Sophie Germain

Template:Short description Template:About Template:Use dmy dates Template:Infobox scientist Marie-Sophie Germain (Template:IPA; 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 Le Blanc). 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.Template:Sfn Because of prejudice against her sex, she was unable to make a career out of mathematics, but she worked independently throughout her life.Template:Sfn Before her death, Gauss had recommended that she be awarded an honorary degree, but that never occurred.<ref>Mackinnon, Nick (1990). "Sophie Germain, or, Was Gauss a feminist?". The Mathematical Gazette 74 (470): 346–351, esp. p. 347.</ref> 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.

Early life

Family

Marie-Sophie Germain was born in a house on Rue Saint-Denis on 1 April 1776, in Paris, France, and baptized the same day.Template:Sfn According to most sources, her father, Ambroise-François, was a wealthy silk merchant,Template:Sfn Template:Sfn Template:Sfn though some believe he was a goldsmith.Template:Sfn In 1789, he was elected as a representative of the bourgeoisie to the États-Généraux, which he saw change into the National Assembly. It is therefore assumed that Sophie witnessed many discussions between her father and his friends on politics and philosophy. Gray proposes that after his political career, Ambroise-François became the director of a bank; in any case, the family remained well-off enough to support Germain throughout her adult life.Template:Sfn

Marie-Sophie had one younger sister, Angélique-Ambroise, and one older sister, Marie-Madeline. Her mother was also named Marie-Madeline, and this plethora of "Maries" may have been the reason she went by Sophie. Germain's nephew Armand-Jacques Lherbette, Marie-Madeline's son, published some of Germain's work after she died (see Work in Philosophy).Template:Sfn

Introduction to mathematics

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.Template:Sfn

Germain thought that if the geometry method, which at that time referred to all of pure mathematics,Template:Sfn could hold such fascination for Archimedes, it was a subject worthy of study.Template:Sfn 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 Template:Lang by Étienne Bézout and Template:Lang by Jacques Antoine-Joseph Cousin. Later, Cousin visited Germain at home, encouraging her in her studies.Template:Sfn

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.Template:Sfn After some time, her mother even secretly supported her.Template:Sfn

École Polytechnique

File:Porton antigua sede École Polytechnique Sainte Geneviève.JPG

Entrance to the historic building of the École Polytechnique

In 1794, when Germain was 18, the École Polytechnique opened.Template:Sfn As a woman, Germain was barred from attending, but the new system of education made the "lecture notes available to all who asked".Template:Sfn The new method also required the students to "submit written observations".Template:Sfn 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,Template:Sfn<ref>Template:Cite web</ref> "fearing", as she later explained to Gauss, "the ridicule attached to a female scientist".Template:Sfn 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,Template:Sfn and he became her mentor.Template:Sfn

Early work in number theory

Correspondence with Legendre

Germain first became interested in number theory in 1798 when Adrien-Marie Legendre published Template:Lang.Template:Sfn After studying the work, she opened correspondence with him on number theory, and later, elasticity. Legendre included some of Germain's work in the Template:Lang to his second edition of the Template:Lang, where he calls it Template:Lang ("very ingenious"). See also Her work on Fermat's Last Theorem below.Template:Sfn

Correspondence with Gauss

Germain's interest in number theory was renewed when she read Carl Friedrich Gauss's monumental work Template:Lang.Template:Sfn After three years of working through the exercises and trying her own proofs for some of the theorems,Template:Sfn she wrote, again under the pseudonym of M. Le Blanc,Template:Sfn to the author himself, who was one year younger than she.Template:Sfn Gauss's replies were mailed to the home of Antoine-Isaac, Baron Silvestre De Sacy, who must have understood Germain's reasons for assuming a masculine pseudonym and agreed to help her conceal her identity.Template:Sfn

The first letter, dated 21 November 1804,Template:Sfn discussed Gauss's Template:Lang 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 number of the form p = 8k + 7.Template:Sfn However, her proof contained a weak assumption, and Gauss's reply did not comment on Germain's proof.Template:Sfn

Around 1807 (sources differ),Template:Sfn during the Napoleonic wars, 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 (Joseph Marie de Pernety), a family friend, requesting that he ensure Gauss's safety. Template:Sfn General Pernety sent the chief of a battalion to meet with Gauss personally to see that he was safe.Template:Sfn As it turned out, Gauss was fine, but he was confused by the mention of Sophie's name.Template:Sfn

Three months after the incident, Germain disclosed her true identity to Gauss.Template:Sfn He replied:Template:Sfn

Template:Blockquote

File:Carl Friedrich Gauss.jpg

1840 painting of Carl Friedrich Gauss; by Gottlieb Biermann after Christian Albrecht Jensen

