Editing Sophie Germain

{{short description|French mathematician, physicist, and philosopher}}
{{About|the mathematician Sophie Germain|the number theory (set, or predicate)|Sophie Germain prime}}
{{Use dmy dates|date=April 2025}}
{{Infobox scientist
| name = Sophie Germain
| image = Germain - Œuvres philosophiques, 1896 (Illustration page 4).jpg
| image_size = 180px
| caption = Sketch of Germain based upon a bust by [[Zacharie Astruc]]
| birth_name = Marie-Sophie Germain
| birth_date = {{birth date|df=yes|1776|4|1|}}
| birth_place = Paris, France
| death_date ={{death date and age|df=yes|1831|6|27|1776|4|1}}
| death_place = Paris, France
| field = Mathematician, physicist, and philosopher
| academic_advisors = [[Carl Friedrich Gauss]] (epistolary correspondent)
| known_for  = [[Elasticity (physics)|Elasticity theory]]<br>[[Number theory]]<br>[[Mean curvature]]<br>[[Sophie Germain prime]]<br>[[Sophie Germain's theorem]]<br>[[Bending of plates#Small deflection of thin rectangular plates|Germain−Lagrange plate equation]]
| footnotes = '''Other name:''' Auguste Antoine Le Blanc
}}
'''Marie-Sophie Germain''' ({{IPA|fr|maʁi sɔfi ʒɛʁmɛ̃|lang}}; 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 [[Leonhard Euler|Euler]], and from correspondence with famous mathematicians such as [[Joseph Louis Lagrange|Lagrange]], [[Adrien-Marie Legendre|Legendre]], and [[Carl Friedrich Gauss|Gauss]] (under the pseudonym of Monsieur Le Blanc). One of the pioneers of [[Elasticity (physics)|elasticity theory]], she won the grand prize from the [[French Academy of Sciences|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.{{sfn|Del Centina|2008|page=373}} Because of prejudice against her sex, she was unable to make a career out of mathematics, but she worked independently throughout her life.{{sfn|Case|Leggett|2005|p=39}} 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 [[French Academy of Sciences|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 [[baptism|baptized]] the same day.{{sfn|Stupuy|1896|p=398}} According to most sources, her father, Ambroise-François, was a wealthy silk merchant,{{sfn|Del Centina|2005|loc=sec. 1}}{{sfn|Gray|1978|p=47}}{{sfn|Moncrief|2002|p=103}} though some believe he was a [[goldsmith]].{{sfn|Gray|2005|p=68}} In 1789, he was elected as a representative of the [[bourgeoisie]] to the [[Estates General of 1789|États-Généraux]], which he saw change into the [[National Assembly (French Revolution)|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.{{sfn|Gray|2005|p=68}}

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 [[Sophie Germain#Work in philosophy|Work in Philosophy]]).{{sfn|Gray|1978|p=47}}

===Introduction to mathematics===
When Germain was 13, [[Storming of the Bastille|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 [[Jean-Étienne Montucla|J. E. Montucla's]] ''L'Histoire des Mathématiques'', and his story of the death of [[Archimedes]] intrigued her.{{sfn|Gray|1978|p=47}}

Germain thought that if the geometry method, which at that time referred to all of pure mathematics,{{sfn|Gray|1978|p=47}} could hold such fascination for Archimedes, it was a subject worthy of study.{{sfn|Ogilvie|1990|p=90}} 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 [[Isaac Newton|Sir Isaac Newton]] and [[Leonhard Euler]]. She also enjoyed {{lang|fr|Traité d'Arithmétique}} by [[Étienne Bézout]] and {{lang|fr|Le Calcul Différentiel}} by [[:fr:Jacques Antoine-Joseph Cousin|Jacques Antoine-Joseph Cousin]]. Later, Cousin visited Germain at home, encouraging her in her studies.{{sfn|Gray|1978|p=48}}

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.{{sfn|Gray|1978|pp=47–48}} After some time, her mother even secretly supported her.{{sfn|Gray|1978|p=48}}

