MarieSophie Germain  

MarieSophie Germain 

Born  April 1, 1776 Paris, France 
Died  June 27, 1831 (aged 55) 
Nationality  French 
Other names  Auguste Antoine Le Blanc 
Occupation  mathematician 
Known for  elasticity theory, differential geometry, and number theory 
MarieSophie Germain (April 1st 1776 – June 27, 1831) was a French mathematician, physicist, and philosopher. Despite initial opposition from her parents and difficulties presented by a genderbiased society, she gained an education from books in her father's library and from correspondence with famous mathematicians such as Lagrange, Legendre, and Gauss. One of the pioneers of elasticity theory, she won the grand prize from the Paris Academy of Sciences for her essay on the subject. Her work on Fermat's Last Theorem provided a foundation for mathematicians exploring the subject for hundreds of years after.^{[1]} Because of her gender, she was unable to make a career out of mathematics, but worked independently throughout her life.^{[2]}
Contents 
MarieSophie Germain was born on April 1, 1776, in Paris, France, in a house on rue St. Denis. According to most sources, her father, AmbroiseFranҫois, was a wealthy silk merchant;^{[3]}^{[4]}^{[5]} however, Mary Gray, chair of the Department of Mathematics and Statistics at American University in Washington, D.C., says that he was a goldsmith.^{[6]} In 1789, he was elected as a representative of the bourgeoisie to the ÉtatsGénéraux, which he saw change into the Constitutional Assembly. It is therefore assumed that Sophie witnessed many discussions between her father and his friends on politics and philosophy.^{[7]} Gray proposes that after his political career, AmbroiseFranҫois became the director of a bank; at least, the family remained welloff enough to support Germain throughout her adult life.^{[8]}
MarieSophie had one younger sister, named AngéliqueAmbroise, and one older sister, named MarieMadeline. Her mother was also named MarieMadeline, and this plethora of Maries may have been the reason she went by Sophie. Germain's nephew ArmandJacques Lherbette, MarieMadeline's son, published some of Germain's work after she died (see Work in Philosophy).^{[9]}
When Germain was 13, the Bastille fell, and the revolutionary atmosphere of the city forced her to stay inside. For entertainment she turned to her father's library.^{[10]} Here she found J. E. Montucla's L'Histoire des Mathématiques, and his story of the death of Archimedes intrigued her.^{[11]}
Germain decided that if geometry, which at that time referred to all of pure mathematics,^{[12]} could hold such fascination for Archimedes, it was a subject worthy of study.^{[13]} So she pored over every math book in her father's library,^{[14]} even teaching herself Latin and Greek so she could read works like those of Sir Isaac Newton and Leonhard Euler. She also enjoyed Traité d'Arithmétique by Étienne Bézout and Le Calcul Différential by Jacques AntoineJoseph Cousin. Later, Cousin visited her in her house, encouraging her in her studies.^{[15]}
Germain's parents did not at all approve of her sudden fascination with mathematics, which was then thought inappropriate for a woman. When night came, they would deny her warm clothes and a fire for her bedroom to try to keep her from studying, but after they left she would take out candles, wrap herself in quilts and do math.^{[16]} As UC Irvine's Women's Studies professor Lynn Osen describes, when her parents found Sophie “asleep at her desk in the morning, the ink frozen in the ink horn and her slate covered with calculations,” they realized that their daughter was serious and relented.^{[17]} After some time, her mother even secretly supported her.^{[18]}
In 1794, when Germain was 18, the École Polytechnique opened.^{[19]} As a woman, Germain was barred from attending, but the new system of education made the “lecture notes available to all who asked."^{[20]} The new method also required the students to “submit written observations."^{[21]} Germain obtained the lecture notes and began sending her work to Joseph Louis Lagrange, a faculty member. She used the name M. LeBlanc ^{[22]}, “fearing,” as she later explained to Gauss, “the ridicule attached to a female scientist."^{[23]} When Lagrange saw the intelligence of M. LeBlanc, he requested a meeting, and thus Sophie was forced to disclose her true identity. Fortunately, Lagrange did not mind that Germain was a woman,^{[24]} and he became her mentor.^{[25]} He too visited her in her home, giving her moral support.^{[26]}
Germain first became interested in number theory in 1798 when AdrienMarie Legendre published Essai sur la théorie des nombres.^{[27]} After studying the work, she opened correspondence with him on number theory, and later, elasticity. Legendre showed some of Germain's work in the Supplément to his second edition of the Théorie des Nombres, where he calls it trés ingénieuse (See Best Work on Fermat's Last Theorem).^{[28]}
