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When she was eighteen (and like all French girls unable to study science at school) she managed to regularly obtain professors' notes from courses at the Ecole Polytechnique. Sophie followed along with the courses and boldly submitted comments and papers to the professors under the assumed name M. leBlanc. One of these professors, the brilliant Joseph-Louis Legrange, was impressed enough to request a meeting. leBlanc revealed her identity as a woman and Legrange, won over by Sophie's intelligence, continued to help her in her work.

A similar result occurred when Germain began corresponding as M. leBlanc with Carl Friedrich Gauss in Prussia, a man widely regarded as the greatest mathematician of his time. Gauss learned that his correspondent was a woman when Sophie prevailed upon her friend, French General Joseph-Marie Pernety to protect Gauss when Napoleon's armies invaded Gauss's home of Brunswick. Pernety complied, identified Gauss's protector as Sophie Germain and the connection was made with his friend Monsieur leBlanc. I am much taken by the ensuing correspondence:

Germain to Gauss, 1807:

In describing the honourable mission I charged him with, M. Pernety informed me that he made my name known to you. This leads me to confess that I am not as completely unknown to you as you might believe, but that fearing the ridicule attached to a female scientist, I have previously taken the name of M. LeBlanc in communicating to you those notes that, no doubt, do not deserve the indulgence with which you have responded.

Gauss to Germain:

But how to describe to you my admiration and astonishment at seeing my esteemed correspondent Monsieur Le Blanc metamorphose himself into this illustrious personage who gives such a brilliant example of what I would find it difficult to believe. A taste for the abstract sciences in general and above all the mysteries of numbers is excessively rare: one is not astonished at it: the enchanting charms of this sublime science reveal 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 superior genius. Indeed nothing could prove to me in so flattering and less equivocal manner that the attractions of this science, which has enriched my life with so many joys, are not chimerical, the predilection with which you have honored it.

Germain's research included important work in number theory on topics including prime numbers (a class of which bear her name) and Fermat's Last Theorem, as well as in the emerging field of solid mechanics, the physics of elasticity and vibration of solid objects, in which her contributions were a critical underpinning of the engineering of large structures such as modern skyscrapers and Paris's own Eiffel Tower.

She never married, and in 1831 she lost a two-year battle with breast cancer and died at the age of fifty-five. In the years preceding her death she and Gauss had renewed their correspondence. Gauss had convinced his employer, the University of Göttingen, to award Germain an honorary doctorate, the degree which she had long since earned but had never been able to receive. Her illness and eventually her death prevented her from traveling to Göttingen to receive her degree. The trip would also have marked the first time she and Gauss ever met.

For a long time Germain was overlooked by historians who denied her her place alongside the better known male giants of science and math, but in recent years she has achieved greater recognition. The net is rich with Sophie Germain information, and I referred to quite a few of these as I wrote this post. In addition to those linked above I recommend the following:

• About.com Women's History: Sophie Germain
• "Marie-Sophie Germain", article by J. J. O'Connor and E. F. Robertson at University of St. Andrews, Scotland, which also provided Germain's letter to Gauss in a link.
• "Sophie Germain" article by Carol Conrad at Wichita State University
• Simon Singh's article available under two titles: "Sophie and the Revolution" (on Singh's own site) and "Math's Hidden Woman" (on the NOVA site).
• Wikipedia entry: Sophie Germain
• Engines of Our Ingenuity, Number 223: Sophie Germain, by John H. Lienhard
• "Sophie Germain", article by Amanda Swift and Larry Riddle at Agnes Scott College
• mathNews: Sophie Germain (University of Waterloo)
• Eric Weisstein's World of Scientific Biography: Sophie Germain

Addendum: Germain went by Sophie instead of Marie or Marie-Sophie because both her mother and one of her sisters were named Marie.