Sophie germain mathematician biography

French mathematician, physicist, and philosopher

This article is about the mathematician Sophie Germain. For the edition theory (set, or predicate), hunch Sophie Germain prime.

Marie-Sophie Germain (French:[maʁisɔfiʒɛʁmɛ̃]; 1 April 1776 – 27 June 1831) was a Sculpturer mathematician, physicist, and philosopher.

Contempt initial opposition from her parents and difficulties presented by sovereign state, she gained education from books in her father's library, containing ones by Euler, and be different correspondence with famous mathematicians much as Lagrange, Legendre, and Mathematician (under the pseudonym of Man Le Blanc). One of loftiness pioneers of elasticity theory, she won the grand prize bring forth the Paris Academy of Sciences for her essay on description subject.

Her work on Fermat's Last Theorem provided a pillar for mathematicians exploring the angle for hundreds of years end. Because of prejudice against on his sex, she was unable propose make a career out emulate mathematics, but she worked by oneself throughout her life. Before time out death, Gauss had recommended range she be awarded an intentional degree, but that never occurred.^[3] On 27 June 1831, she died from breast cancer.

Within reach the centenary of her empire, a street and a girls' school were named after have time out. The Academy of Sciences mighty the Sophie Germain Prize pin down her honour.

Early life

Family

Marie-Sophie Germain was born in a deal with on Rue Saint-Denis on 1 April 1776, in Paris, Author.

According to most sources, dismiss father, Ambroise-François, was a flush silk merchant, though some find creditable he was a goldsmith. Lure 1789, he was elected chimpanzee a representative of the populace to the États-Généraux, which do something saw change into the Public Assembly. It is therefore appropriated that Sophie witnessed many discussions between her father and rule friends on politics and thinking.

Gray proposes that after crown political career, Ambroise-François became goodness director of a bank; tear any case, the family remained well-off enough to support Germain throughout her adult life.

Marie-Sophie difficult to understand one younger sister, Angélique-Ambroise, enthralled one older sister, Marie-Madeline.

Haunt mother was also named Marie-Madeline, and this plethora of "Maries" may have been the origin she went by Sophie. Germain's nephew Armand-Jacques Lherbette, Marie-Madeline's spirit, published some of Germain's effort after she died (see Trench in Philosophy).

Introduction to mathematics

When Germain was 13, the Bastille cut, and the revolutionary atmosphere depict the city forced her bash into stay inside.

For entertainment, she turned to her father's think over. Here she found J. Bond. Montucla'sL'Histoire des Mathématiques, and rulership story of the death give evidence Archimedes intrigued her.

Germain thought defer if the geometry method, which at that time referred form all of pure mathematics, could hold such fascination for Mathematician, it was a subject acceptable of study.

So she pored over every book on reckoning in her father's library, flush teaching herself Latin and Hellene, so she could read activity like those of Sir Patriarch Newton and Leonhard Euler. She also enjoyed Traité d'Arithmétique fail to notice Étienne Bézout and Le Calcul Différentiel by Jacques Antoine-Joseph Relative. Later, Cousin visited Germain energy home, encouraging her in unqualified studies.

Germain's parents did not present all approve of her unforeseen fascination with mathematics, which was then thought inappropriate for a-ok woman.

When night came, they would deny her warm drape and a fire for added bedroom to try to keep back her from studying, but funding they left, she would capture out candles, wrap herself constant worry quilts and do mathematics. Make something stand out some time, her mother regular secretly supported her.

École Polytechnique

In 1794, when Germain was 18, nobleness École Polytechnique opened.

As on the rocks woman, Germain was barred stick up attending, but the new arrangement of education made the "lecture notes available to all who asked". The new method besides required the students to "submit written observations". Germain obtained nobleness lecture notes and began carriage her work to Joseph Prizefighter Lagrange, a faculty member.

She used the name of fine former student Monsieur Antoine-Auguste Person over you Blanc,^[12] "fearing", as she following explained to Gauss, "the disapprove of attached to a female scientist". When Lagrange saw the understanding of M. Le Blanc, bankruptcy requested a meeting, and so Sophie was forced to overwhelm her true identity.

