CLAIRAUT, A.C.(1713-1765) D'ALEMBERT, JEAN-LE-ROND(1717-1783) and LAMBERT, J.H.(1728-1777)

	Alexis Chaude Clairaut was born in Paris in 1713 and died there in 1765. He was a youthful mathematical prodigy, composing in his eleventh year a treatise on curves of the third order. This early paper, and a singularly elegant subsequent one on the differential geometry of twisted curves in space, won him a seat in the French Academy of Sciences at the illegal age of
eighteen. In 1736. he accompanied Pierre Louis Moreau de Maupertuis(1698-1759) on an expedition to Lapland to measure the length of a degree of one of the earth's meridians. The expedition was undertaken to settle a dispute as to the shape of the earth. Newton and Huygens had concluded, from mathematical theory, that the earth is flattened at the poles, The measurement made in Lapland unquestionably confirmed the Newton-Huygens belief and earned Maupertuis the title of "earth flattner." In 1743, after his return to France, Clairaut published his definitive work, Theorie dela figure de la Terre. Clairaut had a brother, three years his junior and known in the history of mathematics only as "le cadet Vlairaut" (1716-1732), who tragically died of smallpox when only sixteen, but who at fourteen read a paper on geometry before the French Academy and at fifteen published a work on geometry. The father of the Clairaut children, Jean Baptiste Clairaut (died soon after 1765), was a teacher of mathematics, a correspondent of the Berlin Academy, and a writer on geometry; he had twenty children of whom only one survied him.
	Jean-le-Rond d'Alembert (1717-1783), like Alexis Clairaut was born in Paris and died in Paris. As a newborn he was abandeoned near the church of Saint Jean-le-Rond and was discovered there by a gendarme who had him gurriedly christened with the name of the place where he was found. Later for reasons not known, the name d'Alembert was added. A scientific rivalry, often unfiriendly, existed between d'Alembert and Clairaut. At the age of twenty-four, d'Alembert was admitted to the French Academy. In 1743, he
published his Traite de dynamique, based upon the great principle of kmetics that now bears his name. It says that the internal actions and reactions of a system of rigid bodies in motion are in equilibrium. D's Alembert showed interest in the foundations of analysis. In 1754, he made the important suggestion that a sound theory of limits was needed to put analysis on a firm foundation, but most of his contemporaries paid little heed to his suggestion. D'Alembert worked so diligently in an effort to prove the fundamental theorem of algebra (that every polynomial equation f(x) = 0 having complex coefficients and of degree n ￣ I has at least one complex root) that the theorem is today known in france as d'Alembert's theorem. D'alembert, like Euler, was broadly educated, with especial knowledge in law, medicine, mathematics, and scence. Sharing many common interests, the two men corresponded with one another on a number of matters. In 1754, D'Alembert became permanent secretary of the French Academy. During his later years, he worked on the great French Encyclopedie, which had been begun by Denis Diderot and himself. D'Alembert died in 1783, the same year in which Euler died. A famous and oft-quoted remark made by D'Alembert (and welf worth citing on occasion in an elementary algebra class)is: "Algebra is generous;she often gives more than is asked of her," He also once aptly remarked: "Geometrical truths are in a way asymptotic to physical truths; that is to say, the latter approach the former indefinitely near without ever reaching them exactly." Perhaps the most perceptive of D'Alembert's comments on mathematics is the following: "I have no doubt that if men lived separate from each other, and could in such a situation occupy themselces about anything but self-preservation, they would prefer the study of the exact sciences to the cultivation of the agreeable arts. It is chiefly on account of others that a man aims at excellence in the latter; it is on his own account that he devotes himself to the former. In a desert island, accordingly, I shoul think that a poet could scarcely be vain, whereas a mathematician might still enjoy the pride of discovery,"
	A little younger than Clairaut and D'Alembert was Johann Heinrich Lambert (1728-1777), born in Mulhouse (Alsace), then part of Swiss territory. Lambert was a mathematician of high quality. As the son of a poor tailor, he was largely self-taught. He possessed a fine imagination, and he established his results with great attention to rigor. In fact, Lambert was the first to prove rigorously that the number 休 is
irrational. Lambert was a many-sided scholar who made noteworthy contributions to the mathematics of numerous other topics, such as descriptive geomentry, the determination of comet orbits, and the theory of projections employed in the making of maps (a much-used one of these projections is now named after him). At one time, he considered plans for a mathematical logic of the sort once outlined by Leibniz. In 1766, he wrote his posthumously published investigation of Euclid's parallel postulate entitled Die Theorie der Parallellinien, a work that places him among the forerunners of the discovery of non-Euclidean geometry (see Section 13-7). For a short time, Lambert was an associate of Euler in the Prussian Academy. It has been said that when Frederick the Great once inquired of Lambert in which science he was most competent, Lambert curtly replied, "All." Lambert died in 1777, the year in which Carl Freidrich Gauss was born.