Adrien-Marie Legendre

Adrien-Marie Legendre (September 18 1752 – January 10 1833) was a French mathematician. He made important contributions to statistics, number theory, abstract algebra and mathematical analysis.

The Legendre crater on the Moon is named after him.

Life
Born in a wealthy family, Legendre studied physics in Paris and later taught at a military academy out of interest, not because of financial need. His earliest work in physics concerned the trajectories of cannonballs, but later he moved more towards mathematics.

In 1782, he was elected a member of the French Academy of Sciences.

Legendre lost his money during the French Revolution. His Éléments de Géométrie was a lucrative book and was much reprinted and translated, but it was his various teaching positions and pensions that kept him at an acceptable standard of living. A mistake in office politics in 1824 led to the loss of his pension and he lived the rest of his years in poverty.

Scientific activity
Most of his work was brought to perfection by others: his work on roots of polynomials inspired Galois theory; Abel's work on elliptic functions was built on Legendre's; some of Gauss' work in statistics and number theory completed that of Legendre. He developed the least squares method, which has broad application in linear regression, signal processing, statistics, and curve fitting. Today, the term "least squares method" is used as a direct translation from the French "méthode des moindres carrés".

In 1830 he gave a proof of Fermat's last theorem for exponent n = 5, which was also proven by Dirichlet in 1828.

In number theory, he conjectured the quadratic reciprocity law, subsequently proved by Gauss; in connection to this, the Legendre symbol is named after him. He also did pioneering work on the distribution of primes, and on the application of analysis to number theory. His 1796 conjecture of the Prime number theorem was rigorously proved by Hadamard and de la Vallée-Poussin in 1898.

Legendre did an impressive amount of work on elliptic functions, including the classification of elliptic integrals, but it took Abel's stroke of genius to study the inverses of Jacobi's functions and solve the problem completely.

He is known for the Legendre transform, which is used to go from the Lagrangian to the Hamiltonian formulation of classical mechanics. In thermodynamics it is also used to obtain the enthalpy and the Helmholtz and Gibbs (free) energies from the internal energy. He is also the namesake of the Legendre polynomials which occur frequently in physics and engineering applications, e.g. electrostatics.

He also wrote the influential Éléments de géométrie in 1794.