===École Polytechnique===
[[Image:Porton antigua sede École Polytechnique Sainte Geneviève.JPG|thumb|right|Entrance to the historic building of the École Polytechnique]]
In 1794, when Germain was 18, the [[École Polytechnique]] opened.{{sfn|Moncrief|2002|p=103}} As a woman, Germain was barred from attending, but the new system of education made the "lecture notes available to all who asked".{{sfn|Gray|1978|p=48}} The new method also required the students to "submit written observations".{{sfn|Gray|2005|p=69}} 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,{{sfn|Gray|1978|p=48}}<ref>{{cite web |url=https://www.pbs.org/wgbh/nova/physics/sophie-germain.html |first=Simon |last=Singh |year=1997 |publisher=WGBH Educational Foundation |title=Math's Hidden Woman |access-date=20 July 2014}}</ref> "fearing", as she later explained to Gauss, "the ridicule attached to a female scientist".{{sfn|Mackinnon|1990|p=348}}  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,{{sfn|Gray|1978|p=48}} and he became her mentor.{{sfn|Moncrief|2002|p=103}}

==Early work in number theory==

===Correspondence with Legendre===
Germain first became interested in [[number theory]] in 1798 when [[Adrien-Marie Legendre]] published {{lang|fr|Essai sur la théorie des nombres}}.{{sfn|Del Centina|2005|loc=sec. 2}} After studying the work, she opened correspondence with him on number theory, and later, [[elasticity (physics)|elasticity]]. Legendre included some of Germain's work in the {{lang|fr|Supplément}} to his second edition of the {{lang|fr|Théorie des Nombres}}, where he calls it {{lang|fr|très ingénieuse}} ("very ingenious"). See also [[#Her work on Fermat's Last Theorem|Her work on Fermat's Last Theorem]] below.{{sfn|Sampson|1990|p=158}}

===Correspondence with Gauss===
Germain's interest in number theory was renewed when she read [[Carl Friedrich Gauss]]'s monumental work {{lang|la|[[Disquisitiones Arithmeticae]]}}.{{sfn|Del Centina|2005|loc=sec. 2}} After three years of working through the exercises and trying her own proofs for some of the theorems,{{sfn|Del Centina|2008|page=352}} she wrote, again under the pseudonym of M. Le Blanc,{{sfn|Gray|1978|p=48}} to the author himself, who was one year younger than she.{{sfn|Sampson|1990|p=157}} Gauss's replies were mailed to the home of [[Antoine Isaac Silvestre de Sacy|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.{{sfn|Musielak|2020|p=201}}

The first letter, dated 21 November 1804,{{sfn|Gray|2005|p=71}} discussed Gauss's {{lang|la|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''&nbsp;=&nbsp;''p''&nbsp;−&nbsp;1, where ''p'' is a [[prime number]] of the form ''p''&nbsp;=&nbsp;8''k''&nbsp;+&nbsp;7.{{sfn|Dickson|1919|p=733}} However, her proof contained a weak assumption, and Gauss's reply did not comment on Germain's proof.{{sfn|Del Centina|2008|page=355}}

Around 1807 (sources differ),{{sfn|Dunnington|1955|p=67}} 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. {{sfn|Gray|1978|p=48}} General Pernety sent the chief of a battalion to meet with Gauss personally to see that he was safe.{{sfn|Dunnington|1955|p=67}} As it turned out, Gauss was fine, but he was confused by the mention of Sophie's name.{{sfn|Dunnington|1955|p=67}}

Three months after the incident, Germain disclosed her true identity to Gauss.{{sfn|Mackinnon|1990|p=348}}  He replied:{{sfn|Mackinnon|1990|p=349}}
{{blockquote|
How can I describe my astonishment and admiration on seeing my esteemed correspondent M. Le Blanc metamorphosed into this celebrated person ... when a woman, because of her sex, our customs and prejudices, encounters infinitely more obstacles than men in familiarising herself with [number theory's] knotty problems, yet overcomes these fetters and penetrates that which is most hidden, she doubtless has the noblest courage, extraordinary talent, and superior genius.
}}[[Image:Carl Friedrich Gauss.jpg|thumb|right|1840 painting of Carl Friedrich Gauss; by Gottlieb Biermann after [[Christian Albrecht Jensen]]]]Gauss's letters to [[Heinrich Wilhelm Matthäus Olbers|Olbers]] show that his praise for Germain was sincere.{{sfn|Dunnington|1955|p=67}}{{sfn|Bell|1937|p=262}} 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.{{sfn|Dickson|1919|p=733}}{{sfn|Waterhouse|1994}}

