rather, as the case may be, “if a motion, without changing a or b, should transform the plane abc into the plane, ... .” And so on. P17–Theorem. If the points a, b, c, d are not coplanar, it is absurd to suppose that the lines ad and bd should both be perpendicular to the plane abc. That is, two lines perpendicular to one and the same plane cannot be drawn from an external point (EUCLID XI.13). Proof. Indeed, by that hypothesis, ad and bd would both be perpendicular to ab by P12,13, contrary to P29§2. P18–Theorem. To a plane  and through a point a on it a perpendicular line can always be erected (EUCLID XI.12). Proof. Let b be a point of  not coincident with a. The existence of distinct points in a plane cannot be denied, since by P20§1 for example, the statement “ is a plane” does not differ from another: “there exist three noncollinear points x, y, z and  is a name given to the plane x yz” (compare P18§1). Also, by postulate XV there certainly exists a point not lying in , and consequently also a plane , different from  and containing a line ab together with . In such a plane  draw the line ad perpendicular to ab (see P6). And then similarly in a plane containing ad but different from  draw ac perpendicular to ad, so that this last ad will be orthogonal to both of the lines ab, ac. Then, if we will rotate the plane abc about the points a, b as hinges (see P4), after which cr and dr, for example, will be the positions of the points c and d, the line adr will turn out to be perpendicular at once to ab and acr, both lying in . See P34§2 and P13.

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P19–Theorem. If a, b are noncoincident points, each line that should pass through the center a of the sphere through b about a meets it at two distinct points. Proof. Having taken a point c that should not belong to the join of a with b (see P16§1), a point d can always be selected outside of the plane abc so that the join of a with d should be normal to both the lines ab and ac, by P18. Then that motion, which rotates the plane abd onto the plane adc holding a and d in place (postulate XVI), will move b to some point of the join of a with c by P31,34§2, without removing it from the sphere b a (see P28–30§1), so that this must meet that join. The rest is already told in P1,2§2. P20–Definition. To say that the points a and b should be symmetric to each other with respect to a line r or with respect to a plane  will be like asserting that these points are both on a line that should meet the line r or the plane  orthogonally at the midpoint of a, b. Thus, by means of a line r or a plane  there is determined a certain representation of points by points (of space into itself), such that each point x of r or of  is associated with the same point—that is, the reflection of x with respect to x (see P3§2)—and an arbitrary point y that lies neither in r nor in  is associated with the symmetric point with respect to the foot of the normal dropped from y to r, or to  (see P28,29§2 and P16,17). This transformation or geometric correspondence is to be called, conforming to custom, axial symmetry with respect to r in the one case, and planar symmetry with respect to  in the other; but the names half-turn or overturning 55 about r, and reflection across  are also suitable. It will be possible to indicate the image of a point z by the symbol z/r or z/ . A planar symmetry certainly cannot be equivalent to a motion, for the reason that all the points of a certain plane  are fixed there, but not all the points that exist, (see P11,20§1 and P1); whereas on the contrary, an arbitrary axial symmetry is not distinguishable from a certain motion, as appears from the following: P21–Theorem. Given the noncollinear points a, b, c, if a motion , altering neither a nor b, should transform c into its symmetric point with respect to the join of a with b, the same motion will also have to change any other point z into its symmetric point z/ab. And such a motion actually exists. See P20. Proof. Indeed, by P5,20,22,24§1, P17§2, and so on, the operation 2 will be a motion that holds a, b, c individually fixed, and hence every point fixed, by P11§1. Therefore, if zr  z, the points z and zr are permuted with each other by . Therefore, by P5–7§2 and so on, their midpoint x  z|zr will have to be fixed and thus will have to lie on ab, for the reason that , by the hypothesis, is a proper motion. It follows by P19,27§2 and so on that either z = zr = x or xz z ab, and thus in each case, zr = z/ab. The motion  is none other than the rotation of the plane abc onto itself about a and b as hinges (see P11§2), and so on. 55

[Pieri’s word was ribaltamento, which is cognate with rabatment, a term familiar in descriptive geometry, which Pieri was teaching. See subsection 9.1.6, page 338. It has also been translated here as rotation.]

8.3 Rotating One Plane onto Another; Properties of Lines, etc. (§3)

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P22–Theorem. Whenever a, b, c should be noncollinear points and p, q, points not on the plane abc nor aligned with a, it will be impossible that both ap and aq be lines normal to the plane abc (EUCLID XI.13). Proof. It can be conceded that ac be perpendicular to ab. We shall prove that an absurdity follows from the hypothesis that each one of ap, aq be perpendicular to each one of ab, ac. Having set br  b/a , cr  c/a, pr  p/a, and qr  q/a , carry out the motion that rotates the plane abc onto itself about a and b as hinges. The points c and cr will come out interchanged; also, consequently, p and pr by P21, and at the same time also q and qr, since that hypothesis induces pr = p/ab and qr = q/ab by P20. But the point b will remain fixed. Afterwards, perform a new rotation of the plane abc onto itself, but this time about a and c as hinges. Again the points p and pr will be interchanged, and at the same time also q and qr, while b will go to br. Therefore, in conclusion, the resultant motion, or product of the two rotations (of the two symmetries, the one with respect to ab, the other with respect to ac) would leave unchanged each one of the points a, p, q, while it would carry b into br. By P11§1, since b = / br, this contradicts the hypothesis that a, p, q should not align. The fact that two distinct perpendiculars to a plane cannot be drawn from the same point on it (P22) serves to exclude the possibility of a fourth dimension in Π, according to the common significance accorded to the dimensions of a body. Among the postulates admitted until now, it seems to be that VIII weighs more in such a conclusion. Reasons would not even be lacking to argue how postulate VIII in itself (or better, together with the preceding I–VII) should impose on the class of points a restriction so that one could say that space does not have more than three dimensions. Whether all that should conflict with or be advantageous to a system of elementary geometry, let the reader decide. Whoever might want, moreover, a fourdimensional geometry without deviating too much from this will be able to begin, for example, by substituting for the definitions of collinear points and of line (P9,12§1) two similar definitions of coplanar points and of plane, and for VII and VIII two similar propositions about three or four points respectively; but this is not the place for more detailed specifications. P23–Theorem. Should points a, b, c be noncollinear, and points d, e external to the plane abc, if the lines ac, ad, and ae are all three perpendicular to the join of a with b, all three will have to lie in one and the same plane (EUCLID XI.5). That is to say, there do not coexist two distinct planes, perpendicular to one and the same line at the same point. See P13. Proof. In the two planes acd and ace, the one and the other perpendicular to ab by the hypothesis, let us construct respectively the lines adr and aer perpendicular to ac (see P6). These lines will also fall orthogonally on ab by P12; therefore, they will both be perpendicular to the plane abc. Therefore, by P22, they will coincide; and with them, also the planes acdr and acer (see P23§1), which are not distinguished from the first ones, acd and ace (see P24§1). And for the reason that just one plane contains all of the lines perpendicular to the given one at a given point in it, one can speak of the plane normal to the line at

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that point as a single individual, uniquely distinct, in the same way that, according to P16–18 and P22, phrases such as “line normal to the plane  at the point a on it or from the external point p” acquire the sense of a proper name, which otherwise they would not possess. P24–Theorem. Two noncoincident planes that have a point in common meet along a line. Proof. Let r and s be the lines perpendicular to those planes  and  at the point a, supposed common to them (see P18); for certain, they cannot be fused into one, as long as the planes do not coincide, by P23. The line t, perpendicular to both lines r and s at the point a (see P18), will lie simultaneously in the planes  and  by P23. In most modern treatments this theorem is proved, together with others not less fundamental, by means of premises about the parts into which a line is said to be divided by any one of its points, or a plane by one of its lines, or space by a plane. I omit some difficulties that are met in wanting to define exactly these concepts pertinent to analysis situs (see the following section). But I do not know how to restrain myself from questioning whether it should not be better to avoid them when one can without effort as here (often better still, introducing clarity and simplicity where not seldom are found obscurity and complication) and they appear every such time, mostly distanced from the offices of elementary geometry rather than necessary.56 P25–Theorem. The locus of a point equidistant from two given points a and b noncoincident with each other will be the plane normal to the join of a with b, at the midpoint of a and b. Proof. This is from P7,8,23, and so on. P26–Theorem. Points a, b, c being noncollinear, the spheres c a and c b will meet at all the points of a circle whose center is the foot of the perpendicular dropped from c to ab, and whose plane is normal at this point to ab. See P13§2. Proof. Let d be that point of ab for which cd z ab (see P28,29§2), and e, a point external to the plane abc such that also de z ab (see P1,6, and so on), so that cde will be the plane normal to ab that can be constructed from the point c or at the point d. If the plane abc is rotated onto the plane abe, turning about the points a and b, the other point c will fall on the line de without leaving any one of the three spheres c a , c b , and c d , so that in each line normal to ab erected from the point d are found points common to the four figures cde, c a , c b , and c d . But vice versa, every point common to any two of these figures will lie in all four, as by now the reader sees without any effort.—To the reader [is left], as well, another P27–Theorem. If a, b, c, d are noncoplanar points and the plane abc should not be normal to the line da, this plane and the sphere a d will meet in a circle whose center is the foot of the normal dropped to the plane abc from the point d. The restriction ad z ª abc can be removed freely.

56

See Peano 1894b, 51–54.

8.3 Rotating One Plane onto Another; Properties of Lines, etc. (§3)

281

P28–Definition. Two figures are to be called congruent to each other, or superposable, whenever there will exist a motion that should represent one figure by the other, point for point, so that in short, the one figure is transformed precisely into the other by virtue of that motion. See P3§1. Such a relation between two given figures f and f r is certainly conversive or symmetric (see P4§1). Each figure is congruent to itself, for the reason that the identity transformation is a motion by P6,7§1. And two figures congruent to one and the same figure are likewise congruent to each ~ f r will be used in place of “f is congruent other by P5§1. The expression f = 57 to f r.” This notion of congruence is very nearly the same as that of motion. No other difference is encountered substantially but this: the one focuses on figures connected and related through it; the other embraces every point and relates the whole space to itself. Thus, our principles can be expressed easily (following the example of PASCH, and with slight modifications of form) by the entities point and congruent ~. And figures: in short, attributing the quality of a primitive idea to the relation = this way would perhaps be preferred, because it is less abstract, in a book for the young. The custom of designating two figures congruent to each other as simply equal, however it be done, will perhaps sooner or later be rejected (not impeding the advantage of using words in a sense commonly accepted since antiquity), because there is no lack of examples of misunderstandings born, so to speak, from the influence of that terminology. P29–Theorem. The lines ab and cd are congruent, where a and b are distinct points, as well as c and d. Thus also are two planes given at pleasure, two pairs of c orthogonal lines, and so on. Proof. Thanks to P6§2 there exists a motion   a  ; let br  b, so that (ab) = cbr by P19§1. If br ó cd, nevertheless the line cd and the sphere brc will meet by P19, for example in the point bO. By P28§1 that is as  c, b  much to say, there exists a motion   c, b  . There is consequently a motion –1 by which the line ab is transformed into cbO: that is to say, into cd, by P4,5,17§1. And so on. P30–Theorem. Let points a, b, c, d be noncoplanar and ad be supposed perpendicular to the plane abc; furthermore, de and df perpendicular to da in the planes abd and acd, respectively. The two pairs of lines (ab, ac) and (de, df ) will then be congruent. In brief, two normal sections of the same “dihedron” 58 are always congruent. Proof. See, for example, Faifofer 1886, 402. Putting o  a*d (see P7§2), we shall construct in the planes abd and acd the lines op and oq perpendicular to ad, and suppose the points p and q equidistant from o, taking q on the sphere p o , for example (see P19). After that, having set m  p*q, the 57

[For Pieri, congruence meant direct congruence. His congruence symbol was w .]

58

[Pieri’s term was diedro, by which he meant union of two distinct planes. This differs from a contemporary description of diedro as a dihedral angle (Faifofer 1886, 401), and from modern definitions of a dihedron as a degenerate polyhedron with two superimposed planar faces, viewed from one side and from the other.]

