Theory of Numbers.—Legendre’s Théorie des nombres and Gauss’s Disquisitiones arithmeticae (1801) are still 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. The fourth edition appeared in 1900. 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 P. de 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 (a/p) which is known as Legendre’s symbol, and denotes the positive or negative unit which is the remainder when a1/2p(−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 (2nd ed., p. 394) (see [Number]).
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 sphéroides 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 − 2h cos φ + h2)−1/2 be expanded in ascending powers of h, and if the general term be denoted by Pnhn, then Pn 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 Isaac 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 Académie 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. See also [Spherical Harmonics].)
Geodesy.—Besides the work upon the geodetical operations connecting Paris and Greenwich, of which Legendre was one of the authors, he published in the Mémoires de l’Académie 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 be 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 des comètes, which is memorable as containing the first published suggestion of the method of least squares (see [Probability]). In the preface Legendre remarks: “La méthode qui me paroît 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 de l’Académie 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 π. This had been first proved by J. H. 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 Géométrie 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.
See Élie de Beaumont, “Memoir de Legendre,” translated by C. A. Alexander, Smithsonian Report (1874).
(J. W. L. G.)
