Germain, Sophie (1776–1831)

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Germain, Sophie (1776–1831)

French mathematician and winner of the French Academy of Sciences' gold medal. Name variations: Marie Germain. Born Marie-Sophie Germain on April 1, 1776, on rue St. Denis, Paris, France; died on June 27, 1831, rue de Savoie, Paris, France; buried in Père Lachaise cemetery, Paris; daughter of Marie-Madeleine (Gruguelin) Germain and Ambroise François Germain; self-taught; never married; no children.

Began study of mathematics (1793–94); École central des travaux opened in Paris (1794); began correspondence with J.L. Lagrange under the pen name M. le Blanc (1794); began correspondence with Carl Friedrich Gauss under pen name M. le Blanc (1804); concerned about Gauss' safety after the French Invasion of Prussia (1807); had her true identity revealed to Gauss (1807); submitted essay to the French Academy of Sciences on vibrating elastic surfaces (1811), essay rejected; submitted second essay and received honorable mention (1813); submitted third essay and awarded grand prize of the French Academy of Sciences (1816); permitted to attend public sessions of the French Academy of Sciences (1822); diagnosed with breast cancer (1829).

Selected publications:

Recherches sur la théorie des surface élastique (Paris: Huzard-Courcier, 1821); Recherches sur la nature, les bornes et l'étendue de la question des surface élastique (Paris: Huzard-Courcier, 1826); Considération generales sur l'État des sciences et des lettres aux différentes époque de leur culture (Armand-Jacques Lherbette, ed. Paris: Huzard-Courcier, 1833); Oeuvres philosophiques de Sophie Germain (H. Stupuy, ed. Paris: Paul Ritti, 1879).

Sophie Germain was born in Paris on April 1, 1776, at a time when French science flourished, as did the growth of educational institutions which catered to it, though women were excluded from admission. Even so, Sophie Germain has been called one of the founders of mathematical physics.

We know little about Germain's mother save her name, Marie-Madeleine Gruguelin . Sophie's father was Ambroise Germain, a prosperous silk merchant, who was elected to the Estates General in 1789 and later became a director of the Bank of France. The Germain home was a frequent meeting place for those interested in liberal reform. Sophie grew up in the final turbulent years of the Ancient Regime and was 13 years old when the Bastille fell. Fortunately, the Germain family was wealthy enough to protect itself from the worst excesses of revolutionary violence.

Sophie spent many solitary hours in the family library, where she read Isaac Newton, Leonhard Euler, and Étienne Bérout's Traité d'Arthmétique. The most influential author of her early education, however, was J.E. Montucla, who describes the death of Archimedes in his book History of Mathematics. During an invasion of his city by the Roman army, Archimedes was so engrossed in the study of a geometrical figure which he had drawn in the sand that, when questioned by a passing Roman soldier, he failed to respond. As a result, he was speared to death. Thus for Germain, mathematics came to represent a repository of truth and knowledge, and Archimedes, the great scientist of antiquity, became her role model.

The choice of mathematics as an area of study was an unpopular one with her family. Although France had a tradition of female intellectuals, there was a common reaction against such pursuits, as typified by Molière's satire La Femme Savante. Fearing that rigorous study would imperil their daughter's health, the Germains fought a desperate rear-guard action to prevent her from undertaking strenuous research. They confiscated the candles in her bedroom, denied her heat, and took away her clothing at night. Not to be outwitted, Germain waited until her parents were asleep before wrapping herself in a quilt and treading her way carefully down to the library by the light of contraband candles. After discovering their daughter asleep one morning, with the ink frozen in the ink well and papers scattered about her covered with calculations, the Germains relented. Thus, during the Reign of Terror (1793–94), Sophie Germain spent the period teaching herself differential calculus.

In 1794, the École central des travaux, later known as the École Polytechnique, was established in Paris for the training of mathematicians and scientists. Women were not admitted. Nevertheless, Germain managed to obtain the lecture notes of various professors. One who sparked her interest was J.L. Lagrange, one of the outstanding mathematicians of the 18th century. Using the new practice of allowing students to hand in written observations at the end of the course, Germain set forth some of her theories under the pen name of M. le Blanc. La grange was suitably impressed. After learning the true identity of his corre spondent, he sought Germain out, praised her ideas, became a lifelong supporter, and introduced her to many of France's leading scientists. Despite this, notes Margaret

Alic , "she was viewed by her contemporaries as a phenomenon, not as a serious student in need of teaching and guidance."

