Sophie Germain

"History immortalized other great female mathematicians who lived before and accomplished as much, but Sophie Germain did it alone. Hypathia had her father, Theon of Alexandria to teach her; Maria Agnessi had her Rampinelli and other instructors; and Emilie Chatelet had Maupertuis and Clairaut as her tutors of mathematics.

Sophie Germain had no teacher."

from the Historical Note in Sophie's Diary (2008)

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Marie-Sophie Germain

(1776 – 1831)

by Dora Musielak

The great mathematician Marie-Sophie Germain was born the first of April of 1776 in Paris, France, She was a daughter of a wealthy merchant. Little is known about her childhood, but it is true that Germain was also thirteen years old when the French Revolution began in 1789. During the siege of the Revolution, Sophie Germain spent her days in her father’s library, discovering and learning mathematics. Her historians tell us that in 1794, after the École Polytechnique was funded, Sophie somehow obtained lecture notes from the greatest mathematicians of the time who were instructors at the École. It is not clear how the young woman acquired the class notes. It is only known that Sophie Germain took a man’s name, M. LeBlanc, and submitted a paper whose originality and insight made Professor Joseph-Louis Lagrange (1736 −1813) take notice. It should be noted that Lagrange’s mathematical analysis was very difficult, even for the students attending his lectures at the École.

Lagrange soon discovered that M. LeBlanc was actually a woman, and, profoundly impressed by her brilliance and resourcefulness, he became her mathematical counselor and advocate. Sophie’s education was, however, disorganized and haphazard, since she never had received the rigorous education that she eagerly desired. Nevertheless, Sophie sought out advice from the greatest mathematicians of her time, and in spite of her shyness, she was bold enough to submit her own ideas and solutions to very difficult mathematical problems.

Sophie wrote to Adrien-Marie Legendre (1752 −1833), another French mathematician, about problems suggested by his Essai sur le Théorie des Nombres, or Essay on the Theory of Numbers (1798). By then Legendre must have known that she was a woman because he was also a professor at the École and Lagrange’s colleague. Legendre must also have seen the genius in the young woman because he corresponded and collaborated with Sophie for a number of years. Legendre included some of her mathematical discoveries in a supplement to the second edition of his Théorie.

Sophie was undoubtedly impressed by the work of German mathematician Carl Frierich Gauss (1777 −1855), who in 1801 published the masterpiece on the theory of numbers, Disquisitiones arithmeticae, or Arithmetical Inquisitions. In 1804, Sophie began to correspond with Gauss, sending him some of her own mathematical analysis. How did she develop the courage to write to him? The only logical answer is that Sophie was seeking acceptance as a genuine mathematician, and apparently she had developed a thorough understanding of the methods presented in Gauss’ dissertation.

Between 1804 and 1809, Sophie wrote many letters to him, initially adopting again the pseudonym “M. LeBlanc” because she feared Gauss would ignore her letters if he knew she was a woman. In these letters she sent proofs of number theory, and Gauss praised her ingenuity and mathematical ability. In fact, Gauss did not know she was a young woman his own age until after the French occupation of his hometown in Germany in 1806. Sophie, fearing for Gauss’s safety, contacted a French commander who was a friend of her family and asked him to inquire about Monsieur Gauss well being.

Eventually Gauss discovered that “M. LeBlanc” was actually a woman. He was truly impressed and even mentioned her mathematical proofs and insight in a letter Gauss wrote to Olbers, one of his colleagues. The correspondence ended in 1809 when Gauss stopped replying to her letters.

Sophie continued working alone, most notably in number theory. At that time, many mathematicians were perplexed trying to prove the last theorem of Fermat. Pierre de Fermat (1601-1665) was a French mathematician who wrote a note on the margin of his book enunciating a mathematical proposition of great significance and declared that he had proved it. His proposition was related to the Diophantine equation in Pythagoras Theorem. Fermat affirmed that it is impossible to find a solution for the family of general equations, xⁿ + yⁿ = zⁿ, where n is greater than 2. This is Fermat’s Last Theorem.

Fermat said that he could prove it. Unfortunately the only thing that Fermat wrote on the margin of his book was: I have a marvelous demonstration to this proposition but the margin is too narrow to contain it.

In the 18th century, Fermat’s Last Theorem was established as the most difficult problem in number theory. The great mathematician Leonhard Euler was only able to arrive to a partial proof. There was no progress until Sophie Germain resumed the search of Fermat’s lost proof.

Germain adopted a more general approach to the proof of the theorem. Her objective was not to prove that a particular equation has no solution, rather she intended to prove it with several equations.

