Marie-Sophie Germain (; 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 Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ...

, and from correspondence with famous mathematicians such as

Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaLegendre, 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 The French Academy of Sciences (French: ''Académie des sciences'') is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research. It was at t ...

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. Because of prejudice against her sex, she was unable to make a career out of mathematics, but she worked independently throughout her life. Before her death, Gauss had recommended that she be awarded an honorary degree, but that never occurred. 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 honor.

Early life

Family

Marie-Sophie Germain was born on April 1, 1776, in Paris, France, in a house on Rue Saint-Denis. According to most sources, her father, Ambroise-François, was a wealthy silk merchant, though some believe he was a

goldsmith A goldsmith is a metalworker who specializes in working with gold and other precious metals. Nowadays they mainly specialize in jewelry-making but historically, goldsmiths have also made silverware, platters, goblets, decorative and servicea ...

. In 1789, he was elected as a representative of the

bourgeoisie The bourgeoisie ( , ) is a social class, equivalent to the middle or upper middle class. They are distinguished from, and traditionally contrasted with, the proletariat by their affluence, and their great cultural and financial capital. Th ...

to the États-Généraux, which he saw change into the

Constitutional Assembly A constituent assembly (also known as a constitutional convention, constitutional congress, or constitutional assembly) is a body assembled for the purpose of drafting or revising a constitution. Members of a constituent assembly may be elected b ...

. 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. Marie-Sophie had one younger sister, named Angélique-Ambroise, and one older sister, named 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).

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 Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientis ...

intrigued her. Germain thought that if the geometry method, which at that time referred to all of pure mathematics, could hold such fascination for Archimedes, it was a subject worthy of study. 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 Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...

. She also enjoyed by

Étienne Bézout Étienne Bézout (; 31 March 1730 – 27 September 1783) was a French mathematician who was born in Nemours, Seine-et-Marne, France, and died in Avon (near Fontainebleau), France. Work In 1758 Bézout was elected an adjoint in mechanics of the ...

and by Jacques Antoine-Joseph Cousin. Later, Cousin visited Germain at home, encouraging her in her studies. 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. After some time, her mother even secretly supported her.

École Polytechnique

In 1794, when Germain was 18, the

École Polytechnique École may refer to: * an elementary school in the French educational stages normally followed by secondary education establishments (collège and lycée) * École (river), a tributary of the Seine flowing in région Île-de-France * École, Savoi ...

opened. As a woman, Germain was barred from attending, but the new system of education made the "lecture notes available to all who asked". The new method also required the students to "submit written observations". 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, "fearing", as she later explained to Gauss, "the ridicule attached to a female scientist". 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, and he became her mentor.

Early work in number theory

Correspondence with Legendre

Germain first became interested in

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...

in 1798 when

Adrien-Marie Legendre Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are name ...

published . After studying the work, she opened correspondence with him on number theory, and later, elasticity. Legendre included some of Germain's work in the to his second edition of the , where he calls it ("very ingenious"). See also Her work on Fermat's Last Theorem below.

Correspondence with Gauss

Germain's interest in number theory was renewed when she read

Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...

' monumental work . After three years of working through the exercises and trying her own proofs for some of the theorems, she wrote, again under the pseudonym of M. Le Blanc, to the author himself, who was one year younger than she. The first letter, dated 21 November 1804, discussed Gauss' 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 A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

of the form ''p'' = 8''k'' + 7. However, her proof contained a weak assumption, and Gauss' reply did not comment on Germain's proof. Around 1807 (sources differ), during the Napoleonic wars, the French were occupying the German town of

Braunschweig Braunschweig () or Brunswick ( , from Low German ''Brunswiek'' , Braunschweig dialect: ''Bronswiek'') is a city in Lower Saxony, Germany, north of the Harz Mountains at the farthest navigable point of the river Oker, which connects it to the ...

, where Gauss lived. Germain, concerned that he might suffer the fate of Archimedes, wrote to General Pernety (), a family friend, requesting that he ensure Gauss' safety. General Pernety sent the chief of a battalion to meet with Gauss personally to see that he was safe. As it turned out, Gauss was fine, but he was confused by the mention of Sophie's name. Three months after the incident, Germain disclosed her true identity to Gauss. He replied:

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 umber theory'sknotty problems, yet overcomes these fetters and penetrates that which is most hidden, she doubtless has the noblest courage, extraordinary talent, and superior genius.

