Sophie Germain

French mathematician

• Born: April 1, 1776
• Birthplace: Paris, France
• Died: June 27, 1831
• Place of death: Paris, France

Germain overcame the limits of a haphazard education and a variety of social and institutional impediments to make fundamental advances in the proof of Fermat’s last theorem and in the physics of elasticity. Those achievements represent the most original and significant contribution to mathematics by any woman before the end of the nineteenth century.

Early Life

Sophie Germain (zhayr-mehn) was born Marie-Sophie Germain, the daughter of Ambroise-François Germain, a prosperous French silk merchant. All that is known of her mother is the latter’s name: Marie-Madeleine Gruguelin. Sophie also had two sisters—Marie-Madeleine, who was six years older, and Angélique-Ambroise, who was three years younger. That she shared the given name “Marie” with her mother and older sister probably explains her lifelong use of “Sophie” by itself. Destined to lead conventional lives of Parisian upper-middle-class women, both of Sophie’s sisters married prominent professional men. In contrast to her sisters, Sophie never married, and her life was anything but conventional.

In 1789, the year of the French Revolution, Sophie’s father was elected to the Estates General that King Louis XVI had been forced to convene. Over the next two years, the family shop and home was a center of political discussion. The earliest account of Sophie’s life by her friend and fellow mathematician Guglielmo Libri-Carucci suggests that she disliked these discussions. Retreating to her father’s study, she became absorbed in Jean-Etienne Montucla’s Histoire des mathématiques (1799). In that book she discovered the story of the famous ancient Greek mathematician Archimedes, who, lost in the beauty of a geometrical demonstration and oblivious to the turmoil of the Roman conquest of Syracuse, was killed when he failed to respond to a soldier’s order.

Ignoring the political drama of the revolution and the social expectations of her family, Sophie was similarly captivated by the study of mathematics. Libri-Carucci’s account of her life depicts concerned parents depriving their daughter of candles and even heat, all in an effort to discourage her new interest, but all to no avail. Sophie soon taught herself Latin so she could read the works of Isaac Newton and Leonhard Euler. Eventually, her parents relented. As her father’s foray into politics ended, the Germain house on the rue St. Denis increasingly played host to a company of scholars.

Life’s Work

The years of the French Revolution constituted one of the most formative periods in the history of mathematics. The center of this ferment was the École Polytechnique, which opened in 1794. Its faculty included a true pantheon of late Enlightenment scientists. However, its classrooms excluded women. Nonetheless, an innovative pedagogy that made professors’ lecture notes public and that invited student observations offered Sophie Germain an opportunity. She obtained the lecture notes from the college and, borrowing the name of a male student, Monsieur Le Blanc, she offered observations on Joseph-Louis Lagrange’s mathematics lectures. It was not long before Lagrange uncovered Germain’s deception, but he was so impressed with her ability he encouraged her to continue her work.

Pursuing an interest in number theory in 1798, Germain initiated a correspondence with Adrien-Marie Legendre, the author of an important recent treatise on the subject, and began to study the Disquisitions arithmetical that Carl Friedrich Gauss published in 1801. Again under the name M. Le Blanc, she began a correspondence with Gauss in 1804. Her identity was revealed only in the aftermath of the French-Prussian battle at Jena in October, 1806. Fearing that Gauss might suffer the same fate as Archimedes, Germain used her family connections in the French military to ensure the safety of the German mathematician.

Germain’s correspondences with Legendre and Gauss are the source of the first of her two major contributions to mathematics. Around the year 1638, the French mathematician Pierre Fermat articulated a theorem while annotating a copy of the third century Greek mathematician Diophantus of Alexandria’s Arithmetica. Fermat observed that, in contrast to the Pythagorean theorem, which holds that the square of the hypotenuse of a right triangle is equal to the sum of the squares of its other two legs, there were no solutions for the equation xⁿ + yⁿ = zⁿ when the factor n is greater than 2. When Fermat published his notes on Diophantus’s Arithmetica in 1670, his observation stimulated important advances in number theory. However, proof of the tantalizing intuition that it was impossible to find a cube that was the sum of two other cubes or indeed to find any number raised to a power greater than 2 that was the sum of two other numbers raised to the same power—Fermat’s so-called “last theorem”—turned out to be one of the most intractable problems in the history of mathematics.

