In 1794, when Sophie was 18, the Ecole Polytechnique was founded in Paris. It was an academy to train mathematicians and scientists for the country, but women were not allowed to enroll. Nevertheless, Sophie was able to obtain the lecture notes for several of the courses and study from them. This gave her the opportunity to learn from many of the prominent mathematicians of those days. Sophie was particularly interested in the teachings of Joseph Louis Lagrange. Under the pseudonym of M. LeBlanc (a former student of Lagrange's), she submitted a paper on analysis to Lagrange at the end of the term. He was quite impressed with the work and wanted to meet the student who had written it. He was amazed that the author of the work was actually a female; he recognized her abilities and became her mentor. With a male to introduce her, Sophie could enter the circle of scientists and mathematicians that she never before could. Up until this point not only had her gender been a hindrance to her, but her social status had been too, because she was not an aristocratic woman but one of the middle class. Lagrange certainly made his colleagues aware that Germain was a girl with mathematical talent and several of them wrote to her; for instance, Monge. But not everyone treated her with the respect she deserved. One case was Jérôme Lalande. She started to talk to him about Laplace's Exposition du système du monde. Lalande told her that she should not be reading such works, rather she should be reading the second edition of his book Astronomie des dames. This "astronomy for ladies" does not contain a single mathematical equation and Germain felt insulted by his suggestion. Lalande sent her a letter of apology but she never forgave him.
However, Germain’s most famous correspondence began in 1804 with the German mathematician Carl Friedrich Gauss. She was intrigued with his work in number theory and sent him some of the results of her work. Again, she used her pseudonym to disguise her true identity; she feared being ignored because of being a woman. It was not until 1807 that Gauss found out who M. LeBlanc truly was. He was thrilled to find that his "pen pal" was a very gifted woman. Gauss was amazed for seeing that all the excellent that he could find in the interchange of letters and discussions came from a female, considered the weak gender. He was not bothered but pleased.
At about this time, in 1808, the French Academy of Sciences announced a contest to explain the "underlying mathematical law" of a German physicist's study on the vibration of elastic surfaces. Sophie was fascinated and set out to explain the law underlying Chladni's study. The Academy set a two-year deadline, and in 1811 Sophie submitted the only entry in the contest. Her lack of formal education was evident in the anonymous paper she submitted, and thus she was not awarded the prize. Lagrange corrected her errors and two years later she again entered the contest which had been extended. She received honorable mention this time. It was in 1816, when she entered for the third time and won with her paper Memoir on the Vibrations of Elastic Plates. After winning the contest, Sophie continued her work on the theory of elasticity publishing several more memoirs. Her work in the theory of elasticity would prove to be very important to the field.
Among her work done during these years is what would be her most important work in number theory. In 1820, Sophie made a statement about the divisibility of solutions to Fermat’s equation, for an odd prime p. She proved that if x, y, and z are integers and if x + y = z then either x, y, or z must be divisible by 5.
The prize from the Academy, however, was of immediate importance because it introduced her into the ranks of the prominent mathematicians of the time. She became the first woman who was not a wife of a member to attend the Academy of Sciences' sessions. She was also praised by the Institut de France and was invited to attend their sessions. Sophie continued working with well-known male mathematicians in the 1820s as an "equal collaborator" to refine her proofs and work in number theory; and also continued to work in philosophy until her death, in 1831.
After a battle with breast cancer, Sophie Germain died at the age of 55 on June 27, 1831 in Paris. Shortly before this Gauss, one of her earliest mentors, had convinced the University of Gottingen to give Sophie an honorary degree. She died before she could receive it.
https://www.agnesscott.edu/lriddle/women/germain.htm#:~:text=Sophie%20Germain%20was%20a%20revolutionary,also%20very%20important%20to%20mathematics.
https://www.britannica.com/biography/Sophie-Germain
https://mathshistory.st-andrews.ac.uk/Biographies/Germain/
https://totallyhistory.com/sophie-germain/
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