Immanuel Kant">
All abstract knowledge, all knowledge which is dry, it is cautioned, must be abandoned to the laborious and solid mind of man. For this reason, women will never learn geometry.
—Immanuel Kant

Then Carl Friedrich Gauss published his masterpiece Disquisitiones Arithmeticae in 1801. Germain was so taken with his theories of cyclotomy and arithmetical forms that in 1804 she sent him a copy of her own calculations, again employing the pen name M. le Blanc. The pair began a lively correspondence. Germain was particularly anxious to learn of Gauss' reaction to Adrien Marie Legendre's Essai sur le Théorie des Nombres. When Napoleon's forces invaded Prussia and besieged Breslau, near Gauss' home, Germain apparently feared that Gauss might suffer a fate similar to that of Archimedes. She interceded on his behalf with General Pernety, a family friend. But Gauss denied knowledge of anyone named Sophie Germain when the general sent a messenger to enquire after Gauss' safety. The misunderstanding was cleared up when Germain admitted that she was not M. le Blanc. Replied Gauss:

The tastes for the abstract sciences in general and above all, for the mysteries of numbers, is very rare: this is not surprising, since the charms of this sublime science in all their beauty reveal themselves only to those who have the courage to fathom them. But when a woman, because of her sex, our customs and prejudices, encounters infinitely more obstacles than men in familiarizing herself with their knotty problems, yet overcomes these fetters and penetrates that which is most hidden, she doubtless has the most noble courage, extraordinary talent, and superior genius.

Gauss' admiration was not reserved for Sophie's ears alone. He also sang her praises to his colleagues, as a letter written to H.W.M. Oblers on July 21, 1807, demonstrates:

Lagrange is warmly interested in astronomy and the higher arithmetic; the two test-theorems (for which the prime 2 is a cubic or a biquadratic residue), which I also communicated to him some time ago, he considers "among the most difficult to prove." But Sophie Germain has sent me the proofs of these; I have not yet been able to go through them, but I believe they are good; at least she has attacked the matter from the right side, only somewhat more diffusely than would be necessary.

Germain's early research focused exclusively on number theory. However, by the turn of the century, the interests of French mathematicians were turning increasingly to the work of Ernst Chladni, a German physicist living and working in Paris. Chladni's research involved the vibration of elastic surfaces. By sprinkling sand on a metal sheet and striking the edge with a violin bow, he conducted experiments on elastic surfaces. The pattern formed by the sand's nodal lines was then recorded on a piece of paper.

Interest in the vibrations of elastic materials dated back to the Pythagoreans, but no mathematical theory existed to explain the phenomenon. Napoleon was greatly impressed by Chladni's research and ordered the French Academy of Sciences to hold an essay competition on the subject. The question to be answered was: formulate a mathematical theory of elastic surfaces and indicate just how it agrees with empirical evidence. French mathematicians refused to enter the contest, following assurances from Lagrange that contemporary mathematical methods were unequal to the task.

Sophie Germain, however, was intrigued by the challenge. She completed her entry in eight months and submitted the results anonymously on September 21, 1811. She was the sole entrant. Lagrange, who was a member of the selection committee, thought that her method of passing from a line to the surface was neither accurate nor complete. Thus, her entry was rejected. The Academy informed her that her mathematical equation was incorrect, although her hypothesis was plausible.

The deadline for entries was extended until October 1813. For the next year and a half Germain worked on her second essay. Adrien Legendre, who also sat on the selection committee, wrote to her on December 4, 1813, having just read the results:

Mademoiselle,
I do not understand the analysis you send me at all; there is certainly an error in the writing or the reasoning, and I am led to believe that you do not have a very clear idea of the operations on double integrals in the calculus of variations. Your explanation of the four points does not satisfy me any more.… There is a great lack of clarity in all this. I will not try to point out to you all the difficulties in a matter that I have not especially studied and that does not attract me; therefore it is useless to offer to meet with you and discuss them.… In any case, there is the possibility of having your research published, reestablishing the correct analysis or down playing it.

In 1813, Germain's essay received an honorable mention from the French Academy of Sciences. Again the deadline for entries was extended. Germain, however, was concerned about the impartiality of some of the judges sitting on the selection committee. In a letter to an unknown correspondent she wrote:

I enjoin your probation of memoir No. 1 carrying this epigram: But by far the greatest obstacle to the progress of science and to the undertaking of new tasks and provinces therein is found in this: that men despair and think things impossible.
If I had found the occasion, I would have consulted you before adopting this quotation, since it has an air of pretentiousness, which hardly suits me, having so many reasons to mistrust my own skills and, indeed, not seeing any strong objection to my theory other than the improbability of having it meet with justice. I fear, however, the influence of opinion that M. Lagrange expressed. Without doubt, the problem has been abandoned only because this grand geometer judged it difficult. Possibly this same prejudgment will mean a condemnation of my work without a reflective examination.

