Sophie Germain's Early Contribution to the Elasticity Theory Marie-Sophie Germain (1776-1831) Was Born in Paris to a Family of the (Increas- Ingly) Rieh Bourgeoisie

HISTORICAL NOTE

Sophie Germain's Early Contribution to the Elasticity Theory Marie-Sophie Germain (1776-1831) was born in Paris to a family of the (increas- ingly) rieh bourgeoisie. She blossomed, in her teens, into a self-taught mathemati- 1 ' v [ J-'iy-J. cian. Her contributions to number theory, ! . Solutions to special cases of the Fermat theorem, are still remembered and refer- ■■ ;<: enced, along with a rieh harvest of anec-

dotes. Her contribution to the develop- K ment of the theory of elasticity is less V » known, though we owe her the first / / c mathematical treatment of the resonance .)_„.. l._ nodes of vibrating membranes. . Germain, by her own aecount, selected the study of mathematics to isolate her seif from mounting social unrest and i' political revolution. Her father had been elected as a representative of the bour geoisie—le tiers etat—to the congress of S - Versailles. It was the prohibition of the

meetings of this assembly that sparked t n tST—j the events of July 14, 1789—the French Revolution. Reportedly, when Germain Fy. i. 4* read an aecount, in her father's library, of Archimedes who was slaughtered by a Roman soldier during the siege of Siracusa because he was too engrossed in : a mathematical problem to notice the events, the young teenager singled out mathematics as a subjeet absorbing Sophie Germain's sketch of an elastic bar and its radius of curvature when bent by an exter enough to distract one's mind from the na! force, taken from her book on the theory of elastic surfaces (1821). unsettling events at hand. Unfortunately, the study of mathemat ics was inappropriate for a merchant's daughter. Germain's parents tried to curb Europe, produced written notes for stu which distributed themselves in fixed her worrisome activities by preventing dents, and requested that Solutions be patterns on membranes made to resonate her studies in every way. When she submitted to the problems. Germain with the arch of a violin. This experiment resorted to study at night, they hid her obtained these notes under the name of duly impressed Napoleon, who approved clothes, forbade a fire, and hid away can- Monsieur LeBlanc. She revealed herseif as a public contest for the theoretical dles. It is said that Germain would rise to a woman when Lagrange desired to meet description of these patterns, with a prize study mathematics in the dead of night, the Student who submitted such brilliant of 3,000 francs. The beginnings of the wrapped in bedeovers, while it was so Solutions. Under the same name of modern theory of elasticity did not thus cold that the ink would freeze in the LeBlanc she initiated a correspondence spring from interest in practical engineer- inkpot. Her family soon desisted. with Gauss, offering Solutions to some of ing problems, but from a desire of find- Although no institutions of higher educa- the problems of his Disquisitiones ing, with the new mathematical tools at tion would admit women as students, Arithmeticae, and asking for advice on hand (the variational methods of those were times when the laws of Soci how to proeeed with her studies. Gauss Lagrange), a Solution to physics problems. ety, and the role of women therein, were was pleased with his correspondent from The call, open for two years, reeeived only being rewritten. While Germain studied Paris, who was able to offer a partial one entry, by Sophie Germain. Gauss's Disquisitiones Arithmeticae, Paris proof of Fermat's theorem. In 1806 This entry contained a basic hypothesis, grew with political chat-rooms, the Napoleonic troops invaded Brunswick namely, that the elasticity was proportional salons, hosted by women. In 1789, Paris where Gauss resided. Germain recom- to the sum of the inverses of the prineipal also witnessed the 12-mile march by mended his safety to General Pernety, a radii of curvature of a surface. Unfor 6,000 women on the royal palace of friend of the family, and it is he who tunately, no proof was offered for the Versailles. This Amazon army, led by the revealed to Gauss the true identity of his hypothesis and the derivation of the subse- Belgian Theroigne de Mericourt, once a Paris correspondent. quently inferred equations was not correct, courtesan, was instrumental in bringing The appreciation for her work allowed so the prize was not awarded, but the con Louis XVI to Paris under direct control of Germain to be invited to attend the test was extended. Lagrange, who was a the French Citizens. demonstration which set the basis for member of the commission that judged the Although women had no possibility of interest in vibrating surfaces. The acousti- entry to the contest, offered the correct par reeeiving a formal education, the Ecole cian Chladni demonstrated in 1808 at the tial differential equation. However, he died Polytechnique, newly founded to consoli- French Academy of Sciences the vibrating two years later and Germain remained date the new role of France in positivist modes of plates with grains of sand, unable to derive Lagrange's equation by