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symmetry with respect to the axis mo (see P20), which is not different from the rotation of the plane opq onto itself about the points m, o as hinges (see P21, and so on), will permute the points a and b, the one with the other, in the same way as the planes abd and acd; and also the lines ab and df with each other, as well as ac and de. Therefore, the pair (ab, ac) is transformed into (df, de) and, if desired, into (de, df ) with the addition of an overturning like that just now used to exchange the lines op and oq with each other. And so on. P31–Theorem. Given that a, b, c should be noncollinear points and e, f points external to the plane abc, in such a way that ae and bf are perpendicular to that plane, it will be necessary that the four points a, b, e, f be coplanar (EUCLID XI.6). Proof. Indeed, if in the plane abe are constructed the line bf r and in the plane abc the lines ad and bdr, all three perpendicular to ab, the pairs (ad, ae) and (bdr, bfr) will turn out to be congruent to each other by P30. From this it arises that bf r z bdr (see P34§2), for the reason that ae z ad by the hypothesis (see P12,13). Therefore, bf r is normal to the plane abc, and consequently bf r = bf by P22, which was to be proved. P32–Theorem. Two triples of points (a, b, c) and (d, e, f ) are congruent to each other if these pairs are so as well, at the same time: (a, b) and (d, e), (b, c) and (e, f ), (a, c) and (d, f ) [EUCLID I.8]. Proof. First of all, I suppose a, b, c noncollinear. ~ (d, e), there must exist a motion   d, e  according to P28; and if Since (a, b) =  a, b  cr  c, it will be necessary that d, e, cr not be collinear. Now from the assumption ~ (d, f ) it will arise that (d, cr) = ~ (d, f ); therefore, cr 0 f and in the same (a, c) = d way, cr 0 fe . Thus, rotating the plane decr about the points d, e [as hinges] onto a plane  containing the points d, e, f, the point cr will fall onto one of the points common to the two circles fd , fe ; thus, by P14§2 it will either fall directly onto f or it will come into coincidence with f after the rotation of  onto itself about the points d, e. The case where a, b, c are collinear remains, but that is left to the reader.

§4 Points internal or external to a sphere. Segments, rays, half-planes, angles, and so on. P1—Definition. Whatever may be the points a and b, this symbol— Sfr(a, b) — denotes the sphere through a or through b, constructed about the midpoint of a, b as center: Sfr(a, b)  aa*b = ba*b  Sfr(b, a). See P28,30§1; P7§2. In the discussion we shall sometimes call it the polar sphere59 of a, b; at other times, “sphere normal to the pair a, b”; and the points a and b will also be known as poles of this

59

[Pieri wrote “sfera polare—o polosfera.”]

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sphere, or as diametrically opposite points. If the points a and b coincide, Sfr(a, b) contracts into one point. P2—Definition. Whenever a and b should be points, we shall say “x is a point internal to the sphere b a” when there exist two noncoincident points on it that should have midpoint x. And it will be said that “z should be a point external to the sphere b a” if there do not exist on it two points, not even coincident, for which the midpoint should be z. P3—Theorem. Any point that should not belong to the sphere b a will have to be internal or external to the sphere; nor can it be the one and the other at once. No point on the sphere b a can be internal or external to it. If points a and b are distinct, the center a of the sphere b a will be internal. But if the points a and b coincide, no point will be internal to the sphere b a ; or rather, every point different from b will be external. And so on. Proof. That no point internal to b a , for example, can lie on b a (whatever else the points a and b might be) emerges from P7,15§2. And so on. P4—Theorem. Provided that a, b, x be points and a, b should not coincide, to say that “x is internal to the sphere b a” is equivalent to admitting one of two cases: that x must coincide with the center a of the sphere, or that the plane perpendicular at x to the join of a with x should meet the sphere at some point different from x. And also these two propositions are equivalent: “x is a point external to the sphere,” and “x does not coincide with a, and the plane perpendicular at x to the join of a with b does not meet the sphere.” According as x is internal or external to b a , each point of the sphere x a will be internal or external to b a . And so on. Proof. If x = / a but on the sphere there exist two distinct points, b and c for example, such that x = b*c, then it is obvious how the line bc should be perpendicular to ax, by P21§2 and so on; and how the plane perpendicular to ax at the point x should contain bc, by P13,23§3: that is to say, should meet the sphere at the points b and c. See also P27§3. The rest [is left] to the reader. P5—Definition. The points of a plane are called internal or external to a given circle in that plane according as they are internal or external to the sphere containing that circle, and having its center on the plane. See P13§2. Therefore, if a plane should contain the points a, b, x (a and b distinct from each other), x will be internal to the circle b a in that plane only when x coincides with a or when the line drawn through x perpendicular to a x in that plane meets the circle elsewhere than at x. See P19,27§3 and so on. P6—Definition. Whenever a and b are points, it will be possible to say that x lies between a and b, or in the interval from the one point to the other, to express that x is aligned with a and with b and, moreover, internal to the polar sphere of a, b. See P1,2 and so on. Therefore, by P3,4 and so on, neither a nor b —and if a = b, no point—lies between a and b; but if a = / b, between these two always lies their midpoint. And so on.

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P7—Definition. Provided only that a, b be points, by segment a, b will be understood the figure consisting of all those points that lie between a and b, or that coincide with a or with b. This figure is represented by the sign *ab*, as in “line terminated by the points a and b,” which are the extremities of the segment, whereas the points that lie between a and b are the points internal to *ab*. And so on. It will be manifest that *ab* = * ba*; that if a = b, the figure contains only one point; and that when a and b do not coincide, the segment a, b is none other than the locus of all points on ab for which it should be possible to construct to this line a normal line that should meet the polar sphere of a, b (see P1–6). All these figures—namely, the polar sphere of a, b, the class of points internal to the sphere b a , the line segment *ab* —are covariants of a and b with respect to every motion; and all have the most essential function in elementary geometry. The notion of terminating line or segment that is generally introduced in the nature of a simple or primitive geometric entity is decomposed here into the elementary concepts point and motion, and reconstructed from these only, with successive merely nominal definitions.60 Perhaps also the use of this notion is excessive in ordinary treatments, and it will be possible to reduce it to a smaller role, as it appears from the content of the preceding sections 6.1–6.3, which are entirely independent of it. POSTULATE XVII P8. Points a, b, c being aligned and distinct, if a plane perpendicular to the line at a point different from a, b, c meets one of the polar spheres of the three pairs ( a, b), ( a, c), and ( b, c), it will also have to meet another. See P1. Indeed, by P7 and so on, this is in brief the same as “Given four points a, b, c, d, collinear and distinct, it will never happen that d should lie in just one of the three segments *ab*, *ac*, and *bc*. POSTULATE XVIII P9. If a, b, c are points and c lies between a and b, no point can lie at once between a and c and between b and c. See P6. Therefore, under this hypothesis a plane perpendicular to the line cannot simultaneously meet outside of it the two polar spheres, of a, c and of b, c. 60

Against our definition of segment (P6,7), one perhaps will be able to counter that it might be less simple than what we should wish in a definition. I admit that no criterion for such simplicity was ever established a priori; but I still concede the fairness and goodness of that observation and respond in this manner. Discovering, for example, that a certain axiom (c) should be a consequence of certain other propositions (a) and (b), they are acknowledged and recognized equally as primitive. Who will want to maintain that proposition (c) is not to be removed from the axioms, just because its deduction based on (a) and on (b) might be expressed less simply than what one wishes to display? And surely the proposals for defining an object by means of other objects, and for deducing a fact from certain other facts, are not so remote from each other and dissimilar that the one should be denied without some reasons that do not contest the other.

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P10–Theorem. If the points a, b, c are such that c should lie in the segment *ab*, the entire segment *ac*, and in the same way *bc*, will be contained in *ab*; and each point internal to *ac* or to *bc* will be internal to *ab*. See P7. Proof. If a = b, no words are needed. Let us suppose that a and b should not coincide. Then every point of *ac* distinct from a and from c will be internal to the one or to the other segment *ab* and *bc* by P8. But by P9 it cannot be internal to *bc*; therefore, it is internal to *ab*. And so on. P11–Theorem. Under the same hypothesis as P10, it cannot happen that b should lie between a and c, nor that a should lie between b and c. Proof. The case a = b, or else a = / b but c coincident with a or with b, is entrusted to the reader. If a and b are distinct points and c internal to *ab*, then, in order that b should lie between points a and c, it would be necessary that each point internal to *ab* be internal to *ac* by P10; whereas c is certainly not internal to *ac*, by P6,7. P12–Theorem. And, in the hypothesis mentioned, *ab* is composed of all the points that belong to *ac* or to *bc*, without distinction, and of only these. Proof. Indeed, if a, b, c are distinct, let a point x be selected between a and b. Then by P8 the plane perpendicular at x to ab will have to meet the one or the other of the two polar spheres of a, c and b, c, and consequently x [will have] to lie in at least one of the two segments *ac* and *bc*, by P7. The rest is said already in P10. P13–Theorem. Provided that a, b be points, the spheres through b with center a and through a with center b intersect. Compare EUCLID I.1. Proof. It can be assumed that a and b should not coincide, because in the case a = b there is nothing to say. Having set m  a*b and ar  a/b, we shall be certain that m 0 *ab* and b 0 *aar* by P6, and consequently that m 0 *aar* by P10; better still, m is internal to this segment. Therefore, the plane orthogonal to the line at m meets the polar sphere of a, ar, namely the sphere a b , at some point different from m, by P7. Let x be such an intersection, so that mx z ab by P12,13§3. Now if the plane ab x is rotated onto itself turning about its points m and x, the points a and b will as a result be exchanged, the one for the other, and the spheres a b and b a permuted, so that the point x, which does not change, will also have to stay on b a . —This is in fact the proof that there exists an equilateral triangle having a given segment *ab* as edge: see P14§5, and so on. P14–Theorem. From supposing points a, b, c noncollinear and (a, c) z (a, b), it follows that a will be internal to the sphere c b and c external to the sphere a b . Proof. That a should be a point internal to c b follows at once from P20§2 and P2. The sphere c b and the join of a with b will meet at points d and dr, which are distinct from each other and from the point a by P19§3 and so on; accordingly, let ar a/b. For the second part of the theorem it is sufficient to prove that d or dr be external to the segment *aar*, considering that d is transformed into c by a motion that does not alter a b at all. Now the points a and ar, being internal to d b , both lie between d and dr by P6. Therefore, by P11 it cannot happen that

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d should lie between a and dr, nor dr between a and d. On the other hand, ar must be found in one of the two segments *ad* and *adr* by P12. Therefore, if ar 0 *adr* and consequently *aar* f *ad* (see P10), it cannot happen that d should belong to the segment *aar*, for the reason that, this being different from the points a and dr, it cannot fall between a and dr. Nor when ar 0 *ad*, hence *aar* f *ad*, will it be possible that dr should belong to *aar*, which would be like wanting dr to lie between a and d. And so on. Hence, almost immediately, as a corollary: a line tangent to the sphere a b —that is, which should meet the sphere in a point, for example a, without meeting it in two separate points (see P22, 30§2)—cannot contain any point internal to the sphere. On the contrary, all of its points will be external, with the exception of just one, the point of contact. Also, this theorem: the segment *ba* perpendicular to *ac* in the hypothesis of P14 will always be less than the oblique segment *bc*. That could of course have been demonstrated in P14, where the definitions of less and greater would be anticipated here (see §5). From P14 one would also deduce that if a, b, c are points and a is internal to c b then c will be external to a b . Proof. If a, b, c are distinct, there exists on c b , by P2, a point cr different from a such that (a, cr) z (a, b). This point, and consequently also c, will be external to a b by P4,14.—But considerations of this sort seem better postponed until the relations less or greater, from which they proceed in a more efficient and straightforward manner. P15–Theorem. From a, b 0 Π and c, d 0 *ab* follows *cd* f *ab*. Proof. Certainly d 0 *ac* or d 0 *bc*, by P12. In the one case, *cd* f *ac*; in the other, *cd* f *bc*; and in both, *cd* f *ab* by P10. P16–Theorem. From supposing a, b 0 Π, c 0 *ab*, and d 0 *ac*, it always arises that c 0 *bd*. Proof. If d were coincident with a or with c, the definition of segment would be sufficient (see P7, and so on). Without doubt, d should always be a point of *ab* (by P10) and therefore c will have to belong to the one or the other segment *ad* or *bd* if not to both, by P12. But by the hypothesis and thanks to P11, c cannot lie between a and d; therefore, in case c be different from these two points, it will have to fall in *bd*. Now indeed, provided that d should not coincide with a or with c, as must be supposed, c will still be different from a, since otherwise d would not be able to belong to *ac*. POSTULATE XIX P17. Given the noncollinear points a, b, c, whenever a line lying in the plane a b c should pass between a and b —that is to say, should meet the join of a with b between these two points—it will also have to pass between the points a and c or between the points b and c , provided, however, it does not contain any of a, b, c. Thus also [postulated] Moritz

