Sophie Germain. Revolutionary mathematician [2 ed.] 9783030383749, 9783030383756

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Springer Biographies

Sophie Germain Revolutionary Mathematician Second Edition

DORA MUSIELAK

Springer Biographies

The books published in the Springer Biographies tell of the life and work of scholars, innovators, and pioneers in all fields of learning and throughout the ages. Prominent scientists and philosophers will feature, but so too will lesser known personalities whose significant contributions deserve greater recognition and whose remarkable life stories will stir and motivate readers. Authored by historians and other academic writers, the volumes describe and analyse the main achievements of their subjects in manner accessible to nonspecialists, interweaving these with salient aspects of the protagonists’ personal lives. Autobiographies and memoirs also fall into the scope of the series.

More information about this series at http://www.springer.com/series/13617

Dora Musielak

Sophie Germain Revolutionary Mathematician Second Edition

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Dora Musielak University of Texas at Arlington Arlington, TX, USA

ISSN 2365-0613 ISSN 2365-0621 (electronic) Springer Biographies ISBN 978-3-030-38374-9 ISBN 978-3-030-38375-6 (eBook) https://doi.org/10.1007/978-3-030-38375-6 Originally published with the title: Prime Mystery: The Life and Mathematics of Sophie Germain 1st edition: © AuthorHouse 2015 2nd edition: © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are solely and exclusively licensed 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. Cover illustration: Sketch according to the bust of Sophie Germain by sculptor Zacharie Astruc (1835–1907). Source Stupuy (1896) This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To my beloved mathematician, for a lifetime of intellectual pleasure.

Preface to the Second Edition

This new edition gives me the opportunity to expand on the mathematical work that Sophie Germain carried out, adding more details extracted from correspondence that had not previously been explained. This edition will be distinguished from the previous one by the new material added and by a reorganization of the subjects. Chapter 3 is new. It focuses exclusively on the research on number theory that Sophie Germain carried out while communicating with Gauss in the years between 1804 and 1809. I added this chapter to highlight Germain’s work on quadratic reciprocity before she matured her plan to prove Fermat’s Last Theorem (FLT). Hence, the section in the original edition devoted to Germain’s contribution to FLT has been redone almost entirely and now is addressed in Chap. 9. Chapter 12 contains new details extracted from the letters written by Carl Bader to Gauss regarding Sophie Germain, and also from her last letter to Gauss in 1829, together with an explanation of Germain’s last papers published in Crelle’s Journal. The remaining chapters have experienced enough changes to add clarity where needed. For example, I translated excerpts from correspondence and from published memoirs, historical pieces that are important to support the story. In addition, this edition contains translations of nine of the 14 letters exchanged between Germain and Gauss. These epistles are supplemented by two other communications that help us shed light on the events that led to the revelation that the Monsieur Le Blanc corresponding with Gauss was actually Sophie Germain. I also added a comprehensive list of References, including her own works and those of others, spanning from 1705 to the present. The reader must be assured that I researched every detail about Sophie Germain’s life and work. The citations in the footnotes are intended to validate every statement made throughout this book. I strived to avoid unsubstantiated claims or embellished stories, in order to portray a real mathematician with her

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share of faults and intellectual triumphs. At the same time, it is my desire that this work will serve as inspiration for readers who may wish to continue exploring the historical archives to uncover more of Sophie Germain’s endeavors. Arlington, TX, USA

Dora Musielak

Preface to the First Edition

In the Age of Enlightenment, scholars were engaged in stimulating scientific work that led them to discover a myriad of novel mathematical methods. New branches of mathematics were born with those discoveries. The names of Euler, Bernoulli, Newton, Lagrange, Gauss, Legendre, Cauchy, and others are attached to these mathematical discoveries. Their efforts produced the powerful mathematical methods we apply today in all scientific disciplines so familiar to students of science and engineering. Among those renown scholars we find the name of Sophie Germain, sometimes linked to Gauss, sometimes hyphened to Lagrange’s. As we learn more about her story, Sophie Germain emerges as an important figure that helped shape the foundation of two branches of mathematics: number theory and mathematical physics. A theorem on number theory related to the proof of Fermat’s Theorem is named after Germain; today, we study her special proposition together with the results by Euler, Legendre, and other number theorists. Before Germain, no other woman had accomplished so much. Surprisingly, she never attended school! Despite her lack of scientific training and despite being excluded from the learned societies, Sophie Germain succeeded in developing her own theorems and in stirring new ideas in others. Sophie Germain grew up in an era of mathematical and social revolution. Her childhood developed during the reign of the tragically infamous Louis XVI and Marie-Antoinette, and she came of age during the most tumultuous years of insurrection against the monarchy. Within months of her thirteenth birthday, the French Revolution exploded violently with the storming of the Bastille. Coincidently, that same year Sophie awoke to the dawn of her own intellectual development, one that would convert her into an unusual and formidable mathematician. Alone, she undertook two of the most important problems of her time, a scientific feat that required incredible ingenuity in order to develop two diametrically different mathematical theories. Sophie Germain was a mathematician, researcher, physicist, and philosopher. And thus, a book about her life must address diverse aspects of her work. I begin with her work in applied mathematics. ix

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Preface to the First Edition

Sophie Germain was influential in the development of the theories of elasticity and vibrations. She derived the first equation that attempted to explain Chladni’s vibrating plates. Overcoming prejudice and numerous obstacles to her scientific endeavors, in 1816 Sophie Germain became the first woman in the history of science to win the Prix de Mathématiques of the French Academy of Sciences. This prestigious contest was established at that time by the Premier Classe des Sciences Physiques et Mathématiques, a branch of the Institut de France, at the request of Napoléon. The story of Sophie Germain’s contribution to elasticity theory and the checkered history of her prize have already been told in Bucciarelli and Dworsky’s fine book of 1980. In retelling it in chapter 6, I make use of my own historical research and also endeavor to give a different perspective on the mathematical ideas that sustained Sophie Germain to develop her own theory. I continue to expound Germain’s research through chapter 7, as I attempt to reconstruct the experiments with vibrating plates that she carried out after she won the prize. I will refer to her last publication to call attention to her experiments with vibrating plates. Seeing her engaged in such research activity suggests that Germain was not just a mathematician; she could very well be considered the first woman research engineer! In number theory, Sophie Germain is the first and only woman to make a substantial contribution to the centuries-old proof of Fermat’s Last Theorem. I will devote chapter 9 to her work in number theory, focusing especially on her theorem. For the most part, this chapter is based on the scholarly article of R. Laubenbacher and David Pengelley. I added additional historical and mathematical background to align their exposition to the context of this book. Another aspect of Germain’s work I review is her philosophy. This is important because I want the reader to hear Sophie’s words, beautifully tinted by her deep love for mathematics and that reveal a bit more of her being. Finally, to better appreciate her background, I feel justified to include short biographical sketches of the mathematicians who helped shape her scientific legacy. With many books and articles published about women mathematicians in history, biographical sketches of Sophie Germain abound. And because information about her childhood and educational beginning is obscure, some of those descriptions are often vague or inaccurate. Of course, Sophie Germain herself made it difficult for her biographers to ascertain the truth about the woman behind the fame. She left no autobiography or intimate letters that can shed light into her feelings about loved ones. Germain did not write about her childhood or her everyday life, focusing instead on describing her mathematical discoveries. Thus, a biographer has a daunting endeavor, trying to piece together what little historical evidence exists to ascertain Sophie Germain’s private life with data about the development of science before and during her time. I have a twofold aim in this book. I wish first to relate some historical developments in mathematical physics and in number theory, and in doing so, attempt to recreate the social and intellectual milieus in which Sophie Germain lived. Second, I hope to explain in easy terms the mathematical ideas and methods that Germain

Preface to the First Edition

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pursued, and on which other contributions have been built. Most biographies of mathematicians are typically devoid of mathematics, focusing instead on details of their personal lives that make for a very attractive reading. I believe that there is room for a book like this, one that will go deeper into the history and the mathematical foundation of an intriguing woman who left us a beautiful scientific legacy. Most of the citations in this book are taken from the originals; some I left in French to retain the subtlety or ambiguity of the expression, while other passages (unless otherwise noted) I translated myself. My goal is to provide a perspective about the social and historical environment that surrounded Sophie Germain, while introducing the reader to some of her scientific endeavors. In many popular books, it is not uncommon to read embellished accounts of the lives of mathematicians. Disregarding the truth, many authors use apocryphal stories as fact, discrediting the biographical information they attempt to present. That leads to conveying to the reader many falsehoods that are repeated in other biographies and disseminated through the Internet. I try to avoid unfounded speculation and only include in this book extracts from the life story of Sophie Germain that are fully documented. I verified historical facts, read her articles, and consulted many historical and scholarly publications. In addition to reviewing Germain’s work, I also attempt to present a hint of the woman through an account of how life evolved in Paris, from her childhood, when France was besieged by its violent revolution, to the last years of her life. I hope that by including snippets of French history will increase our understanding of how Germain’s life unfolded, and help us place in historical context her scientific isolation. I considered not only how Germain’s work relates to the history of mathematics, but also as a means to reveal a glimpse of her temperament. I did not confine this account to chronological order. Rather, I arranged it under different topics, a method which may result in some repetition or may lead me to present an event tinted by the clear eye of retrospect. However, the timeline at the end of the book presents a chronological sequence that may serve to guide us through the major events that shaped Sophie Germain’s fifty-five-year-life. Chapter 11 interrupts the biographical narrative to provide short life-stories of the mentors, friends, and rivals, the erudite men who formed part of Germain’s scientific story. Her own life story is remarkable and worth recounting, at least in part. Of course, there are many questions about her that remain unanswered. As a young woman, Sophie Germain dazzled the Parisian intellectuals when they discovered that she carried out mathematical work in response to lectures by Lagrange at the École Polytechnique. How did Germain learn mathematics on her own before sending her analysis to Lagrange? Women were not admitted to the École. What drove Germain to circumvent the rules? She signed her work with the pseudonym M. Le Blanc; was this assumed name borrowed from a male student she knew? Who was Monsieur Le Blanc? Many have suggested that Lagrange became her teacher of mathematics. Then why did Lagrange not write about Germain to say how he discovered her? Lagrange did not talk about her talents nor did he describe her as a pupil. These and many other questions I raise in chapter 13, inquiries that may serve as the basis for further study.

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Preface to the First Edition

The story of Sophie Germain is particularly important and inspiring because it reminds us of a time when higher education was not a right of women. Learning about her struggles to acquire an education, and to be part of the search for a better understanding of the world, may help us to appreciate the freedoms and opportunities we women enjoy now. I believe this effort is important because it puts in perspective how a person pursues a scientific activity, including the learning that is required, and the tenacity and motivation scientists must have to pursue their work in spite of many obstacles and challenges. This book is intended for anyone who wishes to learn more about Sophie Germain’s work and the ideas she championed. Hers is a truly fascinating mathematical story. I hope that a wide range of readers will find it as interesting and exciting as I do, and this will encourage them to explore the beauty of mathematics in more detail and lead them to discover its many gems. Consequently, I have written neither a conventional biography nor a mathematical study meant solely for the benefit of specialists. In fact, my approach may be controversial to some. At this point I want you to turn the page and begin to read Sophie Germain’s life story. However, I must first acknowledge those who supported this endeavor, the two people who were aware of the first steps I took, and saw me stumbling in my effort to write a book that was accurate in its mathematics and also fair in its historical context. First, I am sincerely indebted to Professor David Pengelley for reading the first drafts of chapter 9, and for his invaluable instruction regarding Germain’s work on number theory. I also thank Professor Dominic Klyve wholeheartedly for the insightful comments he made on several aspects of the book. I ask them both for forgiveness if I didn’t take their advice entirely. However, I will be forever grateful to David and Dominic for their encouragement and words of wisdom during the darkest hours when the feedback from others proved so disheartening. I gratefully acknowledge the Bibliothèque Nationale de France for providing copies of many original materials, and especially for granting me permission to reproduce the sketches drawn by Sophie Germain and archived among her manuscripts. Likewise, I acknowledge The Euler Archive where I found every one of Euler’s publications and guided me through the historical referencing of the works in elasticity theory, sound and vibrating bodies, and a testament of Euler’s contribution in number theory. Finally, I thank my daughters Lauren and Dasi for their smart and thoughtful comments. I hope you will find in this account a source of inspiration. It was an indescribable pleasure for me to write about Sophie Germain and her work, and it is my sincere desire that you will enjoy reading it, too. Dora Musielak

Acknowledgements

I have special people to thank for their thoughtful and unselfish input to this edition. I am grateful to Menso Folkerts, Professor of the History of Science at the University of Munich, Germany, who was kind to send me transcripts of Carl Bader’s original letters to Gauss, and he translated them into English. This helped me to better ascertain the circumstances that prompted Sophie Germain to pen her last letter to Gauss in 1829 and confirmed what I wrote about it in Chap. 12. I owe an immense debt of gratitude to David Pengelley, Emeritus Professor of Mathematics at New Mexico State University. He read critically Chaps. 3 and 9 and provided invaluable comments, helping me to clarify the exposition of the topics in those chapters, especially the details of Sophie Germain’s Theorem. His critique ensured that my retelling of what Germain did to advance Fermat’s theorem is mathematically and historically correct and consistent with his scholarly findings. To Ramon Khanna, Executive Editor at Springer, my most sincere thanks for believing in this project, his careful attention to the manuscript, and for his kind encouragement throughout. And of course, I am grateful to Springer editorial staff, especially Rebecca Sauter, for their expert help in the process of preparing the manuscript for publication. I acknowledge the Bibliothèque Nationale de France (BNF), Göttingen State and University Library, the ETH-Bibliothek Zürich, and The Euler Archive for making available all the memoirs, letters, and the images I use to illustrate this book. Finally, my eternal thanks to my daughters Dasein and Lauren, and to my husband Zdzislaw, for their loving support and knowing, Omnes enim trahimur, et ducimur ad cognitionis et scientiae cupiditatem, in qua excellere pulchrum putamus. Dora Musielak

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Contents

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Unforgettable Childhood . . . . . . . . . Primary Education . . . . . . . . . . . . . . Revolutionary Mathematicians . . . . . . Coming of Age Through the Terror . . Institut de France: Science Above All

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Lessons from l’École Polytechnique . . . . . . . . Lagrange’s Lecture Notes 1797–1799 . . . . . . . M. LeBlanc Metamorphoses into Mlle Germain A Young Scholar Emerges . . . . . . . . . . . . . . .

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Sophie’s Sublime Arithmetica . . . . . . . . . . . . . . . . . . . . . Number Theory: From Diophantus to Gauss . . . . . . . . . . . . Sophie Germain and Carl Friedrich Gauss . . . . . . . . . . . . . How Napoléon’s Invasion Led to Unmasking M. Le Blanc . Sophie Germain Tackles the Law of Quadratic Reciprocity . Gauss: Mathematical Astronomer . . . . . . . . . . . . . . . . . . . . Gauss and Legendre: A Matter of Priority . . . . . . . . . . . . .

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Chladni and His Acoustic Experiments . . . . . . . . . . . . . . . . . . . . . . The Prize of Mathematics, 1809 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Euler and the Bernoullis . . . . . . . . . . . . . Euler and the Mechanics of Elastic Bodies . Foundation of Elasticity Theories . . . . . . . . Sound and Vibrating Bodies . . . . . . . . . . .

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Germain and Her Biharmonic Equation . First Hypothesis . . . . . . . . . . . . . . . . . . . . Second Attempt: More Disappointment . . . Paris in 1814 . . . . . . . . . . . . . . . . . . . . . . Winning the Grand Prix de Mathématiques

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Contents

Confronting a Rival . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Germain-Lagrange Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Experiments with Vibrating Plates . . . . . . . . . . . . . . . . . . . . . . . . . Sophie Germain’s Experimental Research . . . . . . . . . . . . . . . . . . . . . .

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Elasticity Theory After Germain . . . . . . Navier’s Bending Equation . . . . . . . . . . . Cauchy and His Mathematical Formalism Poisson and an Incorrect Prediction . . . . . Poisson-Germain-Navier Public Dispute . . Kirchhoff’s Plate Theory . . . . . . . . . . . . . Ritz Method to Model Chladni’s Plates . .

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Germain and Fermat’s Last Theorem . . . Pierre de Fermat . . . . . . . . . . . . . . . . . . . . Euler and Fermat’s Theorems . . . . . . . . . . Legendre Proposes a Contest to Prove FLT Sophie Germain’s Theorem . . . . . . . . . . . . Unexpected Revelation . . . . . . . . . . . . . . . Germain’s Research to Prove Fermat’s Last Fermat’s Last Theorem After Germain . . . . The Fermat-Wiles Theorem . . . . . . . . . . . . Unsolved Problems in Number Theory . . . .

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10 Pensées de Germain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 11 Friends, Rivals, and Mentors . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . Carl Friedrich Gauss (1777–1855) . . . . . . . . . . . . Joseph-Louis Lagrange (1736–1813) . . . . . . . . . . Adrien-Marie Legendre (1752–1833) . . . . . . . . . . Jean-Baptiste-Joseph Fourier (1768–1830) . . . . . . Siméon-Denis Poisson (1781–1840) . . . . . . . . . . . Claude-Louis-Marie-Henri Navier (1785–1836) . . Jean-Baptiste Joseph Delambre (1749–1822) . . . . Augustin-Louis Cauchy (1789–1857) . . . . . . . . . . Guglielmo Libri, Count de Bagnano (1803–1869) . Leonhard Euler (1707–1783) . . . . . . . . . . . . . . . . Archimedes of Syracuse (c. 287–212 B.C.) . . . . .

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12 The Last Years . . . . . . . . . Reaching Out to Gauss, One Glorious Summer of 1830 . . Germain’s Last Publications

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Contents

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13 Unanswered Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 14 Princess of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 Women and Science Education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Sophie Germain Legacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Germain-Gauss Correspondance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Sophie Germain Timeline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Illustration Credits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

Chapter 1

Unforgettable Childhood

Sophie-Marie Germain was born in Paris on 1 April 1776, ten months after Louis XVI was crowned King of France and his wife Marie Antoinette became Queen. That was a time when France was the most powerful country in Europe, and its cultural influence was such that nobles, monarchs, and the educated people in many other countries often spoke elegant French instead of their native languages. France was also among the most scientifically enlightened nations in the world. The Royal Academy of Sciences in Paris had among its members some of the most eminent and influential mathematicians in history such as Lagrange, Laplace, and Legendre. Unbeknownst to them all, thirteen years later the reign of Louis XVI and Marie Antoinette would crumble, destroyed by a violent social revolution that would change France forever, and when young Sophie would discover her passion for mathematics. She would grow up to work with Legendre and Lagrange in challenging problems of pure and applied mathematics. Sophie was the middle child of Ambroise-François Germain d’Orsanville and his wife Marie-Madeleine Gruguelu. Sophie had two sisters: Marie-Madeleine, who was five years older, and Angélique-Ambroise who was two years younger. Germain’s father was a silk merchant (Merchand de soi en bottes), belonging to the liberal and educated classes. Mr. Germain was the son of Thomas Germain, orfévre,1 sculptor, and architect. The Germains resided on rue Saint-Denis, no 336, an address on the right bank, 400 metres from the Seine River, right in the heart of Paris. Their house was at the level of la Fontaine des Innocents at the intersection with rue au Fers (now rue Berger). Sophie Germain’s early childhood developed during the declining era of the Ancien Régime, a time of radical transition that would shape her life and influence her career. Under the old monarchic, social and political system, the French were subjects of the King as well as members of an estate and province. All rights and

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Stupuy (1896).

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 D. Musielak, Sophie Germain, Springer Biographies, https://doi.org/10.1007/978-3-030-38375-6_1

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1 Unforgettable Childhood

status flowed from the social institutions that were divided into three orders: the clergy (First Estate), the nobles (Second Estate), and the commoners (Third Estate), which included the bourgeoisie class and the peasants. For a long time, the Kingdom of France (Ancien Regime) had experienced many problems and suffered financial difficulties. When Louis XVI ascended the throne, he inherited a kingdom that was nearly bankrupt driven by the opulence of his predecessors, by wars, and by other factors. Harsh winters resulted in failed crops, and the price of bread and other food soared. The French economy began to deteriorate and led to high unemployment, and even hunger. The people were justifiably discontent, seeing the lavish lifestyle of the monarchs in Versailles in drastic contrast with their suffering. Their discontent was fueled by dislike of Marie Antoinette, a foreigner, a Habsburg Archduchess from Austria before she married Louis XVI. The union had been a strategic alliance, a political agreement between the Bourbons (France, Spain, Parma, Napoli and Sicily) and the Habsburg (Austrian Empire dynasty). The French people targeted Marie Antoinette as the main source of their economic problems. Louis XVI attempted to reform the monarchic system in accordance with Enlightenment ideals. The king tried to abolish serfdom (status of peasants under feudalism), remove the taille (a tax levied on the common people), and increase tolerance toward non-Catholic and other minority groups. The French nobility reacted to the proposed reforms with hostility, and opposed their implementation. Louis XVI also supported the North American colonists, who sought independence from Great Britain, and which was realized with the 1783 Treaty of Paris. By 1787, France faced great economic challenges, and discontent among the commoners escalated due to the unfair taxation system. This and many other factors contributed to the ferocious rebellion that exploded in Paris and throughout France in the late 1780s. It can be argued that the rising wealth and influence of the bourgeoisie middle class, an archaic and regressive tax system, and a structure of rank and subordination more suited to the middle ages, all contributed to the downfall of the Ancien Régime. In 1789 the financial crisis intensified, and years of feudal oppression, combined with poor harvests and loss of jobs, had resulted in hunger among the peasants and the workers. All this helped create a French society that was ripe for rebellion. In the countryside, peasants and farmers revolted against their feudal contracts by attacking the manors and estates of their landlords. That same year, Ambroise-François Germain, Sophie’s father, was elected Deputy of the Third Estate, representing the common people at the assembly of the Estates-General in Versailles. The Estates-General of 1789 was the first meeting that the King of France had convened since the Estates-General of 1614, a general assembly representing the French estates of the realm: clergy, nobility, and the commoners. Summoned by King Louis XVI to propose solutions to his government’s financial crisis, the Estates-General sat in Versailles for several weeks in May and June 1789, but it came to an impasse as the three estates clashed over their respective powers. It was

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3

brought to an end when many members of the Third Estate, the group of proletariats (workers, merchants, and peasants), formed themselves into a National Constituent Assembly (Assemblée nationale constituante). Its revolutionary spirit galvanized the nation in a number of violent ways. Being a wealthy merchant, Sophie’s father understood the proposed economic reforms during his tenure as Deputy of the Third Estate. Monsieur Germain d’Orsanville2 was member of the National Assembly from 14 May 1789 to 30 September 1791. During that time, he participated in meetings, advocating economic reforms, including ways to remedy the scarcity of cash, such as the creation of small assignats3 and reduce the public debt. It is reasonable to assume that Monsieur Germain supported the resolutions that resulted into the Constituent Assembly. However, Louis XVI didn’t approve the new Assembly. On 12 July 1789, the King dismissed his very popular Minister of Finances. Soon after a rumor spread in the streets of Paris of an impending counter-attack by the King’s soldiers to destabilize the newly proclaimed parliamentarians. On July 14, mobs of angry citizens stormed the Bastille, a prison that was a symbol of the King’s absolute and arbitrary power. When the prison governor refused them entry, the mob charged, taking arms and gunpowder, and after a violent battle they took hold of the building. The uprising quickly turned into a gory massacre of hundreds of people. The governor was seized and killed, and his head on a spike was paraded through the streets. The storming of the Bastille (Fig. 1.1) marked the beginning of the French Revolution. Sophie Germain was thirteen when this tragic event unfolded. She must have become acutely aware of the social and economical conflicts just by listening to the accounts of her father and his colleagues. After the storming of the Bastille, the revolution ignited. Sophie’s house on rue Saint-Denis was geographically located at the epicenter of the brutal revolts that shook Paris. Les Halles, the famous market from where hundreds of angry women protested against the monarchy, was just steps away from her home; from that market the women marched for several hours to Versailles, armed with pitchforks, to confront the king. The Hôtel de Ville, the city hall, was a six-minute walk from Germain’s residence. This building was the center stage for many violent events during the French Revolution, notably the murder of the last provost of the merchants, Jacques de Flesselles, on 14 July 1789, and the hanging of Foulon de Doué at the Place de Grève just across from the Hôtel de Ville. The birth of the French Revolution was an important and stimulating time for liberal thinkers. We can ascertain that, as a deputy of the National Assembly, Mr. Germain was engaged with important leaders and intellectuals. Was he an ardent revolutionary? Very likely. On 8 October 1790, and on 5 May the following year,

2

Name shown in official documents of the National Assembly meetings. See: https://www.persee. fr/authority/722281. 3 An assignat was a monetary instrument used during the French Revolution.

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Fig. 1.1 Fall of the Bastille Prison, 14 July 1789. Credits see Appendix “Illustration Credits”

Mr. Germain gave speeches to protest against agiotage (the business of exchanging currency).4 However, after the dissolution of the National Assembly, the name of Mr. Germain disappeared from the records. It was suggested that he was later occupied in currency speculation, and for a while he was a bank director. It is probable that Sophie Germain’s father disappeared from the political scene due to health issues or because of his age. In 1791, he was sixty-five. Yet another plausible reason could have been fear for his family and his own safety since the political environment became rather contentious and dangerous. With the power shifting to their side, the revolutionaries began to demand more from the king who lost control of any peaceful negotiations. In September of that year, after long negotiations, Louis XVI reluctantly accepted a new French Constitution. Redefining the organization of the French government, citizenship and the limits to the powers of government, the National Assembly set out to represent the interests of the general will. The new, but short-lived constitution abolished many institutions, which were injurious to liberty and equality of human rights. Family life continued its course, despite the social chaos. Early in 1790, Madeleine Germain, Sophie’s older sister, married M. Lherbette. On 16 September

4

Stupuy (1896), p. 3.

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5

Fig. 1.2 The proclamation of the French constitution on 14 September 1791 (After Louis XVI approved the new Constitution, jubilant Parisians celebrated at the place du Marché des Innocens, not far from Germain’s residence. The Fountain of the Innocents was a traditional place of rendezvous). Credits see Appendix “Illustration Credits”

1791 the couple had a son whom they named Jacques-Amant. Two days before he was born, the neighborhood of Germain’s home was the stage for the jubilant proclamation of the new French constitution (Fig. 1.2). The young Lherbette would grow up and become influential in making Sophie Germain’s philosophical legacy known to the world. Everything was quickly changing in a nation fiercely revolting against the old regime. In the summer of 1792, as distrust of the King and the “aristocratic conspiracy” built up, Parisians began to panic as both Austrian and Prussian troops were moving closer to the city. Believing that the king, or his disliked wife, was giving information to these foreign powers, the Paris Commune planned an attack on the Tuileries Palace, the king’s residence. On the 10th of August, a mob of angry citizens advanced toward the Tuileries to capture Louis XVI. That morning, forty-seven of the forty-eight Parisian sections headed by the National Guard and revolutionary fédérés from Marseilles and Brittany attacked the Tuileries Palace. The king was alerted, and so he fled with his family, seeking refuge in the Legislative Assembly building; the Swiss Guard was left behind to defend the palace. When the enraged people did not find Louis XVI, they murdered not only thousands of the Swiss guards but also anyone within the

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palace grounds that might have been associated with the monarch: cooks, servants, maids, no one was spared. Soon after, the mob moved to the Legislative Assembly building and arrested Louis XVI, Marie-Antoinette and their family, children included. They were taken to the Temple, an ancient fortress in Paris that was used as a prison. Under pressure, the Assembly voted the suspension of the king’s powers. Louis XVI was stripped of all titles and honours. This event, known in French history as the Insurrection of 10 August 1792 (Journée du 10 août 1792, Fig. 1.3), signified the end of the monarchy and began the official trial of the imprisoned king. As the republican movement swelled, French men and women were addressed as “citizens” (citoyen and citoyenne) to support the fevered patriotism and political correctness of the time. By the time the insurrection of August 10 had overthrown the monarchy, “citoyen” and “citoyenne” had become quasi-official terms of address. On 23 September 1792, after the declaration of the French Republic, a leader of the Convention rose to demand that citoyen become the official designation of all Frenchmen. The egalitarian designation citoyen and citoyenne were

Fig. 1.3 This dramatic military scene depicts the insurrection of 10 August 1792 during the attack on the Tuileries Palace, royal residence. This engraving depicts the fight that exploded in the courtyard of the palace after the king and his family fled the royal residence. The fire that ravaged the premises (traces of which were found during excavations in 1990–91) is visible on the left. Nothing survives today: the courtyard is now the Carrousel Gardens, while the Tuileries Palace itself, which ran between the Flore and Marsan pavilions, completely burned down in May 1871 during the La Semaine sanglante (The Bloody Week), another social uprising in Paris. Credits see Appendix “Illustration Credits”

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adopted as universal forms of address, replacing the “aristocratic” designations “Monsieur” and “Madame.” That same year, the king was brought to trial for “crimes against the people.” By then, he was known as “Citoyen Louis Capet,” a nickname in reference to Hugh Capet, the founder of the Capetian dynasty—which the revolutionaries interpreted as the king’s family name. On 21 January 1793, King Louis XVI of France was brutally executed at the guillotine. That same year, on October 16, Queen Marie-Antoinette suffered the same fate; like her husband, she was executed in front of cheering crowds. Sophie lived very close to La Conciergerie, the prison where Marie-Antoinette and many other unfortunate citizens spent their last days before being beheaded. Known as the “antechamber to the guillotine,” the menacing building was about five hundred meters away from Sophie’s home. One can imagine that it was close enough for her to bear witness to many horrid events; the street below her window could have led directly to the dreaded prison.

Primary Education Little is known about Sophie Germain’s early education or how she learned mathematics, an unusual pursuit of young women at the time, especially since there is no record that her bourgeoisie family included a tradition of scholarship. Did she have a private tutor or attend secondary school? This is unlikely. Under the Ancien Régime, Paris had numerous schools, most of which were run by the clergy. For example, the Collège des Quatre Nations, better known as Collège Mazarin, was an institution for young men where Condorcet, d’Alembert, and Legendre received their early education. There were convent schools for girls of the aristocrat classes, but even if Germain had attended such a religious lycée, it is doubtful that she would have learned mathematics there, since girls were not taught the same curriculum as boys. To gain a proper science education, young women had to either impersonate men or study on their own, isolated from the sources of formal education. Sophie Germain did both. The year 1789 was pivotal for Sophie Germain. Hipolite Stupuy wrote in Germain’s first biography5 that when the revolution exploded, Sophie found refuge in her father’s library. It is not difficult to imagine her as a sensitive teenager, finding solace in her studies, immersing herself in books to build a protective shield against the chaos outside her window. The awakening of Sophie’s interest in mathematics is attributed to her encounter with Archimedes while reading the history of mathematics in Montucla’s book. Jean Étienne Montucla was a French historian of mathematics who wrote an

5

Stupuy (1896), p. 5.

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interesting work on the history of squaring the circle,6 a well-known problem proposed by ancient geometers. In 1756, Montucla published Histoire des mathématiques. The first volume was devoted to the history of mathematics from ancient times to the year 1700, while the second volume covered the history of mathematics in the seventeenth century. It is a fascinating book that would have been a source of knowledge and inspiration for any youthful reader interested in mathematics. I believe that when Sophie Germain read Montucla’s history of mathematics, she had already realized her aptitude for the exact sciences. Otherwise, she would not have found the book interesting, or she would not have been touched by the tales about ancient mathematicians. Archimedes, in particular, impressed her deeply. The absorption of Archimedes in his mathematical research is legendary. This preoccupation with mathematics is said to have been the cause of his death, which occurred during the capture of Syracuse by Marcellus in 212 B.C. Montucla wrote about the death of Archimedes at the hands of a Roman soldier.7 One can imagine the young Sophie being profoundly captivated after reading the story. She must have told someone about it because Stupuy credited Archimedes as her idol and placed this event as her initiation into mathematics. He wrote that Germain chose to dedicate herself to this “geometric science so endearing that nothing could distract her from it, not even a threat of death, this science which she barely knew by name, that was the one that suited her; and, at that moment, Sophie took the heroic resolution to give herself to it completely.”8 The story of Archimedes would haunt Sophie Germain, and she carried a fear throughout her adult life for the well-being of noble, extraordinary scholars. At sixteen, Sophie Germain would witness another violent insurrection, a defining event of the French Revolution. On 10 August 1792, a day known in French history as “the insurrection of 10 August 1792,” marked the end of a thousand years of French monarchy. There is no doubt that by then, Sophie Germain was preparing for her work in mathematics. That early study would give her the foundation that led eventually to be discovered by Lagrange and Legendre.

Revolutionary Mathematicians Amid the turmoil of the revolution, France was making an enormous contribution to the fund of knowledge, including the development of modern analysis and mathematical physics. At that time, some of the most distinguished mathematicians of the eighteenth century were working to advance several branches of the exact sciences: Joseph-Louis Lagrange, Gaspard Monge, Pierre-Simon Laplace,

6

Montucla (1754). Montucla (1756), p. 249. 8 Stupuy (1896), p. 6. 7

Revolutionary Mathematicians

9

Adrien-Marie Legendre, and Joseph Fourier. These great erudite men further developed the mathematics of Newton, Leibniz, Euler, the Bernoullis, Fermat, and other mathematicians before them. When Sophie Germain was twelve years old, Lagrange had already expanded the analytical mechanics of Euler. With his Mécanique analytique, Lagrange revealed Newtonian mechanics as a rich field of exploration for mathematicians. The scientific environment in Paris in 1788 was vibrant. The French Royal Academy of Sciences9 counted among its members some of the most famous scientists of the eighteenth century. Then the revolution erupted, and its ideology would have a drastic impact on the French scientific institutions. On 8 August 1793, the National Convention abolished the Royal Academy of Sciences, together with other learned institutions, because of its royalist title and elitist nature. In the midst of the social upheaval, the French savants—the mathematicians and science scholars in those days—were reshaping the sciences. Among their work was the creation of the metric system, a standardized system of measurement so crucial for scientists, engineers, and for everyday life. The metric system that emerged during the French Revolution was essentially the creation of a core of scholars from the Paris Academy of Sciences. All through the Ancien Régime, the privileges of the nobility included the right to control local weights and measures, leading to corruption and irreconcilable disputes among the different estates. On 4 August 1789, three weeks after the storming of the Bastille, the nobility surrendered their privileges. However, the complaints about the lack of uniform weights and measures continued. In 1790, the Constituent Assembly set up a new committee sponsored by the Academy to investigate weights and measures, with the goal of producing a national system that would provide a standard scale for weights, measures, and money. The members were five of the most able savants of the day—Lagrange, Laplace, Monge, the Marquise de Condorcet, and Jean-Charles de Borda. Condorcet, the founder of mathematical social science, expressed his desire that the new system of measure would belong to all the people of the world. After some studies, the committee favoured a unit of length (meter) equal to a ten-millionth of the Earth’s quadrant, the part of the meridian from the North Pole to the Equator, measured at sea level. This monumental enterprise required performing astronomical observations and geodesic surveying to define the length of the meter. It involved grueling triangulation work by the best French astronomers, and rigorous geometric calculations lead by mathematical physicists. The still-ruling king authorized this scientific undertaking. Without much fanfare, in 1792 two astronomers set out to measure the world. Jean-Baptiste-Joseph Delambre led the northern portion of the meridian, from Dunkerque on the upper northern corner of France (10 km from the border with Belgium) to Rodez in southern France. Pierre-François-André Méchain was

9

The Royal Académie des Sciences was established in Paris in 1666 under the patronage of Louis XIV to advise the French government on scientific matters.

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responsible for the southern portion of the meridian, from Rodez to Barcelona in Spain. Their triangulation odysseys would take seven years. When the revolutionary Convention Nationale abolished the Royal Académie des Sciences, it eliminated the Commission of Weights and Measures (8 August 1793). The feared Committee of Public Safety required all groups of scientists to justify their existence. It was thanks to chemist Antoine François, Comte de Fourcroy, that the Commission was allowed to continue. Fourcroy was member of the Convention who argued about the importance of reforming weights and measures for securing the future of France. Borda extolled the decimal system for weights and money, recommending that it be extended to the measures used in astronomy and geography. Astronomers Delambre and Méchain continued their triangulations as rapidly as the troubled times permitted, and the effort to establish the metric system carried on. A new calendar wholly independent of the Gregorian calendar was also among the reforms of the Republican government. Eager to overthrow the oppression of the nobility, and sweep away the Ancien Régime, the National Convention adopted a new descriptive calendar intended to reflect reason, science, and nature. Thus, a commission comprised of scientists, poets, and artists was formed to conceive the new Republican Calendar. Lagrange, Monge, and astronomer Lalande were among those responsible for dividing the French year into twelve months of thirty days, and ten-day weeks (decades). The months, with names such as Messidor (harvest) and Brumaire (fog), were based on the seasons of the year, and do not correspond to the standard months of January through December. For example, Thermidor (heat) corresponded to July 19–August 17. Five or six feast days remained at the end of each year known as complementary days (jours complémentaires) and these were dedicated to vacations and celebrations. Feast days were named in honor of Virtue, Genius, Labor, Opinion and Rewards. During leap year the additional day was called Revolution. The Republican Calendar was officially adopted in France on 24 October 1793, more than one year after the advent of the First Republic (there was no year 1). In addition to Lagrange, mathematician Adrien-Marie Legendre was also involved with the scientific, astronomical, and administrative duties of the French reform of weights and measures. Legendre, who would become a mentor and friend of Sophie Germain, had published a memoir on spherical triangles, which was later successfully used in the geodesic operations. He also developed his method of least squares, an important tool in the reduction of data acquired in the geodesic surveys.

Coming of Age Through the Terror After abolishing the monarchy of Louis XVI, the revolution intensified and became more vicious. The drastic actions taken by the new extremist government culminated with the Reign of Terror (La Terreur in French). This was a period of

Coming of Age Through the Terror

11

violence that occurred between June 1793 and July 1794, when thousands of well-known and nameless victims were guillotined. During this era, France remained in a state of chaos, aggravated by internal political conspiracies that resulted from opposing ruling parties, and by the wars waged with other European countries. Caught up in civil and foreign wars, the revolutionary government decided to make “Terror” the order of the day (September 5 decree) and to take harsh measures against those suspected of being enemies of the Revolution (nobles, priests, hoarders, outspoken individuals). Sophie Germain was seventeen when the Reign of Terror turned Paris into a city of unspoken violence against humanity, with hundreds of people ending at the scaffold. The gory executions were carried out in the Place de la Revolution (former Place Louis XV and current Place de la Concorde); the guillotine stood in the corner near the Hôtel Crillon, not far from the building where today houses the American Embassy. During those years, Sophie Germain must have grown both psychologically and intellectually, engrossed in her studies. She was surrounded by an atmosphere of heightened reality, breathing the city air that must have been a peculiar mixture of fear and elation, idealism and anxiety. Immersing herself in mathematics would have been a way to erect a protective wall around her when death was so near and life was so precious. About this time, the Germain family moved from the house on rue Saint-Denis to the maison de M. d’Egligni on rue Sainte-Croix de la Bretonnerie no. 23. Sophie would reside at this home until about 1821. Scholars were not immune to the political winds that swept the country. During the Reign of Terror, a revolutionary judge publicly stated that France had no need for scientists. Several leading savants succumbed to the bloodshed and intrigue of the French Revolution, and others were suspect and closely watched. In September 1793, the French Committee of Public Safety issued the “law of suspects,”10 ordering the arrest of all foreigners born in enemy countries and requiring that all their property be confiscated. Italian-born Lagrange was among this group, but his friend, chemist Antoine-Laurent Lavoisier, interceded on his behalf. Lavoisier asked the Committee to make an exception for “le célèbre La Grange, le premier des géomètres, qui est né à Turin, mais qui a fait de la France sa patrie adoptive et qui y a fixé depuis sept ans domicile, est inquiet relativement à la exécution de ce décret.”11 [… “the famous Lagrange, the first of the mathematicians, who was born in Turin, but who made France his adoptive homeland where he has resided for seven years, is concerned for the execution of this

10

The Committee of General Security was created to supervise the rounding up of suspects who were first sent to the Revolutionary Tribunal for sentencing. Committees of Surveillance sprung up to catch “suspects” and “foreigners.” The loi des suspects resulted in thousands of people imprisoned. 11 Copy of Lavoisier’s letter dated 7 September 1793 in Œuvres de Lagrange, Vol. 14, p. 314.

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decree.”]. After Lavoisier’s eloquent letter, Lagrange was spared. Other scientists were not, most notably Lavoisier himself and Condorcet. Marie-Jean-Antoine-Nicolas Caritat, marquis de Condorcet, was a leading mathematician and a philosopher. At a young age, he studied mathematics under Jean le Rond d’Alembert and Alexis Clairaut who praised his analytical abilities. At twenty-two, Condorcet published Essai sur le calcul intégral, an important book in the emerging field of calculus, which helped him get elected to the Académie Royale des Sciences in 1769. Condorcet’s career was bright, publishing several important papers on probability and integral calculus, and he corresponded with Leonhard Euler. In 1777, Condorcet became Permanent Secretary of the Académie, holding the post until its abolition in 1793. He touted the egalitarian virtues of the metric system. Condorcet was also an ardent revolutionary who championed the liberal cause during the early years of the French Revolution. Soon, he shifted his focus from mathematics to philosophy and political matters. Condorcet was elected as the Paris representative in the Legislative Assembly, becoming its secretary. As a member of the moderate Girondists, he drafted a new constitution and, at the King’s trial he argued strongly to spare his life. He made many enemies, especially among the more radical political groups, and soon he was accused of treason. When a warrant was issued for his arrest, Condorcet went into hiding. Eventually, he fled from Paris and after three days wandering the countryside, authorities apprehended him. Two days later, on 29 March 1794, Condorcet was found dead in his prison cell. A few weeks later, his colleague Antoine-Laurent Lavoisier was arrested and tried as an “enemy of the people.” Lavoisier was accused of using his position in a private tax collection company (Ferme Générale) to exploit the poor citizens. It did not help that Lavoisier was a leading scientist who had worked to support the revolutionary cause. In fact, he was president of the Bureau de consultation des arts et metiers, a group of academicians established by government decree to report on various inventions and processes of a practical or technical nature. Among his contributions to the work of the Bureau was his proposal for the reformation of public education. Lavoisier was a wealthy aristocrat, and that alone was sufficient for his enemies. Despite the pleas from his colleagues at the Bureau, the eminent French chemist was executed by guillotine on 8 May 1794. The political environment remained problematic for scientists and foreigners. Some have said that Laplace survived the revolution because he compromised with the ruling regime, while most other scientists kept a low profile. The leading academicians remained under scrutiny—even including a célèbre scholar such as Lagrange. On 30 March 1795, he wrote a self-identifier for the government, a required act to remain in their favor.

Institut de France: Science Above All

13

Institut de France: Science Above All After the reign of barbarism, which lasted nearly two years, French Republican leaders realized that the nation needed not just to strengthen all branches of education, but also to push the boundaries of knowledge, and to spread the treasures of science and engineering that would emerge from the great Écoles. On 22 August 1795, the National Convention promulgated a new Constitution, which included the establishment of a National Institute. This organization differed considerably, at least in form, from the old royal Academies suppressed in 1793. The Convention named it Institut National des Sciences et Arts, thus including in its membership artists, mathematicians, and researchers in all areas of science. Under the Act of 1795, the Institute was composed of three major groups called “classes,” the first of which was named Classe des Sciences physiques et mathematiques. Each class was divided into sections: there were, in all, twenty-four sections, each comprising six members, which gave a total of one hundred forty-four chairs.12 After two years of interruption, the old Academy of Sciences was once again reborn to form the First Class of the newly created Institute of France (Premier Classe des sciences physiques et mathematiques); it was divided in ten sections to include all sciences from mathematics, experimental physics, astronomy, chemistry, to medicine and zoology. The mathematics section counted among its founding members Lagrange, Laplace and Legendre. The Institute was installed in the Louvre palace (Fig. 1.4), where the mathematicians and scientists occupied the former premises of the Academy of Sciences. Henceforth, I will refer to the First Class as the Academy of Sciences. The formal Inauguration of the Institut de France was on 4 April 1796. The solemn ceremony of inauguration was held in the Louvre’s room of the caryatids (la salle des Cariatides, Fig. 1.5).13 Five directors, of whom two were part of the Institute, led the ceremony clothed in their great regalia: blue garments with burgundy coat covered with gold embroidery, silk belt, baldric and plumed hats. Diplomatic corps and representatives of all government bodies were also in attendance. Fifteen hundred spectators were packed in the stands, not to mention the choir and an orchestra.14 Following speeches by the president of the Directories,15 the Institute’s president, Pierre Claude François Daunou, member of the class of moral and political sciences, gave the opening speech. Scholars read fifteen memoirs in order to define and establish the nature of this research institution.

12

Franqueville (1895). This room takes its name from the four female figures sculpted by Jean Goujon in 1550 to support the music gallery. Located on the ground floor of the Louvre’s Pierre Lescot’s 16th-century Renaissance wing. Today it houses Roman copies of Greek originals long since disappeared. 14 Franqueville (1895). 15 The Directory (Directoire) was the government of France during the second-to-last stage of the French Revolution. It operated following the National Convention and preceding the Consulate, from 2 November 1795 until 10 November 1799, a period commonly known as the Directory era. Napoléon brought down this form of government when he made himself emperor. 13

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Fig. 1.4 Exterior view of the façade of the Louvre (side of the river). Credits see Appendix “Illustration Credits”

Fig. 1.5 Inaugural session of the Institute, held at the Louvre on 4 April 1796. Credits see Appendix “Illustration Credits”

Institut de France: Science Above All

15

The first meeting of the First Class of the Institute was held on 27 December 1795. Sophie Germain was about to turn twenty. From the first sessions, the members of the Class began to discuss the proposals submitted to them, quickly reached practical solutions, gave the government the advice it asked of them on a host of questions, read the books who were presented to them, examined and discussed the memoirs submitted to them, either by confreres or by foreign correspondents and scholars at the Academy. Those sessions were very long, lasting more than four hours as the reading of long memoirs required. Each of the classes of the Institut de France regularly held their weekly meetings, and all members of the Institute gathered at the beginning of each quarter, in a public session, to hear scholars read important memoirs on various subjects. In the first meeting (27 December 1795), members of the First Class appointed Lagrange as president, Laplace as vice-president, and de Lacépède (a zoologist) was named secretary. On 25 April 1796, Laplace was voted to take the President’s position. From 1795 to 1804, academicians were addressed as citoyen and citoyens (Cn and Cns): “les Cns Laplace et Legendre” and Lagrange was “le Cn Lagrange.” However, foreign scholars continued being addressed as Monsieur. In May 1796, the Institute of France re-established the prestigious award prizes that the former French Academy of Sciences had traditionally bestowed on the most influential mathematicians in history. Members of the First Class decided to award two prizes: one for works in mathematics and the second for work related to the physical sciences. A commission for the award in mathematics was formed with Lagrange, Laplace, Borda Prony and Delambre.16 That same year, Pierre-Simon de Laplace published “The System of the World” (Exposition du système du monde), a semi popular treatment of his work in celestial mechanics that was widely read. The book contains Laplace’s theory on the shape of the Earth, and his “nebular hypothesis”—attributing the origin of our Solar System to cooling and contracting of a gaseous nebula. It introduced Laplace’s most notable and influential celestial mechanics book (Traité de Mécanique Céleste); the first volume appeared in 1799. Between 1797 and 1798, Lagrange published two important memoirs that must have been central to Sophie Germain’s mathematical studies: “Theory of Analytic Functions” (Théorie des fonctions analytiques, June 1797), and “Treatise on the Resolution of Equations of all Degrees” (Traité de la Résolution des équations de tous les degrés, 1798). In the latter work, Lagrange introduces the method of approximating to the real roots of an equation by means of continued fractions, and he states several other fundamental theorems. On 13 July 1798, Legendre published Essai sur la Théorie des nombres. It provided the foundation on number theory that inspired Sophie Germain to pursue her own unique contribution to the proof of Fermat’s Last Theorem.

16

Institut de France. Procès-verbaux. Tome I, p. 46.

Chapter 2

Lessons from l’École Polytechnique

… the unalterable truth of a well-established fact is in harmony with the character of the mathematical sciences. —SOPHIE GERMAIN

Following the brutal Reign of Terror, an emergency council was set up in Paris. Its main task was the creation of a new engineering school called the École centrale des Travaux publics, which had the objective to train engineers, both civilian and military. Four hundred students quickly enrolled, with “revolutionary courses” in mathematics and chemistry as the foundation of their studies.1 The school opened its doors on 21 December 1794, a Sunday.2 The original building was the Hôtel de Lassay, a stately mansion overlooking the Seine River, right next to the Palais Bourbon (Fig. 2.1) (now the National Assembly). The luxurious Hôtel de Lassay (Fig. 2.2),3 which had been confiscated as national property in 1792, housed the new engineering school from 1794 to 1804. In September 1795, by a decree of the Convention the school name changed to École Polytechnique, presumably intended to convey the idea of a plurality of techniques.4 Joseph-Louis Lagrange was the founding professor of analysis, and Gaspard Riche de Prony of mechanics. Gaspard Monge taught descriptive and differential geometry. Monge is considered the father of differential geometry because he introduced the concept of lines of curvature of a surface in 3-space in his famous Application de l’analyse à la géométrie.

1

Grattan-Guinness (2005). 1er nivôse an III (Dimanche, 21 décembre 1794), Ouverture des cours de l’École Centrale des Travaux publics. 3 This building is now residence of the president of the National Assembly. The building located on rue de l’Université faces the Jardin des Tuileries to the east and the Champs-Élysées on the west, in the 7th arrondissement. 4 Grattan-Guinness (2005), p. 233. 2

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 D. Musielak, Sophie Germain, Springer Biographies, https://doi.org/10.1007/978-3-030-38375-6_2

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Fig. 2.1 Hôtel de Lassay, the École Polytechnique first building (1794–1804). Credits see Appendix “Illustration Credits”

Fig. 2.2 The Palais Bourbon (upper left) and the Hôtel de Lassay (lower right), as depicted on the Turgot map of Paris (1739). Auteur: Louis Bretez, cartographer; Claude Lucas, engraver. https://fr. wikipedia.org/wiki/Plan_de_Turgot

The first lecture Lagrange gave at the École was on Sunday, 24 May 1795,5 but his lecture notes were not published until after 1798. As a technical institution, the school had no option for admitting women. Sophie Germain must have realized that 5

Fourcy (1828), p. 74.

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her only alternative was to impersonate a male student so that she could receive an education in mathematics. But how? Her first biographers, Hipolite Stupuy and Guglielmo Libri,6 introduced Sophie Germain to posterity enveloped in a shroud of mystery. Libri wrote in Germain’s obituary notice of 1833, and Stupuy repeated in her biography of 1879, that she obtained the lecture notes known as cahiers (published by the École) of several courses including Fourcroy’s chemistry and Lagrange’s analysis, and then submitted to Lagrange some of her own work, using the name M. Le Blanc to conceal her identity. The lectures were available in print since August 1798 in the Journal de l’École Polytechnique. This journal was established initially to publish the material addressed by professors in their lectures. Each volume was composed of a varying number of topics. The print run was a thousand copies, sometimes more. Moreover, it was customary that, at the end of the course, professors would ask their students to submit their observations about the course. This explains why Sophie sent her work to Lagrange, which must have contained smart, insightful remarks to arouse his interest. He “praised them and when he learned the true name of the author showed his astonishment in the most flattering terms.”7 Neither Stupuy nor Libri provided historical details to explain exactly how or when Sophie obtained the lecture notes or how Lagrange ultimately discovered her true identity. They did not provide dates or extracts of her written observations. Historical evidence suggests that it was around 1797 when Lagrange found out that LeBlanc was actually Sophie Germain. She was twenty-one years old. It is unclear whether Le Blanc was a fictitious name she made up or that of a male student she knew. In Fourcy’s Histoire de l’École Polytechnique,8 I found the list of students enrolled in 1794. It shows “Leblanc (Ant.-Augustin)—1797” followed by the note “Elève démiss.” That student resigned (in 1797). Is this the M. Le Blanc that Sophie impersonated? In any case, after discovering her identity, several members of the Parisian Academy offered Sophie Germain books and other material to facilitate her self-directed study. Although we don’t know which books she had at her disposal, or what curriculum she considered interesting, it is safe to conclude that Sophie Germain had acquired a fundamental mathematical background in order to understand Lagrange’s analysis. Most probably, Germain began to study the cahiers of the École Polytechnique as soon as the lecture notes were available in print in 1798. Lagrange’s first lecture was on numerical analysis on the transformation of fractions, a relatively easy topic compared with the subsequent theory of analytic functions, a course that must have been difficult even for the students attending his

6

Libri was an Italian mathematician who befriended Germain when he visited Paris in 1825. Libri wrote an obituary note found in Germain’s posthumously published work in philosophy, Germain (1833). 7 Germain (1833), p. 12. 8 Fourcy (1828), p. 395.

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lectures. In fact, de Prony reported that his students in mechanical analysis were so unprepared that he had to teach them basic arithmetic.9 To understand Lagrange’s analysis, a student required a solid mastering of algebra. Did Sophie Germain study Euler’s Algebra book? This is one of the earliest textbooks intended to teach foundational algebra, recognized even today as a pedagogical masterpiece for the clarity of the exposition, setting out algebra in the modern form we recognize today. Euler composed this work sometime in 1765, by dictating to a young servant since he was already blind, and gave it the title Vollständige Anleitung zur Algebra (Complete Instruction to Algebra). Algebra begins with the definition of mathematics, and then Euler discusses the nature of numbers and the signs + and − in order to build the fundamental operations of arithmetic and number systems. Euler’s exposition gradually moves towards more abstract topics, systematically developing algebra to a point at which a student could solve polynomial equations of the fourth degree, first by an exact formula and then approximately. The first edition of Euler’s Algebra was published in 1770 by the Royal Academy of Sciences in St. Petersburg. A few years later, the young mathematician Johann (Jean) III Bernoulli (who was director of the Observatory in Berlin) undertook to translate Euler’s book in the French language, and to enrich it with some historical notes. In addition, he asked Lagrange to add a piece to complete the method of undetermined analysis. Lagrange (who was director at the Academy of Berlin at that time), added approximately one hundred pages of work on Diophantine equations. With the title Élémens d’algèbre, the book appeared in Paris in 1774, and it was republished with additions when Sophie Germain was eighteen years old.10 It’s plausible that she studied this work and thus was prepared for Lagrange’s analysis. In fact, the students applying for entrance to the École were required to know arithmetic and algebra, including the resolution of polynomial equations of up to the fourth degree; geometry, including trigonometry, the application of algebra to geometry, and conic sections.

Lagrange’s Lecture Notes 1797–1799 The first issue of the Journal de l’École Polytechnique is dated 1794 but was published in the spring of 1795. It begins with a lecture by Monge on stéréotomie (technique traditionnelle de la coupe des matériaux de construction), and it contains a lecture by Gaspard de Prony dealing with analysis applied to mechanics (Cours d’analyse appliqué à la mécanique). Mastering the material in these cahiers was intended to prepare students for a degree in engineering.

9

See L’Ecole Polytechnique par Charles C. Gillispie, at www.sabix.org/. Euler (1794). This is a French translation of Euler’s Algebra book (1770) by Jean Bernoulli III with contributions by Lagrange. 10

Lagrange’s Lecture Notes 1797–1799

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The second issue of 1795 (published in 1796) contains an announcement by de Prony about a basic course of analysis by Lagrange (Notice sur un cours élémentaire d’analyse fait par Lagrange, par R. Prony). The third issue contains the organization chart of the École, providing the names of all instructors and the subject matter included in the program of study. It begins with lessons on analysis by Gaspard de Prony. The material of Lagrange’s first lectures appeared in the fifth cahier (dated 1797) published in the summer of 1798 (Prairial an VI).11 Lagrange lectured on numerical analysis and the transformation of fractions,12 and added a very short introduction to the principle of virtual work.13 This topic provided the foundation for future expansion on the calculus of variations that Lagrange would apply to the study of the statics and dynamics of mechanical systems. This cahier also contains a lecture from Fourier on statics and the theory of moments, as well as a lecture by Laplace on the motion of bodies under the action of external forces, which he may have taught at the École Normale. The cahier also included an introduction to mechanics by de Prony. Lagrange’s used his first lecture, “Essay on Numerical Analysis on the Transformation of Fractions,” as a mathematical foundation for the analysis that would follow. He gave a unified treatment of subtraction algorithms, expressed in terms of approximations and introduced basic transformations, such as how to C transform the expression Ba
Am  C into Ba ¼ ma  Aa . He made a reference to ascending continued fractions that he attributed to Lambert. Mathematician J. H. Lambert was a colleague of Euler and Lagrange at the Berlin Academy, and today he is better known for providing the first rigorous proof that the number p is irrational. Lagrange also taught infinite series at the beginning of his curriculum. Remarkable, today we also teach those infinite series and count them among the most powerful and useful tools of calculus, especially for science and engineering students. We teach series for manipulating limits and in the study of convergence, but ultimately, we show their power as tools in analyzing differential equations, in developing methods of numerical analysis, in defining new functions, in estimating the behaviour of function, and so on. The following power series is an example of elementary function from Lagrange’s cahier: On sait qu’en nommant e le nombre dont le logarithme hyperbolique est l’unité, on a généralement eu ¼ 1 þ u þ

11

u2 u3 þ þ ; 2 23

Prairial was the ninth month in the sixth year of the French Republican Calendar. It started on May 20 and ended on June 18, 1798. For example, 13 Prairial année VI was equivalent to 1 June 1798. 12 Lagrange (1797a), pp. 93–114. 13 Lagrange (1797b), pp. 115–118.

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2 Lessons from l’École Polytechnique donc si u ¼ 1 ou ¼ 1i , i étant un nombre quelconque entier, la série qui représente la valeur de e1=i sera de la forme dont il s’agit; par conséquent le nombre e1=i sera nécessairement irrationnel.

The factorial notation n! would not be introduced until ten years later by a lesser-known French mathematician, Christian Kramp, a professor of mathematics at Strasbourg. In his Théorie des fonctions analytiques, ninth cahier of the Journal de l’École Polytechnique and published in 1797, Lagrange introduced the first theory of functions of a real variable. According to him, his objective was to give “… the principles of the differential calculus, freed from all consideration of the infinitely small or vanishing quantities, of limits or fluxions, and reduced to the algebraic analysis of finite quantities.” He believed that the ordinary operations of algebra suffice to resolve problems in the theory of curves. For Lagrange, “algebraic analysis” described the more general branch of mathematics that results when a wider class of functions is permitted into algebra. Lagrange’s Théorie is divided in three parts: The first part contains the principles of differential and integral calculus, independent of the use of infinitesimal entities (rendus indépendans des notions d’infiniment petits). The second part addressed the application of those principles to lines and curved surfaces. And in the last part of the notes, Lagrange included the application of calculus of functions to mechanics. Sophie Germain’s written observations, submitted under the name M. Le Blanc —if they were about the above topics—must have been good enough to make Lagrange take notice. Of course, she could have addressed another topic not necessarily found in the material covered by Lagrange. That also would have given him reason to inquire about the author. The next set of Lagrange’s lecture notes were published in 1799. On 26 January, for the opening meeting of the courses at the École, presided by the Minister of the Interior, Lagrange gave a very brief exposition on the theory of analytic functions,14 and a more extensive presentation on the solutions of some problems relative to spherical triangles.15 These were published in cahier 6, tome 2, Journal de l’École Polytechnique dated Thermidor An VII.16 A spherical triangle (sometimes called an Euler triangle) is formed on the surface of a sphere by three circular arcs intersecting pairwise in three vertices. The Legendre theorem on reducing the spherical triangle to the plane triangle, formulated in 1787 and proved by Legendre in 1798, was crucial for the study and application of geodesy, an important topic for engineers. In his presentation on the theory of analytic functions, which Lagrange planned to expose that year “with more detail than he had in the printed book,” he clarified that his theory was designed to avoid the difficulties that occur in the principles of 14

Lagrange (1799a), pp. 232–235. Lagrange (1799b), pp. 270–296. 16 Thermidor is the eleventh month of the French Republican calendar. In this case, 12 Thermidor année VII was 30 of July 1799. 15

Lagrange’s Lecture Notes 1797–1799

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differential calculus and that discourage most of those who undertake its study. Lagrange intended to unite calculus to algebra, which up to then were “viewed as separate sciences.” He discussed his conception of algebraic analysis and critically alluded to Euler’s concept of derivative, saying that “Euler regarde les différentielles comme nulles, ce qui réduit leur rapport à l’expression vague et inintelligible de zéro divisé par zéro.” At the end of 1799, Lagrange resigned from his teaching at the École for health reasons.17 His Leçons sur le calcul des fonctions would not be published until 1801, and a second edition in 1806. In this work, Lagrange included more details on his treatment of the analytical subjects of the Théorie. In 1987, Craig G. Fraser published a study of Lagrange’s algebraic vision of calculus based on the second edition of the Leçons (1806), which contained the most advanced statement of Lagrange’s program of algebraic analysis,18 providing a contrast between Lagrange’s ideas and the modern foundation of the calculus. Sylvestre-François Lacroix, a gifted mathematician, filled Lagrange’s vacant position. Lacroix made a major contribution to the teaching of mathematics throughout France and also in other countries with his important textbooks. His most famous work is the two-volume Traité de calcul différentiel et du calcul intégral (1797–1798). In Volume 1, the young Lacroix introduced for the first time the expression “analytic geometry.” The engineer mathematician Gaspard de Prony was also an important professor of analytical mechanics at the École. The school decided that the courses taught by Lacroix and de Prony should be available only to gifted students and were held only once a week. The regular courses were taught by other, lesser-known instructors. The 1800 issue of the Journal de l’École, published in 1801, was devoted exclusively to Lagrange’s Théorie des fonctions analytiques (Theory of Analytic Functions).19 In this 277-page volume, Lagrange expressed his view that the differential and integral calculus could be based solely on assuming the Taylor expansion of a function in an infinite power series and on algebraic manipulations thereafter. In addition to pure analysis, Lagrange included extensive applications to geometry and mechanics. The volume, which contains 227 sections, is divided in two main parts. In Part 1, Lagrange treated the general theory of functions, starting with the definition of function and giving applications of the binomial formula and development of logarithmic and exponential functions. In this part Lagrange gave an algebraic proof of Taylor’s theorem, and derived L’Hôpital rule. This was followed by methods of approximation and an estimation of the remainder in the Taylor series. The last few sections concentrated on the study of differential equations, singular solutions and series methods. Lagrange also focused on multi-variable calculus and partial differential equations.

17

Schubring (2005), p. 371. Fraser (1987). 19 Lagrange (1801). 18

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Part 2 of the Théorie is divided into applications of the theory of functions to geometry of curves and mechanics, devoting the last forty-two sections to applications to mechanics. According to Ball,20 this work is the extension of an idea contained in a paper Lagrange had sent to the Berlin Memoirs in 1772. Its objective was to substitute for the differential calculus a group of theorems based on the development of algebraic functions in series. In these lectures, Lagrange also tackled the theory of maxima and minima in the ordinary calculus, and then introduced some basic results in the calculus of variations, including an important theorem of Adrien-Marie Legendre regarding the question of sufficiency. This refers to the additional conditions that must hold in order that there be a true maximum or minimum for the proposed solution.21 In the last fifty-three pages, which were devoted exclusively to applications to mechanics, Lagrange began with particle dynamics, including a discussion of the Newtonian motion in a resisting medium. He then derived the conservation laws of momentum, angular momentum, and living forces (forces vives) or vis viva.22 Finally, Lagrange examined the equation of live forces and applied it to problems of elastic impact and machine performance. Lagrange treated calculus of variations on an analytical level much more extensively in the following Lessons on the calculus of functions (Leçons sur le calcul des fonctions, Fig. 2.3), published in 1806. In this lecture, Lagrange stated that “the calculus of functions has the same object as the differential calculus, taken in the broadest sense, but it is not subject to the difficulties which are found in the principles and in the ordinary course of this calculus: it serves more to bind the differential calculus immediately with algebra, of which one can say that it has been presented until now as a separate science.”23 These Leçons were intended to suplement the Théorie des fonctions analytiques. The lessons of Lagrange at the École are considered as the starting-point for the researches of great mathematicians such as Cauchy, Jacobi, and Weierstrass. They also might have been the lessons Germain studied and we can assume that material became her own formal initiation into the world of analysis. We also consider the possibility that Sophie Germain had acquired a basic mathematical foundation earlier and may have studied lecture notes published by the École Normale. This was a school intended to educate teachers where Monge, Lagrange, and Laplace were founding professors of mathematics. On 30 October 1794, the Convention Nationale had issued a decree for the creation of the École Normale Supérieure, stating that, “a normal school shall be established in Paris to which citizens from all parts of the Republic who are already instructed in the most useful sciences shall be summoned in order to learn, under the most skillful

20

Ball (1908), p. 337. Legendre (1786), pp. 7–37. 22 Leibniz coined the term vis viva or “living force.” In Lagrange’s mechanics the live force of the system plays an essential role in dynamics. 23 Lagrange (1804). 21

Lagrange’s Lecture Notes 1797–1799

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Fig. 2.3 First page from Lagrange’s lecture notes on the calculus of functions (Journal de l’Ecole Polytechnique, Cahier XII, 1804)

instructors in every area, the art of teaching.” The school’s main objective was to teach pedagogy to teachers and university professors. The École Normale was in session for just four months, from 20 January to 15 May 1795, when Sophie Germain was nineteen. The curriculum included both lectures and “débats.” In the first meeting or assembly, the professors only lectured; in the debates or discussions, the students could ask questions about the material in

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the course, and the professors could also quiz the students. The École Normale published the lectures as “Séances des écoles normales recueillies par des sténographes et revues par les professeurs.” The printed notes allowed students to study the material covered in the classroom at their own pace. From the first publication of the Séances,24 we learn that, on the first and sixth day of each calendar décade (a ten-day week in the republican calendar), Lagrange and Laplace taught mathematics, while Monge and Hauy lectured on descriptive geometry and physics, respectively. Chemistry, history, geography, and literature were taught in the remaining days of the décade. On 30 January 1795, Lagrange and Laplace presided together, giving the first discussion on mathematics.25 Lagrange opened the debate making some remarks on arithmetic. Since he had nothing written on the subject, Lagrange expounded them “in the same order where they present themselves to him.” He talked about numbers and their manipulation by Archimedes and the arithmetic of Diophantus, and defined prime numbers. Throughout the debate, Laplace interjected with his own comments, and some students asked questions. All in all, this was a rather low-level discussion on some basic aspects related to arithmetic and geometry. The published debates were easy reading, not containing the extensive mathematical analysis that Lagrange taught and that would be the basis for his lecture notes at the École Polytechnique. Nevertheless, if Sophie Germain obtained copies of these transcriptions, she would have had at her disposal additional insights to help her studies.