Although Gauss thought well of Germain, his replies to her letters were often delayed, and he generally did not review her work.{{sfn|Del Centina|2008|page=355}} Eventually his interests turned away from number theory, and in 1809 the letters ceased.{{sfn|Del Centina|2008|page=355}} Despite the friendship of Germain and Gauss, they never met.{{sfn|Gray|1978|p=49}}

==Work in elasticity==

===Germain's first attempt for the Academy Prize===
[[Image:Echladni.jpg|thumb|left|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 [[mathematical analysis|analysis]] deterred all but two contestants, [[Siméon Denis Poisson|Denis Poisson]] and Germain. Then Poisson was elected to the academy, thus becoming a judge instead of a contestant,{{sfn|Petrovich|1999|p=384}} and leaving Germain as the only entrant to the competition.{{sfn|Gray|1978|p=52}}

In 1809 Germain began work. Legendre assisted by giving her equations, references, and current research.{{sfn|Petrovich|1999|p=386}} 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".{{sfn|Petrovich|1999|p=384}} Lagrange was able to use Germain's work to derive an equation that was "correct under special assumptions".{{sfn|Gray|2005|p=71}}

===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.{{sfn|Petrovich|1999|p=384}} Germain's anonymous{{sfn|Gray|2005|p=71}} 1813 submission was still littered with mathematical errors, especially involving [[multiple integral|double integrals]],{{sfn|Gray|1978|p=52}} and it received only an honorable mention because "the fundamental base of the theory [of elastic surfaces] was not established".{{sfn|Petrovich|1999|p=384}} The contest was extended once more, and Germain began work on her third attempt. This time she consulted with Poisson.{{sfn|Gray|2005|p=71}} 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).{{sfn|Gray|1978|p=52}}

Germain submitted her third paper, "{{lang|fr|Recherches sur la théorie des surfaces élastiques}}",{{sfn|Gray|2005|p=71}} under her own name, and on 8 January 1816{{sfn|Gray|1978|p=52}} she became the first woman to win a prize from the Paris Academy of Sciences.{{sfn|Petrovich|1999|p=385}} She did not appear at the ceremony to receive her award.{{sfn|Gray|2005|p=71}} Although Germain had at last been awarded the {{lang|fr|prix extraordinaire}},{{sfn|Del Centina|2008|page=355}} the academy was still not fully satisfied.{{sfn|Ogilvie|1990|p=91}} Germain had derived the correct [[differential equation]] (a special case of the [[Kirchhoff–Love plate theory#Isotropic plates|Kirchhoff–Love equation]]),{{sfn|Ullmann|2007|p=31}} but her method did not predict experimental results with great accuracy, as she had relied on an incorrect equation from Euler,{{sfn|Gray|2005|p=71}} which led to incorrect boundary conditions.{{sfn|Ullmann|2007|p=31}} 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 ''N''<sup>2</sup> is a constant.{{sfn|Gray|2005|p=71}}<ref>{{cite book|title=A History of the Theory of Elasticity and of the Strength of Materials: Volume 1|author=Isaac Todhunter|editor=Karl Pearson|isbn=978-1108070423 |publisher=Cambridge University Press|year=2014|page=153}}</ref><ref>This is equation (B) in Germain's own book. {{cite book|title=Recherches sur la théorie des surfaces élastiques|url=https://archive.org/details/TO0E039736_TO0324_PNI-1705_000000|author=Sophie Germain|year=1821|page=[https://archive.org/details/TO0E039736_TO0324_PNI-1705_000000/page/27 27]}}</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.{{sfn|Petrovich|1999|p=386}}