Gauss's letters to Olbers show that his praise for Germain was sincere.Template:Sfn Template:Sfn In the same 1807 letter, Germain claimed that if <math>x^n + y^n</math> is of the form <math>h^2 + nf^2</math>, then <math>x + y</math> is also of that form. Gauss replied with a counterexample: <math>15^{11} + 8^{11}</math> can be written as <math>h^2 + 11 f^2</math>, but <math>15 + 8</math> cannot.Template:Sfn Template:Sfn

Although Gauss thought well of Germain, his replies to her letters were often delayed, and he generally did not review her work.Template:Sfn Eventually his interests turned away from number theory, and in 1809 the letters ceased.Template:Sfn Despite the friendship of Germain and Gauss, they never met.Template:Sfn

Work in elasticity

Germain's first attempt for the Academy Prize

File:Echladni.jpg

Ernst Florens Friedrich Chladni

When Germain's correspondence with Gauss ceased, she took interest in a contest sponsored by the Paris Academy of Sciences concerning Ernst Chladni's experiments with vibrating metal plates. 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". 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. Then Poisson was elected to the academy, thus becoming a judge instead of a contestant,Template:Sfn and leaving Germain as the only entrant to the competition.Template:Sfn

In 1809 Germain began work. Legendre assisted by giving her equations, references, and current research.Template:Sfn 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".Template:Sfn Lagrange was able to use Germain's work to derive an equation that was "correct under special assumptions".Template:Sfn

Subsequent attempts for the Prize

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

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

<math>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,</math>

where N² is a constant.Template:Sfn<ref>Template:Cite book</ref><ref>This is equation (B) in Germain's own book. Template:Cite book</ref>

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.Template:Sfn

Later work in elasticity

File:Germain - Recherches sur la théorie des surfaces élastiques, 1821 - 723413.tiff

Template:Lang, 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.Template:Sfn

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.Template:Sfn

One further work of Germain's on elasticity was published posthumously in 1831, her "Template:Lang". She used the mean curvature in her research (see Honors in number theory).Template:Sfn

Later work in number theory

Renewed interest

Germain's best work was in number theory,Template:Sfn and her most significant contribution to number theory dealt with Fermat's Last Theorem.Template:Sfn In 1815, after the elasticity contest, the academy offered a prize for a proof of Fermat's Last Theorem.Template:Sfn It reawakened Germain's interest in number theory, and she wrote to Gauss again after ten years of no correspondence.Template:Sfn

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.Template:Sfn She outlined a strategy for a general proof of Fermat's Last Theorem, including a proof for a special case.Template:Sfn 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.Template:Sfn

Her work on Fermat's Last Theorem

File:Pierre de Fermat.jpg

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":Template:Sfn Template:Blockquote

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".Template:Sfn 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.Template:Sfn

In an unpublished manuscript titled Template:Lang,Template:Sfn Germain showed that any counterexamples to Fermat's theorem for p > 5 must be numbers "whose size frightens the imagination",Template:Sfn around 40 digits long.Template:Sfn 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).Template:Sfn Germain also proved or nearly proved several results that were attributed to Lagrange or were rediscovered years later.Template:Sfn Del Centina states that "after almost two hundred years her ideas were still central",Template:Sfn but ultimately her method did not work.Template:Sfn

Work in philosophy

In addition to mathematics, Germain studied philosophy and psychology.Template:Sfn 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.Template:Sfn

Two of her philosophical works, Template:Lang and Template:Lang, were published, both posthumously. This was due in part to the efforts of Lherbette, her nephew, who collected her philosophical writings and published them.Template:Sfn Template:Lang is a collection of personal notes on scientific subjects (the writings of Tycho, Newton, and Laplace), aphorisms, and philosophical reflections.Template:Sfn Template:Sfn In Template:Lang, the work admired by Comte, Germain argues that there are no substantive differences between the sciences and the humanities.Template:Sfn

Final years

File:Germain - Œuvres philosophiques, 1896 (page 343 crop).jpg

Photograph of Sophie Germain's death mask, showing intentional deformation of the skull practiced in early modern France

In 1829 Germain learned that she had breast cancer. Despite the pain,Template:Sfn she continued to work. In 1831 Crelle's Journal published her paper on the curvature of elastic surfaces and "a note about finding Template:Math and Template:Math in <math>\tfrac{4(x^p - 1)}{x - 1} = y^2 \pm pz^2</math>".Template:Sfn Mary Gray records: "She also published in Template:Lang an examination of principles which led to the discovery of the laws of equilibrium and movement of elastic solids."Template:Sfn On 27 June 1831, she died in the house at 13 rue de Savoie.Template:Sfn

Despite Germain's intellectual achievements, her death certificate lists her as a "Template:Lang"Template:Sfn (property holder),Template:Sfn not a "Template:Lang".Template:Sfn 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".Template:Sfn

Assessments

Contemporary assessments

Vesna Petrovich found that the educated world's response to the publication in 1821 of Germain's prize-winning essay "ranged from polite to indifferent".Template:Sfn Yet, some critics had high praise for it. Of her essay in 1821, 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".Template:Sfn Claude-Louis Navier sent Germain a note calling it "a work so remarkable that quite few men could read it, and only one woman could write it."Template:Sfn Template:Sfn

Germain's contemporaries also had good things to say relating to her work in mathematics. Gauss certainly thought highly of her and recognized that European culture presented special difficulties to a woman in mathematics (see Correspondence with Gauss).