M. LeBlanc Metamorphoses into Mlle Germain Sophie Germain dazzled the Parisian intellectuals who were astonished to discover a woman mathematician. The novelty was not only that a young lady had carried out mathematical work in response to Lagrange’s lectures at the École, but also that she had sent her remarks under the assumed name M. Le Blanc. I believe that Sophie Germain was discovered in late 1797. My conclusion is based on the following facts: (1) Lagrange started teaching at the École later in 1795, but his lecture notes were not available outside the classroom until 1798. (2) If Lagrange knew Sophie Germain before that time, she would not have sent her work to him under an assumed name. Moreover, the only student named Le Blanc enrolled at the École did not leave the school until 1797. It seems unlikely that Germain would have used this student’s name if he were still taking classes. (3) Letters from admirers and academicians who tried to help Sophie Germain are dated late 1797 and after, suggesting that Lagrange learned about her around that time.

Séances des écoles normales recueillies par des sténographes et revues par les professeurs, tome I, 1795. pp. 13–14. 25 Ibid. pp. 3, 33. 24

M. LeBlanc Metamorphoses into Mlle Germain

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In 1797, J. A. J. Cousin published the mathematics book that he allegedly sent to Sophie Germain (perhaps the same year or later) to offer his “unconditional help in pursuit of her studies.” This elementary text would not have been challenging enough for Germain, an intelligent adult of twenty-one. Besides, by then she had already responded to Lagrange’s lectures in analysis, which were more advanced. Professor Cousin had learned about Germain precisely because Lagrange had just discovered her. One more fact supports my assertion. In late 1797, Joseph-Jérôme Lefrançais de Lalande, a famous astronomer and prominent figure in Parisian intellectual society, insulted Germain by casting doubt on her knowledge or questioning her intelligence. Lalande, who was Director of the Paris Observatory, was known for his lecturing and writing in astronomy, including a popular science book for women titled Astronomie des dames (1785). At the recommendation of his friend J. A. J. Cousin, the sixty-five-year-old Lalande went to visit Sophie in early November. We don’t know what words Lalande uttered, but the tone could have been demeaning or insulting, because from then on, Sophie refused to be associated with the popular astronomer. While we don’t know exactly what he said to Germain in person, Lalande later apologized in a letter dated 4 November 1797, which we here reproduce in the original orthography [then in free English translation]: Il était difficile, Mademoiselle, de me faire sentir plus que vous ne l’avez fait hier, l’indiscrétion de ma visite et l’improbation de mes hommages, mais il m’était difficile de le prévoir. Je ne puis même encore le comprendre, et le concilier avec les talents que mon ami Cousin m’a annoncé. Il me reste à vous faire des excusés de mon imprudence; on apprend à tout âge, et les leçons d’une personne aussi aimable et aussi spirituelle que vous, se retiennent plus que les autres. Vous m’avez dit que vous aviez lu le Système du monde de Laplace, mais que vous ne vouliez pas lire mon Abrégé d’astronomie; comme je crois que vous n’auriez pas entendu l’un sans l’autre, je n’y vois d’autre explication que le projet formé de me témoigner l’indignation la plus prononcée, et c’est l’objet de mes excuses et de mes regrets.26 [It would be difficult, Mademoiselle, for anyone to make me feel as you did yesterday, for the indiscretion of my visit and the disapproval of my respects to you, but it was difficult for me to predict it. I still cannot understand or reconcile it with the talents that my friend Cousin told me (about you). All that it remains for me is to apologize for my imprudence; one learns at any age, and the lessons of someone as amiable and as spiritual as you are, remain longer than those of others. You told me that you had read Laplace’s Système du monde, but that you do not want to read my abridged work on astronomy (Abrégé d’astronomie); as I believe that you cannot understand one without the other, I do not see another explanation other than my suggestion caused you the greatest indignation, and this is the subject of my apologies and my regrets.]

From this note I surmise that, during the visit, Lalande recommended that she read his little book for ladies. If Sophie Germain had already mastered mathematics, the simplified book that he suggested must have been intellectually beneath her. I suppose that Sophie would be angered by Lalande’s patronizing attitude, and by

26

Stupuy (1896), p. 392.

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his presumption that she knew so little that she needed to read his overly simplified book on astronomy in order to understand Laplace’s “System of the World.”27 Lalande may have offended Sophie Germain with more than his obnoxious attitude. He was described by Adler28 as being “… an extremely ugly man, and proud of it. His aubergine-shaped skull and shock of straggly hair trailing behind him like a comet’s tail made him the favorite of portraitists and caricaturists. He claimed to stand five feet tall, but precise as he was at calculating the heights of stars, he seems to have exaggerated his own altitude on earth. He loved women, especially brilliant women, and promoted them in word and deed.” Before the publication of the third edition of his Traité d’astronomie, in 1791, Lalande was elected Director of the Collège de France. One of his first acts in this position was to admit women to all classes.29 According to Adler, Lalande was “… a feminist who propositioned young women…” Lalande made contributions to the development of the mathematical sciences, although this was not through scientific innovation. His importance was as a teacher, supporting his students such as Delambre and Méchain; by his accurate observations, which helped provide evidence to support Newton’s theory of gravitation with results on the three-body problem; and his successful popularizing of astronomy. Lalande died on 4 April 1807. Of course, one cannot fault Lalande for his way of thinking. At the end of the eighteenth century, astronomy had become a bureaucratic science that employed underpaid young women as calculators. Those women were called “calculators” because they performed the laborious calculations of the positions and motions of astronomical objects. There are many examples in the history of astronomy of daughters, wives, or sisters becoming assistants to their male relatives. The female assistants performed work that required intelligent pattern-recognition capabilities, and some of these women were as capable astronomers as the males. However, with a few exceptions, most of those women astronomers did not enjoy the same status as the male stargazers did. Only men were believed to be capable of performing the “creative” scientific work that astronomy required. Thus, when Lalande visited Germain and suggested that she read his Astronomie des dames, perhaps the astronomer was implying that she could assist in the Paris Observatory, since she understood numbers, and thus that she could “calculate.” But women like Sophie Germain were rare. She aspired to be more like Gauss, Lagrange, and Legendre; she wanted to be a mathematician, a true savant of science. With that first letter to Gauss—or perhaps the first remarks she submitted to Lagrange—Sophie Germain launched herself on a very different kind of career, that of a mathematical scholar. It is clear that Germain wanted to be a scientifique. She

27

In 1796, Laplace published Exposition du système du monde (The System of the World), a popular treatment of his work in celestial mechanics. It includes Laplace’s “nebular hypothesis” to explain the origin of the solar system. 28 Alder (2002). 29 http://www-history.mcs.st-and.ac.uk/Biographies/Lalande.html.

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considered herself a learned person, a researcher engaged in developing scientific theories that were aimed at uncovering the mysteries of the world through the application of science.

A Young Scholar Emerges Sophie Germain’s first biographer asserted that, after being discovered, she became acquainted with “all known scientists of the time.”30 Stupuy wrote that some of these scholars communicated their work to her while others visited her in an effort to help. Among those academicians were J. A. J. Cousin, a professor at the College of France, and Gaspard Monge, professor at the École Polytechnique, also took an interest in Germain’s studies. There is no record to indicate that she took advantage of their offer. Besides, no casual interaction with them could allow for adequate instruction of the type a student could receive at the Écoles. No wonder, someone wrote, “her education was disorganized, and the type of problems her admirers offered Germain were uneven. Because of this, her overall exposure to mathematics was unsystematic and lacked rigor. Instead of an orderly curriculum, she received books, lecture notes, and trivial mathematical problems.”31 From this conjecture, it may appear that Germain learned mathematics mainly from the material that those well-meaning admirers sent her when she was already twenty years or older. Is that how it really happened—or did she learn on her own much earlier? Surely, in order to carry out analysis work for Lagrange, Sophie Germain had to have already known enough mathematics. No biography has explained this inconsistency. Having more questions than answers, many years ago I composed a mathematical novel32 inspired by Sophie Germain, in an attempt to present a plausible scenario that would help me understand how a bright young girl could have learned mathematics prior to entering the world of Lagrange’s analysis in the first place. What happened after it was discovered that M. Le Blanc was Mademoiselle Germain? The available evidence suggests that Sophie continued her studies, guided by her own interests and perhaps by well-meaning scholars who recommended their books. There is no historical record to help us state with certainty, but I believe that it was actually Legendre who came to her aid, won her trust, and became her mentor. What is evident is that the humble, well-mannered mathematician did take Sophie Germain seriously. As we shall see in the following chapters, there is no doubt that Sophie Germain was an exceptional mathematician. In order to achieve the level of mastery to tackle two of the most important mathematical challenges of her time, she had to have developed her analytical skills at a young age. I conjecture

30

Stupuy (1896), p. 2. Bucciarelli and Dworsky (1980). 32 Musielak (2012). 31

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that, before she approached the professors at the École, Sophie Germain must have learned mathematics on her own much earlier, perhaps from 1789 to 1796, when she was between the ages of thirteen and twenty. Sophie probably used Montucla’s “History of Mathematics,” which contained basic concepts of geometry, to build her foundation. My mathematical novel33 puts into perspective how Germain could have developed the essential strategies to study and pursue her own mathematical research. Then, after exhausting the books in her father’s library or those sent to her, Germain could have obtained the published cahiers from the new Écoles, studied them, wrote reports on what she learned, and then penned letters to Lagrange, Legendre, and others to share her insights— initially signing as M. Le Blanc.  Sophie Germain was born to witness and participate in one of the most exciting intellectual revolutions that advanced the sciences and gave birth to engineering as an applied science. The curriculum at the École Polytechnique, the French model of engineering education, was dominated by mathematical analysis. Created to train civilian and military engineers, the École also instructed the founders of mathematical physics such as Biot, Cauchy, Navier, and Poisson. Despite the social chaos that reigned in France during Germain’s adolescence, many advances in applied science originated at that time. On 7 April 1795 the metric system was formally defined in French law. On 22 October of the same year the work of the commission (since reconstituted as a three-man agence temporaire directed by Legendre) became part of the newly formed National Institute of Arts and Science, and under the Directory government, it was transferred to the Office for Weights and Measures under the Minister of the Interior. Three years later, on 15 November 1798, Delambre and Méchain returned from their triangulation expeditions after completing the survey of the Dunkirk-Barcelona meridian. Their data was analyzed and a prototype meter constructed from platinum with a length of 443.296 lignes.34 At the same time, a prototype kilogram was constructed with the mass of a cube of water at 4 °C, each side of the cube being 0.1 m. The prototype meter was presented to the French legislative assemblies on 22 June 1799. A month later, in a formal ceremony the definitive standards of the metric system—the platinum meter and the platinum kilogram—were deposited in the French National Archives. That same year, Lagrange retired from teaching, but continued his research as a member of the Paris Academy. Unfortunately, scenes of grief and terror around Paris often troubled these peaceful triumphs of science and engineering. At the same time that Lagrange was

33

Ibid. A ligne (from the French word “line”) was a measure of length used prior to the definition of the meter. The standardized conversion for a ligne is 2.2558291 mm.

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A Young Scholar Emerges

31

giving his first lecture at the École, for example, the Convention was attacked by a furious mob.35 The five days 20–24 May 1795 are known in history as L’insurrection du 1er prairial an III, or Insurrection of 1 Prairial Year III. During that violent popular revolt, Monge went into hiding for two months. A deputy, Jean-Bertrand Feraud, who opposed the entry of the mob to the Tuileries, was struck down and his head was severed and paraded on a pike. The Constitution of Year III (August 1795) dissolved the National Convention and established a unique new governmental system for France—a bicameral legislature led by a five-member known as Directoire executif. This government did not restore peace. The political and social turmoil affected the École since its students were not exempt from their service in the National Guard, and had to take up arms, along with the citizens of Paris, to protect the Government against the faction that strove to re-enter power.36 No wonder Lagrange retired from teaching and other academicians had to hide. On 9 November 1799, Paris awoke to a coup d’état. The young General Napoléon Bonaparte overthrew the Directoire government, and thanks to his political skills, he was elected First Consul of the French Republic. Napoléon was clearly the highest authority, and he became a fairly absolutist ruler of the powerful nation. A month later, a law was passed confirming the meter and the kilogram as the only legal standards for measuring length and mass in France. It had taken ten years, and the effort of some of the most prominent French scientists to obtain the accurate determination of these standards. With Napoléon, post-Revolutionary France entered a new era. Influenced by the ideals of the Enlightenment, France was a world leading power, and French was the lingua franca of educated Europe. In 1800, Paris was the largest city in continental Europe. With a population of 547,000, it was the sixth largest city in the world, but it was still a medieval-looking city that had been devastated by ten years of revolution and anarchy. When Napoléon seized power, he was determined to make it a city full of splendor. As First Consul, Napoléon took up residence with his wife Josephine at the Petit Luxembourg Palace in November 1799, but soon after the emperor was ready to move to the Tuileries palace. On 2 July 1800, Napoléon made his change of residence the occasion of a splendid military display. He had spared no expense in renovating the interiors of his palatial home. The emperor wanted to join the Louvre and Tuileries palaces and to create an imperial city worthy of his magnificence. At a special meeting of the First Class on 30 January 1803, Napoléon Bonaparte reorganized the Institut de France. The First Class was divided into six sections: geometry, mechanics, astronomy, geography and navigation, general physics, and chemistry.37 Lagrange, Laplace, and Legendre were among the members of the

35

Fourcy (1828), p. 75. Ibid. 37 Institut de France. Procès-verbaux. Tome II, p. 621. 36

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geometry section. Bonaparte himself was a member of the mechanics section along with Monge and de Prony. In the new organization the secretaries would become perpetual. Jean-Baptiste Delambre was named perpetual secretary of mathematics (Secrétaire perpétuel pour les Sciences Mathématiques). Napoléon fixed the salary of the perpetual secretaries of the Institute at 6000 francs; it was said that when someone remarked that it was too much, Napoléon replied, “The perpetual secretary must be enabled to receive at dinner all the learned foreigners who visit the capital.” On 18 May 1804, both the French people and the French Senate voted to change the consulate to an empire and to make Napoléon the Emperor of the French. With incredible splendor and at considerable expense, his coronation ceremony took place on December 2 in the Cathedral of Notre-Dame in Paris. The arrogant Napoléon did not allow Pope Pius VII to crown him. Instead, he placed the crown on his own head and then crowned his wife Josephine as the new empress. In 1805, Napoléon crowned himself again, this time with the iron circlet that symbolized his rule over Italy. The French Emperor was also influential in the reversal of many of the social reforms imposed by his predecessors. By May of 1804, the designation Monsieur reappeared, and citoyen was completed abandoned by the end of the year. Napoléon abolished the Republican Calendar on 1 January 1806. As ruler, Napoléon started to recreate an aristocracy, a long French tradition that the Revolution had eliminated. The polished manners of the old French nobility once more became the fashion. It was at the salon of Josephine in Luxembourg that the title Madame, which the revolutionary government had replaced with citoyenne, began to be heard once again. Madame de Laplace, the wife of the famous mathematician, was counted among the circle of acquaintances who frequented the salon of the empress. Napoléon Bonaparte is unique among historical leaders for his active interest in science. He had personal knowledge of results and problems in the science and technology of his time. He was not a scientist but he thought of himself as one. As a schoolboy and as a young artilleryman, Bonaparte had developed an interest in science. He entered the École Militaire in 1785 and did exceptionally well in mathematics. In 1805, the Emperor relocated the École Polytechnique to Montagne SainteGeneviève, in the Quartier Latin, and made it a military academy with the motto Pour la Patrie, les Sciences et la Gloire (For the Nation, the sciences and the glory). During Napoléon’s rule, mathematicians and scientists flourished, most prominently Joseph-Louis Lagrange, Gaspard Monge, Pierre-Simon Laplace, Adrien-Marie Legendre, and Jean-Baptiste-Joseph Fourier, and Claude-Louis Berthollet. After Lavoisier, Berthollet was one of the most distinguished of the French chemists, and he became a close friend of the emperor. In 1808, Napoléon began granting titles of nobility to people who served him particularly well, including men of science like Lagrange and Laplace, to whom he gave the title of Count of the French Empire.

A Young Scholar Emerges

33

On the advice of a commission of trusted scholars, Emperor Bonaparte decided to resurrect the academies while leaving them as part of the Institute of France. On 31 January 1803, the Institute was re-organized into four classes corresponding to the academies that the Revolution had suppressed, including the Première Classe: Sciences Physiques et Mathématiques, which corresponded to the Académie Royale des Sciences. It was in this First Class of the Institut that Lagrange, Laplace, Legendre, Poisson, and other well-known scholars were attached. Interestingly, Napoléon was also member of the Première Classe, in the section of Mechanics together with Monge and de Prony. Taking a glimpse at the social climate of the time, it is now easier to understand why Sophie Germain took a man’s name to gain access to her mathematical education. Even Napoléon Bonaparte, who was a visionary of sorts, did not believe in the education of women. His reform of the school system in 1802 was specifically designed for boys, and his thoughts on the subject—as recorded in a note of 1807 about a proposed public school for girls—were thoroughly conservative and backwards. Napoléon wrote: What we ask of education is not that girls should think, but that they should believe… I want the place to produce, not women of charm, but women of virtue: they must be attractive because they have high principles and warm hearts, not because they are witty or amusing… But the main thing is to keep them all occupied, for three-quarters of the year, working with their hands. They must learn to make stocking, shirts, and embroidery, and to do all kinds of women’s work.38

This was not the kind of education Germain sought. By the time she was a young woman, Sophie Germain had acquired habits of precise methodical thought. She was eager to learn mathematics, physics, and other exact sciences. Although the original curriculum of the École Polytechnique had placed emphasis on engineering or the practical sciences, by 1799 it was modified to accentuate the pure sciences. This explains how people like Poisson, Navier, and Cauchy—all engineering graduates of the École—were able to push the frontiers of the exact sciences. Had she been a male, Germain would have had the same educational privileges as they did. It is a pity that the doors of the École Polytechnique were shut to women. However, that did not stop her. After being discovered by Lagrange, Sophie Germain continued learning on her own. Some time ago, during a lecture, a student asked me, could Sophie Germain have dressed in men’s garments and sneaked into the École Polytechnique in order to have access to the education she desperately wanted? This is highly unlikely. From what is known about her character and the social conventions that ruled her family life, Sophie would not have dared to leave the house unaccompanied, let alone disguised in the masculine dress of the students attending the École. Even when Sophie Germain was a grown woman, acquaintances had to request written permission from her mother to visit her or, as Fourier did, had to include her mother when inviting Sophie to dinner. 38

Bonaparte, Correspondence, 15, no. 12585, dated 15 May 1807.

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One wonders, did the endorsement of mathematics and science by the young and powerful emperor Napoléon inspire Sophie Germain, or help to promote her ascent in the world of science? Perhaps not. By the time Napoléon became the ruler of France, Germain was twenty-eight years old, already displaying her skills as a mathematician. From her correspondence, we can ascertain that, by 1804, Sophie Germain had already mastered number theory. In 1798, Adrien-Marie Legendre had published Essai sur le théorie des nombres, (Essay on the Theory of Numbers), the first treatise to bring together, in a harmonious whole, the almost totality of knowledge in number theory at that time, including results obtained by Fermat, Euler, Lagrange and Legendre’s own personal discoveries. Her curiosity for the theorems of Fermat must have been sparked by some remarks that she read in Legendre’s Essai. Through Legendre, Lagrange, or perhaps by her own discovery, Germain came into possession of the Disquisitiones arithmeticae (Arithmetical Research), the masterpiece in number theory authored by German mathematician Carl Friedrich Gauss in 1801. She studied this astonishingly original treatise, suggesting that she knew Latin, which was the lingua franca of the learned world for centuries, since the Disquisitiones was not translated into French until 1807. Germain was also familiar with Euler and had studied Lagrange’s work, in particular the lecture notes from the École Polytechnique, the Mécanique analytique (1788), and she had read memoirs on planetary motion by Legendre and Laplace. However, number theory arose in Sophie Germain a genuine passion, and to it she devoted the next few years of unrelentless study, as we shall see next.

Chapter 3

Sophie’s Sublime Arithmetica

La langue mathématique est celle de la raison dans toute sa pureté; elle interdit la divagation … [“Mathematics is the language of reason in all its purity; it forbids straying …”]. —SOPHIE GERMAIN

It is the autumn of 1804. Sophie Germain is twenty-eight, a woman ready to assert her worth as a serious scholar. By her own account, she had been studying the theory of numbers and had built the necessary skills to understand the theorems in Legendre’s Essai sur la théorie des nombres. Now Germain began courting the affirmation of her mathematical efforts from the pre-eminent authority in the purest of mathematical science: Gauss. Three years earlier, Gauss had captivated Germain with a new approach to study “higher arithmetic” or arithmeticae sublimiori, as Gauss called it. This is the branch of mathematics dedicated to the general study of the proper, particular properties of the integers, which now we call number theory. On 21 November 1804, Sophie Germain penned a letter to Gauss, reverting to using her pseudonym M. Le Blanc: “Monsieur, Your Disquisitiones arithmeticae has been the object of my admiration and of my studies for a long time. The last chapter of this book contains, among other remarkable things, the beautiful theorem …. I enclose two proofs of this generalization…”1 And at that moment, those written words marked the beginning of one of the most alluring stories in the history of mathematics. In the span of five years, Germain wrote to Gauss eight letters, each replete with samples of her work. Considering herself an enthusiastic amateur mathematician (amateur euthousiaste), she asked Gauss for his opinion on her proofs. Gauss responded to four of those letters, providing some feedback, as she requested it, and also sharing some bits of other mathematical ideas of his own. Before we examine the contents of the Germain-Gauss correspondence, let us begin with a quick review of number theory and its main proponents, in order to put in perspective why this branch of mathematics fascinated Sophie Germain throughout her life. 1

Germain-Gauss Correspondence, Appendix in this book. Letter 1.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 D. Musielak, Sophie Germain, Springer Biographies, https://doi.org/10.1007/978-3-030-38375-6_3

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Number Theory: From Diophantus to Gauss At the turn of the nineteenth century, the theory of numbers was about to enter a golden era. Number theory is devoted to the study of the integers—that is, the class of whole numbers and zero. The positive integers are called the natural numbers, and a special class of them are called prime. By definition, a prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A captivating characteristic of the theory of numbers is that it includes many simply stated theorems and easily understood problems whose proofs are either unknown or incredibly challenging. The long history of elementary number theory is fascinating. The earliest surviving records of the explicit study of prime numbers come from the ancient Greeks. Pythagoras and Euclid were perhaps the first to treat arithmetic as a science with theorems and proofs to establish the properties of numbers. Euclid devoted three of the thirteen chapters of his Elements to number theory. These contain important theorems about primes, including the infinitude of primes (Euclid’s Theorem), and the essence of the fundamental theorem of arithmetic.2 Paulo Ribenboim divided the theory of prime numbers into four main inquiries: How many prime numbers are there? How can one produce them? How can one recognize them? and, How are the primes distributed among the natural numbers?3 The answer to the first question dates back to antiquity, when Euclid proved that there are infinitely many primes. Answers to the remaining questions have sparked new ideas and developed number theory from its roots in antiquity to its present state. Let us begin with the work of Diophantus, for it was the source of inspiration of many great number theorists. Number theory includes the study of the integer solutions to polynomial equations with integer coefficients, which are called Diophantine equations after the Greek mathematician Diophantus of Alexandria. In the third century A.D., Diophantus wrote a series of books published under one title, Arithmetica, which deals with algebraic equations and their solutions. Diophantus began with a pedagogical lecture on the elements of algebra, introducing a systematic, albeit rather primitive, algebraic notation. A second part of the work contained a large number of exercises, and the general propositions related to these were rather important to the development of number theory. Arithmetica included a collection of 130 algebraic problems, giving numerical solutions of determinate equations (those with a unique solution) and indeterminate equations. The method for solving them is known as Diophantine analysis. Formally, a Diophantine equation is an indeterminate polynomial equation that allows the variables to take integer values only. For example, ax þ by ¼ c is a linear equation (in two variables) in which a, b, and c are integers, and the solutions 2

Euclid’s key result is that if a prime divides a product, then it divides one of the factors; from this the uniqueness theorem is deduced. See: Pengelley and Richman (2006). 3 Ribenboim (1994).

Number Theory: From Diophantus to Gauss

37

sought are x, y integers. These equations define an algebraic curve, algebraic surface, or more general objects. In his Arithmetica, Diophantus considered the basic problems of elementary number theory. For example, Problem 8 from Book II is “to divide a given square number into two squares.” His solution is as follows4: Let the given square be 16; let x2 be one of the required squares and ð2x  4Þ2 the other square. The resulting equality is x2 þ 4x2  16x þ 16 ¼ 16, which requires x ¼ 16=5. Therefore, the squares are 256=25 and 144=25: We observe that in this case solutions are rational. Over thirteen centuries elapsed until French mathematician Pierre de Fermat discovered Diophantus5 and his Arithmetic and made number theory interesting. Then Leonhard Euler came along, plunging number theory into the mainstream of mathematics. After Euler, what was known on number theory was found in his memoirs and in the works of Euclid, Diophantus, Fermat, and Lagrange. We shall find more in Chap. 9 about the extensive work that Fermat and Euler did, as it relates to Sophie Germain’s own mathematical efforts. In 1785, Adrien-Marie Legendre made his first significant contribution in the field of the theory of numbers when he presented to the French Academy of Sciences a memoir that synthesized all the results to date. His Recherches d’analyse indéterminée (Researches in Indeterminate Analysis) contained the theorem of quadratic reciprocity (théorème de réciprocité), known also as the loi de Legendre. The law of quadratic reciprocity is a theorem that gives the rule of reciprocity that exists between any two unequal, odd prime numbers. This theorem gives conditions for the solvability of quadratic equations. In 1798, when Sophie Germain was an independent student of mathematics, Legendre published an important treatise entitled Essai sur la théorie des nombres (Essay on the Theory of Numbers).6 This work would influence Gauss and introduce Sophie Germain to the rigorous study of numbers. Legendre was at the forefront of research in this branch of mathematics and would mentor Germain as she found this study fascinating. The Essai contained the most important results in number theory, including results from the research of Euler and Lagrange, as well as Legendre’s own personal discoveries and extensive tables to provide numerical evidence of many theorems. Legendre wrote in his preface: “I do not offer this as a complete treatise, but merely as an essay intended to show the present state of the theory.”7 Interestingly, although Legendre included several theorems of Fermat, he did not

4

Cox (1993). We know very little about Diophantus, aside from the fact that he resided in Alexandria. The only hint of his personal life comes from a riddle, which he composed in the terminology of his own algebraic problems. Diophantus was a contemporary of Julian the Apostate who reigned from 361 to 363 AD. 6 Legendre (1798). 7 Ibid. On pages ix–x, Legendre wrote: «… je le donne non comme un traité complet, mais simplement comme un essai qui fera connoître à-peu-près l’état actuel de la science …». 5

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make the last theorem a central topic. At that time, the designation “Fermat’s Last Theorem” was not in use. In fact, there were still two of Fermat’s conjectures yet to be proven. In the Essai, Legendre also reformulated Euler’s law of quadratic reciprocity, using what is now called the Legendre symbol, and presented a proof of it. The
 Legendre symbol is a function

p q

, which is defined to be equal to 1; depending

on whether p is a quadratic residue or non-residue of an integer q relatively prime to p. Unfortunately, Legendre’s proof was not complete. The Essai was rather important both for the new results that it provided and for giving the synthesis of the field, which Sophie Germain studied diligently to conceive ideas of her own. This treatise had a profound influence on the mathematical culture of the period. The Essai also served as a reference for the original work that was conceived by the twenty-year old German mathematician Carl Friedrich Gauss. In the summer of 1801, Gauss published Disquisitiones arithmeticae, where he addressed both elementary number theory and parts of what now is called algebraic number theory. In the preface, Gauss explained how he came upon “an extraordinary arithmetic truth.” He wrote: The purpose of this volume, whose publication I promised five years ago, is to present my investigations into the field of Higher Arithmetic. Lest anyone be surprised that the contents here go back over many first principles and that many results had been given energetic attention by other authors, I must explain to the reader that when I first turned to this type of inquiry in the beginning of 1795, I was unaware of the more recent discoveries in the field and was without the means of discovering them. What happened was this. Engaged in other work I chanced upon an extraordinary arithmetic truth (if I am not mistaken, it was the theorem of art. 108). Since I considered it so beautiful in itself and since I suspected its connection with even more profound results, I concentrated all my efforts in order to understand the principles on which it depended and to obtain a rigorous proof. When I succeeded in this, I was so attracted by these questions that I could not let them be.8

Gauss divided his 665-page dissertation into seven sections. In the first section, he introduced a new mathematical concept and new notation that changed the practice of number theory for students such as Sophie Germain. Gauss began with article 1: Si numerus a numerorum b, c differentiam metitur, b et c secundum a congrui dicuntur, sin minus, incongrui: ipsum a modulum appellamus. Uterque numerorum b, c, priori in casu alterius residuum, in posteriori vero nonresiduum vocatur. [Art. 1: If the number a measures the difference of the numbers b, c, then b and c are said to be congruent according to a; if not, incongruent; this a we call the modulus. Each of the numbers b, c are called a residue of the other in the first case, a nonresidue in the second.] In the following article, Gauss introduced his new notation: b  c ðmod aÞ, stating that the congruence of numbers would be given by the symbol  and, when necessary, he would put modulo in parenthesis, for example, −16  9 (mod 5). In a

8

Gauss (1801).

Number Theory: From Diophantus to Gauss

39

footnote, Gauss emphasized that he adopted this symbol to differentiate between the notions of equality and congruence. The theory of congruences plays a central role in modern number theory. Congruence is a statement about divisibility (having the same remainder when divided by a specified integer) placed into a formal mathematical framework. Suppose a is a positive integer. If b, c are integers such that a divides ðb  cÞ or ajðb  cÞ; we say that b is congruent to c modulo a and denote this by b  cðmod aÞ. If a does not divide ðb  cÞ then b; c are incongruent modulo a. The modulus of a congruence b  cðmod aÞ is the number a. It is the “base” with respect to which congruence is computed. For example, instead of saying, “When 39 is divided by 7 it leaves a reminder 4,” it is more convenient to write, 39  4 (mod 7), and we say that “39 is congruent to 4 mod 7.” The term mod stands for modulus. Gauss used the remainder of section 1 to make basic observations on convenient sets of residues modulo a and on the compatibility of congruences with the arithmetic operations. The theorem of art. 108 that Gauss mentioned in the preface relates to the case when 1 is a quadratic residue. He stated: 1 is residue of all prime numbers of the form 4n þ 1, and non-residue of all prime numbers of the form 4n þ 3. Or we can say, let p be an odd prime. Then, 1 is a quadratic residue modulo p iff p  1 ðmod 4Þ. Legendre introduced the Disquisitiones arithmeticae of Gauss to the French during a regular meeting of the Academy of Sciences on 26 January 1802.9 Two years letter, the Geometry Section of the Academy (led by Lagrange) voted to name Gauss official Correspondant of the Institut de France, giving him a distinguished status.10 Lagrange was clearly impressed by the work of this young German mathematician. On 31 May 1804, Lagrange wrote to Gauss: “Your Disquisitiones have placed you immediately in the ranks of the foremost geometers” (vos Disquisitiones, vous ont mis tout de suite au rang des premiers géomètres).11 How did Sophie Germain become aware of Gauss’s work? Perhaps through Lagrange, although it is more likely that Legendre introduced Sophie to the Disquisitiones, perceiving her talent for the abstract concepts in his Essai. Let us remember, she was discovered around the time when Legendre published his book. Thus, it just seems natural to believe that Legendre would have recommended Gauss’s dissertation to advance her studies.

9

Institut de France. Procès-verbaux. Tome II, p. 457. Institut de France. Procès-verbaux. Tome III, p. 59. 11 In the nineteenth century, a mathematician was called géomètre (geometer). See Lagrange’s letter at https://gauss.adw-goe.de/handle/gauss/3552. 10

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Sophie Germain and Carl Friedrich Gauss Sophie Germain began to correspond with Gauss, under the assumed name “Le Blanc.” In the first letter, dated 21 November 1804, she wrote: “I enclose two proofs of this generalization with my letter.” Germain referred to a “beautiful theorem” that she thought could be generalized. She wrote: “The last chapter of this book includes, among other remarkable things, n 1Þ the beautiful theorem contained in equation 4 ðxx1 ¼ Y 2  nZ 2 ; I believe it can be ns ðx 1Þ generalized as 4 x1 ¼ Y 2  nZ 2 ; n is always a prime number and s any other number. I enclose in my letter two demonstrations of this generalization. After finding the first, I searched how the method that you used in art. 357 could be applied to the case I had considered. I did this work with all the more pleasure for it gave me the opportunity to familiarize myself with this method, which I have no doubt in your hands is an instrument for new discoveries.”12 This was an important result for Germain, which she would publish twenty-seven years later (see Chap. 12). She also communicated that she “had managed to prove” [Je crois être parvenu à prouver] Fermat’s equation xn þ yn 6¼ zn . We shall return to this topic in Chap. 9 to explain what Germain meant. Gauss wrote his treatise in Latin. Thus, when Sophie Germain began to study it, she must have been proficient in the scholarly language in order to understand his innovative work in number theory. The French translated edition of the Disquisitiones13 was introduced to the Parisian Academy in January 1807, long after Germain had studied the original. By then she had to have mastered Legendre’s Essai sufficiently to understand the methods presented in the Disquisitiones. She also was able to elucidate her own ideas. Germain ended the letter by asking Gauss to send his reply to the address of M. Silvestre de Sacy, a member of the Institut de France. She signed simply, “Le Blanc.” Who was Silvestre de Sacy and how did Germain convince him to hide her identity and to receive her correspondence addressed to Le Blanc? A member of the class of history and ancient literature at the Institut de France,14 Silvestre de Sacy was forty-six years old (Germain was twenty-eight). He resided on rue Hautefeuille across the Siene River, a brisk twenty-minute walk from her own residence.15 Not surprisingly, when Gauss replied on 16 June 1805, he addressed her as “Monsieur.” He apologized for the six-month delay in responding and wrote about “research to which he devoted the most beautiful part of his youth, and which had been the source of his enjoyment.”16 After writing about his beloved Arithmetic, for

12

Germain-Gauss Correspondence, Letter 1. Translated as Recherches arithmétiques par A.-C.-M. Poullet-Delisle. Paris: Courcier, 1807. 14 The Institute was reorganized and its members were assigned to four classes. Institut de France. Procès-verbaux. Tome II, pp. 621–624. 15 The Germain family lived on rue Sainte-Croix-de-la-Bretonnerie, no. 23. 16 Germain-Gauss Correspondence, Letter 2. 13

Sophie Germain and Carl Friedrich Gauss

41

which now he had little time, Gauss wrote how pleased he was that Arithmetic acquired a rather clever friend in him (Germain). Gauss assumed he was responding to Monsieur Le Blanc. He was especially delighted with her new demonstration for primes, including her result that “2 is [quadratic] residue or nonresidue of primes of certain form,”17 but he remarked that her proof “seemed to be an isolated case and not applicable to other numbers.” He was alluding to an observation Germain made in reference to art. 112 of his Disquisitiones, showing that “Ainsi 2 est résidu des nombres premiers de la forme 8n þ 3; et non résidu de ceux de la forme 8n þ 5.” He added: I have often regarded with admiration the singular enchainment of arithmetic truths. For example, the theorem that I call fundamental (article 131) and the particular theorems concerning residues 1, ± 2, intertwine with a host of other truths, where they would never have been sought. In addition to the two proofs I have given in my work, I am in possession of two or three others, which at least do not yield to those in regard to elegance.

Here Gauss was referring to the law of quadratic reciprocity, which he called the Fundamental Theorem because it contains in itself all the theory of quadratic residues. In his private diary, he called it the Theorema Aureum, or the Golden Theorem, one that had captivated him for years. Gauss published six different proofs during his lifetime. Gauss did not comment on Sophie Germain’s results related to the (last) theorem of Fermat. In the following year, still signing as Le Blanc, Sophie Germain wrote two more letters to Gauss, including more samples of her work. Gauss was clearly impressed by her command of the subject because he wrote to his friend H. W. M. Olbers: “I am amazed that M. Le Blanc has completely mastered my Disq. Arith., and has sent me very respectable communications about them.”18 As she later admitted, she had adopted the pseudonym M. Le Blanc out of fear that Gauss would ignore her letter if he knew that it contained the work of a woman.

How Napoléon’s Invasion Led to Unmasking M. Le Blanc In September 1806, after Prussia declared war against France, Napoléon unleashed all his forces east of the Rhine, deploying the corps of the Grande Armée along the frontier of southern Saxony. Concerned about the well-being of her idol mathematician, Sophie Germain appealed to General-de-Brigade Joseph-Marie Pernety to ensure the safety of Gauss during the hostile invasion. Evidently, Sophie had feared that Gauss would be killed

Ibid. Gauss wrote: “Surtout votre nouvelle démonstration pour les nombres premiers, dont 2 est résidu ou non résidu, m’à extrémement plu.” 18 Letter dated 3 September 1805. See copy (in German) at https://gauss.adw-goe.de/handle/gauss/ 731. 17

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by a soldier just as Archimedes was centuries before, and had asked General Pernety for special protection. General Pernety, who led the Napoleonic troops to occupy Prussia, was leading the capture of Breslau in Lower Silesia (Wrocław, now in Poland).19 To comply with her request, Pernety sent officer Chantel, his chief of battalion, to Brunswick to facilitate a special protection for Gauss. On November 27, Chantel went looking for Gauss and found him in his home with his wife and child. He informed Gauss that General Pernety, who was busy with the encampment in Breslau, had sent him at the instance of “Demoiselle” Sophie Germain in Paris to inquire after Gauss and, if necessary, to offer his protection. Gauss was perplexed, as he had no idea who the young lady was nor the French general. “Il me parut un peu confus [he seemed a little confused],” Chantel reported to his general shortly after this scene.20 In all Paris, Gauss knew only one lady, Madame Lalande, who was related to the famous astronomer. Chantel wrote: “I asked him if he wanted to write to Paris, to give me the letter, that you were responsible for having it forwarded to its destination. He did not answer yes or no to this offer. I departed his house, leaving him with his wife and child.” Gauss had simply thanked the officer and his general for the kind attention shown to him. A day before Christmas, General Pernety wrote to Sophie from Breslau to tell her about the commission she had entrusted him to do, assuring her that “Je désire qu’elle satisfasse vos vœux pour cet émule d’Archimède, mieux traité que lui, comme vous le verrez.”21 [“I want to satisfy your wishes, for this disciple of Archimedes to be treated better than him, as you will see.”] Pernety enclosed Chantel’s letter. Germain knew at that instant that she had to reveal her true identity to Gauss. At once, she penned a letter to explain the reason why she had adopted the name Le Blanc to communicate her work to him: she feared the ridicule attached to the title of femme savant, or woman scholar. It was fortunate that Sophie had sent an ambassador to look after Gauss’s well-being. Because a month earlier, Gauss had witnessed the horrible end of Carl Wilhelm Ferdinand, Duke of Brunswick-Lüneburg, his patron and friend, at the hands of the French invaders. Gauss was sad and at the same time resentful because the Duke of Brunswick, whom he truly loved and appreciated for financially supporting his education and early career, had lost the battle against Napoléon’s troops. Gauss lived with his young wife and infant son (born 3 months earlier) at Steinweg 22, just opposite the Castle gate. On October 25, he saw the departure of 19

Capture of Breslau in November and December 1806 during the Fourth Coalition wars. The French troops arrived on December 1806 in front of the city of Breslau, and began the siege of the city on December 9. The city surrendered on 29 December 1806. 20 Stupuy (1896), p. 267. Letter A in Germain-Gauss Correspondence, Appendix in this book. 21 Letter from General Pernety to Sophie Germain. Letter B in Germain-Gauss Correspondence, Appendix in this book.

How Napoléon’s Invasion Led to Unmasking M. Le Blanc

43

the wonded, dying Duke. “Gauss was quiet and speechless and bore this great sorrow without words and sounds of complaint … poignant grief gained control of him, accompanied by rancor against the invaders of Germany, in whom he also hated the enemy of his beloved prince. Personal reasons for hatred of Napoléon soon arose.”22 The Duke of Brunswick died on 10 November 1806, leaving his protégé without financial support. Gauss and his young family remained in Brunswick. It was not until the next year that Gauss accepted the call from Göttingen University to teach astronomy and run the observatory there.

Sophie Germain Tackles the Law of Quadratic Reciprocity Gauss received the letter from Sophie Germain and her confession, one she ended with a plea, “I hope that the singularity of this, which I make an admission today, will not deprive me of the honor which you have granted me under an assumed name, and that you will not disdain to devote a few moments to send me news from you.”23 Still hopeful, Sophie Germain enclosed four pages of her own mathematical research. Would he take it seriously? Gauss was pleasantly surprised. He replied on 30 April 1807: But how can I describe my admiration and astonishment at seeing my correspondent, M. Leblanc, metamorphosed into this illustrious personage, who gives such a brilliant example of what I would have difficulty believing? The taste for the abstract sciences in general and especially for the mysteries of numbers is very rare … the enchanting charms of this sublime science reveal themselves in all their beauty only to those who have the courage to deepen it. But when a person of this sex, who by our manners and our prejudices, must meet infinitely more obstacles and difficulties, than men, to become familiar with its thorny researches, nevertheless knows how to overcome these obstacles and to penetrate this that they have moreover hidden, it is undoubtedly necessary, that it has the most noble courage, talents quite extraordinary, the superior genius.24

After such glowing remarks, Gauss gave a critique of two theorems that Germain had stated too generally: If the sum of the nth powers of any two numbers is of the form h2 þ nf 2 , then the sum of these numbers themselves will be of the same form. If one of the factors of the formula j2 þ nz2 (n a prime) is of the form ð1; 0; nÞ, the other necessarily belongs to the same form.

Then Gauss wrote: “… I managed to add to an entirely new branch of analysis. It is la théorie des résidus cubiques et des résidus biquarrés (the theory of cubic residues and bi-quadratic residues), which is brought to a degree of perfection, 22

Dunnington (2004). Germain-Gauss Correspondence, Appendix in this book. Letter 6 dated 20 February 1807. 24 Ibid. Letter 7 dated 30 April 1807. 23

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equal to that reached by the theory of quadratic residues. I place this theory, which represents a new day on the quadratic residues, among the most remarkable research I have ever done. I could not give you an idea without writing an express memoir.” To illustrate his theory, Gauss added these two theorems: I. Let p be a prime number of the form 3n þ 1. I say that 2 ð þ 2 and  2Þ is cubic residue of p, if p can be reduced to the form x2 þ 27y2 ; then 2 is non-cubic residue of p, if 4p can be reduced to this form. For ex., 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97. You find that 31 = 4 + 27; 43 = 16 + 27; 2  43 ðmod 31Þ; 2  ð93 Þðmod 43Þ. II. Let p be a prime number of the form 8n þ 1. I say that þ 2 and 2 are biquadratic residues or non-biquadratic residues of p, following that p is or not of the form x2 þ 64y2 . For ex., among the numbers 17, 41, 73, 89, 97, 113, 137 you only find that 73 = 9 + 64; 89 = 25 + 64; 113 = 49 + 64; and 254  2ðmod 73Þ; 54  2ðmod 89Þ; 4 20  2ðmod 113Þ.

Gauss noted, “The proofs of these theorems and those more general are intimately connected to delicate research.”25 By delicate Gauss probably meant that the proofs required intricate hypotheses, all of which must be precisely stated in order to reach the desired conclusion. In other words, a mathematical research is “delicate” because it must be approached with the right arguments and/or tools. The form of these statements was unprecedented. The first is about cubic reciprocity, that the congruence x3  hð mod pÞ exists. In fact, after he found his first proof, Gauss continued to seek more proofs of quadratic reciprocity, hoping to find methods that would generalize to the cubic and biquadratic situations. (When do x3  hð mod pÞ or x4  hð mod pÞ have solutions?) The last precious jewel in Gauss’s communication26 was another proposition concerning quadratic residues: “Let p be a prime number. Let p  1 numbers less than p separated into two classes. A: 1 1; 2; 3; 4. . . ðp  1Þ 2 and B: 1 1 1 ðp þ 1Þ; ðp þ 3Þ; ðp þ 5Þ. . .:p  1 2 2 2

Let a be any number not divisible by p. Multiply all the numbers A by; take the smallest residue according to the modulo p, being, among its residues, a belonging Les démonstrations de ces théorèmes et de ceux qui sont plus généraux sont intimement liées à des recherches délicates. 26 Germain-Gauss Correspondence, Letter 7 (dated 30 April 1807). 25

Sophie Germain Tackles the Law of Quadratic Reciprocity

45

to A, and ɛ belonging to B, such that a þ e ¼ 12 ðp  1Þ. I say that a is quadratic residue of p when ɛ is even, and not residue when ɛ is odd.” Gauss added: “there are several notable consequences to this proposition; among other things, it gives the idea of extending by induction, by which we gather special cases of the fundamental theorem as far as we like, which could not be done by the methods outlined in art. 106–124.” This proposition is known today as Gauss’s Lemma and the number ɛ is called the characteristic number. Note Gauss use of the term induction. Mathematicians use induction, a technique for proving a theorem (or an equation), to prove, for example, a statement about an arbitrary number n by first proving it is true when n is 1 and then assuming it is true for n = k and showing it is true for n = k + 1. When Gauss stated he obtained a result “by induction,” he meant that he developed the statement of a result by generalizing from examples, not that he was proving the result. Hence, in his Disquisitiones, and the letters he wrote to Germain, he provided many examples to clarify each demonstration. Gauss did not add a proof for Germain “as not to deprive you of the pleasure of developing it yourself, if you find it worthy to occupy a few moments of your leisure time.” And knowing that she was seriously studying his Disquisitiones, Gauss admitted to some typographical errors such as that on page 144, art. 139, line 3: instead of a N p it should read a R p. We note that Gauss wrote a R p to denote that a is a quadratic residue ð mod pÞ, and a N p to denote that a is a quadratic non-residue. The modern notation is clear in the following definition: We say that an integer h which is not divisible by p is a quadratic residue modulo p if there exists x such that x2  hð mod pÞ; we say that h is a quadratic non-residue if no such x exists. Using the Legendre symbol, we write 8   < 1 if h is a quadratic residue modulo p h ¼ 1 if h is a quadratic non-residue modulo p : p 0 if h  0ð mod pÞ Hence, we can rewrite Gauss’s Lemma (included in his letter to Germain) as follows: Let p be an odd prime, and set  A¼

 p1 1; 2; . . .; ; 2

  pþ1 ; . . .; p  1 B 2

Let a be an integer not divisible by p, and let eða; pÞ be the number of integers in the list p1 2
 whose least residue ð mod pÞ lies in B. Then ap ¼ ð1Þeða;pÞ . a  1; a  2; . . .; a 

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Fig. 3.1 Germain’s Proof to Gauss’s theorem. Letter 8 dated 27 June 1807. Credits see Appendix “Illustration Credits”

Encouraged by Gauss’s glowing remarks and his propositions on the law of reciprocity, Germain redoubled her mathematical efforts, and she responded with a long letter on 27 June 1807, to which she appended proofs of each one of the aftermentioned propositions.27 As shown in Fig. 3.1, Sophie Germain proved Gauss’s Lemma in just 13 lines.28

27

Germain-Gauss Correspondence. Letter 8. Copy of the Germain’s letter preserved Universitätsbibliothek in Göttingen, Germany.

28

at

the

Niedersächsische

Staats-

und

Sophie Germain Tackles the Law of Quadratic Reciprocity

47

In the body of that letter, Germain stated the following propositions on power residues, begging for his approval: 1: p étant un nombre premier, si q est un nombre premier à p − 1, touts les nombres de la série 1; 2; . . .; p  1 seront résidus puissance qieme ð mod pÞ. 2: Si l’on a au contraire p − 1 = qs, il y aura, parmi les qs nombres 1; 2; . . .; qs résidus et (q − 1)s non résidus, puissance qieme ð mod pÞ: 3: Le produit de deux résidus puissance est en général le produit de a  rm par b  rn est ou n’est pas résidu puissance qieme ð mod pÞ suivant que m + n est ou n’est pas  0ðmod qÞ. 4: Si l’on désigne par 2m ; q; q0 ; . . . les différents facteurs de k dans ieme p = 2 k + 1, −1 sera résidu puissance ð2m1 Þ ; qieme ; q0 ieme ð mod pÞ.
i ieme i 5: Pour les nombres premiers 22 þ 1, 2 est résidu 22 i1 puissance. Sophie Germain explained each one of her propositions and gave examples to make her analysis more convincing. On 19 January 1808, Gauss sent Sophie Germain a poignant message. Apologizing for his lateness in responding to her previous letter, Gauss explained why: This neglect is, for the most part, a result of the changes that have occurred in my situation. I changed my home, to accept the position of professor of astronomy at Göttingen that had been offered to me for a long time. I tell you nothing of the unfortunate circumstances which at last determined me to take this step, or the new annoyances to which I find myself exposed here; I hope that the interposition of the Institute I have used will put an end to it. Let us contemplate now only the beautiful perspective I have of being able, with greater ease, at least in the following, to watch over my arithmetical works, and to publish them successively in the memoirs of the Göttingen Society.29

Gauss also mentioned his work on the calculation of planetary orbits and asked her forgiveness that this time he could not dwell more on the beautiful proofs of his arithmetic theorems. Nonetheless, Gauss ended with these words: “Always be very happy, my dear friend, that your rare qualities of mind and heart deserve it, and continue from time to time to renew to me the sweet assurance that I can count myself among the number of your friends, of which I shall be always proud.” In 1808, upon publishing his third proof of quadratic reciprocity,30 where he used his lemma, Gauss sent Germain a copy. On 19 March 1809, she wrote back to admit to an error she made proving theorem II, which she corrected by considering a system of six congruences.31 In the end, she ceded to the superiority of his proofs (found in the 1808 memoir she had just read). A year later, Germain sent him a very lengthy summary of her work related to the “beautiful theory of cubic and 29

Germain-Gauss Correspondence. Letter 9 (dated 19 January 1808). Gauss (1808). 31 Germain-Gauss Correspondence. Letter 10, dated 19 March 1808. 30

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bi-quadratic residues”, which was the topic that most inspired her.32 It appears that Gauss did not reply to either letter. When he wrote that important missive in 1807, Gauss closed with remarks about the discovery of a new planet (an asteroid by today’s standards) by his astronomer colleague and friend Wilhelm Olbers. Gauss wrote: “I have just completed an extensive work on the methods, which are my own, to determine the orbits of the planets.” He was clearly proud to tell her that he had calculated the orbit of the new celestial body he would name “Vesta,” after the virgin goddess of home and hearth from Roman mythology.33 Gauss did not brag about his fame or the theory of planetary motion that he had developed since 1801. He simply shared with Sophie his excitement about the discovery. Let us take a step back and see Gauss not as the premier number theorist who inspired Sophie Germain but as the famous astronomer that he became immediately after he published his Disquisitiones.

Gauss: Mathematical Astronomer Gauss developed interest in astronomy since he was a student in Göttingen from 1795 to 1798, stimulated by reading Monatliche Correspondenz (Monthly Correspondence). This was the world’s only astronomical journal at that time, which Lalande described as the depot of astronomy for every part of Europe. It was in the Monatliche where Gauss first read about Giuseppe Piazzi’s discovery of the minor planet Ceres on 1 January 1801. Using limited observational data, the young mathematician quickly derived a theory of orbital motion which helped other astronomers to detect the faint celestial object. Then, using more astronomical observations, Gauss improved his theory to such degree that it became possible to ascertain the effects of planetary perturbations that disturb a body’s orbit from the Keplerian (ideal) elliptical form. In November 1801, Gauss computed the orbit of Ceres. Gauss’s method—based on the inverse-square distance law of gravitational attraction—was much different from any that had been used before. He assumed that the orbit had to be a conic, without concerning about the form of the assumed unperturbed orbit. The December issue of the Monatliche published Gauss’s predictions. They were so precise that stargazers were able to relocate Ceres where Gauss projected it would be; on New Year’s Eve, this was confirmed by Wilhelm Olbers. Almost overnight, the reputation of the young mathematician was firmly established throughout Europe. Laplace, Lalande, Burckhart and other French astronomers were very impressed with Gauss and his astronomical analysis.

32

Germain-Gauss Correspondence. Letter 11, dated 22 May 1809. von Zach, Franz X. (1807). Monatliche correspondenz zur beförderung der erd- und himmels-kunde. 15. p. 507. Vesta was discovered on 29 March 1807.

33

Gauss: Mathematical Astronomer

49

After Olbers discovered Pallas (another minor planet) in 1802, Gauss developed a new and more rigorous numerical approach by making use of his mathematical theory of interpolation and his method of least-squares. While calculating the orbits of the newly discovered asteroids (believed to be planets at that time), Gauss completed in 1806 a German version of the book he had been preparing, but he could not find a publisher. Eventually, a publisher agreed if Gauss would translate his memoir into Latin. He did, and in 1809, the crowning achievement in astronomy for Gauss became his Theoria motus corporum coelestium in sectionibus conicis solem ambientium (Theory of the motion of the heavenly bodies moving about the sun in conic sections). Gauss wrote in the preface that “scarcely any trace of resemblance remains between the method in which the orbit of Ceres was first computed and the form given in this work.”34

Gauss and Legendre: A Matter of Priority The Theoria motus corporum coelestium led to a bitter dispute between Gauss and Legendre on the matter of priority for the discovery of the method of least squares.35 This bitter debate did not concern Sophie Germain’s work directly. However, this unpleasant situation may have clouded her relationship with Gauss. The method of least squares (MLS) combines independent observations on a single quantity by forming their arithmetic mean. This method was developed independently by Legendre and by Gauss to provide an accurate description of the behavior of celestial bodies. MLS involves sets of equations in which there are more equations than unknowns. The term “least squares” is used to mean that the overall solution minimizes the sum of the squares of the residuals made in the results of every single equation. In March 1805, Legendre first named and published MLS in his New methods for the determination of cometary orbits.36 Described as an algebraic procedure for fitting linear equations to data, Legendre named it Méthode des moindres quarrés (method of least squares) and demonstrated it by analyzing the same data as Laplace for the shape of the Earth. Legendre’s method was immediately recognized by leading astronomers and geodesists.

34

A complete historical account of the discovery of the asteroids, Gauss’s work to develop his theory of planetary motion, including his communications with astronomers, is given by C. J. Cunningham, “Investigating the Origin of the Asteroids and Early Findings on Vesta.” Springer (2017). 35 Plackett (1972). 36 Institut de France. Procès-verbaux. Tome III, p. 188. On 11 March 1805, Legendre presented his memoir Nouvelles méthodes pour la détermination des orbites des comètes, Paris, 1805, (XIII) in-4.

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On 26 May 1809, Germain received from Gauss a gift copy of his Theoria motus, which she happily shared with Legendre. He in turn became upset when he read Gauss’s statement: On the other hand our principle, which we have made use of since the year 1795, has lately been published by LEGENDRE in the work Nouvelles methodes pour la determination des orbites des cometes, Paris 1806, where several other properties of this principle have been explained, which, for the sake of brevity, we here omit.37

What enraged Legendre were the words principum nostrum, Latin for “our principle” used by the young German mathematician, which implied that he (Gauss) had discovered first his (Legendre’s) méthode des moindres quarrés. Legendre wrote to Gauss on 31 May 1809: I imagine Monsieur, that Mademoiselle Germain will have delivered the message which she kindly agreed to take, which was to thank you very much for the paper which you were kind enough to send me on the summation of a number of series…. A few days ago, Mlle. Germain received from Germany your Theoria motus corporum coelestium…. It was with pleasure that I saw that in the course of your meditations you had hit on the same method which I had called Méthode des moindres quarrés in my memoir on comets … I confess to you that I do attach some value to this little find. I will therefore not conceal from you, Monsieur, that I felt some regret to see that in citing my memoir on p. 221 you say principium nostrum quo jam inde ab anno 1795 usi sumus etc. There is no discovery that one cannot claim for oneself by saying that one had found the same thing some years previously…38

Three years earlier, when Gauss learned that Legendre had published a new method to determine planetary orbits, he wrote to Olbers: Much of what was original in my method, particularly in its first form, I find again also in this book. It seems to be my fate to compete with Legendre in almost all my theoretical works. So, it is in the higher arithmetic, in the researches on transcendental functions connected with the rectification of the ellipse, in the fundamentals of geometry, and now again here. Thus, for example, the principle which I have used since 1794, that the sum of squares must be minimized for the best representation of several quantities which cannot all be represented exactly, is also used in Legendre’s work and is most thoroughly developed.39

The dispute between Legendre and Gauss lasted many years and it involved Laplace, Olbers, and others who took sides, just as it happens when individuals debate scientific priority. That poetic message Gauss wrote to Sophie on 19 January 1808, may have been a deliberate farewell. In addition to shifting the focus of his research to astronomy and other topics in applied mathematics, Gauss was grieving and had many other personal issues to worry about. In October 1809, he lost his beloved wife and was left alone to care for three small children, including a baby born a month earlier.

37

Gauss (1857). Plackett (1972), pp. 242–243. 39 Ibid. p. 241. Gauss letter to Olbers is dated on 30 July 1806. 38

Gauss and Legendre: A Matter of Priority

51

These circumstances and the dispute with Legendre help us understand and find a plausible reason why Gauss did not continue responding to Germain.  By studying the extensive proofs she sent to Gauss between 1804 and 1809, one can appreciate how deeply Sophie Germain understood the concepts in his Disquisitiones, and how she had mastered it long before any other mathematician. Her extensive notes attached to letters to Gauss provide evidence of her ability to prove theorems related to the theory of power residues. She also studied the theory of ternary quadratic forms, Section V, which contains Gauss’ deep and powerful analysis of quadratic forms.40 By 1809, Sophie Germain had studied the Essai sur la theories des nombres and especially the Disquisitiones arithmeticae so assiduously that she knew enough of number theory to demonstrate theorems and conceive her own mathematical theories. She was ready now to prove to Legendre, Gauss, and to the scientific world, that she was more than a student or an amateur euthousiaste … Sophie Germain had earned the distinction of being called a mathematician.

40

Germain-Gauss Correspondence. See letter 3 dated 21 July 1805.

Chapter 4

Chladni and His Acoustic Experiments

… we do not even have the differential equations of motion for this kind of vibrations … and the mere search for these equations would offer mathematicians a very interesting subject of reflexion, which could also contribute to the progress of physics and of analysis. —CHLADNI

In early 1809, a peculiar acoustics demonstration by Chladni, a German physicist, arose in Sophie Germain her intellectual curiosity. At that time, the learned community in Paris was once again vibrant, spurred by the support of Napoléon. The first graduates from the École Polytechnique were working alongside senior researchers, paving new roads in applied mathematics and physics. Scientists from abroad were descending in Paris to learn from the French and to share with them their own scientific discoveries. Acoustics and elasticity theories were emerging. In 1777, a year after Sophie Germain was born, Ernst Chladni had made an astonishing discovery: he observed that when he excited a metal plate with the bow of his violin, he could make sounds of different pitch, depending on where he touched the plate with the bow. The plate itself was fixed only in the center. Chladni then sprinkled sandy powder on the surface and strummed the edges with the bow; for each pitch, a striking sand pattern formed on the vibrating surface. Ernst Chladni had discovered the various modes of free vibrations, manifested through the regular patterns formed by the sandy powder on the plates after the induction of vibration. He observed that the powder accumulated along the nodal lines, those places on the plate where no vertical displacements occurred. Ten years later, Chladni described his technique to make sound visible in a book titled “Discoveries in the Theory of Sound.” He included drawings of the powder figures that formed on the vibrating plates. Those patterns are now called Chladni figures. The discovery of the sound figures aroused the curiosity of lay people and the scientific interest of researchers. In 1791, Chladni began to tour half of Europe carrying in his own coach the musical instruments he had designed. He gave public lectures on the physics of sound, demonstrating the sand figures on vibrating plates

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 D. Musielak, Sophie Germain, Springer Biographies, https://doi.org/10.1007/978-3-030-38375-6_4

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and also showing and playing the clavicylinder and other musical devices.1 The clavicylinder, a keyboard instrument producing its tones through the friction of metal rods against a set of glass cylinders, was a redesign of the “musical cylinder” or string phone designed by Robert Hooke.2 German composer Felix Mendelssohn-Bartholdy described the clavicylinder as having “a tone like that of a very gentle oboe.” Chladni’s exhibitions drew large crowds; he delighted the people with the sounds and sights of his musical devices. In 1802, Chladni published his breakthrough work on acoustics Die Akustik,3 a book where he compiled, commented, and expanded numerous articles on acoustics found on his travels across Europe and depicted the figures he observed on his vibrating plates (Fig. 4.1). In 1806, Chladni set off again from Wittenberg (Germany) on a long journey through Europe. He arrived in Paris at the end of 1808 to give public demonstrations of his musical instruments and the vibrating plates with the curious sand patterns. The scientific community welcomed the German scientist. On 19 December 1808, at a meeting of the Class of Physical Sciences and Mathematics of the Institute of France, Chladni presented the clavicylinder and gave an overview of his work related to sound. Members of the Class of Fine Arts (Beaux-Arts) were also in attendance. Prior to this lecture, Monge had introduced Chladni’s research in acoustics. On a February evening in 1809, Dr. Chladni drove in a carriage and stopped at the Tuileries Palace, the official residence of the French emperor, to show his vibrating plates (Fig. 4.2). He was accompanied by the leading French scholars: mathematician Pierre-Simon de Laplace, comte de l’Empire and chancellor of the Senate; naturalist Etienne, comte de Lacépède, great chancellor of the Legion of Honour; chemist Claude-Louis Berthollet, senator of the empire; as well as Alexander von Humboldt, a famous Prussian geographer, naturalist and explorer.4 Other French researchers in attendance were Siméon-Denis Poisson, Claude-Louis Navier, Jean-Baptiste Biot, and Félix Savart. Navier, who had graduated three years earlier from the École Nationale des Ponts et Chaussées, had just returned to Paris at the request of the Corps of Engineers to edit his late granduncle’s engineering work. Navier and Poisson would become important contributors—along with Sophie Germain—to the development of the mathematical theories to explain Chladni’s experiments.

1

Ullmann (2007). Robert Hooke was an English scientist who made contributions to many different fields including mathematics, optics, mechanics, architecture and astronomy. In 1660 he discovered an instance of Hooke’s law while working on designs for the balance springs of clocks. However, he only announced the general law of elasticity in his lecture Of Spring, which he gave in 1678. 3 Chladni (1802). 4 Stöckmann (2007). 2

4 Chladni and His Acoustic Experiments

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Fig. 4.1 a Frontispiece of Die Akustik, first monograph devoted to experimental acoustics by E. F. F. Chladni, 1802. b Pages from Chladni’s Die Akustik, showing the patterns created by the sand as it moves to the nodes or nodal lines of the vibrating plate excited to its resonant frequencies

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4 Chladni and His Acoustic Experiments

Fig. 4.2 Chladni demonstrates the vibrating plates to Napoléon. Credits see Appendix “Illustration Credits”

At that time, Savart was performing important acoustics experiments. His colleague Biot, who was also educated at the École Polytechnique and was now professor of mathematical physics at the Collège de France, had already carried out important research on the vibration of surfaces. The German visitor made an impression on Napoléon Bonaparte. To set the stage on his presentation, Chladni played some musical pieces on the clavicylinder. Then he demonstrated the vibrating plates, showing the emperor the peculiar sand figures on them. As Chladni later reported, “Napoléon showed much interest in my experiments and explanations, and asked me, as an expert in mathematical questions, to explain all topics thoroughly, so that I could not take the matter too easy.

4 Chladni and His Acoustic Experiments

57

He was well informed that one is not yet able to apply a calculation to areas curved in more than one direction, and that, if one were successful in this respect, it could be useful for applications to other subjects as well.”5 Napoléon Bonaparte had studied mathematics; he graduated in 1785 from the Parisian École Royale Militaire, where Laplace was one of his professors. Thus, the French emperor was able to appreciate the need for additional research in acoustics phenomena to expand Chladni’s work. The next morning, he gave Chladni a gratuity of 6000 francs. On 18 March 1809, Gaspard de Prony and Jean-Baptiste Delambre reported Chladni’s research to the Class of Mathematics6 and recommended that his treatise, which was written in German, be translated into French. In November of the same year, Chladni’s book was published as Traité d’Acoustique.7 The German researcher remained in France until 1810.8

The Prize of Mathematics, 1809 The problem of the vibrating plates was one of great scientific and practical interest. In his seminal book, Chladni addressed his own research, and he also summarized the current state of knowledge, including the theories of elasticity and vibration derived by Euler and the Bernoullis. We address those theories in the next chapter. Chladni alluded to Poisson, who had recently read Euler’s memoirs at the Institute of France. Chladni also referred to the work of Barnabé Brisson and Jean-Baptiste Biot, both civil engineers who were working at the school of roads and bridges. Biot, who was rather good in mathematics, was studying the movement of flat, vibrating surfaces by considering the elasticity in the direction of the plane. In 1800, he had published his research on vibrations of surfaces.9 Biot had attempted, unsuccessfully, to derive the partial differential equations to explain the vibrating plates. At that time, the design and building of new structures and bridges in France was giving a strong impetus toward more rigorous analytical investigations of plate problems subjected to in-plane forces. During the meetings held between 19 September 1808 and 13 March 1809, the members of the Institute of France discussed Chladni’s experimental research. Because there was no suitable theory to explain his results, the academicists 5

Ibid. Rapport adopté par la Classe des Sciences mathématiques et physiques, et par celle des beaux arts, dans les séances du 13 février et 18 mars 1809, sur l’ouvrage de M. Chaldni relatif à la théorie su son. (De Prony, rapporteur). Journal de physique, de chimie et d’histoire naturelle, Volume 68, Avril 1809. 7 Chladni (1809). 8 Ullmann (2007). 9 Biot. Recherches sur l’intégration des équations différentielles partielles, et sur les vibrations des surfaces. Lu le 21 mai 1800 (1er prairial an VIII). Mémoires de l’Institut national des sciences et arts (T. 3, 21–111). 6

58

4 Chladni and His Acoustic Experiments

proposed a prize competition to spur mathematical research that would explain Chladni’s work. The announcement for the Prix de Mathématiques stated in part: “The Class proposes for the topic of the prize to develop the mathematical theory of the vibration of elastic surfaces, and compare it with experience.” It required that the mathematics agree with Chladni’s experiments. The deadline for the contest was set at 1 October 1811. The prize, a medal of gold valued at 3000 francs, was to be awarded to the winner at the public session of the First Class, on the first Monday of January 1812. The judging panel was comprised of Legendre, Laplace, Lagrange, Lacroix, and Étienne-Louis Malus, a young engineer and mathematician who would die a year later. In 1809, the description of Chladni’s figures was rather qualitative, since there was no valid theory to describe the behaviour of two-dimensional vibrating plates. At that time, the analytical work to derive it was considered too formidable for most mathematicians. In fact, Lagrange allegedly stated that the mathematical methods available were inadequate to derive the theory.10 Sophie Germain, however, accepted the challenge. With insufficient mathematical background and working outside the confines of the Institute, Germain spent the next two years trying to derive the theories of elasticity and acoustics, basing her analysis on that of Euler.

Public Announcement for the Mathematics Competition, Paris 1809.

10

Stupuy (1896), p. 345.

