# Adrien-Marie Legendre

**Adrien-Marie Legendre** (Paris, 18 September 1752 – Paris, 10 January 1833) was an important French mathematician whose name lives on in the Legendre polynomials and associated Legendre functions.

## Life and works

Legendre came from a wealthy family and studied mathematics and physics at the Collège Mazarin in Paris. In 1770, at the age of 18, Legendre obtained a degree in mathematics and physics at this college. After his graduation Legendre, being of independent means, first concentrated on his research, and after five years he accepted a position to teach at the École Militaire. He stayed there from 1775 until 1785. In 1782 he won a prize of the Berlin Academy with theoretical work on the trajectories of projectiles; his essay was entitled *Recherches sur la trajectoire des projectiles dans les milieux résistants* (Research on the trajectories of projectiles in resisting media). This essay established Legendre as an important mathematician and drew the attention of Lagrange who was then in Berlin.

Legendre next determined the gravitational potential due to an ellipsoidal, axial-symmetric, mass distribution at an arbitrary exterior point. The concept of potential (force integrated over path) was introduced around the same time (1782) by Laplace. In his work Legendre made a McLaurin expansion of the potential. Taking derivatives with respect to distance he obtained an expansion of the force in ordinary Legendre polynomials and taking derivatives with respect to latitude angle he obtained associated Legendre functions for *m* = 1 (linear in the sine of latitude angle). Legendre submitted his results to the Académie des Sciences in January 1783 and his work was much appreciated as appears from his appointment as an adjoint to this prestigious Academy a few months later (March 30). The work was published^{[1]} after Laplace had announced similar results, which gave rise to a priority dispute. However, it is now generally accepted that credit for the Legendre functions is indeed due to Legendre.

In the same period Legendre published on celestial mechanics with papers such as *Recherches sur la figure des planètes* (Research on the shape of the planets, 1784) in which a closed form of the *n*th order ordinary Legendre polynomial appears. In addition he worked on number theory and elliptic functions.

In 1787 Legendre became a member of a team that performed a triangulation survey between the Paris and Greenwich observatories, which was done in connection with measurements on the size of the Earth. This work resulted in Legendres election to the Royal Society of London in 1787.

After the French revolution of 1789 Legendre became in 1791 a member of a committee with the task to standardize the weights and measures. The committee chose the *grave* (1/1000 of a kilogram) as unit of mass. Further the metre was fixed to the length of a prototype bar which was supposed to be 1/10 000 000 of the length of the meridian of Paris from the north pole to the equator.

In the aftermath of the revolution the Academy of Sciences was closed (1793) and Legendre lost his family wealth during the upheaval. Nevertheless, he married at this time, which did not keep him from publishing (1794) *Eléments de géométrie* which was modeled after Euclid's elements; it greatly rearranged and simplified the propositions of Euclid's classical work. Legendre's *Elements of Geometry* was to become the leading elementary text in most of Europe for around 100 years. His work was a great success in the United States, undergoing numerous translations starting in 1819; one such translation went through 33 editions.

In 1795 the Académie des Sciences was reopened as the Institut National des Sciences et des Arts. Each section of the Institut contained six places, and Legendre was one of the six in the mathematics section. In 1803 Napoleon reorganized the Institut and a geometry section was created and Legendre was put into this section.

In 1806 Legendre published a book on the orbits of comets and in an appendix Legendre gave the least squares method of fitting a curve to observational data. Gauss published the least squares method in 1809 and, while acknowledging that it appeared in Legendre's book, Gauss still claimed priority for himself. This greatly hurt Legendre who fought for many years to have his priority recognized.

Legendre continued his work on elliptic functions and his major work on the topic *Exercices du Calcul Intégral* appeared in three volumes in 1811, 1817, and 1819. Later he revised and improved this work in *Traité des Fonctions Elliptiques* again in three volumes of 1825, 1826, and 1830.

In 1824 Legendre refused to vote for the government's candidate for the French Academy of Sciences, which was re-instituted in 1816 by the French king Louis XVIII, a brother of Louis XVI. As a result of his refusal, Legendre lost his pension from the École Militaire, where he had served from 1799 to 1815 as the mathematics examiner for graduating artillery students. Nine years later he died in poverty.