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.
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.''