The Prize of Mathematics, 1809

59

Today, some historians regard Ernst Chladni as the father of acoustics for his seminal experimental work on vibrations. He discovered a simple algebraic relation for approximating the modal frequencies of the free oscillations of plates and other bodies. This relation, known as Chladni’s law, relates the frequency of modes of vibration for flat circular surfaces with fixed center as a function of the numbers m of diametric (linear) nodes and n of radial (circular) nodes, f ¼ C ðm þ 2nÞp where C and p depend on the plate’s properties. In addition to inventing a method for visualizing the patterns of vibrations, Chladni was also inventor of musical instruments and a respected musician who performed popular works by famous composers. He conducted research into tuning forks, paving the way for their perfection as musical and scientific instruments. This study resulted in the clavicylinder. Another musical instrument he invented was called the Euphonium, which was made of glass and steel rods that emitted unique sounds by being rubbed with the moistened finger. Chladni brought science to the lay people by touring Europe and giving popular lectures on sound and vibration of plates. His experimental research led Chladni to estimate sound velocities in different gases by placing those gases in an organ pipe and measuring the characteristics of the sounds that emerged when the pipe was played. Ernest Chladni died on 3 April 1827 in Breslau, Lower Silesia, known today as Wrocław, a city in southwestern Poland.

Chapter 5

Euler and the Bernoullis

That among all curves of the same length which not only pass through the points A and B, but are also tangent to given straight lines at these points, that curve be determined in which the value of R BA Rds2 be a minimum. —EULER, 1744

Sophie Germain set out to derive the mathematical theory to describe the complex phenomena manifested on Chladni’s vibrating plates. To do that, Germain sought to obtain a clear understanding of the theories advanced by Euler, the Bernoullis, d’Alembert, and Lagrange, and she tried to extend and improve their analysis. This was a daunting task. Her predecessors had worked for many years to formulate the mathematical foundation for elasticity that was in place in 1809. The basic ideas can be traced to the sixteenth century when Leonardo da Vinci considered the elasticity of beams. Later, in 1638, Galileo Galilei studied the resistance and flexure of solid bodies, and in 1678, Robert Hooke discovered that “the force applied on any springy body is in the same proportion with its extension.” This became known as Hooke’s law. In modern terms, Hooke’s law states that the extension (elastic deformation) of a coiled spring is in direct proportion to the load applied to it, which in mathematical form is simply F ¼ kx, where F is the applied force, x is the extension of the spring or deformation of the elastic body subjected to the force F, and k is the spring constant. As we know, Hooke’s law only holds if the extension of the spring is sufficiently small. If it becomes too large, then the spring deforms permanently, or even breaks. In this case, Hooke’s law is no longer applicable. Between 1694 and 1705, Jakob (Jacques) I Bernoulli considered the form of a bent, elastic thin layer of material. In correspondence with mathematician Gottfried Wilhelm von Leibnitz regarding applications of the infinitesimal calculus, Bernoulli investigated the deflection of cantilever beams, determining the elastic curve which is formed by an elastic rod fixed at one end and bent by a weight (load) applied to

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 D. Musielak, Sophie Germain, Springer Biographies, https://doi.org/10.1007/978-3-030-38375-6_5

61

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5 Euler and the Bernoullis

the other.1 He discovered that the curvature of an elastic beam at any point is proportional to the bending moment at that point. Bernoulli first published his preliminary results in 1694 and a more comprehensive treatise in 1705, the same year he died. Years later, Daniel Bernoulli and Leonhard Euler advanced the theories much further. Euler’s contributions to the mechanics of elastic bodies were fundamental: he posed and solved the main problems of elasticity theory, resistance of materials, and structural mechanics. Euler studied the bending, stability, and vibrations of elastic bodies such as strings, rods, beams, and membranes. His mathematical theories were the basis of the work that Sophie Germain and her contemporaries carried out to extend them to two-dimensional vibrating plates. We must note that Euler’s work on elastic bodies was influenced by the research of his friend Daniel Bernoulli. However, although both worked in similar problems, they used very different methods. Considered a physicist, Daniel used observations from experiments to support his conclusions, and shared his results with Euler. Daniel and Leonhard had a close relationship, and for several years the two carried on a lively correspondence to communicate their mutual interests in questions of the theory of elastic vibrations. Thus, the foundation of the research with vibrating elastic plates greatly benefited from the work of these two friends. Did Germain possess such insight? Let’s keep in mind that, working alone and outside the academic world, she did not have access to all available publications that could be useful to her research. Here I provide highlights of Euler’s main works in order to put in perspective the state of knowledge that was available at that time.

Euler and the Mechanics of Elastic Bodies Leonhard Euler started his enormously productive career in St. Petersburg, Russia, when he was barely twenty years old. His mathematical research included number theory, infinitesimal analysis, and differential equations, which were vital to the foundations of calculus, and essential to rational mechanics.2 Before his twenty-ninth birthday, Euler wrote Mechanica (Mechanica sive motus scientia analytice exposita),3 his first important book, in which he presented Newtonian dynamics in the form of mathematical analysis for the first time. Mechanica is Euler’s first publication in which e appeared, and in which Euler used for the circumference of the circle of unit diameter the symbol p we use today. In 1744, Euler published a monumental work that gave birth to variational calculus and became the foundation for the mathematical theories of elasticity. In

1

Bernoulli I (1705). Calinger (1996). 3 Euler (1736). 2

Euler and the Mechanics of Elastic Bodies

63

this book, Methodus inveniendi lineas curvas maximi minive,4 Euler examined end-curves that cut a family of geodesics so that they have equal length and showed that it must be orthogonal to the geodesics. He formulated the variational problem in a general way, identified standard forms of solution, and provided a systematic technique to derive them. Euler included one hundred special problems that served to illustrate his new method. The procedure started with the now-called Euler differential equation as the first necessary condition, and it led to the principle of least action, which Euler had discovered prior to 1743.5 Euler’s new variational approach would play a very important role in the development of elasticity theory.

Foundation of Elasticity Theories Between 1732 and 1775, Euler wrote forty-two essays and letters addressing various aspects of elasticity. In 1742, Daniel Bernoulli suggested to Euler that he apply calculus of variations to derive the equations of elastic curves. At the end of the letter, Daniel wrote: “Since no one is so completely the master of the isoperimetric method (variational calculus) as you R are, you will very easily solve the following problem in which it is required that ds=R2 shall be a minimum.”6 He was referring to the integral of the functional representing the strain energy of a bent bar (neglecting a constant factor), where ds is the axial length of the beam element and R is its curvature. Taking Daniel’s suggestion, Euler applied his variational method to derive the differential equation for elastic curves.7 Daniel Bernoulli was the son of Johann I Bernoulli, younger brother to Jakob I. His most important work considered the basic properties of fluid flow, pressure, density and velocity, and gave the Bernoulli principle. Daniel Bernoulli conducted extensive research on the mechanics of flexible and elastic bodies, and in 1728, he derived their equilibrium curves. In addition, Daniel determined the shape taken by a perfectly flexible thread when acted upon by forces of which one component is vertical to the curve and the other is parallel to a given direction.8 In the process, Daniel derived an entire series of curves named by their shape, such as the velaria (shape of a sail filled with wind), catenaria (curve that an idealized hanging chain or cable assumes when supported at its ends), lintearia (a flexible rectangular sheet with two sides fixed horizontally and filled with a heavy liquid), and the elastica, which gave its name to elasticity.

4

Euler (1744). Goldstine (1980), p. 67. 6 Correspondence from Daniel Bernoulli to Euler. Quoted statement (in Latin) is at the end of the letter dated 20 October 1742. Letter XXVI in Fuss (1843), and online at The Euler Archive. 7 Euler (1744). An English translation of the first selections of this book, along with some brief commentary, is found in Struik (1969) pp. 399–406. 8 Straub (1990). 5

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5 Euler and the Bernoullis

Fig. 5.1 Sketch to represent the Euler-Bernoulli Beam Theory

The main results obtained by Euler on beam bending are contained in Methodus inveniendi. Euler believed in what he called “effective causes” and “final causes,” thus developing both equilibrium- and energy-based methods to analyze diverse problems. To find the deflection equation for an elastic bar, Euler sought an expression for strain energy, the equation derived by Daniel Bernoulli. Euler used his excellent method to derive the correct equation for elastic surfaces that Jakob I Bernoulli had sought, and he integrated it (Fig. 5.1). Euler therefore obtained the formula for the deflection h of the end of a cantilever beam: h ¼ PL2 =3C, where P is the load, h is the angle made by the neutral surface with the x-axis at the point P, and C is a constant that Euler called the “absolute elasticity,” saying that it depended on the elastic properties of the material.9 Using the “method of final causes,” Euler considered the buckling of straight bars under an axial load and derived the equation for the load,10 P ¼ Cp2 =4L2 . Euler determined that a short column under self-weight or an applied load P at the top was fully in compression, but in a long column, there was bending. The limit at which the behaviour of the column changed was identified as critical load, Pcr ¼ EI ðp=2LÞ2 , where E is the modulus of elasticity, I the moment of inertia of the cross section, and L the length of the column. He identified the limiting load at which the performance of the column changed as critical load, Pcr ¼ EI ðp=2LÞ2 . Euler also addressed the bending of bars that have an initial curvature 1=R0 and derived the equation for the axial force as C ð1=R  1=R0 Þ ¼ Px. In the first appendix11 of his Methodous inveniendi, Euler referred to Daniel Bernoulli’s experimental work and then, using his variational method, 4 Euler derived the equation Ek 2 ddx4y ¼ My af ; and gave a solution of the general form 2

Aex=c þ Bex=c þ C sin x=c þ D cos x=c, where c is the constant defined as c4 ¼ EkMaf : 9

Timoshenko (1983). Ibid., p. 34. 11 Euler (1744) Additamentum 1, De curvis elasticis, pp. 245–310. Equation appears in p. 285. 10

Foundation of Elasticity Theories

65

He assumed the behaviour of the elastic bar was dependent on the size of its bending rigidity, which Euler denoted by Ek2 . This work resulted in the Euler–Bernoulli beam theory, a simplification of the linear theory of elasticity that provides a means of calculating the load-carrying and deflection characteristics of beams. The Euler-Bernoulli equation describes the relationship between the beam’s deflection w ¼ wð xÞ and the applied load P:   d2 d2w EI 2 ¼ P dx dx2 where the curve w(x) describes the deflection w of the beam at some position x (the beam is modeled as moving in one-dimension), and P is a distributed load. In 1757, Euler presented Sur la force des colonnes (Concerning the strength of columns),12 a paper containing one of his most important contributions to the theory of elasticity. He had sought the smallest force that suffices to give the least curvature to a column, when applied to one end parallel to its axis, the other being fixed. 2 Euler found that the force must be at least equal to p2 Ek a2 , where a is the length of the column and Ek 2 is the column’s moment of stiffness (moment du ressort). The moment of stiffness multiplied by the curvature at any point of the bent beam is the measure of the moment about that point of the force applied to the beam. Euler introduced what is now known as Young’s modulus of linear elasticity. Also known as modulus of elasticity, Young’s module is a material property that describes its stiffness, one of the most important properties of solid materials. In an undated memoir with the title De oscillationibus annulorum elasticorum (“The oscillations of elastic rings”),13 which was published in his Opera Postuma, Euler derived the formula M ¼ EI=r for the bending moment acting on a beam. Between 1772 and 1774, Euler studied the vibratory motion of elastic laminates, solving many new types of vibrations not treated before. In two separate memoirs, Euler presented the fully explicit theory in solving the equation k4 y0000 ¼ y using circular and exponential functions. Euler focused on the calculation of frequencies and nodal ratios for vibrating bars. Sophie Germain began her own research by studying these memoirs, as we shall see in the next chapter. At this point is worth mentioning that description of the mechanics of solids came largely from observation and experiments. And thus, it is interesting that Euler was able to provide the mathematical theories, much before researchers were able to measure with high precision the outcomes that he had predicted. In fact, it was more than fifty years after Euler that British researcher Thomas Young described the characterization of elasticity, the concept that now we know as Young’s modulus E. In his 1807 Lectures on Natural Philosophy and the Mechanical Arts, Young stated: “We may easily observe that if we compress a 12

Euler (1757). Euler (1862).

13

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5 Euler and the Bernoullis

piece of elastic gum in any direction, it extends itself in other directions; and if we extend it in length, its breadth and thickness are diminished.”14 This behaviour of elastic materials was characterized by the so-called squeeze– stretch ratio. Young described the concepts of stress (the external pressure), and strain, which is the resulting fractional distortion (change in length as a ratio of the original length, ΔL/L). However, Young did not define as such the modulus that bears his name. Eventually, the stress-strain ratio became known as Young’s module; it allowed prediction of the strain in a body subject to a known stress (and vice versa).

Sound and Vibrating Bodies Euler’s work in acoustics went back to the beginning of his career when he was nineteen years old. As part of his first application to the Physics chair of the University of Basel, Euler wrote Dissertatio physica de sono, an 18-page dissertation on the physics of sound. He began with his theory of what makes up the atmosphere, basing his ideas on the theory of elasticity he learned from Johann I Bernoulli, his former professor and mentor. Euler gave a formula for the speed of sound propagation, from which he obtained numerical values for the speed of sound in air. He also compared the sounds produced by the vibrating cords with those generated by the wind instruments. Euler did not get the position in Basel. However, Johann I Bernoulli’s eldest sons, Daniel and Niklauss, opened a door for him in Russia. Euler accepted the invitation of Empress Catherine I to join the Russian Academy of Sciences in St. Petersburg. In 1731, Euler wrote a rather comprehensive book on the theory of music entitled: Tractatus de musica. This work was published in 1739 as Tentamen novae theoriae musicae ex certissimis harmoniae principiis dilucide expositae (An attempt at a new theory of music, exposed in all clearness according to the most well-founded principles of harmony). Euler’s interest in music led him to develop the mathematical foundation of phenomena related to vibrations of elastic bodies. In his work related to vibrations of strings, rods, drums, and membranes, Euler studied resonance with the model of forced harmonic vibrations of a harmonic oscillator. He assumed that the vibrations were small. This allowed him to confine the study to isochronous motions, that is, where the period and frequency are independent of the amplitude and the initial phase of the motion. Euler used d’Alembert’s principle,15 but in a more refined form.

14 Young (1807). See especially Lecture 13, On Passive Strength and Friction, pp. 109–113; Definition of squeeze–stretch ratio, p. 105. 15 D’Alembert’s principle is an alternative form of Newton’s second law of motion. This principle reduces a problem in dynamics to a problem of statics.

Sound and Vibrating Bodies

67

Euler presented his first paper on vibrating flexible bodies to the Academy of St. Petersburg in 1735. It was later published as “On the smallest oscillations of rigid and flexible bodies. A new and easy method.”16 In this work, Euler integrated the equation k 4 y000 ¼ y; using power series. By then Euler had already studied the problem of finding curves which are formed by an elastic strip (in 1732), studied the oscillations of flexible wires (in early 1735), and developed a theory on the propagation of sound. In 1740, Euler developed a differential equation for the transverse vibration of a bar—the same differential equation obtained by Daniel Bernoulli— which models the displacement of the beam in the y direction, which is perpendicular to the longitudinal coordinate x of the bar. In 1747, Jean le Rond d’Alembert published a study, in which he derived the partial differential equation of a vibrating string,17 stretched under tension between fixed end points x ¼ 0 and x ¼ L on a horizontal x axis: @ 2 yðx; tÞ @ 2 yðx; tÞ ¼ : @t2 @x2 A vibrating string is a taut filament or cord, such as a guitar string, undergoing a planar vibratory motion. D’Alembert constructed a general solution consisting of two arbitrary functions f and g: yðx; tÞ ¼ f ðx þ tÞ þ gðx  tÞ; and applying the boundary conditions yð0; tÞ ¼ yðL; tÞ; he reduced the solution to the form yðx; tÞ ¼ f ðx þ tÞ þ f ðx  tÞ: The function f was required to be periodic, odd, and everywhere differentiable, since f ðx  tÞ is just f ð xÞ translated to the left or right by an amount t. In 1748, Leonhard Euler presented his own analysis of the vibrating string.18 He derived a more general wave equation 1 @ 2 yðx; tÞ @ 2 yðx; tÞ ¼ ; c2 @t2 @x2 where c is a fixed constant, which accounts for the tension on the string and its mass, and he gave its solution as yðx; tÞ ¼ f ðx þ ctÞ þ gðx  ctÞ:

16

Euler (1740). D’Alembert (1747). 18 Euler (1748–1749). 17

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5 Euler and the Bernoullis

Applying the boundary condition, Euler obtained yðx; tÞ ¼ f ðx þ ctÞ þ f ðx  ctÞ: The only difference between Euler’s and d’Alembert’s analysis was on the specification of the function f . Euler saw f as deduced solely from the initial conditions: if Y ð xÞ and V ð xÞ are the initial positions and velocity of the string, then   Z 1 1 x þ ct yðx; tÞ ¼ Y ðx þ ctÞ þ Y ðx  ctÞ þ V ðsÞds 2 c xct To understand Euler’s reasoning, let’s consider a string of length L with fixed ends. Suppose we start the string vibrating by plucking (that is, pulling it aside a small distance h at the center and letting it go). Then we have the shape of the string at t ¼ 0, that is its displacement y0 ¼ f ð xÞ as depicted in Fig. 5.2, where the function f is defined as ( 2h f ð xÞ ¼

L

if 0\x\ L2

x

2h L ðL

 xÞ

if L2 \x\L

From his observation of the plucked string, Euler believed that the functions Y ð xÞ and V ð xÞ need not be functions in the mathematical sense. He thought they could be any curve drawn by hand in the interval 0  x  L. Thus, Euler made allowance for curves with corners as solutions to his wave equation. This must have infuriated d’Alembert who only sought a pure general solution to his equation and had no interest in the physical reality. Euler, on the other hand, used a physical observation to impose a mathematical condition.19 In such analysis, it was important for Euler to tie together the mathematics and the physics. In 1753, Daniel Bernoulli presented his analysis of the vibrating string, which was based on the wave’s fundamental amplitude: yð xÞ ¼ A sin

px L

:

For Bernoulli, the solution must be a sum of this fundamental amplitude and higher harmonics, i.e.,     pct 2px 2pct cos þ A2 sin cos þ  yðx; tÞ ¼ A1 sin L L L L px

Note that the principle of superposition was not known at that time.

19

Wheeler and Crummet (1987).

Sound and Vibrating Bodies

69

Fig. 5.2 Plucked string model

In this manner, Daniel defined the simple nodes and the frequencies of oscillation of a vibrating string. He showed that the movements of strings of musical instruments are composed of an infinite number of harmonic vibrations all superimposed on the string.20 Daniel Bernoulli was the first to derive the differential equation governing lateral vibrations of prismatic bars, an equation that Euler integrated. Daniel also put to test his analysis, writing to Euler to communicate that “these oscillations arise freely, and I have determined various conditions, and have performed a great many beautiful experiments on the position of the knot points and the pitch of the tone, which fit beautifully with the theory.”21 D’Alembert continued his critique of Euler’s approach, especially due to the use of functions with corners (plucked string), which in his opinion needed to be periodic, odd, and everywhere differentiable. In the process, d’Alembert found a new technique to derive his equation, which was the first application of the method of separation of variables. D’Alembert questioned Euler’s interpretation of the wave equation for a plucked string. He did not view the string motion as a compound motion of separate distinct modes. Neither Euler nor d’Alembert realized that the function f can be odd or even, depending on how it is extended along the negative axis, and that the solution need only be for the interval 0  x  L. In 1759, Joseph-Louis Lagrange carried out his own analysis of the vibrating string as part of his research on sound propagation. Although he sided with Euler’s solution, his own approach did not use the wave equation. Instead, Lagrange considered the string as a collection of n equally spaced, point masses, connected by a light cord. This resulted in a set of n equations of the form d 2 yk ¼ c2 ðyk1  2yk þ yk þ 1 Þ: dt2 Lagrange sought an integral solution, as he thought that the trigonometric series was not sufficiently general. The matter remained unresolved. In 1807, two years before the prize competition on Chladni’s vibrating plates, Joseph Fourier discovered his famous series while studying the heat diffusion 20

Straub (1970–1990). Daniel Bernoulli to Euler. Letter LVII dated 7 October 1753 in Fuss (1843) and online at the Euler Archive.

21

70

5 Euler and the Bernoullis

equation. Initially Lagrange objected to Fourier’s results, in particular questioning the convergence of the series. Eventually, Fourier was able to demonstrate the convergence for one of his solutions. Cauchy and others after him established the subsequent mathematical rigor required to determine the conditions under which the general series solution would converge. But let’s go back to Euler, as his work had a major influence in the mathematical theory of vibrating elastic plates derived by Sophie Germain. In 1761, Euler presented the first known treatise devoted to the science of vibrating membranes; it was published five years later as De motu vibratorio tympanorum (On the motion of vibrations in drums).22 Euler considered a perfectly flexible membrane as being composed of two systems of strings perpendicular to each other, i.e., elastic filaments parallel to the x- and y-axes, and evenly spaced a tiny distance d apart. With such a model, Euler derived the partial differential equation for the two-dimensional rectangular vibrating membrane: @2z @2z @2z ¼ a2 2 þ b2 2 2 @t @x @y where zðx; y; tÞ denotes deflections and a and b are constants; the term on the left, Euler said, arose from the principle of mechanics. The projection on the z-axis of 2 @2z @2 z the forces causing the filaments to flex are respectively sx d2 @x 2 , and sy d @y2 , where s is the tension force. Euler’s expression is equivalent to the classical two-dimensional wave equation, which governs the displacement of a vibrating membrane of mass r (such as a rectangular drumhead) stretched uniformly under a tension s per unit length. Denoting the displacement of the membrane out of the x; y plane as wðx; y; tÞ, and pffiffiffiffiffiffiffiffi assuming c  s=r, it yields  2  @ w @2w @2w c þ : ¼ @x2 @y2 @t2 2

The solution of the two-dimensional wave equation, when applied to a rectangular vibrating membrane of sides a and b, is wðx; y; tÞ ¼

1 X 1 X n¼1 m¼1

Amn sin

mpx npy sin cos xmn t; a b

where the frequencies of the vibrating membranes are given by xmn

22

Euler (1766).

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 n2 ¼ pc þ 2: a2 b

Sound and Vibrating Bodies

71

The initial condition wðx; y; 0Þ ¼ f ðx; yÞ requires that f ðx; yÞ ¼

1 X 1 X n¼1 m¼1

Amn sin

mpx npy sin a b

on the rectangular membrane. This expression is an example of a double Fourier series. Euler also examined the partial differential equation for the vibration of a circular drum and obtained the displacement of the membrane as a function of t, r, and u (in polar coordinates). He defined the solution in terms of a certain ordinary differential equation now known as Bessel’s equation, thus giving the solution in terms of Bessel functions. By 1779, Euler had considered the transverse vibration of elastic rods, membranes, and rings. All in all, he published at least 42 papers related to the mechanics of elastic bodies. However, Euler’s formulation and analysis extended to thin plates foretold only a small part of the curious phenomena observed by Chladni. The younger mathematician of the Bernoulli family had already attempted to find a theoretical explanation for Chladni’s experiments on sound. Jacques II (or Jakob II) Bernoulli was the youngest son of Johann II, and Daniel’s nephew.23 After joining the St. Petersburg Academy of Sciences, Bernoulli wrote important works on mathematical physics (elasticity, hydrostatics and ballistics). After Chladni published his “Discoveries in the Theory of Sound,” the twenty-nine-year-old Jacques II Bernoulli sought to derive a theoretical basis for Chladni’s experiments. He noted that the previous work had addressed only vibrations of bodies which can be regarded as having only one dimension, strings and elastic laminae, but even in this case it was clear that Euler’s theory did not match Chladni’s results. In 1789, Jacques II Bernoulli published the first memoir on the vibration of elastic plates.24 He represented a plate as a system of mutually perpendicular strips at right angles to one another, each strip regarded as functioning as a beam. To obtain an equation for the vibrations, Bernoulli assumed that a curved surface (a vibrating plate) may be considered as made up of an infinite number of curves of simple curvature. He divided the plate into two series of annuli (just as Euler divided his bells) at right angles to each other, and considered the separate motions of these annuli. However, the governing differential equation that Jacques II Bernoulli derived for the bending of the plate was incomplete—it did not contain the middle mixed-derivative term. Moreover, his theory did not explain Chladni’s experiments. It would take thirty more years to arrive at the correct equation. 23

Jacques II Bernoulli married a granddaughter of Euler in St Petersburg. He died in 1789 by drowning in the Neva River. 24 Bernoulli II (1789).

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5 Euler and the Bernoullis

Thus, when the Institut de France announced the competition of 1809, the analytical work of Euler and that of the Bernoullis provided the scientific basis for developing the theory of plate vibrations, which required both the combined effect of bending or deflections and vibrations on a two-dimensional elastic plate. Euler, in effect, paved the way for Sophie Germain and her mathematical understanding of calculus of variations, which Lagrange had extended and formalized in his applications to mechanics. She used Euler’s approach to establish her hypothesis of the strain-energy integral, and that would lead her to the model of the deflection of thin elastic plates caused by vibrations. The impetus on the mathematical work that followed was due to this remarkable woman, who embarked on a competitive quest to derive the formidable scientific theory to explain Chladni’s vibrating plates, as we shall see next.

Chapter 6

Germain and Her Biharmonic Equation

Toute équation est une égalité. Que sont les propriétés d’une courbe? une égalité entre les produits, au les combinaisons de certaines lignes droites renfermées et bornées par cette courbe. [“Every equation is an equality. What are the properties of a curve? an equality between the products, to the combinations of certain straight lines enclosed and bounded by this curve.”] —SOPHIE GERMAIN

What prompted Sophie Germain to enter the prize competition to derive a theory for vibrating surfaces? Did she see the contest as a source of mathematical knowledge and sought to advance her own intellectual development? In science, mathematical contests have been used for centuries to solve problems and to stimulate research or to give more interest to a given area of study. The contests issued by the learned academies in Sophie Germain’s time, most notably those in Berlin, Paris, and St. Petersburg, had the objective to influence the direction of research and to solve an outstanding problem. This drew attention to key problems and offered substantial rewards for solving them. Moreover, the topics chosen for competitions required perfect insight into the state of an entire discipline or posed a fundamental unsolved problem.1 Thus, the invitation was typically not addressed to young aspirants of science but to the leading savants—Euler, the Bernoullis, Lagrange, d’Alembert, Legendre—who willingly accepted it, and the result of their efforts advanced the sciences. In some cases, there is evidence that a topic of a contest may have been set with someone in mind. In the case of the Institut de France, the 1809 prize competition had the primary objective of deriving a mathematical theory to explain the phenomenon demonstrated experimentally by Chladni. It has been suggested that Laplace hoped that this contest would help advance Poisson’s career.2 Indeed, Laplace had been helping his protégé to secure several important positions immediately after graduation: in 1800 Poisson became répétiteur at the

1

Grey (2006), p. 6. Ibid., p. 10.

2

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 D. Musielak, Sophie Germain, Springer Biographies, https://doi.org/10.1007/978-3-030-38375-6_6

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École Polytechnique, and two years later he was promoted to deputy professor. In 1806, Poisson was appointed to the professorship vacated by Fourier. Poisson was a member of the Société d’Arcueil, an elite group of scientists led by Laplace and Berthollet who met regularly at their country homes in Arcueil (then a village 3 miles south of Paris) to discuss the latest research. The Society of Arcueil published important findings in their own journal, Mémoires de Physique et de Chimie de la Société d’Arcueil. The senior members of such prestigious group encouraged and helped their young protégés to further their careers. The Society ensured the election of its younger members to the First Class of the Institut de France. Be that as it may, Poisson did not enter the 1809 prize contest. This is surprising since he was working to develop a mathematical theory of elasticity and became Germain’s opponent. Let us first focus on the initial research that Sophie Germain carried out to develop her own theory.

First Hypothesis Sophie Germain admitted that the demonstrations of Chladni’s experiments aroused her scientific curiosity.3 She began to study Euler’s memoirs, initially not with the intention to contribute to the extraordinary prize proposed by the Institute, but with the sole desire to appreciate the difficulties of the problem described in the program. In the first days of January 1811, Sophie Germain communicated with mathematician Adrien-Marie Legendre to discuss her hypothesis.4 It is clear that she wrote to bounce ideas and ask questions related to both her studies of Euler’s memoirs and her own hypotheses. Legendre provided clarifications and gave her basic explanations to guide her through the lengthy derivations found in Euler’s papers. Written in Latin, some parts of the memoirs were rather obscure for a novice. From his responses, it seems that Legendre was well disposed to guide Sophie Germain through her research. In an undated letter written to respond to Germain’s inquiry concerning an equation found in Euler’s memoir, Legendre explained at length the algebraic manipulations she may have found puzzling. Legendre referred to Investigatio motuum, quibus laminae et virgae elasticae contremiscunt, a paper where Euler provided the mathematical theory of vibrating rods. Legendre continued the explanations in two more letters dated January 19 and 28, 1811. It is clear that by now Germain had formulated a basic hypothesis and was working to refine the details of her analysis. Issues related to the manipulation of exponentials and expansion of trigonometric functions seemed to be unclear to

3

Germain (1821), p. v. Stupuy (1896), p. 291.

4

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her, and the amiable Legendre was supplying the mathematical background to help her navigate Euler’s derivations, illuminating those areas that were obscure. On 21 September 1811, ten days before the deadline, Sophie Germain, now thirty-five, submitted her anonymous memoir. We can picture Paris at that time, beautiful with the soft glow of autumn. Whether that Saturday was sunny or not, Sophie didn’t say. But perhaps the crisp air of the morning gave her a sense of courage. This was a bold step to take even for a well-educated mathematician. We can only imagine Sophie, folding the pages of her manuscript, pouring the wax seal, and entrusting her precious work to a courier who would deliver her competing memoir to the Institute of France. Sophie Germain wrote in 1821: “Neither the feeling of my inability, neither the insufficient [knowledge of] calculus, nor the little time which I had left until the time of the contest, could prevent me from addressing a Report to the Institute, in which I proposed the theory that I had conceived. I felt at the time that this theory was worthy of attention, and I eagerly submitted it to the judgment of the Academy.”5 Under the rules of the competition, entries had to be anonymous, identified only by an epigraph. Sophie chose the words Effectum naturalium ejusdem generis eœdem sunt causœ,6 a fragment from Newton’s Principia, Rule 2. Loosely translated, it means, “Therefore to the same natural effects we must, as far as possible, assign the same causes.” At the regular meeting (on 7 October 1811) of the Class of Phyical Sciences and Mathematics it was announced that one entry was submitted for the prix extraordinaire. Germain didn’t know that hers was the only entry in the contest until Legendre told her. She must have been anxiously waiting for the outcome, and reached out to her mentor. In a letter dated October 22th, Legendre responded: “Mademoiselle, Votre Mémoire n’est pas perdu; il est le seul qu’on ait reçu sur la question des vibrations des surfaces.”7 [Miss, your memoir is not lost; it is the only one received on the problem of vibrating surfaces.] He added that five judges, him included, were appointed to review it. Legendre ended the letter by advising her to remain silent until the final judgment, implying that their exchange created a conflict of interest. In her first attempt, Sophie Germain developed an insightful hypothesis, obtaining an equation for the motion of the vibrating plates. She used Euler’s theoretical foundation to develop her own theory. Germain thought that the sum of the principal curvatures of the plate, when bent, would play the same role in the theory of plates as the curvature of the elastic central-line in Euler’s theory of rods. Thus, she proposed to regard the work done in bending as proportional to the integral of the square of the sum of the principal curvatures taken over the surface. In other words, Germain assumed that the strain-energy of a plate was represented by this integral:

5

Germain (1821), p. vi. Stupuy (1896), p. 298. 7 Ibid. 6

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 ZZ  1 1 2 þ 0 ds A r r where r and r′ are the principal radii of curvature of the bended vibrating surface, and A is a constant. From this assumption and Euler’s principle of virtual work, Germain deduced the equation of flexural vibration in the form now generally admitted. It was shown later that the formula assumed for the work done in bending was incorrect.8 As the weeks went by, Germain must have felt insecure. From a letter written by Legendre on November 10, I ascertain that Germain had sent him an appendix and asked that he attach it to her competing memoir. Wondering if it was sufficient to establish the theory to explain Chaldni’s experiments, perhaps Germain had added more details to her analysis, hoping to strengthen her results. Legendre responded that “Les commissaires jugeront ensuite s’ils doivent tenir compte on non de ce supplément.” [The judges must determine whether to consider this supplement.] Her waiting was over on December 4, when a letter from Legendre told Sophie that her essay did not win.9 She had made a mistake in calculating the variation of her integral. Germain did derive a fourth-order partial differential equation, but it was not the correct one. She had studied Euler’s memoirs, including his De Motu vibratorio tympanorum (on the motion of vibrations in drums),10 and Tentamen de Sono campanarum (an effort on the sound of bells)11; both works were cited in the public announcement for the prize competition. Sophie later admitted to having taken a formula derived by Euler without carefully examining it. Legendre explained in his letter why her main equation was not correct, “even assuming the hypothesis that that elasticity at each point can be represented by 1 1 y þ y0.” He wrote: “Lagrange found that, in this case, the real equation should be of 4  2 4 d4 z the form ddt2z þ k 2 ddx4z þ 2 dxd2 dyz 2 þ dy ¼ 0; assuming that z is very small.”12 The 4 variable z represented the deflection of the plate. A note written by Lagrange in 1811, though not made public until years later, explained in a sussinct manner what happened when he studied the first memoir submitted by Germain: Note communiquée aux Commissaires pour le prix de la surface élastique (décembre 1811). L’équation fondamentale pour le mouvement de la surface vibrante ne me paraît pas exacte, et la manière dont on cherche à la déduire de celle d’une lame élastique, en passant

8

Love (1906), p. 6. Ibid., p. 300. 10 Euler (1761). 11 Euler (1760). 12 Stupuy (1896), p. 300. 9

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d’une ligne à une surface, me parait peu juste. Lorsque les z sont très-petits, l’équation se réduit à

 6  @2z @ z @6z þ gELc þ ¼0 @t2 @x4 @y2 @y4 @x2

mais en adoptant, comme l’auteur, 1=r þ 1=r0 pour la mesure de la courbure de la surface, que l’élasticité tend à diminuer, et à laquelle on la suppose proportionnelle, je trouve dans le cas de z très-petit une équation de la forme

 4  @2z @4z @4z 2 @ z þ k þ 2 þ ¼0 @t2 @x4 @y2 @x2 @y4

qui est différent de la précédent.13

Lagrange stated in this note that the fundamental equation for the motion of the vibrating surface did not seem correct, but “the manner in which it was sought, deduced from that of an elastic curve, passing from a line to a surface, seems just right.” He wrote that, for very small z, the equation reduces to the equation by Germain, but “by adopting, as the author did, 1r þ r01 for the measurement of the curvature of the surface, that the elasticity tends to decrease, and the assumed proportionality, I find an equation of the form in the case of z very-small (the amended equation), which is different from the previous one.” As Legendre had advised her, Germain probably consulted Lagrange’s Mécanique analytique, but it is unclear whether she grasped all of the subleties of this analysis. Although Lagrange’s equations provide a systematic way to formulate the equations of motion of a mechanical system or a (flexible) structural system with multiple degrees of freedom, which had some bearing on the basis for developing the general theory of vibration, his approach to equilibrium of an elastic lamina was inadequate. For example, when Legendre referred Germain to the Mechanique analytique, page 148, he expected that she would follow Lagrange’s analysis14 with indeterminate multipliers. Moreover, Lagrange had assumed without enough explanation that the internal force of the lamina called into action is of a certain kind, but in the problem he treated he had omitted a part of the force. Legendre also mentioned in that same letter (4 December 1811) that Jean-Baptiste Biot, who had learned about her memoir, claimed to have found the

13

Note in the Annales de Chimie, Vol. 39, 1828, p. 149, part of Navier’s remarks regarding an article published by Poisson. 14 Letter from Legendre to Germain dated 4 Dec 1811. In Stupuy (1896), p. 302.

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true equation of the vibrating elastic surface. Biot insisted that he had shown it to Laplace a long time before, an equation which was not of the form as that found by Lagrange based on her assumptions. And indeed, in May 1800, Biot had submitted to the Academy a memoir entitled Recherches analytiques sur les vibrations des surfaces and it was assigned to Laplace and Lacroix to examine it.15 However, Biot’s equation was incorrect and Legendre must have seen it years earlier in order to have reason to doubt it. Legendre closed his letter to Sophie Germain with these encouraging words: “J’imagine que la même question sera posée avec un nouveau délai; ainsi miséricorde n’est pas perdue. Au contraire, il faut plus que jamais songer à emporter la palme.” [I imagine that the same question will be posed with a new deadline; Thus all hope is not lost. On the contrary, it should be stronger than ever to hope to win the award.]

Second Attempt: More Disappointment In January 1812, the Institute proposed the same problem for the next competition, giving 1 October 1813 as the deadline for submitting new results. This was a politically difficult time for France. Napoléon was waging war with many nations, in an attempt to control the whole of Europe and build and expand a powerful French Empire. By late 1812, the emperor’s army had entered a deserted Moscow. The Prussians and Austrians deserted the Grande Armée. Meanwhile, in France, coups were attempted against Napoléon. He left his broken army and returned to Paris to assert his authority. However, the allied armies began to move against the French troops. On 5 March 1813, Prussia declared war on France, joining an Austrian, British, Russian, and Swedish alliance. In April Lagrange died, just a week after Napoléon had honored him with the Grand Croix of the Ordre Impérial de la Réunion. There is no record to tell us how Sophie Germain reacted, or how she mourned the loss of the great mathematician. Later that month, Napoléon arrived in Germany and took the offensive, and in early May 1813, he won a victory against the Alliance at Lützen. This was followed with victories at Bautzen and Dresden. On 21 June 1813, the Duke of Wellington defeated the French troops in Spain at Vitoria, removing Spain from the French Empire. On 12 August 1813, Austria declared war on France. Sophie Germain sought to escape from this political turmoil by immersing herself in her research, refining her hypothesis for the second competition. On 23 September 1813, Germain submitted her memoir, the only entry in the contest. This time, she used the equation that Lagrange had amended, since it was, after all, the equation that she should have obtained from her initial analysis.

15

Institut de France. Procès-verbaux. Tome II, p. 169.

Second Attempt: More Disappointment

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This time, Sophie Germain gave the equation for the vibration of a plane surface as  N

2

 @4z @4z @4z @2z þ 2 þ ¼ 0; þ @x4 @y2 @x2 @y4 @t2

where N 2 is a constant, which contains as a factor the fourth power of the thickness of the vibrating plate. Germain also derived an equation for the vibration of a circular ring,  N

2

 @4r 1 @2r @2r  ¼0 þ @s4 a2 @s2 @t2

where a is the radius of the circle. This has a striking resemblance to Euler’s (incorrect) equation derived when he attempted to model the vibrations of a bell, which Euler divided into annuli by vertical and horizontal sections.16 Unfortunately, Sophie Germain could not provide a satisfactory derivation of the equation for the vibration of a plane elastic surface. She wrote in her self-published ninety-six-page paper: “I sent, before the 1st October 1813, a memoir in which the already known equation is given, and also the conditions of the specified ends using the assumption that had provided the equation.”17 This paper concluded with an attempt to compare experiments and her mathematical theory. From Chladni’s experiments, Sophie Germain began with the realization that stable surfaces can vibrate with the most striking wave patterns. Germain’s explanation of the wave patterns was this: at a given point ða; bÞ of a vibrating plate, the structural force exerted by the plate to restore it to a plane is proportional to the average of the maximal and minimal planar curvatures of the surface. The mean curvature of the surface thus captures geometrically the elastic force exerted by the plate. Defining the curvature as k ðaÞ, where a is the angle between 0 and 2p, with k1 representing its maximum value, and k2 its minimum value, Germain concluded that the elastic force of a surface at the point ða; bÞ is proportional to ðk1 þ k2 Þ=2, the mean or average curvature. Germain was not able to obtain a solution for either equation or correlate her experiments to reach an agreement with Chladni’s patterns, as required by the contest. Nonetheless, Sophie Germain was the first to correctly identify the elastic force of a thin plate under stress.18 By referring to the linear case, Germain meant one where the forces of elasticity are supposed to be proportional to the reciprocal ratio of the radius of curvature of a simple curve. Early in December 1813, Legendre wrote to Germain seemingly annoyed and also disappointed on her work. He started the letter by saying, “I do not understand 16

Euler (1760). Germain (1821), p. vi. 18 Lodder (2003). 17

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the analysis that you sent me; there is certainly an error, either in writing, or in reasoning…”. It is clear that Germain had sent him additional details of her analysis, but instead of clarifying her mathematical methods, or her conclusions, it made things worse because the end result was not satisfactory to him. However, Legendre admitted that hers was the correct equation of the vibrating plate, and that her explanation of the phenomena was acceptable. As a good mentor, Legendre ended the letter with words of encouragement: “Si la commission de l’Institut était de cet avis, vous pourriez an mains être mentionnée honorablement.”19 [If the commission is of the same opinion, your work deserves an honorable mention.] The judges of the contest must have agreed. Starting with the correct equation to model the vibration of the plates, Germain’s analysis could not be faulted. The physical hypothesis was also justified; in fact, by then, Poisson had arrived at the same general equation, though by a very different method.20 Therefore, the panel of judges deemed Germain’s second mathematical memoir worthy of an honorable mention. Once again, the contest was reopened; this time, the Institut made it clear that it required experimental demonstration of the equation, establishing 1 October 1815 as the deadline. While Germain worked on her third essay, attempting to fulfill the requirements of the last contest, Paris was again in a state of chaos.

Paris in 1814 On the first day of 1814, the Coalition armies (Russian, Prussian, and Austrian) entered France. At daybreak on March 30, while Sophie worked on her third mathematical essay, French soldiers and students from the École Polytechnique battled with the Coalition armies at the gates of Paris. All morning, Parisians watched the battle through telescopes. The artillery of the French Guard was firing continuously. One can imagine the disquieting sounds coming through Sophie’s windows, the deafening booms from cannons firing intermingled with the galloping horses carrying grenadiers to resupply ammunition. On the second day of fighting, Russians and Prussian soldiers surrounded a group of voltigeurs of the Young Guard at St.-Denis, Sophie’s childhood neighborhood. They had run out of ammunition, and they had to surrender. Tsar Alexander I, King of Russia, had directed the main Coalition armies to march on to the city; meanwhile, a Russian general with ten thousand cavalrymen rode toward Saint Pizier, where Napoléon was fighting with the Austrian allies. Napoléon realized too late that it was not the main army. By this time, Russian and Prussian armies were entering Paris.

19

Stupuy (1896), p. 305. Poisson (1814).

20

Paris in 1814

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The French surrendered on 31 March 1814. At 11:00 a.m., the Coalition armies triumphantly entered the French capital through the Pantin Gate. Tsar Alexander I was at the head of the Coalition Army; on one side was Frederick William III, King of Prussia, and on his other side was Schwarzenberg representing the Emperor of Austria, whose daughter Marie Louise was Napoléon’s wife. A huge crowd of more than one thousand Russian, Austrian, Prussian, and British officers followed. Russian soldiers shouted “Paris! Paris!” and broke their ranks to press forward to see the glorious city. The French National Guard was lined up on either side of the avenue, making way for the men they had been fighting the day before. Parisians climbed up into trees, on top of carriages and onto rooftops, while handkerchiefs waved from the windows, and a shower of white lilies fell from every story upon the victorious enemy. Every well-dressed man in the streets wore a white cockade, the symbol of the Bourbons, who were Napoléon’s deadly enemies. When the horses stopped across the Arc de Triomphe, Tzar Alexander proclaimed in a loud voice: “I do not come as an enemy. I come to bring you peace and commerce!” The Parisians cheered, and someone took a step forward and cried: “We’ve been waiting for you a long time!” The monarch replied: “If I didn’t come sooner it is the bravery of French troops that is to blame!” The Parisians shouted: “Long live Alexander! Long live the Allies!” On the same day, the Russian tsar was presented with the keys to the French capital. One can imagine that on that Thursday, it would have been difficult for any Parisian to remain oblivious to this historical event. Until then, no foreign army had reached their magnificent city in nearly four hundred years. Napoléon abdicated a few days later. In the spring of 1814, Paris was fully controlled by the Allies. A huge bonfire was lighted in the Court of Invalides so that hundreds of standards captured from the Allies by French troops “were given to the flames.” The reaction of Parisians to the occupying forces varied. Some were angry and hostile, while others were welcoming. The French royalists were mad with joy, parading the streets shouting, “Long live the Bourbons!” Some Parisians feared that the invading soldiers would loot and burn their city. It was reported that during the occupation, “the British were looked down, the Prussians were hated, but the Russians succeeded in creating a friendly relationship with the French.” On the 3rd of May, Louis XVIII entered Paris solemnly. Beside him was his niece, Marie-Antoinette’s daughter, who was the sole survivor of the former royal family (Fig. 6.1). The Comtesse de Boigne recalled that historic homecoming: “The procession was escorted by the old Imperial Guard. Its aspect was imposing, but it froze us. It marched quickly, silent, and gloomy. With a single glance it checked our outbursts of affection. … The silence became immense, and nothing could be heard but the monotonous tramp of its quick striking into our very hearts.”21

21

Adèle d’Osmond, Comtesse de Boigne, Memoirs of the Comtesse de Boigne, Volume 1, 1781– 1815, Helen Marx Books; 2 edition (April 2000), p. 123. She was a writer known for her memoirs describing life under the July Monarchy.

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Fig. 6.1 Louis XVIII entering Paris on 3 May 1814. Credits see Appendix “Illustration Credits”

Aristocrats and commoners welcomed the new King of France and Navarre. On August 29, Parisians congregated in a joyful feast, which was given to Louis XVIII in celebration of the restoration of the Bourbon throne. Meanwhile, the hostilities in the scientific world were brewing. Siméon-Denise Poisson was determined to sabotage Germain’s efforts. During the meeting of the Class of Mathematics on 8 August 1814, Poisson stood to read his thesis on the mathematical theory of elastic surfaces. Legendre objected, saying that the Class should not hear a memoir on the same topic of the competition before the prize would be awarded. Poisson argued that the memoir he was about to read was such that it would not affect the prize. Hence, he continued reading. Nevertheless, Legendre’s objection was referred to a Committee which would be nominated for a vote in the next session. This vote never happened, as no one raised the issue again. And that is how Poisson introduced his own mathematical theory on elastic surfaces,22 less than a year after Sophie Germain had developed her own. Not surprisingly, Poisson arrived at the same equation by considering the case of a nearly plane surface with no applied forces. He also obtained an elasticity constant similar to Germain’s. However, we wonder, why was Poisson permitted to remain as a judge in the panel for the prize competition? Legendre’s objection was 22

Poisson (1814).

Paris in 1814

83

justified; yet, no one in the Class seemed to be bothered by the glaring conflict of interest. At the regular meeting on 14 November 1814, Carnot and Prony reported on the memoir of Binet relating to analytical expressions of elasticity and stiffness of double curvature curves. Jacques Philippe Marie Binet made significant contributions to number theory, and the mathematical foundations of matrix algebra, ideas which Arthur Cayley and other mathematicians later expanded. In his memoir on the theory of the conjugate axis and of the moment of inertia of bodies, Binet enumerated the principle now known as Binet’s theorem. Regarding elasticity, Binet observed in Lagrange’s formulation the omission of the moment around the tangent to the elastic fibre, i.e. the moment or torque component that develops a twisting reaction in bending the elastic rod. Two years later, Poisson added a term equal to this torsional moment multiplied by the cosine of one of the angles that the tangent makes with the three coordinates. Upon the return of Louis XVIII in the spring of 1814, Napoléon was exiled to Elba. In February 1815, the obstinate emperor returned to power for one hundred days, while the king fled to Belgium. With the monarch gone, Napoléon moved back into the Tuileries. The period that is historically known as the Hundred Days had begun. In June, during the Battle of Waterloo, Napoléon was defeated by the British and taken to St Helena. On July 8, the Bourbon rule was restored when Louis XVIII returned to Paris.

Winning the Grand Prix de Mathématiques By the deadline of the third prize competition, Sophie Germain, who was now thirty-nine, had gained greater confidence in her work; this time, she submitted a new mathematical analysis under her own name. The motto she attached to her memoir was most appropriate: Felix qui potuit rerum cognoscere causas, from Virgil (Géorgiques, liv. II), which I translate as, “Fortunate one who is able to know the causes of things.” Sophie Germain was indeed fortunate. This time, her third essay won the prize of mathematics. At the regular meeting on Tuesday 26 December 1815, the judging commission proposed to award the prize to the only competing essay on vibration of surfaces.23 The note attached to her submission was opened and, not surprisingly, they found Sophie Germain’s name. The manner in which she was officially notified did not harmonize with the accolades Germain should have received from the Institut de France. A letter dated 6 January 1816, and signed by Jean-Baptiste Delambre, secrétaire perpétuel for the mathematical sciences of the Institut, is rather perplexing. It states (Fig. 6.2), in part,

23

Institut de France. Procès-verbaux. Tome V, p. 595.

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Fig. 6.2 Letter from Delambre to Germain, 6 January 1816. Credits see Appendix “Illustration Credits” M. Delambre a l’honneur de présenter ses hommages à Mlle. Germain et de lui envoyer deux billets d’Institut, présumant bien que ses amis lui en demanderont plus qu’elle n’en aura à distribuer, si, comme il le suppose, elle en a reçu hier ou aujourd’hui. Mais M. Delambre ayant appris par M. Sédillot que Mlle Germain n’en avait pas encore reçu hier soir, il craint qu’il n’y ait eu quelque oubli, et la prie, dans ce cas, d’avoir recours à lui, parce que les billets imprimés étant épuisés, il peut y suppléer par un billet à la main pour autant de personnes qu’il conviendra à Mlle Germain de lui en indiquer. [Mr. Delambre has the honor to present his compliments to Mademoiselle Germain and sends her two

Winning the Grand Prix de Mathématiques

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Institute tickets, although presuming her friends will ask for more than she has to distribute, if, as he supposes, she has received them yesterday or today. But Mr. Delambre having learned from Mr. Sedillot24 that Mademoiselle Germain had not yet received them last night, he fears that there was some oversight, and requests, in this case, to resort to him, because the tickets are printed out, he may supply a ticket in hand for as many people as it will suit Ms. Germain to tell him.]

How was Germain supposed to react to such an ill-conceived letter, when she, the guest of honor, was made to beg for tickets to attend her own award ceremony? It was two days before the meeting, and Delambre had learned from a Mr. Sedillot that Sophie had not received the admission tickets! Was this an oversight, as he said? At their public meeting on 8 January 1816, the First Class (mathematics and physical sciences) of the Institute of France announced that Sophie Germain had won the grand prix of mathematics. It was an evening affair that began at half past five and continued till half past seven.25 The President of the Institut first announced the winners of the different contests: Sophie Germain for her memoir on vibration of plates, and Augustin-Louis Cauchy who had won for his work on wave theory. Alas! Sophie Germain was not in attendance. The weekly newspaper Journal des Débats noted the historical event in these terms: La classe des sciences mathématiques et physiques de l’Institut a tenu aujourd’hui sa séance publique, devant une assemblée fort nombreuse qu’avait attirée sans doute le désir de voir une virtuose d’un genre tout nouveau, Mlle Sophie Germain, à qui le prix des lames élastiques devoir être décerné. L’attente du public a été trompée: cette demoiselle n’est point venue recevoir une palme que son sexe n’avait paient encore cueillie en France.26

[The class of mathematical and physical sciences of the Institute held its public session today, a very large assembly that attracted without doubt those desiring to see a virtuoso of a new kind, Miss Sophie Germain, to whom the prize for elastic membranes was to be awarded. The expectation of the public was deceived: the young lady did not go to take the trophy that no one of her gender has ever received in France.] Did Germain ever receive the entrance tickets from the Institute? If she did, perhaps she felt insulted by Delambre’s insensitive letter. Or perhaps she was overly timid, insecure in public. Maybe Sophie Germain was simply satisfied to have achieved her goal. However, her struggle for recognition of her mathematical work was just beginning.

A graduate of the École, Jean Jacques Emmanuel Sédillot (1777–1832) was an astronomer at the Bureau des Longitudes, where he assisted Delambre and Laplace in their research. 25 Bugge (2003), p. 89. 26 Journal des Débats, Mardi 9 janvier 1816, p. 2. 24

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6 Germain and Her Biharmonic Equation

Confronting a Rival Winning this coveted prize, which was one of Sophie Germain’s greatest accomplishments, must have felt bittersweet. The circumstances surrounding her winning were not conducive to celebration. Not only had the Institut had blundered in telling her she won, Sophie Germain also learned that the review committee (Poisson, Laplace, Legendre, Poinsot, and Biot) had not been satisfied with her hypothesis and the manner in which she explained it, even though her differential equation was correct. Sophie Germain interpreted this as a disapproval of her work, when she had been expecting praise for her mathematical theory. The criticism and the comparison of her analytical derivation of the equation to that made by Poisson was too much to bear. She wrote to Poisson: From the judgment pronounced by the class it appears that I had been deceived by the demonstration which was acceptable to you, but it did not explain the nature of the error I made. M. Hallé,27 to whom I have expressed my desire to know what mistake is in my proof, has kindly offered to ask you to clear up my doubts. I do not believe I am mistaken in the way in which the general equation has been deduced from the hypothesis; it must be, therefore, the hypothesis itself that has not been satisfactorily justified. In order to avoid the trouble of revisiting the demonstration, I have reproduced in the note attached the arguments on which it is based. I wrote them mid-margin, so that it would be easier for you to mark where you think my chain of reasoning is interrupted. The more respect I have for your judgment, the more importance I must attach to obtaining the clarification I ask from you.28

Understandably, Sophie Germain was upset and wanted clarification on “the nature of the error” that she had made. This was not an unreasonable request, as she was asking for a referee report. She included an attachment (la note ci-jointe), in which she elaborated further on her analysis, leaving large margins so that it would be easier for Poisson to mark the places where he found a mistake. Poisson was not inclined to referee her analysis; he replied on 15 January 1816: “The reproach that the commission made of your memoir relates less to the assumption as to the manner in which you applied the calculus to this assumption. The result to which this calculation has led you does not agree with mine, except in the single case where the surface deviates infinitely little from a plane, either in the state of balance, or of movement.”29 As it turns out, Poisson’s equation was similar in form to Germain’s equation after Lagrange had corrected it. Poisson’s response can be interpreted as communicating that Germain’s analysis was deficient and lacked mathematical rigor. He also abstained from giving her the critique that she requested, and he merely repeated why the award had been given to

27

J.-Noël Hallé (1754–1822) was a medical doctor, perhaps a friend of Sophie Germain’s family. He was a member of the Institut de France since 1795. 28 Stupuy (1896), pp. 307–308. 29 Ibid., p. 310.

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her with such reservation. He ended the letter with the offer to send her a copy of his own memoir when it appeared in print. This must have hurt Germain, but it also challenged her more; she was not ready to give up. Meanwhile, on 21 March 1816, Louis XVIII issued a royal ordinance to reorganize the Institute of France into societies, “in order to revive their former glory and that they have gained ever since.” At a special session on 27 March, the Class of Physical Sciences and Mathematics read the king’s decree and restored its name as Royal Academy of Sciences; it was comprised of sixty-five titular members, ten free members, eight associates, and one hundred correspondents. As an Academy, it was organized into twelve sections, beginning with the Geometry Section made up of Laplace, Legendre, Lacroix, Biot, Poinsot, and Ampere. Interestingly, Cauchy was assigned to the Mechanics Section, while Poisson was assigned to Physics. Delambre remained as Perpetual Secretary. Jean-Baptiste-Joseph Fourier was elected to the Academy of Sciences in 1817. A year later, Fourier published his own view of vibrating elastic surfaces (Note relative aux vibrations des surfaces élastiques et au mouvement des ondes)30 but his interest was in developing the theory of heat. Fourier’s Théorie Analytique de la Chaleur (1822) is one of the most important books published in the nineteenth century. In this work, Fourier developed the theory of the series known by his name, and which is now applied it to the solution of boundary-value problems with partial differential equations.

The Germain-Lagrange Equation Years after winning the prize, Sophie Germain admitted to making a mistake. It is evident that Germain’s theoretical base was sound, and her hypothesis was right, since it led to the correct equation (Fig. 6.3), despite her mathematical clumsiness; otherwise, Lagrange would not have been able to derive it. Of course, as Germain herself later admitted, the agreement between theory and observation was not close. And that is why she attempted to refine her analysis and to further explain how it related to the vibrating plate experiments. The year 1821 was important and memorable for Sophie Germain. In the spring, having revised and synthesized her earlier mathematical work, Germain self-published, at her own expense, the first scientific summary of her research on the theory of elastic surfaces, which she titled Research on the Theory of Elastic Surfaces.31 In the preface she introduced the problem as stated in the prize competition, and presented the history of her endeavor. In this introduction, Germain referred to Poisson but without using his name. She cited the memoir on elastic

30

Bulletin des Sciences par la Société Philomatique de Paris (1818). Germain (1821), p. v.

31

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6 Germain and Her Biharmonic Equation

Fig. 6.3 Equation of elastic vibrating plate according to Germain

surfaces that he had presented at the Academy, clearly attempting to claim priority over her own theory and the main equation. She wrote: In August 1814, a member of the class read a memoir on elastic surfaces. The author adopts a new hypothesis; he treats it with the talent which characterizes all his works; and, confining himself to the case of naturally plane surfaces, he arrives at the general equation of this kind of surface; but at another time it sets the conditions under which the limits must be bound.32

Germain firmly stressed that her own hypothesis was more correct, and that it led to the same equation, the one she had derived years earlier. Setting the record straight, she seems to claim priority over Poisson with her first derived equation and that resulted in the prize for advancing the theory of elasticity. Regarding her mistake, Germain wrote that, fortunately, one of the judges, M. Lagrange, had understood her hypothesis and obtained the equation that she should have obtained “if she had complied with the rules of calculus.”33 In July (1821), Germain sent a copy of her paper to the Academy. From a letter by Fourier it appears that Legendre had asked him to review her work. And, in fact, in the preface Sophie acknowledged that she requested counsel from Fourier.34 At the meeting of the Class of Mathematics on July 16, it was recorded that Germain had sent a copy of her Mémoire sur les Surfaces élastiques. No public comment on Germain’s work was ever made. The first congratulatory letter that Germain received for her publication was from her friend and mentor Legendre, who was the one who advised her to publish her research. On July 23 he wrote to Germain, politely disengaging himself from the responsibility of giving a scientific critique and merely pointing out a typographical error (a misspelling of the Latin word campanarum). However, being her mentor, Legendre felt that he had the right to chastise Sophie for the tone she used to refer to the unnamed Poisson. He wrote: “… If there is anything to criticize you, it is the way you praise the mathematician with whom you disagree. I hope he responds

32

Ibid. Ibid., p. vi. 34 Ibid., pp. viii–ix. 33

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with dignity to this civilized assault …”35 That was harsh. But then again, Poisson was never nice to Germain. On the same day, Germain received a letter from Delambre, in his role as secrétaire perpétuel de l’Académie, thanking her for the memoir and simply conveying that it would be deposited in the library of the Institut. Germain also received a letter from Cauchy, but instead of commenting on the equation she had derived or the essence of her mathematical research, as the eager author surely expected, he responded by sending a copy of his freshly printed volume, Cours d’Analyse. On August 2, Navier sent Sophie a short note full of disconcerting flattery, saying that he appreciated “the merits of something so remarkable that few men could read, and only one woman could [write it.]”36 If Sophie expected validation for her mathematical research from the scholars at the Institute, they did not respond. To add to her disappointment, she closed the year in mourning. Her father passed away on 15 December 1821. This sad event may have contributed to a reassessment in the family affairs. Starting in 1823, her correspondence was addressed to Rue de Braque No. 4, maison de M. Geoffroy médecin.37 Geoffroy was her brother in-law, a medical doctor married to her younger sister Angélique. In addition to Legendre, Sophie Germain had a friend in Joseph Fourier, a scholar with whom she could converse about her work. Thus, it is quite likely that he understood the rebuff she must have felt after the cold reception of her memoir. Sophie was a loyal friend to him. In 1822, she wrote letters to members of the Institute, asking them to vote for Fourier, who was seeking the position of secrétaire perpétuel, left vacant when Delambre died. Lacking the professional critique from a referee, Germain’s scientific work and professional progress were more challenging. After all, she did not have a membership to participate in the scientific activities of the class of mathematics and physics of the Institute, and she could not engage in truly meaningful discussions with other researchers. Not counting Fourier and Legendre, other academicians may not have developed a professional rapport with Germain. When she won the prize, Sophie Germain was almost forty years old. Poisson and Navier were five and nine years younger, respectively. Or perhaps Poisson and Navier were so preoccupied proving themselves professionally that they could not admit that Sophie’s ideas were unique and insightful. In their quest to undo one another, Poisson and Navier had become scientific rivals. The appointment of Fourier as perpetual secretary of the class of mathematics of the Institute in November 1822 may have given Germain the false sense of Stupuy (1896), p. 314. Legendre’s wrote: «vous proposez votre opinion de la manière la plus modeste et, si l’on avait quelque chose à vous reprocher, ce serait les compliments dont en quelque sorte vous accablez le géomètre dont vous combattez l’opinion. Puisse-t-il répondre dignement à cet assaut de civilité; c’est ce que je désire plus que je n’espère.» 36 Stupuy (1896), pp. 315–317. French text: “qu’il le mérite un écrit aussi remarquable, que bien peu d’hommes peuvent lire, et qu’une seule femme pouvait faire.” 37 Correspondance de Sophie GERMAIN. p. 27. 35

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approval, perhaps believing that her chances for being admitted to his scientific circle had improved. However, having social contact with Fourier could also have helped Germain to gain a better perspective of what was going on at the Institute of France, especially since Fourier was rather fond of conversation. During this time period, Louis XVIII ruled as King of France and Navarre. On 29 September 1820, the birth of the last heir to the branch of the Bourbons had created extraordinary popular fervor in France. The baby’s father was the Duke of Berry. He had been assassinated in February of the same year on the steps of the Opera House while helping his pregnant wife into a carriage. Named Henri, Duc de Bordeaux, the baby was presented to the people of France in the arms of Louis XVIII, who was his great-uncle. His nickname was “The Miracle Child.” On 1 May 1821, the king endowed the baptism of the Duke of Bordeaux with all of the solemnity and pomp of a coronation, as the child carried on his shoulders the future of the Bourbons and of the restored monarchy. For the Christening at Notre-Dame, King Louis XVIII invested a considerable sum for the execution of a Berline coach, a sumptuous and ornamented vehicle that was particularly rich in its decoration. The body was painted with the royal coats of arms of France and Navarre. Jubilant Parisians lined the streets to witness the splendor of the procession. Sophie Germain self-published again.38 In 1824, Sophie Germain penned another essay, hoping perhaps to prove once and for all that her work was worthy, and in March she sent it to the Institute. At their regular meeting on March 8, the class of mathematics and physics reported having received Germain’s memoir with the title “Effects due to the greater or lesser thickness of elastic plates.”39 A commission composed of Laplace, de Prony, and Poisson was appointed to review it. Fourier wrote to inform Sophie Germain that he had asked M. Cuvier to read her memoir, adding that a commission was appointed to write a report, just as she had requested.40 Cuvier was the perpetual secretary of the physics class.41 Fourier assured her that if Poisson “opposed the results of her research, he would have to yield to her experience.” Fourier added a postscript to clarify that she would receive the minutes of the meeting in a separate letter. However, the published meeting minutes of the Academy of Sciences42 contain nothing about Sophie Germain’s work or give indication that her memoir was ever read or reviewed. The Mémoires de l’Académie des sciences de l’Institut de France for 1824 include papers by Poisson, Navier, and Cauchy. The last section of this document is the History of the Academy where the perpetual secretaries (Fourier

38

Germain (1824). Effets dus à l’épaisseur plus ou moins grande des plaques élastiques. 40 Stupuy (1896), pp. 320–322. 41 Jean Léopold Nicolas Frédéric or Georges Cuvier (1769–1832) was a French naturalist and zoologist. Cuvier was Permanent Secretary of Physical Sciences of the Academy since 1803. 42 Institut de France. Procès-verbaux. Tome VIII, p. 35. 39

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and Cuvier) wrote an overview of the memoirs received, read, or published. They did not mention any work by Germain nor her letter. In retrospect, it should have been evident to Sophie that Fourier’s promise was somehow insincere, considering that Cuvier usually reviewed works in chemistry, geology, physiology, botany, anatomy, medicine and surgery, areas that at the time were considered part of “physics.” Of course, one cannot judge Fourier and his actions. Maybe he had good intentions and wished to help Sophie Germain. In the preface of her first paper, Germain acknowledged that Fourier had said that he preferred a purely geometric approach to demonstrate her hypothesis, and he proposed that she use Jakob Bernoulli’s model of a straight beam. It is unclear whether this advice was sincere or relevant, since Bernoulli’s approach was not directly applicable to her quest for a theory on the vibration of plates. Also, by suggesting that she follow a “geometric” approach, was Fourier implying that this would be easier for her or that her method was incorrect? However, despite giving her this advice, Fourier must have found a polite way to remain disengaged from Germain’s research. Moreover, the moment-curvature relationship that Bernoulli had developed was incorrect. He had followed Mariotte’s assumption that the neutral axis would be located at the bottom of the beam, which is on the compressive face. Edme Mariotte was a French scientist better known for his extensive work in Thermodynamics. Assuming that plane sections remained plane, Bernoulli equated the net bending moment across the section to the external bending moment, and derived the relationship Cr ¼ px, where r is the radius of curvature and C is a stiffness factor, which is one order of magnitude wrong. Later, Daniel Bernoulli and Euler derived the correct form of the equation.43 Germain may not have realized that. In any event, she may have followed Fourier’s advice and attempted to find a geometric relationship for the curvature. In her first self-published memoir,44 Sophie Germain discussed how she had assumed that the elastic constant was proportional to the square of the thickness of the plate, just as Euler had done. She referred to Savart’s experiments, attempting to compare her results with his. She ended that first paper by asking for validation for her work: “… d’ailleurs, mon but, en publiant ce Mémoire, a été de consulter l’opinion des géomètres, non-seulement sur la légitimité de l’hypothèse que j’ai cherché à démontrer, mais encore sur les applications dont la théorie me paraît susceptible.” [… indeed, my aim in publishing this memoir was to consult the opinion of mathematicians, not only on the legitimacy of the assumption I have tried to demonstrate, but even on applications whose theory seems to me so disposed.]

43

Timoshenko (1983). Germain (1821).