Three years later, in 1816, Sophie Germain's third essay was awarded the grand prize. She exposed the laws of vibrating elastic surfaces by describing a fourth-order partial differential equation which explained what was happening in Chladni's research.

For her efforts, she was awarded a one kilogram gold medal, worth 3,000 francs. To the disappointment of the public, however, Germain refused to participate in the awards ceremony, which was held on January 8, 1816. One can only speculate as to the reason. It should be noted, however, that Germain was not allowed to attend public sessions of the French Academy of Sciences until 1822, when Joseph Fourier was elected permanent secretary.

Winning the grand prize was the highlight of Sophie Germain's career. As a result, she was welcomed into mathematical circles and met Augustin L. Cauchy, André Marie Ampère, M.H. Navier and Siméon D. Poisson. A published version of Germain's work appeared in 1821. It was celebrated by Navier, who was so impressed with Sophie Germain's powers of analysis that he wrote of her research, "It is a work which few men are able to read and which only a woman was able to write." The Baron de Prony, a member of the French Academy, referred to Germain as the Hypatia of the 19th century.

Germain continued to pursue research on elasticity and published several other works on the topic. The most significant of these included an essay on the qualities and binding abilities of elastic surfaces and another work which explored the mysteries of the curvature of elastic surfaces. Her best work, however, was in the field of number theory. She proved Pierre de Fermat's Last Theorem. In 1909, the American algebraist Leonard E. Dickson generalized her theorem to primes less than 1,700, and, more recently, Barkley Roser extended the upper range to 41,000,000.

The interests of Sophie Germain were varied—from chemistry to physics, from geography to history. Like her predecessors, such as Maria Gaetana Agnesi and Émilie du Châtelet , Germain was interested in philosophy and hoped to apply scientific methods to the study of psychology. The thrust of her musings entailed the thesis that human behavior could be analyzed and predicted with the mathematical precision which contemporary scientists used to define the natural world. This philosophical stance, which emphasized the holistic nature of the physical and moral order of the world, anticipated many of the features of Auguste Comte's positivistic approach.

Despite an extensive correspondence, Gauss and Germain never met, but Gauss demonstrated his enduring respect by recommending that the University of Göttingen award her an honorary doctorate. Before she could travel to Göttingen, however, Sophie Germain died of breast cancer in Paris on June 27, 1831, at the age of 55. Ironically, Germain's death certificate listed her profession, not as a mathematician or a scientist, but merely as a landlord.

Guillaume Libri wrote in her obituary that she "carried throughout everything" a "forgetfulness of self; in the science which she cultivated with entire self denial, without dreaming of the advantages that success would procure, applauding even, on occasion, the sight of her ideas fertilized by others who had seized them; saying it was not important where an idea came from but only how far it could go, and [she was happy], as long as her ideas bore their fruit for science without furthering the reputation, which she disdained; and [she] proclaimed ludicrous the glory of the bourgeois, [calling fame] the small place which we occupy in the minds of others." Now, a commemorative plaque appears outside her house at 13 rue de Savoie in Paris.

Sophie Germain is principally remembered for her work on elasticity, although her confirmation of Fermat's Last Theorem was by far her best work. In either case, her research has stood the test of time. As Alic points out, however, "Germain's … work was all being done within a community that excluded her so completely that she did not even realise what was happening. It was her sex, not her mathematical ability, that was the determining factor."

Unlike other women who had husbands, friends, and even lovers, through whom access to the scientific community could be gained, Sophie Germain was bereft of such support. Thus, much of her work suffered from the professional isolation in which it was conducted. As Louis Bucciarelli and Nancy Dworsky noted:

Every conversation was a formal social event requiring lettres of invitation, planning for transportation, requests for permission. Sophie Germain could not stop to chat with friends at meetings of the Institute nor get into a serious conversation over cigars and brandy after dinner.

Germain was a woman outside the male scientific community. Self-educated, she never benefited from the educational opportunities which France offered. Although her work furthered our understanding of elasticity, when the Eiffel Tower was built her name was not included at the base of the structure along with those scientists whose research had made the construction of this Paris landmark possible. As well, Sophie Germain was denied membership to the French Academy of Sciences. She shares this sad distinction with such eminent scientists as Maria Agnesi, Sophia Kovalevskaya , and Nobel Prize-winner, Marie Curie .