**Marie-Sophie Germain** was the middle daughter of Ambroise-François, a prosperous silk-merchant, and Marie-Madelaine Gruguelin. Sophie's home was a meeting place for those interested in liberal reforms and she was exposed to political and philosophical discussions during her early years.

At the age of thirteen, Sophie read an account of the death of Archimedes at the hands of a Roman soldier. She was moved by this story and decided that she too must become a mathematician. Sophie pursued her studies, teaching herself Latin and Greek. She read Newton and Euler at night while wrapped in blankets as her parents slept - they had taken away her fire, her light and her clothes in an attempt to force her away from her books. Eventually her parents lessened their opposition to her studies, and although Germain neither married nor obtained a professional position, her father supported her financially throughout her life.

Sophie obtained lecture notes for many courses from École Polytechnique. At the end of Lagrange's lecture course on analysis, using the pseudonym M. LeBlanc, Sophie submitted a paper whose originality and insight made Lagrange look for its author. When he discovered "M. LeBlanc" was a woman, his respect for her work remained and he became her sponsor and mathematical counsellor. Sophie's education was, however, disorganised and haphazard and she never received the professional training which she wanted.

Germain wrote to Legendre about problems suggested by his 1798 *Essai sur le Théorie des Nombres*, and the subsequent Legendre - Germain correspondence became virtually a collaboration. Legendre included some of her discoveries in a supplement to the second edition of the *Théorie.* Several of her letters were later published in her *Oeuvres Philosophique de Sophie Germain*.

However, Germain's most famous correspondence was with Gauss. She had developed a thorough understanding of the methods presented in his 1801 *Disquisitiones Arithmeticae*. Between 1804 and 1809 she wrote a dozen letters to him, initially adopting again the pseudonym "M. LeBlanc" because she feared being ignored because she was a woman. During their correspondence, Gauss gave her number theory proofs high praise, an evaluation he repeated in letters to his colleagues. Germain's true identity was revealed to Gauss only after the 1806 French occupation of his hometown of Braunschweig. Recalling Archimedes' fate and fearing for Gauss's safety, she contacted a French commander who was a friend of her family. When Gauss learnt that the intervention was due to Germain, who was also "M. LeBlanc", he gave her even more praise.

Among her work done during this period is work on Fermat's Last Theorem and a theorem which has become known as Germain's Theorem. This was to remain the most important result related to Fermat's Last Theorem from 1738 until the contributions of Kummer in 1840.

In 1808, the German physicist Ernst F F Chladni had visited Paris where he had conducted experiments on vibrating plates, 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 for all entries was set.

Most mathematicians did not attempt to solve the problem, because Lagrange had said that the mathematical methods available were inadequate to solve it. Germain, however, spent the next decade attempting to derive a theory of elasticity, competing and collaborating with some of the most eminent mathematicians and physicists.

In fact, Germain was the only entrant in the contest in 1811, but her work did not win the award. She had not derived her hypothesis from principles of physics, nor could she have done so at the time because she had not had training in analysis and the calculus of variations. Her work did spark new insights, however. Lagrange, who was one of the judges in the contest, corrected the errors in Germain's calculations and came up with an equation that he believed might describe Chladni's patterns.

The contest deadline was extended by two years, and again Germain submitted the only entry. She demonstrated that Lagrange's equation did yield Chladni's patterns in several cases, but could not give a satisfactory derivation of Lagrange's equation from physical principles. For this work she received an honourable mention.

Germain's third attempt in the re-opened contest of 1815 was deemed worthy of the prize of a medal of one kilogram of gold, although deficiencies in its mathematical rigour remained. To public disappointment, she did not appear as anticipated at the award ceremony. Though this was the high point in her scientific career, it has been suggested 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 .

Certainly Poisson, her chief rival on the subject of elasticity and also a judge of the contest, sent a laconic and formal acknowledgement of her work, avoided any serious discussion with her and ignored her in public.

As one biographer phrases it:

Although it was Germain who first attempted to solve a difficult problem, when others of more training, ability and contact built upon her work, and elasticity became an important scientific topic, she was closed out. Women were simply not taken seriously.

Germain attempted to extend her research, in a paper submitted in 1825 to a commission of the Institut de France, whose members included Poisson, Gaspard de Prony and Laplace. The work suffered from a number of deficiencies, but rather than reporting them to the author, the commission simply ignored the paper. It was recovered from de Prony's papers and published in 1880.

Germain continued to work in mathematics and philosophy until her death. Before her death, she outlined a philosophical essay which was published posthumously as *Considérations générales sur l'état des sciences et des lettres* in the *Oeuvres philosophiques*. Her paper was highly praised by August Comte. She was stricken with breast cancer in 1829 but, undeterred by that and the fighting of the 1830 revolution, she completed papers on number theory and on the curvature of surfaces (1831).

Germain died in June 1831, and her death certificate listed her not as mathematician or scientist, but *rentier* (property holder).

**Article by:** *J J O'Connor* and *E F Robertson*

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