Gauss' letters to Olbers show that his praise for Germain was sincere. In the same 1807 letter, Germain claimed that if

x^n + y^n

is of the form

h^2 + nf^2

, then

x + y

is also of that form. Gauss replied with a counterexample:

15^ + 8^

can be written as

h^2 + 11 f^2

, but

15 + 8

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

Work in elasticity

Germain's first attempt for the Academy Prize

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 Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...

deterred all but two contestants, Denis Poisson and Germain. Then Poisson was elected to the Academy, thus becoming a judge instead of a contestant, and leaving Germain as the only entrant to the competition. In 1809 Germain began work. Legendre assisted by giving her equations, references, and current research. 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". Lagrange was able to use Germain's work to derive an equation that was "correct under special assumptions".

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. Germain's anonymous 1813 submission was still littered with mathematical errors, especially involving double integrals, and it received only an honorable mention because "the fundamental base of the theory f elastic surfaceswas not established". The contest was extended once more, and Germain began work on her third attempt. This time she consulted with Poisson. 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). Germain submitted her third paper, "", under her own name, and on 8 January 1816 she became the first woman to win a prize from the Paris Academy of Sciences. She did not appear at the ceremony to receive her award. Although Germain had at last been awarded the , the Academy was still not fully satisfied. Germain had derived the correct

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...

(a special case of the Kirchhoff–Love equation), but her method did not predict experimental results with great accuracy, as she had relied on an incorrect equation from Euler, which led to incorrect boundary conditions. Here is Germain's final equation for the vibration of a plane lamina: :

N^2\left(\frac + 2\frac + \frac\right) + \frac = 0,

where ''N''² is a constant. 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 Jean-Baptiste Joseph Fourier (; ; 21 March 1768 – 16 May 1830) was a French mathematician and physicist born in Auxerre and best known for initiating the investigation of Fourier series, which eventually developed into Fourier analysis and ha ...

, a secretary of the Academy, who obtained tickets to the sessions for her.

Later work in elasticity

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. 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 Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. H ...

, who had been appointed to review her work, recommended her to publish it, and she followed his advice. One further work of Germain's on elasticity was published posthumously in 1831, her "". She used the mean curvature in her research (see Honors in number theory).

Later work in number theory

Renewed interest

Germain's best work was in number theory, and her most significant contribution to number theory dealt with Fermat's Last Theorem. In 1815, after the elasticity contest, the Academy offered a prize for a proof of Fermat's Last Theorem. It reawakened Germain's interest in number theory, and she wrote to Gauss again after ten years of no correspondence. 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. She outlined a strategy for a general proof of Fermat's Last Theorem, including a proof for a special case. 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.

Her work on Fermat's Last Theorem

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 In number theory, Sophie Germain's theorem is a statement about the divisibility of solutions to the equation x^p + y^p = z^p of Fermat's Last Theorem for odd prime p. Formal statement Specifically, Sophie Germain proved that at least one of the ...

Let ''p'' be an odd prime. If there exists an auxiliary prime ''P'' = 2''Np'' + 1 (''N'' is any positive integer not divisible by 3) such that: # if ''x''^''p'' + ''y''^''p'' + ''z''^''p'' ≡ 0 ( mod ''P''), then ''P'' divides ''xyz'', and # ''p'' is not a ''p''-th power residue (mod ''P''). Then the first case of Fermat's Last Theorem holds true for ''p''.

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". 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. In an unpublished manuscript titled , Germain showed that any counterexamples to Fermat's theorem for ''p'' > 5 must be numbers "whose size frightens the imagination", around 40 digits long. Germain did not publish this work. Her brilliant theorem is known only because of the footnote in Legendre's treatise on number theory, where he used it to prove Fermat's Last Theorem for ''p'' = 5 (see Correspondence with Legendre). Germain also proved or nearly proved several results that were attributed to Lagrange or were rediscovered years later. Del Centina states that "after almost two hundred years her ideas were still central", but ultimately her method did not work.