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PASCH, in Grundsatz IV concerning flat surfaces.61 In a more concise and polished form: “In the plane of the noncollinear points a, b, c there does not exist a line that should meet only one of the three segments *ab*, *ac*, and *bc*.” 62 P18–Theorem. If a, b, c are collinear points, one of three assertions is true: that c belongs to *ab*, or that b belongs to *ac*, or that a belongs to *bc*. Proof. We shall treat the case that a, b, c be distinct, showing that from c ó *ab* and b ó *ac* it follows that a 0 *bc*. Let d be a point outside the line (see P16§1), and e a point lying between b and d, for example their midpoint (see P6). The line ce passes between b and d, but not between a and b; therefore, it will pass between d and a by P17, cutting the segment *ad*, for example at f. Because f lies between a and d, whereas b does not lie between d and e (see P11), the line bf must pass between a and e by P17, cutting the segment *ae*, for example at g. And because g lies between a and e, but b not between a and c, the point f will have to fall between c and e by P17. Finally, the same P17, with respect to the points e, b, c and to the line da, makes it evident that the point a must lie between b and c, which was to be proved. P19–Theorem. If a point d lies between the points a and b and between the points a and c, it cannot be between b and c. Proof. It emerges that a, b as well as a, c are distinct, and that the conclusion is verified if b = c (see P6). Therefore, let b be different from c. Now, if c falls between a and b, or b between a and c, it emerges that the conclusion is still satisfied, because no point can lie at once between a and c and between b and c in the one case, or between a and b and between b and c in the other, by P9. It remains to suppose that a [falls] between b and c (see P18), for the reason that the points a, b, c are distinct; but this contradicts the fact that d is at once between b and a and between c and a (see P9). P20–Theorem. Points a, b, c being collinear and distinct, any point d that should be external to *ab* but internal to *bc* will be internal to *ac*. Proof. Of the three hypotheses c 0 *ab*, b 0 *ac*, and a 0 *bc* (see P18), the first is to be excluded at once, because it would yield *bc* f *ab* by P10, and therefore d 0 *ab*, contrary to the assumption. The second causes *bc* to be contained in *ac* by P10, thus verifying the conclusion, for the reason that d 0 *bc* and does not coincide with b or with c. The last implies that *bc* be the union of the segments *ab* and *ac* by P12, or rather that *bc* – *ab* = *ac*: 63 here again, consequently, it follows that d 0 *ac*. And so on. As in P17, and thus in the following P21–24, it will be easy to perceive the bestknown property of a triangle relative to its boundary. 61

Pasch 1882b, 21.

62

Had we merely stipulated a, b, c as distinct instead of noncollinear points, principle XVII would of course remain affirmed.

63

[This equation should be |bc| – |ab| = |ac| – {a}, since Pieri’s segments are closed. This does not negate the conclusion.]

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P21–Theorem. If the points a, b, c are noncollinear, no line of the plane abc can pass at once between b and c, between c and a, and between a and b. Or rather, three points ar, br, cr situated respectively between b and c, [between] c and a, and [between] a and b, will never be collinear. Proof.64 If the point ar could lie between br and cr, the line bc would have to pass between br and a or between cr and a by P17, hence c, to lie between br and a, or b between cr and a, contrary to P11. In the same way it is proved that br does not lie between ar and cr, nor cr between ar and br; therefore, the points ar, br, cr are certainly not collinear, by P18. P22–Theorem. If the points a, b, c are not collinear, and should the points ar and br be taken in the segments *bc* and *ca*, there will have to exist a point common to the segments *aar* and *bbr*. Proof. Suppose ar different from b and from c, and br different from c and from a. Postulate XIX, asserted with respect to the points b, c, br and to the line aar, ensures, considering also P11, that this line cuts the segment *bbr* at one point. Just as well, the line bbr will have to intersect segment *aar* at one point. But these two points certainly will coincide, the one and the other being common to both of the lines aar and bbr, which cannot coincide, by P17§1 and so on. P23–Theorem. Given the noncollinear points a, b, c and given a point d in the plane abc but not in any of ab, bc, or ca, it will be necessary that the line da should pass between the points b and c, or else db between c and a, or cd between a and b. Proof. It is known from the definition of the plane abc (P20§1) that one of the three pairs of lines ad, bc and bd, ca and cd, ab must have an intersection point. Therefore, we can suppose, for example, that cd should meet with ab at the point dr. Then, by P18, it will be possible to consider these hypotheses about dr: dr 0 *ab*, b 0 *adr*, and a 0 *bdr*. If dr 0 *ab*, the line cd passes between the points a and b. If b 0 *adr* while d 0 *cdr*, the line ad passes between b  d  and c in accordance with  a  P17 and P11. If b 0 *adr* but d ó *cdr*, then bd  d  passes between a and c, for the same reason, by  b  P17. Finally, if a 0 *bdr*, all will go as before, except for exchanging the points a and b with each other. P24–Theorem. But if ad will pass between b and c, and bd between c and a, the remaining cd will pass between a and b. Proof. If the lines ad and bd will cut the segments *bc* and *ca* at ar and br, the point d will be internal to the segment *aar* by P22, while c will be external to *bar* by P11. From here the  a  conclusion follows, by  c  P17. Reported here are the definitions of extension of a given segment to the right or to the left, of half-line or ray, of half-plane, and of convex plane angle, chiefly following the cited work of Giuseppe PEANO.65

64

See Pasch 1882b, 25.

65

[Peano 1894b.]

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P25–Theorem. If a and b are noncoincident points, the shadow of a from b, or the extension of *ab* on the side of a, is the class of points for each one of which, x for example, the point a lies in the segment *bx*.66 The point a and the reflection of b across a belong to the shadow of a from b, and the points b and a/ b to the shadow of b from a (see P7§2; P6,7). P26–Theorem. Whenever a and b should be distinct points, there does not exist any point common to the two extensions of *ab*, nor any point that lies in *ab* and in the extension on the side of b, if b be excluded; but the segment *ab* with its extensions, the shadow of a from b and the shadow of b from a, constitute the line ab. Proof. The last matter is clearly addressed in P18, in the presence of P25. It emerges from P7,11 that a contradiction should arise from the assumption, “x belongs to the shadow of a from b and to the shadow of b from a,” that is to say, by P25, from a 0 *b x* together with b 0 *a x*. And in P11 it is also stated that a point x in *ab* should coincide with b whenever b 0 *ax*, if it is considered that, by the hypothesis, b 0 *ax* cannot happen with x = a, and that x 0 *ab* and b 0 *a x* cannot both hold if a, b, x are distinct, by P11. P27–Definition. By means of noncoincident points a and b there will also be given the “half-line from a through b,” or “ray from a toward b,” which is the class of points lying in *ab* or on the extension of *ab* on the side of b. See P7,25. Let the point a be the origin of this figure, which, as the line terminated just at a, can be designated by *ab. By what preceded, the shadow of b from a and the shadow of a from b are rays; and if ar  a/ b, the shadow of b from a does not differ from the ray *bar. Even more: if c is a point of the half-line from a through b, the two rays *ab and *ac will coincide.67 P28–Theorem. Given at pleasure a line r and a point a on it, there exist two rays contained in r, the one and the other emanating from the point a as origin, and that together constitute the line, but do not have any point in common different from that origin. Take in r a point b not coincident with a, and the point br  b/ a : then *ab and *abr will be two such half-lines. Proof. If these should have had a point in common (not counting a) they would coincide by P27, whereas the point br, which belongs to the ray *abr, does not belong to *ab (see P6,7,11, and so on). If an arbitrary point x of the line lies in *bbr*, it will be in the one or in the other ray *abr or *ab, depending on whether it will fall (see P12) in *abr* or in *ab*. And if it does not lie in *bbr*, it will be in the one or the other, depend x, b  ing on whether (see P18) br 0 *b x* or b 0 *brx*, in the manner of  c, d  P20,  b, x, b    P20, and so on. b, c, d 



66

[This sentence is a definition.]

67

Here for example, and more often as follows, the desire to shorten the path leads us to suppress any scheme of deduction for certain facts that are considered easier to demonstrate than certain others. Here this license is excused, even if the choice will not always seem (nor actually be) the most appropriate among all.

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Therefore, these phrases acquire a precise meaning: “the points c and d are situated on the same side of a,” or else “on opposite sides with respect to a,” it being possible to express with them how the points c and d are both in only one of these two rays, *ab and *abr, or else the one in one ray, for example *ab, and the other in the other ray *abr. Two events like these will never occur simultaneously if both the points c and d should be different from the point a; but one of the two relations will hold in each case. Briefly: the point a breaks the line r into two half-lines *ab and *abr, which can be called the two sides of a on the line. And so on. P29–Theorem. Two rays given at pleasure are always congruent to each other; and if they coincide they have the same origin. See P28§3. Proof. If a, b, c, d 0 Π, a = / b,  c, cd  and c = / d, there will exist a motion   a, ab  by P29§3; and since the attribute of half-line is an invariant of motion (like the attribute of segment), it will be necessary that (*ab) = *c( b). For that reason, if b 0 *cd then certainly (*ab) = *cd will hold, by P27; and if b ó *cd, the symmetry  with respect to an axis perpendicular at c to the line cd (see P6,20,21§3) will carry the point b onto the ray *cd, so that the motion  will cause the ray *ab to be superposed on *cd. If two rays *ab and *arb should actually have been able to coincide without having the same origin, b being a point common to them different from the origins a and ar, then either ar 0 *ab* or b 0 *aar* by P7,27. In the one case, each point x between a and ar would certainly not be in *arb* nor such that b 0 *arx*, by P9–11; consequently, while lying in *ab, it would not lie in *arb. In the other case any point y different from a, ar, b for which ar 0 *a y*, and consequently such that  y, a, b  ar 0 *b y* by reason of  b, c, d  P16, would not be in *arb* nor such that b 0 *ary*, by P11; consequently, although belonging to *ab, it would not belong to *arb because b 0 *a y* by P10. P30–Definition. If r is a line, and p a point outside it, the shadow of r from p should be the class of points x, for each one of which the segment *px* meets r. The halfplane from r toward p, or *rp, should be the class of points y such that r should not meet * p y*,68 or should meet it at y. It is left to the reader to demonstrate how a half-plane should also be the shadow of r from p —for example, setting pr  p/ r (see P20§3), how the shadow of r from p should be precisely equal to the half-plane *rpr, and so on (see P17,21)—and furthermore, how the half-planes *rp and *rq should coincide whenever q should be a point of the half-plane *rp but not belong to r. And so on. P31–Theorem. If a line r be drawn in a plane , the plane will be divided into two halfplanes not having any point in common outside r; but indeed such that, taken together, they constitute the plane: these are the two sides of r in the plane . In brief, the intersection of the two half-planes will be equal to r and the union, to . Two half-planes, even if they should lie in different planes, are always congru68

[Pieri defined alternative phrases, semipiano “r per p” and “da r verso p”; this translation uses the second. In error, he wrote *a y* instead of * p y*.]

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ent to each other. And so on. Proof. There exists in  some point not lying on r (see P20§1 and so on): let it be p for example. And the reflection of p across r having been called pr, those two parts of  will be precisely *r p and *r pr. The rest [is left] to the reader. Compare P28,29. P32–Theorem. Given the noncollinear points a, b, c, any plane  that should pass between the points a and b —that is to say, that should meet the segment *ab* without containing a nor b —also passes between b and c, or will pass between c and a, or will pass through c. Compare P17. Proof. It is sufficient to consider that a plane that should meet the line ab must meet the plane abc along a line, by P24§3; and so on.—One can therefore speak of the two sides of a plane , considering that, as happens for a plane with respect to a line that lies in it, and for a line with respect to a point, space is divided by the plane  into two distinct figures. These are (in the preceding hypothesis) the half-space from  toward a, and the half-space from  toward b,69 called again, if one wishes, the shadow of  from b and the shadow of  from a: the first being the locus of each point x for which it happens that *a x* should not meet , or meets it at x, hence also the locus of a point x such that *bx* should always meet , and so on. Compare P28,31.70 P33–Definition. If a, b, c are points, a different from b and from c, by the convex angle of the rays *ab from a toward b and *ac from a toward c will be understood the figure containing a and all those points different from it, for each one of which, be it x for example, the half-line *ax from a toward x should meet the segment *bc*. See P7,27. That figure will also come to be designated by ^ a Abc. The vertex of the angle ^ a Abc is the point a; the edges, the half-lines or rays *ab and *ac. Any point of the angle that does not lie on either edge is called internal to the angle. If the point d will be internal, the entire ray *ad will be inside the angle. If the points a, b, c are collinear, the angle ^ a Abc is reduced to a half-line or to a line, according as the points b and c should lie on the same side a Acb, and so on. or on opposite sides of a. It is surely evident that ^ a Abc = ^ With this, the “plane angle” is still not introduced in all its customary generality, as sooner or later it is demanded, for example, that more angles given arbitrarily be accounted for. Only the convex plane angle is defined (the only one that might be present, after all, in the first books of EUCLID); and the adjective “convex,” in that 69

[Pieri defined alternative phrases, semispazio “ per a” and “da  verso a”; this translation uses the second.]

