# Adrien-Marie Legendre

# Adrien-Marie Legendre

**1752-1833**

**French Mathematician**

Though he was overshadowed by his older contemporaries Joseph-Louis Lagrange (1736-1813) and Pierre-Simon de Laplace (1749-1827), Adrien-Marie Legendre played a significant role in the world of eighteenth-century mathematics. Not only was he the author of the highly popular *Eléments de géométrie* (1794), but in the field of research he founded the theory of elliptic functions and contributed to the understanding of number theory and celestial mechanics.

Born in Paris on September 18, 1752, Legendre was the son of wealthy parents. He studied at the Collège Mazarin in Paris, and in 1770, when he was 18 years old, defended his theses in mathematics and physics. Four years later he published a work on mechanics, the first of many publications. In 1775, Legendre took a position at the Ecole Militaire in Paris, where he would remain for five years.

Legendre first distinguished himself internationally when in 1782 his essay on projectile paths and air resistance won a prize from the Berlin Academy. This attracted the attention of both Lagrange, who took an interest in the much younger man, and Laplace, who would become a jealous rival.

By 1783, Legendre was ready to present his first paper before the Académie Royale des Sciences in Paris. The topic was the gravitational attractions of planetary spheroids—i.e., sphere-like celestial objects that are not perfectly round--and the paper helped win him election to the Académie later that year. A year later, another paper on celestial mechanics introduced what came to be known as "Legendre polynomials," solutions to a specific type of differential equation that would prove highly useful in applied mathematics.

From the 1780s onward, Legendre concerned himself primarily with two areas of interest, number theory and elliptic functions. He also participated in a 1787 geodesic survey conducted jointly by observatories in Paris and Greenwich, England, that resulted in a theorem stating the properties of a triangle when moved from the surface of a sphere to that of a plane.

The outbreak of the French Revolution in 1789 led to the suppression of the Académie Royale, and this had a negative impact on Legendre's career. He had recently married Marquerite Couhin (the couple had no children), and his financial situation became threatened as his modest inheritance began to diminish. He took a number of jobs in public service during the years that followed, first by serving on a commission that in 1791 converted the measurement of angles to the decimal system. By 1799, he had taken the place of Laplace as examiner in mathematics for artillery students.

Following his publication of *Eléments de géométrie,* which popularized aspects of the *Elements* by Euclid (c. 325-c. 250 b.c.), Legendre issued several appendices to his highly successful book. Among these was a proof of the irrationality of π and its square. His *Essai sur la théorie des nombres,* in which he discussed number theory, was first published in 1789, but again, Legendre continued to add appendices over the course of later years.

In 1811, Legendre published the first volume of *Exercises de calcul intégral,* and during the period from 1825 to 1828 brought out the three volumes of his most significant work, *Traité des fonctions elliptiques.* The latter expanded on earlier work in elliptic function research, but with the later efforts of Karl Gustav Jacob Jacobi (1804-1851) and Niels Henrik Abel (1802-1829), Legendre realized he had approached the problem incorrectly, failing to recognize the potential of elliptic function research to be considered as a distinct branch of analysis. In his last
years, he published supplements to the seminal work, discussing corrections by the younger mathematicians and humbly paying homage to their improvements on his work. Legendre died in Paris on January 9, 1833.

**JUDSON KNIGHT**

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**Adrien-Marie Legendre**