Throughout much of history, women never had much opportunity to study mathematics. The Establishment, made up of men, perpetuated the myth that women weren't up to understanding anything scientific, and if a scientific or mathematically-minded woman came along, she would receive little support. Nonetheless, some women were able to advance in the fields of mathematical and scientific learning, and Sophie Germain was one of them.
The Death of Archimedes
Sophie Germain was born on 1 April, 1776, in Paris. Her father was a silk merchant, and quite successful. When she was 13 years old and browsing through her father's library she came across the story of Archimedes' death. Archimedes (the Greek mathematician famed among other things for jumping out of his bath and running down the street crying 'Eureka!' when he had solved an important problem) was killed by a Roman soldier when the Romans sacked Syracuse, where he lived, around 212BC. It is claimed that he was engrossed in some geometrical figure he had drawn on the sand, and when he refused to do immediately what the soldier told him, he was killed. Germain was highly impressed that someone was willing to die for mathematics - she thought the subject itself must be important, and was worthy of study.
Unfortunately for Germain, however, studying mathematics was not a considered a ladylike thing to do1. Her family was initially opposed to her studying the work of Newton and Euler until late into the night, and confiscated her clothes and candles so that she wouldn't be able to. However, Germain kept a secret supply of candles, and eventually her family came around to the idea that she was always going to study mathematics. Germain never married; her father supported her research throughout her life.
The Mysterious Monsieur Le Blanc
In 1794, the Ecole Polytechnique was founded in Paris. One of its students was Monsieur Antoine-August Le Blanc, who wasn't very mathematically gifted, and left Paris without letting the school authorities know. They continued to print material for him, and expected to receive work from him. Using the lecture notes meant for Monsieur Le Blanc, Germain, who couldn't have attended the Ecole Polytechnique on account of being female, was able to study some courses there and even submit some work.
Unsurprisingly (or surprisingly, depending on how you choose to look at it), the people marking the work of 'Le Blanc' soon realised that he had gone from being not very good at mathematics to presenting intelligent and insightful solutions to the problems posed to him. A paper Germain had written on analysis caught the attention of JL Lagrange. He decided he wanted to meet the student who had submitted the paper, and was surprised to find that 'Monsieur Le Blanc' was female! Lagrange, recognising her talent, became her mentor.
Germain's greatest contribution to mathematics was her work on prime numbers, inspired by trying to prove Fermat's Last Theorem. This says that for n greater than 2, there are no solutions to the following equation:
an+bn=cn where a, b, c and n are integers (whole numbers).
Despite seeming very simple, this is an incredibly complex problem in number theory that has only recently been proved. Germain wished to discuss the problem with another number theorist, the German mathematician Carl Friedrich Gauss (one of the greatest mathematicians of all time). She didn't want him to know that the questions were coming from a woman, though, in case he ignored her, so she used the pseudonym Le Blanc again. She and Gauss had a very fruitful correspondence, and Gauss would never have known that the letters he was receiving didn't come from a man if it hadn't been for Napoleon.
Sophie Germain lived during a period of political turmoil. The French Revolution took place when she was quite young, and Napoleon Bonaparte took power in 1799. By 1806, Napoleon was invading German cities and Germain, remembering Archimedes's fate, was afraid for the life of Gauss. She asked her friend, a general, to spare his life, which he did, telling Gauss that he had to thank one Mademoiselle Germain. Of course, Gauss had no idea who this was, or why she knew of him, so Germain had to come clean. Gauss was very impressed with her, however, all the more so because he knew what kind of hurdles she would have had to overcome being a woman.
Germain's work on Fermat's last theorem led to Legendre and Dirichlet proving it for the case of n=5.
Work on Elasticity
Algebra is but written geometry, and geometry is but figured algebra.
Sophie Germain - Memoir sur les surfaces élastiques
Germain's later work was mainly on the subject of elasticity. In 1809, the French Academy of Sciences offered a prize to anyone who could come up with a mathematical theory of elasticity. The deadline was in 1811, and Germain was the only person who entered. Her entry was incomplete, however, and Laplace helped her to correct her errors. Finally, in 1815, after the deadline had been extended twice, the Academy awarded her the prize.
Germain continued to work in mathematics, and also in philosophy, until she died of breast cancer on 27 June, 1831. Gauss had persuaded the university of Gottingen, where he was a professor, to award her an honorary degree, but unfortunately she died before she could receive it. Even after her work on elasticity she still faced prejudice. While Jean-Baptiste-Joseph Fourier (of Fourier Theory fame) enabled her to attend the proceedings of the French Academy of Sciences, when she submitted a paper to the Institut de France it was ignored. Even on her death certificate she was not listed as a mathematician.