44

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In the memoir she sent to the Institut in 1824, Germain noted that Navier had determined the constant to be proportional to the third power. However, she insisted that her analysis yielded a fourth power, saying: “J’ai reconnu, comme je le dirai plus bas, que ce n’est ni la première ni la seconde, mais plutôt la quatrième puissance de l’épaisseur, qui doit entrer dans la valeur du coefficient constant.” [I have recognized, as I will say below, that it is neither the first nor the second, but rather the fourth power of the thickness, which must enter into the value of the constant coefficient.]45 What happened to that essay that Sophie Germain submitted to the Academy in 1824? Years after Gaspard de Prony died, Germain’s memoir was found in his archives, and it was finally published in 1880.46 Once again, Sophie Germain self-published in 1826 a paper titled Remarques sur la nature, les bornes et l’étendue de la question des surfaces élastiques et équation générale de ces surfaces. [“Remarks on the nature, the bounds and the scope of the question on elastique plates and the general equation of these plates”]. Here she expanded her explanations, hoping to better justify her hypothesis regarding the forces of elasticity. Germain sent copies to Cauchy and to the Academy. On July 24, she received a letter from the Perpetual Secretary (Fourier) to acknowledge receiving the memoir, saying that Cauchy was designated to give a verbal report on it, and adding that it’d be placed in the Institute’s library.47 A day earlier, Germain had received a short note from Cauchy thanking her for sending him a copy. Again, no verbal report was ever made of Germain’s work. Germain’s struggle to match the mathematical theory to the experiments is evident in her papers (Fig. 6.4). She was aware of her deficiencies and she eagerly sought a referee to help her shape her ideas. At the end of her first paper she wrote: I feel how much more needs to be done before I can say that the experiment confirms the correctness of the theory that has been deduced from the hypothesis I have proposed. In spite of the imperfection of this work, I desired to submit it to the judgment of the reader; it will show him at least what part one could draw from the formulas, and what are the difficulties that one encounters in their application, especially because of the so-called accidental complication that experience presents in the first cases of vibration. I also thought that a theory which, by its nature, is applicable to a certain order of phenomena, should not be presented in isolation: moreover, my purpose, in publishing this Memoir, was to consult the opinion of mathematicians, not only on the legitimacy of the hypothesis I have tried to demonstrate, but also on the applications whose theory seems to me susceptible.

Then she turned to experimental research to test her theory. It may seem inconceivable that a woman, working at home without the resources typically found in a research laboratory, would attempt such activity. Yet, Sophie Germain performed experiments with vibrating plates that only well-trained engineers such as 45

Germain (1880), p. 16. Germain (1880). 47 Correspondance de Sophie GERMAIN. The same letters reproduced by Stupuy (pp. 326–328) give the year as 1823 instead of 1826. 46

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Fig. 6.4 Title pages of Sophie Germain’s Publications in 1821 and 1826

Savart and Biot, and Chladni himself, had to carry out. We explore the nature of those experiments in the next chapter. Nowadays, there is a consensus among many authors that Sophie Germain was the first to propose the classical theory of thin plates, which results in the formulation of a non-homogenous biharmonic equation for the transverse deflection of the plate, characterized by a fourth-order partial differential equation. The term biharmonic indicates that the function describing the transport processes satisfies Laplace’s equation twice explicitly. And although it is generally attributed to Lagrange the statement that the dynamic flexural behavior of a plate is characterized by a fourth-order partial differential equation, it was Germain who conducted the initial stage of the fundamental analysis, and whose hypothesis led to the development of the correct governing equation.48 In view of the crucial role that Sophie Germain played in its derivation, it would be unfair and ahistorical to leave her name out of the first time-dependent inhomogeneous biharmonic equation that models the normal vibrations of two-dimensional plates. Thus, to do justice to both, I will refer to the biharmonic equation of thin plates as the Germain-Lagrange equation.

48

For the modern derivation of the biharmonic equation governing elastic thin plates, we consulted Selvadurai (2000). He refers to the biharmonic equation governing flexure of thin plates as the “Germain-Poisson-Kirchhoff thin plate equation.”

Chapter 7

Experiments with Vibrating Plates

Il nous reste à faire connaître les résultats de l’expérience à l’égard de l’influence qu’a sur les sons l’inégale répartition de l’épaisseur entre les différents points de la lame vibrante. —SOPHIE GERMAIN

When Sophie Germain attempted to develop a theory for vibrating plates of variable thickness, she was aware of the complexity and importance of the problem. She had to build especial plates to carry out her own experiments. Her memoir of 1825 begins with a review of the relevant literature, citing papers by Euler, Bernoulli, Lagrange, Chladni, Poisson, Navier, Savart, and Italian physicist Giordano Riccati. It is evident that Germain remained abreast of scientific developments in her area of research; she read the papers presented at the Paris Academy of Sciences, especially those by Poisson and Navier, and she provided her own commentaries about their results.1 Germain needed to solve the governing fourth-order partial differential equation with variable coefficients in w in order to describe the bending of thin plates with variable thickness. As we know, a closed form solution of such an equation is possible only in very special cases. Today’s engineers analyze plates of variable thickness using approximate and numerical methods such as the variational approach (the Ritz method), finite element methods, and the small parameter method. In the course of her research, Sophie conducted experiments to understand the nature of the vibration and elasticity of her plates, trying to reconcile her hypothesis to the sand patterns in Chladni’s plates. In her 1825 paper, Germain outlined the sound experiments, noting how difficult it was to procure plates with a thickness that varied according to the law that was the subject of her research. She expressed gratefulness to a M. Arcet for introducing her to M. Moulfarine, a skilled mechanic who was willing to undertake the work of making the required plates.2 1

Germain (1880). Ibid., p. 47.

2

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 D. Musielak, Sophie Germain, Springer Biographies, https://doi.org/10.1007/978-3-030-38375-6_7

95

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Experiments with Vibrating Plates

It appears that Sophie Germain designed the plates herself and conceived a way to test them. The experiments must have been grueling and expensive. In a letter to Arcet, Germain asked him to find a workman who could give the glass plates the different curvatures that she needed. She added: “I spent 100 francs last year without obtaining anything but shapeless fragments of annealed glass because it became impossible to cut [a plate] with a diamond without breaking it in pieces.”3

Sophie Germain’s Experimental Research There are insufficient details regarding the experimental work that Sophie Germain carried out to prove her mathematical hypotheses. In the following description, I attempt to reconstruct what she did. Germain would have begun with a square plate held by a suitable clamp at its center (Fig. 7.1). She had to sprinkle fine sand over the plate, and then apply some small damping on the middle point of one of its edges, perhaps by touching it with her fingernail. Finally, she had to draw a bow across the edge of the plate, near one of its corners. As the plate vibrated, the sand would move away from certain parts of the plate’s surface, collecting along two nodal lines and dividing the large square into four smaller ones, as in figure A. This division of the plate corresponds to the sound with the deepest tone. Sophie Germain must have observed exactly that, because she wrote: Take for example the nodal figures that belong to the first case of vibration given by the formulas … z ¼ cos

ppx qpy cos ; A A

z ¼ cos

pps qpr cos : A0 A0

If p ¼ 1, q ¼ 1, the figures reduce to the two nodal lines perpendicular and parallel to the edge. On a plate of constant thickness, the two lines intersect at the center of the surface, and each of them ends, along on the same surface equidistant from the two opposite corners.4

After sprinkling sand over the surface again, Germain would dampen one of the corners of the plate and excite it by drawing the bow across the middle of one of its sides. This would cause the sand to dance over the surface and arrange itself into two sharply defined ridges along its diagonals, as shown in Fig. 7.2, sketch B. The note produced here is a fifth above the last. Then dampening two other points on the edge and drawing the bow across the middle of the opposite side of the plate, Germain would obtain a higher-pitched note than in either of the former cases. The mode of plate vibration would look like the one in Fig. 7.2, sketch C.

3

Ibid. Ibid., p. 50.

4

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Fig. 7.1 Making a plate vibrate with a violin bow. Credits see Appendix “Illustration Credits”

Fig. 7.2 Figures drawn by Sophie Germain. Source Bibliothèque Nationale de France

The signs + and − in the figures denote that the two squares on which the nodal lines occur are always moving in opposite directions. When the squares marked + are above the average level of the plate, those marked − are below it; and when those marked − are above the average level, those marked + are below it. The nodal lines mark the boundaries of these opposing motions. They are the places of transition from one motion to the other, and are therefore unaffected by either.

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What an enthralling experience it must have been for Sophie Germain to watch the beautiful patterns of sand forming on the vibrating plate, while listening to the distinct sounds generated by the vibration. She mentioned using plates of glass and might have also used metal plates, which probably would have been easier for her to mount and would be resistant to breakage. From her experiments, Germain concluded: “One can see, § 9, that the position of the two nodal lines is found exactly in line with what the formulas tell us.”5 She also reviewed the acoustic experiments of Félix Savart and Charles Wheatstone, and she tried to incorporate their methodologies. British scientist Wheatstone published in the August 1823 of the Annals of Philosophy an article with the title “New Experiments in Sound,” which was read at the Paris Academy by French engineer Dominique François Jean Arago. Wheatstone believed that more minute vibrations must be present in order to account for the varieties or “timbre” which are found in notes of the same pitch. He then demonstrated the existence of these minute vibrations by covering a sounding plate with water. Sophie Germain followed Wheatstone’s example and covered her own plates with water, emphasizing that “experiments of this kind must be regarded as a test against those of Mr. Chladni.” In September 1823, Germain wrote a long letter to the Institute addressing Wheatstone’s experiments with vibrating metal plates. The letter, likely addressed to Fourier who was her friend and the Perpetual Secretary, is not in the Archives of the Academy of Sciences in Paris. After reading it at the meeting of September 1, someone was supposed to archive it since “Fourier and Arago would take a special knowledge of the subject of this letter, and would report to the Academy.” However, no report was ever made. Sophie Germain continued to work alone in her effort to explain Chladni’s plates, while Poisson and Navier advanced their theories, stimulated by the design and building of bridges and other architectural structures. There is no doubt that Sophie was obstinate and intellectually driven. Inspired by Wheatstone, she extended her research and, in 1825, she submitted a new paper to the Institut de France. The memoir was assigned to a review commission including Poisson, Gaspar de Prony, and Laplace. The commission ignored Sophie’s work, which went unrecognized for fifty-five years; her memoir summarizing her own research concerning plate thickness effects on the theory of elasticity was finally published in 1880.6 A copy of Germain’s September 1823 letter was found in the library of the British Museum, in London, and it was published in 1880 as an appendix to Germain’s memoir.7

5

Ibid., p. 52. Ibid. 7 Ibid., p. 60. 6

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Historian Andrea Del Centina8 discovered in Florence’s Biblioteca Moreniana many handwritten documents of Sophie Germain that include reports of experiments, as well as her remarks on papers by Cauchy and Navier. This material contains further details of Germain’s experimental work. Caught in the dispute for priority between Navier and Poisson, Sophie Germain published in the 1828 issue of Annales de chimie et de physique an essay in which she defended her view on the problem of elastic surfaces.9 Here she recommended her hypothesis, insisting that it was better than the attempt to construct a theory of the action of molecular forces, as Poisson had done. Despite Germain’s research efforts, which must have inspired new ideas in others, even later scholars were not ready to bestow on a woman full credit for her contributions. One of her critics was Isaac Todhunter, an English mathematician who in 1886 examined Germain’s contribution to elasticity.10 Todhunter was not kind in reviewing her work, chastising Germain for her mathematical mistakes. He assessed her analysis under the (incorrect) assumption that she had been a pupil of Lagrange.11 However, Lagrange was not her teacher, nor did he work with her directly on this effort. It is true that she studied Lagrange’s lecture notes and learned mathematics from his memoirs and other books at her disposal. But she did not have the benefit of Lagrange’s direct instruction. On the other hand, for the prize competition, Sophie refined her analysis with help from discussions she had with Legendre by mail. Legendre explained mathematical concepts, pointed out her mistakes, and clarified some important parts in Euler’s memoirs, alerting her to issues directly relevant to her own work. Thus, Legendre must be considered her true mentor and teacher. Todhunter also suggested that Germain had not derived her hypothesis from physical principles, and his opinion was that “the judges must have been far from severe.”12 After evaluating some of her mathematical derivation, Todhunter concluded that it lacked the necessary rigor owing to her deficiency in analysis and the calculus of variations. But then again, neither one of the other competing mathematicians formulated a complete variational principle (Lagrangian), which would have given them a powerful tool for the calculations of stress fields or deformations. In all fairness, Todhunter ended his Chap. 3 with the statement: “Sophie Germain with all her vagaries at least succeeded in finally establishing the equation for the normal vibrations of a plate.” Indeed! And I wonder just how a woman without schooling had the insight to derive her theory independently from the leading mathematicians of her time.

8

Del Centina (2005), p. 14. Germain (1828), pp. 123–131. 10 Todhunter (1886). 11 Ibid. “The lady does not appear to have paid that attention to the Calculus of Variations which might have been expected from the pupil and friend of its great inventor Lagrange”, pp. 156–157. 12 Ibid. 9

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To solve the questions posed by Chladni’s experiments would require the solution of the linear eigenvalue problem associated with the biharmonic equation under homogeneous boundary conditions. These boundary conditions result from the assumption that no forces are applied at the edges of the plate. At the time, those ideas were not fully understood as such. And thus, Sophie Germain had the daunting task of deriving an equation without the modern advances in partial differential equations, eigenvalue methods, and the refinement of the variational calculus that took many great people to develop. Moreover, we infer a striking result when we compare the Germain-Lagrange equation  N

2

 @4w @4w @4w @2w þ 2 þ ¼0 þ @x4 @y2 @x2 @y4 @t2

to the governing equation of free vibrations of an isotropic rectangular plate of constant thickness h, adopting the same variable w for the deflection of the plate,13 Dr2 r2 wðx; y; tÞ þ qh

@2w ðx; y; tÞ ¼ 0; @t2

where q is the mass density of the material; the constant D is the flexural rigidity of the plate, D ¼ Eh3 =12ð1  m2 Þ; and E and m are the modulus of elasticity and Poisson’s ratio, respectively. The latter is the negative ratio of transverse to axial strain as defined by Poisson in 1820. Sophie Germain’s constant N 2 , which accounts for the flexural rigidity of the plate, contained a fourth power factor of the thickness, which is an order of magnitude higher than the flexural rigidity constant of the plate. In her 1824 memoir,14 she discussed how she had assumed that the elasticity constant was proportional to the square of the thickness of the elastic plate, as Euler had done. Then she noted that Navier determined the constant to be proportional to the third power of the thickness, referring to his 1820 memoir (presented to the Academy in August).15 However, she insisted that her analysis yielded to a fourth power, saying: “j’ai reconnu, comme je le dirai plus bas, que ce n’est ni la première ni la seconde, mais plutôt la quatrième puissance de l’épaisseur, qui doit entrer dans la valeur du coefficient constant.” [I recognized, as I say below, that this is neither the first nor the second, but rather the fourth power of the thickness, which must enter in the value of the constant coefficient.] In her preliminary equation, Sophie Germain was missing the twisting moment Mxy ¼ Myx , which is directly proportional to the curvature with respect to the x and @2 w y axes, vxy ¼  @x@y , and which defines the warping of the middle surface at a point 13

Ventsel and Krauthammer (2001), Eq. (9.3). Germain (1880). 15 Navier (1823). 14

Sophie Germain’s Experimental Research

101

with coordinates x and y. This accounts for the term that Lagrange added to amend her equation. The only term missing in her equation was the mass per unit area of the plate element, qh, in the acceleration term, which arises from Newton’s second law. Poisson introduced the ratio m in the governing equation of elasticity. This coefficient involves stresses and strains on materials. Poisson observed that when a sample of material is stretched in one direction it tends to get thinner in the other two directions. He found the ratio of the relative contraction strain, or transverse strain normal to the applied load, to the relative extension strain, or axial strain in the direction of the applied load. The analysis that led to the Germain-Lagrange equation required the establishment of basic assumptions analogous to Kirchhoff’s hypotheses, which in turn are associated with the simple bending theory of beams. These assumptions are needed in order to reduce a three-dimensional plate problem to a two-dimensional one. It also required a deep physical insight to realize that a flat plate develops shear forces, bending and twisting moments to resist transverse loads. Germain thus needed to master a wide range of scientific topics. She required an understanding of Euler’s treatises, most of which were written in Latin. Germain needed a working knowledge of Lagrange’s mechanics and deep acquaintance with various branches of analysis. This in itself would be a great accomplishment, even for someone with a formal education. Sophie Germain undertook research that included analysis and performing diverse experiments with vibrating plates, attempting to duplicate those of Chladni. She performed similar experimental work as some of her contemporaries, many of whom were educated as engineers. An engineer applies scientific knowledge, mathematics, and ingenuity to develop solutions for technical problems. The term engineer derives from the Latin roots ingeniare (to contrive, devise) and ingenium (cleverness). Moreover, engineers are grounded in applied sciences, and those with doctoral degrees or who have more advanced knowledge of mathematics usually work in research and development endeavors. While conducting her experiments with vibrating plates, Germain was clearly engaged in research, design, and experimentation, establishing a link between the applied science and her mathematical theories. If not engineer, she met the definition of a mathematical physicist. There were numerous, unsuccessful attempts by others to reconcile their mathematical analysis to Chladni’s experiments with the vibrating plates. The problem proved to be obstinate and rather challenging. Poisson, Navier, Fourier, and Cauchy continued research in this area for many years, pursuing the full mathematical theory. And now, two hundred years after Sophie Germain developed the first equation for the motion of two-dimensional vibrating surfaces, engineers and scientists continue research on vibrating plates, developing models to predict the structural performance of large structures. In fact, analytical and numerical methods for solving linear and nonlinear plate and shell problems are an integral part of the education of aerospace, civil, and mechanical engineers, because thin-walled structures (plates and shells) are encountered in those branches of engineering. Thin

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plate and shell structures, when correctly designed, can support large loads. They are typically utilized in aerospace structures such as aircraft and spacecraft, in which lightweight is critical requirement. The elastic properties and the vibration characteristics of these systems are very important for their stability. And thus, because of her (unexpected!) contribution to engineering science, we may be justified in considering Sophie Germain a mathematical physicist, and even the first woman research engineer.

Chapter 8

Elasticity Theory After Germain

…je me suis occupée, à diverses reprises, delà théorie des surfaces élastiques. J’ai multiplié les expériences, les calculs et les réflexions. J’avouerai que j’ai toujours cru voir de nouveaux motifs pour tenir à mon opinion. —SOPHIE GERMAIN

The mathematical theory of elastic vibrating plates originated in 1811, when Sophie Germain first developed the first valid hypothesis. This led to the first fourth-order partial differential equation now known as the Germain-Lagrange equation. Much more work had yet to be done, of course. The governing equation for the equilibrium of thin elastic plates was derived from that basic formulation. Following that work, research intensified, yielding the basic theories of structural engineering and that ultimately led to understanding Chladni’s vibrating plates. Three Frenchmen in particular emerged in this scientific story, trailing in the wake of Sophie Germain’s winning memoir. Let us highlight some of the work that Germain’s contemporaries did after her.

Navier’s Bending Equation Four years after Sophie Germain won the grand prize, Claude-Louis Navier introduced the general equations of equilibrium and motion that must hold at every point of the interior of a body, as well as those that must hold at every point of its surface.1 Navier, a bright engineer educated at the École Polytechnique, presented on 21 August 1820 a memoir in which he treated the flexure of elastic plates

1

Todhunter (1886), p. 133.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 D. Musielak, Sophie Germain, Springer Biographies, https://doi.org/10.1007/978-3-030-38375-6_8

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(Flexion des plans elastiques).2 In this paper, Navier gave the plate thickness in the general plate equation as a function of flexural rigidity D. Navier’s bending equation for thin plates in static equilibrium gives the deflection wðx; yÞ as @4w @4w @4w p þ 2 þ ¼ @x4 @x2 @y2 @y4 D where p is a vertical distributed load of intensity pðx; yÞ applied to the upper surface, and D is the flexural rigidity of the plate. This equation is considered the first satisfactory theory of the bending of plates,3 the same theory that Jacques II Bernoulli failed to derive in 1788. In his 1820 paper, Navier introduced a method of solution, which transformed the above differential biharmonic equation (plate equation of static equilibrium) into algebraic expressions by use of Fourier trigonometric series.4 This is known as Navier’s solution method or double series solution, and it converges sufficiently rapidly for calculating the deflections. However, it was found years later that Navier’s basic method is unsuitable for calculating the bending moments and stresses because the series for the second derivatives obtained by differentiating the series converge extremely slowly. In fact, these series for bending moment—and thus, for stresses and for the shear forces— diverge directly at the singular point, i.e., the point of application of a concentrated force. In 1826, Navier established the elastic modulus as a property of materials.

Cauchy and His Mathematical Formalism The basis of the mathematical theory of elasticity was set forth by the work of Navier. But it was actually Cauchy who, using those results, established the fundamental equations of elastic bodies. That is, Cauchy derived the equations of static and dynamic equilibrium and the relations between displacements and strains, which are valid for any deformable solid bodies in the form still used today. Augustin-Louis Cauchy was one of the commissioners appointed to review Navier’s memoir on elasticity. He made an important contribution to the developing field with his work on the equilibrium of rods and elastic membranes and on waves in elastic media. Cauchy’s research required that he develop entirely new mathematical techniques such as the Fourier transforms.

2

On 4 September 1820, de Prony, Fourier, and Girar gave an extensive verbal report on Navier’s work and adopted his conclusions, Procès-verbaux. Tome VII, pp. 84–88. 3 Ventsel and Krauthammer (2001). 4 Navier (1823).

Cauchy and His Mathematical Formalism

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During the meeting of the Academy on 30 September 1822, Cauchy proposed a complete elementary theory of elasticity including six stress components.5 Presumably, he already had obtained the general equations for the internal equilibrium of a solid body. In his general linear theory of elasticity,6 Cauchy used stress-strain models to analyze the behaviour of various kinds of surfaces and solids.7 Expanding all the characteristic quantities into series in powers of distance from a middle surface, Cauchy retained only terms of the first order of smallness. In this way, he obtained the governing differential equation for deflections, which coincides completely with the Germain–Lagrange equation. The continuum hypothesis stemming from Germain’s work on vibrating plates, which Cauchy formalized in 1823, led to the system of stress–strain Cauchy relationships, now considered “the fundamental mathematical apparatus of elasticity theory.”8 After publishing her second paper in 1826, in which she attempted to present a clarified view of her analysis, Sophie Germain sent a copy to the Academy. Cauchy was appointed to make a verbal report on it.9 She wrote a long letter to Cauchy on 18 July 1826, to expand on her ideas and also to ask pointed questions. Germain thought that the experiments performed by Félix Savart were inadequate to derive the conditions governing the motion of solid elastic bodies. She believed that it would require a more extensive mathematical theory, such as the one Cauchy was developing. In fact, Cauchy published Savart’s results,10 and used them to verify his mathematical theory. In the same letter, Sophie Germain requested a copy of his publication. She had begged M. Ampère11 to ask Cauchy where she could find it, but Ampère “did not do it.” She was frustrated. Her words suggest that she believed that her ideas related to the motion of a surface were relevant, suggesting to Cauchy that he “take up this area of research”12 so that her own work would take on a real importance. Some historians have speculated that Cauchy encouraged Germain to self-publish her work merely to relieve the Academy of the embarrassment of having to deal with the memoir she sent for review. This yields a rather unflattering portrait of the academicians but one has doubts. Because, based on her modus operandi, it is more likely that when Germain learned about Cauchy’s analytical work, she wrote to him, expecting that he would respond

5

Cauchy (1823), pp. 9–13. Cauchy (1828), p. 328. 7 The Princeton Companion to Mathematics (2008), p. 758. 8 Greaves (2012). 9 Bucciarelli and Dworsky (1980), pp. 106–107. 10 Todhunter (1886), p. 356. 11 André-Marie Ampère (1775–1836), physicist and mathematician, a professor of mathematics at the École Polytechnique since 1809. In 1824, he was elected chair in experimental physics at the Collège de France. 12 Bucciarelli and Dworsky (1980), p. 106. 6

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to express his ideas and compare them with hers. But that did not happen. Although Cauchy was polite and answered her letter, instead of engaging on a scientific discussion, he just referred Germain to his own paper (published in 1823). Cauchy didn’t say what he thought of her work, and there are no letters to indicate whether he ever addressed her extensive scientific inquiry. However, Germain had perceived the importance of Cauchy’s work, the breadth of which was analyzed by Isaac Todhunter13 in 1886. In his historical account of the development of elasticity, Todhunter dedicated a full chapter to Cauchy’s mathematical theory.14 He demonstrated that, for the case of an elastic plate in a state of motion, Cauchy’s formulation yields the Germain-Lagrange equation, the foundation for the research that won Sophie Germain the prize of mathematics in 1816. I see this as the validation she sought.

Poisson and an Incorrect Prediction Before his controversial Théorie mathématique des surfaces élastiques (1814), Poisson had published his famous book, Traité de Mécanique (1811)15 where he addressed the mechanics of materials. The two-volume Mécanique became the standard text; it provided a clear treatment of mechanics based on his course notes at the École Polytechnique. In this work, Poisson described the way materials react to external forces. He defined the ratio of the size change of a material in the direction perpendicular to the applied force versus the expanded length in the direction of the force. Poisson found that the larger the ratio, the larger the effect: rubber, for example, has a larger ratio than concrete. This ratio is now known as Poisson’s ratio. In March 1823, Poisson presented an essay on motion propagation in elastic fluids (Propagation du mouvement dans les fluides élastiques). As we noted before, in March 1824 Poisson was assigned to examine Sophie Germain’s essay on the effects of elastic plates thickness (Effets dus à l’épaisseur plus ou moins grande des plaques élastiques). In 1827, Poisson published a short note addressing the elasticity of brass, prompted by the experiments conducted by French Engineer Cagniar de la Tour. The note,16 published in the Annals de chimie et de physique, started as follows: Let a be the length of an elastic string which has everywhere the same thickness; let b the area of the normal section to its length, and therefore ab is its volume. Suppose we make it undergo a small extension, so that its length becomes að1 þ aÞ, where a is a very small fraction; at the same time the string will narrow: and if we denote by bð1  bÞ what will

13

Todhunter (1886). Ibid., Chapter V, pp. 319–376. 15 Poisson (1811). 16 Poisson (1827), pp. 384–387. 14

Poisson and an Incorrect Prediction

107

become of the section of the normal section, b being also a very small fraction, its new volume will be very nearly abð1 þ a  bÞ. Now, according to the theory of elastic bodies that I will expose in a future Memoir, one must have: 1 b ¼ a; 2 from which it follows that by the extension of an elastic string, its volume is increased, according to the ratio of 1 þ 12 a to unity, and its density diminished according to the inverse ratio. This result is in perfect accord with an experiment recently given by M. Cagniard-Latour to the Academy, of which here is the description.

In this note, Poisson proposes that if a is the equilibrium length of a rod and b its cross-sectional area, then if a increases to að1 þ aÞ, and b decreases to bð1  bÞ under tensile stress, the volume increases to abð1 þ a  bÞ. He says that he has deduced from the theory of molecular interactions that, for a solid body composed of molecules simply held together by central forces on a crystalline lattice, b ¼ 12 a. Poisson clearly regarded the value of b=a ¼ 12 as a constant. Poisson then went on to describe the experiment of M. Cagniard-Latour [Cagniard de la Tour], since it agreed with his theory. Poisson assumed b ¼ 12 a from his theory of elasticity, but he didn’t realize that in the case of rupture of a bar, the limits of elasticity have been exceeded. In fact, the section of the bar does not uniformly diminish but it reaches a condition of plasticity before rupture. In 2012, G. N. Greaves explored Poisson’s ratio, starting with the controversy concerning its magnitude and uniqueness in the context of the molecular and continuum hypotheses competing in the development of elasticity theory in the nineteenth century.17 He then gives its place in the development of materials science and engineering in the twentieth century, and concludes with its recent re-emergence as a universal metric for the mechanical performance of materials on any length scale.

Poisson-Germain-Navier Public Dispute At the regular meeting of the Academy on 21 April 1828, Poisson read his essay on the equilibrium and movement of elastic bodies.18 But Poisson did not mention the contributions of Navier, Cauchy, or of Sophie Germain, which sparked a public and bitter debate. Navier in particular—perhaps trying to rebuild his reputation after the bridge he designed collapsed in 1826—was quite assertive in his fight with Poisson for scientific precedence.

17

Greaves (2012). Poisson (1829), p. 357.

18

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After Poisson read his memoir, Navier charged that the results were the same he had published years earlier.19 Indeed, Navier had developed the physical principle specific to the problem and had established the differential equations containing the laws of equilibrium and the internal movement of molecules of elastic bodies. Navier asked the members of the Academy to recognize the priority of his results and add them to the minutes of the meeting. Cauchy also renewed his claim voiced earlier. After making various remarks about his results, Poisson indicated that citations to Navier’s work would be reflected in the memoir to be delivered to the printer. Was Sophie Germain privy to this discussion at the Academy? Shortly after, she was drawn into the vortex of a scientific dispute unleashed by Navier. In Volume 38 of the Annales de Chimie et de Physique of the same year, Germain published a short essay stating her views on the laws of equilibrium and movement of elastic bodies. She began her review by establishing her own priority, quoting important points she made in her two memoirs published in 1821 and 1826. She then referred to Poisson, again without mentioning his name. “Seulement, après avoir rapporté le passage suivant, qui exprime la pensée de l’auteur dont je combattais l’hypothèse: « Or, quelle que soit la cause de cette qualité de la matière (l’élasticité), elle consiste en une tendance des corps à se repousser mutuellement. … » , j’ai cru pouvoir ajouter que la qualité de la matière dont il s’agit dans ce passage me semblait être plutôt l’expansibilité que l’élasticité elle telle que nous l’observons dans les corps solides.” [But, having reported the following passage, which expresses the thought of the author whose hypothesis I was fighting: “However, whatever the cause of this quality of matter (the elasticity), it consists of a tendency of the bodies to repel each other. …”, I thought I could add that the quality of the matter in question in this passage seemed to me to be due to expansion (expansibilité) rather than to elasticity, as we observe it in solid bodies.]20

The beautiful conclusion of her essay is rather philosophical: “Nous venons de montrer comment les notions hypothétiques ne définissent pas la question, mais la dénaturent. Ne nous en étonnons pas; la langue des calculs ne saurait ajouter à la certitude des idées qui lui sont confiées. S’il est dans la destinée des vues conjecturales de présenter sans cesse de nouveaux aspects, on peut encore dire que leur existence précaire et fugitive semble contraster avec l’appareil imposant des formules analytiques; tandis qu’au contraire la vérité inaltérable d’un fait bien constaté est en harmonie avec le caractère des sciences mathématiques.” [We have just shown how the hypothetical notions do not define the question, but distort it. Do not be surprised; the language of calculus cannot add to the certainty of the ideas entrusted to it. If it is in the destiny of conjectural views to constantly present new aspects, we can still say that their precarious and fugitive existence seems to contrast with the imposing apparatus of analytic formulas; while, on the contrary, the unalterable truth of a well-established fact is in harmony with the character of the mathematical sciences.]

19

Institut de France. Procès-verbaux. Tome IX, p. 55. Germain (1828).

20

Poisson-Germain-Navier Public Dispute

109

In the same volume of the Annales where Germain’s essay was published, Navier wrote a bitter note chastising Poisson.21 He stressed that, although Poisson mentioned the work of the anciens géomètres in his memoir of April 1828, he had not properly acknowledged Navier’s research findings, which were presented to the Academy in 1821 and published in 1822 and 1827. Navier summarized his main results to assert his contributions and precedence over Poisson. Halfway through his article, Navier referred to the fundamental equation that Lagrange had derived from the principle conceived by Sophie Germain. This is how Navier stated this fact: Indeed, these investigations have, in truth, given an equation whose form did not differ, as regards to the terms that depend on the resistance of the plane [surface] to the flexion, from another equation found by Lagrange, according to a principle conceived by mademoiselle Sophie Germain. This principle, ingenious and true, consists in looking at the forces which resist flexion as being, for each point of the plane, proportional to the sum of the inverse values of the two principal radii of curvature: it remains only (which is not difficult) to show how this geometric notion results from the physical nature of molecular actions.22

In the next volume of the same journal, Navier included another statement addressing the article by Poisson. One can imagine what followed. Poisson responded to the comments made by Navier by writing a letter to Dominique Arago, who was one of the editors of the Annales. In this letter, the arrogant Poisson defended his actions and wrote that “researchers can now compare my thesis to that undertaken on the same subject; and if this comparison gives rise to some useful observations, then I will be glad for the developments that may result from what I proposed in this work …” Arago should have helped clarify all claims, but he did not. This public dispute went on for several months. In the Bulletin universel des sciences of 1829, Navier summarized the debate, concluding with some remarks that also helped to establish Sophie Germain’s contributions. In the July 1828 issue of the Annales of chimie, Navier had stated that Lagrange found an equation for the laws of bending and vibration of elastic plates, according to a principle devised by Sophie Germain. In August, Poisson denied that Lagrange would have inferred from “such assumption the equation for the vibrations of elastic plates.” In October Navier replied, citing the extracted note that Lagrange wrote, stating how he derived the equation by adopting Sophie Germain’s hypothesis. To this, Poisson replied that “it is true that Lagrange derived the equation for the vibrations of elastic plates” but he implied that it had been done prior to the 2nd edition of the Mécanique analytique. In January 1829, Navier clarified that Lagrange’s note was dated December 1811, and the first volume of the second edition of the Mécanique had appeared in September 1811. Also, a supplement to the manuscript had been printed long after

Ibid., pp. 304–314. Note relative à d’article intitulé: Mémoire sur l’équilibre et le mouvement des Corps élastiques, page 337 du tome précédent. 22 Ibid., p. 309. 21

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the death of Lagrange. Therefore, nobody could deny that Lagrange had derived the equation in question after his book was published. This also confirmed that the equation arose from Sophie Germain’s original work. Navier was exasperated with Arago. From the note it seems that Arago had taken the stance that Poisson had acknowledged the error, but he was not eager to report the omission, because he waited until January 1829 to publish the clarification. Ignoring Lagrange and Germain altogether, Poisson denied Navier’s claims of priority on the grounds that “a result derived by false reasoning is not derived at all.” Moreover, Poisson contended that, despite the correctness of the final results, Navier’s analysis was wholly faulty.23 Poisson published his work in 1829 without adding any references to the contributions of Navier, Cauchy, or Germain. His Mémoire sur l’équilibre et le mouvement des corps élastiques included a five-page addition to complete equilibrium and motion analysis of elastic bodies.24 Contemporary researchers consider Navier the founder of modern structural analysis. At the meeting of the Academy on 12 October 1829, Poisson read an important memoir presenting the general equations for equilibrium and motion of elastic solids and fluids (Équations générales de l’équilibre et du mouvement des corps solides élastiques et des fluides).25 In this treatise, Poisson expanded the Germain–Lagrange plate equation to the solution of a plate under static loading. In this solution, the plate flexural rigidity D was set equal to a constant term.26 However, Poisson suggested setting up three boundary conditions for any point on a free boundary. Boundary conditions are the known conditions on the surfaces of the plate which must be prescribed in advance in order to obtain the solution of the governing equation. At that time, this was a subject of controversy that none of the leading French scientists could resolve. The next outstanding breakthrough in the theory of plates occurred thirty-seven years after Sophie Germain had conceived her equation for the vibrations of elastic plates, and only then were the required boundary conditions formally defined. Of course, now we know that for a thin plate in equilibrium, two boundary conditions must be prescribed, i.e., the deflection w must vanish at the edge, and the tangent plane at every point of the edge must remain fixed when the plate is bent. But this was yet to be conceived.

Kirchhoff’s Plate Theory In 1850, the German Prussian physicist Gustav Robert Kirchhoff published his famous paper on the equilibrium and motion of an elastic disk.27 He was twenty-six. Using variational calculus, Kirchhoff solved some problems that 23

Truesdell (1953), p. 456. Poisson (1829). 25 Institut de France. Procès-verbaux. Tome IX, p. 328. 26 Ventsel and Krauthammer (2001). 27 Kirchhoff (1850), pp. 51–88. 24

Kirchhoff’s Plate Theory

111

remained from the analysis carried out by Germain, Poisson, Navier, and Cauchy. He derived with scientific strictness the plate theory named after him. Kirchhoff began his research using Sophie Germain’s hypothesis as a reference (Fig. 8.1). He obtained a differential equation for plates identical to that given by Poisson in 1829, apart from the factor for Poisson’s ratio. Germain had arrived at the correct differential equation via two errors which, to a certain extent, cancel

Fig. 8.1 First page of Kirchhoff’s 1850 paper

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each other out: on the one hand the product of curvature in x- and y-directions (Gaussian curvature) was missing in her hypothesis, and on the other, her plate analysis resulted in incorrect boundary conditions.28 In deriving his plate theory, Kirchhoff considered the Gaussian curvature and reduced Poisson’s three boundary conditions to two. He noted that Poisson had reached the same partial differential equation as Sophie Germain’s hypothesis, but using three different boundary conditions. Kirchhoff asserted: “I will prove that generally these cannot be satisfied simultaneously. From this it follows that also according to Poisson’s theory a plate subjected to out-of-plane loading need not generally have an equilibrium position. However, I shall present the proof only after I have derived the two boundary conditions to be used instead of Poisson’s three because the proof naturally follows from the considerations from which I wish to derive the boundary conditions.”29 The theory that Kirchhoff developed began with two independent basic assumptions that are now widely accepted in the plate-bending theory and are known as ‘‘Kirchhoff’s hypotheses.’’ These basic assumptions provide needed insight into the bending theory, starting with the notion that the plate is made of material that is elastic, homogeneous, and isotropic. The second hypothesis is that the plate is initially flat, and that the smallest lateral dimension of the plate is at least ten times larger than its thickness. Using these assumptions, Kirchhoff simplified the energy functional of elasticity theory for bent plates. By requiring that the plate be stationary, he obtained the Germain-Lagrange equation as the Euler equation. He also concluded that only two boundary conditions apply on the edge of the plate. Kirchhoff used the integral theorems of Gauss and Green, which were not available to Sophie Germain. He started from the usual assumptions of the theory, deduced the expression for the potential of the deflected plate, and found by the calculus of variations not only the differential equation but also the correct boundary conditions. He obtained a double integral from that analysis, which resulted in the differential equation for an elastic plate (biharmonic equation) plus a curvilinear integral along the contour line of the midsurface (boundary curve of elastic plate), from which conclusions are drawn via the boundary conditions.30 Kirchhoff found that, at the free edge, the normal stress is equal to zero, and that the shear stresses must be replaced by a single relation (for an element of the boundary perpendicular to the x-axis). If the plate is assumed to be clamped on all 31 sides, the boundary conditions would be w = 0, and @w @n ¼ 0 on the edge. In 1883,

28

Kurrer and Ramm (2012), p. 533. Translation by Karl-Eugen Kurrer (2012) from Kirchhoff’s paper, p. 54. 30 Kurrer and Ramm (2012), p. 534. 31 The first boundary condition of a Dirichlet boundary condition, named after Johann Peter Gustav Lejeune Dirichlet; the normal derivative is known as the Neumann boundary condition after Carl Gottfried Neumann (1832–1925). 29

Kirchhoff’s Plate Theory

113

Lord Kelvin and Tait32 provided additional understanding concerning the boundary equations by converting twisting moments along the edge of a plate into shearing forces. Thus, it was determined that the edges of the plate are subject to only two forces: shear and moment. Kirchhoff solved the Chladni’s plate problem for the special case of a circular plate, which, due to symmetry, was much easier to handle.33 But for other configurations, the partial differential eigenvalue problem with the free boundary conditions was still too difficult to solve. Nonetheless, Kirchhoff also discovered the frequency equation of plates, and he introduced virtual displacement methods in the solution of plate problems. With his bending plate theory, Kirchhoff contributed to the physical clarity of the theory of elasticity of the nineteenth century. Moreover, since the calculation of bending moments and shear forces using Navier’s approach is not acceptable (due of slow convergence of the series), in 1899, M. Levy developed a better method for solving rectangular plate bending problems using single Fourier series.34 To perform numerical calculations, Levy’s method is simpler to implement, and it is applicable to plates with various types of boundary conditions.

Ritz Method to Model Chladni’s Plates A century after Sophie Germain began her research, the deformation of an elastic plate under an external force remained as a very difficult problem. The French Academy of Sciences announced the Prix Vaillant for 1907 requiring participants to perfect the Germain-Lagrange equation. Once again, the French Academy of Sciences challenged mathematicians to solve the problem presented by the equilibrium of the built-in (encastrée) elastic plates, almost one hundred years after the first contest. Swiss physicist Walter Ritz had worked with many such problems in his doctoral thesis, so this contest sparked his interest. He submitted a memoir, anticipating that he had a good chance to win this competition. Ritz was a classical physicist best known for his work in electrodynamics of moving bodies.35 To develop a mathematical model for Chladni figures on a vibrating plate, Ritz modified Kirchhoff’s equations, as Chladni figures on a square plate correspond to eigenpairs (eigenvalues and corresponding eigenfunctions) of the bi-harmonic operator. Thus, Ritz developed a method used for the computation of approximate solutions of operator eigenvalue equations and partial differential equations.

32

Thomson and Tait (1883). Gander and Wanner (2011). 34 Levy (1899), p. 219. 35 Forman (1975). 33

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In 1909, Ritz presented his method to compute Chladni’s figures: instead of trying to solve the partial differential eigenvalue problem directly, he proposed to use the principle of energy minimization, from which he derived the correct equations.36 Ritz considered a special functional,37 and from this minimization problem he could obtain again the partial differential eigenvalue problem. Even though Ritz explained his new method on the concrete example of Chladni’s figures on a square plate, he pointed out that it is completely general and could be applied to plates of arbitrary shapes.38

Prix Vaillant (4000 fr.): Perfectionner en un point important le problème d’Analyse relatif à l’équilibre des plaques élastiques encastrées, c’est-à-dire le problème de l’intégration de l’équation 

 @4u @4u @4u þ 2 2 2 þ 4 ¼ f ðx; yÞ 4 @x @y @x @y

avec les conditions que la fonction u et sa dérivée suivant la normale au contour de la plaque soient nulles. Examiner plus spécialement le cas d’un contour rectangulaire. Les Mémoires devront être envoyés au Secrétariat avant le 1er janvier 1907. The announcement of the French contest, as it appeared in Journal für Mathematik und Physik.

Ritz’s method is based on a linear expansion of the solution and determines the expansion coefficients by a variational procedure, which is why this approach is known as linear variation method. This mathematical approach enabled him later to solve difficult problems in applied sciences. Walter Ritz thereby revolutionized the variational calculus and became one of the fathers of modern computational mathematics.39

36

Gander and Wanner (2011), p. 18. Ritz (1909), pp. 737–786. 38 Gander and Wanner (2011), p. 19. 39 Ibid. 37

Chapter 9

Germain and Fermat’s Last Theorem

… le géomètre porte une attention soutenue vers l’idée heureuse qui dirige ses recherches. Toutes les forces de son intelligence seront employées à dérouler la chaîne des vérités contenues dans cette vérité première … [“… the mathematician pays close attention to the happy idea that directs his research. All the forces of his intelligence will be employed in unfolding the chain of truths contained in this first truth …]”. —SOPHIE GERMAIN

On the same day Sophie Germain won the prize for her mathematical work on vibrating plates, the commissioners of the Academy of Sciences announced the (last) theorem of Fermat as the topic for the 1818 contest of mathematics. A happy coincidence? Germain kept her feelings secret, or at least there are no written words to shed light on her intellectual delight, as the new competition would carry her back to her first love—number theory. It would give her a fresh impetus to pursue once again the proof of the célèbre équation de Fermat, as she called it, an effort she had begun years earlier. The topic must have been chosen by Legendre. Two weeks before, he had reported on Cauchy’s complete proof of the theorem of Fermat on polygonal numbers. And then there was one remaining theorem of Fermat yet to be proven. To pay homage to “one of the scholars who had most honored France, and at the same time to provide mathematicians with the opportunity to develop this part of Science,” on 26 December 1815 the academicians issued for the prize of mathematics the general proof of Fermat’s Last Theorem. The prize was set as a gold medal valued at three thousand francs, which would be awarded in January 1818. Did Sophie Germain decide at that moment to enter the competition? If so, what was her plan? In her first letter to Gauss in 1804, Germain stated that she could prove that xn þ yn ¼ zn is impossible if n ¼ p  1; where p is a prime of the form 8k þ 7: How would that approach lead her to a general proof for all n? Now, eleven years later, would she follow the same strategy?

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 D. Musielak, Sophie Germain, Springer Biographies, https://doi.org/10.1007/978-3-030-38375-6_9

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In order to better appreciate Sophie Germain’s work, we first need to understand the background against which she worked. Let us begin with Fermat, the amateur mathematician who left behind some fascinating mysteries and inspired a young woman to prove something he said, just as he had inspired Euler.

Pierre de Fermat After Euclid and Diophantus, little or no progress was made in the development of number theory. Then, in the seventeenth century, Pierre de Fermat rediscovered it and brought it to the forefront of research. Fermat, a contemporary of René Descartes and his rival in mathematical ability, was a lawyer and government official1 who pursued mathematics in his spare time. Pierre de Fermat was born on 17 August 1601 in Beaumont-de-Lomagne in southern France, a town located about 60 km from Toulouse. Fermat (Fig. 9.1) was a man of great erudition with a passion for mathematics. He made contributions to analytic geometry and to infinitesimal analysis, developing a method to solve problems of maxima and minima. Fermat’s name, however, became legendary in number theory, for his contributions and for the problems he posed for others to solve. In 1621, French mathematician Claude Gaspard Bachet de Méziriac translated from the original Greek the available books of Diophantus’s Arithmetica. It was the first time this work was available in Latin, the language of scholarship. Bachet was the first European author to discuss the resolution of indeterminate equations by continuous fractions. He also worked on number theory and discovered a method to build magic squares (carrés magiques) for recreational mathematics. Bachet made extensive commentaries on the problems of Diophantus, many that had appeared in Bombelli’s Algebra. The Arithmetica became the foundation for Fermat’s research in number theory. He studied perfect and amicable numbers, Pythagorean triads, divisibility, and he was especially drawn to problems involving prime numbers. Fascinated by these concepts, Fermat stated many theorems, some of which he proved by a method he invented called “infinite descent”—a type of proof by contradiction which relies on the fact that the natural numbers are well ordered and that there are only a finite number of them that are smaller than any given one. Infinite descent proves that certain properties or relations are impossible for whole numbers by proving that if they held for any numbers they would hold for some smaller numbers; then, by the same argument, they would hold for some numbers that were smaller still, and so forth ad infinitum, which is impossible because a sequence of positive whole numbers cannot decrease indefinitely.2

1

www-history.mcs.st-andrews.ac.uk/Biographies/Fermat.html. Edwards (1977), p. 8.

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Fig. 9.1 Pierre de Fermat. Portrait engraved by François Poilly (1623–1693). Credits see Appendix “Illustration Credits”

Fermat communicated his mathematical research in letters to men of learning such as Pierre de Carcavy,3 Bernard Frenicle de Bessy,4 and Marin Mersenne, sharing some results and conjecturing many others, but in those letters Fermat provided little or no proof of his assertions. Carcavy and Frenicle were two French amateur mathematicians who are mainly known for their correspondence with Fermat. Mersenne was a Jesuit monk mathematician who promoted scientific ideas, mainly through correspondence with Fermat and other scholars of his time. Diophantus posed this question in problem III, 19 of his Arithmetica: which primes are sums of two squares? In 1660, Fermat responded: those of the form 4n þ 1. However, he did not explain why, and it is not known if he had a proof. Fermat also stated that primes of the form 4n  1 cannot be written as the sum of two squares in any manner whatever. Thus, for example, 5 and 13, since 5 ¼ 12 þ 22 and 13 ¼ 22 þ 32 . On the other hand, 3 and 11 cannot be so decomposed. This was a remarkable insight about the property of prime numbers, one that required a proof but that Fermat did not provide. Similarly, in 1638 Fermat asserted that every whole number can be expressed as the sum of four or fewer squares, claiming to have a proof but, if he did, he did not share it with anyone. That was typical of him. He was very secretive about his

3

Pierre de Carcavy (1600/1603–1684). Mathematician, secretary of the royal library of Louis XIV. Bernard Frénicle de Bessy died in Paris in 1674.

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mathematical research. A letter he wrote to Frenicle in 1640 illustrates the cryptic style Fermat used to communicate his work. Claiming that all integers of the form n 22 þ 1 are prime, Fermat wrote: … I do not have the exact proof, but I have excluded such a large number of divisors by infallible proofs, and I have such a strong insight, which is the foundation for my thought, that it would be difficult for me to retract it.5 n

Convinced that all numbers of the form 22 þ 1 are prime, Fermat verified this up to n = 4 (or 216 þ 1). But in 1732, Euler proved him wrong. In October 1640, Fermat wrote to Frenicle to communicate this proposition: “Every prime number evenly divides one of the powers minus one of any progression in which the exponent of the given power is a factor of the given prime number minus one; and after one has found the first power which satisfies this property, all those numbers having exponents that are multiples of the exponent of the first satisfy all of the same properties.”6 In mathematical notation, letting p denote a prime number, the formula ap1  1 can always be divided by p, unless a can be divided by p. Fermat asserted that if p is prime and a is any whole number, then p divides evenly into ap  a. Thus, if p ¼ 7 and a ¼ 12, one can be sure that 7 is a divisor of 35; 831; 796. Fermat did not prove his assertion, saying simply that “this proposition is generally true for all progressions and for all prime numbers; the proof of which I would send to you, if I were not afraid to be too long.” This is known today as “Fermat’s Little Theorem,” and Euler proved it ninety-four years later.7 Fermat must have been struck by the sheer simplicity of Pythagoras’s theorem and studied Pythagorean triples, i.e., a triple of positive integers x, y, and z such that a right triangle exists with sides x; y and hypotenuse z. By the Pythagorean theorem, this is equivalent to finding positive integers x, y, and z satisfying x2 þ y2 ¼ z2 : As we learn in elementary school, the smallest Pythagorean triple is ðx; y; zÞ ¼ ð3; 4; 5Þ. Fermat considered a variant of Pythagoras’s equation: x3 þ y 3 ¼ z 3 ; and he asserted that this equation did not have non-zero whole number solutions but gave no proof. On the other hand, Fermat did prove8 that the equations x4  y4 ¼ z2 and x4 þ y4 ¼ z4 have no solutions in integers all different from 0. After Fermat died, his son Samuel discovered that he had made extensive notes on the margins of Arithmetica, his favorite book. In those scribbled notes (written in

5

Letter from Fermat to Frenicle dated August 1640. Translated by Amanda Bergeron and David Zhao. Available at The Euler Archive. 6 Ibid. 7 Fermat’s little theorem is the basis for a primality test, a check to determine if a number is prime or composite. 8 Ribenboim (1979), p. 2.

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Latin), Fermat left behind a testament to his brilliant thinking and the seeds of his significant contributions to number theory. Appreciating the importance of his father’s handwritten records, Samuel de Fermat published them in 1670 as Observations on Diophantus, including his father’s correspondence and other manuscripts.9 On the margin of page 51, Book 2 of Diophantus’s Arithmetica, Fermat wrote this enigmatic note next to problem 8: It is impossible to divide a cube into two cubes, or a fourth power into two fourth powers, or in general any power greater than the second into powers of the same degree: of this I have discovered a very wonderful demonstration [demonstrationem mirabilem sane detexi]. This margin is too narrow to contain it.

What Fermat meant is that if n is a positive integer n [ 2, the equation zn ¼ x þ yn has no non-zero integer solutions for x, y and z, i.e., no three integers x, y and z exist, such that xyz 6¼ 0, which satisfy the equation. Fermat’s declaration was rather intriguing. It was known as a theorem because Fermat claimed to have proved it. However, no proof was ever found. n

Euler and Fermat’s Theorems Early in the eighteenth century, number theory was still regarded as a minor subject, largely of recreational interest. Mathematicians were more interested in applying the new infinitesimal calculus invented by Newton and Leibniz to solve a variety of problems in physics and astronomy. After Fermat, Leonhard Euler gave luster and depth to number theory, ushering it into the mainstream of mathematics. Euler was the most prolific mathematician ever—and one of the most influential—and when he turned his attention to number theory, the subject attained a higher status. Combining pure genius, a superhuman ability to perform complex mathematical analysis and computations in his head, and enormous intellectual sensibility, Euler was precisely the right mathematician to deal with the unproven assertions of Fermat. It was Christian Goldbach who (in a letter dated December 1729) introduced the twenty-two-year old Euler to the work of Fermat. In the postscript of that letter, Goldbach wrote: x1

P.S. Notane Tibi est Fermatii observatio omnes numeros hujus formulae 22 þ 1 nempe 3, 5, 17, etc. esse primos, quam tamen ipse fatebatur sc demonstrare non posse, et post eum nemo, quod sciam, demonstravit. [P.S. Note Fermat’s observation that all numbers of this x1 form 22 þ 1, that is 3, 5, 17, etc. are primes, but he confessed that he did not have a proof, and after him, no one, to my knowledge, has proved it.]10

9

Oeuvres de Fermat. Bibliothèque Nationale de France. http://gallica.bnf.fr. Fuss (1843), Letter II dated 1 December 1729, p. 10.

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The terse postscript from his mentor sparked the young man’s interest. Euler responded immediately that he could not find a proof either, Nihil prorsus invenire potui, quod ad Fermatianam observationem spectaret. But Euler was not totally convinced.11 Two years later, he disproved Fermat’s assertion. At twenty, Euler had joined the Russian Academy of Sciences in St. Petersburg. The young mathematician was under pressure to establish his career, fulfilling the many and varied requirements of his position, working in many different areas of applied mathematics, physics, geography, and astronomy. Christian Goldbach was historian and professor of mathematics. As the secretary of the Academy from December 1725 through January 1728, Goldbach became Euler’s mentor. After moving to Moscow in 1729, Goldbach continued his elocuent mathematical conversations with Euler through extensive letters, sharing his ideas and motivating the young mathematician. At first, Euler was curious about Fermat’s assertions and perhaps he took this work as a mathematical pastime. In two letters to Goldbach dated June 4 and 25, 1730, Euler mentioned the divisibility of numbers of the form 2n  1 and 2n þ 1. 5 He also noted that 22 þ 1 is composite.12 In other words, numbers of the form n 22 þ 1 are not all prime, as Fermat had claimed in 1640. One can easily show that Fermat’s formula fails for n ¼ 5; 6, and other values of n. For n ¼ 5, we get 232 þ 1 ¼ 4294967297; a number that is divided by 641, and n therefore is not prime. Nowdays, primes of the form 22 þ 1 are known as Fermat primes. In 1732, Euler presented a memoir to the Russian Academy showing that the fifth Fermat number is not prime, and he included six new theorems of his own (without proof).13 The first of these six propositions is the so-called Fermat’s Little Theorem we found earlier.14 Mathematicians call this “little” to distinguish it from Fermat’s Last Theorem. In 1644, Marin Mersenne published Cogitata Physica-Mathematica. In the preface, he stated that the numbers 2n  1 were prime for n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257, and were composite for all other positive integers n \ 257. His conjecture was incorrect, because 267  1 and 2257  1 are composite and Euler confirmed that 231  1 is prime. Nowdays, when the number 2n  1 is prime it is said to be a Mersenne prime and is denoted Mn . In 1747, Euler related Mersenne primes to perfect numbers, proving that when 2n  1 is prime, the formula 2n1 ð2n  1Þ will yield all the even perfect numbers. In 1742, Goldbach wrote to Euler: “Every integer which can be written as the sum of two primes, can also be written as the sum of as many primes as one wishes 11

Ibid. Letter III dated 8 January 1730, p. 18. [Sed nondum prorsus persuasus sum, quomodo sola x inductione id inferre legitime potuerit, cum certus sim ipsum numeris in formula 22 loco x substituendis nec ad senarium quidem pervenisse.] 12 Matvievskaya and Ozhigova (2007), p. 131. 13 Euler (1732). 14 Sandifer (2003).

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until all terms are units.” Almost as an afterthought, Goldbach added in the margin: “Every integer greater than 2 can be written as the sum of three primes.” These statements led to the Goldbach conjecture, which states that every even integer n [ 2 is expressible as the sum of two primes. Today, an even positive integer that can be expressed as the sum of two primes is called a Goldbach number. Hence, we can state Goldbach’s conjecture by saying: “all even integers greater than or equal to 4 are Goldbach numbers.” When Euler started to work on number theory, he did not know which claims Fermat had actually proved, and among his many statements, he did not know which were true.15 Initially, Euler began by proving or disproving every one of Fermat’s assertions and conjectures one after another, except two. As it happens, Euler’s proof of Fermat’s assertion regarding the equation xn þ yn ¼ zn , which he called “a very beautiful theorem,” was just a partial proof: for the cases n ¼ 3; and n ¼ 4: In solving the problem of Fermat, one could attempt to prove that one equation in the infinite set has no solutions and then extrapolate to all remaining equations. Fermat had described a “proof” for the specific case of n ¼ 4, which he incorporated into the proof of a completely different problem. Using the method of infinite descent, Fermat assumed that there was a hypothetical solution x ¼ X1 ; y ¼ Y1 ; z ¼ Z1 : By examining the properties of ðX1 ; Y1 ; Z1 Þ, Fermat could demonstrate that if this solution did exist then there would have to be a smaller solution ðX2 ; Y2 ; Z2 Þ. Then, by inspecting this new solution, Fermat could show there would be an even smaller solution ðX3 ; Y3 ; Z3 Þ, and so on. This is the essence of the infinite descent method of solutions, which in principle could continue forever, generating even smaller numbers. Because x, y, and z must be whole integer numbers, the infinite descent is impossible since there must be a smallest possible solution. This contradiction proves that the initial assumption must be false. Euler’s initial approach was to build a general proof of Fermat’s assertion that would build up to n = infinity, and build down to n ¼ 3: In 1738, Euler presented “The proofs of some arithmetic theorems,”16 a memoir which contained his proof for Fermat’s equation for the case n ¼ 4: In it, Theorem 1 states that the sum of two biquadratics, a4 þ b4 , is never a square, unless one of the biquadratics vanishes. Euler’s proof was based on the method of infinite descent, a technique Fermat had used as well, although Euler, perhaps unaware of it, did not call it so. Sandifer noted that this theorem is not exactly the case n ¼ 4 of Fermat’s equation, but since any biquadratic is also a square, it follows immediately.17 For the case n ¼ 3, Euler showed a simple proof in his famous Algebra book,18 a demonstration that mathematicians say is flawed. Be that as it may, when Euler 15

Sandifer (2007), p. 284. Euler (1738). 17 Sandifer (2007), p. 287. 18 Euler (1770). 16

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wrote to Goldbach in August 1753 to say that he had proved the theorem of Fermat for the cases n ¼ 3 and n ¼ 4, he stated: “for these two cases are so different from each other, that I see no possibility of deriving from them a general proof for an þ bn 6¼ cn if n [ 2.” Euler added, “But one can see quite clearly by extension that the greater the n, the more difficult the demonstrations must be.”19 Euler was absolutely right. He had advanced the proof of Fermat’s Last Theorem, just a bit. That is how the theorem of Fermat stood when Sophie Germain encountered it. A proof of the general case xn þ yn 6¼ zn would require development of new mathematical methods, and more than two centuries of efforts to be discovered. But Germain did not know that, or that Euler had warned that the greater the coefficient n, the more difficult the proof would be. Hence, Germain attempted to prove Fermat’s assertion, perhaps driven to solve the puzzle by the sheer simplicity of the Diophantine equation, or lured by Fermat’s words that he had discovered “a very wonderful demonstration.”

Legendre Proposes a Contest to Prove FLT I believe that Legendre’s Essai sur la théorie des nombres introduced Sophie Germain to the study of numbers, and especially to Fermat’s theorem. Legendre wrote about the significance of this important assertion and gave a clue to his readers: We have demonstrated in this paragraph, that the equation x3  y3 ¼ z3 is impossible, as well as the equation x4  y4 ¼ z2 and even more so x4  y4 ¼ z4 . Also, Fermat assured us (ed. cit. Dioph pag. 61) that the equation xn þ yn ¼ zn , is generally impossible, when n exceeds 2; but this proposition, after the case of n ¼ 4, is among those that remain to be proven, and that is why the methods that we have just exposed appear insufficient. Moreover, it is easy to see that the proposition could be demonstrated in general, if it was for the case where n is a prime number.20

On 11 December 1815, Legendre and Arago reported on Cauchy’s Démonstration complète du théorème de Fermat sur les nombres polygones.21 In this important memoir, the young mathematician had proved an assertion of Fermat regarding polygonal numbers, which received much praise from the academicians. Polygonal numbers, e.g., triangular numbers, squares, pentagonal numbers, and so on, take their names from arrangements of number-representing dots in the geometric shapes of the successive regular polygons. These are the “forms with many sides” that the Pythagoreans introduced to Classical Greece in the sixth century BCE. For example, a square number (a perfect square) is a figurate number

19

Fuss (1842), p. 618. Legendre (1798), p. 410. He presented a copy to the Institut on 13 July 1798. 21 Institut de France. Procès-verbaux. Tome V, p. 592. 20

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of the form Sn ¼ n2 , where n is an integer. The square numbers for n ¼ 0; 1; 2; . . . are 0, 1, 4, 9, 16, 25, 36, 49, … Fermat stated that every positive integer can be written as the sum of three or fewer triangular numbers, and as the sum of four or fewer square numbers, and as the sum of five or fewer pentagonal numbers, and so on. This became Fermat’s polygonal number theorem. Although he claimed to have a proof, it has never been found. Fermat’s polygonal theorem: every nonnegative integer is the sum of m þ 2 polygonal numbers of order m þ 2, where m  1. The polygonal numbers of order m þ 2 are the integers pm ðkÞ ¼ m2 ðk2  k Þ þ k for k ¼ 0; 1; 2; . . . For m ¼ 2, Lagrange proved that every nonnegative integer is the sum of four squares p2 ðk Þ ¼ k2 : In his Disquisitiones, Gauss proved that every nonnegative integer is the sum of three triangular numbers p1 ðkÞ ¼ ðk2 þ kÞ=2; or, equivalently, that every positive integer n  3ðmod 8Þ is the sum of three odd squares. In his 1815 memoir, Cauchy proved Fermat’s assertion for all m  3: Cauchy’s proof was the catalyst that revived the interest in Fermat’s remaining assertion. The day after Christmas 1815, a Commission at the French Academy of Sciences—led by Legendre—proposed Fermat’s (last) Theorem as the new topic for the Prize of Mathematics for the year 1818. The academicians noted that, since Cauchy had just demonstrated the theorem on polygonal numbers, just one assertion of Fermat remained unproven. They wrote: “A proof of this theorem for the case of fourth power was given by Fermat in one of his marginal notes on Diophantus [Arithmetica]. Euler then showed for the third power case in a similar manner; but the proof remains to find for the subsequent powers, or only for those whose exponent is a prime number, because from this single case all the others are deduced.”22 Finding the general proof of the Théorème de Fermat carried a two-year deadline, and the winner would receive a gold medal valued at three thousand francs. In February 1816, Legendre published the First Supplement to his Essai sur la theorie des nombres. The 63-page long paper is divided in three chapters, the second dealing with the general proof of Fermat’s theorem on polygonal numbers, and several analogous theorems. Finally, Legendre included two new methods for the approximate resolution of numerical equations. Perhaps this was his way to provide inspiration or tools to those working on the last unproven asssertion of Fermat. Although Gauss had developed a methodology that would be important for the proof (at that time), he did not consider participating in the contest. On 21 March 1816, Gauss wrote to Olbers about it: “I am very much obliged for your news concerning the Paris prize. But I confess that Fermat’s theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of.”23 22

Ibid., p. 596. Ribenboim (1979), p. 3.

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Sophie Germain didn’t say what drove her to attempt to prove this theorem that was one of the most challenging problems in mathematics. Maybe Sophie was struck by its simplicity, or perhaps she was fascinated by Fermat’s assertion because it was easy for her to understand. One thing is sure, Germain had devised her own method for the proof, and those preliminary results she submitted for Gauss’s judgment attest to that. Of course, there was the matter of her intellectual obsession. In 1816, she may have been determined (again) to prove something of lasting importance in mathematics. There is no official record to ascertain whether Sophie Germain made a submission to the contest. But we cannot be sure. At the meeting of the Academy on 15 December 1817 it was noted that an entry to the contest was received, identified with the epigraph Labor omnia vincit.24 On 5 January 1818, a Commission led by Legendre was named to review the competing memoirs. On 26 January, another anonymous entry was received.25 In March 1818, the commissioners determined that the entries contained nothing that deserved the award. The contest was renewed. Once again, at the public meeting on 16 March 1818, the Academy announced the prize of mathematics to demonstrate the remaining theorem of Fermat. The deadline for submitting the proof was set as 1 January 1820. On 22 February 1819, the Academy received two memoirs sent by Prosper Coste, an engineer graduated from the École Polytechnique, one of which addressed, unsuccessfully, a generalization of Fermat’s theorem.26 While working on her own proof, Sophie Germain found an unexpected visitor at her door. It was a friend of Gauss, German-Danish astronomer Heinrich Christian Schumacher. On 10 May 1819, Schumacher wrote to Gauss: “Excuse the haste of this letter. I am happy to have reached Miss Germain and shall eat with her on Wednesday. Further details [will follow] orally when I will rush through Göttingen, where unfortunately I can stay only for a few moments.”27 Schumacher must have brought much joy to Sophie Germain, and perhaps it renewed her hope of hearing again from Gauss. Following their dinner, she entrusted a letter to Schumacher to deliver to Gauss. This is perhaps one of the most significant letters that Germain wrote about her work, containing her strategy for obtaining a general proof of Fermat’s Last Theorem. However, the scientific contents of Germain’s letter remained hidden for over 186 years, until it was 24

Institut de France. Procès-verbaux. Tome VI, p. 242. The motto is a phrase adapted from Virgil’s Georgics translated as “Work conquers all.”. 25 Institut de France. Procès-verbaux. Tome VI, pp. 242, 262. 26 Généralisation du théorème de Fermat sur les doubles égalités. Institut de France. Procèsverbaux. Tome VI, p. 418. 27 Schumacher wrote: “Verzeihen Sie die Eile dieses Briefes. Ich bin glücklich zu Mlle. Germain durchgedrungen und soll Mittwochen bei ihr essen. Das nähere mündlich auf meinem Durchfluge durch Göttingen, wo ich leider nur einige Augenblicke bleiben kann.” Peters, C.A.F., Ed. 1860. Briefwechsel zwischen C.F. Gauss und H.C. Schumacher, erster Band. Altona, Esch. p. 159.