70

If a, b, c, d as well as ar, br, cr, dr are noncoplanar points, there must be a motion that transforms a into ar, the ray from a toward b into the ray from ar toward br, and the half-plane from ab toward c into the half-plane from arbr toward cr (see P29,30, P4§3, P11§2, and so on). If that occurs for two motions,  and  for example, it will be found that a = a, b = b, c = c, and consequently d = d, by P11§1 and so on. Now, if the point d will be on the same side as the point dr with respect to the plane arbrcr, one will be able to say that the senses a, b, c, d and ar, br, cr, dr are equal, or that the tetrahedra *abcd* and *arbrcrdr* have the same chirality [verso]; the contrary, if the points dr and d will lie on opposite sides of the plane arbrcr. The sense of a tetrahedron can also be defined that way.

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it excludes every ambiguity, leaves entirely unprejudiced every future extension of the concept of a “plane angle.” Later, it will also be possible to define the concave angle of the rays *ab and *ac; and finally also the improper angle generated by a rotation greater in amplitude than 360°; and all that without contradiction of any sort. If, for example, the points a, b, c are not collinear, the concave angle of the rays *ab and *ac can already be defined as the locus of a point x coincident with a or else lying in the plane abc in such a way that the ray *ax should not meet the segment *bc* except, at most, at its extremities.71 P34–Theorem. If a, b, c are noncollinear points and br and cr, two points given in the rays *ab and *ac respectively, but both different from the point a, the angle ^ a Abrcr will coincide with the angle ^ a Abc. Proof. Each line r passing through a that should meet *bc* or *brcr* also meets *bcr* by P17 and so on, considering that a will thus be external to *bbr* as well as *ccr*, and then r meets *brcr* or *bc*, respectively, for the same reason. But P17 also tells us that the points of intersection with *bc* and *brcr* will always be on the same ray of r (by the hypothesis): that is to say, both on that side of a where the point of intersection with *bcr* lies. See P27,28. P35–Theorem. Given the noncollinear points a, b, c, and taking at pleasure a point d in the shadow of a from b, and a point e in the shadow of a from c, excluding only a, the opposite vertical angles ^ a Abc and ^ a Ade are congruent to each other (EUCLID I.15). Proof. The sphere b a intersects the four rays *ab, *ac, *ad, and *ae at four points b, cr, dr, and er, distinct from each other and from the point a, by P19§3,P7§2,P6,7,11,28, and so on. If m is the midpoint between b and er, the rotation of the plane abc onto itself about the points m, a (certainly distinct from each other) will exchange the points b and er, the one with the other, in the same way as the points cr and dr, and therefore the angles ^ a Abcr and ^ a Adrer as ^ well, the one with the other, which are not different from the given angles a Abc and ^ a Ade (see P34 and so on).

§5 Relation less than or greater than between two segments or between two angles. Triangle is introduced. Congruence of triangles and other propositions of the first and third books of Euclid. The notion of “terminated line” (segment, ray), which has a role so relevant in the propositions of the preceding section, does not appear in any of the geometrical facts that were discussed in sections 8.1, 8.2, and 8.3. Whoever would attribute a certain weight to such a disparity, and form from it a prominence that it does not properly have from 71

[In error, Pieri wrote *ab* instead of *bc* in this sentence.]

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only the deductive-hypothetical view, would be able to call “descriptive properties,” for example, all those that are deduced from postulates I–XVI, and the others, “segmental properties,” where the next postulates, XVII, XVIII, and so on, play a role. Distinctions similar to this could be made in good numbers for every deductive science, and are made here, for example, when we speak of “properties independent from the continuity of the line, from parallelism, and so on.” P1—Theorem. Having presumed that a, b, c, d are points and *ab* and *cd* are congruent to each other, any motion that superposes the one onto the other will have to transform the extremities of the one into those of the other; and therefore, the pair (a, b) will be congruent to the pair (c, d), and so also to the pair (d, c). Thus also, if two convex plane angles are congruent to each other, it is always possible to move the one onto the other in such a manner that edges coincide with edges, in the order that one prefers. Given P28§3, one could also say merely, “two segments, or two convex plane angles, cannot coincide as long as the extreme points, or else the edges, of the one are different from the extreme points or the edges, of the other.” Compare P29§4. Proof. Let it be allowed us to suppose a = / b, and as a consequence c= / d. If the motion  should transform the one segment *ab* into the other one, *cd*, so that *ab* = *cd*, the points ar  a and br  b, as belonging to the figure *ab*, will certainly also be found in the other one, *cd*. Therefore br, for example, will fall into the segment *car*, or else into the segment *ard*, by P12§4; and consequently either c does not lie between ar and br, or d does not lie there, by P11§4. But for *arbr* = *cd* to happen, it will be necessary that c as well as d should belong to *arbr*, in which case, when c should not fall between ar and br, it will be necessary by P7§4 that it coincide with ar or else with br, and the same should be said with respect to d. It follows that at least one of the points c and d will coincide with one of ar and br. The rest [is left] to the reader. P2—Definition. If a, b, c, d are points, a different from b, the assertion that “the segment *ab* should be less than the segment *cd*, or that *cd* would be greater than *ab* — *ab* < *cd*, or *cd* > *ab* —is a concise form of expressing how “there should exist a motion that should represent one of the extremities a or b of the segment *ab* by one of the extremities c or d of the segment *cd*, and the other extremity of *ab* by a point lying between c and d;” or rather—which is the same—“there exists between c and d a point x such that *ab* should be congruent to *c x* or to *d x*. See P6§4. When the points a and b coincide, the phrase *ab* < *cd*, or the other one, *cd* > *ab*, will serve only to express that “the points c and d are distinct.” Under this definition one can collect, in the guise of an almost immediate corollary, the following facts: “if c should be a point of *ab* not coincident with b, it will always be the case that *ac* < *ab*” and “whenever *ab* should be less than *cd*, there will always exist a motion capable of inserting the one segment *ab* into the other, even if it be prescribed which of the extremities a and b we wish to be carried to c, or which of the extremities c and d is to represent a.” Proof. Indeed, if it is possible to transport a onto c and b between c and d with a motion , for example, a can also be carried onto

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d by following  by a half-turn around the midpoint c*d (see P6,7§2),72 without b departing from the segment between the points c and d. —If two of the three segments , ,  are congruent, for example  and , the same relation among ~, or < that can peradventure hold between the two segments  and  will >, = have to occur again between the pair  and . —And so on. P3—Theorem. Given at pleasure four points a, b, c, d, one of three cases holds— *ab* will be congruent to *cd*, or *ab* less than *cd*, or *ab* greater than *cd* —and each one of these three cases will exclude the other two. And each motion that superposes the two rays *ab and *cd will cause b to be transformed into a point internal or external to *cd* or coincident with d, according as *ab* should be less or greater than, or congruent to, *cd*. Proof. I restrict myself to ~ *cd*, no motion  can exist assuming a = / b and c = / d. If, for example, *ab* = by which a should be transformed into c and b should come to a point br belonging to *cd* but not coincident with d: because if it existed, having seen that  c, d  a motion   a, b  is always given as well by the hypothesis (see P28§3,P1), there would also exist a motion –1 transforming the point d into br without altering the other extremity c; and moreover, d would coincide with the point br or with the point br/c by P3–5§2. But indeed, the one or the other event is contrary to the assumption that br should belong to *cd* without coinciding with d (see ~ *cd* is P6,7,11§4 and so on). Thus, it rests proved that the condition *ab* = incompatible with the condition *ab* < *cd* and with the other one, *cd* < *ab*,  c, d, a, b  the substitution  a, b, c, d  being permissible here. With similar reasoning is also proved the incompatibility of the hypotheses *ab* < *cd* and *cd* < *ab*, which  c, b   a, d , b  would involve the existence of motions such as   a, b  and   c, d, b  for example, with br lying between c and d, and d1 between a and b, and b1 between d1 and a. Now, the motion , for example, would carry b to b1, holding a fixed, so that b1 = b or b1 = b/a , whereas neither one of these conclusions is reconcilable with the premise that b1 should fall between a and d and d1 , between a and b. And so on. Concerning the first part of the theorem, it is sufficient to carry the ray *ab onto the other one, *cd, in accordance with P29§4; after that, if b will not fall on d, it will be necessary that it should arrive at a point internal to *cd* or else in the extension beyond d, by P27§4. And so on. '1

1

P4—Theorem. Let a, b, c, d, e, f be given points. If *ab* is less than *cd* and *cd* less than *e f *, then *ab* is less than *e f *. Proof. If, for example, the motion  represents a by c, and b by a point br between c and d (see P2), and the motion  similarly represents c by f, and d by dO between f and e, then the point br is carried by  to73 a point bO between dO and f, considering that lying between two points is a covariant property of them with respect to each motion. Therefore, bO will lie between the points f and e by P10§4; consequently, the

72

[The axis of the half-turn can be any line perpendicular to cd at that midpoint.]

73

[In error, Pieri wrote  instead of  here.]

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segment *cbr*, congruent to *ab*, will be less than the segment *e f * by P2. And so on. P5—Theorem. Whatever be the points a, b, c, and according as *ac* should be less than, greater than, or congruent to *ab*, the point c will be internal, external, or belonging to the sphere ba . See P2§4. Proof. If b = / a and *ab* < *ac*, for example, there will have to exist a motion that, not affecting a, should by P2 transform b into some point br lying between c and a, so that c should be external to the segment *abr* by P11§4. On the other hand, c will be external to *abO* if bO br/a , lying on the ray *abr, which does not have points in common with the ray *abO except a, by P28§4; but c is different from the points a, br, bO. Therefore, c is external to *brbO* by P6,12§4: that is as much to say, external to the sphere ba , by P4,7§4. If, instead, we put c = / a and *ac* < *ab*, a motion must exist that, keeping a on a, should carry c between the points a and b, for example to cr. Now, such a point cr will always be between b and b/a ; therefore cr, and consequently also c, will be internal to the polar sphere of the points b and b/a by P6§4 and so on: that is as much to say, to the sphere ba . P6—Theorem. Whatever be the points a, b, c, if c will be internal to the sphere through b about a, by contrast b will be external to the sphere ca ; and vice versa. And each point internal to ca will also be internal to ba . Proof. Like this, from P2,4,5 and so on. P7—Theorem. And each line that should pass through the internal point c must meet the sphere ba at two distinct points. Proof. If that line passes through a, or if it is perpendicular to ca, refer to P19§3 or P4§4. If not, dropping the perpendicular ad from the center onto the line, the foot turns out to be internal to the sphere ca by P14§4, hence internal to ba by P6. Therefore, by P4§4, it is true that cd, perpendicular to da, should meet the sphere ba twice. P8—Theorem. If a, b are points, any line whatever will contain points external to the sphere ba . Proof. It can be conceded that b = / a. If the line is tangent to the sphere, that is, meets it at one point, for example c, and not elsewhere, each point different from c on the line will be external to the sphere by P30§2 and P14§4. And if a point d on the line is internal to the sphere, so that *ad* < *ab* by P5, then having taken an external point at pleasure, the point ar  a/b for example, *ab* will moreover be less than *aar*; therefore, *ad* < *aar* by P4 and d will consequently be internal to the sphere ara by P5. This and the given line therefore meet by P7 and the common points are external to ba , like ar, by P4§4. P9—Theorem. The points a and b being distinct, c a point external to ba , and  a plane passing through all three, it proves possible to draw from that point two lines in this plane tangent to the sphere (EUCLID III.17). Proof. Any one of the points, say for example d, at which the line ca meets the sphere ba (see P19 §3), will be internal to the sphere ca (see P6), so that the perpendicular directed to the line ca from the point d in the plane  meets the sphere ca at two points e and er.