===Later work in elasticity===
[[File:Germain - Recherches sur la théorie des surfaces élastiques, 1821 - 723413.tiff|thumb|{{lang|fr|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.{{sfn|Gray|2005|p=71}}

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.{{sfn|Del Centina|2005|loc=sec. 4}}

One further work of Germain's on elasticity was published posthumously in 1831, her "{{lang|fr|Mémoire sur la courbure des surfaces}}". She used the [[mean curvature]] in her research (see [[Sophien|Honors in number theory]]).{{sfn|Gray|2005|p=71}}

==Later work in number theory==

===Renewed interest===
Germain's best work was in number theory,{{sfn|Del Centina|2005|loc=sec. 1}} and her most significant contribution to number theory dealt with Fermat's Last Theorem.{{sfn|Sampson|1990|p=158}} In 1815, after the elasticity contest, the academy offered a prize for a proof of Fermat's Last Theorem.{{sfn|Del Centina|2008|page=357}} It reawakened Germain's interest in number theory, and she wrote to Gauss again after ten years of no correspondence.{{sfn|Del Centina|2005|loc=sec. 2}}

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.{{sfn|Del Centina|2008|page=357}} She outlined a strategy for a general proof of Fermat's Last Theorem, including a proof for a special case.{{sfn|Del Centina|2008|pp=356–357}} 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.{{sfn|Del Centina|2008|page=362}}

===Her work on Fermat's Last Theorem===
[[Image:Pierre de Fermat.jpg|thumb|right|[[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]]":{{sfn|Del Centina|2008|page=349}}
{{blockquote|1=
Let ''p'' be an odd prime. If there exists an auxiliary prime ''P''&nbsp;=&nbsp;2''Np''&nbsp;+&nbsp;1 (''N'' is any positive integer not divisible by 3) such that:
# if ''x''<sup>''p''</sup>&nbsp;+&nbsp;''y''<sup>''p''</sup>&nbsp;+&nbsp;''z''<sup>''p''</sup>&nbsp;≡&nbsp;0 ([[modular arithmetic|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''.{{sfn|Del Centina|2008|page=372}}
}}

Germain used this result to prove the first case of Fermat's Last Theorem for all odd primes ''p''&nbsp;<&nbsp;100, but according to Andrea Del Centina, "she had actually shown that it holds for every exponent ''p''&nbsp;<&nbsp;197".{{sfn|Del Centina|2008|page=372}} [[Leonard Eugene Dickson|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.{{sfn|Dickson|1919|p=763}}

In an unpublished manuscript titled {{lang|fr|Remarque sur l'impossibilité de satisfaire en nombres entiers a l'équation x<sup>p</sup>&nbsp;+&nbsp;y<sup>p</sup>&nbsp;{{=}}&nbsp;z<sup>p</sup>}},{{sfn|Del Centina|2008|page=349}} Germain showed that any counterexamples to Fermat's theorem for ''p''&nbsp;>&nbsp;5 must be numbers "whose size frightens the imagination",{{sfn|Cipra|2008|p=899}} around 40 digits long.{{sfn|Del Centina|2008|page=371}}  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''&nbsp;=&nbsp;5 (see [[Sophie Germain#Correspondence with Legendre|Correspondence with Legendre]]).{{sfn|Cipra|2008|p=899}}  Germain also proved or nearly proved several results that were attributed to Lagrange or were rediscovered years later.{{sfn|Del Centina|2008|page=373}} Del Centina states that "after almost two hundred years her ideas were still central",{{sfn|Del Centina|2008|page=373}} but ultimately her method did not work.{{sfn|Cipra|2008|p=899}}

==Work in philosophy==
In addition to mathematics, Germain studied philosophy and [[psychology]].{{sfn|Gray|1978|p=48}} 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]].{{sfn|Gray|2005|p=73}}