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discovered in 2005. Sadly, we don’t know what Gauss thought of her approach, but soon we shall see the essence of what Germain wrote to him and her unique analysis. In November 1819, the Academy received another memoir on Fermat’s theorem with this motto: Omnia gravia sunt dum ignores; ubi cognoveris, facilia.28 Three more entries were received, the last one on 27 December had the epigraph Aggrediar non tam perficiendi etc.29 Laplace, Poisson, Legendre, Poinsot, and Lacroix were to judge them.30 On 24 January 1820, the Commission received an anonymous request from a person who was working on the problem of Fermat, asking for an extension of the deadline set for the competition. It was denied. Again, none of the competing essays produced new results to demonstrate Fermat’s theorem. Did Sophie Germain submit one of these unnamed memoirs? Among the Papiers de Sophie Germain31 archived at the Bibliotèque de France, there are several undated pages that show she was working on a proof. On 6 March 1820, by recommendation of the Commission of judges, the Academy decided that the prize would not be given and that the topic would be withdrawn from the competition. The sum intended for the prize on Fermat’s theorem was given to works in astronomy.32 However, this did not stop further attempts. On 5 April 1824, Pierre Laurent Frizon submitted to the Academy of Sciences a memoir with the title Démonstration du théorème de Fermat [Proof of Fermat’s theorem]. Legendre and Cauchy, who were assigned to review it, reported that although it contained a new approach and Frizon was very knowledgable in the field, he did not prove the theorem. On 18 July 1825, Legendre and Lacroix gave a report on a memoir submitted by Lejeune Dirichlet in which he attempted Fermat’s Last Theorem for exponent n ¼ 5: Johann P. Gustav Lejeune Dirichlet was a twenty-year old Prussian student at the Collège de France. As Legendre noted, Dirichlet had “failed to demonstrate the impossibility of the equation x5  y5 ¼ z5 included in Fermat’s theorem, he has at least succeeded in proving the impossibility of an infinity of other analogous equations, such as33 x5  y5 ¼ 4z5 ; x5  y5 ¼ 16z5 ; etc.” Nonetheless, Legendre and Lacroix were very impressed and because Dirichlet’s memoir contained “new From Lactantius: “Those aspiring to perfection, must arrive there through the most unpleasant difficulties.” 29 From Cicero’s Orator: “I will take it on, not so much out of hope of completing it as out of willingness to try.” 30 Institut de France. Procès-verbaux. Tome VI, pp. 507, 521; Tome VII, p. 8. 31 Papiers de Sophie GERMAIN. Français 9115. A collection of Germain’s handwritten notes showing her research and analysis. 32 The Institute split the 3000 francs between French astronomer M.-C. Damoiseau, and Italian astronomers F. Carlini and G. Plana, who developed new Lunar Tables; see Institut de France. Procès-verbaux. Tome VII, p. 18. 33 Institut de France. Procès-verbaux. Tome VIII, p. 241. 28

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results in a difficult subject, and until now little cultivated”, it was approved to be published in the Recueil des Savants étrangers. Two months later, Legendre published a Second Supplement to his Essai, a 40-page article exclusively devoted to undetermined analysis and particularly focused on Fermat’s theorem.34 Legendre stated in the preface: Although the Academy of Sciences, wishing to honor the memory of Fermat, proposed for the subject of one of its mathematical prizes, the demonstration of the theorem of which we speak, the contest, extended even beyond the ordinary term, did not produce any results. It therefore seems that a particular difficulty is attached to this problem and we still lack the special principle that would be necessary to solve it. Waiting for a happy chance to find this principle, as Fermat had conceived it, the lovers of the theory of numbers may see with pleasure that the case of the fifth power has been rigorously demonstrated.

Almost immediately, on 14 Novembre, Dirichlet submitted to the Academy an addition to his original paper where he proved FLT for n ¼ 5, using an approach similar to that which Legendre used, and which Dirichlet presented in such a way as “to show the great analogy that it had with the method exposed in his previous memoir.”35 The most surprising result in Legendre’s Second Supplement was an assertion attributed to Sophie Germain, the independent mathematician who had kept private her achievement. In the footnote on page 17, Legendre noted that a proposition applied to Fermat’s theorem was due to Mademoiselle Germain; this proposition would later be named Sophie Germain’s Theorem.

Sophie Germain’s Theorem After Euler, all that remained was to prove Fermat’s Last Theorem for odd primes n. Consider this: if an odd prime p divides n, then we can make the reduction ðxq Þp þ ðyq Þp ¼ ðzq Þp , and redefining the arguments we obtain the Fermat equation xp þ yp ¼ z p If no odd prime divides n, then n is a power of 2, so 4jn and, in this case, the above equation works with 4 in place of p. Since the case n ¼ 4 was already proved, it is sufficient to prove the theorem by considering odd prime powers only. Hence, Fermat’s Last Theorem can be split into two cases: Case 1: where exponent p does not divide xyz, or p-xyz; Case 2: where exponent p does divide xyz, or pjxyz.

34

Legendre (1825). Dirichlet (1825). The Addition with the final proof was attached to page 12 of the original memoir.

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For prime exponent p, Case 1 asserts that there do not exist nonzero, pairwise relatively prime integers x, y and z such that xp þ yp þ zp ¼ 0 and p does not divide xyz. Through a very clever analysis, Sophie Germain proved that, if p is an odd prime satisfying two special conditions, then Case 1 of Fermat’s theorem holds for p. This Germain verified for p\100: In the footnote, Legendre wrote: “This ingenious demonstration is due to Miss Sophie Germain, who cultivates with success the physical and mathematical sciences, as she proved with the award from the Academy that she won for her work on the vibrations of elastic membranes. One owes to her also the proposition of article 13 and that one which concerns the particular form of prime divisors of a, given in article 11.”36 Legendre showed how Germain’s theorem applied to the proof of Fermat’s theorem and included his own proof for exponent n = 5. Let’s now examine her theorem, using modern language. Sophie Germain’s Theorem For an odd prime exponent p, if there exists an auxiliary prime h such that there are no two nonzero consecutive p th powers modulo h, nor is p itself a p th power modulo h, then in any solution to the Fermat equation zp ¼ xp þ yp , one of x, y, or z must be divisible by p2 . By producing a valid auxiliary prime, Sophie Germain’s Theorem can be applied for many prime exponents to eliminate the existence of solutions to the Fermat equation involving numbers not divisible by the exponent p. This elimination is today called Case 1 of FLT. For example, to prove that any solution to the equation z3 ¼ x3 þ y3 (prime exponent p ¼ 3) would have to have one of x, y, or z divisible by p2 ¼ 9, it requires to show that (with auxiliary prime h ¼ 13) no pair of the nonzero cubic residues 1, 5, 8, 12 modulo 13 are consecutive, and p ¼ 3 is not itself among the residues. In the same fashion, we can look for solutions for any other exponent of Fermat’s equation. And indeed, this is what Legendre did in the Second Supplement. He verified Germain’s hypotheses and generated a table to show, for p\100, the validity of the p-th power residues modulo h for a single auxiliary prime h chosen for each value of p. He continued to develop more theoretical means of verifying the hypotheses of Germain’s Theorem and to demonstrate that any solutions to the Fermat equation for certain exponents would have to be extremely large.37 Sophie Germain’s proposition is the first general result about arbitrary exponents for Fermat’s Last Theorem. Her theorem automatically proves Case 1 whenever 2p þ 1 is prime. Today, these numbers are known as Germain primes.

36

Legendre (1825), pp. 13–14. For a proof of Sophie Germain’s theorem, see Ribenboim (1999), chap. IV.

37

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There are many primes p for which 2p þ 1 is also prime, but it is unknown whether there are infinitely many such primes. The smallest Germain prime is p ¼ 2 since 2  2 þ 1 ¼ 5; which is prime. The next is p ¼ 3 because 2  3 þ 1 ¼ 7, also prime, and the next are 11 and 23. Note that Germain primes include 2, the only even prime that is known today. As of January 2007, the largest Germain prime was 48047305725  2172403  1:

Unexpected Revelation Let us return to Germain’s letter to Gauss in 1819 where she outlined her strategy for obtaining a general proof of Fermat’s Last Theorem. Del Centina38 published this letter in its entirety, and Laubenbacher and Pengelley39 made a full assessment of Sophie Germain’s grand plan as she outlined it. After explaining to Gauss that this idea had been in her mind ever since reading his Disquisitiones, Sophie Germain shared the analysis she had carried out to prove the theorem of Fermat. To summarize her effort, Germain begins: Voici ce que ja’i trouvé (“Here is what I have found”) (Fig. 9.2) and then she outlined her grand plan for a general proof, including auxiliary theorems and intermediate results that were needed to advance it. Her plan included a unique approach for producing, for each odd prime exponent p, an infinite sequence of qualifying auxiliary primes, which would prove the famous theorem.40 Germain wrote that she had recognized a connection between the theory of residues and Fermat’s equation, which in fact she had told Gauss about in her correspondence ten years earlier. This letter is dated 12 May 1819, several months before the Academy of Sciences would withdraw the contest for proving Fermat’s theorem. Thus, one can surmise that Sophie Germain was aiming for the prize. She needed a referee, who better than Gauss to assess her proof? Sophie Germain asked Gauss for a critique: “I will be very grateful if you are kind to tell me what you think about the steps I’ve followed.”41 No letter has been found to determine whether or not Gauss replied. Working independently, Del Centina and Laubenbacher and Pengelley studied this letter along with other undated notes, which revealed the depth and scope of the work Sophie Germain carried out in her effort to prove Fermat’s Last Theorem. In July 1819, Sophie wrote to Louis Poinsot42 in reference to work he did in number theory and that she found relevant for her proof. Poinsot was a professor at

38

Del Centina (2008), pp. 349–392. Laubenbacher and Pengelley (2010). 40 Ibid., pp. 641–92. 41 Del Centina (2008), p. 362. 42 Del Centina (2005), p. 62. 39

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Fig. 9.2 Excerpt of Germain’s letter to Gauss on 12 May 1819. Letter 13

the École Polytechnique who studied Diophantine equations and found ways to express primes as the difference of two squares and primitive roots. In a paper43 presented to the Parisian Academy in 1817, Poinsot developed simultaneously the congruence xn  1ðmod pÞ and the equation xn  1 ¼ 0 to give an analytic representation of power residues by means of imaginary roots of unity.44 Germain believed that Poinsot’s approach was useful to search for the pth power primitive residues.

43

Poinsot (1817), p. 381. Del Centina (2008), p. 361.

44

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Fig. 9.3 Germain’s proof of z2n ¼ x2n þ y2n

For Sophie Germain, this epoch was characterized by intense mathematical research (Fig. 9.3) and a closer relationship with Legendre. A letter from him dated 31 December 1819 responds to analysis she sent him, addressing an equation which seems to lead nowhere. For some reason, Legendre discouraged Sophie from pursuing her research approach, warning her “… since I first spoke to you about it, the opinion I had that it could succeed is now weakened, and that, on the whole, I think it will be as sterile as the others.”45 Sophie was not easily discouraged.

Germain’s Research to Prove Fermat’s Last Theorem In 1830 Legendre published a third edition of his Essai, renamed Théorie des nombres,46 but he did not mention Sophie Germain’s extensive work aiming to prove Fermat’s Last Theorem. This omission helps us to understand why, for

45

Stupuy (1896), pp. 310–311. Legendre (1830).

46

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almost two centuries, it was assumed that her contribution to FLT was just as Legendre summarized it in the footnote of his Second Supplement in 1825. However, the reevaluation of her manuscripts and her correspondence with Legendre and Gauss by Reinhard Laubenbacher and David Pengelley47 indicates otherwise. They carried out a comprehensive study of Sophie Germain’s work, publishing in 2010 a reassessment of her “grand plan” for proving Fermat’s Last Theorem. Laubenbacher and Pengelley provide the most comprehensive evaluation of Sophie Germain’s contribution to proving FLT to date. They discovered a wealth of results in these manuscripts beyond the single theorem for Case 1 for which Germain was known. They analyzed the supporting algorithms that she invented and found that they were based on ideas and results that other number theorists discovered independently much later.48 Sophie Germain’s Theorem is simply a small part of her big program, a piece that could be applied separately as an independent theorem. Germain’s objective was to prove the theorem for exponent p by producing an infinite sequence of qualifying auxiliary primes of the form h ¼ 2Np þ 1. She developed methods to validate a qualifying requirement for infinitely many auxiliary primes, called the Non-Consecutivity (N-C) condition: There do not exist two nonzero consecutive pth power residues, modulo h. Starting from Fermat’s equation xp þ yp ¼ zp , p a prime number, Germain claimed that if this equation were possible, then every prime number of the form 2Np þ 1 (N being any integer), for which there are no two consecutive pth power residues in the sequence of natural numbers, necessarily divides one of the numbers x, y, and z. To prove Fermat’s Last Theorem, Germain intended to prove that there were infinitely many such qualifying numbers 2Np þ 1, so the Fermat equation would be impossible. Germain’s plan for proving Fermat’s Last Theorem for exponent p required that she develop methods to validate the qualifying condition for infinitely many auxiliary primes of the form h ¼ 2Np þ 1. Sophie Germain developed an algorithm verifying the condition within certain ranges, and outlined an induction on auxiliaries to carry her plan forward.49 To establish this condition for various N and p, she carried out her analysis over many pages, including the general consequences of nonzero consecutive pth power residues modulo a prime h ¼ 2Np þ 1, where N is never a multiple of 3. Germain included a summary table with her results verifying Condition N-C for auxiliary primes h using relevant values of N  10 and primes 2\p\100, confident that she could easily extend the range. Sophie Germain conceived a sophisticated plan to prove Fermat’s Last Theorem, which, as noted by Laubenbacher and Pengelley, used the multiplicative structure in a cyclic prime field and a set (group) of transformations of consecutive pth powers.

47

Laubenbacher and Pengelley (2010), pp. 641–92. Ibid. 49 Ibid., pp. 657, 682. 48

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However, Germain’s plan cannot be carried to completion, as we will discuss shortly. Moreover, Laubenbacher and Pengelley also found an undated letter to Legendre in which Germain actually proved that her plan fails for p ¼ 3: For any prime h of the form 6a þ 1, with h [ 13, there are (nonzero) consecutive cubic residues. In other words, the N-C condition fails for h ¼ 2Np þ 1 when p ¼ 3 and N [ 2, so the only valid auxiliary primes for p ¼ 3 for the N-C condition are h ¼ 7 and 13. Sophie Germain also attempted to prove Fermat’s Last Theorem by related means for other special forms of the exponent, but none of these approaches succeeded.50 She knew that she had not proved the theorem even for a single exponent. Hence, she wanted to show for specific exponents that any possible solutions to the Fermat equation would have to be extremely large. In her letter to Gauss, she stated that any possible solutions would consist of numbers “whose size frightens the imagination.” Germain stated, proved, and applied her “large size theorem” to show for p ¼ 5, the valid auxiliary primes h ¼ 11; 41; 71; 101 indicate that any solution to the Fermat equation would force a solution number to have at least 39 decimal digits. Gauss could have told her that this theorem and its applications were not valid. But it appears that he did not reply to her letter. The details of Sophie Germain’s analysis and a comprehensive evaluation of her proofs are given by Laubenbacher and Pengelley. In the end, Sophie Germain knew the futility of her plan. In an undated letter to Legendre (Fig. 9.4), Germain proved that her plan cannot work for prime p ¼ 3: Laubenbacher and Pengelley elucidated her argument, translated her analysis, and verified her claim. They found her proof rather remarkable, especially since she performed it over the course of one night. As they concluded, Sophie Germain was an impressive number theorist. When did Germain know that her plan for proving Fermat’s Last Theorem was hopeless? In 1829, Italian mathematician Guglielmo Libri asserted that seeking infinitely many auxiliary primes would not work. He proved for n ¼ 3 and n ¼ 4; and stated for any n, that for the auxiliary primes exceeding an assignable limit, Germain’s congruence has solutions prime to p, so that it is futile to attempt to prove Fermat’s theorem by trying to show that one of the unknowns is divisible by an infinitude of primes. Libri claimed that this result was already in two of his earlier memoirs presented to the Parisian Academy of Sciences in 1823 and 1825. From this claim, it was concluded that close followers of the Academy should have been aware by 1825 that Libri’s work would doom the auxiliary prime approach to FLT.51 But, is this really how it happened?

50

Ibid., p. 679. Ibid., p. 662.

51

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Fig. 9.4 Sophie Germain’s undated letter to Legendre with her proof that the plan fails for p = 3. From Laubenbacher and Pengelley (2010)

There is no record of a review of Libri’s work in 1823. On 9 August 1824, during a meeting of the Academy, Cauchy gave a verbal report on Libri’s memoir.52 Théorie des nombres is divided into three parts. In part 1, Libri showed a method to provide whole solutions of indeterminate equations. In part 2, titled “theory of congruences,” Libri determined the number of solutions of a congruence

52

Institute de France. Procès-verbaux. Tome VIII, pp. 121–123.

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on the basis of these same roots. In part 3, Libri established analytical formulas with which he expressed the number of solutions of an indeterminate equation.53 From Cauchy’s overview we can infer that Libri established “in a very simple way Euler’s formula relating to the divisors of numbers.” Libri also proved that the relationships between the coefficients of algebraic equations and their roots extend to congruences in which all roots are real. He inferred from this principle Wilson’s theorem, which states that a natural number n [ 1 is a prime number if and only if ðn  1Þ!  1ðmod nÞ: In other words, Wilson’s theorem asserts that the factorial is one less than a multiple of n exactly when n is a prime number. Then again, Cauchy did not mention Fermat’s Last Theorem or its proof, nor did he note that Libri had determined that “for exponents 3 and 4, there can be at most finitely many auxiliary primes satisfying the non-consecutive (N-C) condition.” Cauchy would have emphasized such crucial result if indeed it were in that memoir. Even if he had obtained it, as Libri claimed, Sophie Germain would not have known about that result before 1824. During his first visit to Paris, Libri presented his work to the Academy at a meeting on 13 June 1825.54 Again, he made no reference to analysis related to auxiliary primes. A month later, Dirichlet proved the impossibility of an infinity of analogous equations to Fermat’s (as we saw earlier), and in November he finally proved FLT for exponent n ¼ 5: Fermat’s theorem was an exciting topic of discussion among academicians when Libri and Sophie Germain met in May 1825. It is conceivable that during their scientific exchange, Germain shared with him the results of her proof. The existence of some of Sophie Germain’s manuscripts in an Italian library today suggests that Libri had them in his possession. After he left France, Libri wrote to Germain asking her to speak to Fourier in order to expedite the report on the memoir that he had submitted six months earlier.55 On 13 March 1826, Fourier and Cauchy gave a verbal report of Libri’s paper.56 Cauchy stated that Libri “had shown the method by which he determined equal roots of numerical equations that contain a single unknown and can be extended to an indeterminate equation; and then he established principles specifically applicable to equations that are linear over one of the unknowns, and that Gauss named congruences.”57 Again, this report does not assert that the attempts of others to prove Fermat’s Last Theorem by finding infinitely many such auxiliaries are in vain. This conclusion would have made a huge impact in the work of Germain, and Legendre would have noted it in his “Theory of Numbers” published in 1830. It is

53

Ibid. Ibid., p. 223. 55 Del Centina (2005), p. 8. 56 Institut de France. Procès-verbaux. Tome VIII, p. 358. 57 Mémoires de l’Académie des sciences de l’Institut de France (1830). Tome IX, Année 1826. Chez Firmin Didot Frères, libraries, Paris. pp. xix–xxx. 54

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unclear why he did not include in that edition the method that Sophie Germain had pioneered. Germain kept waiting for validation from Gauss, the mathematician she had asked repeatedly for a critique of her work and who never really told her what he thought of her proof. In the end, Germain knew that her effort was futile, and that the general proof of Fermat’s theorem was out of reach. Otherwise she would have published it.

Fermat’s Last Theorem After Germain The elegant theorem that Sophie Germain developed is an important contribution to the proof of Fermat’s Last Theorem. Her work stood in the midst of the so-called Golden Age of number theory, a period of time linking Euler to Kummer.58 Led by Euler, the work of Lagrange, Legendre, and Gauss had a great effect on the development of the new techniques, which were later to be used successfully in the study of Fermat’s Last Theorem. The new mathematics include Kummer’s theory of ideal factors, a concept he developed in order to deal with the higher reciprocity laws, and Dirichlet’s analytical formula, which is used for the number of classes of binary quadratic forms with given determinant.59 Dirichlet created the field of analytic number theory. As we discovered earlier, while a student in Paris in 1825, Dirichlet proved FLT for n ¼ 5: In 1832, he published a proof for n ¼ 14. Dirichlet also proved that in any arithmetic progression with first-term coprime to the difference there are infinitely many primes. Euler had stated that every arithmetic progression beginning with 1 contains an infinite number of primes. Legendre first conjectured the theorem in this form in his unsuccessful proofs of quadratic reciprocity, and Dirichlet proved it in 1837 using the so-called Dirichlet L-series. The proof is modeled on Euler’s earlier work relating the Riemann zeta function to the distribution of primes. The Riemann zeta function is an extremely important special function that arises in definite integration, and is intimately related with very deep results surrounding the prime number theorem. The Riemann zeta function is defined over the complex plane for one complex variable. Riemann founded the study of this function in 1859. Prussian mathematician Ernst Eduard Kummer extended the results about the integers to other integral domains by introducing the concept of an ideal number. Since attempts to prove Fermat’s Last Theorem broke down because the unique factorization of integers did not extend to other rings of complex numbers, in 1843 Kummer tried to restore the uniqueness of factorization by introducing ideal numbers.

58

Edwards (1977), p. 61. Ibid., p. 60.

59

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In 1850, the Académie des Sciences de Paris offered a golden medal and a prize of 3000 francs for the proof of Fermat’s Last Theorem. Cauchy was among the judges. There was no winner among the eleven contestants. However, Cauchy wrote a report in 1856 in which he praised the work by M. Kummer. Seven years later, the French Academy awarded him the Grand Prize of Mathematics for this effort. Kummer’s work became the basis of later attempts to prove Fermat’s Last Theorem. He developed criteria for proving the theorem for certain irregular primes. Although there are some gaps in his proofs, Kummer’s contribution is important because his concept of an ideal number allowed the development of ring theory and much of abstract algebra. After Sophie Germain, Legendre, Dirichlet, and Kummer, many undertook the quest to discover their own proof. They tried many approaches and discovered along the way other gems of mathematics. Many books were written about the progress made. Academies and scientific foundations offered prizes for anyone who could demonstrate Fermat’s assertion. Thousands of incorrect and partial proofs were obtained. Paulo Ribenboim collected references to wrong proofs.60 However, all this work was not in vain. All those attempts gave number theory the tools to develop and enrich it, making it one of the most sophisticated and beautiful branches of mathematics. Finally, in 1994, mathematician Andrew Wiles of Princeton University developed the complete proof. The announcement of a proof of Fermat’s Last Theorem caused a sensation, not just within the mathematics community but also among the general public. For the first time in history, a mathematician made front-page news. Articles in popular magazines were published and TV documentaries were produced dedicated to telling the story of a puzzling mathematical assertion that had originated in the 1630s. Even non-mathematicians were curious. One thing was clear: proving Fermat’s Last Theorem was a major scientific breakthrough. It required mathematical analysis so challenging that it had stumped scholars for more than three hundred years. The name Sophie Germain appeared in a few articles, making the story even more intriguing. Number theorists know Germain’s Theorem, but not everybody was familiar with her story.

The Fermat-Wiles Theorem Andrew Wiles first announced his proof of Fermat’s Last Theorem in 1993, but his former student Richard Taylor showed that it was incomplete and contained some errors. A year later, Wiles obtained the complete proof with Taylor’s collaboration.

60

Ribenboim (1999), Appendix A.

The Fermat-Wiles Theorem

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Their papers were published together in 1995 in a special volume of the Annals of Mathematics.61,62 The conjecture of Fermat became the Theorem of Fermat-Wiles. We can only imagine the elation Wiles must have felt. Not only did he find the mathematical proof to the most enigmatic and challenging proposition from three centuries earlier, he was awarded a $50,000 prize. Perhaps more importantly, Andrew Wiles won a coveted place in the history of mathematics. The complete proof of FLT is of great sophistication—so much so that the mathematical details are difficult even for professional mathematicians, unless they specialize in number theory and possess an exceptionally broad mastery of their field. It took Andrew Wiles many years and a multifaceted and exhaustive analysis to complete this work. Even though Fermat implied that it would be a long demonstration (since he could not write it down in the margin of the page), when the problem was finally solved, the full proof took 100 pages of dense mathematical logic, and it required sophisticated theories and methods that were not known at the time of Fermat. Ah, not only was the proof difficult and its advanced methods required specialized knowledge, it also took a superior mind and a determined genius to carry it out to fruition. That was British mathematician Andrew Wiles. He revealed that, as a 10-year-old on the way home from school, he read the story of Fermat’s Last Theorem. “I knew from that moment that I would never let it go. I had to solve it.” Indeed! It took him decades of focused work infused with frustration and disappointment. When he finally found the proof in 1993, he uncovered it for all to see. Wiles was forty years old at that time. In 2000, he was appointed Knight Commander of the Order of the British Empire. Sir Andrew John Wiles is the recipient of the 2016 Abel Prize and the 2017 Medal bestowed by the Royal Society. In 2018, Wiles was appointed as the first Regius Professor of Mathematics at Oxford. We must note that the actual proof of the theorem of Fermat-Wiles is very indirect, and it involves two branches of mathematics, which appear to have nothing to do with each other. It involves elliptic curves and modular forms, very modern and sophisticated concepts. It also uses analysis done previously by Japanese mathematicians Yutaka Taniyama and Goro Shimura who developed the modularity theorem, stating that all elliptic curves are modular; it is also known as the Taniyama–Shimura–Weil conjecture. Elliptic curves are of the form y2 ¼ x3 þ ax2 þ bx þ c; where a, b, c are integers. We seek integer solutions (extremely hard to do), and if we find them, can we prove that there are others or none? The proof also required using Hecke algebra, which is generated by Hecke operators. In the theory of modular forms, a Hecke operator, named after German-Polish mathematician Erich Hecke, is a certain kind of averaging operator that plays a significant role in the structure of vector spaces of modular forms and 61

Wiles (1995). Taylor and Wiles (1995).

62

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more general automorphic representations. A modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition.63 These topics are in the realm understood by specialists.64 Let us not forget, however, Sophie Germain was among the very few who succeeded in advancing the proof of Fermat’s Last Theorem in the nineteenth century. She made some important progress after Euler. Germain invented unique algorithms to develop her proof, which are based on ideas and results discovered independently only much later by other number theorists. To date, the analysis and evaluation of Sophie Germain’s manuscripts is incomplete. Although the results that were published to date help to answer many questions, they also raise numerous new ones, opening the door for further study of Germain’s extensive notes. The reader is urged to consult the study published by Reinhard Laubenbacher and David Pengelley to obtain the full details of the work carried out by Sophie Germain to prove Fermat’s Last Theorem.

Unsolved Problems in Number Theory Up until the mid-twentieth century, number theory was considered the purest branch of mathematics, with no direct applications to the real world. However, that perception changed overnight when computer technology revealed that number theory could provide unexpected answers to real-world problems such as in cryptography. At the same time, computer technology has enabled number theorists to make unimaginable advances in factoring large numbers, determining primes, testing assertions and conjectures, and solving numerical problems once considered impossible for our human computing power. In fact, modern number theory is far different from what it was even in the last century. Today, number theorists specialize in elementary number theory, algebraic number theory, analytic number theory, geometric number theory, or probabilistic number theory. These specializations reflect the methods they use to deal with the problems concerning the integers. Number theory is a vast and fascinating field of mathematics. Although there are many simply stated problems in number theory, some of which are easily understood even by school children, their proofs are either unknown or extremely difficult. For example, many fundamental questions about Mersenne primes remain unresolved. It is not even known whether the set of Mersenne primes is finite or

63

http://mathworld.wolfram.com/ModularForm.html. For a treatment of number theory where these concepts are expounded, see for example, Jean-Pierre Serre (1973).

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infinite. Another interesting problem is the twin prime conjecture, which states that there are infinitely many pairs of primes whose difference is 2. Other conjectures deal with the question of whether an infinite number of prime numbers subject to certain constraints exists. It is conjectured that there are infinitely many Fibonacci primes and infinitely many Mersenne primes. Are there infinitely many Fermat primes? How many Sophie Germain prime numbers are there? Questions like these continue to contribute to the development of various branches of number theory, and contemporary researchers focus on the study of analytic or algebraic aspects of numbers. Perhaps the best-known conjecture is Goldbach’s conjecture, first stated in 1740, which proposes that every even integer greater than 2 can be expressed as the sum of two primes. Despite its apparent simplicity, Goldbach’s conjecture remains an open problem. In recent times, some steps have been taken toward finding particular proofs. In 2005, the Portuguese mathematician Tomas Oliveira e Silva showed Goldbach’s Conjecture to be true for numbers up to 3  1017 : Yes, this is a very large number, but is not sufficient to declare that the conjecture is fully proved. It is also conjectured65 that the number of Sophie Germain primes less than x is proportional to logx2 x : If true, this would have some implications for computational number theory (encryption). Questions like those continue to contribute to the development of various branches of number theory. The infinitude and order of prime numbers is another problem that continues to mystify mathematicians. As Euler said once, “Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate.”66 For me, the most intriguing riddle is in Pierre de Fermat’s own words. “I have a truly marvelous demonstration of this proposition, which this margin is too narrow to contain.” What happened to his proof? It is plausible that Fermat wrote that note prematurely, after he had proven the case for n ¼ 3, and just as it had happened with the proof of his little theorem, Fermat must have assumed that his initial approach would work for all n. We will never know. Of course, Andrew Wiles has proved that there are no sets of positive integers that make xn þ yn ¼ zn true for all n [ 2. And it took him more than a hundred pages of sophisticated mathematics to show it.

65

Davenport (2008), p. 202. Havil (2010), p. 163.

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Chapter 10

Pensées de Germain

… le Créateur de l’univers n’a pas commencé; l’idée qu’il ne doit pas finir est presque symétrique de la première. [… the Creator of the universe did not begin; the idea that he must not end is almost symmetrical to the first.] —SOPHIE GERMAIN

Sophie Germain was a mathematician, a physicist, and a philosopher. She left us a legacy that portrays a deeply sensitive woman who was curious about the world and studied diverse subjects such as astronomy, chemistry, history, and geography. In her last years, Germain penned an exquisite philosophical composition, her words colored by her sensibility and her penetrating thoughts. Her nephew, Jacques-Amand Lherbette, published this essay posthumously in 1833. Although not intended for publication, Lherbette felt compelled to make those manuscripts public to honor Germain’s memory. A practicing lawyer,1 Lherbette became executor of Germain’s scientific and philosophical manuscripts. He must have recognized his aunt’s importance and wanted to ensure that her philosophical legacy endured. In her essay General considerations on the state of the sciences and literature at different times of our culture,2 Germain wrote her ideas about how she perceived the sciences and literature, and related the intellectual processes of the two. Her philosophical focus stemmed from a historical overview that she used to examine the character and nature of science. The key concept that unifies her text is the “analogy” that she believed allows one to sort and discover the laws of the universe. In this memoir, Germain traced the history of human intellectual development in order to study society and the connections between science and art. She discussed the similarities between artistic and scientific endeavors. Germain argued that, although it is undeniable that the impression produced by an artistic presentation is different than that produced by the study of a mathematical text, she believed that there are still underlying rules which science, literature, and art must follow in order to be interpreted as great or beautiful. 1

http://www.assemblee-nationale.fr/sycomore/fiche.asp?num_dept=9419. Germain (1833). © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 D. Musielak, Sophie Germain, Springer Biographies, https://doi.org/10.1007/978-3-030-38375-6_10

2

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She believed genius and eloquence are pleasing to us because they reveal important relations between subjects that we previously had not seen. Saying that it is to this unexpected order that we respond, Germain must have alluded to the sense of beauty and elegance that we perceive in the rhythmic construct of a poem, the esthetic lines of a sculpture, or the logical cadence in the mathematical proof of a theorem. Drawing a parallel between literature and mathematics, she stated that “the formulas replace a sentence; they may be more or less elegant. Analysis speaks to our eyes. Therefore, instead of harmony of sounds, it must contain (among its various elements) order and simplicity.” This is similar to our concept of an elegant mathematical proof. Germain’s thought process is beautiful and pure. She wrote: “Ah ! n’en doutons plus, les sciences, les lettres et les beaux-arts sont nés d’un seul et même sentiment. Ils ont reproduit, suivant les moyens qui constituent l’essence de chacun d’eux, des copies sans cesse renouvelées de ce modèle inné, type universel de vérité, si fortement empreint dans les esprits supérieurs.”3 [Ah! no doubt, the sciences, literature and the fine arts were born of one and the same feeling. They reproduced, according to the means that are the essence of each of them, copies of their constantly renewed innate style, a universal type of truth, that is so strongly imprinted in superior minds.] Sophie Germain wrote from her own experience as a mathematical scholar. In her words, “The language of calculus may give rise to corrections of its own; because it also has its style, and all writers do not write it with the same degree of perfection.” [La langue des calculs peut donner lieu à des corrections qui lui sont propres; car elle a aussi son style, et tous les auteurs ne l’écrivent pas avec le même degré de perfection.]4 The philosophy of Germain reflects what she had learned in the sciences as well as her personal spiritual beliefs. In the 1842 edition of his Cours de philosophie positive, French philosopher of science Auguste Comte praised Germain’s philosophical work. He found in her Considérations a scholarly philosophy both wise and strong, and found affinity between her writing and his own way of thinking regarding the intellectual development of mankind.5 Auguste Comte is the founder of positivism, a philosophical and political movement very popular in the second half of the nineteenth century. The movement fell from grace during the twentieth century, eclipsed by neopositivism. However, because Comte developed successively a philosophy of mathematics, a philosophy of physics, a philosophy of chemistry, and a philosophy of biology, he is considered today the first philosopher of science in the modern sense. His focus on the social dimension of science resonates in many respects with current points of view.

3

Stupuy (1896), p. 90. Ibid., p. 86. 5 Comte (1835), p. 604. 4

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Growing up in the wake of the French Revolution, Comte rejected religion and royalty, focusing instead on the study of society, which he named “sociology.” He divided the subject into two categories: the forces holding society together (social statics) and those driving social change (social dynamics). His ideas and his use of scientific methods greatly advanced the field. There are some similarities between the ideas of Sophie Germain and those of Comte. In fact, Stupuy suggested that perhaps Germain was the true founder of sociology.6 Several scholars examined this claim and concluded that, without diminishing the importance of her philosophical work, Germain could not have influenced the ideas of Comte, since he had published his Cours de philosophie positive before Germain’s philosophical essay became public. French philosopher and archaeologist Felix Ravaisson included in his Rapport sur la philosophie en France au XIX siècle a comparison between the philosophy of Sophie Germain and the positivism of Auguste Comte. Referring to Germain as a woman of great knowledge and a penetrating mind, Ravaisson wrote: “she indicated, with a remarkable correctness of thought and expression, a point of view, not yet considered by sociologist Herbert Spencer, which explains both how we should consider things and how we should finally understand them.”7 Sophie Germain believed that there is within us a deep feeling of unity, order and proportion that guides our judgments, saying “We have: in morality, the rule of property; in the intellect, the knowledge of the truth; in matters of pure pleasure, the sense of beauty.”8 And, after having indicated that there may be contradictions in the theories according to which there are relative truths, she revealed that there is necessarily a type that we consider, compare, and measure until we find the truth in the consciousness we have of our own being. Stupuy published a collection of her Pensées, fragments of Germain’s ideas about life and about science. In those beautiful statements, the mathematician gave us a glimpse of her soul, her ideals, and her beliefs in a creator of the universe and the human spirit. She also revealed her interest in astronomy, as the names of Copernicus, Kepler, “Tycho,” and Galileo are ever-present in her prose. Sophie Germain was an admirer of men of science, especially Gauss, Euler, Lagrange, and Newton. For her, “Euler made a perfect application of mathematics to physics.” [Euler fit une application heureuse de la géométrie à la physique.] Lagrange, she said, “solved it in his own way, with his genius and by a profound and ingenious analysis” [il l’a résolu à sa manière, avec son génie et par une analyse profonde et ingénieuse. She thought Newton was a modest and secluded person who did great things with great simplicity. She believed that pride (regarding our own scientific endeavors) arises from a feeling of mediocrity, and the admission of our weakness. In her own words:

6

Stupuy (1896), pp. 375–377. Ravaisson (1889), pp. 71–72. 8 Stupuy (1896), p. 90. 7

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La simplicité de Newton, sa modestie, naissaient de sa supériorité. On s’en étonne, en considérant cette supériorité même. Les hommes de cet ordre font facilement des choses difficiles. Comment admireraient-ils des œuvres qui leur ont si peu coûté ? Ce n’est point un paradoxe de dire que la vanité ne naît point de la facilité du travail et de la rectitude des idées. Il faut avoir eu souvent tort pour s’enorgueillir d’avoir raison. Les hommes ne s’applaudissent que quand ils sont surpris de leurs productions; ils attachent un grand prix au fruit des efforts pénibles. L’orgueil est le sentiment de la médiocrité, et l’aveu de notre faiblesse.

Her nephew, M. Lherbett, assumed that Germain wrote down her philosophical essay during her last few months of life, when the excruciating pain of cancer prevented her from conducting mathematical research. It is also possible that, as imperfect as her manuscript was when his family discovered it, such a deep philosophical essay could have been written a long time before her illness. In Sophie’s Diary,9 (a mathematical novel inspired by Germain), I attempted to present a portrait of an introverted youngster, an avid reader who was acquainted with the works of the philosophers of the Enlightenment. The fictional character in the story read Blaise Pascal’s Pensées on her sixteenth birthday. I also introduced an encounter of the fictional Sophie with the marquis de Condorcet when she was fourteen, at the time when she became aware of the social exclusion of women desiring an education, especially in mathematics. Could the philosophy of Pascal or of Condorcet have influenced the real Sophie Germain? Pascal’s Pensées is a compelling picture of the “thoughts” of a seventeenth century mathematician, physicist, and religious thinker. However, Pascal intended to publish a book defending Christianity. Those powerful thoughts were part of his discussion about the great wonder and beauty of the human condition, the incarnation, God, the meaning of life, revelation, and the paradoxes of Christianity. Pascal passionately argued for the Christian faith, using both discussion and his famous “Wager.” Perhaps because he was a mathematician, Pascal’s ideas and arguments were sometimes developed and intricate, and, at other times, abrupt and mysterious. Hence, his Pensées became a startling and powerful book in which one could discover new, profound insights, not just about Christianity, but also about our own human nature. Condorcet, on the other hand, was a great advocate of social reform, especially regarding education. Marie-Jean-Antoine-Nicolas de Caritat, marquis de Condorcet, was a mathematician who was most often referred to as one of the last philosophes. He was a proponent of human rights who boldly advocated for the education and the rights of women. Condorcet moved from his work in mathematics (mainly probability) into public service, expecting to apply to social and political affairs a scientific model that he termed a “social arithmetic.” In 1790, Condorcet published his essay “On the Admission of Women to the Rights of Citizenship.” Because of his political views and writings, Condorcet was arrested on March 27, 1794, and imprisoned in Bourgla-Reine, where he was found dead in his prison cell on March 29; the cause of his death remains unknown. 9

Musielak (2012).

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Germain was sixteen when this happened and must have been aware of his writings. In 1795, after her husband’s death, Madame de Condorcet published the Esquisse, and in 1799, his Éloges des academicians (Eulogies of Academicians). She worked assiduously to defend Condorcet’s reputation and, despite becoming destitute after the French revolutionaries confiscated their family wealth, Madame Condorcet published her husband’s complete works, which appeared from 1801 to 1804. I don’t believe that Sophie Germain’s philosophy was an attempt to reform society, nor did it have the objective to defend Christianity. Rather, she wrote what she thought about science, about those who had made major contributions to science, and about her personal beliefs, which reflected her spirituality and understanding of nature. Sophie Germain referred to the philosophy of Immanuel Kant when raising her own questions related to logic and human reason. Kant’s fundamental idea is “critical philosophy,” arguing that the human understanding is the source of the general laws of nature that structure all our experience; and that human reason gives itself the moral law, which is our basis for belief in God, freedom, and immortality. For Kant, therefore, scientific knowledge, morality, and religious belief are mutually consistent and secure because they all rest on the same foundation of human autonomy. Germain asked, “Our question includes that of Kant, which could be thus expressed: our logic is that of the absolute reason, is it only suitable for human reason?”10 Stupuy suggested that Sophie Germain was personally acquainted with German philosopher Georg Wilhelm Friedrich Hegel, whose historicist and idealist account of reality revolutionized European philosophy. Stupuy argued that, since Germain knew well the philosophy of Kant, she must know that of Hegel: “Maybe she had a personal relationship with him during his trip to Paris in 1827.”11 In fact, during that trip, Hegel stayed near the Palais de Luxembourg and was in contact with many intellectuals. Hegel wrote to his wife: “Paris is the capital of the civilized world.”12 But he never wrote about meeting Sophie Germain. Thus, we cannot accept Stupuy’s presumption. This, of course, is not important. What matters to us is Sophie Germain’s legacy. With her philosophical essays, she expressed her sensibility as a mathematician. She believed that a similar emotion inspired the sciences, literature, and the fine arts. Her prose was full of correlations between mathematics and literature, saying that the language of calculus also has its own style, and “all authors do not write it with the same degree of perfection.” When she wrote that “the formulas replace the sentence; they may be more or less elegant. Analysis speaks to the eyes,”13 Germain was expressing her love for mathematics and describing how she was able to discern its harmony. As a true mathematician, she saw how the language of mathematics “has a special kind of charm that draws some people to its study.”

10

Stupuy (1896), p. 109. Ibid., p. 378. 12 Hegel’s Biography by Canfora/Froeb at www.hegel.net/en/hegelbio.htm. 11

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“Voilà l’empire de la géométrie. C’est alors qu’elle est grande, qu’elle est vaste comme l’univers ! Ouvrage miraculeux de la raison humaine, les hommes y ont concentré toutes les idées d’ordre et de rectitude qu’ils ont reçues du ciel. Si elle a ses limites comme l’esprit humain, elle s’est toujours élevée avec lui, et tient de sa hauteur la double immensité, qui s’applique à tous les temps et à tous les lieux, mesurant également et les espaces de la durée fugitive, et ceux de la matière présente et visible.”14 [That’s the empire of geometry. While it is large, it is as vast as the universe! Miraculous work of human reason, men concentrated all ideas of order and rectitude they received from heaven. If it has its limits as the human mind, it still soars with him, and takes its height double immensity, which applies to all times and all places, also measuring and fleeting, and those of the matter present and visible.]

Sophie Germain’s philosophical work was also a testament to her belief in God, and such conviction was revealed very clearly in her writings: L’Être des êtres est immatériel, il a créé toutes choses, et il agit sur elles. Il a créé l’univers, il existait donc avant cet univers, which loosely translated means, “The Being of beings is intangible, he created all things, and it is on all of them. He created the universe, therefore he existed before this universe.” She thought that faith and science were united: “The Holy Scripture does not predict any posterity with respect to science, and God has used his genius in this kind of revelation.”15

13

Stupuy (1896). Ibid., p. 222. 15 Ibid., p. 203. L’Écriture sainte ne prévient point la postérité à l’égard des sciences, et Dieu n’a employé dans ce genre d’autre révélation que celle du génie. 14

Chapter 11

Friends, Rivals, and Mentors

Through this chaos of thoughts, the genius distinguishes a simple idea; his choice is irrevocably fixed, he knows that this idea will be fruitful. —SOPHIE GERMAIN

Introduction Sophie Germain was ill at ease by unwelcome attention. She may have been rude and even quick-tempered with those who treated her condescendingly. At the peak of her career, she displayed arrogance unexpected in a woman of her time, and she fought back with those who did not take her work seriously. As a young woman, Sophie Germain must have been painfully shy, socially awkward, and introverted. As an adult, her character can be ascertained through well-documented relationships she had with a number of professional acquaintances, most notably the dislike she had of astronomer Lalande and the unpleasant rivalry that she developed with Poisson. We also have evidence of her collegial relationships with Legendre, Fourier, and Cauchy, the respect and admiration she had for Gauss, as well as the affectionate friendship she had in her last years with the much-younger Italian scholar Libri. We do not know who she considered to be her best friends, but after rebuffing some admirers when she was young, Sophie always reached out to those individuals with whom she chose to become acquainted. Above all, Sophie Germain sought to associate with those engaged in the pursuit of science, such as Gauss, who was perhaps the most inaccessible (to her) of all mathematicians. They never met, but they corresponded, exchanging mathematical ideas in their letters. Fourier sought Germain socially beginning in 1816, inviting her to dinners at his apartments and also visiting her at her family home. She was already forty and he was forty-eight. No doubt Fourier had many tales to share with Germain over dinner, for he was known to entertain his guests with recollections of his exciting life in Egypt. Libri also befriended her after their first meeting in Paris in 1825, and

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 D. Musielak, Sophie Germain, Springer Biographies, https://doi.org/10.1007/978-3-030-38375-6_11

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it is likely that he visited her home. These friendships developed within the context of their shared scientific work. Sophie Germain was associated in some manner with other scholars. In 1800, there were many eminent mathematicians and physicists in Paris; most of them were professors, examiners, or graduates from the École Polytechnique and members of the Institut de France. From her correspondence, I form an image of a woman who was not only acquainted with the research work of her contemporaries, but one who also had professional and social contact with many intellectuals besides the famous mathematicians. Because of her scientific work, Sophie Germain earned the respect of Legendre, Lagrange, Gauss, and Fourier. Letters from Delambre, Cauchy, Poinsot, and Navier written between 1820 and 1823 suggest that Sophie was admired by the French pléiade d’hommes supérieurs. French physicist Jean-Baptiste Biot wrote: “Mlle Germain is probably the one of her sex who has most deeply penetrated the science of mathematics, not excepting Mme Du Châtelet, for there was no Clairaut.” (He referred to mathematician Alexis-Claude Clairaut who was tutor to Émilie Marquise Du Châtelet.)1 The following sketches of the mathematicians closest to Germain shed some light on the relationship that she developed with them. The small space I devote to these biographical portrayals will not be nearly sufficient to convey the immensity of their collective contributions to science. However, learning a bit about their lives and work may help us understand the woman who became, in some way, part of their scientific story. Every aspect of the life and work of these individuals has been thoroughly researched and documented. It is for this reason that I feel justified to omit specific references in these short portraits.

Carl Friedrich Gauss (1777–1855) Gauss was a good friend of Sophie Germain, even if they never met. He was a gifted mathematician and physical scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy, and optics. Carl Friedrich Gauss (Fig. 11.1) was born in Braunschweig (Germany), on 30 April 1777, one year after Germain. Unlike her, Gauss had a superb education, and his mathematical genius was discovered in childhood. In fact, there are many unverified anecdotes about his mathematical precocity; some were stories he told himself as a grown man, and thus could have been somehow exaggerated.

1

Paul Ritti included this quote in his review of Germain’s Oeuvres Philosophiques in the 1879 issue of Journal de Savants, p. 307.

Carl Friedrich Gauss (1777–1855)

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Fig. 11.1 Karl Friederich Gauss. Portrait published in Astronomische Nachrichten 1828

After receiving a stipend from the Duke of Brunswick-Wolfenbüttel, young Gauss entered Brunswick Collegium Carolinum in 1792. At eighteen, Gauss left Brunswick to study at Göttingen University. In 1798, he returned to Brunswick, where his advisor was Johann Friedrich Pfaff. Gauss’s dissertation was on the fundamental theorem of algebra, which earned him a doctoral degree from the University of Helmstedt in 1799. For the next year, Gauss prepared the Disquisitiones arithmeticae,2 the classical book on number theory published in the summer of 1801. This work was important to Sophie Germain as she saw in it the tools that led to a new way to proving Fermat’s Last Theorem. The Disquisitiones inspired Sophie Germain to write to Gauss in 1805. He may not have realized how much his work had made an impact on hers. When he replied to Sophie’s first letter, after some very nice remarks about arithmetic, instead of commenting on the results she enclosed, Gauss shared a rather personal situation. He said: “I’m not rich enough to publish at my own expense and submit to the dishonesty of foreign booksellers, as it happened to me on the occasion of the first volume. A Mr. Duprat, for example, who is bookseller for the Bureau des longitudes in Paris, received from me, almost three years ago, copies for the value of six hundred eighty francs; but I have not received a penny from him, and he has not

Gauss dedicated the Disquisitiones to the Duke of Brunswick, who financed its publication and had already financed Gauss’s education, and to whom Gauss felt deeply indebted. 2

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bothered to respond to my letters.” Gauss asked to give him information to engage this man to do his duty. Let’s remember, Sophie Germain had signed her letter as Le Blanc. Gauss thought he was dealing with a gentleman. In November, she (Le Blanc) sent Gauss a response to his inquiry regarding M. Duprat. She reported that he had moved to a small village where he lived from a small income and his successor said to have since completed its payments for which the product was immediately cleared up. She added that, “the general opinion of all the people that I have visited has been that it would be almost impossible to get a refund.” To Sophie Germain, Gauss was the golden, upmost number theorist, the mathematician who understood her passion for the abstract part of mathematics. Although his exceptional work in arithmetic had placed him among the premiers géomètres, after 1801 Gauss had become more famous for his work in celestial mechanics. Gauss’s contributions to theoretical astronomy and to practical astronomy are fully documented. His research from 1801–1818 was mainly on mathematics and the computation of planetary orbits. From 1818 onwards, Gauss worked primarily in observational astronomy, geology, and geomagnetism. Since Sophie Germain had asked for the sequel to the Disquisitiones (she had assumed that he was working on it right then), on 20 August 1805 Gauss sent her a copy of his 1799 doctoral thesis.3 In this work Gauss gave the first proof of the fundamental theorem of algebra: Every polynomial f ð xÞ 2 R½ x with real coefficients can be factored into linear and quadratic factors. Gauss critiqued previous attempts to prove the theorem by d’Alembert, Euler, and others. Gauss thought these proofs were unsatisfactory because they presupposed that the roots of the polynomial could be obtained as complex numbers. However, Gauss’s own proof, using geometric notions, had its own missing parts, which were left to be rigorously proved later.4 Eventually, Gauss discovered that Le Blanc was actually Sophie Germain. As we found in Chap. 3, Gauss was astonished but delighted “at seeing my correspondent, M. Leblanc, metamorphosed into this illustrious personage, who gives such a brilliant example of what I would have difficulty believing. The taste for the abstract sciences in general and especially for the mysteries of numbers is very rare …”.5 Gauss added: “les charmes enchanteurs de cette sublime science ne se décèlent dans toute leur beauté qu’à ceux qui ont le courage de l’approfondir.” [The enchanting charms of this sublime science will be revealed in all their beauty to those who have the courage to deepen it.] These rather poetic words must have been inspiring and encouraging to Sophie Germain. After the death of his benefactor the Duke of Brunswick, Gauss moved to Göttingen with his pregnant wife Johanna and their young son to begin his career as

3

Gauss (1799). Cain (?). 5 Germain-Gauss Correspondence. Letter 7. 4

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professor of astronomy and director of the observatory. They arrived in Gottingen on 21 November 1807. His intellectual interest was now in astronomy, and had little time for number theory. In the summer of 1809, Gauss published Theoria motus corporum coelestium in sectionibus conicis solem ambientium, his major work in celestial mechanics. In a letter to Germain dated 19 January 1808, Gauss apologized for not responding to her sooner, mentioning his move to Göttingen and suggesting that some “unfortunate circumstances” caused him to take the new position. Gauss wrote: “I say nothing of the unfortunate circumstances that finally forced me to take this step, due to new difficulties to which I find myself exposed here. I hope that the interposition of the Institute will come to an end.”6 It is clear that Gauss was going through a difficult time; he added: “My work in arithmetic makes me happy at a time when I only see misfortune and despair around me! It is only in the sciences, the heart of my family, and the correspondence with my dear friends, where I can find comfort.” Despite all adversities and career changes, Gauss respected Sophie Germain and praised her mathematical work. In 1808, he wrote to Olbers: “Sophie Germain has sent me the proofs for the two sample theorems (for which prime numbers is [the number] two a cubic or biquadratic residue) … . I believe they are good.”7 In fact, as we discovered in Chap. 3, Gauss had shared with Germain important theorems on power residues and other ideas from his own research, proving that he took her work seriously. The warm tone used in all his letters suggests that Gauss considered Sophie Germain a friend. In his last letter to her, he delivered an important message: “Soyez toujours aussi heureuse, ma chère amie, que vos rares qualités d’esprit et de cœur le méritent, et continuez de temps en temps de me renouveler la douce assurance que je puis me compter parmi le nombre de vos amis, titre duquel je serai toujours orgueilleux.” (“Be always happy, my dear friend, as your rare qualities of mind and heart deserve, and continue from time to time to renew the fresh assurance that you can count me among your friends, a title of which I’ll be always proud.”) The events that shaped Gauss’s life during this period of time may help explain why he stopped replying to Germain’s letters. In February 1808, his daughter was born, and a short time later, his father died. Meanwhile, Gauss was teaching at the university, carrying out pioneering research, and overseeing the construction of a new astronomical observatory. In October 1809, Gauss lost his beloved wife Johanna, one month after giving birth to a third child. He was devastated and expressed his sorrow and grief to his friend Olbers, with whom he sought solace immediately after burying his wife. Most likely, Sophie knew nothing of Gauss’s personal affairs. All she knew was that Gauss was now deeply involved in his astronomical research at the observatory. In December of 1809, Gauss won the French prize of astronomy established

6

Germain-Gauss Correspondence, Letter 9. Laubenbacher and Pengelley (2010), p. 653.

7

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by Lalande in Paris, for his work on the theory of planets and the means of determining their orbits based on three observations.8 Despite not answering her letters, Gauss continued counting Sophie Germain among his friends, as the following episode suggests. On 14 May 1810, Sophie received a request from Jean-Baptiste Joseph Delambre in his role as trésorier de l’Université Impériale. Delambre had received a letter from Gauss and he was entrusted with a commission on which he (Gauss) engaged him to ask her (Sophie’s) opinion on the purchase of a gift for his wife. In January, Gauss had received the Lalande medal (médaille de Lalande), a prize valued at 500 francs, for his work on the theory of planetary orbits. Instead of money, Gauss desired to have une belle montre à pendule to gift to his wife-to-be. I write “wife-to-be” because, at the time, Gauss was just engaged. In October 1809, Gauss had lost his first wife. Six months later, he asked Friederica Wilhelmine Waldeck to marry him, and the wedding took place on 4 August 1810. Thus, we can deduce that the pendulum clock was a gift Gauss intended for his new bride. In any case, there is no written record to ascertain how Sophie responded to this request or how she helped Delambre select the gift. One is tempted to fill the gap in that story with the conclusion that Sophie Germain may have been delighted to have that privilege; if so, then Gauss should have been grateful. He could have shown more support when she asked for validation of her work related to Fermat’s Last Theorem, but it appears that he did not. Gauss’s extensive scientific contributions to several branches of mathematics have been widely documented. In addition to his Disquisitiones, Gauss composed two memoirs on the theory of biquadratic residues, one in 1825 and another in 1831, which gave the study of number theory an impulse in a new direction. Gauss’s interest in differential geometry led him to publish many papers on the subject. His 1828 Disquisitiones generales circa superficies curva (General Investigations of Curved Surfaces) was perhaps his most important work. Gauss independently discovered Bode’s law, the binomial theorem, and the arithmetic-geometric mean, as well as the law of quadratic reciprocity and the prime number theorem. “Mathematics,” Gauss said, “is the queen of sciences and arithmetic [number theory] is the queen of mathematics. She often condescends to render service to astronomy and other natural sciences, but in all relations, she is entitled to the first rank.” In Men of Mathematics,9 historian of mathematics E.T. Bell called Gauss the “prince of mathematicians.”

8

Gauss won the prize for his essay entitled Théorie des planètes et les moyens de déterminer l’orbite de première apparition d’après trois observations et sans aucune connaissance préliminaire d’aucun des éléments (26 Decembre 1809). 9 Bell (1937), p. 218.

Joseph-Louis Lagrange (1736–1813)

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Joseph-Louis Lagrange (1736–1813) Was Lagrange a mentor to Germain, her teacher of mathematics? There is no historical record to suggest such relationship ever existed. To be associated with Lagrange in any way would be the highest honor for any aspiring mathematician. Pity, we do not know what Sophie Germain thought about her acquaintance with the man who was one of the greatest minds of her time and whose name will be forever attached to hers. The name Lagrange is familiar to every student of mathematics, engineering, and physics because he made many and significant contributions to all fields of analysis, number theory, and classical and celestial mechanics. His most famous work is the Mécanique analytique, the most important book on classical mechanics since Newton. In his Mécanique, Lagrange developed the principle of virtual work, a powerful tool for deriving the static and dynamic equations of multibody systems. Lagrange’s name is linked to Euler’s for their pioneering work in variational calculus. Numerous concepts from mathematics and physics are named after him: Lagrange polynomials, the Lagrange multiplier, the Euler-Lagrange equation, the Lagrangian, the Lagrange invariant, Lagrangian coordinates, and Lagrangian points, to name a few. Joseph-Louis Lagrange was born in Turin, which is in the northwest part of Italy near France and Switzerland. He was a gifted mathematician and began writing articles at eighteen. Lagrange studied the tautochrone, contributing substantially to the new branch of mathematics which Euler named the calculus of variations. At the age of nineteen, Lagrange developed his general method of dealing with isoperimetric problems and sent it Euler, who was then at the Prussian Academy of Sciences in Berlin. The elder scholar was impressed with the elegance of Lagrange’s approach. In fact, Euler delayed the publication of his own memoir on the same topic to allow Lagrange the opportunity to finish his and claim priority. At that time, Berlin was part of the Kingdom of Prussia. In 1772, under King Friedrich II (Frederick the Great), this area consisted of the provinces of Brandenburg, Pomerania, Danzig (now Gdansk, Poland), West Prussia, and East Prussia (modern day East Germany, northern Poland, and a small portion of the Soviet Union). Before returning to St. Petersburg, Euler recommended that Lagrange take the position Euler was vacating as director of the mathematical department at the Berlin Academy. Lagrange accepted, and on 6 November 1766, he began his scientific research there, earning a salary of 6,000 francs. Shortly after arriving in Berlin, Lagrange married his cousin, Vittoria Conti. She died in 1783 after a long illness, sending Lagrange into a deep depression. Lagrange was forty when Sophie Germain was born, unaware that one day his life would intersect with that of a young woman who would resort to taking a false name with the sole purpose of studying his mathematics. In 1787, King Frederick II died, and the fifty-one-year-old Lagrange must have felt that there was nothing more to tie him to Berlin. Because he was in the prime of his career, the most prestigious royal academies of sciences in the world offered

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Lagrange attractive positions. He accepted the invitation of Louis XVI, King of France, and soon joined the Royal Academy of Sciences in Paris, of which he had been a foreign member for fifteen years. The French court and the academicians gave Lagrange a warm welcome, and he was housed in a comfortable apartment in the Louvre. By changing his title to that of veteran pensionary, the Academy gave him the right of voting in all their deliberations. But this welcoming environment was not sufficient to dispel Lagrange’s melancholy, and at the beginning of his career in France, he did not add anything to his mathematical discoveries. Even the printed copy of his Mécanique analytique (1788), on which he had worked for a quarter of a century, lay for more than two years unopened on his desk.10 Let us take a glimpse at the sad-looking Lagrange in the portrait (Fig. 11.2). He appears dejected, drained of vitality, exhausted of what? Lagrange shows signs of deep depression. The French Revolution first roused him out of his lethargy. In 1792, Lagrange married Renee-Francoise-Adelaide Le Monnier, the twenty-five-year-old daughter, granddaughter, and niece of members of the Academy of Sciences.11 He was fifty-six. Despite the difference in ages, Madame Lagrange was a devoted wife to whom he became affectionately attached. Germain was nine years younger than his new bride and, when they met, Lagrange may have felt a paternalistic affection for the aspiring scholar as well. However, it is rather peculiar that Lagrange did not mention Sophie Germain in any of his existing correspondence. Lagrange was a kind and gentle person. He was described as being of medium height and slightly formed, with pale blue eyes and a colorless complexion. He was nervous and timid; he detested controversy, and, to avoid it, willingly allowed others to take credit for what he had done.12 In his monumental work Mécanique analytique (1788), Lagrange gave a comprehensive account of what is known today as Lagrangian mechanics. He used the principle of virtual work in conjunction with the so-called Lagrangian multiplier to solve problems of statics. For the treatment of dynamics, Lagrange introduced a third concept known as d’Alembert’s Principle in order to develop the equations of motion. With this work, Lagrange unified the entire science of mechanics using only three concepts and algebraic operations. In the second edition (1811), Lagrange introduced the general notion of a surface integral. Lagrange viewed himself almost exclusively as a pure mathematician who sought mathematical elegance. In the preface of his Mécanique, he wrote: “No diagram will be found in this work. The methods that I explain in it require neither construction nor geometrical or mechanical arguments, but only the algebraic operations inherent to a regular and uniform process. Those who love Analysis will, with joy, see mechanics become a new branch of it and will be grateful to me for thus having extended its field.”

10

Ball (1908). Lagrange Oeuvres, p. XLVII. 12 Ball (1908), pp. 401–412. 11

Joseph-Louis Lagrange (1736–1813)

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Fig. 11.2 Joseph-Louis Lagrange. Lithograph by François-Séraphin Delpech (1778–1825)

In the Mécanique, Lagrange summarized all the work done in analytical mechanics since the time of Newton. He laid the foundation for variational dynamics and he generalized the Principle of Least Action, which states that nature chooses the most economical path for moving bodies, a concept that was first formulated by Pierre de Maupertuis. Although perhaps it was not well understood by the scientific community when his book was published, Lagrange also presented a new and very powerful tool for deriving equations of motion, now called “Lagrange’s equations.” In 1810, when he undertook the new edition of the Mécanique, Lagrange was already advanced in years. The intellectual and physical demands of the task caused the elder scholar to fall ill with fatigue. Madame Lagrange tended to her ailing husband with tenderness and much affection. She urged him to slow down and take care of himself, but Lagrange was eager to finish this work. The first volume of the second edition appeared in 1811. At the beginning of 1813, Lagrange, who was seventy-seven years old, was not well. On February 22, at the meeting of the Class of Mathematics, he presented a new edition of his Théorie des fonctions analytiques. Although he remained active,

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his health was quickly deteriorating. At the end of March, Lagrange developed a fever; he lost his appetite, had trouble sleeping, and felt faint. He could not attend the meeting at the Academy on April 5. Gaspard Monge, B.-G.-É. Lacépède, and J.-A. Chaptal went to visit him on the morning of April 8. This would be Lagrange’s last conversation with his colleagues. Chaptal was a chemist. Napoléon awarded him the Grand Cross of the Legion of Honor and made him treasurer to the Senate. Lacepède was a leading politician and a naturalist. In 1803, Napoléon appointed him Great Chancellor of the Legion of Honor. Chaptal wrote about the last visit to Lagrange, describing a cordial and gentle man who was ready to leave this earth. To the old and wise Lagrange, “Death is not to be feared, and when it comes without violence, it is the last function, which is neither painful nor disagreeable.” He said Lagrange talked about his life and career, saying, “I have performed my task. I have acquired some celebrity in mathematics, I have hated no one, I have done no ill; it is now proper to finish.” Lagrange promised them that, upon recovering his strength, they would meet and dine at Lacepède’s country house. He promised to tell them more about his life. According to Chaptal, during this last conversation—lasting more than two hours— Lagrange’s memory often failed him. He made vain efforts to recall names and dates, “but his discourse was always connected, full of strong thoughts and bold expressions.” On 10 April 1813, at three-quarters past nine o’clock in the morning, the great scholar died. Three days later, Lagrange’s remains were carried to the Panthéon. His close friend Laplace gave the funeral oration on behalf of the Senate, and Lacépède spoke in the name of the Institute of France. Various universities of the kingdom of Italy held memorial services, but nothing was done in Berlin, since Prussia had joined the coalition against France. Napoléon ordered the acquisition of Lagrange’s papers, which were turned over to the Institute of France. Lagrange’s scientific work is rather extensive. His Œuvres complètes were published in fourteen volumes from 1867 to 1892.13 Important didactic works include the Mécanique analytique (1788), with second edition in two volumes (1811, 1815), reprinted as Œuvres 11 and 12; and Théorie des fonctions analytiques (1797), with a second edition in 1813, and reprinted as Œuvres 9. Leçons sur le calcul des fonctions (1801) was reissued in 1804 in the Journal de l’École Polytechnique, 12 cahier, tome 5; the second edition of 1806 includes “un traiti complet du calcul des variations,” and was reprinted as Œuvres 10. Napoléon named Lagrange to the Legion of Honour and, in 1808, made him a Count of the Empire. Shortly before his death, he had received another honor. On 3 April 1813, Lagrange was awarded the Grand Croix of the Ordre Impérial de la Réunion.

13

Joseph Louis de Lagrange—Œuvres complètes, tome 1. Available online at http://portail. mathdoc.fr/cgi-bin/oetoc?id=OE_LAGRANGE__1.

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Adrien-Marie Legendre (1752–1833) I consider Adrien-Marie Legendre the true mentor of Sophie Germain, her teacher of mathematics and perhaps the truest of her friends. He was twenty-four years older than Germain and was well educated. Legendre respected Germain’s intellect enough to give her the opportunity to contribute to his most important book on number theory, Essai sur la théorie des nombres. He also helped her plow through Euler’s manuscripts as she developed her theory of elasticity. There is no record to establish how Germain established contact with Legendre. However, since Legendre was Examinateur at the École Polytechnique in 1799, it is conceivable that Germain approached him via a letter to ask questions about his Essai, a book she must have studied diligently to learn number theory. I cannot conclude when their relationship became a true collaboration, but it is evident that, through the years, they had important scientific discussions through the exchange of letters.14 Legendre took her work seriously and saw that her first attempt to prove Fermat’s Last Theorem merited its place as a theorem in his book. Legendre was born to a wealthy family and studied in Paris at the Collége Mazarin where he learned mathematics from Abbé Joseph-Francois Marie. He graduated in 1770, having written important mathematical essays that his teacher included in a Traité de mécanique (1774). Sophie Germain must have found a kinship in the amiable savant. Poisson described Legendre in the eulogy: “Our colleague has often expressed the desire that, in speaking of him, it would only be the matter of his works, which are, in fact, his entire life.”15 And thus we respect Legendre’s wishes and enumerate a few of his achievements. With his paper on the trajectory of projectiles, the young Legendre won a prize from the Berlin Academy of Sciences in 1782, which brought him to the attention of Lagrange, who was the Director of Mathematics there. Lagrange must have been impressed; he wrote to Laplace asking for more information about the prize-winning mathematician. From 1775 to 1789, Legendre taught at the École Militaire in Paris, where his appointment was made on the advice of d’Alembert. His research was devoted to number theory, statistics, mathematical analysis, and algebra. The five books Legendre published indicate the areas of his greatest contributions: Eléments de géométrie (1794), Essai sur la théorie des nombres (1798, 1808, 1825), Nouvelles méthodes pour la détermination des orbites des comètes (1806), Exercices de calcul intégral (1811, 1817, 1819), and Traité des fonctions elliptiques (1825, 1826, 1830). In addition, his major work on elliptic integrals provided basic analytical tools for mathematical physics. Legendre gave a simple proof that p is irrational as well as the first proof that p2 is irrational. 14

Stupuy (1896), pp. 287–306. Discours prononcé aux funérailles de M Legendre par M. Poisson. Moniteur universel (20 Jan 1833), 162.