Fermat himself had offered a proof of the case n = 4. In 1738, Euler proved the case n = 3. Germain demonstrated that the conditions for the case in which n is an odd prime number, such that 2p + 1 was also prime, were so stringent as to be virtually impossible. In short, she moved beyond proofs of singular cases to a general strategy for many cases. Modern mathematics honors her achievement by referring to those prime numbers p in which 2p + 1 is also prime as “Sophie Germain primes.”

During 1807-1808, Germain’s interests turned to applied mathematics. In 1809, the Institute of France announced a two-year prize competition for a mathematical theory of elasticity. Two years later, Germain submitted the only entry in the competition. The institute acknowledged that her “experiments presented ingenious results” but judged the rigor of her analysis inadequate and renewed the competition. Working with Lagrange, Germain again submitted the only entry in 1813. This time the institute awarded her an honorable mention but questioned the derivation of her equations from established principles of physics. Only after a second extension did the institute finally award Germain the prize, making her the first women ever to achieve such a public recognition in mathematics.

Germain did not attend the award ceremony on January 8, 1816. Some historians speculate that she wished to avoid the attendant notoriety that the event would bring to her. Other historians suggest that she was angry that the institute had made its award to her with reservations and that one of her judges, Siméon-Denis Poisson, was using her work without attribution for his own alternative theory of elasticity.

During the 1820’s, Germain’s friendship with the mathematician Joseph Fourier, another rival of Poisson, played a role in allowing Germain to be the first woman who was not a wife of a member to attend sessions of the French Academy of Sciences. By this time, she was increasingly writing on the philosophy of science. In her essay, “General Considerations on the Condition of the Arts and Sciences at Different Stages of their Cultivation,” she—much as her younger contemporary Auguste Comte—argued that just as the natural sciences proceeded from observation and classification of phenomena through generalization and mathematical systemization, so similar progress was possible in the human sciences. Germain herself, however, was unable to contribute to this progress. On June 27, 1831, she died in Paris after a two-year battle with breast cancer.

Significance

Shortly before Sophie Germain died, Gauss recommended her to the University of Göttingen for an honorary degree, but his request was refused. Germain never received a university degree, she never attended university classes, and, indeed, never seems to have had much formal education at all. Nevertheless, in number theory, she took what at that time was the single most significant step forward in the proof of Fermat’s last theorem—a theorem that was finally proved only in 1994. Likewise, in applied mathematics, it was not the model of the impeccably rigorous scientific insider Poisson that provided the foundation for the physics of elasticity but the model of the self-taught scientific outsider Sophie Germain.

The life of Sophie Germain is eloquent testimony to the difficulties women have historically faced in pursuing scientific careers. Through sheer determination, Germain was able to transcend social expectations. In the end, her parents’ generosity provided her enough income to live independently. She was also able to challenge some of the institutional barriers to women in science. Through what Libri-Carucci aptly called her courage, Germain secured the support of many of the most eminent professional mathematicians of her time. However, subtle pressures always weighed on her while she worked. For example, when she was preparing her theory of elasticity and had to visit the École Polytechnique or Institute of France, she needed formal invitations and escorts. When she submitted a paper to the Academy of Sciences extending her theory in 1825, her submission was simply ignored.

Posthumous recognition has come to Germain almost as reluctantly. In 1889, when the opening of Paris’s Eiffel Tower commemorated seventy-two people whose contributions to the mathematics of elasticity made possible the tower’s construction, there was no mention of Sophie Germain. Thanks to the increase in interest in women’s history in late twentieth century Europe, Germain’s name now appears on Parisian schools and streets. However, in many standard histories of mathematics and science, the achievements of Sophie Germain are still overlooked.

Bibliography

Bucciarelli, Louis L., and Nancy Dworsky. Sophie Germain: An Essay in the History of the Theory of Elasticity. Dordrecht, Netherlands: D. Reidel, 1980. This is the only extended study in English of Germain’s achievements, but Bucciarelli and Dworsky present a lifeless Germain, and their presentation of the theory of elasticity is too technical for average readers.

Dahan Dalmédico Amy. “Sophie Germain.” Scientific American 265, no. 6 (December, 1991): 116-122. A clear, if brief, summary of Germain’s career by a leading historian of mathematics.

Petrovich, Vesna Crnjanski. “Women and the Paris Academy of Sciences.” Eighteenth-Century Studies 22, no. 3 (1999): 383-390. Places Germain in the context of the strategies that eighteenth and early nineteenth century women used to achieve recognition for their scientific work.

Singh, Simon. Fermat’s Enigma. New York: Doubleday-Anchor, 1997. This lucid introduction to the history of Fermat’s last theorem includes an appreciative discussion of Germain’s contribution.