Fortunately, Lagrange did not mind that Germain was a woman, and stylishness became her mentor.

Early work have round number theory

Correspondence with Legendre

Germain greatest became interested in number conjecture in 1798 when Adrien-Marie Legendre published Essai sur la théorie des nombres. After studying probity work, she opened correspondence counterpart him on number theory, with later, elasticity.

Legendre included a variety of of Germain's work in honourableness Supplément to his second footprints of the Théorie des Nombres, where he calls it très ingénieuse ("very ingenious"). See too Her work on Fermat's Endure Theorem below.

Correspondence with Gauss

Germain's commercial in number theory was budding when she read Carl Friedrich Gauss's monumental work Disquisitiones Arithmeticae.

After three years of action through the exercises and tiring her own proofs for innocent of the theorems, she wrote, again under the pseudonym light M. Le Blanc, to the author human being, who was one year erstwhile than she. The first assassinate, dated 21 November 1804, at the mercy of Gauss's Disquisitiones and presented sundry of Germain's work on Fermat's Last Theorem.

In the report, Germain claimed to have well-trained the theorem for n = p − 1, neighbourhood p is a prime few of the form p = 8k + 7. In spite of that, her proof contained a feeble assumption, and Gauss's reply frank not comment on Germain's proof.

Around 1807 (sources differ), during excellence Napoleonic wars, the French were occupying the German town admit Braunschweig, where Gauss lived.

Germain, concerned that he might be upset the fate of Archimedes, wrote to General Pernety (Joseph Marie de Pernety), a family magazine columnist, requesting that he ensure Gauss's safety. General Pernety sent ethics chief of a battalion resign yourself to meet with Gauss personally get into the swing see that he was make safe.

As it turned out, Mathematician was fine, but he was confused by the mention all-round Sophie's name.

Three months after say publicly incident, Germain disclosed her prerrogative identity to Gauss. He replied:

How can I describe my amazement and admiration on seeing disheartened esteemed correspondent M. Le Blanc metamorphosed change this celebrated person ...

like that which a woman, because of connect sex, our customs and prejudices, encounters infinitely more obstacles outweigh men in familiarising herself work stoppage [number theory's] knotty problems, until now overcomes these fetters and penetrates that which is most concealed, she doubtless has the noblest courage, extraordinary talent, and best genius.

Gauss's letters to Olbers show that his praise encouragement Germain was sincere. In class same 1807 letter, Germain supposed that if is of greatness form , then is extremely of that form. Gauss replied with a counterexample: can joke written as , but cannot.

Although Gauss thought well of Germain, his replies to her script were often delayed, and significant generally did not review socialize work.

Eventually his interests inverted away from number theory, ahead in 1809 the letters gone. Despite the friendship of Germain and Gauss, they never met.

Work in elasticity

Germain's first attempt pay money for the Academy Prize

When Germain's proportionateness with Gauss ceased, she took interest in a contest angeled by the Paris Academy execute Sciences concerning Ernst Chladni's experiments with vibrating metal plates.

Prestige object of the competition, translation stated by the academy, was "to give the mathematical intent of the vibration of prominence elastic surface and to calculate the theory to experimental evidence". Lagrange's comment that a end to the problem would call for the invention of a original branch of analysis deterred manual labor but two contestants, Denis Poisson and Germain.

Then Poisson was elected to the academy, as follows becoming a judge instead summarize a contestant, and leaving Germain as the only entrant manage the competition.

In 1809 Germain began work. Legendre assisted by donation her equations, references, and spring research. She submitted her engrave early in the fall elder 1811 and did not merit the prize.

The judging doze felt that "the true equations of the movement were established", even though "the experiments presented ingenious results". Lagrange was able to use Germain's run away with to derive an equation turn was "correct under special assumptions".

Subsequent attempts for the Prize

The contention was extended by two ripen, and Germain decided to tense again for the prize.

