## Revolutionary Pursuits

## Circa May 1816: Germain Forms Theory of Elastic Surfaces

This image is taken from "Sophie Germain—Revolutionary Mathematician" at http://www.sdsc.edu/ ScienceWomen/germain.html |

She became interested in mathematics as a child. One day, while browsing in her father's library, she came across an account of the death of the Greek mathematician Archimedes. Legend has it that when the Roman army invaded his home city of Syracuse, Archimedes was so engrossed in studying a geometric figure that he failed to respond to a soldier's questioning, and was unceremoniously speared to death. The young Sophie concluded that if someone could be so consumed by a geometric problem, it must be the most fascinating subject in the world.

She taught herself the basics of number theory and calculus without the aid of a tutor, and began studying Newton and Euler at night. But her parents objected to her learning such an "unfeminine" subject and did their best to dissuade her. Her father confiscated her candles and clothing and removed all heating. But Germain persevered. She kept a secret cache of candles and wrapped herself in bedclothes to keep warm while she secretly studied, even though it was often so cold that the ink froze in the inkwell. Eventually her parents relented, and her father supported her for the rest of her life, since she never married.

In 1794, when Germain was 18, the Ecole Polytechnique opened in Paris to train future mathematicians and scientists. But it was open only to men. Undaunted, Germain assumed the identity of a former student, Antoine August LeBlanc. Every week she would submit answers to problems under her new pseudonym. The course supervisor, Joseph Louis Lagrange, soon noticed the marked improvement in a student who had previously shown abysmal mathematical skills, and requested a meeting. Germain was forced to reveal her true identity, but far from being appalled, Lagrance was impressed, and become her mentor.

Among the problems that caught Germain's interest in her early 20s was Fermat's Last Theorem, first mentioned by the mathematician in 1637. This is the statement that for arbitrary positive integers x, y, and z, and for any integer *n* greater than 2, the equation

x^{n} + y^{n} = z^{n} has no solutions. The proof was still undiscovered more than 150 years later. [In fact, it wasn't proved until 1994.]

Encouraged by a second mentor, the German mathematician Carl Friedrich Gauss, Germain adopted a new, more general approach to the problem: instead of proving one particular equation had no solutions, she outlined a calculation which focused on the case that the index *n* in Fermat's theorem was what is now called a Germain prime number-that is, a prime number n for which *2n*+1 is also a prime. In that case, she was able to show, except for the special case that the integers x, y and z were multiples of n, that Fermat's theorem was true. This partial result encouraged other mathematicians to think that Fermat's Last Theorem for the case that n is a Germain prime could be fully solved.

Germain then turned to physics, intrigued by the experiments of German physicist F. F. Chladni on vibrating plates. Chladni produced curious patterns on small glass plates covered with sand and played them, as though the plates were violins, by using a bow. "The sand moved about until it reached the nodes, and the array of patterns resulting from the 'playing' of different notes caused great excitement among the Parisian polymaths.

It was the first scientific visualization of two dimensional harmonic motion." (*From the reference given in the caption*). So the Institut de France sponsored a competition with the challenge of formulating a mathematical theory of elastic surfaces in keeping with the empirical evidence observed by Chladni.

Most mathematicians didn't even bother to enter, believing the existing mathematical models were inadequate to solve the problem. But Germain spent the next 10 years attempting to solve it, submitting three separate papers-the only entrant in the competition all three years. She failed to win anything in her first two submissions, but on her third attempt, submitted around May 1816, the judges deemed her paper entitled "Memoir on the Vibrations of Elastic Plates" worthy of a prize, despite pointing out some remaining mathematical deficiencies.

Germain refused to attend the award ceremony. She apparently felt the judges did not fully appreciate her work, and that the scientific community as a whole did not show her the respect she believed she had clearly earned.

Germain had a point. Poisson, her chief rival on the subject of elasticity, was one of the judges and technically a colleague, yet he pointedly avoided having serious discussions with her, and snubbed her in public. Nonetheless, Germain became the first woman to attend lectures at the French Academy of Sciences who was not the wife of a member-the highest honor that body had ever conferred on a woman.

Late in her life, Gauss arranged to have the University of Gottingen award Germain an honorary degree. Sadly, she died of breast cancer a mere few weeks before the honor could be bestowed upon her. She was 55. Her death certificate identified her not as a mathematician, but as a "single woman with no profession."

Germain was a true social revolutionary, in keeping with someone who was born the year the American Revolution began, and started learning calculus 13 years later at the onset of the French Revolution. She succeeded in becoming a celebrated mathematician despite the social prejudices of her era.

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