In 1794, when Germain was 18, the École Polytechnique opened. As a woman, Germain was barred from attending, but the new system of education made the "lecture notes available to all who asked." The new method also required the students to "submit written observations." Germain obtained the lecture notes and began sending her work to Joseph Louis Lagrange, a faculty member. She used the name M. LeBlanc, "fearing," as she later explained to Gauss, "the ridicule attached to a female scientist." When Lagrange saw the intelligence of M. LeBlanc, he requested a meeting, and thus Sophie was forced to disclose her true identity. Fortunately, Lagrange did not mind that Germain was a woman, and he became her mentor.
Around 1807, the French were occupying the German town of Braunschweig, where Gauss lived. Germain, concerned for his safety, wrote to General Pernety, a family friend, requesting that he ensure Gauss' safety. Pernety sent a chief of a battalion to meet with Gauss personally to see that he was safe. As it turned out, Gauss was fine, but he was confused by the mention of Sophie's name.
Three months after the incident, Germain disclosed her true identity to Gauss. He replied
How can I describe my astonishment and admiration on seeing my esteemed correspondent M leBlanc metamorphosed into this celebrated person ... when a woman, because of her sex, our customs and prejudices, encounters infinitely more obstacles than men in familiarising herself with [number theory's] knotty problems, yet overcomes these fetters and penetrates that which is most hidden, she doubtless has the most noble courage, extraordinary talent, and superior genius.Germain became interested in a contest sponsored by the Paris Academy of Sciences concerning Ernst Chladni's experiments with vibrating metal plates. The object of the competition, as stated by the Academy, was "to give the mathematical theory of the vibration of an elastic surface and to compare the theory to experimental evidence." After two failed attempts at winning, Germain submitted her third paper, "Recherches sur la théorie des surfaces élastique" under her own name, and on Jan. 8, 1816, she became the first woman to win a prize from the Paris Academy of Sciences. She did not appear at the ceremony to receive her award. After winning the Academy contest, she was still not able to attend its sessions because of the Academy's tradition of excluding women other than the wives of members. Seven years later this tradition was broken when she made friends with Joseph Fourier, a secretary of the Academy, who obtained tickets to the sessions for her.
In addition to mathematics, Germain studied philosophy and psychology. She wanted to classify facts and generalize them into laws that could form a system of psychology and sociology, which were then just coming into existence. Her philosophy was highly praised by Auguste Comte.
On June 27 of 1831, Germain died, of breast cancer. Despite Germain's intellectual achievements, her death certificate lists her as a “rentière – annuitant” (property holder, not a "mathematicienne." But her work was not unappreciated by everyone. When the matter of honorary degrees came up at the University of Göttingen six years after Germain's death, Gauss lamented, "[Germain] proved to the world that even a woman can accomplish something worthwhile in the most rigorous and abstract of the sciences and for that reason would well have deserved an honorary degree."
Two of her philosophical works, "Pensées diverses" and "Considérations générales sur l'état des sciences et des letteres aux différentes epoques de leur culture," were published, both posthumously. This was due in part to the efforts of Lherbette, her nephew, who collected her philosophical writings and published them. Pensées is a history of science and mathematics with Sophie's commentary. In Considérations, the work admired by Comte, Sophie argues that there are no differences between the sciences and the humanities.