Thursday, September 18, 2014

Legendre's Elements of Geometry

1820 watercolor portrait of French mathematicians Adrien-Marie Legendre and Joseph Fourier
On September 18, 1752, French mathematician Adrien-Marie Legendre was born. He is best known for his contributions in number theory, celestial mechanics and elliptic functions. It was in a paper on celestial mechanics concerning the motion of planets (1784) that he first introduced the Legendre Polynomials. Moreover, he served as director of the of the Bureau des Longitudes, standardizing French weights and measures.

Adrien-Marie Legendre was born in Paris to a pretty wealthy family. He was admitted to the Collège Mazarin in Paris, and defended his thesis in physics and mathematics in 1770. Afterwards, he continued his career as a teacher at various institutions. In 1782, the Berlin Academy awarded Legendre a prize for his treatise on projectiles in resistant media. He became a member of the Académie des Sciences in 1783 and an associé in 1785. In 1789 he was elected a Fellow of the Royal Society. From 1799 to 1815, Legendre served as mathematics examiner for graduating artillery students at the École Militaire and was made an officer of the Légion d'Honneur.

Legendre developed the least squares method, which has broad application in linear regression, signal processing, statistics, and curve fitting. This word was published in 1806 as part of his book on the paths of comets. In 1830 he gave a proof of Fermat's last theorem for exponent n = 5, which was also proven by Lejeune Dirichlet two years earlier. Legendre conjectured the quadratic reciprocity law which was subsequently proved by Gauss and he pioneered with several works on the distribution of primes, and on the application of analysis to number theory. His 1798 conjecture of the Prime number theorem was rigorously proved by Hadamard and de la Vallée-Poussin in 1896. The scientist became known for the Legendre transformation, 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 energies from the internal energy. His best known work remains Éléments de géométrie which was published in 1794 and was the leading elementary text on the topic for around 100 years.

At yovisto, you may be interested in a video lecture on 'From one to many geometries' by Professor Flood at Gresham College.



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