Two of her philosophical works, {{lang|fr|Pensées diverses}} and {{lang|fr|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.{{sfn|Gray|1978|p=53}}  {{lang|fr|Pensées}} is a collection of personal notes on scientific subjects (the writings of Tycho, Newton, and Laplace), aphorisms, and philosophical reflections.{{sfn|Del Centina|Fiocca|2012|p=591}}{{sfn|Stupuy|1896|p=[https://gallica.bnf.fr/ark:/12148/bpt6k2032890/f69.item.texteImage 60]}} In {{lang|fr|Considérations}}, the work admired by Comte, Germain argues that there are no substantive differences between the sciences and the [[humanities]].{{sfn|Ogilvie|1990|p=92}}

==Final years==
[[Image:Germain - Œuvres philosophiques, 1896 (page 343 crop).jpg|thumb|right|Photograph of Sophie Germain's [[death mask]], showing [[artificial cranial deformation|intentional deformation of the skull]] practiced in early modern France]]

In 1829 Germain learned that she had breast cancer. Despite the pain,{{sfn|Del Centina|2005|loc=sec. 5–6}} she continued to work. In 1831 ''[[Crelle's Journal]]'' published her paper on the [[curvature]] of elastic surfaces and "a note about finding {{math|''y''}} and {{math|''z''}} in <math>\tfrac{4(x^p - 1)}{x - 1} = y^2 \pm pz^2</math>".{{sfn|Gray|2005|p=71}} Mary Gray records: "She also published in {{lang|fr|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."{{sfn|Gray|2005|p=71}} On 27 June 1831, she died in the house at 13 rue de Savoie.{{sfn|Gray|1978|p=49}}

Despite Germain's intellectual achievements, her death certificate lists her as a "{{lang|fr|rentière – annuitant}}"{{sfn|Mozans|1913|p=156}} (property holder),{{sfn|Gray|1978|p=50}} not a "{{lang|fr|mathématicienne}}".{{sfn|Mozans|1913|p=156}} 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".{{sfn|Mackinnon|1990|p=347}}

==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".{{sfn|Petrovich|1999|p=385}} 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".{{sfn|Gray|1978|p=49}} [[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."{{sfn|Stupuy|1896|p=[https://gallica.bnf.fr/ark:/12148/bpt6k2032890/f326.item.texteImage 317]}}{{sfn|Musielak|2020|p=89}}

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 [[Sophie Germain#Correspondence with Gauss|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."{{sfn|Gray|1978|p=51}} Petrovich adds: "This proved to be a major handicap when she could no longer be regarded as a young [[child prodigy|prodigy]] to be admired but was judged by her peer mathematicians."{{sfn|Petrovich|1999|p=384–385}}

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."{{sfn|Gray|1978|p=52}} When the [[Eiffel Tower]] was built and engraved with the names of [[list of the 72 names on the Eiffel Tower|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.{{sfn|Mozans|1913|p=156}}

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 {{lang|la|Disquisitiones}}, and her work of that period that has come down to us seems to touch only on rather superficial matters."{{sfn|Sampson|1990|p=158}} 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."{{sfn|Gray|1978|p=50}} 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",{{sfn|Ogilvie|1990|p=92}} and, as Petrovich proposes, it may have been her very lack of training that gave her unique insights and approaches.{{sfn|Petrovich|1999|p=385}} 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."{{sfn|Petrovich|1999|p=386}}

==Honors==

===Memorials===
[[File:Grave, Sophie Germain.jpg|thumb|right|Grave of Sophie Germain in Père Lachaise Cemetery]]
Germain's resting place in the [[Père Lachaise Cemetery]] in Paris is marked by a gravestone.{{sfn|Gray|1978|p=49}}{{sfn|Gray|2005|p=68}} <!-- The grave seems "OK" in the image, and the lower plaque includes a reference to the girls' school named after her, so I assume it has been kept up or replaced since the centennial mentioned next. --> At the centennial celebration of her life, a street and [[Lycée Sophie Germain|a girls' school]] were named after her, and a plaque was placed at the house where she died.{{sfn|Gray|2005|p=68}} 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, France|National Museum of Natural History]], which was erected in the main courtyard of the school on 2 August 1890.{{sfn|Musielak|2020|p=204}} There are no known likenesses of Germain made from life, nor contemporary verbal descriptions of her appearance.{{sfn|Musielak|2020|p=204}}