Germain's interest in number theory was renewed when she read Carl Friedrich Gauss' monumental work Disquisitiones Arithmeticae.^{[29]} After three years of working through the exercises and trying her own proofs for some of the theorems,^{[30]} she wrote, again under the pseudonym of M. LeBlanc,^{[31]} to the author himself, who was one year younger than she.^{[32]} The first letter, dated 21 November 1804,^{[33]} discussed Gauss' Disquisitiones and presented some of Germain's work on Fermat's Last Theorem. In the letter, Germain claimed to have proved the theorem for n = p – 1, where p is a prime of the form p = 8k + 7;^{[34]} however, her proof contained a weak assumption.^{[35]} Gauss' reply did not comment on Germain's proof.^{[36]}
Around 1807 (sources differ^{[37]}^{[38]}), the French were occupying the German town of Braunschweig, where Gauss lived. Germain, concerned that he might suffer the fate of Archimedes, wrote to General Pernety, a family friend, requesting that he ensure Gauss' safety.^{[39]} General Pernety sent a chief of a battalion to meet with Gauss personally to see that he was safe.^{[40]} As it turned out, Gauss was fine,^{[41]} but he was confused by the mention of Sophie's name.^{[42]}
Three months after the incident, Germain disclosed her true identity to Gauss.^{[43]} He replied,
How can I describe my astonishment and admiration on seeing my esteemed correspondent M leBlanc metamorphosed into this celebrated person. . . when a woman, because of her sex, our customs and prejudices, encounters infinitely more obstacles than men in familiarising [sic] herself with [number theory's] 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.^{[44]}
Gauss' letters to Olbers show that his praise for Germain was sincere.^{[45]}^{[46]} In the same 1807 letter, Sophie claimed that if x^{n} + y^{n} is of the form h^{2} + nf^{2}, then x + y is also of that form. Gauss replied with a counterexample: 15^{11} + 8^{11} can be written as h^{2} + 11f^{2}, but 15 + 8 cannot.^{[47]}
Although Gauss thought well of Germain, his replies to her letters were often delayed, and he generally did not review her work.^{[48]} Eventually his interests turned away from number theory,^{[49]} and in 1809 the letters ceased.^{[50]} Despite the friendship of Germain and Gauss, they never met.^{[51]}
When Germain's correspondence with Gauss ceased, she took interest in a contest^{[52]} sponsored by the Paris Academy of Sciences concerning Ernst Chladni's experiments with vibrating metal plates.^{[53]} The object of the competition, as stated by the Academy, was “to give the mathematical theory of the vibration of an elastic surface and to compare the theory to experimental evidence."^{[54]} Lagrange's comment that a solution to the problem would require the invention of a new branch of analysis deterred all but two contestants, Denis Poisson and Germain.^{[55]} Then Poisson was elected to the Academy, thus becoming a judge instead of a contestant,^{[56]} and leaving Germain as the only entrant to the competition.^{[57]}
In 1809 Germain began work. Legendre assisted by giving her equations, references, and current research.^{[58]} She submitted her paper early in the fall of 1811, and did not win the prize. The judging commission felt that “the true equations of the movement were not established,” even though “the experiments presented ingenious results.”^{[59]} Lagrange was able to use Germain's work to derive an equation that was “correct under special assumptions.”^{[60]}
The contest was extended by two years, and Germain decided to try again for the prize. At first Legendre continued to offer support, but then he refused all help.^{[61]} Germain's anonymous^{[62]} 1813 submission was still littered with mathematical errors, especially involving double integrals,^{[63]} and it received only an honorable mention because “the fundamental base of the theory [of elastic surfaces] was not established."^{[64]} The contest was extended once more, and Germain began work on her third attempt. This time she consulted with Poisson.^{[65]} In 1814 he published his own work on elasticity, and did not acknowledge Germain's help (although he had worked with her on the subject and, as a judge on the Academy commission, had had access to her work).^{[66]}
Germain submitted her third paper, “Recherches sur la théorie des surfaces élastique”^{[67]} under her own name, and on 8 January 1816^{[68]} she became the first woman to win a prize from the Paris Academy of Sciences.^{[69]} She did not appear at the ceremony to receive her award.^{[70]} Although Germain had at last been awarded the prix extraordinaire,^{[71]} the Academy was still not fully satisfied.^{[72]} Sophie had derived the correct differential equation,^{[73]} but her method did not predict experimental results with great accuracy, as she had relied on an incorrect equation from Euler,^{[74]} which led to incorrect boundary conditions.^{[75]} Here is Germain's final equation:
where N^{2} is a constant.^{[76]}
After winning the Academy contest, she was still not able to attend its sessions because of the Academy's tradition of excluding women other than the wives of members. Seven years later this tradition was broken when she made friends with Joseph Fourier, a secretary of the Academy, and he got her tickets to the sessions.^{[77]}
Germain published her prizewinning essay at her own expense in 1821, mostly because she wanted to present her work in opposition to that of Poisson. In the essay she pointed out some of the errors in her method.^{[78]}
