LEGENDRE, ADRIEN-MARIE, an analyst, whose name must follow those of Lagrange and Laplace in the enumeration of the power ful school which existed in France at the time of the revolution, was born at Paris in 1753, end died there January 10, 1833. Of his personal life we can only now say that it was passed in strenuous and successful exertions for the advancement of mathematical science and of its applications. He never filled any political post, or took any marked part in public matters he was, we believe, no favourite of any government, and his scientific fame did not procure him more than a very moderate competency. The writings of M. Legendre consist of various papers in the ' Memoirs' of the Academy of Sciences, and several separate writings of which we shall give a alight account.
The first appearance of Legendre as a mathematician was in 1782 as the writer of two papers, one on the motion of resisted projectiles, the other on the attraction of spheroids, which gained prizes from the academies of Berlin and Paris, and a place in the former as the suc cessor of D'Alembert. In a memoir on double integrals, published in the volume for 1788 (though presented at the end of 1799), he digested a method of transforming an integral with two variables to one depend ing upon other variables, which he applied to the question of the attraction of spheroids. He was the first who extended the solution of this question by the aid of modern analysis : it being not a little remarkable, that this problem in the year 1773 required the power of Lagrange to show that even as much could be done with it by the modern analysis as had been effected with the ancient methods by Newton and Maclaurin. Various other memoirs by Legendre refer either to points of the integral calculus, or to his geodetical operations.
In 1787 he was appointed one of the commissioners for connecting the observatories of Greenwich and Paris by a chain of triangles. Cassini de Thury had memorialised the British government on the expediency of this step : the execution of which was committed to General Roy on the English side, and to Legendre, Cassini, and Mechain on the French. Much of the work was completed in 1787, and a memoir of Legendre, published in the volume for that year, upon some theoret ical points, contains one of those simple and beautiful theorems which carry the name of their inventors with them for ever. It is the cele brated proposition relative to the 'spherical excess' of a small spherical triangle. An account of the actual triangles constructed in his survey is contained in the volume for 1788. When the grand French arc of the meridian was completed, Laplace and Legendre were employed to deduce the form of the spheroid which agreed most nearly with all the observations. In the construction of the large trigonometrical tables (which still remain unpublished) he contributed some simpli fying theorems. In 1806 he published his 'Nouvelles 316thodes pour la D6termination des Orbitea des Cometes; in which he gives a method the peculiarity of which then was that it allowed of the correction of the original observations at any part of the process. It may be doubtful
whether the method itself was an improvement upon those which were then in use; and if it were, it is still superseded by others posterior to it. But this tract is further remarkable by its containing the first proposal to employ the method of least squares. Whether Legendre had seen the hint of Cotes or not, ho made a proposal of great ingenuity, and introduced, as a matter of practical convenience, a method which was afterwards shown by Laplace to be entitled to confidence on the strictest grounds of principle.
Legendre applied himself at an early period of his life to the develop ment of those integrals on which the determivation of the arcs of an ellipse and hyperbola depend. In the 'Memoirs' of the Academy for 1786 are two papers on the subject written by him. His Exereicea du Calcul Integral,' published in 1811, contain, among other matters of high curiosity, an extended view of the same subject. He continued to devote himself assiduously to the cultivation of this new branch of science, and in 1825 and 1826 he produced the two volumes of his Traits des Fonctions Elliptiques et des Integrnles Euleriennea,' eon taing a digested system, with extensive tables for the computation of the integrals. The work was hardly published when the discoveries of Messrs. Abel and Jacobi appeared. Theso mathematicians, both then very young, had begun by looking at the subject iu another point of view, and had produced results which would have materially simpli fied a large part of the work of Legeudre, if he had had the good fortune to find them. With a spirit which will always be one of the brightest parts of his reputation, Legendre immediately set about to add the new discoveries to his own work; and in 1828 and subsequent years appeared three supplements, in which they are presented in a manner symmetrical with the preceding part of the work, and with the fullest acknowledgment of their value and of the merit of their authors.
To Legendre is also due the collection of the results obtained upon the theory of numbers, a subject to which he made very remarkable additions. The second edition of his 'Thdorie des Nombres' was published in 1808, and the third in 1830.
The best known of Legcndre's works is, as might be supposed, his 'Elements of Geometry,' of which Sir David Brewster gave an English translation in 1824, from the eleventh edition : Legendre published his twelfth edition in 1823. Of the finished elegance and power of this very remarkable work it is not easy to speak in adequate terms : and next to the Elements of Euclid, it ought to hold the highest place among writings of the kind. But it would not be difficult to show that much of the rigour of Euclid has been sacrificed, and though those who determine to abandon the latter cannot do better than substitute Legendre's work, we hope that in this country the old Greek will maintain his ground at least until a substitute can be found who shall give equal rigour of demonstration, as well as greater elegance of form.