# Germain, Sophie

# Germain, Sophie

(*b*. Paris, France, 1 April 1776; *d*. Paris, 27 June 1831)

*mathemtics*.

Sophie Germain, France’s greatest female mathematician prior to the present ear, was the the daugther of Ambroise-François Germain and Marie-Madeleine Gruguelu. Her father was for a time deputy to the State-General (later the Constituent Assembly). In his speeches he referred to himself as a merchant and ardently defended the rights of the Third Estate, which he represented, Somewhat later he became one of the directors of the Bank of France. His extensive library enabled his daughter to educate herself at home. Thus it was that, at age thirteen, Sophie read an account of the death of Archimedes at the hands of a Roman soldier. The great scientist of antiquity became her hero, and she conceived the idea that she too must become a mathematician. After teaching herself Latin and Greek, she read Newton and Euler despite her parent’s opposition to a career in mathematics.

The Germain library sufficed until Sophie was eighteen. At that time she was able to obtain the lecture notes of courses at the recently organized École Polytechnique, in particular the *cahiers* of Lagrange’s lectures on analysis. Students at the school were expected to prepare end-of-term reports. Pretending to be a student there and using the pseudonym Le Balanc, Sophie Germain wrote a paper on analysis and sent it to Lagrange. He was stounded at its originally, praised it publicly, sought out its author, and thus discovered that M. Le Blanc was Mlle. Germain, From then on, he became her sponsor and mathemtical counselor.

Correspondence with great scholars became the means by which she obtained ther higher education in mathematics, literature, biology, and philosophy, She wrote to Legendre about problems suggested by his 1798 *Théorie des nombres*. The subsequent Legendre-Germain correspondence was so voluminous that it was virtually a collaboration, and Legendre included some of her discoveries in a supplement to the second edition of the *Théorie*. In the interim she had read Gauss’s *Disquisitiones arithmeticate* and, under the pseudonym of Le Blanc, engaged in corrrespondent with its author.

That Sophie Germain was no ivory-tower mathematician became evident in 1807, when French troops were occupying Hanover. Recalling Archimedes’ fate and fearing for Gausss’s safety, she addressed an inquiry to the French commander, General Pernety, who was a friend of the Germain family. AS a result accorded even more praise to her number-theoretic proofs.

One of Sophie Germain’s theorems is related to the baffling and still unsolved problem of obtaining a general proof for “Fermat’s last theorem,” which is the conjecture that *X*^{n} + *Y*^{n} = *Z*^{n} has no positive integral solutions if *n* is an integer greater than 2. To prove the theorem, one need only establish its truth for *n* = 4 (accomplished by Fermat himself) and for all values of *n* that are odd primes. Euler proved it for *n* = 3 and Legendre for *n*= 5. Sophie Germain’s contribution was to show the impossibility of postive integral solutions if *x, y, z* are prime to one another and to *n*, where *n* is any prime less than generalized her theorem to all primes less than 1,700, and more recectly Barkley Rosser extended the upper limit to 41,000,000. In his history of the theory of numbers, Dickson describes her other discoveries in the higher arithmetic.

Parallel with and subsquent to her pure mathematical research, she also made contributions to the applied mathematics of acoustics and elasticity. This came about in the follwing manner. In 1808 the German physicist E. F. F. Chladniu visited Paris, where he conducted experiments on vibrating plates. He exhibited the so-called Chladniu figures, which can be produced when a metal or glass plate of any regular shape, the most or glass plate of any of the circle, is placed in a horizontal position and fastened at its center to a supporting stand. Sand is scattered lightly over the plate, which is then set in vibration by drawing a violin bow rapidly up and down along the edge of the plate. The sand is thrown from the moving points to those which remain at rest (the nodes), forming the nodal lines or curves constituting the Chladnui figures.

Chladni’s results were picturesque, but their chief effect on French mathematicians was to emphasize that there was no pure mathematical model for such phenomena. Hence, in 1811 the Académie des Sciences offered a prize for the best answere to the following challenge: Formulate a mathematical theory of elastic surfaces and indicated just how it agrees with empirical evidence.

Most mathematicians did not attempt to solve the problem because Lagrange assured them that the mathematical methods available were inadequate for the task. Neverthless, Sophie Germain submitted an anonymous memoir. No prize was awarded to any one; but Lagrange, using her fundamental hypotheses, was able to deduce the correct partial differential equation for the vibrations of elastic plates. In 1813 the Academy reopened the contest, and Sophie Germain offered a revised paper which included the question of experimental verification. That memoir received an honorable mention. When, in 1816, the third and final contest was held, a paper bearing her own name and treating vibrations of general curved as well as plane elastic surfaces was awarded the grand prize—the high point in her scientific career.