In 1826 she submitted a revised version of her 1821 essay to the Academy. According to Andrea del Centina, a math professor at the University of Ferrara in Italy, the revision included attempts to clarify her work by “introducing certain simplifying hypotheses."^{[79]} This put the Academy in an awkward position, as they felt the paper to be “inadequate and trivial,” but they did not want to “treat her as a professional colleague, as they would any man, by simply rejecting the work.”^{[80]} So AugustinLouis Cauchy, who had been appointed to review her work, recommended she publish it, and she followed his advice.^{[81]}
One further work of Germain's on elasticity was published posthumously in 1831: her “Memoir sur la courbure des surfaces.” She used the mean curvature in her research (see Honors in Number Theory).^{[82]}
Germain's best work was in number theory,^{[83]} and her most significant contribution to number theory dealt with Fermat's Last Theorem.^{[84]} In 1815, after the elasticity contest, the Academy offered a prize for a proof of Fermat's Last Theorem.^{[85]} It reawakened Germain's interest in number theory, and she wrote to Gauss again after ten years of no correspondence.^{[86]}
In the letter, Germain said that number theory was her preferred field, and that it was in her mind all the time she was studying elasticity.^{[87]} She outlined a strategy for a general proof of Fermat's Last Theorem, including a proof for a special case (see Best Work on Fermat's Last Theorem).^{[88]} Germain's letter to Gauss contained the first substantial progress toward a proof in 200 years.^{[89]} She asked Gauss if her approach to the theorem was worth pursuing. Gauss never answered.^{[90]}
Fermat's Last Theorem is commonly divided into two cases. Case 1 involves all p that do not divide any of x, y, or z. Case 2 includes all p that divide at least one of x, y, or z. Germain proposed the following, commonly called “Sophie Germain's Theorem”:^{[91]}
Let p be an odd prime. If there exists an auxiliary prime P = 2Np + 1such that:
Then the first case of [Fermat's Last Theorem] holds true for p.^{[92]}
 if x^{p} + y^{p} + z^{p} = 0 (mod P) then p divides xyz, and
 p is not a p^{th} power residue (mod P).
Germain used this result to prove Fermat's Last Theorem for all odd primes p<100, but according to Andrea del Centina, “she had actually shown that it holds for every exponent p<197.”^{[93]} L. E. Dickson later used Germain's theorem to prove Fermat's Last Theorem for odd primes less than 1700.^{[94]}
In an unpublished manuscript entitled Remarque sur l’impossibilité de satisfaire en nombres entiers a l’équation x^{p} + y^{p} = z^{p},^{[95]} Germain showed that any counterexamples to Fermat's theorem for p>5 must be numbers “whose size frightens the imagination,”^{[96]} around 40 digits long.^{[97]} Sophie did not publish this work. Her brilliant theorem is known only because of the footnote in Legendre's treatise on number theory, where he used it to prove Fermat's Last Theorem for p = 5 (see Correspondence with Legendre).^{[98]} Germain also proved or nearly proved several results that were attributed to Lagrange or were rediscovered years later.^{[99]} Del Centina states that “after almost two hundred years her ideas were still central”^{[100]}, but ultimately her method did not work.^{[101]}
In addition to math, Germain studied philosophy and psychology.^{[102]} She wanted to classify facts and generalize them into laws that could form a system of psychology and sociology, which were then just coming into existence. Her philosophy was highly praised by Auguste Comte.^{[103]}
Two of her philosophical works, Pensées diverses and Considérations générales sur l'état des sciences et des letteres aux différentes epoques de leur culture,^{[104]} were published, both posthumously. This was due in part to the efforts of Lherbette, her nephew, who collected her philosophical writings and published them.^{[105]} Pensées is a history of science and mathematics with Sophie's commentary.^{[106]} In Considérations, the work admired by Comte, Sophie argues that there are no differences between the sciences and the humanities.^{[107]}
In 1829 Germain learned she had breast cancer. Despite the pain,^{[108]} she continued to work. In 1831 Crelle's Journal published her paper on the curvature of elastic surfaces and “a note about finding y and z in ."^{[109]} And American University's Gray records, “She also published in Annales de chimie et de physique an examination of principles which led to the discovery of the laws of equilibrium and movement of elastic solids."^{[110]} On June 27 of 1831, she died in the house at 13 rue de Savoie.^{[111]}
Despite Germain's intellectual achievements, her death certificate lists her as a “rentière – annuitant”^{[112]} (property holder^{[113]}), not a “mathematicienne."^{[114]} But her work was not unappreciated by everyone. When the matter of honorary degrees came up at the University of Göttingen six years after Germain's death, Gauss lamented, “[Germain] proved to the world that even a woman can accomplish something worthwhile in the most rigorous and abstract of the sciences and for that reason would well have deserved an honorary degree."^{[115]}