15

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When Laplace was promoted, on 30 March 1783, Legendre took his position as adjunct in the Académie des Sciences, and two years later he was promoted to associate. In 1787, Legendre was appointed to the international team to perform measurements of the Earth, which involved a triangulation survey between the Paris and Greenwich observatories in London, working with other researchers at the Royal Observatory at Greenwich. This work led him to publish a memoir that contains Legendre’s theorem on spherical triangles.16 The same year, he became a member of the Royal Society of London. Like most scholars of his time, Legendre favored the revolutionary ideas that became the basis of modern society; but he remained detached from the excesses that bloodied the French revolution. Legendre went into hiding during the reign of terror when thousands of innocent people were persecuted and many were imprisoned. It was during that time that he fell in love with a young woman, mademoiselle Marguerite-Claudine Couhin, whom he married shortly after. Much younger than her husband, Madame Legendre supported the work of her husband with her devotion and attentiveness; she constantly showed herself as a model of instruction, of grace and kindness.17 During the stormy years of the revolution, Legendre’s scientific career flourished. In May 1791, he was appointed to the committee to standardize weights and measures. This work gave us the metric system obtained by performing astronomical observations and triangulations to define the length of the meter. During this time, Legendre was also working on his Eléments de géométrie, a book published in 1794 in which he introduced a novel concept of symmetry in solid geometry. This textbook became as important as Euclid’s and was the leading elementary geometry book in the world for many years. Legendre also collaborated with Gaspard de Prony in the Tables du cadastre, beginning in 1793. This effort was part of the French reform in the units of weights and measures. It required construction of tables of logarithms based on the more convenient decimal division of the angles, which would become the most accurate tables ever created. This was a daunting computing task that required between seventy and eighty human computers to perform the simplest operations: additions and subtractions of differences. These tables were completed in 1796 but were never printed. Meanwhile, because the revolutionary government closed the Academy of Sciences in 1793, Legendre had lost his salary. He and his family experienced much financial hardship. In 1808, Legendre published a new edition of the Essai, in part to correct some errors of the first edition and also to respond to criticism by Gauss. In the book’s preface, Legendre wrote that he had made corrections and had enlarged the book considerably. Legendre must have resented the manner in which Gauss, a much younger scholar, had claimed credit for the law of reciprocity. Referring to Gauss, Legendre wrote: “[Gauss’s Disquisitions] contains many things analogous to those 16

Legendre (1787). Élie de Beaumont (1861), p. 22.

17

Adrien-Marie Legendre (1752–1833)

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which are treated in the Essay on The Theory of Numbers, published in 1798. It contains in particular a direct and very ingenious demonstration of the law of reciprocity already quoted; demonstration that we proposed to treat in more detail in this second edition.”18 After Cauchy obtained the first proof of Fermat’s theorem on polygonal numbers, in February 1816 Legendre published the first supplement to the Essai where he dealt with the general proof and several analogous theorems. To treat undetermined analysis and particularly Fermat’s (last) theorem, Legendre published in 1825 a Second Supplement, where he included Sophie Germain’s theorem and also the partial proofs for FLT by him and Dirichlet for the case n ¼ 5. In 1830, Legendre published a third, expanded edition in two volumes under the title Théorie des nombres. It has been said that the theory of numbers is the branch of mathematics where Legendre displayed more originality and insight. Yet, he made contributions to many other areas of mathematics and in orbital mechanics where he was in competition with Gauss, as we saw in Chap. 3. It seems that Legendre never forgave Gauss for “appropriating the discoveries of others.” As Legendre wrote twenty-one years later, when he was seventy-five19: “… How can M. Gauss have dared to tell you that the greater part of your [referring to Jacobi] theorems were known to him and that he discovered them as early as 1808?… This extreme impertinence is incredible on the part of a man who has sufficient personal merit to have no need of appropriating the discoveries of others… . But this is the same man who, in 1801, wished to attribute to himself the discovery of the law of reciprocity published in 1785 and who wanted to appropriate in 1809 the method of least squares published in 1805. Other examples will be found in other places, but a man of honour should refrain from imitating them.” To be fair, Legendre used Gauss’s proof of quadratic reciprocity in the 1808 edition of his Essai sur la théorie des nombres, giving proper credit to Gauss. Legendre also added his estimate for the prime counting function pðnÞ, which gives the number of primes less than some integer n, pðnÞ  ln nnþ B, where B ¼ 1:08366; sometimes called Legendre’s constant. Again, Gauss would claim that he had obtained the law for the asymptotic distribution of primes before Legendre, but it was Legendre who first brought these ideas to the attention of mathematicians. The method of least squares is due to Legendre. This procedure is used to determine the best-fit line to data we use today. The method appeared for the first time in 1806 in Legendre’s Nouvelles méthodes pour la détermination des orbites des comètes.20 In the preface, Legendre remarked: “The method that seems to me the simplest and the most general is to make minimum the sum of squared errors,… and that I call method of least-squares;” and in the appendix, Legendre explained 18

Legendre (1808). Letter to Carl Gustav Jacob Jacobi dated 30 November 1827. 20 Legendre (1805). 19

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the application of the method: “Of all the principles that can be proposed for this object, I think that there is none more general, more exact, or an easier application than the one we have used in previous research, and which consists in making minimum the sum of the squares of the errors.”21 Legendre was proposing his method of least squares (Méthode des moindres quarrés) only as a convenient process for treating observations, without reference to the theory of probability. But shortly afterwards, Gauss claimed to have used the method as early as 1795. Legendre furiously fought for priority in the discovery of the method of least squares. In 1820, he publicly denounced how Gauss had claimed priority and had dismissed Legendre’s work published in 1805. In the appendix of his treatise to determine the orbits of comets, Legendre had presented (and named) the method of least squares. Four years later, Gauss wrote in his Theoria Motus Corporum Coelestium22 a note suggesting that he had discovered the method of least squares and had used “since the year 1795.” This enraged Legendre and led him to question (in writing) the claims by Gauss. Let us remember that Gauss was born in 1777, and thus he would have been seventeen years old when (supposedly) he made the discovery of “his principle” (or method of least squares). Legendre was amply justified. In mathematics, as in all other sciences, no discovery can be claimed without supplying the appropriate evidence by citing the article or book where it was first published. Besides, this was not the first time Gauss had claimed priority over something or failed to give credit to others, especially to Legendre. Many concepts in mathematics are named after Legendre. For example, Legendre polynomials, the Legendre formula, and the Legendre transform. Legendre polynomials, sometimes called Legendre functions of the first kind, are the most general solutions to the Legendre differential equation. The Legendre differential equation is a second-order ordinary differential equation. Legendre introduced the polynomials in 1782 as the coefficients in the expansion of the Newtonian potential. The Legendre transform is an operation that transforms one real-valued function of a real variable into another. It is an important tool in theoretical physics that plays a crucial role in classical mechanics, statistical mechanics, and thermodynamics. Legendre’s formula counts the number of positive integers less than or equal to a number which are not divisible by any of the first primes. The Legendre symbol, introduced in his 1798 Essai sur la théorie des nombres is a symbol he used while attempting to prove the law of quadratic reciprocity. It is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo a 21

A year later, Legendre published a follow-up to this work, after French astronomer Alexis Bouvard discovered a comet on 20 October 1805. This was an opportunity for Legendre to test his method once again. 22 Translated as “Theory of Motion of the Heavenly Bodies Moving about the Sun in Conic Sections.”

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prime number p: its value on a (nonzero) quadratic residue mod p is 1 and on a quadratic non-residue is −1. Legendre was a soft-spoken gentleman and a man of principle. In 1824, the seventy-two-year-old scholar refused to vote for the government’s candidate to join the Institut National.23 As a result of his refusal, his pension was terminated. Once again, he experienced some hardships. With the change in government in 1828, the pension was partially reinstated. In 1831, Legendre was honored as officer of the Légion d’Honneur. On 31 December 1832, Arago announced at the Academy that Legendre was rather ill. He died on 10 January 1833. He was eighty-one. Poisson gave the funeral oration, remarking that Legendre worked until the end of his life, still trying to refine his mathematical methods: “M. Legendre had that in common with most of the mathematicians who preceded him, that his labors ended only with his life. The last volume of our memoirs still contains a memoir from him, on a difficult question of number theory; and shortly before the illness which led him to the grave, he obtained the most recent observations of short period comets, which he would use to apply and perfect his methods.”24 Other important works of Legendre include: Mémoire sur l’intégration de quelques équation aux différentielles partielles (1787), Mémoire sur les intégrales particulaires des équations différentielles (1790), Exposé des opérations faites en France (1787), and Pour la conjonction des observatoires de Paris et de Greenwich (1792) in collaboration with Cassini and Méchain, as well as Mémoire sur les transcendantes elliptiques (1793), Nouvelle théorie des parallèles (1803), Exercices de calcul intégral sur divers ordres de transcendantes et sur les quadratures (1816–1817), Traite des fonctions elliptiques et des intégrales eulériennes (1827), and various memoirs published in the Recueils of the Academy. There is no visual rendering of Legendre’s face. Articles and books about Legendre and the history of mathematics written before 2005 contain an image that was believed to be his portrait. But a few years ago, two students at the University of Strasbourg discovered that the face belongs to another man. In Changing Faces: The Mistaken Portrait of Legendre, Peter Duren explains in detail the sequence of events that brought the truth to light.25 Thus, because the “true portrait” is just a caricature of Legendre, I chose to illustrate this entry with the frontispiece of his most important books (Fig. 11.3).

23

http://www-groups.dcs.st-and.ac.uk/*history/Biographies/Legendre.html. Élie de Beaumont (1861). 25 Duren (2009). 24

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Fig. 11.3 Title pages of Legendre’s important works

Jean-Baptiste-Joseph Fourier (1768–1830) Joseph Fourier, another distinguished French mathematician and physicist known for his pioneering work on the propagation of heat, expressed a great admiration for Sophie Germain; he was eight years her senior. Like Sophie, Fourier never married, which is perhaps the principal trait that he shared with her, aside from their mutual interest in mathematics. In addition to developing the mathematical theory of heat transfer, Fourier is famous for his work on the representation of functions by trigonometric series. The son of a tailor, Jean-Baptiste-Joseph Fourier (Fig. 11.4) became a teacher of mathematics at age sixteen at the military school in Auxerre. He later joined the faculty at the École Normale at Paris in the year of its founding (1795) when he was twenty-seven. His talent for teaching was rewarded with the offer of the Chair of Analysis at the École Polytechnique, and in 1807, he was elected to the French Academy of Sciences. His outspoken criticism of corruption in 1794, during the French Revolution, led Maximilien Robespierre to issue a decree demanding his arrest. Robespierre was one of the most influential leaders of the French Revolution, known for sending thousands of people to their death at the guillotine. Fourier traveled to Paris to plead his case but was denied. On his return to Auxerre, where he lived at that time, the Comité de surveillance révolutionnaire issued another order for his arrest. Eight days later, Fourier was imprisoned; that could have meant a swift end for the scholar and teacher, just as was the fate of Antoine-Laurent de Lavoisier, father of

Jean-Baptiste-Joseph Fourier (1768–1830)

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Fig. 11.4 J.-B. Joseph Fourier. Engraving by Louis-Léopold Boilly (1761– 1845)

modern chemistry. However, Robespierre was executed on July 28, 1794, and Fourier was immediately released. In 1798, Napoléon selected Fourier for the expedition to Egypt, where he stayed until 1801. In November of the same year, he returned to Paris with Napoléon, and Fourier resumed teaching at the École Polytechnique.26 Shortly after, in February 1802, the emperor appointed Fourier as Prefect of the Department of Isère, a region in the east of France. Fourier carried out several civil engineering projects such as draining marshes and constructing roads. In 1808, Napoléon rewarded him for his work with the title of baron. From Isère, Fourier wrote his famous monograph on heat diffusion, which he presented to the Institut de France in 1807. In January 1812, Fourier won the Institut prize competition on heat diffusion and began a third version of his work as a book, which was published in 1822 as Théorie analytique de la chaleur. To study the transfer of heat in solid bodies, Fourier approached the problem with the notion of a solution consisting of an infinite sum of sine and cosine functions. In this form, Fourier provided a method for representing a discontinuous periodic function by an infinite series.

26

Fourcy (1828).

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Fourier then provided the method to solve the coefficients of the series involving integrals of the function f ð xÞ and investigated conditions under which the series converges to the given function. For this reason, it is now called a Fourier series. His contributions are contained in the Œuvres complètes.27 In 1817, Fourier was elected to the Royal Academy of Sciences and, following the death of Delambre in August 1822, he took the position of secrétaire perpetual de l’Académie in November. In his biography of Sophie Germain, Stupuy included a letter from Fourier, thanking her for writing to his colleagues (Legendre, Desfontaines, and de Jussien) in support of his appointment as the successor of Delambre.28 It is obvious that Germain was not shy to ask them to vote favorably on behalf of her friend. Whether Germain’s endorsement had any influence on the votes is not known. Nonetheless, M. le baron Fourier was given the position of Permanent Secretary of the Class of Mathematics and Physics in The Royal Academy of Sciences. In this official capacity, Fourier invited Sophie Germain to the public meetings. He sent her two entrance tickets, and offered her one of the reserved seats in the center of the hall.29 That was a privilege reserved for important guests. Fourier was eccentric. He wore heavy coats, even in the hot summer, because he thought it would help him guard against attacks of rheumatism. He would say with a smile, “One would assume I am corpulent, be assured, however, that there is much to deduct from this opinion.” He used as analogy the Egyptian mummies to imply that he was not plump, saying that if he “were subjected to the operation of disembowelment, the residue would be found to be a very slender body.”30 According to his friend Arago, the small townhouse of Fourier was also intensely heated year round and “the currents of air to which one was exposed, resembled the burning wind of the desert.”31 The friendly relationship between Sophie Germain and Fourier is evident in correspondence they maintained from 1816 until his death. It is not known when they met for the first time or how, but it is clear through the letters exchanged that Fourier had a great respect for Germain and sought her company outside the professional relationship they developed. Fourier had a heart condition that manifested after his expedition in Egypt. On 4 May 1830, he fell while descending a flight of stairs. This accident may have aggravated his illness, and two weeks later, it reached its crescendo. On May 16, Fourier collapsed in bed fully dressed, with a high fever, and asked his doctor to remain by his side. But soon this plea was replaced by an urgent cry: Vite, vite, du vinaigre, je m’évanouis!32 and Fourier died.

27

http://portail.mathdoc.fr/cgi-bin/oetoc?id=OE_FOURIER__2. Stupuy (1896), p. 319. 29 Ibid. p. 323. 30 Arago (1833), p. CXXXVI. 31 Ibid. 32 Ibid. 28

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Siméon-Denis Poisson (1781–1840) Poisson was no friend of Sophie Germain. Four years younger and a rising star when they came into professional contact, Poisson was too arrogant to consider Germain his peer. She was a competitor in the quest to derive a theory that he must have thought was too formidable for an amateur like her, or that he believed was his own. Born in Pithiviers, a small village south of Paris where his father had an administrative position, Siméon-Denis Poisson could have become a physician. But his destiny was in mathematics and physics. One day, he found (amongst the official papers sent to his father) a copy of the questions set for admission for the École Polytechnique. He must have been attracted by the rigor and mathematical nature of the questions because, at the age of seventeen, Poisson took the entrance exam at the École and passed. Almost immediately, Laplace discovered his analytic abilities. Lagrange, whose lectures on the theory of functions Poisson attended at the École, also recognized the youth’s talent and became his friend for life. Laplace regarded Poisson almost as his son and used every opportunity to support his career. The elder scholars became Poisson’s advisors and friends. Coincidently, Sophie Germain became acquainted with Lagrange around the same time that Poisson started his studies at the École. In 1799, Poisson wrote a memoir on the number of integrals of an equation of finite differences, which was examined by S. F. Lacroix and Adrien-Marie Legendre. It must have been a good paper because Legendre recommended that it be published in the Recueil des savants étrangers, a great honor for a youth of eighteen. Less than two years after enrolling at the École Polytechnique, Poisson published this essay and another one on Bezout’s method of elimination. Upon graduation in 1800, Poisson became a repétitéur (lecturer) at the École, and, with the support and recommendation of his influential mentors—especially Laplace—he continued throughout his life to hold various government scientific posts and professorships. He was promoted to professeur suppléant in 1802. In 1806, Poisson took the position that Fourier had vacated when Napoléon sent him to Grenoble. In 1806, Poisson (Fig. 11.5) attempted to be elected to the Institute of France; he had the support of Laplace, Lagrange, Lacroix, Legendre, and Biot, who recommended him for a place in the Mathematics Section.33 In 1808, the twenty-seven-year-old mathematics professor became astronomer to the Bureau des Longitudes. When the Faculté des Sciences was established in 1809, he was appointed professeur de la mécanique rationnelle. In 1812, Poisson became a member of the Institute of France. The same year, he won the grand prix of the Institute for his mathematical work on electricity. In 1815, he published a work on heat that drew the criticism of Fourier: “Poisson has too 33

The MacTutor History Biographies/Poisson.html.

of

Mathematics,

http://www-groups.dcs.st-and.ac.uk/*history/

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Fig. 11.5 Siméon-Denis Poisson. Lithograph by François-Séraphin Delpech (1778–1825)

much talent to apply it to the work of others. To use it to discover what is already known is to waste it …” Fourier made valid objections to Poisson’s arguments, which the younger man later corrected. At thirty-six, Poisson married Nancy de Bardi, an orphan born in England to émigré parents. By then, he had been examiner at the military school at St. Cyr since 1815 and examiner at the École Polytechnique since 1816. He later became councillor of the university (1820), and then succeeded Laplace as géomètre at the Bureau des Longitudes in 1827. Poisson published his work on heat transfer in 1820, 1821, and 1823 and influenced Sadi Carnot. Poisson also was the last opponent of the wave theory of light and was proven wrong on that matter by Augustin-Jean Fresnel. The main contributions of Poisson to mathematics include his correction of Laplace’s second order partial differential equation for potential, known today as Poisson’s equation. It is a partial differential equation of elliptic type with a wide range of applications in electrostatics, mechanical engineering, and theoretical physics. Poisson’s equation is usually written as r2 u ¼ f , where ∇ is the Laplace operator, and f is a function.

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Another of his contributions to mathematical physics is the Poisson’s ratio (m), the elasticity constant that relates the transverse contraction strain to longitudinal extension strain in the direction of stretching force. Poisson published between 300 and 400 mathematical works in all. Arago remembered Poisson as saying, “La vie n’est bonne qu’à deux choses: à faire des mathématiques et à les professer.” [“Life is good for only two things: discovering mathematics and teaching mathematics.”] This was a personal belief that drove Poisson to work in a wide range of topics.

Claude-Louis-Marie-Henri Navier (1785–1836) Claude-Louis Navier was an engineer and mathematician who made important contributions to the general theory of elasticity. An admirer of Sophie Germain, Navier could have collaborated with her if either had accepted the commonality of their efforts or if they had appreciated the different approaches each one followed. Or perhaps Navier, like others educated in the sciences and higher mathematics, disregarded the intuition and foresight that Sophie Germain used to derive her mathematical theories. Claude-Louis Navier (Fig. 11.6) was born on 15 February 1785 in Dijon, a city located about 300 km southeast from Paris. His father, a lawyer and member of the National Assembly in Paris during the French Revolution, died when Navier was a child of eight. At fourteen, Navier found a second father in an uncle, Emiland Gauthey, a civil engineer at the Corps des Ponts et Chaussées. The education of Navier, now directed by his uncle, focused on the cultivation of the sciences that a good engineer must possess. At seventeen, Navier took the admission examination at the École Polytechnique where he studied until 1804. He then enrolled at the École Nationale des Ponts et Chaussées, from which he graduated in 1806. At the Polytechnique, Navier learned analysis from Fourier who influenced his research and became a friend for life. In 1816, Navier published a note on the life of Archimedes,34 as he had a great admiration for the ancient Greek mathematician and engineer, just as Germain did. Navier also wrote extensively on topics of engineering. In 1826, he published an important book on the resistance of solid bodies,35 which summarized the lectures he gave at the École des Ponts et Chaussées on the applications to mechanics. The name Navier is well known to students of engineering and applied physics due to his many contributions, including the formulation of the general theory of elasticity in a mathematically usable form (1821), which made it available to engineering for the first time. In 1819, Navier determined the zero line of mechanical stress, correcting Galileo’s incorrect results. In 1826, he established the

34

Navier (1816), p. 45. Navier (1826).

35

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Fig. 11.6 Claude-Louis Navier. From a bust at the École des Ponts et Chauseés

elastic modulus as a property of materials independent of the second moment of area. For these and other contributions. Navier is considered to be the founder of modern structural analysis. At the peak of his career, Navier was known for his research of suspension bridges, which was synthesized in a two-volume treatise published in 1823. He gained the attention of both lay and professional people with his impressive design for the Pont des Invalides over the Seine.36 The 155-m bridge was based on Navier’s theory and it was to be a state-of-the-art engineering achievement. However, despite its meticulous design, disaster struck and the bridge collapsed. On the night of September 6–7 of 1826, the buttress of the bridge in the right bank “cracked” due to flooding caused by a broken water pipe from a nearby pumping station.37 Although the damage was repairable, because of politics and financial issues, the work was terminated and the bridge was dismantled. Navier’s reputation was tarnished for a while.

36

Cannone and Friedlander (2003). Ibid.

37

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Navier published his work on the elastic theory of beams in 1826. His more lasting contributions, however, are his fluid equations. Navier published an article38 in 1822 in which he derived the partial differential equations of motion for a viscous fluid. These were a modification of the Euler momentum equations in which he included the effects of attraction and repulsion between neighboring molecules. He did this as a preliminary step toward a realistic theory of elasticity.39 The concept of shear stress in a fluid was still undefined. And yet, Navier was able to derive the famous momentum equations for viscous flows by modifying Euler’s equations to take into account the forces between the molecules in the fluid. Navier obtained the correct equations that we use today to describe fluid motion involving friction. His equations are now known as the Navier–Stokes equations because British mathematician George G. Stokes independently derived them. The Navier-Stokes equations, the only hydrodynamic equations that are compatible with the isotropy and linearity of the stress-strain relation, are now regarded as the universal basis of fluid mechanics. The application of the Navier-Stokes equations is far-reaching; we use them to model the flowfield in rocket engines, the weather, ocean currents, water flow in a pipe, air flow around airplane wings, blades of wind turbines, blood and body fluids flow, and even the motion of stars inside a galaxy. However, Navier’s proof of 1822 was not initially given the importance it has today. Navier’s momentum equation was rediscovered or re-derived at least four times, by Cauchy in 1823, by Poisson in 1829, by Saint-Venant in 1837, and by Stokes in 1845.40 The Navier–Stokes equations are also of great interest in mathematics because there is no proof that, in three-dimensions, solutions always exist (existence) or that, if they do exist, they do not contain any singularity (or infinity or discontinuity) (smoothness). This is one of the seven most important open problems in mathematics. Currently, the Clay Mathematics Institute offers a US $1,000,000 prize for a solution or a counter-example of Navier’s equation. In his History of Aerodynamics, Anderson wrote41: “Navier became recognized as a scholar of engineering science. He edited the works of his granduncle, which represented the traditional empirical approach to many applications in civil engineering. In that process, on the basis of his own research in theoretical mechanics, Navier added a somewhat analytical flavor to the works of Gauthey. That, in combination with textbooks that Navier wrote for practicing engineers, introduced the basic principles of engineering science to a field that previously had been almost completely empirical.” Navier was elected to the Académie des Sciences in Paris in 1824 and became Chevalier of the Legion of Honour in 1831. Navier died unexpectadely on 23 August 1836. Because of his thoughtful character, Navier had many friends. The

38

Navier (1822), pp. 389–440. Darrigol (2002). 40 Ibid. 41 Anderson (1997). 39

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funeral procession included countless engineering students and his many colleagues. Emmery, Coriolis, and Raucourt gave the memorial speeches, brief but full of sensitivity and respect.

Jean-Baptiste Joseph Delambre (1749–1822) Delambre was the Perpetual Secretary of the Institute of France when Sophie Germain won the prize of mathematics. They were acquainted and most likely met in person a number of times. Jean-Baptiste-Joseph Delambre was a renowned astronomer who produced tables of the location of planets and their moons. He was a student and later colleague of astronomer Jérôme Lalande; in 1800, Delambre became president of the French Bureau des Longitudes. This scientific organization was founded on 25 June 1795 and charged with the improvement of nautical navigation, standardization of timekeeping, geodesy, and astronomical observation. Lagrange, Legendre, and Laplace were also members of this institution. Delambre (Fig. 11.7) led the northern portion of the meridian expedition in 1792–99 that formalized the meter. This scientific mission to measure the piece of the meridian arc which ran from Dunkerque through Paris to Barcelona (Spain) had

Fig. 11.7 J.-B. Joseph Delambre. Engraving by Julien Leopold Boilly (1796– 1874)

Jean-Baptiste Joseph Delambre (1749–1822)

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as its main objective to define the meter, the new standard of measure, as one ten-millionth of the distance from the North Pole to the equator. Delambre’s colleague, astronomer Pierre-François-André Méchain, led the measurements of the southern portion of the meridian. Well-appreciated by Napoléon Bonaparte, Delambre was appointed Perpetual Secretary of the Institute of France for the division of mathematical sciences in 1801, a position he held until his death. In 1809, he received the prize for his book Base du système métrique décimal, the foundation of the metric system, as the best scientific publication of the decade. The Base was the official account of the meridian expedition. After a relationship of several years, Delambre married Elizabeth Vineet Leblanc de Pommard in 1804; she was the mother of his assistant, Charles de Pommard. Madame Delambre was an intelligent and highly educated woman. Delambre must have been more sensitive to the female intellectual capacity as compared with the views of others that prevailed at that time. Just as Méchain and Lalande did, Delambre must have also cherished his wife’s scientific expertise. After the premature death of her son, she studied mathematics in order to help her husband with astronomical calculations. What did Delambre think of Sophie Germain? Being married to a well-educated and intelligent woman, he could have treated Germain with the same respect. However, we cannot expect that he would have understood her struggle to be accepted by her scientific peers. Delambre lectured at the College of France, where he was chairman of astronomy beginning in 1807. After the fall of the Napoleonic Empire in 1815, Delambre preserved his academic and government positions because he had maintained political neutrality during the Revolution. In the last years of his career, he concentrated on the study of the history of astronomy and published a number of books in this topic. Delambre died on 19 August 1822, at age seventy-three.

Augustin-Louis Cauchy (1789–1857) Thirteen years younger than Sophie Germain and highly educated by the best mathematicians in France, Cauchy was a rising star when their paths intersected. Cauchy was neither friend nor colleague of Germain, although she made an attempt to establish a scientific conversation when she learned of his work on elasticity. An early pioneer of analysis, Augustin-Louis Cauchy (Fig. 11.8) was born in Paris one month after the French Revolution exploded. His father, Louis-François Cauchy, was a French government officer, a devout Roman Catholic, and a strict royalist. During the Reign of Terror (the most violent era of the revolution), Catholic royalists were persecuted. Fearing for their lives, the Cauchy family moved to Arcueil, a quiet southern suburb of Paris where Laplace also resided and Lagrange visited. Cauchy was four or five. It was in Arcueil that his father began

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Fig. 11.8 Augustin-Louis Cauchy. Lithograph by Zéphirin Belliard after a painting by Jean Roller

his instruction. Life was rather difficult for the family, as food was scarce and they lacked other basic necessities. The Cauchys returned to Paris when the political conflicts stabilized after the execution of Maximilien Robespierre, the dictateur sanguinaire (bloodthirsty dictator). Cauchy’s father secured a new bureaucratic job and quickly regained his place in the government. When Napoléon Bonaparte rose to power in 1799, Mr. Cauchy was promoted to Secretary-General of the Senate, working directly under mathematician Laplace. Cauchy’s parents were very good friends with Lagrange and Laplace. The two scholars introduced the young Cauchy to mathematics. In the fall of 1802, when he was thirteen, Cauchy enrolled at the École Centrale du Panthéon, the best secondary school of Paris at that time. At sixteen, Cauchy entered the École Polytechnique, and in 1807 he continued his engineering studies at the École des Ponts et Chaussées. Laplace and Lagrange had become his academic advisors and treated Cauchy as their son. Upon graduation, Cauchy first performed the functions of an engineer and worked in the project of the canal de L’Ourcq (aqueduct of the Saint-Denis Street). Then he was hired to work on the design and construction of the Saint-Cloud Bridge. In 1810, when he was twenty-one, Napoléon sent Cauchy to Cherbourg, a port city about 300 km from Paris where the harbor was being fortified to prevent British naval incursions. During his stay in Cherbourg, Cauchy continued his

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studies of mathematics. The young engineer had taken along the most important books for him: Lagrange’s Traité des Fonctions and Laplace’s Mécanique céleste. He also took Virgil and Imitation of Christ, a book Cauchy read daily to remind him of his Christian faith.42 After three years of grueling civil engineering work, Cauchy’s health began to deteriorate and he had to return to Paris. On the advice of his mentors (Lagrange and Laplace), Cauchy renounced his career of engineer and devoted himself exclusively to mathematics.43 In 1818, Cauchy married Aloise de Bure, the daughter of a family of publishers and booksellers who published the majority of his works. The day after his twenty-fifth birthday, 22 August 1814, Cauchy presented to the Institute de France his work on definite integrals, which was to become the basis of the theory of complex functions we study today. This extensive treatise was examined by Legendre and Lacroix, who recommended in their report that it be published in the Mémoires présentés à l’Institut des Sciences, Lettres et Arts, par divers savants. However, due to the reorganization of the Institut after the restoration of the Bourbon king, Cauchy’s paper had to wait until 1827, when it appeared in the first volume of the replacement periodical, the Mémoires présentés par divers savans à l’Académie Royale des Sciences de l’Institut de France. Cauchy’s prize essay (1815) on the theory of waves was also published in this issue.44 Germain and Cauchy crossed paths at a crucial moment of their careers in January 1816. During the same session where Germain was awarded the prix des lames élastiques, Cauchy received the prix de la théorie des ondes for his extensive work on wave propagation at the surface of a liquid,45 a memoir which was written in 1815. Interestingly, this was the same topic that Poisson was actively pursuing at the time. Cauchy published this work in 1827 as a three-hundred-page text with a number of additional notes at the end. His results are now classics in hydrodynamics. Cauchy did important work in number theory. In 1815, he demonstrated one of Fermat’s unproven claims on polygonal numbers.46 The Fermat polygonal number theorem, found in additive number theory, states that every positive integer is a sum of at most n n-gonal numbers. That is, every positive number can be written as the sum of three or fewer triangular numbers, and as the sum of four or fewer square numbers, and as the sum of five or fewer pentagonal numbers, and so on. Three such representations of the number 17, for example, are: 17 = 10 + 6 + 1 (triangular numbers); 17 = 16 + 1 (square numbers); and 17 = 12 + 5 (pentagonal

42

Valson (1868), p. 27. Ibid, p. 42. 44 Smithies (2008), p. 24. 45 Cauchy (1827). 46 Cauchy (1815), p. 177. 43

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numbers). Fermat had stated the theorem without proof, promising Mersenne to write it in a separate work. Fermat never did, and thus it remained unproven along with the last theorem. A professor at the École Polytechnique between 1816 and 1830, Cauchy was known for his rigorous analysis teaching, which many students found extremely difficult. He published his lecture notes as comprehensive textbooks.47 In his Cours d’analyse, Cauchy provided the foundation for calculus—essentially as it is taught today—by developing the concepts of limits and continuity. He introduced the “epsilon-delta” definition for limits (epsilon for “error” and delta for “difference”). Cauchy developed the theory of complex functions, introducing integral theorems and establishing the calculus of residues. A devout Catholic, Cauchy’s attitude toward his religion caused him many problems, including strained relations with his colleagues, who criticized Cauchy for “bringing religion into his scientific work.”48 In July 1830, his faith was tested. The second French Revolution or Trois Glorieuses shook Paris, forcing the abdication of Charles X on August 2. Cauchy, like many others supporters of the king, must have feared for his life and stayed away from participating in the meetings of the Academy. On August 9, Louis-Philippe assumed the title of King of the French. Cauchy lost his positions when he refused to swear an oath of allegiance to the new regime. In September, he went to Switzerland, helping to establish there the Académie Helvétique. From Geneva, he sent a memoir on the calculus and the calculus of variations, which was read at the Parisian Academy on 4 July 1831. That same year, he went to Italy where the King of Piedmont gave him the chair of theoretical physics. His tenure in Turin did not last long. In 1833, Cauchy left for Prague in order to follow the exiled King Charles X and to tutor his grandson. Cauchy returned to Paris in 1838 and took his position at the Academy.49 However, because he continued to refuse the oath of allegiance, he was not allowed to teach and did not get the needed support to assume positions at the Collège de France and the Bureau des Longitudes. After the rule of King Louis Philippe ended in 1848, he regained his university position. Cauchy continued his research on differential equations and applications to mathematical physics, even as his relationship with many colleagues became more strained. In 1850, Cauchy’s work dominated the output of the French Academy of Sciences. Seven of the fourteen memoirs published that year were his. Cauchy died on 23 May 1857, in Sceaux (near Paris). His daughter wrote: “Having remained fully alert, in complete control of his mental powers, until 3:30 a.m., my father suddenly uttered the blessed names of Jesus, Mary, and Joseph. For the first time,

These notes are in his Œuvres complètes, Série 1 and Série 2 available at http://portail.mathdoc. fr/cgi-bin/oetoc?id=OE_CAUCHY_1_1. 48 http://www-groups.dcs.st-and.ac.uk/*history/Biographies/Cauchy.html. 49 Ibid. 47

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he seemed to be aware of the gravity of his condition. At about four o’clock, his soul went to God. He met his death with such calm that made us ashamed of our unhappiness.”50

Guglielmo Libri, Count de Bagnano (1803–1869) Italian mathematician Guglielmo Libri was a good friend of Sophie Germain. Their relationship was born out of their mutual interest in number theory. They met in person for the first time in the spring of 1825, when he visited Paris: Sophie was forty-nine and Libri was twenty-three. Guglielmo Libri, Count of Bagnano (Fig. 11.9), was born in Florence in 1803, and was enrolled in the University of Pisa at the age of fourteen. His career began at seventeen, while he was still an undergraduate. He published his first paper, Memoria sopra la teoria dei numeri in 1820, the year he graduated from the University of Pisa. In 1823, Libri published a short paper on diverse aspects of analysis.51 That same year, he was appointed to the chair of Mathematical Physics at Pisa, and a year later he was given the title of Professor. That Libri had professional ambitions was amply demonstrated. At the end of 1824, he visited Paris where he was introduced to the King of France by the Tuscany ambassador. Laplace, Poisson, Cauchy, Ampère, Fourier, and Arago gave the young scholar a warm welcome. In May 1825, Libri met Sophie Germain and the two became good friends. In the summer of 1830, Libri returned to France, and the friendship with Sophie Germain blossomed. They both met German mathematician and publisher August Leopold Crelle, who was also visiting Paris. In Chap. 12 we shall elaborate further on these events, as they were important in the life and career of Sophie Germain. Despite his noble upbringing, Libri sympathized with the French who sought to overthrow the Bourbon king, and it was reported that he took an active part during the three-day (les trois Glorieuses) July 1830 revolution against Charles X. The twenty-seven-year-old scholar had come to Paris in the midst of the Italian unification movement. That same year, the revolutionary sentiment in favor of a unified Italy was strong, and a series of insurrections laid the groundwork for the creation of one nation along the Italian peninsula. In France, Louis-Philippe—who became King of the French when Charles X was ousted—had promised Italian patriot Ciro Menotti that he would intervene if Austria tried to interfere in Italy with troops. Fearing he would lose his throne, Louis-Philippe did not, however, intervene in Menotti’s planned uprising. The Duke of Modena abandoned his Carbonari supporters, arrested Menotti and other conspirators in 1831, and once again conquered his duchy with help from the

50

Valson (1868). Libri (1823).

51

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Fig. 11.9 Guglielmo Libri. Lithograph by A. N. Noël

Austrian troops. Menotti was executed, and the idea of an Italian revolution centered in Modena faded. Libri had returned to Italy in November 1830. In February 1831, still full with the spirit of the French July revolution, Libri took part in the plot to force a constitution from the Grand Duke of Tuscany. The plot failed, his conspirators accused Libri, and he was forced into exile. Libri returned to France, arriving in Marseilles on the 21st of March 1831.52 Libri admired Sophie Germain for her virtues as a scholar and for her charming and benevolent nature. Years after her death, he described Germain with these affectionate words: Sa conversation avait un cachet tout particulier. Les caractères frappant en étaient un tact sûr pour saisir à l’instant l’idée-mère, et arriver à la conséquence finale, en franchissant les intermédiaires; une plaisanterie, dont la forme gracieuse et légère voilait toujours une pensée juste et profonde; une habitude, qui lui venait de la variété de ses études, de rapprochements constants entre l’ordre physique et l’ordre moral, qu’elle regardait comme assujettis aux mêmes lois … Telle fut cette femme supérieure, qui de toutes a poussé le plus loin les études mathématiques:la seule, à notre avis, qui leur ait fait faire des progrès réels.

52

Del Centina (2005), p. 14.

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In 1833, Libri became a French citizen. At a regular meeting on the first day of April, Libri was elected member of the Académie des Sciences to succeed Legendre who had died three months earlier.53 Some academicians resented the appointment because Libri was still considered a foreigner. His rather arrogant nature increased their antipathy. Arago, on the other hand, was a very good friend. Libri’s early work was on mathematical physics and dealt with the theory of heat. He also made a small contribution to number theory. However, his best work during the 1830s and 1840s was undoubtedly his work on the history of mathematics. From 1838 to 1841, Libri published four volumes of Histoire des sciences mathématiques en Italie, depuis la rénaissanace des lettres jusqu’à la fin du dix-septième siècle. It is known that Guglielmo Libri owned one of the largest private libraries in Europe, including manuscripts that he took, without permission, from many archives. “Libri had loved books, printed or manuscript, since his youth, but in Paris he developed the bibliomania for which he is known to history.”54 It is no secret that Libri bought at auctions, but he also “borrowed” manuscripts from the archives of the Academy of Sciences, the College of France, from other institutions, and from people he knew. Simply stated, Libri was a thief of scientific manuscripts. In a 2012 article,55 Gerald Alexanderson noted that, in February 2010, the New York Times reported the return of a letter of Descartes dated 27 May 1641, from the Haverford College Library in Pennsylvania to France. It was a letter Libri had stolen from the Institut de France in the 1840s. In a letter dated 1829 and found by historian Grattan-Guinnes in 1984 in the Biblioteca Riccardiana in Florence,56 Sophie Germain wrote that, according to Fourier —then Permanent Secretary of the Institute—“some people steal documents from the Academy of Sciences.” In addition, Germain said that the letters previously archived at the Observatory have been “very gallantly placed at the disposal of women attending the courses of astronomy,” suggesting that it was in the albums of the ladies (albums des dames) rather than in the archives of the Institute “that Libri could have found what he was looking for.”57 We also believe that Libri collected Germain’s manuscripts via her nephew, Amand-Jacques Lherbette, who also knew and trusted him. At the regular meeting of the Academy on 20 February 1832, a very suggestive entry reads: “On lit une lettre dans laquelle M. Libri explique comment il est arrivé à la conviction que les manuscrits de Fermat peuvent ne pas être perdus sans

Institut de France. Procès-verbaux. Tome X, p. 238: “Sur l’invitation de M. le Président, M. Libri prend place parmi les Membres.” 54 Del Centina (2006), p. 5. 55 Alexanderson (2012), pp. 327–331. 56 Grattan-Guinnes (1984), pp. 75–76. 57 Ibid. 53

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retour.” [A letter was read in which Mr. Libri explains how he arrived at the conviction that the manuscripts of Fermat may not be lost forever.] This suggests that at that time, some valuable manuscripts of Fermat were already missing. However, the academicians believed his statement (who would doubt a charming foreign scholar?) and “M. Libri sera invité, au nom de l’Académie, à poursuivre ses intéressantes recherches.”58 In 1848, Libri was charged with having committed thefts from French public libraries. Sophie Germain was not alive to witness the sad end for the career of her friend. Libri fled to London, taking along a great number of books and manuscripts. French authorities found additional papers and letters in Libri’s apartment at the Sorbonne where he was teaching. In London, Libri continued collecting books and important manuscripts. By September 1868, his poor health caused him to return to Florence. He shipped twenty crates containing his library, which weighed two tons! Libri arrived in Florence in December after a long, troubled journey. He died on 28 September 1869. Despite his character flaws, we are now indebted to Libri for his bibliomania. As Alexanderson remarked, thanks to Libri, Germain’s manuscripts ended up in the collections of the Bibliothèque Nationale in Paris and in the Biblioteca Moreniana in Florence. “It is largely the survival of these manuscripts that has made possible some of the recent scholarship on Germain and the discoveries of her work on Fermat’s Last Theorem.”59

Leonhard Euler (1707–1783) Genius mathematician Leonhard Euler was Sophie Germain’s teacher. Of course, he couldn’t teach her in the traditional sense, as they never met—for Germain was born seven years before Euler’s death. However, no one can dispute that Euler’s theories formed the basis for her analytical work. She studied his memoirs and learned the fundamental concepts needed to derive her own equations to explain the vibrations in Chladni’s plates. And since Euler had also advanced Arithmetica60 and proved many of Fermat’s assertions, we believe that Germain also studied his approach to proving Fermat’s Last Theorem.

58

Institut de France. Procès-verbaux. Tome X, p. 25. Alexanderson (2012), p. 329. 60 Euler published his first work on number theory when he was twenty-five years old. 59

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Fig. 11.10 Leonhard Euler. Painting by Jakob Emanuel Handmann (1718–1781)

Euler was born on 15 April 1707, in Basel, Switzerland.61 He was the son of a Protestant minister who had studied mathematics under Jakob Bernoulli, the first of the famous family of outstanding mathematicians. Leonhard Euler himself was a favorite pupil of Johann I Bernoulli, the younger brother of Jakob, and became friends with Johann’s sons, Niklaus II and Daniel. At thirteen, Euler entered the University of Basel and studied theology. At sixteen, he earned the academic degree of magister in philosophy. After that, he also studied medicine.62 But all along, Leonhard Euler (Fig. 11.10) was more interested in mathematics. At an early age, he received basic instruction from his father who then hired a private tutor to teach Leonhard mathematics. In his first year at the University of Basel, young Euler attended the freshman course of Johann I Bernoulli, which included geometry and arithmetic. He spent his free time on mathematical studies, excelling in this endeavor and arousing the interest of his teacher. Bernoulli became Euler’s mentor, instructing and guiding the young boy outside the university’s

61 Many interesting facts about Euler’s life are found in Emil A. Fellmann (2007). Leonhard Euler, Birkhäuser-Verlag, Switzerland. For references about his work I recommend two excellent books: C. Edward Sandifer, The Early Mathematics of Leonhard Euler, The MAA Tercentenary Euler Celebration, (MAA, 2007), and Robert Burn’s translation Euler and Modern Science, Eds. N. N. Bogolyubov, G. K. Mikhailov, and A. P. Yushkevich, (MAA 2007). 62 Fellmann (2007).

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classroom. This close contact with a great master and his sons was important to Euler’s intellectual development and future career. In 1727, the same year Isaac Newton died, the twenty-year-old Leonhard Euler began his career in St. Petersburg, Russia. With the recommendation of Daniel Bernoulli, who was a professor there, Euler received and accepted an invitation to join the recently created Royal Russian Academy of Sciences. For fourteen years, Euler devoted himself to the general promotion of science in Russia, writing many important papers in diverse areas of pure mathematics and mathematical physics. In 1730, Euler became professor of physics, and three years later he became professor of mathematics. In 1741, Euler accepted an invitation from Frederick II the Great, King of Prussia, to join and assist in reorganizing the Berlin Academy of Sciences. Euler spent twenty-five years in Berlin, during which time he published his first book of astronomy and dozens of articles related to his research in number theory, elasticity, and acoustics, among other areas of study. Euler was Director of the Mathematical Class of the Berlin Academy when he formulated the main problems of the calculus of variations and developed general methods for their solutions. During the Seven Years War, 1760 was a particularly difficult year for Prussia, especially as the hostilities spilled over the capital, affecting its habitants, Euler included. The Raid on Berlin took place in October when Austrian and Russian forces occupied the capital, left vulnerable by King Frederick’s decision to concentrate his forces in Silesia for several days. During this siege Euler’s house was burned down. When the Russian commander learned about it, he apologized to Euler and gave orders that he be compensated. Meanwhile, a rumor started that the Prussian soldiers led by Frederick were fast approaching to rescue Berlin. This scared the foreign commanders and, on the 12th of October, they withdrew their troops from the city. At the Berlin Academy, Euler had an uneasy relationship with King Frederick. This must have influenced his decision to return to St. Petersburg, for in 1766, he gladly accepted Empress Catherine II’s invitation to rejoin the Russian Academy. Euler had an extraordinary memory and the ability to perform complex mathematical analysis and computations in his head without the benefit of pencil and paper. His enormous power of memorization was paired with a rare power of concentration—noise and hustle in his immediate vicinity rarely disturbed him in his mental work.63 These traits proved to be crucial for accomplishing his brilliant scholarly work later in life, after he lost his eyesight. Engaged in almost every area of mathematics, Euler was a prolific mathematical writer who published hundreds of scientific papers. The Opera Omnia, published by Birkhäuser and the Euler Commission of Switzerland, contains most of Euler’s works. Publication began in 1911, and to date seventy-six volumes have been published, comprising most of Euler’s works. According to Sandifer,64 Euler wrote

63

Ibid. Sandifer (2005).

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about 800 books and papers. The “official” number of entries in Eneström’s index is 866, including a number of letters and unfinished manuscripts. The Euler Archives currently houses 834 works. During his amazing and productive career, Euler won twelve international academy prizes, not counting the eight prizes of his sons Johann Albrecht (7) and Karl (1), which could be considered his as well.65 Not even blindness hindered Euler’s genius and scholarly productivity. While still a young man, his vision had been severely and irreversibly impaired. Early in 1735, Euler suffered a nearly fatal fever, and three years later—when he was barely thirty-one years old—he lost vision in his right eye. In a letter to his friend Goldbach on the 21st of August 1740, Euler admitted: “Geography is fatal to me …” adding, “You know that I have lost an eye and [the other] currently may be in the same danger.”66 His close colleague and grandson-in-law, Paul Nicholas Fuss, wrote in the Éloge that “three days of intense astronomical calculations connected with geographical work underlay the loss in the right eye and began a course leading to Euler’s total blindness in 1767.” By 1771, after failed eye surgeries, Euler was completely blind. Nevertheless, aided by his amazing memory and having practiced writing on a large slate when his sight was failing him, Euler continued to publish his results by dictating them to his assistants. Interestingly, Euler published the largest number of papers in his sixties, during the years when he experienced the worst problems with cataracts in his left eye. Later, after becoming completely blind, Euler wrote as many or perhaps more papers as when he had full eyesight. Euler enjoyed a rather conventional family life. He married Katharina Gsell on 7 January 1734. She was the daughter of a Swiss painter who taught at the gymnasium. The young couple purchased a comfortable house on the banks of the Neva, not far from the Academy. Their first child, Johann Albrecht, was born on November 27, 1734; Euler had twelve more children, but only three sons and two daughters survived early childhood. Johann Albrecht Euler grew up to be a mathematician and became his father’s assistant and collaborator. In 1776, three years after the death of his first wife, Euler remarried, choosing the sister of his father-in-law as his new companion. Of his thirty-eight grandchildren, twenty-six were living at the time of his death. It was widely known that, when the genius mathematician resided in Berlin, he gave lessons to a young girl, Friederike Charlotte von Brandenburg-Schwedt, also known as the Princess Charlotte Ludovica Luisa,67 or Princesa d’Anhalt-Dessau.68 The fifteen-year-old girl was related to King Frederick II. These lessons became Lettres à une princesse d’Allemagne, a three-volume book published after Euler returned to St. Petersburg (Russia), between 1769 and 1773. Over the course of 234 letters, Euler laid out the basics of a dozen disciplines, written in language

65

Fellman (2007), p. 136. Calinger (1996), p. 155. 67 Calinger (1976), p. 213. 68 Condorcet (1842). 66

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understood by the young pupil. He taught the reader about almost everything: history and physics, astronomy and optics, logic and music, electricity and magnetism, theology and philosophy.69 The book was published in France in 1787, when Sophie Germain was a child of eleven. The preface of the book included an Éloge to Euler, written by Condorcet as a tribute to the great master. The “Letters of Euler to a German Princess on Different Subjects in Physics and Philosophy” were published throughout Europe and became immensely popular. As Condorcet remarked, “This is a work inestimable for the singularly clear light in which he has displayed the most important truths of mechanics, of physical-astronomy, of optics, and of the theory of sound.”70 Euler wrote these letters for the instruction of a young and sensible female like Sophie Germain, and it is quite possible that, as a young woman, Sophie studied these Letters as diligently as the German princess, making Euler her first teacher of science. It is also apparent from her amazing grasp of mathematics at the beginning of her career that Germain consulted Euler’s memoirs. No doubt, either Lagrange or Legendre advised Germain to “Read Euler, read Euler, he is the master of us all!” Just as Laplace used to say to his students. About Euler’s teaching, Condorcet said it best: “All the noted mathematicians who live today are his pupils: there is no one who has not formed himself by the reading of his works, who has not received from him the formulas, the methods which he employs; who is not directed and supported by the genius of Euler and his discoveries.”71 Years later Gauss also said emphatically: “Studying the works of Euler remains the best school in the various fields of mathematics and cannot be substituted by anything else.”72 Indeed, Sophie Germain also received from Euler the formulas, the methods, and the guidance to carry out her analytical work. About one-sixth of Euler’s published memoirs include results, methods, or applications of number theory. This body of work prepared the way for Gauss, Legendre, Sophie Germain, and a number of other theorists that followed him. In 1750, Euler gave the proof of a theorem of Fermat that every number of the form 4n + 1 can be given as the sum of two squares.73 Euler made many and important contributions to number theory, proving and disproving many of Fermat’s assertions. Of the over 800 published works, 96 of them are in this subject, and it fills four of the 29 volumes of Euler’s mathematical works. We owe to Euler many of the mathematical symbols in use today, among them i, p, f ð xÞ, and e. The story of the number e is long and fascinating. Euler gave it the notation e in a letter he wrote to his friend, Christian Goldbach in 1731. Euler made

69

Klyve (2010). Condorcet (1842). 71 Ibid. 72 Fellmann (2007), p. 136. 73 Euler (1750), pp. 328–337. 70

Leonhard Euler (1707–1783)

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various discoveries regarding e in the following years, but it was not until 1748 when Euler published his famous book Introductio in Analysin infinitorum that he gave a full treatment of the ideas surrounding e, giving the definition as an infinite series and showing that e is the limit of ð1 þ 1=nÞn as n tends to infinity. Without explaining how he calculated it, Euler gave an astonishing approximation for e to eighteen decimal places: e = 2.718281828459045235. This would have required adding about twenty terms in the series. In an incredible stroke of genius, Euler found a connection between infinite series, e, and p, and from that he discovered many striking identities. One, perhaps the most elegant mathematical expression, is this: eip þ 1 ¼ 0; which I believe has a profound meaning and is interconnected with the nature of our beautiful universe. Euler was not just a mathematician—he was a poet of analysis, a priest of science, a philosopher and a mystic. And perhaps because he was blind, Euler saw beyond the mundane and was able to contemplate the universe from a place no one else has reached—and maybe, just maybe, Euler gazed at the face of the Almighty.74 Overall, Euler published over one hundred memoirs in which he recovered and proved Pierre Fermat’s conjectures, studied Diophantine equations of degree 2, and formulated many major principles for the field. Euler also made enormous progress toward the prime number theorem and introduced the law of quadratic reciprocity (without a proof), which are the two fundamental theorems of number theory. Between Pierre de Fermat and Sophie Germain, Euler stood as a powerful link in her quest to prove Fermat’s Last Theorem. Euler’s death was befitting the great mathematician. On the 18th of September 1783, the seventy-six-year-old Euler spent the morning working in his usual way. He gave a lesson to a grandchild, and during lunch he discussed with his assistants Fuss and Lexell the orbit of the planet Uranus discovered on 13 March 1781 by Herschel. Around five o’clock, while talking with his grandson, all of a sudden, the pipe Euler was smoking slipped from his hand, Meine Pfeife! [my pipe!] he exclaimed; he bent over but could not retrieve it. Euler grabbed his forehead and moments later, with the words Ich sterbe! (I am dying!), he lost consciousness.75 Euler had suffered a stroke and did not regain consciousness. At about eleven o’clock at night, the master mathematician died. As Condorcet described most eloquently, Euler “il cessa de calculer et de vivre.”

74

Musielak (2018). Fellmann (2007), p. 131.

75

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Archimedes of Syracuse (c. 287–212 B.C.) Archimedes provided the inspiration for Sophie Germain to become a mathematician. After reading about his work in geometry, his passion for the science, and his tragic death, Germain was impressed and perhaps found an affinity with the great Greek geometrician. Thus, Archimedes will always be part of Sophie’s story. What we say here about Archimedes of Syracuse is probably what Germain read in Montucla’s Historie des mathématiques. Archimedes left a legacy in geometry and mechanics and wrote, among others, a treatise titled the Measurement of a Circle in which he gave a proof of a relationship between the area of a circle and its circumference. Montucla added that Archimedes determined “the ratio of the cir1 cumference of any circle to its diameter is less than 3 10 70 ; or 3 and 7, but greater than 10 76 3 71.” Archimedes (Fig. 11.11) was known as “the wise one,” and “the great geometer.” Even today, his contributions to mathematics and engineering continue to amaze us. He was born in 287 B.C. in the port of Syracuse, Sicily in the colony of Magna Grecia. In his amazing book The Sand Reckoner, in which he attempted to determine the number of grains of sand that fit in the universe, Archimedes spoke of his father, Phidias, as an astronomer who investigated the sizes and distances of the sun and moon. As a young man, Archimedes may have spent some time in Egypt, and some historians believe that he studied with the pupils of Euclid in Alexandria. It is there that he may have developed a friendship with Eratosthenes for whom he wrote one of his most famous works, the Method, and through him he addressed the cattle problem (described below) to the mathematicians of Alexandria. The Method provides a glimpse into the thinking that led Archimedes to many of his well-known geometry results, including the determination of the area of a parabola, the area and volume of a sphere, and the volume of an ellipsoid. Proposition 2 in the Method, for example, determines the volume of a sphere, a theorem that Archimedes considered to be his greatest achievement. Some historians have written that Archimedes, whose understanding of such matters as levers and centers of gravity was particularly insightful, was able to envision ways to “weigh” various geometric figures against one another so as to successfully compare their areas or volumes. In this manner, his method can be viewed as a basic mathematical proof. Despite the ingenuity and logic of his demonstrations, Archimedes considered them as mere casual calculations, preferring to publish only results with a more formal double indirect proof, now known in mathematics as the “method of exhaustion.” In the Method, Archimedes stated that “… certain things first became clear to me by a mechanical method, although they had to be proved by geometry afterwards because their investigation by the said method did not furnish an actual proof. But it

76

Montucla (1756), p. 234.

Archimedes of Syracuse (c. 287–212 B.C.)

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Fig. 11.11 Archimedes of Syracuse. Image credit: Bibliothèque nationale de France

is of course easier, when we have previously acquired, by the method, some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge.” Written in the form of an epigram, the cattle problem is a mathematical challenge: “Compute, o friend, the number of oxen of the sun, giving thy mind thereto, if thou has a share of wisdom.” Archimedes then goes on to describe, in whimsically poetic language, a certain herd of cattle, consisting of four types, with bulls and cows of each type. The number of cattle in each of the eight categories is not given, but these numbers are related by nine simple conditions, which Archimedes specifies. For example, one of these conditions is that the number of white bulls is equal to the number of yellow bulls plus five-sixths of the number of black bulls. The problem is to determine the number of cattle of each category, and thence the size of the herd. What is required is the smallest possible number, since the nine conditions do not imply a unique answer. On Floating Bodies is considered the first known work on hydrostatics, written around 250 B.C.; it survives only partly in Greek. Archimedes aimed to determine the positions that various solids will assume when floating in a fluid, according to their form and the variation in their specific gravities. He found the law of equilibrium of fluids and proved that water will adopt a spherical form around a center of gravity. This work contains the concept, which became known as Archimedes’

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principle: “Any body wholly or partially immersed in a fluid experiences an upward force (buoyancy) equal to the weight of the fluid displaced.” Archimedes discovered that a submerged object displaces a volume of water equal to the object’s own volume. As children, I enjoyed reading a story in which Archimedes stepped into a bathtub and noticed that the water level rose, concluding that the volume of water displaced must be equal to the volume of the part of his body he had submerged. The jubilant Archimedes ran through the streets shouting, “Eureka!” “Eureka!” In his work On the Sphere and Cylinder, Archimedes proved that the ratio of the volume of a sphere to the volume of the cylinder that contains it is 2:3. In that same work, he also proved that the ratio of the surface area of a sphere to the surface area of the cylinder that contains it, together with its circular ends, is also 2:3. Archimedes invented many engineering devices including war machines that were used in the defense of Syracuse, compound pulley systems, a planetarium, and possibly the water screw; legend attributes to him the idea of burning mirrors as a device of warfare. His mechanical inventions won him legendary fame. After discovering the solution of the problem to move a given weight by a given force, allegedly he said to King Hiero: “Give me a place to stand on and I can move the earth.” When asked for a practical demonstration, Archimedes built a machine by which, with the use of only one arm, he drew out of the dock a large ship, laden with goods and people, which the combined strength of the Syracusans could scarcely move. Impressed, the king proclaimed: “Archimedes is to be believed in everything he might say.” Archimedes was known to get so engrossed in his mathematical work that sometimes he forgot to eat. In fact, his preoccupation with mathematics was the cause of his death. In the massacre that followed the capture of Syracuse by Marcellus in 212 B.C., Archimedes was so intent in deriving a theorem, hunched over a mathematical figure, that he was oblivious to the battle around him. And when ordered by a Roman soldier to attend the victorious general, he did not comply, arguing that first he had to solve his mathematical problem. This enraged the soldier who raised his sword, killing the great geometer. No blame was attached to the Roman general Marcellus, since he had given orders to spare the house of the mathematician. In the midst of his triumph, he lamented the death of Archimedes, provided him with an honorable burial, and befriended his surviving relatives. In accordance with the wish of Archimedes, his tomb was inscribed with the figure of his favorite theorem: “A cylinder enclosing a sphere, giving the proportion by which the containing solid exceeds the contained.” When the Roman philosopher Cicero was in Sicily in 75 B.C., he discovered the neglected and forgotten tomb of Archimedes near the Agrigentine Gate and piously restored it.

Chapter 12

The Last Years

L’infini est le gouffre où se perdent nos pensées. [Infinity is the abyss where our thoughts are lost.] —SOPHIE GERMAIN

In 1824, France had reverted to the splendor of the Bourbon monarchy. On September 16, the ailing King Louis XVIII died. His brother succeeded him to the throne as King Charles X of France. In the first few months of his rule, the government passed a series of laws that bolstered the power of the nobility and clergy, which met with particular public disapproval. This reign dramatized the failure of the Bourbons, after their restoration, to reconcile the tradition of the monarchy by divine right with the democratic spirit produced in the wake of the 1789 Revolution. At the same time, Paris was the mathematical center of the world. And while Sophie Germain continued her efforts to establish her priority and contribution to the mathematical theories of elastic vibrating plates, she returned with greater ardor to her research in number theory. During the last years of her life, Sophie became more determined than ever to assert her role as a mathematician. In May 1823, Fourier had provided Sophie Germain with entrance tickets to the public meetings of the Academy of Sciences that he presided as newly elected Secrétaire perpétuel.1 It is conceivable that she attended those gatherings in order to stay abreast of scientific developments. This would have given her opportunities to meet the young, gifted mathematicians such as Dirichlet, Abel, and Libri, who came to France seeking to establish contact with the leading French scholars and to make their own mark. During this era, the adolescent prodigy Evariste Galois was about to emerge as a revolutionary mathematician whose ideas would become vital to the development of algebraic number theory. Today, Galois is known for producing a method we use to determine when we can solve a general equation by radicals, and for his development of early group theory.

1

Stupuy (1896), pp. 323–324.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 D. Musielak, Sophie Germain, Springer Biographies, https://doi.org/10.1007/978-3-030-38375-6_12

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It was in 1824 when the Italian mathematician Guglielmo Libri appeared in Sophie Germain’s world. In January, the Parisian Academy received Libri’s memoir on the theory of numbers, and on August 9, Cauchy provided a verbal report on it.2 Cauchy wrote: “The author deals with several theorems on the forms of numbers, and it shows, for example, that every integer is the sum of four positive rational cubics. He deals with the theory of congruences, and established in a very simple way Euler’s formula relating to the divisors of the numbers, and several forms of the same kind. He proves that the relationships that exist between the coefficients of algebraic equations and their roots extend to congruences in which all roots are real. He inferred from this principle the theorem of Wilson and several other related to prime numbers.” This would have been an important reference, relevant to Germain’s own research. Libri arrived in Paris shortly before Sophie Germain met him in person for the first time. The encounter occurred on Thursday 13 May 1825, at an evening party hosted by François Arago held at the observatory. The Observatoire de Paris is a magnificent stone building just south of the Luxembourg Gardens. The social interlude must have been rather enjoyable for Germain and Libri. The day after, the young aristocrat wrote to his mother: “Finally yesterday night I met Mademoiselle Germain who won the mathematical prize at the Institute some years ago. I talked with her for about two hours, she has an impressive personality.”3 The aura around Germain must have been indeed radiant and perhaps even blinding to any man not accustomed to conversing with such an intellectual woman. Libri must have been smitten by Sophie Germain, whom I can imagine was a formidable woman who could speak of theorems and proofs as fluently as he did. Years later, Libri portrayed Sophie Germain in these terms: “Her manner was graceful and she had a mild sense of humor, which concealed an exact and profound thought, this arose from the wide range of her studies. She was able to compare and find similarities between physical and moral order, which she considered were subjected to the same laws.”4 Libri described her conversation as witty and charming, saying that Mademoiselle Germain “could seize an original idea and immediately would derive its final consequences, crossing over all intermediate concepts.” Sophie Germain liked the young mathematician; she invited him to her home for lunch. The two met several times that summer,5 and their scientific relationship blossomed into a warm friendship. Libri wrote that he was a “foreigner to her country, but not to her affection and her work.” [Étranger à son pays, mais non à son affection et aux objets de ses travaux.]6 After he left Paris, Germain and Libri remained close in correspondence.

2

Institut de France. Procès-verbaux. Tome VIII. pp. 121–123. Del Centina (2005), p. 6. 4 Libri (1833). 5 Del Centina (2005), p. 6. 6 Libri (1833). 3

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The year 1825 was historically significant for Sophie Germain. In September, her mentor and friend Legendre presented before the Academy his Research on indeterminated analysis and Fermat’s theorem,7 which contains a footnote on page 17, noting Sophie Germain’s Theorem. Although Legendre did not state it, this was part of Germain’s larger effort to prove the theorem of Fermat. In September, Legendre published the Second Supplement to the Essai, which included those results.8 In 1826, Sophie Germain published an essay on the nature and the understanding of elastic surfaces and the general equation that she had derived.9 As we described elsewhere, this was Germain’s attempt to establish her priority during the public debate between Navier and Poisson. In this paper, Germain reviewed the principles that lead to the understanding of the laws governing the equilibrium and movement of elastic solids. Germain refuted Poisson’s theory, in which he assumed the molecular actions as the forces, descending rapidly with distance. Germain sought to establish that assumptions (about the intimate constitution of the body) are useless and even harmful in the question of elastic bodies, and that it is sufficient to resolve the problems of this kind. Navier wrote that, “As to the comments of Mr. Poisson, according to which it would not be allowed to represent the forces resulting from the molecular actions by defined integrals, we do not share this opinion.” The same year, Norwegian mathematician Niels Henrik Abel arrived in Paris. On October 30, he presented to the Academy his work on transcendent functions in which he introduced his main theorem, today known as the “Abel theorem.”10 Abel was twenty-four and eager for the acclaim of the Parisian Academy. The referees to evaluate Abel’s work were Legendre and Cauchy. Legendre, who for four decades had worked on elliptic functions, was better qualified to appreciate Abel’s theory. However, perhaps due to Legendre’s advanced age and poor health, the task of reviewing the memoir was given to Cauchy. It seems that Cauchy was too busy with his own research to pay attention to that of others. Thus, Abel’s manuscript remained forgotten for months. While in Paris, Abel contracted tuberculosis. In December 1826, when his stipend ran out, he returned to Norway without the referee report he needed. At home his circumstances were not favorable, and Abel had to work as a substitute teacher to earn some money. Although he continued writing and publishing important papers, Abel’s luck and health had not improved. In a note begging a friend for a loan, Abel described himself as being “as poor as a church mouse.” His friend Crelle was trying to find a position for Abel, and, in Paris, Legendre and Poisson wrote to the king of Sweden, imploring some help for the young mathematician. The king never replied.

7

Legendre (1827), p. 17. Legendre (1825). 9 Germain (1826). 10 Del Centina (2006), p. 1. 8

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August Leopold Crelle was a German mathematician who had just founded Journal für die reine und angewandte Mathematik that same year. This journal of mathematics became so well known that its authors and readers simply called it Crelle’s Journal. In this periodical, we find important papers by Abel, Dirichlet, Kummer, Libri, and by Sophie Germain. For the 1828 Christmas holiday, Abel traveled by sled to Froland to visit his fiancée, even though the winter was extremely harsh. He became seriously ill on the journey and, although a temporary improvement allowed the couple to enjoy the holiday together, Abel died four months later. At the meeting on 22 June 1829, Legendre informed the Academy of Abel’s death. Sophie Germain may have met Abel, if not in person, at least through Legendre and Fourier or at a séance of the Institut; she wrote to Libri about Abel, lamenting his demise.11

Reaching Out to Gauss, One Last Time In the spring of 1829, Sophie Germain (then almost fifty-three) received unexpected news from Gauss through a German student named Carl (Karl) Bader.12 The young man (twenty years old) studied at the École Polytechnique and, as he wrote to Gauss on 10 February, since arriving to Paris (at the end of September 1828) he was in contact with Arago and Poisson and was applying to the École des ponts et chaussées. He wrote that Germain was very kind to him and that she was very intelligent, witty (Die Mlle Sophie Germain behandelt mich mit vieler Güte. Sie ist eine ausgezeichnet geistreiche Dame). Bader shared with Germain papers that Gauss had recently published: “Theory of biquadratic residues”,13 and “Research on curved surfaces.”14 The latter memoir addressed Gauss’s work in geodesics. It contains new ideas such as what is known today as the Gaussian curvature, a subject which resulted from his interest in differential geometry and his work in astronomy. From Bader’s letter we learn that Germain, who had been ill over the entire wintertime, was still working on her theory of elastic surfaces, which led to her own concept for the curvature of surfaces. Apparently, she intended to submit a new memoir to the Academy (see Chaps. 6 and 8). Bader mentioned that Gauss should have received Sophie Germain’s earlier papers (Recherches sur la théorie des surfaces élastiques; Remarques sur la nature, les bornes et l’étendue, etc.; Examen des principes qui peuvent conduire à la connaissance, etc.). Sophie Germain asked Bader to inquire about it, because she had sent over these papers to Gauss through the Minister of Foreign Affairs.