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Let f and f r be the feet of the perpendiculars dropped from c to the lines ae and aer. If the plane  is rotated onto itself about the points a and c*e (distinct from each other) as hinges (see P11§2), the points c and e will as a result be exchanged with each other, and the lines ca and ea permuted. Thus, the line ed perpendicular to ca is transformed into the line c f perpendicular to ea by P29§2, then the points d and f exchanged with each other as well. And therefore f, like d, will stay on the sphere ba , which is not changed, and c f will be tangent just like ed (see P22,§2). Similarly, c f r is tangent; and the reader can see that these two lines c f r and c f cannot coincide. P10–Theorem. Having assumed that a, b, c are points, c lying between a and b, it follows that the reflection of a across c must be between the point a and the reflection of a across b. See P3§2. That is, granted the ordinary notion of the sum of two segments, as established in P20: if one segment is less than another, also the double of the first is less than the double of the second. Proof. Setting ar  a/b and cr  c/b, and seeing that the points a, b, c are distinct and that c 0 *ab* and b 0 *aar*, so that b, c 0 *aar* by P10§4 and b ó *ac* by P11§4, it follows that b 0 *car* by P12§4, and *bar* f *car*. Now, a motion that should represent a by ar and b by itself (see P1,3§2) will change c into cr; therefore cr 0 *bar*, and consequently cr 0 *car*. And a motion that should permute c with ar (see P4,6§2) will have to transform cr into some point cO for which *ccO* should be congruent to *arcr*, and hence also to *ac*, and therefore, into a point that, if it does not coincide with a, necessarily coincides with the point a/c by P2,3 and so on. But it does not coincide with a, having to lie in *car*, while a does not lie there (inasmuch as c should be between the points a and ar). Therefore, cO = a/c , hence the point a/c will be in *car* and consequently in *aar*, for the reason that c 0 *aar*. P11–Theorem. Under the same hypothesis, the polar sphere of a, b and the sphere through c about a intersect. Proof. Let m be the midpoint of a, b and n, of a and c. It is necessary that n should be between m and a; if not, m would lie internal to *an* (having seen that m, n, a are distinct and that m, n 0 *ab*) and consequently b would lie between the points a and c by P10, contrary to the hypothesis of P11. Therefore, m is external to the sphere na by P2,5; so there exists on na some point d such that (d, a) z (d, m) (see P9 and so on). Now, the point a/d will lie on the sphere ma by P20§2: that is, the polar sphere of a  a, n  and b; as again on the sphere ca , provided that a motion of the sort  a, d  a a represents the point /d by the point /n coincident with c. This last, together with P13§4, now makes certain the existence of a triple ( x, y, z) of points subject to the condition that the segments * x y*, * xz*, and * yz* should all be congruent to a given segment *ab*: or else such that * x y* and * xz* should be congruent to *ab*, but * yz* congruent to another segment less than the double of *ab*. That there should exist a triangle with edges congruent to three given segments, each one less than the sum of the other two, is demonstrated in P19§6. It is known that these facts, inasmuch as they depend on the existence or

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not of points common to two circles, are imperfectly proved in the Euclidean text, but here this is attained. (See EUCLID I.1 and I.22.) P12–Definition. Of two convex plane angles, ^ a Abc and ^ d Ae f for example, where a, b, ... , f should be points, a, b, c noncollinear, and d not coincident with e or with f, it will be said that the first is less than the second or that this is greater than the ^Abc < ^ first (a d Ae f or ^ d Ae f > ^ a Abc) whenever there will exist a motion superposing one edge of the first onto one edge of the second, and the other edge of the first onto a ray internal to the second. (See P29,33§4.) Or rather, which is the same: “if there will exist a point x internal to the angle ^ d Ae f and such that the angle ^ a Abc should be congruent to the angle ^ d Ae x or to the angle ^ d Af x”. (See P28§3.) If on the contrary the points a, b, c are collinear, provided that b and c be different from ^Abc is less than ^ the point a, with the sentence “a d Ae f,” or “ ^ d Ae f is greater than ^ a Abc,” it is affirmed, having supposed b and c on the same side of a on the line, that d, e, f are not collinear or else that d lies between the points e and f; while if b and c should be on opposite sides of a, [those sentences are always false].74 Compare P2. It is proved rather easily (with a reversal of the edges similar to that executed in the demonstration of P9 for the angle ^ a Ace) that whenever a motion superposing the edge *ab of the first onto the edge *de of the second is possible, a motion superposing the edges *ab and *d f is not impossible, holding firm in both cases the condition that the edge *ac should have to fall internally in ^ d Ae f. P13–Theorem. Whenever a, b, ... , f should be points, a different from b and from c, just as d is from e and from f , then one of three cases holds—the angle ^ a Abc is less than, or is greater than, or is congruent to the angle ^ d Ae f —but of these cases no two can hold at once. And if, in addition, g, h, i should be other points as ~^ above, the supposition ^ a Abc = d Ae f with ^ a Abc ^ g Ahi necessitates ^ d Ae f ^ g Ahi ^ d Ae f < ^ g Ahi draw along with respectively, and the two conditions ^ a Abc < d Ae f and ^ them the other one, ^ a Abc < ^ g Ahi. And so on. Proof. Let the reader see, escorted by P33,34§4 and by P2,3,4,12 and so on. P14–Definition. Points a, b, c being arbitrary, the figure comprised of those segments that have one extremity at a and the other extremity in the segment *bc* is to be called the triangle a, b, c and more often, just *abc*. From P11,17,22§4 it follows that each point that should belong to the figure thus defined must also be found in the figure that arises from putting in place of a and of *bc* the elements b and *ca*, and vice versa, so that the figures *abc*, *acb*, *bca*, *bac*, *cab*, and *cba* will coincide in one and the same triangle, of which the segments *ab*, *bc*, and *ca* are edges, and the points a, b, c, vertices. The union of the three edges will be the boundary or periphery of the triangle. Any point of the figure that should not be on the boundary will be internal to the triangle; each point of the plane abc will be external, as long as it is excluded from the figure. The angles of the triangle will be ^ a Abc, ^ b Aca, ^ c Aab. And so on.

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[The clause in brackets replaces a duplication of one in the original that was evidently an editorial error.]

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P15–Theorem. Whenever a, b, c should be noncollinear points, the triangle *abc* coincides with the figure common to any two of the angles ^ a Abc, ^ b Aca, and ^ c Aab. That ^ ^ is to say, *abc* = ^ a Abc 1 b Aca = b Aca 1 ^ c Aab = ^ c Aab 1 ^ a Abc = ^ a Abc 1 ^ b Aca 1 ^ c Aab. Proof. Like this, by P14, chiefly by virtue of P22,33§4. P16–Theorem. If a, b, c as well as d, e, f should be noncollinear points, and the edges *ab* and *ac* and the angle ^ a Abc of the triangle *abc* should be congruent respectively to the edges *de* and *d f * and to the angle ^ d Ae f of the triangle *de f *, the third edge *bc* will also be congruent to the third edge *e f *, and the triangle congruent to the triangle; and of the remaining angles these will be congruent with each other: ^ b Aac with ^ e Adf and ^ c Aab with ^ f Ade, namely those ~^ that include the congruent edges. (See EUCLID I.4.) Proof. For ^ a Abc = d Ae f to hold, there will have to exist a motion  that should represent a by d, *ab by *de, and *ac by *df (see P1 and so on). Now the point b will not be able to fall between the points d and e, nor the point e to be between the points d and b, for the reason that by the hypothesis, the edge *ab* is neither less nor greater than the edge *de* by P2,3. On the other hand, if b were not to coincide with e, it would be possible to verify precisely the one or the other of those two cases, because b and e would be distinct from each other and from the point d, and situated on the same side of it, by P18,27,28§4. Therefore, b = e; and in the same way, c = f, and so on. One other case of congruence between two triangles (EUCLID I.8) can hold, through the demonstration in P32§3, if one considers that the congruence of two segments also implies the congruence of the two pairs of extreme points (see P1).75 P17–Theorem. If a, b, c are noncollinear points, the convex angle ^ a Abc will be the locus of the points common to the two half-planes of ab toward c and of ac toward b. That is to say, ^ a Abc = *(ab)c 1 *(ac)b. (See P28,30,33§4.) Proof. That each point of the angle should be common to the two half-planes emerges immediately from the cited definition and from P11,17 §4, thanks to which the assumption that x be a point lying between b and c, and y a point of the ray *a x but different from the points a and x, will ensure that the line ab will not be able to pass between c and y, nor ca between b and y. Now let z be a point common to the two halfplanes but external to the lines ab, ac, bc. For the reason that by the hypothesis the line ab does not pass between c and z, nor the line ca between b and z, it will be necessary that the line az should pass between b and c, in accordance  z with  c  P23§4: nor will it be possible that a should lie between z and the point zr common to az and *bc*, because the line ba cannot pass between z and zr without likewise passing between c and z, by P11§4 and  za,,zb  P17§4. Therefore,   z 0^ a Abc.

75

[P16 and P32§3 are the familiar SAS and SSS congruence theorems. See subsection 9.3.1, page 423.]

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P18–Theorem. Again, a, b, c being noncollinear points, and having taken a point, let it be d for example, different from c on the extension of the edge *bc* on the side of c, the angle ^ c Aad, external to the triangle *abc*, will be greater than each one of the opposite internal angles ^ a Abc and ^ b Aca (EUCLID I.16). Proof. Set e  a*c and f  b/e . The point f will lie in the half-plane *(ac)d since the line ac, passing between b and d and between b and f, cannot pass between d and f, by P21§4. And at the same time, it will lie in the half-plane *(cd)a, because the line cd could not pass between the points f and a without passing between the points a and e or else between the points f and e, by P17§4. Therefore, f 0 ^ c Aad by P17: in ~ *ec*, fact f is internal to ^ c Aad. Thus, ^ c Aaf < ^ c Aad by P12. But from *ea* = ~ ~ ~ ^ ^ ^ ^ *eb* = *e f *, and e Aab = e Ac f (see P35§4), it follows that a Abe = c Afe by P16, ~^ ~^ while on the other hand it is known that ^ a Abe = a Abc and ^ c Afe = c Aaf by ^ P33,34§4; therefore, ^ a Abc < ^ c Aad by P13. It remains to prove how b Aca should be less than ^ c Aad, but that is left to the reader. P19–Theorem. Given the triangles *abc* and *def*, if the angles ^ b Aca and ^ c Aab of the one should be congruent respectively to the angles ^ e Af d and ^ f Ade of the other, and furthermore, the edges *bc* and *ef* included by the congruent angles should be congruent to each other, or else the edges *ab* and *de* that are opposite the congruent angles, the remaining angle ^ a Abc will be congruent to the remaining angle ^ d Ae f; and those of the remaining edges that are opposite congruent angles will be congruent to each other: that is to say, *ab* to *de* and *ac* to *d f * ~ *e f *. By the hypothesis there will exist a (EUCLID I.26). Proof. First, let *bc* = motion  transforming the pair of rays *ba and *bc into the pair *ed and *e f, by P1, and therefore, also such that c should coincide with f, as well as b with e, by the reasons just given in the proof of P16. Now, if the point a should not coincide with d, it would be necessary to suppose it internal to the segment *de* or else in the extension of this beyond d, by P18,25,27§4. But in the one case we would have ^ c Aab < ^ f Ade and in the other, ^ c Aab > ^ f Ade, by P12; therefore, it is necessary that a coincide with d, by P13; and so on. Having instead supposed the two edges *ab* and *de* congruent to each other, if we make the angle ^ b Aca coincide with the angle ^ e Ad f as before, the point a will come to d. It will not be possible that c should come to a point between f and e, nor that f remain included between the new position of c and the point e, since in the one case the ^ Ade, congruent to the angle ^ angle c c Aab, would as a result be greater than the angle ^ f Ade, and in the other, less (see P18). And so on.

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§6 Sum of two segments. Other properties of triangles, circles, and so on. Continuity of a line. P1—Definition. Given at pleasure the points a, b, c, and having selected three points ar, br, cr so that br should lie in *arcr* and that *arbr* and *brcr* should be congruent respectively to *ab* and *bc*, the attribute of being a sum of the two segments *ab* and *bc* belongs to any segment congruent to *arcr*, and is given only to these. In substance, it is like the class of all the segments congruent to *arcr* being called the sum of *ab* and *bc*.76 But more often, conforming to custom here, we shall also call an arbitrary segment of the indicated class the sum of the two segments. Let the reader show that it is as much to add *bc* to *ab* as *ab* to *bc*, because in both cases segments are obtained that are congruent to each other (commutative property); and that if , ,  are segments and  less than, greater than, or congruent to , the sum of  and  is respectively less than, greater than, or congruent to the sum of  and . And so on. By now we could reproduce all propositions of the first and third books of EUCLID that do not depend on the parallel axiom77 and thus belong, as all the preceding ones, to the so-called Lobachevskian geometry no less than to the Euclidean. Only proposition I.22 of the first book does not appear obtainable from the premises that we have adopted until now; therefore, for this let us refer to our last principle, XX. But concerning the rest, almost nothing is to be added to or removed from the text of the Elements except certain easy modifications of form in the statements of certain propositions, such as I.13, I.14, and I.17, where the sum of two angles is discussed,78 and particularly those in which, under the denomination of problems, it is proposed to “find,” “construct,” “distinguish,” “adapt,” “carry back,” and so on, certain points or figures from certain other given points or figures; and desired to certify in brief how, starting from such and such elements, such and such figures should prove to be determined, and how these might be derived from those, so as to be able to affirm their existence whenever it should be necessary for deductive purposes. That this should truly be the function of the problems in Euclid—that is to say, purely demonstrative and not practical—appears from the examination of the Euclidean proofs, where the problems are always invoked to legitimize the intervention, in the logical operations or transformations, of certain elements extraneous to the hypotheses, so that they properly have the value of existential

76

[This is one of the first examples of conversion of a definition by abstraction into a nominal definition in terms of equivalence classes. Pieri introduced that technique. See section 2.3.]