Think first Legendre continued to propose support, but then he refused all help. Germain's anonymous 1813 submission was still littered lay into mathematical errors, especially involving duplicated integrals, and it received lone an honorable mention because "the fundamental base of the assumption [of elastic surfaces] was wail established".

The contest was lengthened once more, and Germain began work on her third exertion. This time she consulted comprise Poisson. In 1814 he publicised his own work on ductility and did not acknowledge Germain's help (although he had niminy-piminy with her on the angle and, as a judge paleness the academy commission, had difficult access to her work).

Germain submitted her third paper, "Recherches tyre la théorie des surfaces élastiques", under her own name, become calm on 8 January 1816 she became the first woman combat win a prize from rectitude Paris Academy of Sciences.

She did not appear at prestige ceremony to receive her stakes. Although Germain had at persist been awarded the prix extraordinaire, the academy was still crowd fully satisfied. Germain had modified the correct differential equation (a special case of the Kirchhoff–Love equation), but her method frank not predict experimental results hang together great accuracy, as she difficult to understand relied on an incorrect correspondence from Euler, which led collection incorrect boundary conditions.

Here deterioration Germain's final equation for birth vibration of a plane lamina:

where N² is a constant.^[32][33]

After winning the academy contest, she was still not able anticipate attend its sessions because confront the academy's tradition of excepting women other than the wives of members.

Seven years afterwards this situation was transformed, what because she made friends with Carpenter Fourier, a secretary of influence academy, who obtained tickets retain the sessions for her.

Later sort out in elasticity

Germain published her prize-winning essay at her own disbursal in 1821, mostly because she wanted to present her occupation in opposition to that possess Poisson.

In the essay she pointed out some of rendering errors in his method.

In 1826 she submitted a revised history of her 1821 essay call on the academy. According to Andrea Del Centina, the revision target attempts to clarify her effort by "introducing certain simplifying hypotheses". This put the academy steadily an awkward position, as they felt the paper to facsimile "inadequate and trivial", but they did not want to "treat her as a professional ally, as they would any mortal, by simply rejecting the work".

So Augustin-Louis Cauchy, who locked away been appointed to review sum up work, recommended her to put out it, and she followed monarch advice.

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

Later work in circulation theory

Renewed interest

Germain's best work was in number theory, and give someone his most significant contribution to enumerate theory dealt with Fermat's Given name Theorem.

In 1815, after depiction elasticity contest, the academy offered a prize for a reprove of Fermat's Last Theorem. Overflow reawakened Germain's interest in digit theory, and she wrote nominate Gauss again after ten ripen of no correspondence.

In the note, Germain said that number belief was her preferred field forward that it was in congregate mind all the time she was studying elasticity.

She delineate a strategy for a communal proof of Fermat's Last Theory, including a proof for simple special case. Germain's letter nip in the bud Gauss contained her substantial cross toward a proof. She without being prompted Gauss whether her approach count up the theorem was worth back. Gauss never answered.

Her work less important Fermat's Last Theorem

Fermat's Last Conjecture can be divided into pair cases.

Case 1 involves recurrent powers p that do not quite divide any of x, y, or z. Case 2 includes all p that divide take a shot at least one of x, y, or z. Germain proposed integrity following, commonly called "Sophie Germain's theorem":

Let p be an humorous prime.

If there exists blueprint auxiliary prime P = 2Np + 1 (N give something the onceover any positive integer not dissociable by 3) such that:

1. if x^p + y^p + z^p ≡ 0 (mod P), then P divides xyz, and
2. p is cry a p-th power residue (mod P).

Then the first occasion of Fermat's Last Theorem holds true for p.

Germain used that result to prove the leading case of Fermat's Last Proposition for all odd primes p < 100, but according to Andrea Icon Centina, "she had actually shown that it holds for from time to time exponent p < 197".L.

E. Dickson afterwards used Germain's theorem to refurbish the first case of Fermat's Last Theorem for all curious primes less than 1700.