11

Grattan-Guinness (1984), pp. 75–76. Menso Folkerts (personal communication, September 2019). 13 Gauss (1828a). 14 Gauss (1828b). 12

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Bader was preparing to return to Freiburg (his hometown in Southern Germany) on 1 April. Before the young man left Paris, Germain gave him a letter and a parcel to deliver to Gauss. On 27 April, Bader wrote to Gauss from Freiburg to inform him that his traveling companion, Herr von Oetteler of Braunschweig, would have the honor of giving Germain’s packet to him. Bader added that Germain instructed him to tell Gauss that she would be happy if, after having read her treatise, he’d agree with her. Bader surmised that Poisson and Navier, with whom she disputed her theory, did not treat her with the respect that she deserved. [Die Herren Poisson und Navier, mit welchen sie über die von ihr entwickelte Theorie im Streite liegt, behandeln sie nicht immer auf eine ihrer würdige Weise.] Bader was probably referring to the Poisson-Germain-Navier public dispute we encountered in Chap. 8, a debate for priority which had started in 1828 and it was still raging. Her letter to Gauss in 1829 would be the last.15 First, Germain thanked him for sending her the memoir on biquadratic residues, and she expressed regret for “having being deprived of the scholarly correspondence” she had treasured. Then, upon examining Gauss’s memoir on superficies curvas, Germain expressed astonishment and satisfaction to learn that him, a renowned mathematician, almost simultaneously had the same idea as hers and to which “no one else had paid attention.” Of course, Germain admitted that there existed an essential difference between Gauss’s research and hers, because Gauss’s idea was entirely geometric and hers, on the contrary, was based on mechanics and “only involved geometry to establish the identity of the forces” [of elasticity], which had been the object of her research for more than 20 years. Sophie Germain attempted to elucidate her reasoning in comparing the curvature of a surface to that of a certain sphere. As we saw in previous chapters, in her elasticity theory of plates, Germain introduced the concepts of principal curvatures at a point p of a surface S, which correspond to the maxima and minima curvature of the curves on S through p, and that of the mean curvature, and not the total curvature as Gauss defined, which now is called Gaussian curvature. It is clear that Germain wanted Gauss to validate her reasoning and, believing that her idea was sound, expected that Gauss would agree with her (she had directed Bader to ask him), since she was also using a reference sphere that cuts a surface, a sphere of mean curvature Germain called moyenne courbure. Germain added that she was working to prove, “in a superior manner compared to what she had already published,” that whatever the shape of the element of the surface (whatever the manner in which the curvature is distributed about the point of tangency), the force that would be employed to eliminate the curvature of the element remained constant. The last paragraph is rich in sad pathos. Germain regreted being deprived of the privilege that she would enjoy from Gauss’s scholarly conversations (savante

15

Germain-Gauss Correspondence, Letter 14 dated 28 March 1829.

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conversations). She closed the letter: “I regret further not being able to submit to your judgement many ideas that I have not published and that would take too long to write [in the letter].” Maybe through a published memoir she would now reach out to Gauss, the extraordinary grand géomètre she most admired. We may plausibly imagine at this time Germain may have reflected on her legacy to posterity. This letter betrays her need for a scholar friend with whom to exchange ideas and who’d come to her defense in the matter of priority that others were questioning. It is clear from Bader’s letters to Gauss that Germain was isolated in her intellectual fight. Despite having won a prix de mathématiques for her research with vibrating plates and also having being recognized by Legendre for her theorem in number theory, she still had to fight for her rightful place among the Parisian academicians. Sadly, 1829 was the year when her illness darkened the light shining over her. At the dawn of 1830, Sophie was confronted with her own mortality. For the next eighteen months, she suffered from a terminal illness that kept her in agonizing physical pain. The tone and content of a long letter she wrote to Libri on 8 February 1830 suggest that their friendship had become more personal, as she now shared her feelings. Germain began by telling Libri that she was ill, and that dealing with the painful disease left her little time to work on the intellectual projects she had planned to complete.

Glorious Summer of 1830 For Sophie Germain, that hot, dry summer in 1830 must have been full of conflicting emotions. Once again, she met with the young mathematician Libri, right after she had lost her good friend Fourier who died on May 16. The following day, the new edition of Legendre’s Théorie des nombres appeared, but her name was no longer there. In early June, Libri arrived in Paris. He renewed his friendship with Arago, who had taken over Fourier’s position as perpetual secretary of the Academy of Sciences. On June 21, at the regular meeting of the Academy, Libri presented his memoir on mathematics and physics, and on July 12 he read a note containing, he said, a formula that gives numbers—directly and in a general way—the primitive roots of any prime. Poisson and Cauchy were assigned to review this work. It is likely that Sophie Germain was present at both sessions despite her illness. It was at this time that she met others who shared her intellectual interests. At fifty-four, Sophie Germain was finally becoming secure in her status as mathematician. But as the summer turned warmer, she was about to relive some of the terrors of a new social conflict similar to those she had witnessed as a child of thirteen. It all began when, on 9 July 1830, the unpopular King Charles X announced that he would henceforth govern by ordinances, increasing discontent among those who were against the Bourbon rule. On July 25, from Saint-Cloud, the king signed the

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famous July Ordinances, increasing restrictions on civil liberties, introducing censorship of the press, excluding the middle-class from future elections, and dissolving the current Chamber of Deputies. The ordinances, published in the newspaper the following day, angered many citizens and they began to protest against the king. On July 26, the members of the Academy of Sciences held their Monday public meeting, oblivious perhaps to the riots that began to break out in the proletarian neighborhoods of Paris. The following day, Parisians started erecting barricades throughout the city, and more ferocious riots erupted. Charles X fled the capital, and the July revolution erupted violently. Fearful that the excesses of the 1789 revolution were about to be repeated, the government deputies made Lafayette head of a restored National Guard, charged with keeping order. On Tuesday, although Paris seemed to have quieted down, large crowds had assembled in various places. All the shops were closed. In the afternoon, the commanders of the troops of the First Military division of Paris and the Garde Royale concentrated their troops on the Place du Carrousel facing the Tuileries, the Place Vendôme, and the Place de la Bastille. Military patrols were established in order to maintain order and protect shops from looters. As the sun set, the fighting began. Parisians attacked the soldiers in the streets, some from the windows of their homes. At first, soldiers fired warning shots into the air, but before the night was over, twenty-one citizens were killed. Rioters shouted, Mort aux Ministres! À bas les aristocrates! (“Death to the ministers! Down with the aristocrats!”) The Pont Neuf (three hundred meters from Germain’s residence) was dark, while the sound of cannon and gunfire was booming louder. The rioting lasted well into the night until most of the street lamps had been destroyed. By Thursday, royalists were nowhere to be found. In only two days thousands of barricades had been erected throughout the city. The tricolor flag of the revolutionaries—the people’s flag—flew over buildings. After noon, the rioters took the Tuileries Palace and the Swiss Guards ran away. By mid-afternoon the Hôtel de Ville had been captured. A few hours later, a provisional government was in place. Although some disturbances occurred throughout Paris, the bloody insurrection ended that Thursday night. During those three days (July 27–29) known to the French as les Trois Glorieuses,16 many students from the École Polytechnique, hostile towards Charles X, actively participated in the battles with the police and the National Guard. It has been said that Libri also took an active part. His friend François Arago was a member of the provisional government, and he remained active in politics for many more years afterward. Charles X abdicated in favor of his grandson, but he was not allowed to take the throne. On August 9, the Duke of Orleans became Louis Philippe, King of the French. Also known as the “Citizen King”, the new monarch established the

16

The July Column or Colonne de Juillet, located on Place de la Bastille, commemorates the events of the Three Glorious Days of 1830.

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principle of national sovereignty over the principle of the divine right. His new Charter was a compromise between the opposition to Charles X and the Republicans. Many supporters of the king had fled and his ministers were taken and tried. Cauchy, who was a staunch adherent of the Bourbons, followed Charles X into exile. Parisians quieted down and life went back to normal. On September 27, Libri read a memoir related to the solution of a class of algebraic equations (Résolution d’une classe d’équations algébriques).17 Sophie Germain encountered Evariste Galois at the Academy of Sciences, when he was a nineteen-year-old student at the École Normale. Galois must have been somehow disrespectful, because Germain wrote to Libri about him saying: “He continues the injurious behaviour of which he gave you a sample after your best lecture at the Academy.” She added: “despite his impertinence, he displays a good disposition.” What do these words mean? That Galois misbehaved during Libri’s talk? Galois was an ardent revolutionary who participated in the riots of the July Revolution of 1830. Would this have had anything to do with his brazen conduct? Libri also took part in the July revolution; he was so active that a newspaper reported that he “fought bravely.” No record exists to tell us what Sophie Germain thought of the social fight or of her friend’s participation. Libri returned to Italy in November.

Germain’s Last Publications Sophie Germain came into contact with German mathematician August Leopolde Crelle that summer of 1830. Crelle went to Paris to study the teaching methods used by the French. His objective was to bring the model of the École Polytechnique to Germany, for he believed this was the route to train high quality teachers. An undated note suggests that Sophie invited Libri and Crelle to dine with her.18 Sophie Germain must have made an impression on Crelle because he published her last two memoirs (Fig. 11.1). In the January 1831 issue of Crelle’s Journal, we find one paper related to prime numbers,19 and the other addressing the curvature of surfaces.20 Almost immediately, French mathematician Jean Hachette published a review of Germain’s paper on curvature of surfaces.21 Hachette, a member of the Academy of Sciences, stated that Germain had wrongly attributed to Meusnier the formula used to determine the radius of curvature of an oblique surface. However, Hachette

17

Institut de France. Procès-verbaux. Tome IX, p. 511. Del Centina (2005), p. 12. 19 Germain (1831b), pp. 201–204. 20 Germain (1831a), pp. 1–29. 21 Hachette (1831), pp. 17–19. 18

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noted, Germain had identified Meusnier as a former pupil of the École Polytechnique, which was incorrect since Meusnier had died in 1793, before the École was founded. Hachette clarified that the formula in question was due to Charles Dupin, which had been published in 1807. Charles Dupin was a mathematician who made contributions to differential geometry and in particular invented the Dupin indicatrix, a concept that gives an indication of the local behaviour of a surface up to the terms of degree two. A pupil of Monge and collaborator, Jean Hachette also wanted to ensure that the reader knew of Monge’s discoveries, and especially Monge’s theorem of the lines of curve, because Germain had omitted that reference in her memoir on curvature. The second paper published in the same volume of Crelle’s journal was Germain’s comment on the irreducible prime number equation with which she began to correspond with Gauss in 1804. She had observed, in the final Section 7 on cyclotomy of his Disquisitiones, le beau théorème in the equation that she believed could be generalized. This is an equation that decomposes gradually into an increasing number of factors in such a way that the coefficients of these factors can be determined by equations of as low a degree as possible, until one arrives at simple factors, i.e., at the roots. Gauss proved that if n is any prime greater than 2, n
1Þ then 4 ðxx
1 ¼ Y 2  nZ 2 , where Y and Z are polynomials in x with integer coefficients; the sign on the right is + or −, depending on whether n is of the form 4k þ 3 or 4k þ 1. With the title “Note on how the values of y and z are decomposed
in the equation 2

xp
1


1Þ 4 ðxx
1 ¼ y2  pz2 and those of Y 0 and Z 0 in the equation 4 x
1 ¼ Y 02  pZ 02 ,”22 Germain extended Gauss’s result to the case of the more general equation: s ðxn
1Þ 4 x
1 ¼ Y 2  nZ 2 , where s is any positive integer. This was the essence of her 4-page paper, the last souvenir of her work in number theory (Fig. 12.1). Concidently, on 11 October 1830 Legendre read at a meeting of the Academy his memoir focused on the same irreducible equation that Gauss had derived in his Disquisitiones. Legendre’s memoir is titled, “The Determination of functions Y and Z that satisfy the equation 4ðxn
1Þ ¼ ðx
1ÞðY 2  nZ 2 Þ, where n is a prime number 4i  1:” This paper was later published in the Memoires of the Academy.23 Why Legendre did not mention Germain’s generalization? Most importantly, why did Germain choose to publish the irreducible equation that Gauss derived and not her research attempting to prove Fermat’s theorem? By this time, Sophie Germain was suffering intensely. Her illness was progressing rapidly; however, holding in her hands the crisp copies of her published papers, which Crelle had sent her, gave Germain the strength to write to Libri. In her letter dated 2 February 1831, Germain discussed the content of the above theorem, and also told him that she had changed doctors; the new physician assured p

22

Germain (1831b). Legendre (1832).

23

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The Last Years

Fig. 12.1 First page of Sophie Germain’s papers published in Crelle’s journal in January 1831

her that she would get better.24 However, Germain suffered horribly and it became impossible for her to do any work. When Germain learned that Galois had been expelled from the École Normale, she communicated this news to Libri, adding, “one hears that he is becoming totally insane.”25 Of course this event occurred much before Galois’s descent into the drunken madness that drove him to imprisonment, to falling in love, and to fight in the famous duel that ended his life prematurely the following year. In the first months of 1831, Paris shook again with civil unrest. Violent demonstrations were fueled by fears of foreign invasion and by heated political debates among people opposing the repressing laws imposed by the new regime. The brutal revolts were reminiscent of the violence experienced during the revolution of 1789. The government sent soldiers and the National Guard to dissolve the angry crowds. In May, fire hoses were used for the first time as crowd control techniques. Sophie Germain must have been impervious to all that while she waited in seclusion for her own end. In a letter dated 18 April 1831, Sophie shared with Libri

24

Del Centina (2005), pp. 11–12. Bucciarelli and Dworsky (1980), p. 122.

25

Germain’s Last Publications

197

her physical suffering, saying: “A prompt death would be a relief to me because I suffer from unbelievable pain, which leaves me not a moment’s rest.”26 Sophie was suffering immensely. Her last letter to Libri dated 17 May 1831, is heart-wrenching. Sophie wrote: “Je suis malade, Monsieur et très malade, j’ai fait beaucoup d’efforts pendant votre séjour ici pour ne pas vous fermer ma porte, mais le mal est bien augmenté depuis et je ne peut plus aujourd’hui ni recevoir des visites ni m’occuper. Je suis aux prises avec d’horrible souffrances ma vie est un vrai supplice aucune saison ne peut améliorer mon sort on me dit qu’avec beaucoup de tems et des soins je pourrai retrouver quelque repos.”27 [I am ill, Monsieur, and feel very sick; I made great efforts during your stay here not to close my door to you, but my pain had increased and I could not receive visitors. I am experiencing horrible suffering, my life is a real torment, no season can improve my fate, I am told with much time and care I can find some repose.] As she lay in her deathbed on 14 June 1831, a riot exploded on rue Saint-Denis, her childhood neighborhood. It degenerated into an open battle against the National Guard. The deadly revolts continued through June 15 and 16. The agonizing pain of her illness shielded Germain from that social revolution raging around her. It did not matter now. Sophie Germain died at one in the morning, on Monday June 27. Who attended her funeral? Legendre—her mentor, friend, and scientific collaborator—was already an old man of seventy-nine. He was still active in research and was working on the third and final supplement of his Traité des fonctions elliptiques et des intégrales eulériennes, which he completed in 1832. Her friend Fourier had died a year earlier. Navier, De Prony, Girard, Poisson, Biot, Poinsot, Arago, Savart, Cauchy, and Lacroix were acquainted with Sophie socially and knew her scientific work. Did they go to pay their respects? Cauchy had left Paris in September 1830 and was in Geneva at the time of her death. One would expect that Arago, as the secretary of the Academy of Sciences,28 would have attended in an official capacity, representing the organization that had bestowed on Sophie Germain its highest award twenty-five years earlier. Sadly, no one at the Academy reported Sophie Germain’s death.

26

Ibid., pp. 121–122. Del Centina (2005). 28 He took Fourier’s position on 7 June 1830. 27

Chapter 13

Unanswered Questions

Time keeps only the works that defend against it. —SOPHIE GERMAIN

There are many unanswered questions about Sophie Germain. Who taught her basic mathematics when she was a child? What exactly drove her to take a man’s name to communicate her work to the leading mathematicians of her time? Why did she pursue research in mathematical physics while her desire was to work in number theory? What gave her the inspiration and the extraordinary courage to compete in the most prestigious mathematical prize in Europe? Answering these and many other questions would help us paint a better portrait of the woman behind the mathematics. The letters she wrote shine a bright light on her intellect, her mathematical ideas and results, which she wanted to share with the recipients, but she said nothing about her personal life or her feelings for others. Other historians assume that Lagrange became Germain’s mathematical counselor. However, she never said he was, nor did he mention her as such. It is true that she studied his published lectures. He also corrected the initial equation that Germain submitted in her 1811 competition memoir, but this was done as part of Lagrange’s duty as judge. No letters have been found to make me believe that there was a working relationship between Germain and Lagrange before she was discovered or afterwards. Lagrange did not write about Sophie in correspondence with others, as Gauss did, and Lagrange did not mention Germain’s work in his memoirs as Legendre did. The following is the only written note found from Lagrange to Germain, dated 17 germinal (mercredi).1 This date from the Republican calendar corresponds to 6 April 1799 (17 germinal an VII). Sophie Germain was twenty-three years old. Lagrange présente ses respects à Mademoiselle Germain; étant de retour de la campagne où il a passé quelques jours, il se fait un devoir de la prévenir qu’il sera à ses ordres le 19 et le 20 (vendredi et samedi); il ne sortira pas ces jours-là, à moins que Mademoiselle Germain n’anime mieux qu’il vienne chez elle, auquel cas il la prie de vouloir bien l’en

See Œuvres de Lagrange, Vol. 14, p. 286.

1

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 D. Musielak, Sophie Germain, Springer Biographies, https://doi.org/10.1007/978-3-030-38375-6_13

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Unanswered Questions

avertir.2 [Lagrange presents his respects to Miss Germain; being back from the countryside where he spent a few days, a duty that prevented him to be at her disposal on the 19 and 20 (Friday and Saturday); he will not come out these days, unless Mademoiselle Germain prefers to come to his home, in which case he requests to let him know.]

From this short note, one can surmise that the relationship between Lagrange and Germain was not close. He was courteous, but he did not show the affectionate tone he used to communicate with others close to him. For instance, a few weeks before he died, while weak from the illness that took his life, Lagrange wrote to a Mademoiselle Julia de Cheux de Saint-Clair in a most tender manner. Let us compare the tone and choice of words Lagrange used to address Sophie with those he wrote to Julia in a letter dated 13 March 1813: Mademoiselle [Julia], J’ai reçu avec autant de plaisir que de reconnaissance, comme une marque flatteuse de votre bon souvenir, le beau présent que M. de Chaulieu m’a apporté de votre part. J’ai attendu son retour pour vous en remercier et vous envoyer en même temps un petit cadeau, que je vous prie d’accepter comme un faible hommage de mes sentiments pour vous, et comme un témoignage du désir que j’ai de conserver ceux dont vous voulez bien m’honorer. J’ai l’honneur de vous offrir l’assurance de mon tendre respect. [Mademoiselle, I received with as much pleasure as recognition, a flattering symbol of your good reminiscences, the beautiful present that Mr. de Chaulieu brought me on your behalf. I waited his return to thank you and send you a small gift at the same time, I ask you to accept it as a small tribute of my feelings for you, and as a testimony of the desire I have to retain those with which you honor me. I have the honour to offer you the assurance of my tender respect.]

No affectionate letter like this has been found from Lagrange to Sophie Germain. In 1799, Lagrange was already sixty-three and had retired from teaching at the École for health reasons. From what is known about Sophie Germain’s social life when she was young, one is inclined to believe that, if she visited Lagrange, she would have gone accompanied by her mother. Or perhaps Lagrange visited Germain at her home for what could have been a brief interchange. Either way, a meeting this formal would not have provided the proper environment to teach her. Sophie Germain was eager to establish contact with the leading research mathematicians and was rather good at approaching them in writing. Thus, one would expect that there would be many written notes from her to Lagrange. None has been found to date. What Sophie Germain thought of Lagrange is found in her Pensées Divers, where she recorded her ideas about science. She wrote, referring to Lagrange’s theory on the three-body problem: “Lagrange is satisfied, in regards to the action of the nearest planets, to give the method and the formulas; but this method is limited, and it is here that, alongside the genius of the individual, the weakness and the insufficiency of the means of the species are marked.”3 There is nothing personal in these words to suggest that Lagrange was her mentor or friend.

Cette lettre, dont l’original existe à la Bibliothèque national Acquisitions nouvelles, no. 4073, fo. 22), a été publiée par M. Charles Henry dans la Revue philosophique, t. VIII, p. 633, année 1879. 3 Stupuy (1896), p. 224. 2

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Unanswered Questions

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Sophie Germain did have several friends willing to help her in a more personal matter. The first letters from Gauss, addressed to M. Le Blanc, were mailed to the residence of Monsieur Silvestre De Sacy. It is evident that Germain knew De Sacy well enough to ask him to be her accomplice. In turn, Silvestre De Sacy understood her resons to use a man’s name and was willing to keep it secret. No biographer before has addressed this unusual arrangement. Antoine-Isaac, Baron Silvestre De Sacy, was a French linguist and orientalist. In 1781, he was appointed councillor in the cour des monnaies. In 1791, he became commissary-general. The following year, opposing the revolutionary ideas, de Sacy retired from public service and lived in seclusion in a country home near Paris. In 1795, he became professor of Arabic in the new school of Eastern languages. In 1799, he published a book in principes of grammar, which he dedicated to his young sons. Napoléon gave Silvestre De Sacy the title of Knight of Empire in 1809 and made him Baron of the Empire in 1813. De Sacy became secrétaire perpétuel de l’Académie des Inscriptions et belles-lettres in 1833. Sophie Germain began her career in mathematics in number theory, the purest of branches, and perhaps the most beautiful to someone of her exquisite sensibility. Germain’s first obsession was the proof of Fermat’s Last Theorem, which was the reason she communicated with Gauss. In 1808, Gauss stopped answering her letters, and shortly after Germain became involved in the study of elasticity, a branch of science that required an advanced knowledge of classical mechanics and variational calculus. Was the feeling of being ignored by Gauss a factor in her decision to engage in a different area of research? In 1808, German physicist Ernest Chladni visited Paris and gave public demonstrations of his vibrating plates. Did Sophie Germain attend one of those public shows? By her own admission, she became “fascinated by the sight of Chladni’s experiments.” This suggests that she attended one of his demonstrations. Did Germain ever talk with Chladni or communicated with him in writing while she was conducting her own experiments? It is unlikely. However, she did contact Charles Wheatstone in London while conducting her research with vibrating plates. Now, let us imagine Sophie Germain in 1809. She was thirty-three and desired to make a contribution to the science she had studied for some years now. That year the First Class at the Institute of France announced the contest for the prix extraordinarie de mathématiques. How did Germain find out about this? The topic of the contest must have been interesting enough to put her studies of number theory aside to begin conducting research on a rather different branch of mathematics. Years later, she admitted that she had studied Euler’s memoir with the sole desire to appreciate the difficulties of the problem described in the program. Initially Germain had no intention of submitting an entry to the contest. But she did. And she won the prize on 8 January 1816. For unknown reasons, she did not attend the public meeting to receive the prize. What happened to the gold medal Sophie Germain won? Did Delambre, the Perpetual Secretary of the Institute at the time, go to her residence to deliver the medal himself?

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Unanswered Questions

There is a mysterious receipt among Sophie Germain’s letters archived at the Bibliothèque Nationale de France. This note acknowledges receiving a memoir submitted for a prize competition proposed by the Academy for the year 1822: Institut Royal de France—Paris, le 8 novembre 1821 J’ai reçu une Mémoire pour concours au prix des mathématiques proposé pour l’Académie Royal des sciences pour l’année 1822, sur le meilleur ouvrage, on mémoire des mathématiques pures ou appliquées, qui aura paru, ou qui aura été communiqué à l’Académie, dans l’espace des deux années qui sous accordée cause concurrents ayant pour épigraphe: Illecebris harum quaestionum ita fui implicatus, ut eas deserere non potuerim, Gauss, que j’ai numéroté I, et dont j’ai délivré le présent récépissé.4 [I received a memoir for the prize competition of mathematics proposed by the Royal Academy of Sciences for the year 1822, for the best work on pure or applied mathematics, which will be published, or which has been communicated to the Academy, in the space of two years which, under given competing cause with the epigraph: Illecebris harum quaestionum ita fled implicatus, ut eas deserere non potuerim, Gauss. I numbered it I, and for which the present receipt is issued.]

Is this receipt for a memoir that Sophie Germain submitted and that has not been accounted for to date? Was this for work she did to prove the theorem of Fermat? It is quite possible. The most telling sign is the epigraph chosen to identify its author, which is a fragment of a quote by Gauss from the preface of his Disquisitiones arithmeticae, page ix. The full quote is Quod postquam tandem ex voto successisset, illecebris harum quaestionum ita fui implicatus, ut eas deserere non potuerim. [“When I succeed in this, I was so attracted by these questions that it was impossible for me to abandon them.”] Gauss wrote those words when referring to an extraordinary arithmetic truth (the theorem of art. 108 on the theory of residues), which he considered very beautiful and connected to more profound results, saying that he concentrated all his efforts to understand the principles on which it depended and to obtain a rigorous proof. Sophie Germain knew intimately Gauss’s Disquisitiones and those words must have resonated with her. It is not surprising if she would use that sentence to attach to her own work. However, no record exists that a memoir of hers was reviewed by the Academy of Sciences in 1821 or 1822. However, this would not be surprising, as it had happened before. When she sent her paper on the theory of vibrating plates to the Academy, despite the promise by Fourier that M. Cuvier would read it, as she had requested, nobody bothered to mention it again; her memoir was discovered in 1880 among De Prony’s papers. Becoming the first woman to win a coveted prize in mathematics made Sophie Germain well known in Paris and in the international scientific circles. Why did none of her contemporaries, other than Legendre and Libri, mention Sophie Germain’s contributions to mathematics? Why did Arago, for instance—who wrote extensively about Laplace, Fourier, and other scholars of his time—not include her

4

Correspondance de Sophie GERMAIN.

13

Unanswered Questions

203

or at least mention her name? In the last period of her life, Sophie Germain had an active social life and met many prominent people. Libri, for example, wrote to his mother about meeting Sophie at a party given by Arago at the Paris Observatory in 1825. Arago knew about her prize of mathematics, the highest honor and the greatest accomplishment for any scholar. I find it peculiar that Arago did not write about that. She studied the cahiers from the École Polytechnique, but did Sophie Germain ever attend the public lectures at the Collège de France? The classes were free and attended by many women. The Collège Royal was an institution of higher education open to female students since 1791, when astronomer Lalande became its director. On 13 July 1795, the school was renamed the Collège de France. It offered a comprehensive curriculum at a high level, including courses in a wide range of topics, from mathematics, physics, and astronomy, to art and geography. Faculty included J.A.J. Cousin, the author of Leçons sur le calcul differentiel et le calcul integral, and who taught physics and lectured on the Analysis infinitorum with applications to mechanics.5 Danish astronomer Thomas Bugge reported that, during his visit to Paris in 1798–99, he saw at the Collège “a party of ladies who, by clapping of hands, assisted in applauding the speakers and particularly at Cousin.”6 Was Sophie Germain among those ladies? At that time, she was a young woman of twenty-three, and she could have studied the topics taught at the Collège. But nothing indicates that at such age Sophie had the freedom to leave her parents’ home. From the first known letter written by Fourier to Germain, when she was forty years old, it is evident that an invitation to lunch was first requested of her mother, and it was implied that Madame Germain would accompany her grown daughter. Were her parents so overprotective that they prevented Germain from developing the social relationships needed to enhance her professional associations? Sophie Germain was thirteen years old when the French Revolution exploded in Paris and she came of age during the Terror. When she was twenty-eight, Napoléon Bonaparte took over, ruling as emperor for a period of more than ten years, and the Bourbons returned to France when she was a grown woman. How did these events affect Germain’s character? Was she apolitical? Was she a revolutionary or a royalist? Between the coup d’état of Napoléon and the second revolution of 1830, Paris experienced extremes in its political and economic states. When the Bourbon king returned to France in 1815, he was confronted with political disunity among the French people. There were liberals and royalists, and the liberals were divided into Republicans and Bonapartists. Some sympathizers welcomed the restoration of the monarchy, while others expected a reformed, constitutional monarchy. Still other people hoped to restore the glorious Ancien Régime. The first Restoration king, Louis XVIII, brother of Louis XVI, took a moderate path, avoiding the extremes of

5

Bugge (2003), p. 61. Ibid., p. 63.

6

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Unanswered Questions

the liberals and at the same time hoping to strengthen the restored monarchy. Nothing in Sophie Germain’s letters suggests a political sentiment, and thus we do not know what she thought of Bonaparte or the new king. Libri wrote a tribute to Germain in 1833. The six-page synopsis of her life and work was kind to the deceased and was somewhat embellished (as eulogies often are), but it was devoid of personal details. This is not surprising, since Libri had met Sophie Germain only a few times, and most of what he had learned came from her letters, in which she did not share intimate aspects of her childhood or her feelings. In fact, the letters she wrote were typically used to communicate only her mathematical work. Sophie Germain was born a Catholic,7 and her father and maternal uncle were church wardens, suggesting a close connection to her faith. Indeed, Germain’s writings reveal a deep spirituality and a belief in a Creator of the Universe (Createur de l’univers).8 She alluded to God in several paragraphs of her philosophical mussings. Another mystery veiling Germain’s story is her appearance. What did she look like? Was she tall or petite? Was she dark or fair? Why did none of her associates or admirers ever describe her countenance? It is strange that none of those who met her in person gave a visual rendering of the woman, not even Libri. There is no known portrait of Sophie Germain and no verbal depiction that can help us visualize the woman. To date, the only physical portrayal is the death mask (at the National Museum of Natural History) from which a bust was made by Zechariah Astrue, commissioned by the City Council of Paris. On 2 August 1890, the statue was erected in the main courtyard of the Lycée Sophie Germain. The sketch portrait published by Stupuy was drawn from the statute. Hence, this depiction may not necessarily resemble the living woman. Perhaps this tenuous visual rendering of Sophie Germain’s face is symbolic and appropriate. And thus, we shall continue to emphasize only Sophie Germain’s mathematical work, the product of her fine intellect.

7

Stupuy (1896), p. 398. Ibid., p. 96: Le Createur de l’univers n’a pas commencé; il ne doit pas finir: il est éternel.

8

Chapter 14

Princess of Mathematics

A taste for the abstract sciences in general and above all the mysteries of numbers is extremely rare: it is not a subject which strikes everyone; the enchanting charms of this sublime science reveal themselves only to those who have the courage to go deeply into it. But when a person of the sex which, according to our customs and prejudices, must encounter infinitely more difficulties than men to familiarize herself with these thorny researches, succeeds nevertheless in surmounting these obstacles and penetrating the most obscure parts of them, then without doubt she must have the noblest courage, quite extraordinary talents, and a superior genius. —GAUSS to GERMAIN

The leading number theorists at the turn of the nineteenth century were Legendre and Gauss, and right between these two great men we always find Sophie Germain. She carried out original work in number theory, for which Gauss was genuinely impressed. When he discovered her identity in 1807, Gauss wrote: “The scholarly notes, all your letters are so richly written, they give me a thousand pleasures. I studied them carefully, and I admire the ease with which you have penetrated all branches of Arithmetic, and the sagacity with which you generalize and perfect them.”1 Sophie Germain was perhaps the first mathematician to master Gauss’s Disquisitiones Arithmeticae. In her research notes one can observe a deep knowledge, starting with topics on congruences, which she used in her attempt to prove Fermat’s Last Theorem, and in her proof of the quadratic residue behavior of the prime 2. Germain also mastered the difficult Section 5 [Des formes et des équations du second degré], and specifically the composition of forms and the theory of ternary forms.2

1

Germain-Gauss Correspondence, Letter 7. Gauss wrote those words after he discovered that Monsieur LeBlanc was actually Mademoiselle Germain. 2 Goldstein and Schappacher (2007), p. 20. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 D. Musielak, Sophie Germain, Springer Biographies, https://doi.org/10.1007/978-3-030-38375-6_14

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Legendre supported Sophie Germain’s intellectual efforts and gave her a place in the history of mathematics by including her original theorem in his book on number theory. Yet, by her own admission, Germain feared “the ridicule attached to the title of woman scholar.”3 That is why, initially, she used the pseudonym of M. Le Blanc to communicate her mathematical works to Gauss and Lagrange. Where did that fear come from? It is true that, in the eighteenth and nineteenth centuries, women were still considered the “weaker sex” and incapable of deep intellectual work. However, many women through history excelled in the sciences and mathematics. Hypatia of Alexandria, for example, born in the fourth century, is considered the first woman scholar of considerable merit. What we know today about Hypatia was also written in the scientific literature that Sophie Germain had at her disposal. She must have read about Hypatia in Montucla’s Historie des mathématiques. Montucla first mentioned “la savante Hypatia”4 as a commentator of Diophantus’s work and provided a summary of Hypatia’s contributions, her life, and her tragic death.5 Hypatia was the daughter of the mathematician Theon who taught her mathematics and astronomy and engaged her in the computation of astronomical tables. It is believed that Hypatia devised an astrolabe, an instrument for measuring how high the north star is above the horizon to determine latitude. The astrolabe was also used for locating and predicting the positions of the Sun, Moon, planets, and stars. This was probably the first such device ever made. She acquired a comprehensive knowledge of mathematics and philosophy, and around the year 380 AD, Hypatia succeeded her father as teacher of mathematics and astronomy, becoming the head of the Alexandrian school. She taught the ideas of the Greek philosopher Plato, and the mathematics of Euclid and Diophantus. The best-known achievement of Hypatia was her commentaries on important books such as Euclid’s Elements and the Arithmetica of Diophantus. Scholars of her era respected Hypatia for her intelligence and vast knowledge. One of her students was philosopher Synesius of Cyrene, who became “the good bishop of Ptolemais” in the Cyrenaica. Synesius left behind a number of texts and letters that provide us with information about daily life in Alexandria and about the Christianization of the Roman Empire. His last letter to Hypatia, whom he called “the excellent teacher,” still exists. Her intellectual strength and success as teacher became Hypatia’s ruin, and her end was tragic. Amid the intrigue and corruption of Alexandria, Hypatia lived a pure life. Though much esteemed by the Alexandrians, she became an object of envy among the religious zealots, who considered her philosophical doctrines “heathen.”

In her letter to Gauss dated 20 Feb 1807, Sophie wrote: «le ridicule attaché au titre de femme savante». 4 Montucla (1756), pp. 315, 318. 5 Ibid., p. 326. 3

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During this time, the theologian Cyril became the bishop of Alexandria. He was an ill-tempered, quarrelsome, hasty, and violent man who fought with whomever he considered heathens and heretics. He detested Orestes, who was prefect of the city and a friend of Hypatia. Cyril regarded them both heretics. He communicated his hatred to the lower clergy, and especially to certain savage monks from the desert. The monks attacked Hypatia in the streets as she returned from a lecture. The learned woman was dragged and subjected to unspeakable torture. Her body was then torn to pieces and burned. This most horrendous act of savage bigotry occurred in the year 415. By the seventeenth century, after Copernicus unleashed a scientific revolution with his heliocentric model of the universe, scores of women were engaged in the study of the stars. Several of them, such as Caroline Lucretia Herschel and Maria Kunitz, made important contributions to astronomy. In France in the first half of the eighteenth century, the most notable female scholar was Gabrielle Émilie Le Tonnelier de Breteuil, Marquise du Châtelet, known for her study of Newton’s Principia.6 Although she did not develop theorems nor discover scientific principles, du Châtelet was a remarkable philosopher of science and a devotee of Newtonian mechanics. She translated Principia from Latin into French, adding her own views on the principles of Newton. Gabrielle Émilie le Tonnelier de Breteuil was born on 12 December 1706 in Paris. Her father, the Baron de Breteuil, was principal secretary and introducer of ambassadors to Louis XIV. As a child, Gabrielle Émilie had a high aptitude for languages. By the time she was twelve years old, she could read, write, and speak fluent German, Latin, and Greek. At age nineteen, she married the Marquis Du Châtelet, becoming Marquise du Châtelet. They had three children. When her husband joined a military regiment in Lorraine, the young marquise returned to Paris. She left her children behind, entrusted to nannies, as was the custom of noble families. When Du Châtelet’s own daughter was twelve, she was sent to a convent, where she remained until a suitable marriage was arranged for her. After a life of frivolity, the Marquise Du Châtelet returned to her studies. Among her friends was mathematician and philosopher Pierre-Louis Moreau de Maupertuis, known today for his mathematical principle of least action. He advocated Newton’s theories, to the chagrin of his colleagues in the Academy of Sciences. Maupertuis took Émilie as a student to teach her mathematics. After Maupertuis, his friend Alexis-Claude Clairaut gave her lessons. Clairaut was a mathematical genius; at the age of twelve, he submitted a paper to the Royal Academy of Sciences in Paris. At eighteen, Clairaut was admitted as a member of the Academy with an exemption from the King. When she was twenty-eight years old, Émilie Du Châtelet met philosopher Voltaire. They fell madly in love, and eventually, the two moved to her Château de Cirey to write about the philosophy of science. Their collaboration was as close as

6

Musielak (2014).

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their friendship. In the Introduction to the Elements of the Philosophy of Newton, Voltaire wrote that he and Émilie worked together in the book. At the time, Émilie du Châtelet was the only woman in France openly discussing the physics of Isaac Newton, whose theories were still not fully accepted by the French scholars. In 1744, Madame du Châtelet submitted an essay on the nature and propagation of fire7 to compete for the Academy prize. She did not win (Euler was the winner), but her paper was published among the winning memoirs. Émilie du Châtelet continued her study of mathematics to prepare for her translation of Newton’s Principia, which included a short exposition of the system of the world and an explanation of the main astronomical phenomena from Newtonian principles. Sadly, she died before her translation was published. Although Émilie du Châtelet’s efforts were limited to commentary and synthesis, her publications contributed to the progress made by Newtonian science in the first part of the eighteenth century. She was elected and inscribed in the register of members of the Academy of the city of Bologna on 1 April 1746. Very proud of this recognition, Madame du Châtelet felt equal to Laura Bassi and Maria Gaetana Agnesi, two women of science whom she admired. Laura Bassi was an Italian physicist who was initiated into mathematics and Newtonian physics by Gabriele Manfredi. In 1732, at age twenty-one, Bassi became the second woman to earn a doctorate and the first to hold a teaching post at a European university (Bologna). In 1776, she was professor of physics. Maria Gaetana Agnesi was an Italian mathematician. She wrote Instituzioni analitiche ad uso della gioventù italiana (1748), a textbook highly recommended in France for the instruction of differential calculus. In two huge volumes, Agnesi provided a comprehensive and systematic treatment of algebra and analysis, and new developments in calculus. In the eighteenth century, young women of the nobility were educated in diverse subjects, including science in some cases. For example, in 1760, mathematician Leonhard Euler began to write beautiful Letters to a German Princess on Diverse Subjects of Physics and Philosophy. The princess was a fifteen-year old named Friederike Charlotte von Brandenburg-Schwedt. Over the course of two years, Euler wrote two hundred-thirty-four letters to teach Princess Friederike Charlotte almost everything—physical science, astronomy, music, logic, theology, and philosophy. He explained many important concepts of physical science and provided their philosophical background in clear and concise manner that would be understandable to the young girl. In Euler and the German Princess,8 I discuss the nature and the scope of the topics that Euler addressed in those letters to Princess Friederike Charlotte. He wrote about gravity, astronomy and the laws of motion, the nature of sound and light, electricity and magnetism, always using a clear manner and using an effective pedagogical style without invoking equations or formulas. In that amazing book,

7

The title was Dissertation sur la nature et la propagation du feu. Musielak (2014).

8

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209

Euler thoroughly explained all the topics that should be included in an introductory textbook of physics of the time, including beautiful illustrations to clarify some concepts. In the same century of the Enlightenment, two French women astronomers were particularly known: Nicole-Reine Lepante, and Marie-Jeanne-Amélie Harlay de Lalande, the illegitimate daughter of famous astronomer Jérôme de Lalande. Madame Lepante was perhaps the most celebrated of all female astronomers in eighteenth century France. She was born in Paris on 5 January 1723. Her father was in the service to the nobility residing at the Luxembourg palace, and he may have helped to define the education and professional future of his daughter. At twenty-five, Nicole-Reine married M. Lepante, a watchmaker and “horologer to the king.” The Lepante were close friends with Lalande, who worked at the observatory in the Luxembourg palace. At the observatory, Madame Lepante observed and computed star positions. Her greatest work was in connection with the return of Comet Halley, observed in 1757. After its appearances in 1607 and 1682, the comet’s orbital period was believed to be 75 years. Lalande invited Clairaut, who had obtained a solution for the three-body problem, to apply his mathematical method to predict the exact date of the comet’s return. Clairaut began to work on the three-body problem in 1745, in particular on the problem of the Moon’s orbit. First, he concluded that Newton’s theory of gravity was incorrect, and that the inverse square law did not hold. He spent several years trying to convince the scientific world of this. Nonetheless, Clairaut knew that the reckoning of Halley’s comet required much more accurate approximations than had the problem of the moon. “If Madame Lepante will help me,” Clairaut allegedly said, “I might venture it, for besides her I know of no one who could render any assistance.”9 Lepante agreed. She followed the comet step by step, night after night. At each step, the combined disturbing influence of all the then-known planets had to be computed. The work lasted eighteen months. On 14 November 1758, Clairaut presented the results of their collaboration to the French Academy of Sciences. They had succeeded in their predictions. Other notable achievements attributed to Madame Lepante include the calculation of the ephemerides—tables of the positions of astronomical objects in the sky at a given time—for the Royal Academy of Sciences in Paris. She died in 1788. In 1790, Jérôme de Lalande published Astronomie des dames (Astronomy for Ladies), declaring that this book was a model for all women because of their high intellectual qualities. And in fact, women studied and worked with Lalande and the astronomers in the Paris observatory. Women in other parts of Europe were also known to assist their husbands or other male relatives in the pursuit of scientific research.

9

Musielak (2018).

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Louise du Pierry (or Dupiery), an expert in calculations of eclipses, continued Lepante’s work and computed tables for the lengths of day and night10 and tables of refraction for the latitude of Paris. She was the first woman appointed as professor of astronomy at the University of Paris in 1789, when Sophie Germain was thirteen years old. Lalande wrote that he dedicated his Astronomie des Dames to du Pierry, noting that she was “the most educated he knew.” A contemporary of Sophie Germain was astronomer Amélie Harlay de Laland who was born in Paris circa 1773. In 1788, she married astronomer Michel Lefrançais de Lalande, nephew of Jérôme de Lalande. In 1793, Madame Lefrançais prepared hourly tables for the French Navy. This work required huge calculations and earned its author an award and one of the medals given by the Lycée des Arts de Paris to distinguished researchers and artists. Madame Lefrançais cataloged more than ten thousand stars, a task that required thirty-six calculations per star, and performed other important astronomical work. When Sophie Germain was a teenager, Marie Anne Pierrette Paulze Lavoisier, wife and laboratory assistant of chemist mathematician Antoine Lavoisier, was known as an active partner in his research. She learned English to translate the technical works of chemists Priestley and Cavendish. Madame Paulze Lavoisier was a highly skilled draftswoman, engraver, and painter who had studied under the artist Louis David. She illustrated her husband’s Traité Elémentaire de Chimie, first published in 1789. The carefully drawn plates show her working as laboratory assistant to chemist Lavoisier. For example, an engraving shows Madame Lavoisier taking notes during one of his experiments on respiration.11 The drawing is rather realistic, depicting the lady sitting at a table, quill in hand. Her head is turned to observe the experiment, waiting to write down the measurements as they are called out by her husband or by another assistant. Madame Lavoisier also wrote the plan for what experiments were performed at the Arsenal laboratory on a particular day. Although she did not publish, her name appears as translator in some important works of chemistry. In the book published by her husband, there is no credit to her, although the plates are signed Paulze Lavoisier sculpsit to testify to her contribution. After her husband’s perished at the guillotine in 1794, Madame Lavoisier was arrested and spent sixty-five days in prison. Emerging from what must have been a torturous experience, grieving for her husband and fearing for her own life, Paulze Lavoisier recovered Antoine Lavoisier’s confiscated books and worked tenaciously to keep his works in print. Closer in age to Germain was Gabrielle Biot, the wife of physicist Jean-Baptiste Biot since 1797. While she was not a mathematician herself, Biot instructed her in science and mathematics. A competent linguist, Madame Biot collaborated with her

10

Tables de la durée du jour et de la nuit, calculées par Mme DU PIERRY. Paris, 1782. Hoffmann, R., Mme. Lavoisier, www.americanscientist.org/issues/pub/mme-lavoisier.

11

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Princess of Mathematics

211

husband in translating E.G. Fischer’s physics textbook from German into French.12 It was chemist Claude-Louis Berthollet who had asked Biot to make the translation, which was published as Physique mécanique in 1806. However, in line with the practice of the time, the book records Biot himself as the translator rather than his wife. Madame Biot was the daughter of Antoine François Brisson of Beauvais, inspecteur général du commerce et des manufactures, and sister of Barnabé Brisson, one of Biot’s classmate at the École Polytechnique. Madame Biot was just five years younger than Sophie Germain. It is rather likely that they knew each other. In 1817, a year after Sophie Germain won the prize of mathematics, Mary Fairfax Somerville visited Paris. Somerville was a famous Scottish science writer who published a popular account of Laplace’s Mécanique Céleste. She dined at Arcueil with Laplace, her “heroe.” Somerville had studied mathematics and astronomy, and was nominated to join the Royal Astronomical Society at the same time as Caroline Herschel. Somerville other works include On the Connexion of the Physical Sciences (1834), and Physical Geography (1848), which was used as a textbook for many years. The visit of Somerville was major social event for the Society of Arcueil.

Women and Science Education With rare exceptions, French women in the nineteenth century were not free to pursue scientific work, and their formal education was left to their own devices. At the onset of the Revolution, the reforms of the Constituent Assembly led to rapid disintegration of the French educational system. The Collège Mazarin was closed in 1790 and converted into a prison. When Sophie Germain was sixteen, Lavoisier, Condorcet, and other scientists participated in the debate on public education. Even though they all advocated that “education should be open to all and universal, and that the individual should be able to study whatever he wished,” the men of science were not thinking of young women like Sophie Germain. It is difficult to understand this reasoning, since these scholars had wives who were also involved in their scientific pursuits. On 3 September 1792, the Constituent Assembly decided to organize “a system of public education available to all citizens, free as to the choice of subjects indispensable to all men.”13 They charged Charles Maurice de Talleyrand-Périgord, a prominent member of the National Assembly, to assess the state of education and report to the Committee on Public Education. Talleyrand asked mathematicians Laplace and Condorcet for advice, and asked Lavoisier for a critique of his report. Condorcet, who was president of the Committee on Public Education, had

“Biot, Jean-Baptiste.” Complete Dictionary of Scientific Biography. 2008. Encyclopedia.com. www.encyclopedia.com/topic/Jean_Baptiste_Biot.aspx. 13 Poirier (1998), p. 337. 12

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presented a report to the Assembly on 20 April 1792. It included recommendation for one hundred fourteen institutes to offer a more complete education to eighty thousand male students from fifteen to eighteen years old for a four-year program. Nothing was planned for girls at this level.14 Regarding the education in applied science suggested by Lavoisier, the curriculum planned for girls was deplorable: They will learn the arts that they have been exclusively destined to exercise; they will be taught everything concerning needlework, spinning, and knitting. They will be instructed in food preparation, household management, caring for the sick, and the physical education of children. The principles of moral philosophy will also be developed for them and they will be given some notions of history and local geography. In a word, they will be given the basic principles of what makes up the beautiful in the arts of taste and the basic amenities of life.15

Years later, Napoléon reshaped the French educational system, but, again, it did not benefit women. Since the establishment of the Consulate, Bonaparte wanted to regain control of education and reign over the progress of private “secular” or religious education after the signing of Concordat. The Law of 1st May 1802 (11 Floreal X) established the creation of schools and organized primary, secondary, and superior schools. In August 1800, shortly after becoming consul, Napoléon initiated the French Civil Code. Like many philosophers and the majority of active revolutionaries, Napoléon favoured a state system of public education. The curriculum would be secular and schools would be managed under the direction of the state instead of the Church. Women, of course, were excluded. “Marriage is their whole destination,” Napoléon once wrote. He thought that women did not need education, saying: “all they need is religion.” The emperor followed Rousseau, who believed that women were vastly inferior to men. This philosophy was the basis of the Napoleonic Code regarding women. Through the Napoleonic Code, the legal right of men to control women was affirmed. Although the basic revolutionary gains—equality before the law, freedom of religion, and the abolition of feudalism—remained, the Code ensured that married women in particular owed their husband obedience. Married women were forbidden from selling, giving, mortgaging, or buying property. After the proclamation of the Empire, Napoléon renewed his interest in education, convinced of the importance of education to instill patriotism in children. His idea was to reaffirm the role of the state in education. Anticipating the needs of its military, one of the first decrees of the emperor (16 July 1804) placed the École Polytechnique under military authority. Napoléon also had the idea to create a kind of congregation of teachers that would take the name “University.” The law of May 1806 focused on three articles, the first of which established that “There will be formed under the name of Imperial University a body devoted 14

Ibid., p. 339. Ibid., p. 343.

15

Women and Science Education

213

exclusively to education and public education throughout the Empire.” On 17 March 1808, the law was approved, establishing the Imperial University (Université impériale). This decree also restored the École Normale, which was closed soon after it was established in 1795. This school had the mission to train high school (male) teachers, both in letters and science. It re-opened in 1810, in the building of the current Lycée Louis-le-Grand in Paris. The debate about the proper education of women continued. The French Code survived, basically unaltered, for more than 150 years. Finally, in 1965, French wives obtained the right to work without their husband’s permission. In 1970, husbands forfeited the rights that came with their status as head of the family. Regarding formal scientific education, let us remember that it was not until 1972 that women were finally admitted to the École Polytechnique.

Sophie Germain Legacy Unlike her female predecessors and contemporaries, Sophie Germain was an impressive mathematician, making contributions to both number theory and the theories of vibration and elasticity. She was able to walk with ease across the bridge between mathematical physics and the world of pure mathematics. Sophie Germain was the first woman in history to win a major mathematical competition, and also the first and only woman to make an important contribution to the proof of Fermat’s Last Theorem. Sophie Germain’s interest in Chladni’s vibrating plates drove her to carry out mathematical analysis involving physical phenomena rather than purely abstract reasoning. Her original hypothesis to explain the elastic vibrating plates served as the starting point for later researchers, most notably Kirchhoff. Sophie Germain’s work in number theory had the goal to prove Fermat’s Last Theorem, for which she developed a grand plan, which included unique analysis and related results such as finding large size of solutions, p2-divisibility of solutions (i.e., Germain’s Theorem, applicable to Case 1), and special forms of the exponent p. She pioneered the method of finding infinitely many auxiliary primes. Contemporary scholars found that Sophie Germain’s unique results are intertwined in her manuscripts, largely because the hypotheses that require verification overlap. For example, the nonconsecutivity condition on pth-power residues modulo an auxiliary prime h, which is found in the statement of Sophie Germain’s Theorem, is also key to her grand plan. Other mathematicians have pursued this approach, even today. Laubenbacher and Pengelley studied in detail Germain’s grand plan to prove Fermat’s Last Theorem and elucidated from her manuscripts the detailed methods she developed, the progress she made, and its difficulties. Finally, the researchers compared Sophie Germain’s methods, together with her explanation and claims, to the work of Gauss and Legendre’s. Their conclusion was astonishing: Sophie Germain’s accomplishments are much broader, deeper, and more significant than

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has been realized. “Sophie Germain was a much more impressive number theorist than anyone has ever known.”16 Thus, I feel amply justified in calling Sophie Germain the Princess of Mathematics. What matters to posterity is that, for the first time in the history of science, a woman had won a prize that had only been bestowed to the most illustrious mathematicians such as Euler, Lagrange, and Cauchy. It was a great accomplishment; yet, her contemporaries rebuffed Sophie Germain. In retrospect, she deserved the accolades, together with Poisson and Navier, for taking upon a scientific challenge and for advancing the required theories of vibrating elastic plates. Sophie Germain lived during a most tumultuous and tragic era in history when her country was ravaged by two bloody revolutions. During her lifetime, France was ruled by four kings and governed by a republican body that was forcefully taken over by a combative emperor with a grandiose sense of power. Sophie Germain earned her claim to immortality through her work in two distinct branches of mathematics. She lovingly carved her name in the annals of number theory, the purest branch, which was called the queen of mathematics by Gauss, the prince of mathematics himself. She was the first woman to undertake the proof of Fermat’s Last Theorem after Euler, work that yielded the Sophie Germain’s Theorem, and the Sophie Germain primes. Sophie Germain, who had not attended a university and worked outside the institutions that were closed to women, was a mathematician and physicist. She was able to apply a method based on variational calculus to derive a fundamental equation in applied physics. Germain deserves a prominent place in the history of mathematics for being the first woman to develop a mathematical theory of the vibrations of elastic plates, inspired by Chladni’s experiments. Although her explanation was incomplete, the prize of mathematics bestowed upon her in 1815 was well deserved. That prize acknowledged that her mathematical work signified essential progress toward the theory of elasticity and vibrations. Germain founded a new field in mathematical physics, and it provided impetus for the research carried out by Poisson, Navier, and Cauchy. Comte wrote, when addressing the topic of acoustics, “The memorable impetus given to science, in this respect, by the genius of an illustrious contemporary (Sophie Germain), whose recent loss is so unfortunate, led, it is true, the mathematicians to consider, in recent times, a case more difficult and closer to reality, the vibration of surfaces.”17 Many years after her death, a commemorative plaque was erected on the building where she lived in 1831. The townhouse is still standing on 13 rue de Savoi (Fig. 14.1), across from the Seine River and not far from rue Saint-Denis where she grew up. The plaque on the right side of the arched entrance to the building reads: “Sophie Germain, philosopher and mathematician. Born in Paris in 1776. Died in this house on 27 June 1831.”

16

Laubenbacher and Pengelley (2010), p. 692. Comte (1864), p. 604.

17

Sophie Germain Legacy

215

Fig. 14.1 House on 13 rue de Savoi where Germain lived in 1831. a Showing the plaque on the front entrance. b Looking up from courtyard inside the building, searching for Sophie’s window

Sophie Germain earned the title of philosopher on the merits of her Considérations générale sur l’état des sciences et des lettres, a philosophical composition that contains her ideas about the general state of sciences and literature. Germain left behind beautiful statements concerning her beliefs in a creator of the universe, the soul, and the human spirit. Gauss and Legendre recognized publicly that Sophie Germain was a mathematician. Gauss wrote to Olbers on 3 September 1805: “through several letters from Le Blanc in Paris who studies my Disquisitiones arithmeticae with true passion, has completely familiarised himself with them, and shared quite a few nice comments about them with me.”18 This Le Blanc was of course Sophie Germain, who would have been flattered to know that the highest-ranking scholar of her time—the one she admired and sought out as her mentor—spoke on those terms about her. France has honored Germain in various ways. A little Parisian street and a high school are named after her. On 1 March 1882, the first high school for young girls (École Primaire Supérieure de Jeunes Filles) was opened with sixty-five students. By the 1970s, the enrollment had increased to sixteen hundred. Up until 1888, it was known as the School of Rue de Jouy; afterwards it was “named after Sophie Germain, mathematician and philosopher who had to instruct herself during a time when the status of women who wanted an education was a handicap.” The Lycée Sophie-Germain is located at 9 Rue de Jouy, in the 4e Arrondissement of Paris; it is a four-minute walk (260 m) from the Pont-Marie over the Seine River. It is now a co-ed school. Sponsored by the Institut de France, the Sophie Germain Foundation was created in 2003. It awards an annual prize of mathematics to young researchers. In 2006, a

18

Patterson (2007), p. 507.

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Fig. 14.2 Commemorative stamp to honor Sophie Germain, 2016

prize in the amount of eight thousand euros was given for scholarship assistance to the students of the Lycée Sophie Germain. In 2016, France issued a postal stamp to commemorate the 240th anniversary of Sophie Germain’s birth. The stamp was conceived by French artist and illustrator Edmond Baudoin, and engraved by French stamp designer Elsa Catelin. As shown in Fig. 14.2, Baudoin paid tribute to the mathematician while at the same time highlighting the woman. There are many unanswered questions about her personal life, her self-induction into the world of mathematics, and the work that she did in private. Did Libri indeed collect her important manuscripts and then lost some of them? What would we find in those notes besides her famous theorem? The mystery of this intriguing woman who impersonated a man to reach out to Lagrange and Gauss will continue to haunt me. One fact comes across crystal clear after examining the history of her life. As isolated as she was, Germain rebelled by refusing to be simply curious about mathematics. With her distinctive research work and contributions to science and mathematics, she opposed the common belief of her time that women were not capable of independent scientific work. Sophie Germain forever changed the notion of the woman scholar, and for all that she earned the title of revolutionary mathematician. Sophie Germain is unique in many respects. Not content with being a casual student, she entered the world of mathematics discovered by the greatest mathematicians of her time, not by accident but because she was bold and reached out to them to share her own insights. Barred from the academies and the centers of learning, Germain conducted her research in seclusion undeterred and passionate. Despite her isolation, her scientific work contributed to the development of number theory and the mathematical theories of elastic vibrating plates. Germain left us an imperishable work that we continue to study.

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To this day, the mystery that surrounds the private life of Sophie Germain remains unsolved. She revealed, however, her soul and her intellect through the scientific contributions. Perhaps Sophie Germain would have preferred that we emphasize only her intellectual strength because that is her true legacy.

Germain-Gauss Correspondance

Letter

Year

Date

1*

1804

21 November

Le Blanc to Gauss

Main Topics

2*

1805

16 June

Gauss to Le Blanc

3

1805

21 July

Le Blanc to Gauss

4*

1805

20 August

Gauss to Le Blanc

5* A* B*

1805 1806 1806

16 November 27 November 23 December

6*

1807

20 February

Le Blanc to Gauss Chantel to Pernety Pernety to Germain Germain to Gauss

7*

1807

30 April

Gauss to Germain

8*

1807

27 June

Germain to Gauss

9*

1808

19 January

Gauss to Germain

10

1808

19 March

Germain to Gauss

11

1809

22 May

Germain to Gauss

Power residues theorems, (last) theorem of Fermat Fundamental theorem (Article 131 of Disquisitiones) Theory of binary forms and reduction of ternary forms (Article 267 of Disquisitiones) Gauss sends copy of his 1799 dissertation Quaternary forms Comission to Brunswick to ensure military protection to Gauss during France war with Prussia Reveals her identity. More theorems and proofs Most important letter Gauss wrote. It includes Gauss’s Lemma and theorems on cubic and quadratic residues Germain proves Gauss’s Lemma and states five propositions of her own Last letter from Gauss, his farewell? Correcting her previous proof of Gauss’s theorem Germain’s work on cubic and bi-quadratic residues (continued)

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 D. Musielak, Sophie Germain, Springer Biographies, https://doi.org/10.1007/978-3-030-38375-6

219

220

Germain-Gauss Correspondance

(continued) Letter

Year

Date

12

1809

26 May

Main Topics Germain to Gauss

Germain thanks Gauss for memoir on planetary orbits 13 1819 12 May Germain to Gauss Her attempt to prove FLTa 14* 1829 28 March Germain to Gauss Germain outlines her idea on curvature of surfaces Note 1 Sophie Germain’s letters are preserved at the Niedersächsische Staats- und Universitätsbibliothek in Göttingen, Germany. Ref: Cod. ms. Gauss Briefe A: Germain, n. 9. Abteilung Handschriften und Seltene Drucke Note 2 Del Centina and Fiocca (2012) published full transcripts of these letters (in French) * Letters translated in the following pages a See Laubenbacher and Pengelley (2010)

Germain-Gauss Correspondance

221

― Letter 1 ― Le Blanc (Germain) to Gauss

Paris,

21 November 1804

Monsieur, Your Disquisitiones arithmeticae has been the object of my admiration and of my studies for a long time. The last chapter of this book contains, among other remarkable things, the beautiful theorem I think it can be

contained in the equation for a prime

generalized as

and any number . I

the first one, I looked for how the method you used in art. 357 could be applied to the case I considered. I did this work with all the more pleasure as it gave me the opportunity to familiarize myself with this method which, I have no doubt, will be in your hands the instrument of new discoveries. I have added to this article some other considerations. The last one whose is related to the famous equation of Fermat and impossibility in integers has only been demonstrated for . I think I have managed to prove this impossibility for where is a prime number of the form of . I take the liberty of submitting these essays to your judgment, hoping that you will not disdain to enlighten with your opinion an amateur enthusiast of the science which you cultivate with such brilliant successes. Nothing compares with the impatience with which I await the sequel of the book that I have in my hands. I was informed that you are working on it right now; I will spare nothing to obtain it as soon as it appears. Unfortunately, the extent of my mind does not respond to the vivacity of my tastes, and I feel that there is a kind of temerity to annoy a man of genius, when [I have] no other title to his attention than the admiration necessarily shared with all its readers. In re-reading the memoir of M. de la Grange (Berlin 1775), I was

This observation is a new proof of the advantage of your method, which, applied to all the values of , gives for each case, values of and independent of trial and error. If, knowing the values of we want to have those of

and

and

in the equation

,

222

Germain-Gauss Correspondance

it is clear that it would suffice to change the signs of all the terms of and which contain powers of , whose exponent is odd. I do not want to exhaust your attention by multiplying the remarks [on topics] which I learned from your book. If I can hope that you welcome these that I have the honor of communicating to you, and that you do not find them entirely unworthy of a reply, please address it to M. Silvestre de Sacy, a member of the National Institute, rue Hautefeuille in Paris. Believe, Monsieur, that I attach a high value to your opinion, and accept the assurance of the profound respect of Your very humble servant and very diligent reader. Le Blanc

NOTE: For the Addendum that Germain attached to this letter, see Del Centina and Fiocca (2012), pp. 638–645. ― Letter 2 ― Gauss to Le Blanc (Germain)

Brunswick,

16 June 1805

Monsieur, I must ask you a thousand times for forgiveness for having waited six months without answering the letter with which you have honored me. Certainly, I would have been eager to testify to you at once how much I am attentive to the interest you take in the researches to which I devoted the most beautiful part of my youth, which were the source of my most delightful enjoyments and which will always be dearer to me than any other science. But I flattered myself from time to time to be able to gain enough leisure time to put in order and communicate to you in writing one or another of my other arithmetic researches, to give you, in a way, the pleasure you have given me with your communications. My hope has been vain. It is mainly because my astronomical occupations, at present, absorb almost all my time. I reserve myself, however, to discuss with you the mysteries of my darling arithmetic, as soon as happily I am able to return. I read with pleasure the things you wished to communicate to me; I am glad that Arithmetic acquires in you a rather skillful friend. I liked especially your new proof for prime numbers for which 2 is [a certain power] residue or non-residue, was extremely pleasing to me; it is very fine, although it seems to be isolated and cannot be applied to other numbers. I have often regarded with admiration the singular enchainment of arithmetic truths. For example, the theorem I call fundamental (article 131) and the particular theorems concerning residues 1, ± 2, intertwine with a host of other truths, where they

Germain-Gauss Correspondance

223

would never have been sought. In addition to the two proofs I have given in my work, I am in possession of two or three others, which do not compare to those in regard to elegance. I note with great regret that the other occupations in which I am engaged now do not allow me at all to devote myself to my beloved arithmetic. It may only be after several years that I can think of publishing the rest of my research, which will easily fill one or two volumes similar to the first. But I would think I had not lived enough, if I died without having completed all the interesting researches that I once gave myself. Moreover, in Germany, the publication of such a work has its difficulties: whatever may be said, the taste for pure mathematics, if one seeks depth, is not too general. Our booksellers do not associate with these kinds of books, and I am not rich enough to publish on my own and subject myself to the dishonesty of foreign booksellers, as it happened to me on the occasion of the first volume. A monsieur Duprat, for instance, who is named bookseller for the Bureau des longitudes in Paris, received from me, almost three years ago, copies for the value of 680 francs; but I never received a penny from him, and he did not even bother to answer my letters. Perhaps you could give me some information as to how we could engage this man to do his duty. Accept, monsieur, the expression of my highest consideration. Ch.-Fr.GAUSS P.S. May I ask you to send the enclosed letter to Orleans? ― Letter 4 ― Gauss to Le Blanc (Germain)

Brunswick,

20 August 1805

I take advantage of the kindness of Mr. Grégoire to offer you, with many thanks for the communications of your last letter, a copy of a petit memoir which I published in 1799 and which probably will be unknown to you. You wanted to know everything I wrote in Latin. This piece is the only one, besides my arithmetical research, and at the same time the one that first appeared, though the publishing of my Disquisitiones would have been carried beyond that. I am now engaged in perfecting some new methods in regard to the calculation of planetary perturbations; these and the methods I used to calculate the elliptical [orbital] elements of the different new planets will probably provide the material for the first work [in this field]. I greet you cordially, Ch. Fr. GAUSS

224

Germain-Gauss Correspondance

[Gauss enclosed a copy of his Demonstratio nova theorematis: omnem functionem algebraicam rationalem integram unins variabilis in factores reales primi vel secundi gradus resolvi posse. Helmstadt 1799.] ― Letter 5 ― Le Blanc (Germain) to Gauss

Paris,

16 November 1805

Monsieur, I am guilty for taking so long in thanking you for the letter you have honored me, and for sending me the memoir you have kindly enclosed. However, it is not my fault: the package was not given to me until eight days ago. M. de Sacy had been on a trip for more than two months, and the parcel was neglected at his home. It is true that, not expecting from you an answer so promptly, I had not taken care to inquire about the letters addressed to me. Your memoir pleased me so much as I already knew about it by a quick reading of one provided to me by one of the scholars to whom you sent it, long ago, and that always having the desire to study it, as one should do all the works that come out of your pen, I had tried in vain to order it from Leipzig where I received for answer that the edition was exhausted. The indulgence you continue to bestow in me encourages me to show you some more of my new research. After having reduced, as you indicate, the ternary forms whose determinant is zero to the binary forms, I have sought if this property did not extend to the quaternary forms, that is to say if these forms were not likely to be reduced to ternary forms when their determinant is zero, and then I examined some other properties of these forms and their adjoints. I believe that is the determinant of a form composed of a number is the determinant of the adjoint of this form. Thus, of variables, for the determinant of the ternary adjoint and that, is the determinant of the quaternary according to my calculations, adjoint. This analogy is probably not sufficient to establish the generality of the proposition of this form. But we see at least that the determinant being composed of products of the order and the is of the same order as the determinant of the adjoint, that is, of the order . These two propositions, that the adjoint is of the order , and , seemed to me the coefficients of the decisive adjoint of the order to be derived from the general nature of the forms and their adjoints. I regard as a favor the permission you grant me to communicate to you my feeble attempts, persuaded that you will be kind enough to

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warn me of errors which might escape me in the kind of research where you are the only enlightened judge that one can consult. The new information I have found about the bookseller Duprat is less than satisfactory. His successor said that he had long since completed his payments, the proceeds of which were immediately disbursed. He has retired to a small village where he has a meager income, and the general opinion of all the people I have consulted is that it would be almost impossible to obtain any money out of him. I did not think it necessary to communicate these results to you, because I do not see that we can take advantage of them, and that I was waiting to write to you again until you would have given me permission to do so. The delay caused by the delivery of your letter deprived me of my earlier thanks and the expression of my deepest respect. Le Blanc

Note: Germain attached eight additional pages of her analysis. ― Letter A ―

Letter from Chantel to General Pernety Brunswick, 27 Novembre 1806 My general, As soon as I arrived in this city, I took care of fulfilling your commission. I asked several people about the house of Monsieur Gauss, with whom I was to inquire on your and Mademoiselle Sophie Germain’s behalf. He replied that he had not had the honor of knowing you or the lady, as he only knew Madame Lalande in Paris. After having spoken of the matter contained in your instruction, he seemed to me a little confused, and charged me to thank you very much for the attention you paid to him. I asked him, if he wanted to write to Paris, to give me the letter, that I would mail it to you, because you were responsible for having it forwarded to its destination. He did not answer yes or no to this proposal. I departed his house, leaving him with his wife and child. I went to General Buisson, the governor of that city, to recommend him, and, above all, that I had the honor of meeting M. Buisson, the old general of the division. This general then assured me that he’d do everything for him, and to invite M. Gauss to dinner. The commander of the place who was there at that moment told me that this man had already been recommended to him by several persons of merit. I left and I went back to Mr. Gauss to invite him to come and dine with

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me at the governor’s place. Having promised it, in an hour from now I will take him and go together. The fact is that he [Gauss] will have all the esteem and kindness from the governor and the commander of the place. On the way I will endeavor to speak to him so that he [Gauss] may write to you on the manner in which I have discharged my mission, and at the same time that he writes to Paris, if he desires; I gave him for this purpose your address. His is: Monsieur doctor Gauss housed at Ritter, Steinweg n° 1917 in Brunswick. He was in good health, and told me that he was a little afraid at the moment when the troops had returned, but that he remained at Brunswick alone. I reassured him, and I do not doubt that the governor and the commander of the place will reassure him much better on this matter. I ran to the post night and day until this moment. This circumstance obliges me to remain here this afternoon; and tomorrow morning I leave to go to my destination. Deign to accept, General, the sentiments of the deepest respect with which I have the honor of being, etc. CHANTEL, Chief of battalion ― Letter B ―

Letter from General Pernety to Sophie Germain Hotel near Breslau, 23 December 1806 Mademoiselle, I cannot answer better the request which your love for scholars made to me than by sending you the letter of the officer of artillery whom I had charged to bring news of M. Gauss, in Brunswick. I wish to fulfill your wishes for this emulator of Archimedes to be better treated than he was, as you will see. I hope to be in better position to be able to carry out more of your interesting commissions. I will certainly do these better than perform the requests to purchases of frivolities (chiffons) from foreign countries, that unfortunately sometimes are wrongly entrusted to me. Here I am holding a siege, hearing and scouring the thunder, burning houses, churches, because the steeples are good sights for the bombs, doing harm to those who did none to me, and whom I do not know; but that’s my profession. In turn, I am overwhelmed by rounds of bullets, shells and bombs, and everything is going well. Ultimately, the obstinate governor of Breslau will perhaps one day do his part, and so he will do well for the city and for us.

