He died at Paris on January 10, 1833, in his eighty-first year, and the discourse at his grave was pronounced by Poisson. He was engaged in mathematical investigations almost up to the time of his death. The last of the three supplements to his Traité des Fonctions Elliptiques was published in 1832, and Poisson in his funeral oration remarked—“M. Legendre a eu cela de commun avec la plupart des géomètres qui l’ont précédé, que ses travaux n’ont fini qu’ avec sa vie. Le dernier volume de nos mémoires renferme encore un mémoire de lui, sur une question difficile de la théorie des nombres; et peu de temps avant la maladie qui l’a conduit au tombeau, il se procura les observations les plus récentes des comètes à courtes périodes, dont il allait se servir pour appliquer et perfectionner ses méthodes.”

Legendre was the author of separate works on elliptic functions, the integral calculus, the theory of numbers, and the elements of geometry, besides numerous papers which were published chiefly in the Mémoires of the French Academy; and it will be convenient, in giving an account of his writings, to consider them under the different subjects which are especially associated with his name.

Elliptic Functions. — This is the subject with which Legendre’s name will always be most closely connected, and his researches upon it extend over a period of more than forty years. His first published writings upon the subject consist of two papers in the Mémoires of the French Academy for 1786 upon elliptic arcs. In 1792 he presented to the Academy a memoir on elliptic transcendents. The contents of these memoirs are included in the first volume of his Exercices de Calcul Intégral (1811). The third volume (1816) contains the very elaborate and now well-known tables of the elliptic integrals which were calculated by Legendre himself, with an account of the mode of their construction. In 1827 appeared the Traité des fonctions elliptiques (2 vols., the first dated 1825, the second 1826); a great part of the first volume agrees very closely with the contents of the Exercices ; the tables, &c., are given in the second volume. Three supplements, relating to the researches of Abel and Jacobi, were published in 1828-32, and form a third volume. Legendre had pursued the subject which would now be called 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 appeared 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. The sudden occurrence, near the close of his long life, of these great discoveries relating to a subject which Legendre had so completely made his own and apparently exhausted, and their ready acceptance by him, form one of the most striking episodes in the history cf mathematics. A very full account of the contents of Legendre’s work and of the results obtained by Abel and Jacobi has been given in the article Infinitesimal Calculus, vol. xiii. pp. 62-72. See also Leslie Ellis’s report “On the Recent Progress of Analysis,” in the Report of the British Association for 1846 (pp. 44 sq.). Ealcrian Integrals and Integral Calculus.— The Exercices de Calcul Intégral consist of three volumes, a great portion of the first and the whole of the third being devoted to elliptic functions. The remainder of the first volume relates to the Eulerian integrals and to quadratures. The second volume (1817) relates to the Eulerian integrals, and to various integrals and series, developments, mechanical problems, &c., connected with the integral calculus; this volume contains also a numerical table of the values of the gamma function. The latter portion of the second volume of the Traité des Fondions Elliptiques (1826) is also devoted to the Eulerian integrals, the table being reproduced. Legendre’s researches connected with the gamma function are of importance, and are well known; the subject was also treated by Gauss in his memoir Disquisitiones generales circa series infinitas (1816), but in a very different manner. The results given in the second volume of the Exercices are of too miscellaneous a character to admit of being briefly described. In 1788 Legendre published a memoir on double integrals, and in 1809 one on definite integrals.

Theory of Numbers.— Legendre’s Théorie des Nombres and Gauss’s Disquisitiones Arithmeticae (1801) are still the standard works upon this subject. The first edition of the former appeared in 1798 under the title Essai sur la Théorie des Nombres ; there was a second edition in 1808; a first supplement was published in 1816, and a second in 1825. The third edition, under the title Théorie des Nombres, appeared in 1830 in two volumes. To Legendre 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 Mémoires of the Academy for 1785, but the demonstration that accompanied it was incomplete. The symbol which is known as Legendre’s symbol, and denotes the positive or negative unit which is the remainder when α*0 ’ ^ 1 > is divided by a prime number p, does not appear in this memoir, but was first used in the Essai sur la Théorie des Nombres. Legendre’s formula x : (log x -1 - 08366) for the approximate number of forms inferior to a given number x was first given by him also in this work (2d ed., p. 394).