In January 2020, [[Satellogic]], a high-resolution [[Earth observation]] imaging and analytics company, launched a [[ÑuSat]] type [[Small satellite|micro-satellite]] named in honor of Sophie Germain.<ref>{{Cite web|url=https://www.space.com/china-long-march-2d-satellites-january-2020-launch-success.html/|title=China lofts 4 satellites into orbit with its second launch of 2020 |website=space.com|date=16 January 2020 |language=en-US|access-date=30 January 2020}}</ref>

===Honors in number theory===
E. Dubouis defined a ''sophien'' of a prime {{mvar|n}} to be a prime {{mvar|θ}} where {{math|1=''θ''&nbsp;=&nbsp;''kn''&nbsp;+&nbsp;1}}, for such {{mvar|n}} that yield {{mvar|θ}} such that {{math|1=''x''<sup>''n''</sup>&nbsp;=&nbsp;''y''<sup>''n''</sup>&nbsp;+&nbsp;1 (mod ''θ'')}} has no solutions when {{mvar|x}} and {{mvar|y}} are prime to {{mvar|n}}.{{sfn|Dickson|1919|p=769}}

A [[Sophie Germain prime]] is a prime {{mvar|p}} such that {{math|2''p''&nbsp;+&nbsp;1}} is also prime.{{sfn|Del Centina|2008|page=372}}

The ''Germain curvature'' (also called [[mean curvature]]) is <math>(k_1 + k_2)/2</math>,{{sfn|Mackinnon|1990|p=347}} where {{math|''k''<sub>1</sub>}} and {{math|''k''<sub>2</sub>}} are the maximum and minimum values of the normal curvature.{{sfn|Gray|2005|p=71}}

[[Sophie Germain's identity]] states that for any {{math|{''x'', ''y''{{)}}}},

: <math>x^4 + 4y^4 = \big((x + y)^2 + y^2\big)\big((x - y)^2 + y^2\big) = (x^2 + 2xy + 2y^2)(x^2 - 2xy + 2y^2).</math>

===Sophie Germain Prize===

The [[Sophie Germain Prize]] ({{langx|fr|link=no|Prix Sophie Germain}}), awarded annually by the Foundation Sophie Germain, is conferred by the [[French Academy of Sciences|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>{{cite web |title=Prix Sophie Germain – Fondation de l'Institut de France |url=http://www.academie-sciences.fr/activite/prix/laureat_germain.pdf |publisher=Institut de France – Académie des sciences |access-date=20 July 2014 |url-status=dead |archive-url=https://web.archive.org/web/20141129202400/http://www.academie-sciences.fr/activite/prix/laureat_germain.pdf |archive-date=29 November 2014 }}</ref>

==Germain in popular culture==
Germain was referenced and quoted in [[David Auburn]]'s 2001 play ''[[Proof (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 [[Proof (2005 film)|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>{{Cite journal |last=Merow |first=Sophia D. |date=September 2019 |title=One Spark Is All You Need: Germain Gets the Hamilton Treatment |url=https://www.ams.org/journals/notices/201908/rnoti-p1309.pdf |journal=Notices of the American Mathematical Society |volume=66 |issue=8 |pages=1309–1311}}</ref>

Sophie Germain is referenced in Season 1 Episode 3 of the 2025 British miniseries ''[[Prime Target (TV series)|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>{{cite web |title=Prime Target{{Snd}}Season 1 Episode 3 Recap {{&}} Review |last=Wheeler |first=Greg |date=29 January 2025 |url=https://www.thereviewgeek.com/primetarget-s1e3review/ |website=The Review Geek}}</ref>

==See also==
*[[Proof of Fermat's Last Theorem for specific exponents]]
*[[Sophie Germain Counter Mode]]
*[[Sophie Germain prime]]
*[[Sophie Germain Prize]]
*[[Sophie Germain's theorem]]
*[[Timeline of women in science]]
*[[Timeline of women in mathematics]]