Sophie’s analysis resulted in her partial solution to Fermat’s Last Theorem, which has become known as “Germain’s Theorem.” She demonstrated the impossibility of solving the equation xⁿ + yⁿ = zⁿ if x, y, z are not divisible by an odd prime n. This remained the most important contribution related to Fermat’s Last Theorem (1738) until the next result contributed by mathematician Kummer a hundred years later.

In addition to number theory, Sophie was attracted and made contributions to mathematical physics. In 1808, the German physicist Ernst F. F. Chladni came to Paris and drew the attention of the scientific community with his experiments on vibrating plates. He sprinkled sand on elastic surfaces, strummed the edges with a bow and noted the resulting patterns, exhibiting the so-called Chladni figures. The Institut de France set a prize competition with the following challenge: formulate a mathematical theory of elastic surfaces and indicate just how it agrees with empirical evidence. A deadline of two years was given for mathematicians to submit such mathematical theory.

The two-dimensional elasticity theory was considered too formidable for most mathematicians and, since Lagrange thought that the mathematical methods available were inadequate, most scientists did not attempt to formulate it. Sophie, however, accepted the challenge and spent the next two years trying to derive a theory of elasticity. In 1811 she was the only entrant in the contest, but her work did not win the award. Her historians conclude that Sophie had not derived her hypothesis from physical principles, and her analysis lacked the necessary rigour owing to her deficiency in analysis and the calculus of variations. Sophie’s essay did generate more interest in the topic and provided needed insight to pursue the theory. Lagrange, who was one of the judges in the contest, amended Sophie’s calculations and developed an equation that could better describe Chladni’s experiments.

Without a winner, the contest deadline was extended by two years. Once more Sophie submitted the only entry in which she demonstrated that Lagrange’s equation yielded Chladni’s patterns in several cases, but she could not give a satisfactory derivation of the equation. Nevertheless, the panel of judges, which included the best mathematicians of her time, deemed Sophie’s second mathematical memoir worthy of a honourable mention.

Once again the contest was re-opened and, in 1815, Sophie finally won the grand prix, a medal of one kilogram of gold grand prize. What should have been Sophie’s greatest achievement and source of pride, however, turned into a bittersweet victory. She received a laconic response from Siméon Denis Poisson, one of the judges and chief rival on the theory of elasticity, who wrote that her analysis still contained deficiencies and lacked mathematical rigour. Sophie did not appear at the award ceremony, to the disappointment of many who wished to meet her. It has been suggested1 that she thought the judges did not fully appreciate her work, and that the scientific community did not show the respect that seemed due to her. In fact, strangely enough, it was reported that Poisson avoided any serious discussion with Sophie and ignored her in public.

It is not difficult to imagine Sophie’s disappointment at the unwelcoming response from the scientists she so desperately sought out. Although she was the first to attempt solving such a challenging problem, and others used her analysis in elasticity to derive their own results, Sophie was not taken as seriously as she deserved to be.

Sophie extended her research and, in 1825, she submitted a paper to a commission of the Institut de France, whose members included Poisson, Gaspard de Prony and Laplace. Sadly, the commission ignored Sophie’s essay and her work went unrecognized until 1880, when it was found among Prony’s papers.

Joseph Fourier (1768 −1830), another great French mathematician who clearly admired Sophie, befriended her and tried to help. In 1823 he wrote to invite her to the meetings at the Academie des Sciences.

She also earned the respect of many others. French physicist Jean-Baptiste Biot wrote that “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” (Emilie Châtelet’s mathematical collaborator)2. Nevertheless, Sophie Germain continued to work alone until her death. Sophie was stricken with breast cancer in 1829 but, undeterred by her illness and the revolution that raged again in 1830, she wrote papers on number theory and on the curvature of surfaces. She outlined a philosophical essay, which her nephew published posthumously as Considérations générale sur l’état des Sciences et des Lettres in the Oeuvres Philosophiques.

Marie-Sophie Germain died in Paris on June 27, 1831. Her death certificate listed her not as mathematician or scientist as she ought to be recognized, but as a rentier, or a person of private means. Years later, however, a plaque was erected on the house she died naming Sophie Germain philosophe et mathématicienne.

This book was written to honor the memory of Sophie Germain, great woman, philosopher, and brillant mathematician who became immortal thanks to her work with prime numbers, her partial solution to Fermat’s Last Theorem, to her studies of mathematical physics, and her contribution to the study of elasticity.

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1 L. L. Bucciarelli and N. Dworsky, Sophie Germain: An Essay in the History of the Theory of Elasticity (Dordrecht - Boston, Mass., 1980).

2 Journal de Savants, March 1817.