Attractions of Ellipsoids.— Legendre was the author of four important memoirs on this subject. In the first of these, entitled “Recherches sur l’attraction des sphéroïdes homogènes,” published in the Mémoires of the Academy for 1785, but communicated to it at an earlier period, Legendre introduces the celebrated expressions which, though frequently called Laplace’s coefficients, are more correctly named after Legendre. The definition of the coefficients is that if (1-2⅛cos0 + Λ 2 )~ i be expanded in ascending powers of h, and if the general term be denoted by P n h n , then B n is of the Legendrian coefficient of the nth order. In this memoir also the function which is now called the potential was, at the suggestion of Laplace, first introduced. 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. Of this memoir Todhunter writes—‘ ‘ We may affirm that no single memoir in the history of our subject can rival this in interest and importance. During forty years the resources of analysis, even in the hands of D’Alembert, Lagrange, and Laplace, had not carried the theory of the attraction of ellipsoids beyond the point which the geometry of Maclaurin had reached. The introduction of the coefficients now called Laplace’s, and their application, commence a new era in mathematical physics.” Legendre’s second memoir was communicated to the Academy in 1784, and relates to the conditions of equilibrium of a mass of rotating fluid in the form of a figure of revolution which does not deviate much from a sphere. The third memoir relates to Laplace’s theorem respecting confocal ellipsoids. Of the fourth memoir Todhunter writes, “It occupies an important position in the history of our subject. The most striking addition which is here made to previous researches consists in the treatment of a planet supposed entirely fluid; the general equation for the form of a stratum is given for the first time and discussed. For the first time we have a correct and convenient expression for Laplace’s nth coefficient.” 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. For the theory of the Legendrian coefficients and the analysis connected with them, the reader is referred to Heine’s Handbuch der Kugelfunctionen (Berlin, 1878), to Todhunter’s Treatise on Laplace's Functions, Larne's Functions, and Bessel's Functions (1875), or to Ferrers’s Spherical Harmonies (1877). It should be mentioned that Legendre’s coefficients have been recently termed· zonal harmonics by some writers.

Geodesy.— Besides the work upon the geodetical operations connecting Paris and Greenwich referred to above, and of which Legendre was one of the authors, he published in the Mémoires of the Academy 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; by means of it a small spherical triangle may he treated as a plane triangle, certain corrections being applied to the angles. Legendre was also the author of a memoir upon triangles drawn upon a spheroid. Legendre’s theorem is a fundamental one in geodesy, and his contributions to the subject are of the greatest importance.

Method of Least Squares.— In 1806 appeared Legendre’s Nouvelles Méthodes pour la détermination des orbites dés Comètes, which is memorable as containing the first published suggestion of the method of least squares. In the preface Legendre remarks, “La méthode qui me paroit la plus simple et la plus générale consiste à rendre minimum la somme des quarrés des erreurs, ... et que j’appelle méthode des moindres quarrés”; and in an appendix in which the application of the method is explained his words are, * ‘ De tous les principes qu’on peut proposer pour cet objet, je pense qu’il n’en est pas de plus général, de plus exact, ni d’une application plus facile que celui dont nous avons fait usage dans les recherches précédentes, et qui consiste à rendre minimum la somme des quarrés des erreurs.” The method was proposed by Legendre only as a convenient process for treating observations, without reference to the theory of probability. It had, however, been applied by Gauss as early as 1795, and the method was fully explained, and the law of facility for the first time given by him in 1809. Laplace also justified the method by means of the principles of the theory of probability; and this led Legendre to republish the part of his Nouvelles Méthodes which related to it in the Mémoires of the Academy for 1810. Thus, although the method of least squares was first formally proposed by Legendre, the theory and algorithm and mathematical foundation of the process are due to Gauss and Laplace. Legendre published two supplements to his Nouvelles Méthodes in 1806 and 1820.

The Elements of Geometry.— Legendre’s name is most widely known on account of his Eléments de géométrie, the most successful of the numerous attempts that have been made to supersede Euclid as a text-book on geometry. It first appeared in 1794, and went through very many editions, and has been translated into almost all languages. An English translation, by Sir David Brewster, from the eleventh French edition, was published in 1823, and is well known in England. The earlier editions did not contain the trigonometry. In one of the notes Legendre gives a proof of the irrationality of τr. This had been first proved by Lambert in the Berlin Memoirs for 1768. Legendre’s proof is similar in principle to Lambert’s, but much simpler. On account of the objections urged against the treatment of parallels in this work, Legendre was induced to publish in 1803 his Nouvelle Théorie des parallèles. His Geometry gave rise in England also to a lengthened discussion on the difficult question of the treatment of the theory of parallels.

It will thus be seen that Legendre’s works have placed him in the very foremost rank in the widely distinct subjects of elliptic functions, theory of numbers, attractions, and geodesy, and have given him a conspicuous position in connexion with the integral calculus and other branches of mathematics. He published a memoir on the integration of partial differential equations and a few others which have not been noticed above, but they relate to subjects with which his name is not especially associated. A good account of the principal works of Legendre is given in the Bibliothèque Universelle de Genève for 1833, pp. 45-82.