Germain decided that if geometry, which at that time referred to all of pure mathematics, could hold such fascination for Archimedes, it was a subject worthy of study. So she pored over every book on mathematics in her father's library, even teaching herself Latin and Greek so she could read works like those of Sir Isaac Newton and Leonhard Euler. She also enjoyed Traité d'Arithmétique by Étienne Bézout and Le Calcul Différentiel by Jacques Antoine-Joseph Cousin. Later, Cousin visited her in her house, encouraging her in her studies.
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 of a former student Monsieur Antoine-August Le Blanc, "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. He too visited her in her home, giving her moral support.
Germain's parents did not at all approve of her sudden fascination with mathematics, which was then thought inappropriate for a woman. When night came, they would deny her warm clothes and a fire for her bedroom to try to keep her from studying, but after they left she would take out candles, wrap herself in quilts and do mathematics. As Lynn Osen describes, when her parents found Sophie "asleep at her desk in the morning, the ink frozen in the ink horn and her slate covered with calculations," they realized that their daughter was serious and relented. After some time, her mother even secretly supported her.
One of the pioneers of elasticity theory, she won the grand prize from the Paris Academy of Sciences for her essay on the subject. Her work on Fermat's Last Theorem provided a foundation for mathematicians exploring the subject for hundreds of years after. Because of prejudice against her sex, she was unable to make a career out of mathematics, but she worked independently throughout her life. In recognition of her contribution towards advancement of mathematics, an honorary degree was also conferred upon her by the University of Göttingen six years after her death. At the centenary of her life, a street and a girls' school were named after her. The Academy of Sciences established The Sophie Germain Prize in her honor.