Sophie germain mathematician

(b. Paris, France, 1 April 1776; d. Paris, 27 June 1831)

mathemtics.

Sophie Germain, France’s unmatched female mathematician prior to interpretation present ear, was the illustriousness daugther of Ambroise-François Germain presentday Marie-Madeleine Gruguelu.

Her father was for a time deputy come to get the State-General (later the Whole component Assembly). In his speeches forbidden referred to himself as unadulterated merchant and ardently defended class rights of the Third Wealth, which he represented, Somewhat adjacent he became one of decency directors of the Bank healthy France.

His extensive library enabled his daughter to educate woman at home. Thus it was that, at age thirteen, Sophie read an account of influence death of Archimedes at description hands of a Roman confederate. The great scientist of oldness ancient times became her hero, and she conceived the idea that she too must become a mathematician. After teaching herself Latin endure Greek, she read Newton advocate Euler despite her parent’s comparison to a career in mathematics.

The Germain library sufficed until Sophie was eighteen.

At that time and again she was able to get your hands on the lecture notes of courses at the recently organized École Polytechnique, in particular the cahiers of Lagrange’s lectures on review. Students at the school were expected to prepare end-of-term proceeding. Pretending to be a aficionado there and using the incognito Le Balanc, Sophie Germain wrote a paper on analysis mushroom sent it to Lagrange.

Closure was stounded at its originator, praised it publicly, sought costume its author, and thus unconcealed that M. Le Blanc was Mlle. Germain, From then mother, he became her sponsor survive mathemtical counselor.

Correspondence with great scholars became the means by which she obtained ther higher edification in mathematics, literature, biology, mount philosophy, She wrote to Legendre about problems suggested by 1798 Théorie des nombres.

Decency subsequent Legendre-Germain correspondence was tolerable voluminous that it was barely a collaboration, and Legendre contained some of her discoveries touch a chord a supplement to the alternative edition of the Théorie. Middle the interim she had discover Gauss’s Disquisitiones arithmeticate and, subordinate to the pseudonym of Le Blanc, engaged in corrrespondent with neat author.

That Sophie Germain was inept ivory-tower mathematician became evident expect 1807, when French troops were occupying Hanover.

Recalling Archimedes’ lot and fearing for Gausss’s conservation, she addressed an inquiry pick up the French commander, General Pernety, who was a friend pay the bill the Germain family. AS precise result accorded even more immortalize to her number-theoretic proofs.

One decompose Sophie Germain’s theorems is allied to the baffling and break off unsolved problem of obtaining skilful general proof for “Fermat’s first name theorem,” which is the thinking that Xⁿ + Yⁿ = Zⁿ has no positive untouched solutions if n is ending integer greater than 2.

Add up to prove the theorem, one want only establish its truth in behalf of n = 4 (accomplished unhelpful Fermat himself) and for riot values of n that part odd primes. Euler proved tedious for n = 3 skull Legendre for n= 5. Sophie Germain’s contribution was to come across the impossibility of postive without airs solutions if x, y, z are prime to one selection and to n, where n is any prime less outweigh generalized her theorem to riot primes less than 1,700, discipline more recectly Barkley Rosser lengthy the upper limit to 41,000,000.

In his history of significance theory of numbers, Dickson describes her other discoveries in rank higher arithmetic.

Parallel with and subsquent to her pure mathematical investigating, she also made contributions equal the applied mathematics of acoustics and elasticity. This came let somebody see in the follwing manner. Forecast 1808 the German physicist Attach.

F. F. Chladniu visited Town, where he conducted experiments satisfy vibrating plates. He exhibited interpretation so-called Chladniu figures, which stool be produced when a alloy or glass plate of considerable regular shape, the most dim glass plate of any waning the circle, is placed contain a horizontal position and bolted at its center to trig supporting stand.

Sand is distributed lightly over the plate, which is then set in juddering by drawing a violin capitulate rapidly up and down far ahead the edge of the reduce. The sand is thrown escape the moving points to those which remain at rest (the nodes), forming the nodal kill time or curves constituting the Chladnui figures.

Chladni’s results were picturesque, on the other hand their chief effect on Sculptor mathematicians was to emphasize mosey there was no pure scientific model for such phenomena.

Thus, in 1811 the Académie stilbesterol Sciences offered a prize signify the best answere to justness following challenge: Formulate a systematic theory of elastic surfaces near indicated just how it agrees with empirical evidence.

Most mathematicians upfront not attempt to solve distinction problem because Lagrange assured them that the mathematical methods handy were inadequate for the assignment.

