RITHMS.) The subject with which Legendre's name is most closely connected is elliptic functions, his researches extending over more than forty years. The last of the three supplements to his Traite des fonctions elliptiques was published in 1832. He died at Paris on Jan. so, 1833, and the discourse at his grave was pro nounced by Poisson.
J. Jacobi, were published in 1828-1832, and form a third volume.
Legendre had pursued the subject of elliptic integrals alone from 1786 to 1827, the results of his labours having been almost entirely neglected by his contemporaries, but his work had scarcely ap peared in 1827 when the discoveries which were independently made by the two young and as yet unknown mathematicians Abel and Jacobi placed the subject on a new basis, and revolutionized it completely. The readiness with which Legendre, who was then seventy-six years of age, welcomed these important researches, that quite overshadowed his own, and included them in successive supplements to his work, does the highest honour to him. (See FUNCTION.) Legendre also made important researches in gamma f unctions.
Theory of Numbers.—Legendre's Theorie des nombres and Gauss's Disquisitions arithmeticae (1801) are still standard works upon this subject. The first edition of the former appeared in 1798 under the title Essai sur la theorie des nombres; there was a second edition in 18°8; a first supplement was published in 1816, and a second in 1825. The third edition, under the title Theorie des nombres, appeared in 1830 in two volumes. To Legen dre is due the theorem known as the law of quadratic reciprocity, the most important general result in the science of numbers which has been discovered since the time of Fermat, and which was called by Gauss the "gem of arithmetic." It was first given by Legendre in the .Memoires of the Academy for 1785, but the accompanying demonstration was incomplete. (See NUMBER.)
Attractions of Ellipsoids.—Legendre was the author of four important memoirs Qn this subject. In the first of these, entitled "Recherches sur l'attraction des spheroides homogenes," pub lished in the Memoires of the Academy for 1785, Legendre intro duces the celebrated expressions which, though frequently called Laplace's coefficients, are more correctly named after Legendre. Legendre shows that Maclaurin's theorem with respect to confocal ellipsoids is true for any position of the external point when the ellipsoids are solids of revolution. (See Todhunter's History of the Mathematical Theories of Attraction and the Figure of the Earth [1873], the twentieth, twenty-second, twenty-fourth and twenty-fifth chapters of which contain a full and complete account of Legendre's four memoirs. See also SPHERICAL HARMONICS.) Geodesy.—Besides the work upon the geodetical operations con necting Paris and Greenwich, of which Legendre was one of the authors, he published in the Memoires de l'Academie for 1787 two papers on trigonometrical operations depending upon the figure of the earth, containing many theorems relating to this subject. The best known of these, which is called Legendre's theorem, is usually given in treatises on spherical trigonometry.
Method of Least Squares.—In 1806 appeared Legendre's Nouvelles Mithodes pour la determination des orbites des cometes, which is memorable as containing the first published suggestion of the method of least squares. (See PROBABILITY.) The Elements of Geometry.—Legendre's name is widely known on account of his Elements de geometrie. It first appeared in 1794, and went through very many editions, and has been trans lated into almost all languages. An English translation, by Sir David Brewster, from the eleventh French edition, was published in 1823. In one of the notes Legendre gives a proof of the irra tionality of 7r. This had been first proved by J. H. Lambert in the Berlin Memoirs for 1768. Legendre's proof is similar in prin ciple to Lambert's, but much simpler. On account of the objec tions urged against the treatment of parallels in this work, Legendre was induced to publish in 1803 his Nouvelle Theorie des paralleles.
An account of the principal works of Legendre is given in the Bibliotheque universelle de Geneve for 1833, PP. 45-82.