Modern assessments

The modern view generally acknowledges that although Germain 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."Template:Sfn 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."Template:Sfn

Notwithstanding the problems with Germain's theory of vibrations, Gray states that "Germain's work was fundamental in the development of a general theory of elasticity."Template:Sfn When the Eiffel Tower was built and engraved with the names of 72 great French scientists, engineers, and mathematicians, Germain's name was not among them; H. J. Mozans conjectured that despite the salience of her work to the tower's construction, she was excluded from this list because she was a woman.Template:Sfn

Concerning her early work in number theory, J. H. Sampson states: "She was clever with formal algebraic manipulations; but there is little evidence that she really understood the Template:Lang, and her work of that period that has come down to us seems to touch only on rather superficial matters."Template:Sfn Gray adds on to say "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."Template:Sfn Yet Marilyn Bailey Ogilvie recognizes that "Sophie Germain's creativity manifested itself in pure and applied mathematics ... [she] provided imaginative and provocative solutions to several important problems",Template:Sfn and, as Petrovich proposes, it may have been her very lack of training that gave her unique insights and approaches.Template:Sfn 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."Template:Sfn

Honors

Memorials

File:Grave, Sophie Germain.jpg

Grave of Sophie Germain in Père Lachaise Cemetery

Germain's resting place in the Père Lachaise Cemetery in Paris is marked by a gravestone.Template:Sfn Template:Sfn At the centennial celebration of her life, a street and a girls' school were named after her, and a plaque was placed at the house where she died.Template:Sfn The City Council of Paris commissioned a bust by Zacharie Astruc, based upon a death mask in the collection of the National Museum of Natural History, which was erected in the main courtyard of the school on 2 August 1890.Template:Sfn There are no known likenesses of Germain made from life, nor contemporary verbal descriptions of her appearance.Template:Sfn

In January 2020, Satellogic, a high-resolution Earth observation imaging and analytics company, launched a ÑuSat type micro-satellite named in honor of Sophie Germain.<ref>Template:Cite web</ref>

Honors in number theory

E. Dubouis defined a sophien of a prime Template:Mvar to be a prime Template:Mvar where Template:Math, for such Template:Mvar that yield Template:Mvar such that Template:Math has no solutions when Template:Mvar and Template:Mvar are prime to Template:Mvar.Template:Sfn

A Sophie Germain prime is a prime Template:Mvar such that Template:Math is also prime.Template:Sfn

The Germain curvature (also called mean curvature) is <math>(k_1 + k_2)/2</math>,Template:Sfn where Template:Math and Template:Math are the maximum and minimum values of the normal curvature.Template:Sfn

Sophie Germain's identity states that for any Template:Math,

Sophie Germain Prize

The Sophie Germain Prize (Template:Langx), awarded annually by the Foundation Sophie Germain, is conferred by the Academy of Sciences in Paris. Its purpose is to honour a French mathematician for research in the foundations of mathematics. This award, in the amount of €8,000, was established in 2003, under the auspices of the Institut de France.<ref>Template:Cite web</ref>

Germain in popular culture

Germain was referenced and quoted in David Auburn's 2001 play Proof. The protagonist is a young struggling female mathematician, Catherine, who found great inspiration in the work of Germain. Germain was also mentioned in John Madden's film adaptation of the same name in a conversation between Catherine (Gwyneth Paltrow) and Hal (Jake Gyllenhaal).

In the fictional work "The Last Theorem" by Arthur C. Clarke and Frederik Pohl, Sophie Germain was credited with inspiring the central character, Ranjit Subramanian, to solve Fermat's Last Theorem.

A musical about Sophie Germain's life, entitled The Limit, premiered at VAULT Festival in London, 2019.<ref>Template:Cite journal</ref>

Sophie Germain is referenced in Season 1 Episode 3 of the 2025 British miniseries Prime Target. A library book entitled Sophie Germain: The Unsolved Riddle (there appears to be no such book) serves as a "mailbox" for a brilliant, deceased prime number theorist named Safiya Zamil to have passed a handwritten note to a future mathematician on a similar prime number quest.<ref>Template:Cite web</ref>