Work in philosophy

In addition to mathematics, Germain studied philosophy and

psychology Psychology is the science, scientific study of mind and behavior. Psychology includes the study of consciousness, conscious and Unconscious mind, unconscious phenomena, including feelings and thoughts. It is an academic discipline of immens ...

. 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 Isidore Marie Auguste François Xavier Comte (; 19 January 1798 – 5 September 1857) was a French philosopher and writer who formulated the doctrine of positivism. He is often regarded as the first philosopher of science in the modern sense ...

. Two of her philosophical works, and , were published, both posthumously. This was due in part to the efforts of Lherbette, her nephew, who collected her philosophical writings and published them. is a history of science and mathematics with Germain's commentary. In , the work admired by Comte, Germain argues that there are no differences between the sciences and the

humanities Humanities are academic disciplines that study aspects of human society and culture. In the Renaissance, the term contrasted with divinity and referred to what is now called classics, the main area of secular study in universities at t ...

Final years

In 1829 Germain learned that she had breast cancer. Despite the pain, she continued to work. In 1831 ''

Crelle's Journal ''Crelle's Journal'', or just ''Crelle'', is the common name for a mathematics journal, the ''Journal für die reine und angewandte Mathematik'' (in English: ''Journal for Pure and Applied Mathematics''). History The journal was founded by Augus ...

'' published her paper on the

curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the can ...

of elastic surfaces and "a note about finding and in

\tfrac = y^2 \pm pz^2

". Mary Gray records: "She also published in an examination of principles which led to the discovery of the laws of equilibrium and movement of elastic solids." On 27 June 1831, she died in the house at 13 rue de Savoie. Despite Germain's intellectual achievements, her death certificate lists her as a "" (property holder), not a "". But her work was not unappreciated by everyone. When the matter of honorary degrees came up at the

University of Göttingen The University of Göttingen, officially the Georg August University of Göttingen, (german: Georg-August-Universität Göttingen, known informally as Georgia Augusta) is a public research university in the city of Göttingen, Germany. Founded ...

in 1837—six years after Germain's death—Gauss lamented: "she ermainproved 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".

Honors

Memorials

Germain's resting place in the

Père Lachaise Cemetery Père Lachaise Cemetery (french: Cimetière du Père-Lachaise ; formerly , "East Cemetery") is the largest cemetery in Paris, France (). With more than 3.5 million visitors annually, it is the most visited necropolis in the world. Notable figure ...

in Paris is marked by a gravestone. 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. The school houses a bust commissioned by the Paris City Council. In January 2020, Satellogic, a high-resolution

Earth observation Earth observation (EO) is the gathering of information about the physical, chemical, and biological systems of the planet Earth. It can be performed via remote-sensing technologies (Earth observation satellites) or through direct-contact sensors ...

imaging and analytics company, launched a ÑuSat type micro-satellite named in honor of Sophie Germain.

Honors in number theory

E. Dubouis defined a ''sophien'' of a prime to be a prime where , for such that yield such that has no solutions when and are prime to . A Sophie Germain prime is a prime such that is also prime. The ''Germain curvature'' (also called mean curvature) is

(k_1 + k_2)/2

, where and are the maximum and minimum values of the normal curvature. ''Sophie Germain's identity'' states that for any }), 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 Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathe ...

. This award, in the amount of €8,000, was established in 2003, under the auspices of the

Institut de France The (; ) is a French learned society, grouping five , including the Académie Française. It was established in 1795 at the direction of the National Convention. Located on the Quai de Conti in the 6th arrondissement of Paris, the institut ...

Citations

References

* reprinted as *Bucciarelli, Louis L; Dworsky, Nancy (1980). ''Sophie Germain: An Essay in the History of the Theory of Elasticity'', D. Reidel:Holland * * * * * * Reprinted as * Reprinted as * * * * * * * * * *

External links

* * *
Sheroes of History; Sophie Germain
at the Kids Love Science project {{DEFAULTSORT:Germain, Sophie 1776 births 1831 deaths 18th-century French mathematicians 19th-century French mathematicians 18th-century French women scientists 19th-century French women scientists French women mathematicians Number theorists Deaths from breast cancer Deaths from cancer in France French physicists French women physicists French women philosophers