77

[Euclid [1908] 1956, book 1, postulate 5.]

78

[Apparently in error, Pieri wrote XVIII for I.17: no sum is discussed in I.18.]

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lemmas, and nothing else.79 The reason will perhaps be more appropriate to a geometry that is purely hypothetical (and, I would say, better conforming to modern taste): to offer them as existential theorems. And we have preferred this in what has preceded. But it is not a thing of more than middling importance, nor meant to take from whoever desires it the capacity to proceed differently. Some propositions of those two books are reported here only as statements because, while they work to our purpose (and it indeed happens that they are used here, although they may not be seen cited any time before closing the present memoir), it is not necessary to make any change to the demonstrations just as they are offered by the Elements in order to be fully in accord with what has preceded. P2—Theorem. Given that a, b, c should be noncollinear points, and the segments *ab* and *ac* congruent to each other, the angles ^ b Aac and ^ c Aab will also be congruent. And extending those two segments from b and c to the points d and e, the angles ^ b Acd and ^ c Abe will also be congruent to each other (EUCLID I.5). Proof. From the hypothesis and from P1§5 we deduce that c should belong to the sphere ba ; and therefore that given m, the midpoint between b and c, the line bc should be perpendicular to ma by P21§2 and so on. Therefore, the rotation of the plane abc onto itself about the points m and a as hinges (see P11§2 and so on) will ensure that the angles ^ b Aac and ^ c Aab be exchanged with each other, and likewise ^ ^ the angles b Acd and c Abe. But it would also be possible here to reason like Euclid at the place cited. P3—Theorem. If the angles ^ c Aab and ^ b Aac of a triangle are congruent to each other (a, b, c being noncollinear points), the edges *ab* and *ac* that are opposite these angles will also be congruent to each other (EUCLID I.6). Proof. If, for example, *ac* were less than *ab*, and therefore congruent to a segment *bd* with d lying between a and b by P1,2§5, it would result from this that the angles ^ b Aca and  c, a, b, c, d  ^ c Abd would also be congruent to each other, thanks to  a, c, d, e, f  P16§5 and to P34§4: that is to say, ^ c Aba and ^ c Abd. But that contradicts P12,13§5. And so on. P4—Theorem. If in the triangle *abc* (again, a, b, c being noncollinear points) it happens that the edge *ab* should be less than the edge *ac*, then the angle ^ c Aab will also be less than the angle ^ b Aac (EUCLID I.18). Proof. From P18§5,P2 and so on. P5—Theorem. And if, vice versa, ^ c Aab should be less than ^ b Aac, also *ab* will be less than *ac* (EUCLID I.19). Proof. From P2,4 and so on. P6—Theorem. Provided that a, b, c be noncollinear points, any segment that is a sum of the two segments *ba* and *ac* will always be greater than *bc* (EUCLID I.20). Proof. From P18§5 and P1,2,5 and so on. 79

Compare Zeuthen 1896.

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8 Pieri’s 1900 Point and Motion Memoir

P7—Theorem. And if d should be a point internal to the triangle *abc*, the sum of the segments *bd* and *dc* will always be less than the sum of the edges *ba* and *ac*, but the angle ^ d Abc is greater than the angle ^ a Abc (EUCLID I.21). Proof. The property reported at the close of P1 is needed here, and P14,18§5, P6, are to be recalled; and so on. P8—Theorem. Again, a, b, c being noncollinear points, let d be the foot of the perpendicular dropped from the point a to the line bc (see P28§2). If it happens that the segment *db* should be less than the segment *dc*, at the same time *ab* will be less than *ac*. And, vice versa, *ab* cannot be less than *ac* if *db* is not less than *dc*.80 Proof. Having supposed that b should not lie in *cd*, let ar and br be the reflections of a and of b with respect to d. The point br is internal to *cd* by P3§5 and so on, and hence internal to the triangle *caar* by P14§5. Therefore, *aar* is less than the sum of *abr* and *brar* by P6, and this in its turn less than any segment belonging to the sum of *ac* and *car*, by P1,7. Now, since the segments *abr* and *brar* are congruent, and the segments *ac* and *car* congruent, as well, by P20§2 and so on, it is possible to invoke P10§5 and conclude that *abr* should be less than *ac* by P3§5. And so on. P9—Theorem. If a, b, c are points and c belongs to the sphere ba , any point lying between b and c will be internal to the sphere. And vice versa, each point internal to the sphere and aligned with the points b and c will lie between these two (EUCLID III.2). Proof. Let d be a point between b and c, so that b = / c, and m be the midpoint between these points. If m = a, it is sufficient to recall P2,5§5. If m = / a, the line ma will be perpendicular to the line bc by P21,27§2 and so on; hence, *md* < *mb* or else *md* < *mc* by P2§5, according as d is in *mb* or else in *mc* (see P12§4 and so on). Therefore, by P8, *ad* is less than *ab* or else than *ac*, which is the same by P2§5, and consequently, d is internal to the sphere by P5§5. Vice versa, if for example the point dr aligned with b and c will be internal to the sphere ba , we would know that *adr* is less than *ab* by P5§5, and consequently *mdr* < *mb* by P8; therefore, dr is internal to the sphere bm, hence internal to *bc* by P7§4. P10–Theorem. If a and b are distinct points and c, like d, should be a point internal or belonging to the sphere ba , then all points that lie between c and d are internal. Proof. Similar to P9.—This is as much as to say that each sphere is a convex figure. The same holds for each line, plane, segment, ray, half-plane, convex angle, triangle, and so on.81 P11–Theorem. If two triangles have two edges congruent to two edges, the one to the other, but the angles included between these edges are not congruent, the third edge will be the greater in the triangle where the angle is greater (EUCLID I.24). Proof. From P33§4; P1,12§5; P2,P5; and so on. 80

Legendre 1802, book 1, proposition XVI.

81

See Peano 1889 passim.

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P12–Theorem. And if two triangles should have two edges congruent to two edges, the one to the other, but the remaining edges not congruent, the angle included by the congruent edges will be the greater in the triangle where the third edge is greater (EUCLID I.25). P13–Theorem. Points a, b, c being noncollinear and c external to the sphere ab , if it happens that a point d belonging to the sphere should be internal to the angle ^ b Aca, necessarily *cd* will be less than *ca* (EUCLID III.8). Proof. From P11 and so on. P14–Theorem. Let a, b, c be noncollinear points, c belong to the sphere ba , and on the circle of the plane abc and the sphere ba , a point d be taken external to the angle ^ a Abc. Any point of the circle, in case it should be internal to angle ^ a Abc, will be ^ internal to the angle d Abc; and vice versa. Proof. Let y, for example, be a point internal to the angle ^ d Abc besides belonging to the sphere ba , and yr, the point common to the half-line *dy and to the segment *bc* (see P33§4 and so on). This point yr is internal to the sphere by P9, and therefore internal to the segment *dy* by P9. Now, the line bc, passing between d and y and not containing a, will have to pass between d and a or else between y and a, by P8,17§4. But, if bc could contain a point lying between d and a, this would be internal to the sphere by P5§5, and therefore internal to *bc* as before, besides lying on the ray *ad, so that d would come to be internal to the angle ^ a Abc, contrary to the hypothesis. Therefore, bc passes between y and a: that is to say, intersects the segment * ya* at a certain point yO, which will be internal to the circle and consequently internal to *bc* by P9. But it is also internal to *ay by P27§4; therefore, y 0 ^ a Abc. And so on. Let us continue with the principle of continuity in a segment: to be precise, a postulate that encompasses a concept more relevant than what is usually called continuity of a line. It is taken in the main from I principii di geometria by Prof. Giuseppe PEANO.82 POSTULATE XX P15. If a and b are distinct points and k is a nonempty figure totally internal to the segment * a b * , there will have to exist such a point x internal to * a b * or coincident with b for which (1) no point of k should lie between x and b, and (2) however the point y be taken between a and x, there always exist points of k lying between y and x or coincident with x. —Such an x can be called the “limit superior (or inferior)83 of the class k.”

82

[Pieri wrote tolte di peso. Peano 1889, 39.] See also Peano 1894b, 74.

83

[The notions limit superior and inferior refer to a binary order relation < on segment *ab*, which Pieri did not introduce here. If a < b, then x could be called “limit superior;” if b < a, then “limit inferior.” Pieri had constructed the analogous projective order relation in 1898c, §7, and would do so for Euclidean geometry in 1908a, §8. Those are translated in section 6.7 and in M&S 2007, §3.8.]

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From this and from the preceding principles can be derived, for segments and plane convex angles, such properties that go by the name of postulate of ARCHIMEDES (axiom V of On the Sphere and Cylinder), as the reader can see, for example, in the work of PEANO, where the deduction unfolds everywhere with regard to segments.84 Here, the given principle is stated, without dwelling on it further. P16–Theorem. Let a and a1 be distinct points, b a point of the extension of *aa1* on the side of a1. Setting a2  a/a1 , and more generally ai  ai –2 /ai –1 for any index i as long as it is an integer and greater than two, there will always exist a positive integer n such that the point b is in the segment *an –1 an*. See P3§2; P7,25§4. Postulate XX, the necessity of which has not as yet been excluded for certain parts of elementary geometry, will immediately serve here to fill (with P18 and P19) a certain gap that one finds in the demonstration of a fundamental theorem of the Elements—that is, proposition I.22.85 P17–Theorem. Given three noncollinear points a, b, c, where c belongs to the sphere ba , and having taken a point d between b and c, the circles ba and db meet in the plane abc and always have a common point within the plane convex angle ^ a Abc. Proof. That they should meet emerges immediately from P11§5, because from *bd* being less than *bc* by the hypothesis and *bc* less than the sum of *ba* and *ac* by P6, it results that *bd* is less than the diameter *bbr* (see P1), where br  b/a, so that this is cut at a point of the sphere db (see P3§5). Now, for example, let xr be a point common to these circles. If xr ó ^ a Abc, nevertheless ^ a Ab xr will be less than ^ a Abc by P12; thus, there will exist a motion that, not altering the common edge *ab, should give for the image of xr a point x in the interior of the angle ^ a Abc (see P12§5); and on the other hand such a transformation cannot change the spheres ba and db , nor the plane abc. P18–Theorem. From the assumption that a, b be distinct points, and c, d points, one internal to *ab* and the other external, it is demonstrated how the sphere cd and the polar sphere of points a and b meet. Proof.86 It can be supposed that d, for example, lies in the extension of *ab* on the side of a. Take a point o on Sfr(a, b), not lying in ab: for example, one of the points common to the given sphere and to the plane perpendicular at c to the line (see P8§4). The join of o with any point p in the interior of *ab* meets Sfr(a, b) again: that is to say, at a point different from o, which we will distinguish from p by means of an accent, thus calling it pr (see P7§5). There certainly exist in the interval *ab* some points p, to each one of which should be coordinated a point pr thus made, so that *dpr* < *dc*. Indeed the

84

[Archimedes [1897] 2002a, 4.] Peano 1894b, 86, 87, and 90.

85

Compare Veronese 1897, 85.

86

[The proof of P18 fills five paragraphs.]