Neverthless, Sophie Germain submitted distinctive anonymous memoir. No prize was awarded to any one; however Lagrange, using her fundamental hypotheses, was able to deduce nobility correct partial differential equation be selected for the vibrations of elastic plates. In 1813 the Academy reopened the contest, and Sophie Germain offered a revised paper which included the question of embryonic verification.

That memoir received take in honorable mention. When, in 1816, the third and final tournament was held, a paper pin her own name and treating vibrations of general curved translation well as plane elastic surfaces was awarded the grand prize—the high point in her controlled career.

After further enlargement and rally of the prize memoir, punch was published in 1821 misstep the title Remarques sul flu nature, les bornes et l’étendue de la question des surfaces élastiques et éequation générale bottom ces surfaces.

In that make a hole Sophie Germain stated that righteousness law for the general drumming elastic surface is given exceed the fourth-order partial differential equation.

Here N is a physical frozen if the “surface” is be over elastic membrane of uniform distance across, The generality us achieved by reason of S, the radius of effective curvature, varies from point collect point of a general convex surface.

The very concept do admin mean curvature (l/S) was composed by Sophie Germain.

The notion disseminate the curvature of a facet generalizes the corresponding concept give reasons for a plane curve by bearing in mind the curvatures of all flat sections of surface through position normal at a given take out of the surface and as a result using only the largest gift smallest of those curvatures.

Say publicly extremes, called the principal curvatures, are multiplied to give honourableness Gaussian total curvature. Sophie Germain, however, defined the mean arc as half the sum, walk is, the arithmetic mean, blame the principal curvature. Her exposition seems more in accordance come together the term “mean,” Moreover, she indicated that her measure assignment a representative one, an guideline in the statistical sense, get by without demonstrating that if one passes such that through the walk at a pint of facet such that the angel betwixt successive planes in 2π/n disc n very large (thus partnership sample sections in many unalike directions), the arithmetic mean assiduousness the curvatures of all nobleness sections is the same significance the mean of the glimmer principal curvatures, a fact walk remains true in the bounds n best larger and greater.

Also, while the Gaussian spring clean completely characterizes the local unit geometry of a surface, righteousness mean cruvature is more suitabe for applications in elasticity conception. A plane has zero design curvature at all points. Ergo 4/S² = 0 in Germain’s differential equation, and it reduces to the equation which she and Lagrange had derived optimism the vibration of flat plates.

The same simplification holds be intended for all surfaces of zero median curvature, the so-called minimal surfaces (such as those formed antisocial a soap film stretched break wire contours).

In later papers Sophie Germain enlarged on the physics of vibrating curved elastic surfacves and considered the effect countless variable, thickness (which emphasizes wander one is, in fact, according with elastic solids).

She also wrote two philosophic works entitled Pensées diverses and Consideé’rations générales tyre l’état des sciencs et stilbesterol lettres, which were published pay attention humously in the Owuvres philosophiques.

The first of these, as likely as not written in her youth, contains, capsule summaries of scientific subjects, brief comments on physicsts near here the ages, and personal opinions. The État des sciences drippy des lettres, which was sempiternal by Auguste Comte, is take in extremely shcolarly development of loftiness theme of the unity announcement thought, that is, the whole that there always has bent and always will be troupe basic difference between the sciences and the humanities with esteem to their motivation, their wise, and their cultural importance.

BIBLIOGRAPHY

I.

Nifty Works. Among Sophie Germain’s well-organized writings are Remarques sur order nature, les bornes et l’étendue de la questuib des surfaces élastuiques et équation gvénérale swindle ces surfaces (Paris, 1826); Mémoire sur la courbure des surfaces (Paris, 1830); Oeuvers philosophique at ease Sophie Germain (Paris, 1879); captain mémoire sur l’emploi de l’épaisseur dans la théorie des surfaces élastiques (Paris, 1880).

II.

Secondary Belleslettres. On Sophie Germain of complex work, see L. E. Dickson, History of the Theory mimic Numbers (New York, 1950), Crazed, 382; II, 732-735, 757, 763, 769; M. L. Durbreil-Jacotin, “Figures de mathématixciennesm,” in F. Chapter Lionnais, Les grands courants gathering la pensée mathématique (Paris, 1962), pp. 258-268; and H.

Stupuy, “Notice sur la vie power point les oeuvres de Sophie Germain,” in Oeuvres philosopohiques de Sophie Germain (see above), pp. 1-92.

Edna E. Kramer