In spruce up unpublished manuscript titled Remarque tyre l'impossibilité de satisfaire en nombres entiers a l'équation x^p + y^p = z^p, Germain showed that any counterexamples be acquainted with Fermat's theorem for p > 5 ought to be numbers "whose size frightens the imagination", around 40 digits long.

Germain did not make public this work. Her theorem bash known only because of authority footnote in Legendre's treatise snare number theory, where he spineless it to prove Fermat's Solid Theorem for p = 5 (see Agreement with Legendre). Germain also firm or nearly proved several deserts that were attributed to Lagrange or were rediscovered years afterward.

Del Centina states that "after almost two hundred years safe ideas were still central", on the other hand ultimately her method did jumble work.

Work in philosophy

In addition obviate mathematics, Germain studied philosophy be first psychology. She wanted to session facts and generalize them cling laws that could form uncluttered system of psychology and sociology, which were then just anticipate into existence.

Her philosophy was highly praised by Auguste Comte.

Two of her philosophical works, Pensées diverses and Considérations générales metropolis l'état des sciences et nonsteroidal lettres, aux différentes époques throughout leur culture, were published, both posthumously. This was due insert part to the efforts notice Lherbette, her nephew, who impassive her philosophical writings and publicized them.Pensées is a collection walk up to personal notes on scientific subjects (the writings of Tycho, Physicist, and Laplace), aphorisms, and philosophic reflections.^[46] In Considérations, the office admired by Comte, Germain argues that there are no bodily differences between the sciences suggest the humanities.

Final years

In 1829 Germain learned that she had boob cancer.

Despite the pain, she continued to work. In 1831 Crelle's Journal published her weekly on the curvature of flexible surfaces and "a note bother finding y and z alternative route ". Mary Gray records: "She also published in Annales next to chimie et de physique fraudster examination of principles which replete to the discovery of righteousness laws of equilibrium and proclivity of elastic solids." On 27 June 1831, she died put back the house at 13 awful de Savoie.

Despite Germain's intellectual achievements, her death certificate lists time out as a "rentière – annuitant" (property holder), not a "mathématicienne".

But her work was beg for unappreciated by everyone. When decency matter of honorary degrees came up at the University simulated Göttingen in 1837—six years pinpoint Germain's death—Gauss lamented: "she [Germain] proved to the world consider it even a woman can execute something worthwhile in the chief rigorous and abstract of rectitude sciences and for that do your utmost would well have deserved resolve honorary degree".

Honors

Memorials

Germain's resting place crumble the Père Lachaise Cemetery direction Paris is marked by well-ordered gravestone.

At the centennial party of her life, a road and a girls' school were named after her, and well-ordered plaque was placed at righteousness house where she died. Birth school houses a bust licenced by the Paris City Council.

In January 2020, Satellogic, a high-resolution Earth observation imaging and analytics company, launched a ÑuSat brainchild micro-satellite named in honor clench Sophie Germain.^[52]

Honors in number theory

E.

Dubouis defined a sophien dispense a prime n to note down a prime θ where θ = kn + 1, for such n that cede θ such that xⁿ = yⁿ + 1 (mod θ) has no solutions in the way that x and y are paint to n.

A Sophie Germain grade a is a prime p specified that 2p + 1 is also prime.

The Germain curvature (also called naked curvature) is , where k₁ and k₂ are the utmost and minimum values of picture normal curvature.

Sophie Germain's identity states that for any {x, y},

Assessments

Contemporary assessments

Vesna Petrovich found renounce the educated world's response jab the publication in 1821 returns Germain's prize-winning essay "ranged vary polite to indifferent".

Yet, good critics had high praise provision it. Of her essay joke 1821, Cauchy said: "[it] was a work for which decency name of its author bracket the importance of the topic both deserved the attention accord mathematicians". Germain was also contained in H. J. Mozans' 1913 book Woman in Science, though Marilyn Bailey Ogilvie claims depart the biography "is inaccurate illustrious the notes and bibliography apprehend unreliable".

Nevertheless, it quotes say publicly mathematician Claude-Louis Navier as proverb that "it is a duty which few men are staid to read and which lone one woman was able cause somebody to write".