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305

sphere ca and the polar sphere of a, b meet outside the line ab by P11§5; in fact, the circle common to these two spheres (see P26§3) has two distinct points on the plane abo, one of which will fall in the half-plane from ab toward o, the other in the complementary half-plane: that is, in the shadow of ab from o. Now, if this other one should be pr, for example, the lines opr and ab will intersect at a point p internal to *opr* by P30,31§4, and hence internal to *ab* by P9, and *dpr* will be less than the sum of *ad* and *apr*: that is, less than *dc* by P1,6. Thus, the preceding P15 can be invoked on the class of the points p to which are owed the points pr where it happens that *dpr* < *dc*. Therefore, there exists, belonging to *ab* but not coincident with a, a point x with respect to which it is true that (1) no point u that lies between x and b will be such that *dur* < *dc*, so that for any u between x and b, *dur* will always be greater than or congruent to *dc*; (2) however a point y be taken between the points a and x, there will always be some point z between y and x, or coincident with x, for which *dzr* < *dc*. It is demonstrated [in the next two paragraphs] that the point xr assigned to x by the previously mentioned projection from o cannot be such that *d xr* should be less than *dc*, nor can it be greater. From that, by P3§5, we should be certain that *d xr* is congruent to *dc*, and thus that xr belongs to cd by P28§1. Make the hypothesis *d xr* < *dc*. The sphere cd will therefore intersect the ray *d xr at such a point i, for which the sum *d xr* plus * xri* will be congruent to *dc*; and the segment * xri* will have to be less than or greater than the segment * xrb*. For example, let it be less than. Then, the midpoint of a and b having been called e, there will be a point ur common to the two circles ixr and be and moreover internal to the angle ^ e Ab xr by P3§5 and P17, and therefore connected by projection from o with a point u between x and b, by P17§4 and P14. On the other hand, the sum of the segments *d xr* and * xrur* will always be greater than the segment *dur* by P6 and so on, except under the hypothesis that ur should coincide with i. Therefore, if ur = / i, one falls into contradiction with (1). But the contradiction does not disappear when ur should coincide with i, because in such a case each point v lying between x and u, and hence also between x and b, must lie between d and u (in fact the line o xr, passing between the points d and i, also passes between d and u by P17§4, for which reason * xu* f *du* by P10§4) and the corresponding vr must fall in the interior of the angle ^ o Adi by P34§4, and hence vr must be internal to the angle ^ e Adi by P14, and as a consequence *dvr* < *di* by P13: that is, *dvr* would be less than *dc*, although v should lie between x and b. Suppose instead * xri* > * xrb*. Then, having taken a point j at pleasure between xr and b, the sum *d xr* plus * xrj* will certainly be less than *dc*, and as before there will exist a point ur common to the circles jxr and e Ab xr and again such that *dur* < *dc*, by P1,6,17, P4§5, be , internal to the angle ^ and so on, which is equally contrary to the condition (1). The hypothesis *d xr* > *dc* can be invalidated similarly, appealing to (2). In fact, from this will follow the existence of a point i situated between the points d and xr and such that *di* should be congruent to *dc*. But, *dc* being greater than *da*, it will not be possible for * xri* to be congruent to * xra*, because *d xr*

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would then be greater than the sum * xra* plus *ad* by P1, contrary to P5. Therefore, let * xri* be less than * xra*. The circle ixr and the given circle a e will meet by P17 at a point yr internal to the angle ^ e Aa xr, and which, projected from o, will produce a point y lying between x and a by P17§4,P14, and so on. Now, if yr is not aligned with the points d and xr, the sum *dyr* plus * xryr*, congruent to the sum *dyr* plus * xri*, will by P6 certainly be greater than *d xr*, which is congruent to the sum *dc* plus * xri*; therefore, *dyr* > *dc* by P1. Then, by force of (2), between x and y there will always be such a point z for which *dzr* < *dc* holds. But it is necessary that y lie between the points a and z (for the reason that it lies between x and a), and hence that yr should be internal to the angle ^ o Aazr and consequently to the angle ^ e Aazr by P14; therefore, *dzr* is greater than *dyr* by P13, and hence greater than *dc* by P4§5, where a little earlier the contrary was stated. We still have to suppose yr aligned with d and xr. Under such a hypothesis, it will have to coincide with i, or with the reflection of i across xr. In the second case, *dyr* would certainly be greater than *d xr*; and thus *dyr* > *dc* as before; in the first case, *dyr* is congruent to *dc*, but *dzr* > *dyr* would equally result from that, and consequently *dzr* > *dc*. It remains that * xri* might be greater than * xra*. But we shall put, in place of i, a point j chosen at pleasure in the interval that is between the points xr and a, and shall find again a point y common to the two circles jxr and a e as well as internal to the angle ^ e Aa xr, so that the sum *dyr* plus * xrj* will be greater than *dzr* and thus also greater than the sum *dc* plus * xri*. Now, since * xri* > * xrj* by P4§5, it follows that the sum *dyr* plus * xri* is greater than the sum *dc* plus * xri*; and thus *dyr* > *dc* as before. Therefore, the same incompatibility is encountered here that we encountered a while ago under the hypothesis * xri* < * xra*. P19–Theorem. There exists a triangle whose edges are congruent to three given segments, where the sum of any two among these should be greater than the remaining one (EUCLID I.22). Proof. Let , ,  be the three given segments. One can suppose ~= ~  being already  < , the truth of the theorem under the hypothesis  = beyond doubt by P13§4. Having taken the points a, b, c, d in such a way that the segments *ab*, *ac*, and *cd* should be congruent respectively to , , and , and the point b should lie between a and c, and the point d on the ray *ca, and having denoted by br the reflection of b across a (whence c ó *bbr*, nor does br 0 *bc*, but b 0 *cbr*) it is necessary that the point d should fall between c and br if it is desired that the sum *ca* plus *ab*, that is to say *cbr*, should be greater than *cd*. But it cannot fall between b and c nor coincide with b if the sum *cd* plus *ba* would also be greater than *ca*, by P1 and so on. Therefore, d will fall in the interior of *bbr* by P12§4; thus, the spheres ba and dc meet, by P18. If x is a common point, the triangle that has for vertices the points a, x, and c has the edges *a x*, *ac*, and *cx*, precisely congruent to , , and . Turin, April 1899.

9 Pieri’s Works on Foundations and Philosophy of Mathematics This chapter discusses Pieri’s individual published works on foundations and philosophy of mathematics, as well as his unpublished but surviving classroom materials on projective and descriptive geometry. Summaries are provided for all of them, and extensive commentary on their reception or impact. They are organized by subject in sections: 9.1 9.2 9.3 9.4

Course Materials and a Translation Foundations of Projective Geometry Foundations of Elementary and Inversive Geometry Arithmetic, Logic, and Philosophy of Science

Each section is divided into subsections describing single works. Their titles were translated and edited to form the subsection titles. Those are followed immediately by copies of their bibliography entries, with the original titles and publication data.1 The level of detail in a subsection depends on its relation to others and the extent to which a description may be available elsewhere. This chapter’s summaries of works by Pieri, including those translated in chapters 4, 6, and 8, highlight their noteworthy features and explain their context and reception. Those translations illustrate Pieri’s precise and detailed arguments, as well as his style of presenting them. The summaries of the other works expose the crucial elements of the theories, sufficient to provide insights about their content and methodologies. All of Pieri’s works known to the present authors by 2007 were listed in M&S 2007, chapter 6. Some were described there in detail. Several more have come to light since then: the review Pieri 1893g, six volumes of notes on lectures by Pieri’s professors, a large number of very brief records of Pieri’s own lectures, and the page of notes displayed in the frontispiece of the present book. They are enumerated in appendix 1. Those relevant to the subject of the present book are described in more detail in this chapter.

1

Many of Pieri’s papers are reprinted in his 1980 collected works. That publication displays its own set of page numbers in addition to the original ones.

© Springer Science+Business Media, LLC, part of Springer Nature 2021 E. A. C. Marchisotto et al., The Legacy of Mario Pieri in Foundations and Philosophy of Mathematics, https://doi.org/10.1007/978-0-8176-4823-7_9

307

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9 Works on Foundations and Philosophy

9.1 Course Materials and a Translation This section summarizes and comments on the following works: 9.1.1 9.1.2 9.1.3 9.1.4 9.1.5 9.1.6

Higher Geometry Lectures by Riccardo De Paolis (Pieri 1883–1884) Geometry of Position by G. K. C. von Staudt (translation: Pieri 1889a) Projective Geometry: Lectures at the Military Academy (Pieri 1891c) Course Records from University Archives (Catania 1901–1908) Projective Geometry: Lectures at Parma (Pieri 1910, 1911c) Descriptive Geometry: Lectures at Parma (Pieri 1912f)

Pieri’s 1883–1884 notes on lectures by De Paolis are in private hands. Pieri’s 1889a translation of Staudt 1847 and Pieri’s 1891c lectures were conventionally published. The remaining three items are held in the university archives in Catania and Parma. With his August 1900 application for the chair at Catania, Pieri submitted a folder of summaries of lectures for his courses at Turin.2 However, the present authors have not yet found any such Turin documents save Pieri 1891c, described in section 9.1.3. Sixteen volumes of Pieri’s handwritten notes taken from lectures on mathematical physics by Enrico Betti at the University of Pisa have been discovered in the Biblioteca Statale di Lucca.3 They are listed in appendix 1 and the bibliography as Pieri 1882–1883, 1883–1884a, 1883–1884b, and 1883–1884c. They are not further described in the present book because their subjects fall outside its scope. 9.1.1 Higher Geometry Lectures by Riccardo De Paolis Pieri, Mario, editor. 1882–1884. Geometria Superiore dalle Lezioni del Pr. Riccardo De Paolis. Handwritten notes taken from lectures by De Paolis at the University of Pisa, bound in twelve fascicles under one cover, approximately 475 pages.

These notes are presently in Pieri’s library in the former house of his sister Gemma Campetti in Sant’Andrea di Compito, near Lucca, where he died.4 Fascicles 1–4 are from 1882–1883; the rest, from 1883–1884.5 The title page of fascicle 11 and the first page of content from fascicle 1 are shown on the previous two pages. From University records it is not clear whether Pieri was actually enrolled as a student in these courses; but the handwriting is his. He probably attended as note-taker. The topics of the lectures are listed starting on page 311. There are twenty-eight figures. 2

See Accademia 1900.

3

See Dell’Aglio and Roero 2012.

4

Gemma’s great-grandson Francesco Campetti graciously permitted the present authors to examine the lectures. Except where noted, they made photocopies. For information about De Paolis, consult the biographical sketch in M&S 2007, 78.

5

The last item in these notes is dated 1885, after Pieri’s graduation.

Mario Pieri

Libretto XI

Lectures on

Higher Geometry (Prof. R. De Paolis) Università di Pisa 1883–84

Part 1 Algebraic Theory of Forms §1 Generalities about Forms and Their Symbolic Representation

9.1.1 Higher Geometry Lectures by Riccardo De Paolis

311

1882 –1883 Part One

Algebraic Theory of Forms

I. II. III. IV. V. VI. VII.

Generalities about Forms and Their Symbolic Representation Transformations of Variables and Forms: Linear Substitution Polar Forms Jacobians, Hessians Forms That Enjoy Invariant Properties6

§ 1–§9 §10–§15 §16–§24 §25–§29 §30–§33 §34–§39 §40 §41–§55

Part Two I.

Symbolic Representation of Invariant Forms

§56–§60

Part Three

II. III. IV. V. VI.

1 1 1 1 1 1 1 2

Foundations for the Theory of Geometric Forms of the First Species

Homogeneous Coordinates of Generating Elements of a Fundamental Form of the First Species II. §61–§65 Projectivity between Fundamental Forms of the First Species—Involution III. §66 Transformation of the Coordinates of the Generating Elements of Fundamental Forms of the First Species IV. §67–§73 Imaginary Elements of Fundamental Forms of the First Species V. §74–§81 Anharmonic Relation of Four Imaginary Elements—Extension of Projectivity and Involution VI. §82–§85 Invariant Functions of Binaries and the Study of Projective Properties of Sets7 of Elements of a Form of the First Species VII. §86–§91 Generalization of Projectivity and of Involution VIII. §92–§93 Linear Systems of Sets IX. §94–§106 Polar Sets X. §107–§111 Compositions8 of Two Binary Forms and Their Geometric Significance XI. §112–§119 Representation of a Binary Form of Degree n by Means of the Sum of nth Powers of Binary and Linear Forms §120 XII. §121–§139 Quadratic Binary Forms XIII. §140–§154 Biquadratic Binary Forms

I.

Fascicle

2 2 2 2 3 3 3 3 3 3 3 4 4 4

Foundations for the Theory of Geometric Forms of the Second Species

§155–§166 Homogeneous Coordinates of Generating Elements of a Fundamental Form of the Second Species §167–§173 Projectivity of Fundamental Forms of the Second Species §174 Transformation of Coordinates of the Generating Elements of a Fundamental Form of the Second Species §175–§178 Imaginary Elements of Fundamental Forms of the Second Species §179 Extensions of Projectivity and Involution §180–§181 The Geometric Entities of the Fundamental Forms of the Second Species

6

This title seems to apply to the following chapter VI as well.

7

The Italian noun gruppo is translated here as set.

8

For composition Pieri wrote scorrimento (Ueberschiebung).

4 4 4 4 4 4

312

9 Works on Foundations and Philosophy

1883 –1884

Fascicle

VII.

