The foundational work of Sophie Germain (1778-1831) on Fermat's Last Theorem, a problem unsolved in mathematics into the late 20th century, stood unmatched for over one hundred years. Though published by a mentor of hers, Adrien-Marie Legendre, it is still referred to in textbooks as Germain's Theorem.

# Sophie Germain

Germain worked alone, which was to her credit, yet contributed in a fundamental way to her limited development as a theorist. Her famed attempt to provide the mystery of Chladni figures with a pure mathematical model was made with no competition or collaboration. The three contests held by the Paris Academie Royale des Sciences from 1811 to 1816, regarding acoustics and elasticity of vibrating plates, never had more than one entry—hers. Each time she offered a new breakthrough: a fundamental hypothesis, an experimentally disprovable claim, and a treatment of curved and planar surfaces. However, even her final prizewinning paper was not published until after her death.

## Taught Herself Mathematics

Marie-Sophie Germain was born April 1, 1776, in Paris to Ambroise-Francois Germain and Marie-Madeleine Gruguelu. Her father served in the States-General and later the Constituent Assembly during the tumultuous Revolutionary period. He was so middle class that nothing is known of his wife but her name. Their eldest and youngest daughters, Marie-Madeleine and Angelique-Ambroise, were destined for marriage with professional men. However, when the fall of the Bastille in 1789 drove the Germains' sensitive middle daughter into hiding in the family library, Marie-Sophie's life path diverged from them all.

From the ages of 13 to 18 Sophie, as she was called to minimize confusion with the other Maries in her immediate family, absorbed herself in the study of pure mathematics. Inspired by reading the legend of Archimedes, purportedly slain while in the depths of geometric meditation by a Roman soldier, Germain sought the ultimate retreat from ugly political realities. In order to read Leonhard Euler and Isaac Newton in their professional languages, she taught herself Latin and Greek as well as geometry, algebra, and calculus. Despite her parents' most desperate measures, she always managed to sneak out at night and read by candlelight. Germain never formally attended any school or gained a degree during her entire life, but she was allowed to read lecture notes circulated in the Ecole Polytechnique. She passed in her papers under the pseudonym "Le Blanc."

## Correspondence School

Another tactic Germain used was to strike up correspondences with such successful mathematicians as Carl Gauss and Legendre. She was welcomed as a marvel and used as a muse by the likes of Jean B. Fourier and Augustin-Louis Cauchy, but her contacts did not develop into the sort of long-term apprenticeship that would have compensated for her lack of access to formal education and university-class libraries. Germain did become a celebrity once she dropped her pseudonym, however. She was the first woman not related to a member by marriage to attend Academie des Sciences meetings, and was also invited to sessions at the Institut de France—another first.

Some interpret Gauss' lack of intervention in Germain's education and eventual silence as a personal rejection of her. Yet this conclusion is not borne out by certain facts indicating Gauss took special notice. In 1810, Gauss was awarded one of his many accolades, a medal from the Institut de France. He refused the monetary component of this award, accepting instead an astronomical clock Germain and the institute's secretary bought for him with part of the prize. Gauss' biographer, G. Waldo Dunnington, reported that this pendulum clock was used by the great man for the rest of his life.

Gauss survived her, expressing at an 1837 celebration that he regretted Germain was not alive to receive an honorary doctorate with the others being feted that day. He alone had lobbied to make her the first such honored female in history. A hint of why Gauss valued her above the men who joined him in the Academie is expressed in a letter he sent to her in 1807, to thank her for intervening on his behalf with the invading French military. A taste for such subjects as mathematics and science is rare enough, he announced, but true intellectual rewards can only be reaped by those who delve into obscurities with a courage that matches their talents.

## No-Man's Land

Germain was such a rarity. She outshone even Joseph-Louis Lagrange by not only showing an interest in prime numbers and considering a few theorems, about which Lagrange had corresponded with Gauss, but already attempting a few proofs. It was this almost reckless attack of the most novel unsolved problems, so typical of her it is considered Germain's weak point by twentieth century historians, that endeared her to Gauss.

Germain's one formal prize, the Institut de France's Gold Medal Prix Extraordinaire of 1816, was awarded to her on her third attempt, despite persistent weaknesses in her arguments. For this unremedied incompleteness, and the fact that she did not attend their public awards ceremony for fear of a scandal, this honor is still not considered fully legitimate. However, the labor and innovation Germain had brought to the subjects she tackled proved of invaluable aid and inspiration to colleagues and other mathematical professionals as late as 1908. In that year, L. E. Dickson, an algebraist, generalized Germain's Theorem to all prime numbers below 1,700, just another small step towards a complete proof of Fermat's Last Theorem.

Germain died childless and unmarried, of untreatable breast cancer on June 27, 1831 in Paris. The responsibility of preparing her writings for posterity was left to a nephew, Armand-Jacques Lherbette, the son of Germain's older sister. Her prescient ideas on the unity of all intellectual disciplines and equal importance of the arts and sciences, as well as her stature as a pioneer in women's history, are amply memorialized in the Ecole Sophie Germain and the rue Germain in Paris. The house on the rue de Savoie in which she spent her last days was also designated a historical landmark.

## Books

Bucciarelli, Louis L., and Nancy Dworsky. *Sophie Germain: An Essay in the History of the Theory of Elasticity* . D. Reidel Publishing Co., 1980.

*Dictionary of Scientific Biography* . Volume V. Edited by Charles Coulston Gillispie. Charles Scribner's Sons, 1972.

Dunnington, G. Waldo. *Carl Friedrich Gauss: Titan of Science.* Exposition Press, 1955.

Mozans, H. J. *Woman in Science* . D. Appleton and Co.

Ogilvie, Marilyn Bailey. *Women in Science* . MIT Press, 1986.

Perl, Teri. *Math Equals: Biographies of Women Mathematicians.* Addison-Wesley Publishing, 1978.

*Women in Mathematics* . MIT Press, 1992.

*Women of Mathematics,* . Edited by Louise S. Grinstein and PaulJ. Campbell. Greenwood Press, 1987.

## Periodicals

*American Mathematical Monthly,* 92: 1985.

*Archive for History of Exact Science,* 41: 1990-91.

*Association for Women in Mathematics Newsletter,* 6: September-October 1976.

*Century,* 48: 1894. *Scientific American,* December 1991.

## Online

"Sophie Germain." *MacTutor History of Mathematics Archives,.* (December 1996). http://www-groups.dcs.st-and.ac.uk/history/index.html

"Sophie Germain." *Biographies of Women Mathematicians.* June 1997.http://www.scottlan.edu/lriddle/women/chronol.htm (July 22, 1997).

"The Ten Largest Known Sophie Germain Primes." *The Largest Known Primes.* 1995-96). http://www.utm.edu/research/primes/largest.html #Sophie.

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