==Citations==
{{reflist}}

==References==
{{Library resources box|by=yes|onlinebooksby=yes|viaf=12440398}}
* {{cite book |last=Bell |first=Eric Temple |author-link=Eric Temple Bell |title=Men of Mathematics |year=1937 |publisher=Simon and Schuster |title-link=Men of Mathematics }} reprinted as {{cite book |last=Bell |first=Eric Temple |title=Men of Mathematics |year=1986 |publisher=Simon and Schuster |isbn=0-671-62818-6}}
*Bucciarelli, Louis L; Dworsky, Nancy (1980). ''Sophie Germain: An Essay in the History of the Theory of Elasticity'', D. Reidel:Holland {{ISBN|978-90-277-1135-9}}
* {{cite book |last1=Case |first1=Bettye Anne|author1-link=Bettye Anne Case |first2=Anne M. |last2=Leggett|author2-link=Anne M. Leggett |title=Complexities: Women in Mathematics|title-link= Complexities: Women in Mathematics |year=2005 |publisher=Princeton University Press |isbn=0-691-11462-5 }}
* {{cite journal  |last= Cipra  |first= Barry A.  |author-link = Barry A. Cipra |title=A Woman Who Counted |journal=Science |year=2008 |volume=319 |issue=5865 |page=899 |doi=10.1126/science.319.5865.899a |pmid= 18276866 |s2cid= 45461806 }}
* {{cite journal |last=Del Centina |first=Andrea |title=Letters of Sophie Germain preserved in Florence |journal=Historia Mathematica |year=2005 |volume=32 |issue=1 |pages=60–75 |doi=10.1016/j.hm.2003.11.001 |doi-access=free }}
* {{cite journal |last=Del Centina |first=Andrea |title=Unpublished manuscripts of Sophie Germain and a revaluation of her work on Fermat's Last Theorem |journal=Archive for History of Exact Sciences |year=2008 |volume=62 |issue=4 |pages=349–392 |doi=10.1007/s00407-007-0016-4 |bibcode=2008AHES...62..349D |s2cid=189782687 }}
* {{cite journal|last1=Del Centina|first1=Andrea|last2=Fiocca|first2=Alessandra|doi=10.1007/s00407-012-0105-x|issue=6|journal=Archive for History of Exact Sciences|jstor=23319292|mr=2984133|pages=585–700|title=The correspondence between Sophie Germain and Carl Friedrich Gauss|volume=66|year=2012|s2cid=121021850}}
* {{cite book |last= Dickson |first=Leonard Eugene |author-link=Leonard Eugene Dickson |title=History of the Theory of Numbers, Volume II: Diophantine Analysis |year=1919 |publisher=Carnegie Institution }} Reprinted as {{cite book|last= Dickson|first=Leonard Eugene|title=History of the Theory of Numbers, Volume II: Diophantine Analysis|year=2013|publisher=Dover Publications|isbn=978-0-486-15460-2}}
* {{cite book |last=Dunnington |first=G. Waldo |title=Carl Friedrich Gauss: Titan of Science. A study of his life and work |year=1955 |publisher=Hafner }} Reprinted as {{cite book |last=Dunnington |first=G. Waldo |author2=Jeremy Gray |author3=Fritz-Egbert Dohse |title=Carl Friedrich Gauss: Titan of Science |year=2004 |publisher=Mathematical Association of America |isbn=978-0-88385-547-8}}
* {{cite book |last=Gray |first=Mary W. |chapter=Sophie Germain |editor1=Bettye Anne Case |editor2=Anne M. Leggett |title=Complexities: Women in Mathematics |publisher=Princeton University Press |year=2005 |pages=68–75 |isbn=0-691-11462-5 }}
* {{cite encyclopedia |last=Gray |first=Mary |title=Sophie Germain (1776–1831) |encyclopedia=Women of Mathematics: A Bibliographic Sourcebook |editor1=Louise S. Grinstein |editor2=Paul Campbell |publisher=Greenwood |year=1978 |isbn=978-0-313-24849-8 |pages=[https://archive.org/details/womenofmathemati0000unse/page/47 47–55] |url=https://archive.org/details/womenofmathemati0000unse/page/47 }}