Germain's resting place in the Père Lachaise Cemetery in Paris is marked by a crumbling gravestone.^{[116]}^{[117]} At the centennial celebration of her life, a street and a girl's school were named after her, and a plaque was placed at the house where she died. The school houses a bust commissioned by the Paris City Council.^{[118]}
E. Dubouis defined a sophein of a prime n to be a prime θ where θ = kn + 1, for such n that yield θ such that x^{n} = y^{n} + 1 (mod θ) has no solutions when x and y are prime to n.^{[119]}
A Sophie Germain prime is a prime p such that 2p + 1 is also prime.^{[120]}
The Germain curvature (also called mean curvature) is ,^{[121]} when k_{1} and k_{2} are the maximum and minimum values of the normal curvature.^{[122]}
Sophie Germain's Identity states that for any {x,y}, then,
Vesna Petrovich, a graduate of the University of Michigan, found that the educated world's response to the 1821 publication of Germain's prizewinning essay “ranged from polite to indifferent."^{[123]} Yet some critics had high praise for it. Of her 1821 essay, Cauchy said, “[it] was a work for which the name of its author and the importance of the subject both deserved the attention of mathematicians."^{[124]} H. J. Mozans, whose biography of Germain “is inaccurate and the notes and bibliography are unreliable,"^{[125]} but is nonetheless interesting, quotes the mathematician ClaudeLouis Navier as saying, “it is a work which few men are able to read and which only one woman was able to write."^{[126]}
Her contemporaries also had good things to say relating to her work in mathematics. Osen relates that “Baron de Prony called her the Hypatia of the nineteenth century,” and “J.J Biot wrote, in the Journal de Savants, that she had probably penetrated the science of mathematics more deeply than any other of her sex."^{[127]} Gauss certainly thought highly of her, and he recognized that European culture presented special difficulties to a woman in mathematics (see Correspondence with Gauss).
The modern view generally acknowledges that although Sophie had great talent as a mathematician, her haphazard education had left her without the strong base she needed to truly excel. As explained by Gray, “Germain's work in elasticity suffered generally from an absence of rigor, which might be attributed to her lack of formal training in the rudiments of analysis."^{[128]} Petrovich adds, “This proved to be a major handicap when she could no longer be regarded as a young prodigy to be admired but was judged by her peer mathematicians.”^{[129]}
Notwithstanding the problems with Sophie's theory of vibrations, Gray states that “Germain's work was fundamental in the development of a general theory of elasticity.”^{[130]} H. J. Mozans writes, however, that when the Eiffel tower was built and the architects inscribed the names of 72 scientists whose fundamental work in elasticity theory had made construction of the tower possible, Germain's name was not among them. “Was she excluded from this list... because she was a woman? It would seem so."^{[131]}
Concerning her early work in number theory, J. H. Sampson, author of “Sophie Germain and the Theory of Numbers,” states, “She was clever with formal algebraic manipulations; but there is little evidence that she really understood the Disquisitiones, and her work of that period that has come down to us seems to touch only on rather superficial matters."^{[132]} Gray adds that “The inclination of sympathetic mathematicians to praise her work rather than to provide substantive criticism from which she might learn was crippling to her mathematical development."^{[133]} Yet Marilyn Bailey Ogilvie, Curator of the History of Science Collections and Professor of the History of Science at the University of Oklahoma recognizes that “Sophie Germain's creativity manifested itself in pure and applied mathematics...[she] provided imaginative and provocative solutions to several important problems,"^{[134]} and, as Petrovich proposes, it may have been her very lack of training that gave her her unique insights and approaches.^{[135]} Louis Bucciarelli and Nancy Dworsky, Germain's biographers, summarize as follows: “All the evidence argues that Sophie Germain had a mathematical brilliance that never reached fruition due to a lack of rigorous training available only to men."^{[136]}
Germain is referenced and quoted in David Auburn's 2001 play Proof. The protagonist is a young struggling female mathematician, who found great inspiration in the work of Germain.
Germain is also mentioned in John Madden's 2005 movie "Proof" in a conversation between Gwyneth Paltrow and Jake Gyllenhaal's characters.
In the fictional work "The Last Theorem" by Arthur C. Clarke and Frederik Pohl, Sophie Germain is credited with inspiring Ranjit Subramanian to solve Fermat's Last Theorem.
MarieSophie Germain (April 1, 1776 – June 27, 1831) was a French mathematician who made important contributions to differential geometry and number theory.
The Sophie Germain prime and Crater Germain on Venus are named after her.
