VI.24 Adrien-Marie Legendre
b. Paris, 1752; d. Paris, 1833
Analysis; theory of attractions; geometry; number theory
Legendre passed his career in Paris and seems to have been largely of independent means. Somewhat younger than LAGRANGE [VI.22] (who was resident there from 1787) and LAPLACE [VI.23], he did not quite match their reputation, though the range of his mathematical interests was comparably wide. His professional appointments were modest; however, in 1799 he took over from Laplace as a graduation examiner at the École Polytechnique and remained there until his retirement in 1816. Additionally, in 1813 he succeeded Lagrange at the Bureau des Longitudes.
Legendre’s early research concerned the shape of Earth and its external attraction to a point. Solutions of the differential equations involved led him to examine properties of the functions that are named after him; he was in rivalry with Laplace, after whom the functions were named during the nineteenth century. His other main concern in analysis, and the longest lasting, was with elliptic integrals. He wrote on them at great length up to a Traité of 1825-28. But in supplements of 1829-32 he acknowledged that his theory had just been eclipsed by the inverse ELLIPTIC FUNCTIONS [V.31] of JACOBI [VI.35] and ABEL [VI.33]. He also studied various other (functions defined as) integrals, including the beta and GAMMA FUNCTIONS [III.31]; solutions to differential equations; and optimization in the CALCULUS OF VARIATIONS [III.94].
Among Legendre’s contributions to numerical mathematics was a beautiful theorem (found in 1789) relating spheroidal triangles (that is, triangles drawn on the surface of a spheroid) to spherical triangles, which was used in the 1790s by J. B. J. Delambre in the triangulation analysis that led to the specification of the meter. His most famous numerical result is the least-squares criterion of curve fitting, proposed in 1805 in connection with determining the orbits of comets. For him the criterion was simply one of minimization; he did not make the connections to probability theory that were soon to be effected by LAPLACE [VI.23] and GAUSS [VI.26].
Legendre’s Essai sur la Théorie des Nombres (1798) was the first monograph on this subject. After reviewing CONTINUED FRACTIONS [III.22] and the theory of equations, he focused upon the algebraic branch, solving various Diophantine equations. Among many properties of integers, he stressed QUADRATIC RECIPROCITY [V.28], and proved various partition theorems concerning quadratic and some higher forms. Little in the book was new; while expanded editions appeared in 1808 and 1830, he had been quickly eclipsed on methods of proof by the Disquisitiones Arithmeticae (1801) of the young Gauss.
For educational use Legendre produced Elements de Géométrie (1794), an account of EUCLIDEAN GEOMETRY [I.3 §6.2] that emulated the same kind of form and organization and standards of proof of the Greek original. He also handled aspects that had lain outside Euclid’s concerns, such as alternatives to the parallel postulate, related numerical issues such as approximations to the value of π, and a lengthy summary of planar and spherical trigonometry. He produced eleven further editions up to 1823 and there were further posthumous editions up until 1839 (which were followed by reprints). It was a very influential book in mathematics education.
Further Reading
de Beaumont, E. 1867. Eloge Historique de Adrien Marie Legendre. Paris: Gauthier-Villars.