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I trust that your health has improved, and that your parents and your sister stay well. Such are, at least, the most sincere wishes of your devoted servant and admirer. J. Pernety ― Letter 6 ― Germain to Gauss

Paris,

20 February 1807

Monsieur, The consideration due to superior men suffices to explain the care I took in begging General Pernety to make known, to whom he would judge proper, that you be entitled to the esteem of every enlightened government. In realizing the honorable mission I had entrusted to him, M. Pernetty told me that he had made you known my name; this circumstance prompts me to confess to you that I am not so entirely unknown to you as you believe; but that, fearing the ridicule attached to the title of learned woman, I borrowed the name of M. Le Blanc to write to you and to communicate to you notes which, doubtless, did not deserve the indulgence with which you were kind enough to answer them. The gratitude I owe you for the encouragement you have given me, by showing me that you count me among the amateurs of the sublime arithmetic of which you have unreaveled its mysteries, was for me a particular reason for seeking news from you in a moment when the conflicts of the war might inspire fears, and I learned with genuine satisfaction that you remained in your home as quietly as circumstances permitted. I fear, however, that the consequences of these events will not deprive us for too long of the works you are preparing in astronomy, and, above all, of the continuation of your arithmetical researches; because this part of science has a particular attraction for me and I always admire with great pleasure the sequence of the truths exposed in your book; unfortunately, the faculty of thinking intelligently is an attribute reserved for a small number of privileged minds, and I am not sure of finding any of the developments which, for you, seem an inevitable consequence of what you have already discovered. I add a note intended to testify to you that I have a taste for analysis that I developed after reading your work, and which inspired me and gave me the confidence of addressing you with my feeble attempts, without any other recommendation to you but rather trusting

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on the benevolence granted by the scholars to the admirers of their work. I hope that the singularity, of this which I make an admission today, will not deprive me of the honor which you have granted me under an assumed name, and which you will not disdain to devote a few moments to send me news from you, believe, Monsieur, I am interested in knowing; and receive the assurance and the sincere admiration with which I have the honor to be, Your very humble servant, Sophie Germain P. S. My address is: Mlle Germain, at her father’s house, rue SainteCroix-de-la-Bretonnerie, no 23, in Paris.

Note: Germain enclosed 4 pages of her proofs. ― Letter 7 ― Gauss to Germain

Brunswick,

30 April 1807

Your letter of 20 February, which did not reach me until the 12th of March, was for me the source of as much pleasure as surprise. How sweet is the acquisition of such a flattering and precious friendship to my heart! The lively interest you have taken in my fate during this deadly war deserves the most sincere gratitude. Assuredly, your letter to General Pernety would have been very useful to me had I been in need of having recourse to special protection from the French Government. Fortunately, the events and the aftermath of the war have not touched me so far, although I am confident that they will have a great influence on the future plan for my life. But how can you describe my admiration and astonishment at seeing my correspondent, M. Leblanc, metamorphosed into this illustrious personage, who gives such a brilliant example of what I would have difficulty believing? The taste for the abstract sciences in general and especially for the mysteries of numbers is very rare: one may be surprised; the enchanting charms of this sublime science reveal themselves in all their beauty only to those who have the courage to deepen it. But when a person of this sex, who by our manners and our prejudices, must meet infinitely more obstacles and difficulties, than men, to become familiar with its thorny researches, nevertheless knows how to overcome these obstacles and to penetrate this that they have moreover hidden, it is undoubtedly necessary, that she has the most noble courage, talents quite extraordinary, a superior genius. In fact, nothing could prove to

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—

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There are several notable consequences to this proposition; among other things, it gives the idea of extending induction, by which we gather special cases of the fundamental theorem as far as we like, which could not be done by the methods outlined in art. 106–124. I have given in my work two rigorous proofs of this famous theorem, and I have three others all entirely different from each other; two of them can each be conducted in two different ways; so, I could argue that I can prove it in seven different ways. The other proofs I prefer for their elegance to the two already included in my work, will be published as soon as I find the opportunity. By the way, in the first proof, which appears in the fourth section, there is a slight error which I did not noticed until it was too late to acknowledge it. We must therefore make the following correction.

I would have answered your letter earlier, but the discovery of a new planet by M. Olbers distracted me a little. In my first attempt to calculate its orbit, I find its movement considerably faster than that of Ceres, Pallas and Juno, namely 978" per day. The inclination of the orbit is 7° 6'. The eccentricity 0.1. This planet is brighter than Ceres, Pallas and Juno, and I hope to find it among the historical celestial observations, perhaps even those of Flamsteed [John, 1646–1719, English astronomer]. I have just completed an extensive work on the methods, which are my own, to determine the orbits of the planets. But even though I wrote it in German, I find it very difficult to hire a bookseller. The war has suspended all commerce, [and] many of our largest publishers have refused. I am now dealing with another who is a little more courageous. If he finds his footing in this business, perhaps he will be encouraged and take the risk of publishing a second volume of my disquisitiones. Continue, Mademoiselle, to favor me with your friendship and your correspondence, which is my pride, and be persuaded, that I am and will always be with the highest esteem, Your most sincere admirer, Ch. Fr. Gauss Brunswick, 30 April 1807, my birth day.

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Note: Two weeks earlier, Olbers wrote to Gauss: “I thank you again for the nice name Vesta that you have given my planet.” ― Letter 8 ― Germain to Gauss

Paris,

27 June 1807

Monsieur, I owe you a thousand thanks for the flattering things with which your last letter is filled, I take them only as an encouragement and certainly, my greatest ambition will always be not to show myself unworthy of the honor you bestow on me in promising me to continue a correspondence to which I have everything to gain. You took the trouble to look at an inverse proposition that I sent you and to point out the mistake I made. I see the truth of your observation and thank you for having given me this advice; if I did not fear to abuse your consideration, I would ask you to do me the same service in the future, as I will always consider it a sign of your kindness. How much pleasure I had reading your three theorems on residues! I have sought their demonstrations, and I attach them to my letter for your judgment, for they would seem to me correct only when they have your approbation. You would not give me greater pleasure in the future than by sending me the first arithmetic propositions that may come from your hand; by trying to prove them, I will acquire the habit of a kind of thought that is for me full of charm; but it would be too difficult if I were to try it on my own strength because, to tell you the truth, I already wanted to examine the residues of the powers higher than the quadratic, but I could not penetrate this theory which remained the object of my curiosity. Here, however, is a small number of propositions I have carried out and that I would not dare to communicate to you if I did not count on the indulgence to which you have accustomed me …

Note: In chapter 3 we listed Germain’s propositions.2 ― Letter 9 ― Gauss to Germain 2

Gottingen

19 January 1808

For transcripts of the entire letter, see Del Centina and Fiocca (2012).

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Thanking you with all my heart for your last letter and the interesting communications that you send me, Mademoiselle, and I beg you a thousand pardons for answering it so late. This neglect is, for the most part, a result of the changes that have occurred in my situation. I changed my home, to accept the position of professor of astronomy at Göttingue that was offered to me a long time ago. I tell you nothing of the unfortunate circumstances which at last forced me to take this step, or the new annoyances to which I find myself exposed here; I hope that the interposition of the Institute I have used will put an end to it. Let us contemplate now only the beautiful perspective I have of being able, with greater ease, at least in the following, to watch over my arithmetical works, and to publish them successively in the memoirs of the Göttingue Society. I have the pleasure of sending you the premises which, as I hope, will give you some pleasure. You will forgive me that this time I cannot dwell more on the beautiful proofs of my arithmetic theorems. I admire the sagacity with which you have been able to advance in such a short time. I hope to be able soon to publish all the theory of which these elegant propositions are part, with a host of other things. May my arithmetic occupations make me happy in a time when I see nothing but misfortune and despair! It is only the sciences, the bosom of one’s family, and the correspondence with one’s beloved friends, where one can compensate and rest from the general affliction. The work on the calculation of planetary orbits, which I mentioned in my last letter, is finally in press. I hope it will be completed in a few months. I did not dread the trouble of translating it into Latin, so that it could find a larger number of readers. Always be very happy, my dear friend, that your rare qualities of mind and heart deserve it, and continue from time to time to renew to me the sweet assurance that I can count myself among the number of your friends, of which I shall be always proud. Ch. Fr. Gauss ― Letter 14 ― Germain to Gauss Monsieur,

Paris

28 March 1829

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I take advantage of the return to you of your learned disciple, Mr. Bader, to thank you for your kind regards and also to send you copies of my new memoirs. I read with great pleasure, your memoir on the biquadratic residues that this young savant gave me from you. It suffices to satisfy my taste for research in arithmetic, and reminds me that I had in the past the honor of receiving several letters from you; believe Monsieur that I deeply regret having been deprived for a long time of those learned communications to which I have never ceased to attach the highest reward. While talking with Monsieur Bader about the current subject of my studies, I gave him the opportunity to speak to me, and consequently to communicate to me, the scholarly memoir in which you compare the curvature of surfaces to that of the sphere. (I would have liked to be able to keep this memoir; I returned it with regret and discretely, for I do not know where to find it). I cannot tell you, Monsieur, how surprised and satisfied I was when I learned that a great geometer had, almost at the same time as myself, the idea of an analogy, which seemed to me so rational that I cannot conceive how I had not thought of it, nor how no one had paid attention to what I have already published in this respect. Independently of the superiority that characterizes everything that comes out of your savant pen, there is an essential difference between your research and mine. Yours, Monsieur, is quite geometrical: mine, on the other hand, is in a sense mechanical, borrowing from geometry only what is necessary to establish, where appropriate, the identity of the forces whose expression has been the almost exclusive object of my research for a long time. I will try to give you a glimpse of how I have been led to compare the curvature of a surface to that of a certain sphere: [Germain devotes the next two and a half pages to explain her ideas.] 3 I regret to be deprived of the privilege that I would enjoy, like Monsieur Bader, of your scholarly conversation: what he derives from it does not surprise me but is for me an object of envy. Regardless of what I could learn from you, I still regret not being able to submit to your judgment many ideas that I have not published and that it would take too long to write them down. Please Monsieur, at least remember me, and accept the assurance of my deepest respect, your servant. Sophie Germain Paris the 28th of March 1829 3

For a transcript of Germain’s full letter, see Del Centina and Fiocca (2012), pp. 693–695.

Sophie Germain Timeline

Year

Event

1776 1789 1789 1793

Marie-Sophie Germain is born on April 1 in Paris, France Germain discovers her passion for mathematics after reading Archimedes’ story The bloody French Revolution begins with the fall of the Bastille Louis XVI, King of France, is executed at the guillotine on 21 January. His wife, Queen Marie-Antoinette follows, beheaded at the guillotine on 16 October First French Republic. This period is characterized by the fall of the monarchy, the establishment of the National Convention and the infamous Reign of Terror, the founding of the Directory and the Thermidorian Reaction, and finally, the creation of the Consulate and Napoléon’s rise to power Germain collects lecture notes from the École Polytechnique and submits her own work to Professor Joseph-Louis Lagrange using the pseudonym “M. Le Blanc” Lagrange discovers her true identity and Sophie becomes known in the intellectual world of Paris. These were the final days of the Directory and the French Revolution, before the rise of Napoléon Bonaparte Legendre publishes Essai sur la théorie des nombres. He attempts to prove the law of quadratic reciprocity Lagrange retires from teaching at the École Polytechnique due to health reasons German mathematician Carl Friedrich Gauss publishes Disquisitiones arithmeticae on 29 September 1801. This work is introduced by Legendre to the Paris Academy of Sciences on 25 January 1802 Germain writes to Gauss concerning his Disquisitiones arithmeticae and attaches her own analysis, signing “Le Blanc.” She states her wish to prove Fermat’s Last Theorem (FLT) Napoléon crowns himself Emperor of the French on 2 December. His reign had begun in May as First Consul of the French First Republic Gauss writes to Olbers about M. Le Blanc’s letter Gauss discovers M. Le Blanc is actually Mlle. Germain and praises her work Ernst Chladni arrives in Paris to perform public demonstrations of musical instruments and vibrating plates. Germain was 32 and probable attended one session (continued)

1792– 1804

1797– 1798

1798 1799 1801

1804

1804 1805 1807 1808

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 D. Musielak, Sophie Germain, Springer Biographies, https://doi.org/10.1007/978-3-030-38375-6

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(continued) Year

1809 1810 1811

1812

1813 1813

1814

1815

1816

1819

1821

Event Sophie Germain receives last letter from Gauss dated January 19 Legendre publishes 2nd edition of Essai sur la théorie des nombres Institut de France issues prize competition to derive theory of Chladni’s vibrating plates. Deadline for submitting essays is 1 October 1811 On May 14 Sophie receives letter from Delambre asking to select a gift as requested by Gauss who desired to give to his bride-to-be On September 21, Germain submits a memoir with her mathematical analysis to the Institute of France as entry for the competition, explaining the vibration patterns demonstrated by Ernest Chladni In January, the Institut’s prize competition on heat diffusion is awarded to Fourier. Germain’s first entry to the competition on elastic vibrating surfaces did not win. The contest was extended one year Lagrange dies on April 11 On September 23, Germain submits her second entry to the contest Legendre writes on December 4 that she may be awarded an honorable mention in recognition to her mathematical theory to explain Chladni’s vibration plates In January the Institut awards Germain an honorable mention for her second entry. The contest is reopened, emphasizing the requirement to match theory and experiments Battle of Paris. On March 31, the foreign Coalition forces march down the Champs Elysées with the Russian tsar at the head of the cortège, followed by the king of Prussia and Schwarzenberg Napoléon reign ends and he is exiled to Elba on 11 April On May 3, Parisians witness the solemn entry of Louis XVIII. The King of France is seated beside his niece, Marie-Antoinette’s daughter and sole survivor of the former royal family Napoléon escapes from Elba on 26 February. He arrives in Paris on 20 March and governed again for a period now called the Hundred Days. The Allied Powers, particularly England and Prussia, attacked and defeated Napoléon at the Battle of Waterloo on June 18. He surrenders in Paris and is exiled to Saint Helena Second restoration of King Louis XVIII begins on 8 July 1815 On December 26, the Commission at the Institut de France proposes to give Sophie Germain the prize Sophie Germain wins the Prize awarded by the Class of Mathematics and Physics of the Institut de France for her mathematical theory of vibrations of general curved and plane elastic surfaces In the same session, the Institut issues a new topic for a prize competition: to prove Fermat’s Last Theorem In February, Legendre publishes first Supplément à l’Essai sur la théorie des nombres In May, Sophie Germain writes to Gauss to say she has resumed her work to prove Fermat’s Last Theorem. She also corresponds with other mathematicians working in number theory Sophie Germain self-publishes her first memoir, Recherches sur les théories des surfaces élastiques (July) On 15 December, Sophie’s father dies at ninety-five (continued)

Sophie Germain Timeline

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(continued) Year

Event

1823

In May, Germain receives the first official invitation to attend the meetings of the Institut de France, signed by Fourier, Perpetual Secretary of the Class of Mathematics and Physics. Two days later, Fourier sends her two entrance tickets On September 1, the Class of Mathematics and Physics reads her letter addressing Wheatstone’s experiments with metal vibrating plates In March, Germain sends a memoir to the Academy of Sciences summarizing her research on the effect of thickness of elastic plates Charles X ascends to the throne of France after his brother King Louis XVIII dies on 16 September Germain meets Italian mathematician Libri on May 13 Legendre presents Recherches sur quelques objets d’analyse indéterminée et particulièrement sur le théorème de Fermat. It contains a footnote on page 17 stating that the proposition is due to Mlle. Germain (now known as Germain’s Theorem). The paper was published in 1827 in the Mémories de l’Académie des Sciences of 1825 In September, Legendre publishes the Second Supplement to the Essai sur la théorie des nombres. The footnote about Germain’s proposition is on pp. 13–14 Sophie Germain publishes her second paper, Remarques sur la nature, les bornes et l’étendue de la question des surfaces élastiques et équation générale de ces surfaces, (July) Sophie Germain publishes Examen des principes de l’analyse employés dans la solution du problème des surfaces élastiques in Annales de chimie et de physique, Tome XXXVIII Sophie Germain is very ill. On February 8, she writes to Libri: “… me croire morte, car, pour volage cela serait trop extraordinaire… Lorsque la grande fièvre a été passée il m’est resté un grand mal d’yeux qui me fessait craindre l’impression de la plus faible lumière…” In the summer, Sophie meets Libri and A. L. Crelle and begins preparing manuscripts for publishing in Crelle’s Journal. Fourier dies on May 16 The Second French Revolution or Trois Glorieuses shakes Paris, causing the abdication of Charles X. On 9 August, Louis-Philippe assumes the title of King of the French Sophie Germain’s papers are published in Crelle’s Journal: one addressing curbature of surfaces, and another paper related to a generalization of the irreducible prime number equation from Gauss’s Disquisitiones Sophie Germain dies at her home on June 27 Germain’s paper, which she submitted to the Academy of Sciences in 1824, was finally published in the Journal de mathématiques pures et appliquées, 3e série, tome 6

1824

1825

1826

1828

1830

1831

1831 1880

Illustration Credits Figures 1.1, 1.2, 1.3, and 1.5: Collection complète des tableaux historiques de la Révolution française en trois volumes: le premier, contenant les titre, frontispice, l’introduction, les neuf gravures et neuf discours préliminaires, depuis l’Assemblée des notables, tenue. A Paris: chez Auber, Editeur, et seul Propriétaire: de l’Imprimerie de Pierre Didot l’aîné, an XI de la République Française M. DCCCII. [1802]. Source ETH-Bibliothek Zürich, Rar 9608, https://doi.org/10.3931/e-rara-49880/ PublicDomainMark Figure 1.4: Tableau historique et pittoresque de Paris. Tome 1/, depuis les Gaulois jusqu’à nos jours. Authors: Saint-Victor, Jacques-Maximilien Benjamin Bins de (1772–1858) et Tourlet, René (17..–1836). Publisher : H. Nicolle et Le Normant (Paris); Publication date: 1808–1811. Source Bibliothèque nationale de France, département Philosophie, histoire, sciences de l’homme, RES-LK7-6091 Figure 1.6: Description de Paris et de ses édifices: avec un précis historique et des observations sur le caractère de leur architecture, et sur les principaux objets d’art et de curiosité qu’ils renferm. A Paris; à Strasbourg; à Londres: chez Treuttel et Würtz, 1818. Source ETH-Bibliothek Zürich, Rar 6718, https://doi.org/10.3931/e-rara-26770/ PublicDomainMark Figures 3.1 and 9.3: Carl Friedrich Gauß Briefwechsel (Carl Friedrich Gauß Letters) Source Akademie der Wissenschaften zu Göttingen (Academy of Sciences and Humanities Göttingen) Permission to reproduce granted by Professor Menso Folkerts Figure 4.2: Contemporary lithography of Ernst Florens Friedrich Chladni and Bonaparte Source Dr. Svend Buhl (www.niger-meteorite-recon.de) Figure 6.1: Entrée dans la Ville de Paris, de sa Majesté Louis XVIII Roi de France et de Navarre, le 4 [lire 3] Mai 1814 : [estampe]. Éditeur: A Paris chez Jean, Rue Saint Jean de Beauvais, N° 10. 1814 Source Bibliothèque nationale de France, département Estampes et photographie, FOL-IB-3 (3) Figure 6.2: Correspondance de Sophie GERMAIN avec les mathématiciens et savants Cauchy, Delambre, Fourier, Gauss, Le Gendre, etc Source Bibliothèque nationale de France. Département des manuscrits. Français 9118

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 D. Musielak, Sophie Germain, Springer Biographies, https://doi.org/10.1007/978-3-030-38375-6

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Illustration Credits

Figure 6.3: Papiers de Sophie GERMAIN.—Recueil de dissertations et problèmes mathématiques et physiques Source Bibliothèque nationale de France. Département des manuscrits. Français 9115 Figure 14.2: France’s postage stamp in 2016 to commemorate the 240th anniversary of Sophie Germain’s birth Source La Poste (France)

Bibliography

Alder, K. (2002). The measure of all things: The seven year odyssey and hidden error that transformed the world. Simon and Schuster. Alexanderson, G. L. (2012, April). Mathematical perspectives. Bulletin of the American Mathematical Society, 49(2) 327–331. Anderson, J. D. (1997). A history of aerodynamics. Cambridge Press. Arago. M. (1833). Éloge Historique de Joseph Fourier. Lu à la séance publique du 18 novembre 1833. Ball, W., & Rouse, W. (1908). Joseph Louis Lagrange (1736–1813). In A short account of the history of mathematics (4th ed). Bell, E. T. (1937). Men of mathematics. New York: Simon and Schuster. Bernoulli, J. I. (1705). Véritable hypothèse de la résistance des solides avec la démonstration de la courbure des corps qui font ressort. Lettre du 12 Mars 1705. Paris: Mém. de la Académie des Sciences. Bernoulli, J. II. (1789). Essai théorétique sur les vibrations des plaques élastiques rectangulieres et libres (Vol. 5, pp. 197–219). Presented to the Academy of St. Peterburg in October 1788. Nova Acta Acad Petropolit. Biedenkapp, G. (1910). Sophie Germain, ein weiblicher Denker, Jena. Biot. (1800). Recherches sur l’intégration des équations différentielles partielles, et sur les vibrations des surfaces. Lu le 21 mai 1800 (1er prairial an VIII). Mémoires de l’Institut national des sciences et arts (T. 3, 21-111). Blay, M., & Halleux, R. (1998). «La science classique, XVIe-XVIIIe siècle», dictionnaire critique. Flammarion. Boncompagni B. (1880). Cinq lettres de Sophie Germain à Charles-Frederic Gauss. Archiv der Mathematik und Physik, 63, 27–31; 66, 3–10. Bucciarelli, L. L., & Dworsky, N. (1980). Sophie Germain. An essay in the history of the theory of elasticity. Dordrecht, Boston, London: D. Reidel Publishing Company. Bugge, T. (2003). Science in France in the revolutionary era. In M. P. Crosland (Ed.). MIT Press Classics. Burn, R. (2007). Euler and modern science. In N. N. Bogolyubov, G. K. Mikhailov, & A. P. Yushkevich (Eds.). Cain, H. (2005) C. F. Gauss’s Proof of the fundamental theorem of algebra. http://math.huji.ac.il/ *ehud/MH/Gauss-HarelCain.pdf. Calinger, R. (1976). Euler’s ‘Letters to a Princess of Germany’ as an expression of his mature scientific outlook. Archive for History of Exact Sciences, 15(3)(23.VII.1976), 211–233. Springer. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 D. Musielak, Sophie Germain, Springer Biographies, https://doi.org/10.1007/978-3-030-38375-6

241

242

Bibliography

Calinger, R. (1996). Leonhard Euler: The first St Petersburg years (1727–1741). Historia Mathematica, 23, 121–166. Cannone, M., & Friedlander, S. (2003, January). Navier: Blow-up and collapse. Notices of the AMS, 50(1). Cauchy, A.-L. (1815). Démonstration de théoréme général de Fermat sur les nombres polygones. Lu à l’Academie, le 13 novembre 1815. Mémoires de la Classe des Sciences Mathématiques et Physiques de l’Institut de France, Années 1813, 1814, 1815. Cauchy, A.-L. (1823). Recherches sur l’équilibre et me mouvement intérieur des corps solides ou fluides, élastiques ou non élastiques. Philomatique (pp. 9–13). Cauchy, A.-L. (1827). Théorie de la propagation des ondes à la surface d’un fuide pesant d’une profondeur indéfinie (Written in 1815 but published in 1827 with additional Notes). Oeuvres, 1 (1), 4–318. Cauchy, A.-L. (1828). Sur l’équilibre le mouvement d’une plaque solide. Exercises Mathemathics, 3. Charpentier, D. (1977, March). Women mathematicians. Two-Year College Mathematics Journal, 8(2), 73–79. Chladni, E. F. (1802). Die Akustik, Leipzig. Chladni, E. F. (1809). Traité de Acoustique. Paris: France. Coolidge, J. L. (1951, March/June). Six female mathematicians. Scripta Mathematica, 17, 20–31. Comte, A. (1835). Cours de Philosophy Positive, La philosophie astronomique et la philosophie de la physique. Tome Deuxième, Trente-Deuxième Leçon. Paris: Bachelier. Condorcet, M. (1842). Éloge d’Euler, in Lettres de L. Euler à une Princesse d’Allemagne sur divers sujets de physique et de philosophie. Paris: Tome Premier. Cox, D. A. (1993, January). Introduction to Fermat's last theorem. The American Mathematical Monthly, 101(1), 3–14. Article based on a lecture given at the Smith College Regional Geometry Institute in 1993. D’Alembert, J. (1747). Recherches sur la courbe que forme une corde tenduë mise en vibration (Researches on the curve that a tense cord forms [when] set into vibration). In Histoire de l’académie royale des sciences et belles lettres de Berlin (Vol. 3, pp. 214–219). Dahan-Dalmédico, A. (1987). Mécanique et théorie des surfaces: les travaux de Sophie Germain Historia Math, 4, 347–365. Dahan-Dalmédico, A. (1990). Aspects de la mathématisation au XIXè siècle. Entre physique et mathématique du continu et mécanique moléculaire, la voie d’A-L Cauchy. Nantes: Thèse Sciences. Dahan-Dalmédico, A. (1991). Sophie Germain. Scientific American, 265, 117–122. Dahan-Dalmédico, A. (1992). Sophie Germain. In Spektrum der Wissenschaft 2 (pp. 80–87). Dahan-Dalmédico, A. (1992). Mathematisations: Augustin-Louis Cauchy et l’Ecole Française. Paris: Editions du Choix. Darrigol, O. (2002). Between hydrodynamics and elasticity theory: The first five births of the Navier-Stokes equation. Archive for History of Exact Sciences, 56(2002), 95–150. Davenport, H. (2008). The higher arithmetic: An introduction to the theory of numbers (8th ed., p. 202). Cambridge University Press. Deakin, M. (1992). Women in mathematics: Fact versus fabulation. Australian Mathematical Society Gazette, 19, 105–114. Del Centina, A. (2005). Letters of Sophie Germain preserved in Florence. Historia Mathematica, 32, 60–75. Del Centina, A. (2006). Abel’s manuscripts in the Libri Collection: Their history and their fate. Historia Mathematica, 33(2), 224–233. Del Centina, A. (2008). Unpublished manuscripts of Sophie Germain and a revaluation of her work on Fermat’s Last Theorem. Archive for History of Exact Sciences, 62, 349–392. Del Centina, A., & Fiocca, A. (2012). The correspondence between Sophie Germain and Carl Frierich Gauss. Archive for History of Exact Sciences, 66, 585–700.

Bibliography

243

de Beaumont, L. E. (1861). Éloge historique de Adrien-Marie Legendre: lu à la séance publique annuelle du 25 mars 1861; Institut impérial de France. Deledicq, A. (1992). Histoires de maths: K. F. Gauss, S. Ramanujan, Sophie Germain, E. Galois et un tableau chronologique de mathématiciens. Paris: Berger Levrault, Art-Culture-Lecture. Deledicq, A., & Izoard, D. (1998). Histoires de maths (pp. 39–41). ACL—Les éditions du Kangourou. Dickson, L. E. (1950). History of the theory of numbers, New York. Dirichlet, P. G. L. (1825). Mémoire sur l’impossibilité de quelques équations indéterminées du cinquième degré, lu à l’Académie royale des sciences (Institut de France). Le 11 juillet 1825. The Addition with the final proof was attached to page 12 of the original memoir. Dubner, H. (1996). Large Sophie Germain primes. Mathematics of Computation of the American Mathematical Society, 65, 393–396. Dunnington, G. W. (2004). Gauss: Titan of Science. Reprint of 1955 edition. Washington, DC: The Mathematical Association of America. Duren, P. (2009). Changing faces: The mistaken portrait of Legendre. Notices of the AMS, 56(11). Edwards, H. M. (1977). Fermat's last theorem: a genetic introduction to algebraic number theory. Springer-Verlag. Euler, L. (1732). Observationes de theoremate quodam Fermatiano aliisque ad numeros primos spectantibus (Observations on a theorem of Fermat and others on looking at prime numbers). Presented to the St. Petersburg Academy on 26 September 1732. E26 at The Euler Archive. Euler, L. (1736). Mechanica sive motus scientia analytice. I. Tomus (Ed.) Opera Omnia: Series 2 (Vol. 1). The Euler Archive. Euler, L. (1738). Theorematum quorundam arithmeticorum demonstrationes. Presented to the St. Petersburg Academy on 23 June 1738, and Additions to it on 16 August 1738. Euler, L. (1740). De minimis oscillationibus corporum tam rigidorum quam flexibilium. Methodus nova et facilis (On the smallest oscillations of rigid and flexible bodies. A new and easy method) (pp. 99–122). Presented to the St. Petersburg Academy on 27 October 1735. Originally published in Commentarii academiae scientiarum Petropolitanae 7, 1740. Opera Omnia: Series 2, Vol. 10, pp. 17–34. Euler, L. (1744a). Methodus inveniendi lineas curvas maximi minive proprietate gaudentes, sive solutio problematis isoperimetrici lattissimo sensu accepti. Originally published as a book in 1744. Opera Omnia: Series 1, Vol. 24. Index Number E65 in The Euler’s Archive. Euler, L. (1744b). Additamentum 1. In De curvis elasticis (pp. 245–310). Euler, L. (1748–1749). Sur la vibration des cordes (On the vibration of strings) (pp. 512–527). According to C. G. J. Jacobi, a Latin treatise with the title: “De vibratione cordarum” was read to the Berlin Academy on 16 May 1748. Originally published in Nova acta eruditorum, 1749. Opera Omnia: Series 2, Vol. 10, pp. 50–62. Euler, L. (1750). Demonstratio theorematis Fermatiani omnem numerum primum formae 4n+1 esse summam duorum quadratorum. Novi Commentarii academiae scientiarum Petropolitanae 5, 1760, pp. 3–13 or Opera Omnia: Series 1, (Vol. 2, pp. 328–337). Euler, L. (1757). Sur la force des colonnes (Concerning the strength of columns) (pp. 252–282). Presented to the Berlin Academy on 1 September 1757. Published in Memoires de l’academie des sciences de Berlin 13. Opera Omnia: Series 2, (Vol. 17, pp. 89–118). Euler, L. (1760). Tentamen de sono campanarum (pp. 261–281), Novi Commentarii academiae scientiarum Petropolitanae 10 (Eneström Index 303). Euler, L. (1761). De motu vibratorio tympanorum (Vol. 10, pp. 243–260) Novi Commentari Acad Petropolit. (Eneström Index 302). Euler, L. (1766). De motu vibratorio tympanorum (On the motion of vibrations in drums) (pp. 243–260). According to C. G. J. Jacobi, a treatise with this title was read to the Berlin Academy on 22 January 1761. E302. Originally published in Novi Commentarii academiae scientiarum Petropolitanae 10, 1766. Opera Omnia: Series 2, (Vol. 10, pp. 344–358). Euler, L. (1862). De oscillationibus annulorum elasticorum (“The oscillations of elastic rings”) (pp. 129–131). Opera Postuma 2, 1862. Opera Omnia: Series 2, (Vol. 11, pp. 378–383).

244

Bibliography

Euler, L. (1770). Vollstandige Anleitung zur Algebra (Complete instruction in algebra, book 1) Saint Petersburg. Opera Omnia: Series 1, Vol. 1, E387. Euler, L. (1794). Élémens d’algèbre, par Léonard Euler, traduits de l’allemand avec des notes et des additions, Contributeurs : Bernoulli, Jean (1744–1807). Traducteur; Lagrange, J.-L. (1736– 1813). Traducteur. Éditeur: Bruyset aîné (Lyon), 1794. Eves, H. (1990). An Introduction to the (chapter 13), Saunders Series. Philadelphia PA: Saunders College Publishing. Fellmann, E. A. (2007). Leonhard Euler. Switzerland: Birkhäuser-Verlag. Ford, D., & Jha, V. (1993). On Wendt’s determinant and Sophie Germain’s theorem. Experimental Math, 2, 113–119. Forman, P. (1975). Dictionary of scientific biography XI, 475. New York: Charles Scribner’s Sons. Fourcy, A. (1828). Histoire de l’École Polytechnique. Paris, Chez l’Autheur, a l’Ecole polytechnique. BnF, département Philosophie, histoire, sciences de l’homme, 8-Z LE SENNE-3903. Franklin, C. L. (1981). Sophie Germain: An unknown mathematician. Century, 48(1894), 946– 949. Franqueville, C. F. (1895). Le Premier Siècle de l’Institut de France, 25 octobre 1795-25 octobre 1895, Tome Premier, par le comte de Franqueville, Paris. BnF: département Littérature et art, 4-Z-1068 (1). Fraser, C. G. (1987). Joseph Louis Lagrange’s algebraic vision of the calculus. Historia Mathematica, 14, 38–53. Friedelmeyer, J.-P. (1997). L’histoire des mathématiques par correspondance (pp. 57–61). L’Ouvert 88. Fuss, P.-H. (Ed.). (1843). Correspondance Mathématique et Physique de quelques célèbres géomètres du XVIIIème siècle (Vol. I). St. Petersburg: Academia Imperial de las Artes de San Petersburgo. Gander, M. J., & Wanner, G. (2011). From Euler, Ritz and Galerkin to modern computing. Preprint, SIAM Review (SIREV). Gauss, F. (1799). Demonstratio nova theorematis: omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse. Helmstadt. Gauss, F. (1801). Disquisitiones arithmeticae (A. A. Clarke & S. J, Trans.). New Haven, Connecticut: Yale University Press. 1966. Gauss, F. (1808). Theorematis arithmetici demonstratio nova, Göttingen Comment. Societatis Regiae Scientiarum, XVI. Gauss, F. (1828a). Theoria residuorum biquadraticorum (pp. 1–32). Commentatio prima. Göttingen: Dietrich. https://reader.digitale-sammlungen.de/en/fs1/object/display/bsb10053471_00001. html. Gauss, F. (1828b). Disquisitiones generales circa superficies curvas. Gottingae: Typis Dieterichianis. https://archive.org/details/disquisitionesg00gausgoog/page/n7/mode/2up. Gauss, F. (1857). Theory of the motion of the heavenly bodies (C. H. Davis, Trans.). Boston: Little and Brown. Genocchi, A. (1884). Teoremi di Sofia Germain inforno a i residui biquadratici. Bolletino di Bibliografia e di storia delle scienze mathematiche e fisiche, 17, 248–251. Genocchi, A., & Realis, S. (1884). Inforno ad una propozione inesalta di Sofia Germain. Bolletino di Bibliografia e di storia delle scienze mathematiche e fisiche, 17, 315–316. Germain, S. (1821). Recherches sur les théories des surfaces élastiques, Paris, Vè Courcier, in-4, avec planches. Germain, S. (1826). Remarques sur la nature, les bornes et l’étendue de la question des surfaces élastiques et équation générale de ces surfaces, Paris, impr. de Huzard-Courcier, in-4, 21 p. Germain, S. (1828). Examen des Principes qui peuvent conduire à la connaissance des lois de l’équilibre et du mouvement des solides élastiques (pp. 123–131). Tome XXXVIII: Annales de chimie et de physique.

Bibliography

245

Germain, S. (1831a). Mémoire sur la courbure des surfaces élastiques. Journal für die reine und angewandte Mathematik, 1831(7). Germain, S. (1831b). Note sur la manière dont se composent les valeurs de y et z dans l’équation 4 etc. Journal für die reine und angewandte Mathematik, 1831(7) 201–204. Germain, S. (1833). Considérations générales sur l’état des sciences et des lettres aux différentes époques de leur culture, Paris, impr. de Lachevardière, in-8, 102 p. BNF, Z-49501. Germain, S. (1880). Mémoire sur l’emploi de l’épaisseur dans la théorie des surfaces élastiques, Paris, Gauthier-Villars, in-4, 64 p. (extrait du journal de mathématiques pures et appliquées, 3è série Tome 6, 1880). Gianni M. (1985). The philosophical works of Sophie Germain. In C. Mangione (Ed.). Scienza e filosofia, Saggi in onore di Ludovico Geymonat (pp. 712–729, 860). Milan: Garzanti Editore S. p.A.. Goldstine, H. H. (1980). A history of the calculus of variations from the 17th through the 19th century. In Studies in the history of mathematics and physical sciences (Vol. 5). Goldstein, C., & Schappacher, N. (2007).“A book in search of a discipline (1801–1860). In C. Goldstein, N. Schappacher, & J. Schwermer (Eds.). The Shaping of Arithmetic after C. F. Gauss’s Disquisitiones arithmeticae (xii + 578 pp., illus). New York: Springer. Grattan-Guiness, I. (1981). Recent researches in French mathematical physics of the early 19th century. Annals of Science, 38(6), 663–690. Grattan-Guiness, I. (1984). Note sur les manuscrits de Libri conservés à Florence. In Revue d’histoire des sciences. Tome 37 n°1, pp. 75–76. Grattan-Guiness, I. (2005). The Ecole polytechnique, 1794–1850: differences over educational purpose and teaching practice. The American Mathematical Monthly, 112(3), 233–250. Gray, M. W. (1976). Sophie Germain: A bicentennial appreciation. AWM Newsletter, 6(6), 10–14. Gray, M. W. (1987). Sophie Germain. In L. S. Grinstein & P. J. Campbell (Eds.), Women of mathematics. A bibliographic sourcebook (pp. 47–56). Westport, Connecticut: Greenwood Press Inc. Greaves, G. N. (2012). Poisson’s ratio over two centuries: challenging hypotheses. Notes and Records of the Royal Society, 67(1), 37–58. Grey, J. (2006). A history of prizes in mathematics. In J. Carlson, A. Jaffe, & A. Wiles (Eds.). The millennium prize problems. Providence, RI: American Mathematical Society and Clay Mathematics Institute. Gunther, S. (1882). “Il carteggio tra Gauss e Sofia Germain,” Bolletino di Bibliografia e di storia delle scienze mathematiche e fisiche, Vol.15, pp.174–175. Also in Zeitsch. Math. Phys., Hist.Lit. Abt., Vol. 26, 1881, pp.19–25. Hachette. (1831). Note sur un Mémoire de Mademoiselle Sophie Germain (pp. 17–19). Bulletin des Sciences Mathematiques, Physiques et Chime, Tome XV. Hauchecorne, B., & Surateau, D. (1996). Des mathématiciens de A à Z. Editeur: Ellipses Marketing. Haven K. (1996). Marvels of science: 50 fascinating 5-minute read, Libraries Unlimited. Grade 3 & up. Havil, J. (2010). Gamma: exploring euler’s constant. Princeton University Press. Heydemann, M.-C. (1974). Histoire de quelques mathématiciennes, Publications mathématiques d’Orsay No. 86-74.16, in P. Samuel: Mathématiques, Mathématiciens et Société. Hill, A.M. (1995, August). Sophie Germain: A mathematical biography. A B.A. thesis presented to the Department of Mathematics and the Honors College of the University of Oregon. Indlekofer, K. H., & Járai, A. (1999). Largest Known Twin Primes and Sophie Germain Primes. Mathematics of Computation of the American Mathematical Society, 68, 1317–1324. Institut de France. Procès-verbaux des séances de l’Académie des sciences : Tome I, 1795–1799 (An IV–VII); Tome II, 1800–1804 (An VIII–XI); Tome III, 1804–1807; Tome IV, 1808–1811; Tome V, 1812–1815; Tome VI, 1816–1819; Tome VII, 1820–1823; Tome VIII, 1824–1827; Tome IX, 1828–1831.

246

Bibliography

Jaroszewska, M. (1991, December 9–13). Portraits of women mathematicians. In Report on the fifth annual EWM meeting (pp. 23–29). France: CIRM, Luminy. Kelley L. (1996). Why were so few mathematicians female? Mathematics Teacher, 89(7), 592– 596. Kirchhoff, G. R. (1850). Über das gleichgewicht und die bewegung einer elastishem scheibe. J. für die Reine und Angewandte Mathematik (also known as Crelle’s Journal), 40(1), 51–88. Klens, U. (1994). Mathematikerinnen im 18. Jahrhundert: Maria G. Agnesi, G.-E. du Châtelet, Sophie Germain. Fallstudien zur Wechselwirkung von Wissenschaft und Philosophie im Zeitalter der Aufklaerung. Forum Frauengeschichte, Bd. 12, Pfaffenweiler. Klyve, D. (2010). The omnipresent savant. In Opusculum, The Euler Society Newsletter (Vol. 2, no 2). Krasner, M. (1940). A propos du critère de Sophie Germain—Furtwängler pour le premier cas du théorème de Fermat. Mathematica (Cluj), 16, 109–114. Kurrer, K.-E., & E. Ramm (2012). The history of the theory of structures: From arch analysis to computational mechanics. Wiley. Ladd-Franklin, C. (1981). “Sophie Germain: An unknown mathematician. In Century (Vol.48, Oct. 1894). Reprinted in AWM Newsletter (Vol. 11, No. 3, May-June, pp. 7–11). Lagrange, J.-L. (1797a). Essai d’analyse numérique sur la transformation des fractions. Journal de l’École polytechnique, Cahier 5, Tome 2, 93–114. Lagrange, J.-L. (1797b). Sur le principe de vitesses virtuelles. Journal de l’École polytechnique, Cahier 5, Tome 2, 115–118. Lagrange, J.-L. (1799a). Discours sur l’objet de la théorie de fonctions analytiques. Journal de l’École polytechnique, Cahier 6, Tome 2, 232–235. Lagrange, J.-L. (1799b). Solutions de quelques problèmes relatifs aux triangles sphériques, avec une analyse complète de ces triangles. Journal de l’École polytechnique, Cahier 6, Tome 2, 270–296. Lagrange, J.-L. (1801). Théorie des fonctions analytiques. Journal de l’Ecole Polytechnique, cahier 9(3), 1–277. Lagrange, J.-L. (1804). Leçons sur le calcul des fonctions. Journal de l’Ecole Polytechnique, Cahier 12. Tome V. Thermidor an XII, Paris. Bibliothèque de l’École polytechnique. Laubenbacher, R., & Pengelley, D. (2010). Voici ce que j’ai trouvé: Sophie Germain’s grand plan to prove Fermat’s Last Theorem. Historia Mathematica, 37, 641–692. Legendre, A.-M. (1786). Sur la manière de distinguer les maxima des minima dans le calcul des variations (pp. 7–37). Mémoires de l’Académie des Sciences de Paris. Legendre, A.-M. (1787). Mémoire sur les opérations trigonométriques dont les résultats dépendent de la figure de la terre. Paris. BnF, département Littérature et art, VZ-441. Legendre, A.-M. (1798). Essai sur la théorie des nombres, Paris, An IV. Legendre, A.-M. (1805). Nouvelles méthodes pour la détermination des orbites des comètes [New methods for the determination of the orbits of comets]. Paris: F. Didot. Legendre, A.-M. (1808). Essai sur la théorie des nombres. Seconde édition, Paris: chez Courcier, Imprimeur-Libraire. Legendre, A.-M. (1825). Essai sur la théorie des nombres, Second supplément. (Paris, Courcier), 40 p. (Bibl. de l’Institut, in-4°, M 572 A(3)). Legendre, A.-M. (1827). Recherches sur quelques objets d’analyse indéterminée et particulièrement sur le théorème de Fermat. Mémoires de l’Académie des sciences de l’Institut de France, Tome VI, Année 1823. Chez Firmin Didot, père et fils, libraries, Paris. Legendre, A.-M. (1830). Théorie des nombres. Paris, Chez Firmin Didot: Troisième édition. Legendre, A.-M. (1832). La détermination des fonctions Y et Z qui satisfont à l’equation 4(xn–1)= (x–1)(Y2±nZ2), n étant un nombre premier 4i1 (pp. 81–99). Tome XI: Mémoires de l’Académie Royale des Sciences. Levy, M. (1899). Memoire sur la theorie des plaques elastiques planes. Journal de Mathématiques Pures et Appliquées, 3, 219. Libri, G. (1823). Mémoire sur divers points d’analyse. Turin: Imprimerie Royale.

Bibliography

247

Lodder, J. (2003). Curvature in the calculus curriculum. The American Mathematical Monthly, 110 (7), 593–605. Love, A. E. H. (1906). A treatise on the mathematical theory of elasticity (p. 6). Cambridge University Press. Martinez, J. A. F. (1946). Sophie Germain. Scientific Monthly, 63, 257–260. Matvievskaya G. P,. & Ozhigova, E. P. (2007). The manuscript materials of Euler in number theory. In N. N. Bogolyubov, G. K. Mikhailov, & A. P. Yushkevich (Eds.), Euler and modern science. The Mathematical Association of America. Micheli, G. (1985). The philosophical works of Sophie Germain (pp. 712–729). Milan: Scienza e filosofia. Montucla, J.-E. (1754). Histoire des recherches sur la quadrature du cercle : avec une addition concernant les problèmes de la duplication du cube et de la trisection de l’angle (Nouv. éd. rev. et corr.). BnF, département Littérature et art, V-47132. Montucla, J.-E. (1756). Histoire des mathématiques. A. Jombert (Paris): Vol. I. BnF, département Littérature et art, V-6029. Mordell, L. J. (1921). Three Lectures on Fermat’s Last Theorem (p. 30) Cambridge University Press. Musielak, D. (2012). Sophie’s diary: A mathematical novel. Washington, D.C.: Mathematical Association of America, MAA, Spectrum Book. Musielak, D. (2014a). The Marquise du Châtelet: A controversial woman of science. Online at www.arxiv.org, ID 1010554. Musielak, D. (2014b). Euler and the German Princess. Online at www.arxiv.org, ID 1010612. Musielak, D. (2018). Euler Celestial analysis. AuthorHouse Books. Musielak, D. (2019). Germain and her fearless attempt to prove fermat’s last theorem. arXiv:1904.03553v3 [math.HO]. Navier, C. L. M. H. (1816). Note sur la vis d’Archimède. Correspondence de l’École polytechnique, tome, III, 45. Navier, C. L. M. H. (1822). Mémoire sur les lois du mouvement des fluides. Mémoires Académie des sciences de l'Institut France, 6, 389–440. Navier, C. L. M. H. (1823). Mémoire sur la flexion des plans élastiques. Bulletin des Sciences de la Société Philomathique de Paris (livraisons de juin et juillet 1823). Navier, C. L. M. H. (1826). De la résistance des corps solides (first ed.). French engineer Barré de Saint-Venant expanded the 1864 edition. Patterson, S. J. (2007). Gauss Sums. In C. Goldstein, N. Schappacher, J. Schwermer (Eds.). The Shaping of Arithmetic after C. F. Gauss’s Disquisitiones arithmeticae (xii + 578 pp., illus). New York: Springer. Pengelley, D., & Richman, F. (2006). Did euclid need the Euclidean algorithm to prove unique factorization? American Mathematical Monthly, 113, 196–205. Perl, T. (1978). Math Equals Biographies of Women Mathematicians + Related Activities. Menlo Park: Addison Wesley. Petrovich, V. C. (1999). Women and the Paris academy of sciences. In Eighteenth-century studies, Constructions of Femininity (Vol. 32, No. 3, pp. 383–390). Spring. Plackett, R. L. (1972). Studies in the history of probability and statistics. XXIX: The discovery of the method of least squares. Biometrika, 59(2), 239–251. Poirier, J. P. (1998). Lavoisier: Chemist, biologist, economist. University of Pennsylvania Press. Poisson, S.-D. (1811). Traité de mécanique. Paris. A second edition issued in 1833. Poisson, S.-D. (1814). Mémoire sur la Théorie mathématique des surfaces élastiques. Read to the Institut on 1 August 1814. Institut de France. Procès-verbaux. Tome V, p. 385. Also published as Mémoire sur les surfaces élastiques, Cl. Sci. Mathém. Phys. Inst. France 167–225. Poisson, S.-D. (1827). Note sur l’Extension des Fils et des Plaques élastiques. Annales de chimie et de physique (Ann. Chim. Phys.), 36, 384–387.

248

Bibliography

Poisson, S.-D. (1829). Mémoire sur l’equilibre et le mouvement des corps élastique. Memoires de l'Academie Royale des Sciences, 8, 357–570. Addition au Mémoire sur l’équilibre et le mouvement des corps élastiques, Ibid. pp. 623–627. Poisson, S.-D. (1833, January 20). Discours prononcé aux funérailles de M Legendre, Moniteur universel 162. Poinsot, L. (1817). Sur l’algèbre et sur la théorie des nombres. Read at the Academy of Sciences on 5 May 1817. Mémoires de la Classe des Sciences Mathématiques et Physiques de l’Institut de France, Années 1813, 1814, 1815. p. 381. Rashed, R. (1988). Sciences à l’époque de la Révolution Française. Recherches historiques, Librairie du Bicentenaire de la Révolution Française. Librairie Scientifique et Technique Albert Blanchard, Paris, p. 474. Ravaisson, F. (1889). Rapport sur la philosophie en France au XIX siècle (1867): suivie du Rapport sur le prix Victor Cousin (le scepticisme dans l’antiquité). Paris: Troisième édition. Ribenboim, P. (1979). 13 Lectures on Fermat’s Last Theorem. New York: Springer Verlag. Ribenboim, P. (1994). Prime number records. The College Mathematics Journal, 25(4) 280–290. Mathematical Association of America. Ribenboim, P. (1999). Fermat’s last theorem for amateurs. Springer-Verlag. Ritti, P. (1879). Journal de Savants. Ritz, W. (1909). Theorie der transversalschwingungen, einer quadratischen platte mit frein randern. Annals of Physic, 28, 737–786. Sampson, J. H. (1990). Sophie Germain and the theory of numbers. The Archive for History of Exact Sciences, 41(2), 157–161. Sandifer, E. (2003). How Euler did it: Fermat’s little theorem. See also Ian Bruce’s translation of Euler’s article E26 at The Euler Archive. Sandifer, E. (2007). The early mathematics of Leonhard Euler. Volume 1 of MAA tercentenary Euler celebration Spectrum, Mathematical Association of America (MAA). Schubring, G. (2005). Conflicts between generalization, rigor, and intuition: Number concepts underlying the development of analysis in 17th–19th century France and Germany. Springer. Selvadurai, A. P. S. (2000). Partial differential equations in mechanics 2: The Biharmonic Equation, Poisson’s Equation. Springer Science & Business Media. Serre, J.-P. (1996). A course in arithmetic. Volume 7 of Graduate Texts in Mathematics. Springer; 1st Corrected ed. 1973. Corr. 3rd printing 1996 edition (October 10, 1996). Smith, S., & Greer L. (1998). Sophie Germain, notable women in mathematics: A biographical dictionary. In C. Morrow & T. Perl (Eds.) (pp. 62–66). Greenwood Press. Simalarides, A. (1990). Sophie Germain’s principle and Lucas numbers. Mathematica Scandinavica, 67, 167–176. Singal, A. R. (1986). Women mathematicians of the past: Some observations. Mathematics Education, 3(1), 9–18. Smithies, F. (2008). Cauchy and the creation of complex function theory. Cambridge University Press; Reissue edition. Straub, H. (1990). Biography in dictionary of scientific biography. New York 1970–1990. Stöckmann, H.-J. (2007). Chladni meets Napoleon. The European Physical Journal Special Topics, 145, 15–23. Stupuy, H. (1896). Oeuvres philosophiques de Sophie Germain; suivies de pensées et de lettres inédites. Et précédées d’une notice sur sa vie et ses œuvres. Paris, P. Ritti, in-18, 375 p. Struik, D. J. (1969). A Source Book in Mathematics, 1200-1800, Harvard University Press. Taylor, R., & Wiles, A. (1995). Ring theoretic properties of certain Hecke algebras. Annals of Mathematics, 141(3), 553–572. https://doi.org/10.2307/2118560. JSTOR 2118560. OCLC 37032255. Tee, G. J. (1983). The pioneering women mathematicians. The Mathematical Intelligencer, 5, 27–36. Terquem, M. (1860). Sophie Germain. Bulletin de bibliographie, d’histoire et de biographie mathématiques, 6, 9–13.

Bibliography

249

Thomas, M., & Kempis, A. (1939). An Appreciation of Sophie Germain. National Mathematics Magazine, 14(2), 81–90. Kelvin, W. T. L., & Tait, P. G. (1883). Treatise on natural philosophy (Vol. 1). Oxford: Clarendon Press. Timoshenko, S. P. (1983). History of strength of materials. Dover Publications. Todhunter, I. (1886). The history of the theory of elasticity. Travail inachevé, est édité et publié à titre posthume en 1886 par Karl Pearson. Truesdell, C. (1953). Notes on the history of the general equations of hydrodynamics. The American Mathematical Monthly, 60(7), 456. Truesdell, C. (1991). Sophie Germain: Fame earned by stubborn error. Bibliografia Italiana Di Storia Delle Matematiche, 11(2), 3–24. Ullmann, D. (2007). Life and work of E.F.F. Chladni. he European Physical Journal Special Topics, 145, 25–32. Valson, C. A. (1868). La vie et les travaux du varon Cauchy. Paris: Gauthrie-Villars. Ventsel, E., & Krauthammer, T. (2001). Thin plates and shells: theory, analysis, and applications. Marcel Dekker, Inc., Eq. (9.3). Waterhouse, W. C. (1994). A counterexample for Germain. American Mathematical Monthly, 101, 140–150. Wheeler, G. F., & Crummet, W. P. (1987). The vibrating string controversy. The American Journal of Physics, 55, 33. Wiles, A. (1995). Modular elliptic curves and Fermat’s Last Theorem. Annals of Mathematics, 141 (3), 443–551. https://doi.org/10.2307/2118559. JSTOR 2118559. OCLC 37032255. Wussing, H., & Arnold, W. (1989). Biographien bedeutender Mathematiker, 4ed Volk und Wissen. Berlin: Volkseigener Verlag. Young, T. (1807). Course of Lectures on Natural Philosophy and the Mechanical Arts (London, 1807; Taylor & Walton, London, 1845): Lecture 13, On Passive Strength and Friction, 1807, pp. 109–113; Definition of squeeze–stretch ratio, p. 105.

Index

A Abel, Niels Henrik (1802–1829), 189, 190 Académie des Sciences de Paris, 136, 158 Agnesi, Maria Gaetana (1718–1799), 208 Arago, Dominique François Jean (1786–1853), 98, 109, 193 Archimedes of Syracuse (c. 287–212 BC) biographical sketch, 184 inspiration for Germain, 7 Arithmetica (Diophantus), 36, 116 Astronomie des dames, 28, 209 Award Judging Panel in 1812, 58 Award Judging Panel in 1816, 86 B Bachet, Claude Gaspard (1581–1638), 116 Bader, Carl (1796–1874), 190 Bernoulli, Daniel (1700–1782), 62–64, 67–69, 91, 180 Bernoulli, Jacques II (1759–1789), 71, 104 Bernoulli, Jakob (Jacques) I (1654–1705), 61 Bernoulli, Johann I (1667–1748), 66, 179 Bernoulli, Johann III (1774–1807), 20 Berthollet, Claude-Louis (1748–1822), 32, 54, 211 Bessey, Bernard Frenicle de (1605 − 1675), 117 Binet, Jacques-Philippe-Marie (1786–1856), 83 Biot, Gabrielle (1781–?), 210 Biot, Jean-Baptiste (1774–1862), 54, 57, 148, 210

C Carcavy, Pierre de (1600–1684), 117 Cauchy, Augustin-Louis (1789–1857) biographical sketch, 171 letter to Germain, 89 theory of elasticity, 104 Châtelet, Marquise du (1706–1749), 148, 207 Chladni, Ernst (1756–1827), 53, 57, 59, 201 Chladni’s law, 59 Citoyen, citoyenne, 15, 32 Classe des Sciences Physiques et Mathematiques, 13 Collège de France, 28, 174, 203 Collège Mazarin, 7, 211 Condorcet, Marie-Jean-Antoine-Nicolas Caritat de (1743–1794), 9, 12, 144 Corps des Ponts et Chaussées, 167 Cousin, Jacques-Antoine-Joseph (1739–1800), 27, 29, 203 Crelle, August Leopold (1780–1855), 175, 190, 194 Cyril, Bishop of Alexandria, 207 D D’Alembert, Jean le Rond (1717–1783), 67 Delambre, Jean-Baptiste (1749–1822), 9, 57 biographical sketch, 170 letter to Germain, 83 Diophantine equation, 36 Diophantus of Alexandria (c. 200–c. 284), 36 Dirichlet, Johann Peter Gustav Lejeune (1805– 1859)

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 D. Musielak, Sophie Germain, Springer Biographies, https://doi.org/10.1007/978-3-030-38375-6

251

252 Dirichlet, Johann Peter Gustav Lejeune (1805– 1859) (cont.) L-series, 135 normal derivatives, 112 proof for n = 14, 135 proof for n = 5, 134 Disquisitiones arithmeticae, 34, 39 Dupin, Charles F. (1784–1873), 195 E École Normale, 24–26, 162, 194, 196, 213 École Polytechnique, 17, 26, 29, 33, 103, 106, 129, 148, 162, 163, 165–167, 172, 194, 203, 211, 213 Euclid, 36, 184 Euler, Leonhard (1707–1783), 62 Algebra, 20 and Fermat’s Little Theorem, 120 biographical sketch, 178 Lettres à une princesse d’Allemagne, 181 Mersenne primes, 120 partial proof of FLT, 121 vibrating string theory, 67 work on number theory, 178 F Fermat, Pierre de (1601–1665), 116, 118, 183 Fermat primes, 118, 139 Fermat’s Last Theorem, 120, 124, 126, 128, 130–132, 134, 136, 137, 149, 152, 157, 178, 183, 201, 213, 214 1818 contest to prove it, 123 1818 Judging Panel, 124 1850 contest to prove it, 136 attempts to prove, 134, 135 Case 1, 127 Euler’s work, 122 for exponent n = 5, 127 Germain attempts to prove, 131, 132 Germain plan, 131 Germain’s theorem, 127 two cases, 126 Fermat’s Little Theorem, 118 First Class of the Institute of France, 33, 58, 201 Fourcroy, Antoine François (1755–1809), 10 Fourier, Jean-Baptiste-Joseph (1768–1830), 8, 32, 87 biographical sketch, 162 Fourier Series, 163 thanks Germain for her endorsement, 164 Théorie analytique de la chaleur, 163 Fundamental Theorem, 41

Index G Galileo Galilei (1564–1642), 61 Galois, Evariste (1811–1832), 187, 194, 196 Gauss, Carl Friedrich (1777–1855), 38 biographical sketch, 148 congruence, 38 correspondant of Institut de France, 39 Disquisitiones arithmeticae, 38, 149 letter to Germain on 30 April 1807, 43 response to Germain in 1805, 40 Theorema Aureum, 41 Theoria motus, 49 Gaussian curvature, 112, 191 Gauss’s Lemma, 45 General Pernety, 42 Germain, Ambroise-François (1726–1821) (father), 1, 2 Germain-Lagrange equation, 93, 100, 101, 103, 105, 106, 110, 112, 113 Germain, Marie-Madeleine Gruguelu (mother), 1 Germain, Sophie 2016 commemorative stamp, 216 Examen des Principes, 1828, 108 first attempt to prove FLT, 115 friendship with Fourier, 89, 147 friendship with Libri, 188 last letter to Gauss in 1829, 191 letter to Gauss with her plan to prove FLT, 128 letter to Poisson, 86 meets Libri in 1825, 134 Mémoire sur l’emploi de l’épaisseur dans la théorie des surfaces élastiques, 1824, 90 Navier views of her theory, 109 proof of Gauss’s Lemma, 46 propositions on power residues, 47 public dispute with Poisson in 1828, 108 Recherches sur la théorie des surfaces élastiques, 1821, 90 Goldbach, Christian (1690–1764), 120, 181, 182 Goldbach Conjecture, 121, 139 H Hachette, Jean-Nicolas-Pierre (1769–1834), 194 Hecke, Enrich (1887–1947), 137 Hegel, Friedrich (1770–1831), 145 Histoire de l’École Polytechnique, 19 Histoire des mathématiques. See Montucla Hooke, Robert (1635–1703), 54, 61

Index Hooke’s law. See Hooke Hypatia of Alexandria (*370– 415), 206 I Institut de France, 13, 85 1809 prize commpetition, 72 First Class, 13 inaguration, 13 letter to Germain in 1816, 83 Mémoires, 91, 173 perpetual secretary of mathematics, 32 prize competitions, 15 Sophie Germain Foundation, 215 J Journal de l’École Polytechnique, 19, 20, 23, 156 K Kant, Immanuel Kant (1724–1804), 145 Kirchhoff, Gustav Robert (1824–1887), 110 Kirchhoff’s hypotheses, 112 Kummer, Ernst Eduard (1810–1893), 135 work on FLT, 136 L Lacroix, Sylvestre-François (1765–1843), 23, 58 Lagrange, Joseph-Louis (1736–1813), 8, 32 analysis professor at École Polytechnique, 17 biographical sketch, 153 corrects Germain’s equation, 76 discovers Sophie Germain, 22 judge in mathematics contest, 58 letter to Sophie Germain, 199 Mécanique analytique, 77, 154 Œuvres complètes’, 156 polygonal numbers, 123 professor at the École Normale, 26 Theory of Analytic Functions, 23 vibrating string, 69 Lalande, Joseph-Jérôme Lefrançais de (1732– 1807), 27 Astronomie des dames, 27 letter to Sophie Germain, 27 Lalande, Marie-Jeanne-Amélie Harlay de (1768–1832), 209, 210 Laplace, Pierre-Simon (1749–1827), 8, 32, 54 Celestial Mechanics, 15 commission to review Germain’s article, 98

253 commission to review Germain’s second memoir, 90 discovers Poisson’s mathematical abilities, 165 funeral oration for Lagrange, 156 introducing Cauchy to mathematics, 172 judge in 1811 prize competition, 58, 73 judge in 1816 prize competition, 86 professor at the École Normale, 21, 24, 26 read Euler, he is the master of us all!, 182 System of the World, 15, 28 Lavoisier, Antoine-Laurent (1743–1794), 11, 12, 210 Lavoisier, Marie Anne Pierrette Paulze (1758– 1836), 210 Le Blanc, 19, 22, 26, 29, 30, 40, 41, 150, 206 Legendre, Adrien-Marie (1752–1833), 7, 8, 24, 32, 37, 74, 165 biographical sketch, 157 Essai sur la théorie des nombres, 157 method of least squares, 10, 159 théorème de réciprocité théorème de réciprocité, 37 Théorie des nombres, 130 Legendre theorem, 22 Leibnitz, Gottfried Wilhelm von (1646–1716), 61 Leonardo da Vinci (1452–1519), 61 Lepante, Nicole-Reine, 209 Lherbette, Jacques-Amant (1791–1864), 141 Libri, Guglielmo (1803–1869), 132, 175, 188 bibliomania, 177 biographical sketch, 175 Sophie Germain biographer, 19 work on number theory, 133 Louis XVI, King of France, ix, 2, 3, 7, 10, 154, 203 Louis XVIII, King of France and Navarre, 90 Lycée Sophie-Germain, 215 M Malus, Étienne-Louis (1775–1812), 58 Marie Antoinette, Queen of France, 7 Mersenne, Marin (1588 − 1648), 117, 120 Mersenne primes, 120, 138, 139 Method (Archimedes), 184 Method of least squares (MLS), 49 Modularity theorem, 137 Monge, Gaspard (1746–1818), 8, 17, 20, 29, 32, 156 Montucla, Jean Étienne (1725–1799), 7, 184

254 N Napoléon Bonaparte, 31–33, 54, 57, 171, 172, 203 Navier, Claude-Louis (1785–1836), 54, 103, 167 biographical sketch, 167 bridge collapse, 168 dispute with Poisson, 108 double series solution, 104 elastic module, 104 founder of structural analysis, 168 Navier-Stokes equation, 169 O Olbers, Heinrich Wilhelm Matthäus, 41, 151 P Paris Observatory, 27, 28 Pierry, Louise du (1747–1830), 210 Poinsot, Louis (1777–1859), 86, 128 Poisson’s ratio, 100, 106 Poisson, Siméon-Denis (1781–1840), 54 and Society of Arcueil, 74 biographical sketch, 165 judge in 1816 prize competition, 86 professor at École Polytechnique, 165 theory of elastic surfaces, 82 Prix de Mathématiques Announcement (1809), 58, 73 Prony, Gaspard Riche de (1755–1839), 17, 21, 57, 90, 98, 158 Pythagoras, 36

Index Q Quadratic reciprocity law, 37, 160, 183 R Reign of Terror, 10, 11, 17, 171 Republican Calendar, 10, 32 Ribenboim, Paulo, 36 Riemann zeta function, 135 Ritz, Walter (1878–1909), 113, 114 S Savart, Félix (1791–1841), 56, 98, 105 Schumacher, Heinrich C. (1780–1850), 124 Silvestre de Sacy, A.-I. (1758–1838), 40, 201 Society of Arcueil, 74, 211 Somerville, Mary Fairfax (1780–1872), 211 Sophie Germain primes, 139 Sophie Germain’s Theorem, 127 Stupuy, Hipolite, 7 Sophie Germain biographer, 19 Synesius of Cyrene, 206 T Todhunter, Isaac, 99 W Wheatstone, Sir Charles (1802–1875), 98 Wiles, Andrew, 137 Wilson’s theorem, 134 Y Young’s module, 65, 66 Young, Thomas (1773–1829), 65