§182–§189 Invariant Functions of Ternaries and the Study of the Projective Properties of the Geometric Entities of a Fundamental Form of the Second Species 5 VIII. §190–§193 Generalities about Lines and Envelopes: Number of Conditions— Points or Lines—That Determine Them 5 IX. §194–§197 Polar Curves 5 X. §198–§205 Singular Points 5 XI. §206–§208 Singular Tangents 5 XII. §209–§232 Relations that Link the Numbers of Singularities of a Curve: Plücker Formulas9 5 XIII. §233–§236 Unicursal10 Curves, Genus of a Curve 6 XIV. §237–§243 Curves of Genus Zero, One, or Two 6 XV. §244–§250 Covariant Curves: Hessian, Steinerian, Cayleyan 6 XVI. §250–§253 11 Jacobian of Two Curves, Pencils of Curves 6 §253–§269 6bis XVII.12 Linear Systems of Curves 7 XVIII. Generalization of the Theory of Polars 7 XIX. Curves Apolar with Respect to a Given Curve 7 XX. Curves of the Third Order 7 XXI. Schema of Inflection Triangles 7 XXII. Reduction of f = 0 to Canonical Form 7 XXIII. Some Properties of Polar Harmonics of Inflections 8 XXIV. Hessian and Cayleyan of a Cubic 8 XXV. Equations of the Jacobian and the Hermite Curve Relative to a Net of Conics 8 XXVI. Associated Cubics and Conics of Battaglini 8 XXVII. Conjugate Points on a Cubic 8 XXVIII. §3–§9 Constructions for the General Curve of the Third Order13 8 §10–§26 9 General Properties of Surfaces §1–§19

25 unnumbered pages of untitled sections, each divided into subsections, either introductory or labeled consecutively in the style “1)”. Section §19 overlaps fascicles 9 and 10.

9 10

Skew Curves–Cayley Formulas §1–§19

15 unnumbered pages of untitled sections

10

§1–§12

17 unnumbered pages of untitled sections

10

[Unorganized Material]

11

Surfaces of the Third Order

32 unnumbered pages. Only three of these notes have titles: • Hessian of a Cubic and Its Double Points (pages 7–8) • Some Remarks on a Note by Zeuthen (1868) on the Principle of Correspondence (pages 12–18) • On the Singularities of a Plane Curve (pages 19–32, dated 1885) 9

The present authors were unable to examine part three, chapter XII, §217–§232 (about 18 pages).

10

For unicursal Pieri wrote punteggiate univocamente.

11

Two successive sections are numbered §250, and §253 overlaps fascicles 6 and 6bis. Today, “6bis” might be written “6r”.

12

In fascicle 7 and most of 8, Pieri provided no § numbers.

13

Something seems missing after the eighth (unnumbered) page of fascicle 8. The ninth page includes the start of §3, but there are no preceding indications of §1 or §2. The following pages include the labels §4, §5, §5, §6–§8. Fascicle 9 starts with §10, probably compensating for that repeated section number.

9.1.2 Geometry of Position by G. K. C. von Staudt

313

In 1892, after De Paolis’s premature death, Corrado Segre wrote, A talented teacher, he possessed a great clarity and elegance of exposition; always orderly, always ready to fill the gaps, to solve the difficulties that arose during the lesson ... And on such occasions he loved to discuss with [students], explaining things in different ways; and sometimes it happened that he would be led in this way to modify the direction or disposition of the theories he expounded. ... It was usual to give two courses each year…. One [was] generally attended by third-year students ... . The second course had a more special character, and in it De Paolis dealt with several topics, . .. often developing very recent works of his own and others.

Segre’s description of further details of these courses indicates that the organization of Pieri’s notes for 1882–1883 was typical and the notes for 1883–1884 represented one of several versions of the course.14 9.1.2 Geometry of Position by G. K. C. von Staudt Pieri, Mario, editor and translator. 1889a. Geometria di posizione di Giorgio Carlo Cristiano v. Staudt. Preceded by a study of the life and works of Staudt by Corrado Segre. Biblioteca matematica 4. Turin: Fratelli Bocca Editori. JFM: Loria.

This annotated translation of Staudt’s 1847 Geometrie der Lage and the first six chapters of his 1856–1860 Beiträge zur Geometrie der Lage was Pieri’s first published work in projective geometry.15 Pieri’s commentary explains, amplifies, and offers justifications for Staudt’s results. Pieri’s most important comments, on Staudt’s attempted proof of the fundamental theorem, are translated here in their entirety. Pieri had earned his doctoral degree in 1884 at the University of Pisa with dissertations on algebraic and differential geometry. In 1886 he became professor of projective and differential geometry at the Royal Military Academy in Turin. He undertook research alongside the noted algebraic geometer Corrado Segre at the nearby University of Turin, who suggested that Pieri translate Staudt’s Geometry of Position into Italian. Segre consulted on the project, and provided the biographical article on Staudt that begins Pieri’s book. Pieri expressed appreciation to Staudt’s son Eduard and son-in-law August Papellier for giving permission to publish the translation.16

14

C. Segre 1892, 425–426.

15

Preliminary forms of parts of this subsection appeared in Marchisotto 2006, §5–§7. There had been no other translation of Staudt’s work until the French one by Philippe Nabonnand (Staudt [1847] 2011). Pieri called his translated chapters of Staudt 1856–1860 Aggiunte (additions) to the main text. He did not include its paragraphs 48–69 (see comments on pages xxiv, 208). Its last page lists and corrects twenty-one typographical errors.

16

C. Segre [1887] 1997, 1889a. The title page of C. Segre 1889a uses the spelling “C. G. C. V. Staudt”. Pieri 1889a, xxvi. Eduard von Staudt was an actuary in the forest service; Papellier was a politician (Thomasius 1867, 6). In 1888, while this project was under way, Pieri began serving concurrently as assistant to Giuseppe Bruno, chair of projective geometry at the University.

Mario Pieri’s 1889 Translation of G. K. C. von Staudt’s 1847 Geometrie der Lage

9.1.2 Geometry of Position by G. K. C. von Staudt

315

Staudt had intended his works to be abstract synthetic treatments of projective geometry, rendered as an autonomous science. In her detailed review of them, the BritishAmerican mathematician Charlotte A. Scott classified Staudt’s development into three principal phases: intuitive, abstracting from observation; constructive, creating through definitions; and formal, “considering all figures resulting from the combinations of the observed and defined elements.” For Scott, §1–§4 of Staudt 1847 and §1–§6 of Staudt 1856–1860 fall in the first phase and include “the usual propositions of projective geometry.” Sections §5 of Staudt 1847 and §6–§13 of Staudt 1856–1860 belong to the second phase, where Staudt “creates his own enlarged universe by formal definition,” and §14–§26 of Staudt 1856–1860 belong to the third phase. According to Scott, he intended to show that his “enlarged domain is a coherent and manageable whole,” where no essential distinction needs to be made between “the elements recognized by the bodily senses and those apprehended by pure intellect.” Scott observed further, “The fact that there is not a single diagram in his two books throws an interesting sidelight on his conviction of this identity of nature.” 17 In the introductory essay to his 1889a translation, Pieri expressed agreement with Staudt’s strategy of excluding diagrams, which would guide Pieri in his own works on foundations of geometry: Many will find fault that in publishing a work of geometry aimed specially at students, equipping it with a suitable series of illustrative plates has been neglected. But to us it appears better advice to remain faithful here, even in this, to the opinion of the author, who published his work without figures, and always seemed averse to the idea of introducing them ... considering that a student, on the one hand, had to get used to conceiving geometrical forms without any mechanical aid, and on the other, with each single proposition, to focusing his attention not on one individual case, but on a collection of cases.—Only after constructing the figure related to a demonstration, by hand for himself and without any preconception, do we believe that the reader can fully master it.18

Pieri would emulate this approach, excluding diagrams in virtually all of his own works on foundations of geometry. Such a strategy had been signaled also by Pasch, who noted that a “theorem is only truly demonstrated if the proof is completely independent of the figure.” 19 Theodor Reye had written in the introduction to his own influential text, after

17

C. Scott 1900a, 308, 370.

18

Pieri 1889a, xxv–xxvi. The ellipsis represents a footnote that points to a concurring remark by Segre in the preceding biographical sketch. Pieri constructed such a figure, Pieri [1889] 2020, for two paragraphs in Staudt 1847: see the frontispiece of the present book. Alongside it he commented, Synthesis of paragraphs 253 and 254 Given a triangle ABC of which the two vertices A and B should lie on a conic k, and a line r in its plane, if any two of the following conditions are satisfied— 1) that the third vertex C should lie on k 2) that the line r should be reciprocal to AB 3) that the line r should cut the edges AC and BC in reciprocal points —the remaining one must also be satisfied; moreover, under the restriction that conditions (2) and (3) should be assumed, the line r should not pass through either of points A, B.

19

Pasch (1882b, 43) also noted that elimination of figures would involve a “sacrifice of time and effort.” That book contains sixty-nine numbered figures.

316

9 Works on Foundations and Philosophy

lauding Staudt’s Geometry of Position for achieving “recognition of geometric truths through direct intuition,” that the work, obviously not written for beginners, also has several qualities, praiseworthy in themselves, that particularly hinder its study: ... terseness of expression, ... and it remains left to the reader to formulate for himself more easily graspable examples for theorems presented in their full generality. ... The presentation becomes so abstract, that a beginner’s strength wanes with its study. These qualities ... seem to have been very hampering to the well-deserved dissemination and recognition of Staudt’s work ... .

Concerning the inclusion of figures, Reye took the same position that Pieri would defend, but only for plane figures. Reye stressed the difficulties readers might face in grasping spatial forms, which he regarded as a primary goal of instruction in geometry. To make the attainment of this goal easier for the reader, I have appended figure tableaux to my lectures. ... By eschewing these means of conceptualization, ... I would have unnecessarily complicated the understanding of my lectures.20

In 1898 Scott reported, Reye’s Geometrie der Lage ... has undoubtedly great merits as a text book; but the lecturer must be prepared to deal with some flagrant logical lapses. ... I was interested to find last year, in discussing the matter with a distinguished Italian geometer, that he has felt precisely this difficulty, so strongly that he has now adopted von Staudt in place of Reye with good results. As he remarked, von Staudt may be difficult, and may throw some hard work on the lecturer, but in arrangement and thought he is quite as interesting as Reye, and he is always absolutely logical.

That Italian geometer has recently been identified as Pieri.21 With his 1889a translation and in all his subsequent research in projective geometry, Pieri sought to build upon Staudt’s vision. In these works, Pieri interpreted mathematics with the aid of his colleague Giuseppe Peano’s mathematical logic. Indeed, Peano’s noted monograph, The Principles of Geometry, Logically Exposed, appeared the same year.22 Pieri envisioned his “clearly marked”additions to Staudt’s text primarily as clarifications for readers new to the material or unaccustomed to this type of study. For example, Pieri’s first clarification, a footnote to paragraph 1, demonstrates his own early interest in axiomatic development as advocated by Peano: Many of the propositions contained in this paragraph and in the following paragraphs 2 and 3 are presented without proof, or rather as true postulates, on which are based the entirety of the work.

Pieri’s second clarification, in a footnote to paragraph 3, illustrates his reliance, in this early stage of his foundational studies, on the intuition of motion:

20

Reye 1866–1899, part 1 (1866), vii–ix.

21

C. Scott 1899, 176–177. Staudt [1847] 2011, Introduction by Nabonnand, 8.

22

Peano 1889. According to the Italian historian of mathematics Aldo Brigaglia (2012, 26), “Pieri reread Staudt using the language and method of Peano.”

9.1.2 Geometry of Position by G. K. C. von Staudt

317

Two pairs of points of a closed line are not separated, or are separated, according as it is possible, or not, proceeding along the line, to go from the one point to the other of the same pair without crossing either point of the other.23

These clarifications illustrate Pieri’s own early interest in axiomatic development, and his reliance at that stage on the intuition of motion. Pieri noted that only three of his additions treated defects in Staudt’s development. Those related to paragraph 106, Staudt’s demonstration of his fundamental theorem about projectivities. Pieri offered a more complete argument to remove a valid objection raised by Felix Klein; this was obtained by means of three different notes inserted successively in paragraphs 102, 103, and 106, and recounting in substance the modifications suggested by Klein and by Gaston Darboux, and introduced already by Reye in the third edition of his Geometry of Position.24

This addition then led to a small change in paragraph 112. With that, Pieri provided the necessary rigor for Staudt’s proof of the fundamental theorem. Pieri’s argument is described under a separate subheading. This was just the beginning for Pieri: he would revisit the fundamental theorem again and again, constructing proofs in purely projective axiomatic systems. On Pieri’s personal copy of his 1889a translation are many notations in his handwriting, both corrections and clarifications. They are described in detail under the final subheading.25 Pieri’s friend and former teacher at Pisa, Riccardo De Paolis, thanked him for sending the lovely gift of a copy of your beautiful translation of the Geometry of Position by Staudt. It will have cost you much effort; but in return you will have the satisfaction of having made something very useful.

In the Jahrbuch über die Fortschritte der Mathematik, Gino Loria reviewed Pieri’s translation with joy, because it brings credit to the nation that gave rise to it. ... It is of the most scrupulous exactness.

Pieri’s translation quickly became a standard reference in Italy for Staudt’s work.26 For example, it was mentioned in correspondence between mathematicians Achille Sannia and Federico Amodeo, and Segre cited it in his influential paper On Some Tendencies in