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variational calculus. Her ability resided equation by Fourier, in the form of series travaux de Sophie Germain," Historia mostly in algebraic manipulations, and she of sines and cosines. Navier, a former Stu Math. 14 (4) (1987) p. 347; L.L. BucdareUi tried to generalize the work of Euler on the dent of Fourier, offered a new expression and N. Dworsky, Sophie Germain: An Essay elasticity of bars, extending it to two for the elastic moment: It was proportion in the History of the Theory of Elasticity (D. dimensions. Germain related elasticity to al to an elastic constant, the cube of the Reidel, Dordrecht, 1980); S. Germain, the sum of the principal curvatures of a plate's thickness (instead of the fourth Recherches sur la theorie des surfaces elas- surface. power of Germain's), and also contained tiques, par Sophie Germain (Mme. V. The objective of the contest was to offer the product of the principal curvatures. Courcier, Paris, 1821); H. Stupuy, "Notice a mathematical explanation oi Chladni's This research in turn stimulated Cauchy sur la vie et les ouvres de Sophie experiment, and Germain successfully to define how the stress of an elastic plate Germain," Oeuvres philosophique de Sophie experimented with laminae. In her sec- depended on the applied strain in terms Germain (Paul Ritti, Paris, 1879), pp. 1-92. ond entry (again, the only one) to the of tensors while introducing a second extended contest, she modified her equa elastic constant. Cauchy, and subsequent- tion and proved that it was correct in a ly Kirchhoff, set the foundations of the number of special cases. The experimen- present theory of elasticity. tal part earned her an honorable mention These were Germain's final years. She from the jury, and led her to publish her fought a Iosing battle against breast Cancer, work—at author's cost—in 1821. Another while continuing to write her thoughts Christine Mirzayan Stimulus to publication was the fact that over the sciences and the arts, "Real superi- Poisson was devoting his attention to the ority is nothing more than the means of Internship Program problem of elasticity, offering another considering difficult problems from a point approach and yet another equation, and of view whence they become easy, where of the National consistently avoided to duly acknowl- the spirit can embrace them and follow edge Germain's preceding and ongoing them without effort." Germain did not live Academies work. Partly to patch up this breach of long enough to ripen the fruits of Gauss's WASHINGTON, DC confidentiality, the contest was called request to the University of Göttingen to again, with a gold medal as a prize. The award her an honorary doctorate. « gold medal was awarded to Germain, CRISTINA P. TANZI but, to the annoyance of those attending The Christine Mirzayan Internship the public ceremony, she decided not to FOR FURTHER READING: A.D. Dalmedico, Program of the National appear in public. "Sophie Germain," Scientific American Academies is designed to engage In 1828, a new approach was given to Dec. (1991) p. 117; A.D. Dalmedico, graduate and postdoctoral science, the Solution of the plate equation: an "Mecanique et theorie des surfaces: les engineering, and law students in science and technology policy and to familiarize them with the inter- Need Nanomechamcal and actions between science, technolo gy, and government. Nanotribologiccd Testing Services? For the year 2000, the internship Nanomechanics Research Laboratory program will commence in both Nanoscale Mechanical and Tribological Characterization January (for 12 weeks, January 18-April 7) and June (for 10 • Nanoindentation: hardness, modulus, fracture toughness weeks, June 5-August 11). • Nanoscratch: critical load, delamination, interfacial adhesion To apply, candidates should • Nanotribological test: lateral force, friction coefficient, submit the application and one wear, and lubrication effect letter of reference. See Website: national-academies.org/ =iipge: "FTfrq Product Evaluation and Failure Analysis m m internship. • Nanomechanical and tribological evaluation Uli.- ,-H ;: The deadline for receipt of materi- • Failure mechanism analysis als is December 1, for the January • Root cause identification 2000 program and March 1, for • Product design improvement Suggestion the June 2000 program.

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