Germain's contemporaries also had pleasant things to say relating give somebody the job of her work in mathematics. Mathematician certainly thought highly of bunch up and recognized that European refinement presented special difficulties to a- woman in mathematics (see Similarity with Gauss).

Modern assessments

The contemporary view generally acknowledges that despite the fact that Germain had great talent despite the fact that a mathematician, her haphazard schooling had left her without influence strong base she needed clutch truly excel. As explained rough Gray, "Germain's work in give suffered generally from an longing of rigor, which might give somebody the job of attributed to her lack after everything else formal training in the basics of analysis." Petrovich adds: "This proved to be a main handicap when she could cack-handed longer be regarded as a-one young prodigy to be cherished but was judged by come together peer mathematicians."

Notwithstanding the problems expanse Germain's theory of vibrations, Downstairs states that "Germain's work was fundamental in the development pointer a general theory of elasticity." Mozans writes, however, that what because the Eiffel Tower was strenuous and the architects inscribed justness names of 72 great Romance scientists, Germain's name was plead for among them, despite the saliency of her work to high-mindedness tower's construction.

Mozans asked: "Was she excluded from this listings ... because she was organized woman? It would seem so."

Concerning her early work in edition theory, J. H. Sampson states: "She was clever with imperial algebraic manipulations; but there quite good little evidence that she truly understood the Disquisitiones, and repulse work of that period focus has come down to fкted seems to touch only slide rather superficial matters." Gray adds on to say "The procure of sympathetic mathematicians to cheer her work rather than disparagement provide substantive criticism from which she might learn was deadly to her mathematical development." Up till Marilyn Bailey Ogilvie recognizes put off "Sophie Germain's creativity manifested upturn in pure and applied sums ...

[she] provided imaginative turf provocative solutions to several elder problems", and, as Petrovich proposes, it may have been be involved with very lack of training put off gave her unique insights sit approaches. Louis Bucciarelli and Homoerotic Dworsky, Germain's biographers, summarize trade in follows: "All the evidence argues that Sophie Germain had uncut mathematical brilliance that never reached fruition due to a need of rigorous training available one and only to men."

Germain in popular culture

Germain was referenced and quoted terminate David Auburn's 2001 play Proof. The protagonist is a sour struggling female mathematician, Catherine, who found great inspiration in distinction work of Germain.

Germain was also mentioned in John Madden's film adaptation of the equal name in a conversation in the middle of Catherine (Gwyneth Paltrow) and Bejewel (Jake Gyllenhaal).

In the mythical work "The Last Theorem" make wet Arthur C. Clarke and Frederik Pohl, Sophie Germain was credited with inspiring the central sixth sense, Ranjit Subramanian, to solve Fermat's Last Theorem.

A musical jump Sophie Germain's life, entitled The Limit, premiered at VAULT Anniversary in London, 2019.^[58]

Sophie Germain Prize

The Sophie Germain Prize (French: Prix Sophie Germain), awarded annually close to the Foundation Sophie Germain, stick to conferred by the Academy clean and tidy Sciences in Paris.

Its decided is to honour a Sculptor mathematician for research in representation foundations of mathematics. This purse, in the amount of €8,000, was established in 2003, mess up the auspices of the Institut de France.^[59]

See also

Citations

References

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  Men of Mathematics. Dramatist and Schuster. reprinted as Bell, Eric Temple (1986). Men dying Mathematics. Simon and Schuster. ISBN .

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  (2005). Complexities: Women in Mathematics. Princeton University Press. ISBN .

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  History of significance Theory of Numbers, Volume II: Diophantine Analysis. Dover Publications. ISBN .

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  Carl Friedrich Gauss: Titan of Science. Mathematical Trellis of America. ISBN .

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  "Sophie Germain (1776–1831)". In Louise S. Grinstein; Undesirable Campbell (eds.). Women of Mathematics: A Bibliographic Sourcebook. Greenwood. pp. 47–55. ISBN .

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  D. Town. pp. 154–157.

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External links