Germain, Sophie, 1776-1831

Marie-Sophie Germain (French: [maʁi sɔfi ʒɛʁmɛ̃]; 1 April 1776 – 27 June 1831) was a French mathematician, physicist, and philosopher. Despite initial opposition from her parents and difficulties presented by society, she gained education from books in her father's library, including ones by Euler, and from correspondence with famous mathematicians such as Lagrange, Legendre, and Gauss (under the pseudonym of Monsieur LeBlanc). 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 prejudice against her sex, she was unable to make a career out of mathematics, but she worked independently throughout her life.[2] Before her death, Gauss had recommended that she be awarded an honorary degree, but that never occurred.[3] On 27 June 1831, she died from breast cancer. At the centenary of her life, a street and a girls’ school were named after her. The Academy of Sciences established the Sophie Germain Prize in her honour.Marie-Sophie Germain was born in a house on Rue Saint-Denis on 1 April 1776, in Paris, France. 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. Here she found J. E. Montucla's L'Histoire des Mathématiques, and his story of the death of Archimedes intrigued her.[5]

Germain thought that if the geometry method, which at that time referred to all of pure mathematics,[5] could hold such fascination for Archimedes, it was a subject worthy of study.[8] So she pored over every book on mathematics in her father's library, 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 Différentiel by Jacques Antoine-Joseph Cousin. Later, Cousin visited Germain at home, encouraging her in her studies.[9]

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 mathematics.[10] After some time, her mother even secretly supported her.[9]

École Polytechnique

Entrance to the historic building of the École Polytechnique
In 1794, when Germain was 18, the École Polytechnique opened.[6] As a woman, Germain was barred from attending, but the new system of education made the "lecture notes available to all who asked".[9] The new method also required the students to "submit written observations".[11] Germain obtained the lecture notes and began sending her work to Joseph Louis Lagrange, a faculty member. She used the name of a former student Monsieur Antoine-Auguste Le Blanc,[9][12] "fearing", as she later explained to Gauss, "the ridicule attached to a female scientist".[13] When Lagrange saw the intelligence of M. Le Blanc, he requested a meeting, and thus Sophie was forced to disclose her true identity. Fortunately, Lagrange did not mind that Germain was a woman,[9] and he became her mentor.[6] Germain first became interested in number theory in 1798 when Adrien-Marie Legendre published Essai sur la théorie des nombres.[14] After studying the work, she opened correspondence with him on number theory, and later, elasticity. Legendre included some of Germain's work in the Supplément to his second edition of the Théorie des Nombres, where he calls it très ingénieuse ("very ingenious"). See also Her work on Fermat's Last Theorem below.[15]

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.[26] Germain's anonymous[18] 1813 submission was still littered with mathematical errors, especially involving double integrals,[27] and it received only an honorable mention because "the fundamental base of the theory [of elastic surfaces] was not established".[26] The contest was extended once more, and Germain began work on her third attempt. This time she consulted with Poisson.[18] 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).[27]

Germain submitted her third paper, "Recherches sur la théorie des surfaces élastiques",[18] under her own name, and on 8 January 1816[27] she became the first woman to win a prize from the Paris Academy of Sciences.[29] She did not appear at the ceremony to receive her award.[18] Although Germain had at last been awarded the prix extraordinaire,[20] the Academy was still not fully satisfied.[30] Germain had derived the correct differential equation (a special case of the Kirchhoff–Love equation),[31] but her method did not predict experimental results with great accuracy, as she had relied on an incorrect equation from Euler,[18] which led to incorrect boundary conditions.[31] Here is Germain's final equation for the vibration of a plane lamina:

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{\displaystyle N^{2}\left({\frac {\partial ^{4}z}{\partial x^{4}}}+2{\frac {\partial ^{4}z}{\partial x^{2}\partial y^{2}}}+{\frac {\partial ^{4}z}{\partial y^{4}}}\right)+{\frac {\partial ^{2}z}{\partial t^{2}}}=0,}
where N2 is a constant.[18][32][33]

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 situation was transformed, when she made friends with Joseph Fourier, a secretary of the Academy, who obtained tickets to the sessions for her.[28]

Later work in elasticity

Recherches sur la théorie des surfaces élastiques, 1821
Germain published her prize-winning 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 his method.[18]

In 1826 she submitted a revised version of her 1821 essay to the Academy. According to Andrea Del Centina, the revision included attempts to clarify her work by "introducing certain simplifying hypotheses". 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". So Augustin-Louis Cauchy, who had been appointed to review her work, recommended her to publish it, and she followed his advice.[34]

One further work of Germain's on elasticity was published posthumously in 1831, her "Mémoire sur la courbure des surfaces". She used the mean curvature in her research (see Honors in number theory).[18]

Later work in number theory
Renewed interest
Germain's best work was in number theory,[4] and her most significant contribution to number theory dealt with Fermat's Last Theorem.[15] In 1815, after the elasticity contest, the Academy offered a prize for a proof of Fermat's Last Theorem.[35] It reawakened Germain's interest in number theory, and she wrote to Gauss again after ten years of no correspondence.[14]

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.[35] She outlined a strategy for a general proof of Fermat's Last Theorem, including a proof for a special case.[36] Germain's letter to Gauss contained her substantial progress toward a proof. She asked Gauss whether her approach to the theorem was worth pursuing. Gauss never answered.[37]

Her work on Fermat's Last Theorem

Pierre de Fermat
Fermat's Last Theorem can be divided into two cases. Case 1 involves all powers 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":[38]

Let p be an odd prime. If there exists an auxiliary prime P = 2Np + 1 (N is any positive integer not divisible by 3) such that:

if xp + yp + zp ≡ 0 (mod P), then P divides xyz, and
p is not a p-th power residue (mod P).
Then the first case of Fermat's Last Theorem holds true for p.[39]

Germain used this result to prove the first case of 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".[39] L. E. Dickson later used Germain's theorem to prove the first case of Fermat's Last Theorem for all odd primes less than 1700.[40]

In an unpublished manuscript titled Remarque sur l'impossibilité de satisfaire en nombres entiers a l'équation xp + yp = zp,[38] Germain showed that any counterexamples to Fermat's theorem for p > 5 must be numbers "whose size frightens the imagination",[41] around 40 digits long.[42] Germain did not publish this work. Her 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).[41] Germain also proved or nearly proved several results that were attributed to Lagrange or were rediscovered years later.[1] Del Centina states that "after almost two hundred years her ideas were still central",[1] but ultimately her method did not work.[41]

Work in philosophy
In addition to mathematics, Germain studied philosophy and psychology.[9] 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.[43]

Two of her philosophical works, Pensées diverses and Considérations générales sur l'état des sciences et des lettres, aux différentes époques de leur culture, were published, both posthumously. This was due in part to the efforts of Lherbette, her nephew, who collected her philosophical writings and published them.[44] Pensées is a history of science and mathematics with Germain's commentary.[45] In Considérations, the work admired by Comte, Germain argues that there are no differences between the sciences and the humanities.[46]

Final years
In 1829 Germain learned that she had breast cancer. Despite the pain,[47] 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
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{\displaystyle {\tfrac {4(x^{p}-1)}{x-1}}=y^{2}\pm pz^{2}}".[18] Mary 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."[18] On 27 June 1831, she died in the house at 13 rue de Savoie.[25]

Despite Germain's intellectual achievements, her death certificate lists her as a "rentière – annuitant"[48] (property holder),[49] not a "mathématicienne".[48] But her work was not unappreciated by everyone. When the matter of honorary degrees came up at the University of Göttingen in 1837—six years after Germain's death—Gauss lamented: "she [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".[50]