After further enlargement and improvement of the prize memoir, it was published in 1821 under the title *Remarques sul la nature, les bornes et l’étendue de la question des surfaces élastiques et éequation générale de ces surfaces*. In that work Sophie Germain stated that the law for the general vibrating elastic surface is given by the fourth-order partial differential equation.

Here *N* is a physical constant if the “surface” is an elastic membrane of uniform thickness, The generality us achieved because *S*, the radius of mean curvature, varies from point to point of a general curved surface. The very concept of mean curvature (*l/S*) was created by Sophie Germain.

The notion of the curvature of a surface generalizes the corresponding concept for a plane curve by considering the curvatures of all plane sections of surface through the normal at a given point of the surface and then using only the largest and smallest of those curvatures. The extremes, called the principal curvatures, are multiplied to give the Gaussian total curvature. Sophie Germain, however, defined the mean curvature as half the sum, that is, the arithmetic mean, of the principal curvature. Her definition seems more in accordance with the term “mean,” Moreover, she indicated that her measure is a representative one, an average in the statistical sense, by demonstrating that if one passes such that through the normal at a pint of surface such that the angel between successive planes in 2π/*n* where *n* very large (thus yielding sample sections in many different directions), the arithmetic mean of the curvatures of all the sections is the same as the mean of the two principal curvatures, a fact that remains true in the limits *n* best larger and larger. Also, while the Gaussian curvature completely characterizes the local metric geometry of a surface, the mean cruvature is more suitabe for applications in elasticity theory. A plane has zero mean curvature at all points. Hence 4/*S*^{2} = 0 in Germain’s differential equation, and it reduces to the equation which she and Lagrange had derived for the vibration of flat plates. The same simplification holds for all surfaces of zero mean curvature, the so-called minimal surfaces (such as those formed by a soap film stretched from wire contours).

In later papers Sophie Germain enlarged on the physics of vibrating curved elastic surfacves and considered the effect of variable, thickness (which emphasizes that one is, in fact, dealing with elastic solids).

She also wrote two philosophic works entitled *Pensées diverses and Consideé’rations générales sur l’état des sciencs et des lettres*, which were published post humously in the *Owuvres philosophiques*. The first of these, probably written in her youth, contains, capsule summaries of scientific subjects, brief comments on physicsts throughout the ages, and personal opinions. The *État des sciences et des lettres*, which was praised by Auguste Comte, is an extremely shcolarly development of the theme of the unity of thought, that is, the idea that there always has been and always will be not basic difference between the sciences and the humanities with respect to their motivation, their methodology, and their cultural importance.

## BIBLIOGRAPHY

I. Original Works. Among Sophie Germain’s scientific writings are *Remarques sur la nature, les bornes et l’étendue de la questuib des surfaces élastuiques et équation gvénérale de ces surfaces* (Paris, 1826); *Mémoire sur la courbure des surfaces* (Paris, 1830); *Oeuvers philosophique de Sophie Germain* (Paris, 1879); and *mémoire sur l’emploi de l’épaisseur dans la théorie des surfaces élastiques* (Paris, 1880).

II. Secondary Literature. On Sophie Germain of her work, see L. E. Dickson, *History of the Theory of Numbers* (New York, 1950), I, 382; II, 732-735, 757, 763, 769; M. L. Durbreil-Jacotin, “Figures de mathématixciennesm,” in F. Le Lionnais, *Les grands courants de la pensée mathématique* (Paris, 1962), pp. 258-268; and H. Stupuy, “Notice sur la vie et les oeuvres de Sophie Germain,” in *Oeuvres philosopohiques de Sophie Germain* (see above), pp. 1-92.

Edna E. Kramer

# Sophie Germain

# Sophie Germain

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.

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. □

# Sophie Germain

# Sophie Germain

**1776-1831**

**French Mathematician**

Among the achievements of Sophie Germain's career was her attempt at a proof of Fermat's Last Theorem. Her successes were all the more remarkable in light of the fact that she was largely self-taught and practiced at the fringes of a mathematical establishment largely closed to women at that time.