* {{cite journal |last=Mackinnon |first=Nick |title=Sophie Germain, or, Was Gauss a feminist? |journal=The Mathematical Gazette |year=1990 |volume=74 |issue=470 |pages=346–351 |doi=10.2307/3618130 |jstor=3618130 |s2cid=126102577 }}
* {{cite book |last=Moncrief |first= J. William |editor=Barry Max Brandenberger |title=Mathematics, Volume 2: Macmillan Science Library |year=2002 |publisher=Macmillan Reference USA |isbn=978-0-02-865563-5 |chapter=Germain, Sophie }}
* {{cite book |last=Mozans |first=H. J. |author-link=John Augustine Zahm |title=Women in Science: With an Introductory Chapter on Women's Long Struggle for Things of the Mind |url=https://archive.org/details/womaninscience00zahmrich |publisher=D. Appleton |year=1913 |pages=[https://archive.org/details/womaninscience00zahmrich/page/154 154]–157}}
* {{cite book |last=Musielak |first=Dora |title=Sophie Germain: Revolutionary Mathematician |edition=2nd |publisher=Springer Biographies |location=Cham |year=2020 |isbn=978-3-030-38374-9}}
* {{cite book |last=Ogilvie |first=Marilyn Bailey |title=Women in Science: Antiquity Through the Nineteenth Century: a Biographical Dictionary with Annotated Bibliography |year=1990 |publisher=MIT Press |isbn=978-0-262-65038-0 |url-access=registration |url=https://archive.org/details/womeninscience00mari }}
* {{cite journal |last=Petrovich |first=Vesna Crnjanski |title=Women and the Paris Academy of Sciences |journal=Eighteenth-Century Studies |year=1999 |volume= 32 |issue=3 |pages=383–390 |jstor=30053914 |doi=10.1353/ecs.1999.0022 |s2cid=162272331 }}
* {{cite journal |last=Sampson |first=J. H. |title=Sophie Germain and the Theory of Numbers |journal=Archive for History of Exact Sciences |year=1990 |volume=41 |issue=2 |pages=157–161 |doi=10.1007/BF00411862 |jstor=41133883 |s2cid=123148132 }}
* {{Cite book |editor-last=Stupuy |editor-first=Hippolyte |title=Œuvres philosophiques de Sophie Germain |publisher=Firmin-Didot |year=1896 |edition=2nd |location=Paris |language=French |trans-title=Philosophical Works of Sophie Germain}}
* {{cite journal |last=Ullmann |first=D. |title=Life and work of E.F.F. Chladni |journal=European Physical Journal ST |year=2007 |volume=145 |issue=1 |pages=25–32 |doi=10.1140/epjst/e2007-00145-4 |bibcode=2007EPJST.145...25U |s2cid=121813715 }}
* {{cite journal |last=Waterhouse |first=William C. |author-link=William C. Waterhouse |title=A counterexample for Germain |journal=American Mathematical Monthly |year=1994 |volume=101 |issue=2 |pages=140–150 |doi=10.2307/2324363 |jstor=2324363 }}

==External links==
*{{MacTutor Biography|id=Germain}}
*{{MathGenealogy|id=55175}}
*[http://sheroesofhistory.wordpress.com/2014/03/27/badass-wimmin-of-history-presents-sophie-germain/ Sheroes of History; Sophie Germain]
*[http://kids.love.science/grandpa_benny_goes_to_school.html Sophie Germain in the illustrated story] at the Kids Love Science project
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[[Category:19th-century French women scientists]]
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[[Category:French women mathematicians]]
[[Category:French number theorists]]
[[Category:Deaths from breast cancer in France]]
[[Category:French physicists]]
[[Category:French women physicists]]
[[Category:French women philosophers]]
[[Category:18th-century women mathematicians]]
[[Category:19th-century women mathematicians]]