Little is known about Legendre’s early life except that his family wealth allowed him to study physics and mathematics, beginning in 1770, at the Collège Mazarin (Collège des Quatre-Nations) in Paris and that, at least until the French Revolution, he did not have to work. Nevertheless, Legendre taught mathematics at the École Militaire in Paris from 1775 to 1780. In 1782 he won a prize offered by the Berlin Academy of Sciences for his effort to “determine the curve described by cannonballs and bombs, taking into consideration the resistance of air[, and] give rules for obtaining the ranges corresponding to different initial velocities and to different angles of projection.” The next year he presented research on celestial mechanics to the French Academy of Sciences, and he was soon rewarded with membership. In 1787 he joined the French team, led by Jacques-Dominique Cassini and Pierre Mechain, in the geodetic measurements jointly conducted with the Royal Greenwich Observatory in London. At this time he also became a member of the British Royal Society. In 1791 he was named along with Cassini and Mechain to a special committee to develop the metric system and, in particular, to conduct the necessary measurements to determine the standard metre. He also worked on projects to produce logarithmic and trigonometric tables.
The Academy of Sciences was forced to close in 1793 during the French Revolution, and Legendre lost his family wealth during the upheaval. Nevertheless, he married at this time. The following year he published Éléments de géométrie (Elements of Geometry), a reorganization and simplification of the propositions from Euclid’s Elements that was widely adopted in Europe, even though it is full of fallacious attempts to defend the parallel postulate. Legendre also gave a simple proof that π is irrational, as well as the first proof that π2 is irrational, and he conjectured that π is not the root of any algebraic equation of finite degree with rational coefficients (i.e., π is a transcendental number). His Éléments was even more pedagogically influential in the United States, undergoing numerous translations starting in 1819; one such translation went through some 33 editions. The French Academy of Sciences was reopened in 1795 as the Institut Nationale des Sciences et des Arts, and Legendre was installed in the mathematics section. When Napoleon reorganized the institute in 1803, Legendre was retained in the new geometry section. In 1824 he refused to endorse the government’s candidate for the Institut and lost his pension from the École Militaire, where he had served from 1799 to 1815 as the mathematics examiner for graduating artillery students.
Legendre’s Nouvelles méthodes pour la détermination des orbites des comètes (1806; “New Methods for the Determination of Comet Orbits”) contains the first comprehensive treatment of the method of least squares, although priority for its discovery is shared with his German rival Carl Friedrich Gauss.
In 1786 Legendre took up research on elliptic integrals. In his most important work, Traité des fonctions elliptiques (1825–37; “Treatise on Elliptic Functions”), he reduced elliptic integrals to three standard forms now known by his name. He also compiled tables of the values of his elliptic integrals and showed how they can be used to solve important problems in mechanics and dynamics. Shortly after his work appeared, the independent discoveries of Niels Henrik Abel and Carl Jacobi completely revolutionized the subject of elliptic integrals.
Legendre published his own researches in number theory and those of his predecessors in a systematic form under the title Théorie des nombres, 2 vol. (1830). This work included his flawed proof of the law of quadratic reciprocity. The law was regarded by Gauss, the greatest mathematician of the day, as the most important general result in number theory since the work of Pierre de Fermat in the 17th century. Gauss also gave the first rigorous proof of the law.