She was born Marie-Sophie Germain, the middle child of Ambroise-François and Marie-Madeleine Germain, in Paris on April 1, 1776. Later, she would drop the "Marie" because both her mother and her older sister also bore that name in hyphenated form.

In time, both of her sisters would marry successful men, but Sophie's own future—which held neither marriage nor children—was influenced by events that took place when she was 13 years old. That year was when the French Revolution began, and, although her father served in the revolutionary legislative assembly, the tumultuous events of 1789 took their toll on Sophie. She quite literally hid in the family library, where, for the next six years, she undertook a campaign of self-education that would have been impressive even under conditions of peace. Not only did she teach herself Latin and Greek but she also managed to learn algebra, geometry, and calculus.

Germain's formal education was slight but she submitted papers—under the pseudonym Monsieur Le Blanc (Mr. White)—for a course taught by Joseph-Louis Lagrange (1736-1813) at the Ecole Polytechnique. So great was Lagrange's respect for the work of "le Blanc" that he sought out the mystery man and, when he discovered that le Blanc was a woman, he extolled her abilities.

Lagrange was but one of many distinguished mathematicians who, to one degree or another, took Germain under their wings. She carried on a lengthy correspondence with Carl Friedrich Gauss (1777-1855) and in 1806 used her family's political connections to help protect the German mathematician from Napoleonic soldiers invading his homeland. Germain was also acquainted with Augustin-Louis Cauchy (1789-1857) and Jean-Baptiste-Joseph Fourier (1769-1830).

Her correspondence with Adrien-Marie Legendre (1752-1833) led to a collaboration of sorts, and Legendre in 1830 published Germain's proof of Fermat's Last Theorem in *Théorie des nombres.* The proof, like all attempts through the early 1990s, turned out to be incorrect, but Germain nonetheless advanced the understanding of the problem.

Between 1811 and 1815, Germain was the sole contender in three contests sponsored by the Institut de France, which challenged contestants to "formulate a mathematical theory of elastic surfaces and indicate just how it agrees with empirical evidence." Germain's first two solutions were rejected but, although the third had some questionable aspects, it nonetheless won the prize.

Germain became the first woman not married to a member of the Académie des Sciences to attend its meetings, a measure of the respect that her efforts had gained for her. She died of breast cancer in Paris on June 27, 1831.

**JUDSON KNIGHT**

# Germain, Sophie

# Germain, Sophie

*French Mathematician 1776–1831*

Sophie Germain is remembered for her work in the theory of numbers and in mathematical physics. Germain was born in Paris to a father who was a wealthy silk merchant. She educated herself by studying books in her father's library, including the works of Sir Isaac Newton and the writings of mathematician Leonhard Euler.

When the École Polytechnique opened in 1794, even though women were not allowed to attend as regular students, Germain obtained lecture notes for courses and submitted papers using the pseudonym M. LeBlanc. One of the instructors, noted scientist Joseph-Louis Lagrange, became her mentor.

In 1804 Germain began to correspond with German mathematician Carl Friedrich Gauss, sending him discoveries she made in number theory. Among these was a limited proof of Fermat's Last Theorem, her best known contribution to mathematics. This theorem was finally proved in 1994 using her approach. Germain also corresponded with mathematician Adrien Marie Legendre, who used her suggestions in one of his publications.

In mathematical physics, Germain is known for her work in acoustics and elasticity. She won a prize from the French Academy of Sciences in 1816 for the development of mathematical models for the vibration of elastic surfaces. Subsequently, she was invited to attend sessions of the Academy of Sciences and the Institut de France, but because she was a woman, she could never join either group.

see also Euler, Leonhard; Fermat's Last Theorem; Newton, Sir Isaac.

*J. William Moncrief*

## Bibliography

Gray, Mary W. "Sophie Germain (1776–1831)," in *Women of Mathematics.* Edited by Louise S. Grinstein and Paul J. Campbell. New York: Greenwood Press, 1987.

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

# Sophie Germain

# Sophie Germain

**1776-1831**

French mathematician who contributed to the studies of acoustics, elasticity, and the theory of numbers. Born in a time when women were not permitted to attend schools of higher learning, Germain advanced her education early on by reading whatever she could in her father's library. She mastered calculus in this manner, then took correspondence courses (under the pseudonym of M. Leblanc) from the Ecole Polytechnique in Paris, where women were not accepted. Eventually, she became so highly regarded by prominent mathematicians of her era that she was recommended for an honorary Ph.D from the University of Göttingen. However, she died before her degree was presented.

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