Adrien-Marie Legendre was born in Paris on 18 September 1752 to a wealthy family. He received his education at the Collège Mazarin in Paris, and defended his thesis in physics and mathematics in 1770. He taught at the École Militaire in Paris from 1775 to 1780 and at the École Normale from 1795. At the same time, he was associated with the Bureau des Longitudes. In 1782, the Berlin Academy awarded Legendre a prize for his treatise on projectiles in resistant media. This treatise also brought him to the attention of Lagrange.
The Académie des Sciences made Legendre an adjoint member in 1783 and an associé in 1785. In 1789 he was elected a Fellow of the Royal Society.
He assisted with the Anglo-French Survey (1784–1790) to calculate the precise distance between the Paris Observatory and the Royal Greenwich Observatory by means of trigonometry. To this end in 1787 he visited Dover and London together with Dominique, comte de Cassini and Pierre Méchain. The three also visited william Herschel, the discoverer of the planet Uranus.
Legendre lost his private fortune in 1793 during the French Revolution. That year, he also married Marguerite-Claudine Couhin, who helped him put his affairs in order. In 1795 Legendre became one of six members of the mathematics section of the reconstituted Académie des Sciences, renamed the Institut National des Sciences et des Arts. Later, in 1803, Napoleon reorganized the Institut National, and Legendre became a member of the Geometry section. From 1799 to 1812, Legendre served as mathematics examiner for graduating artillery students at the École Militaire and from 1799 to 1815 he served as permanent mathematics examiner for the École Polytechnique. In 1824, Legendre was denied his pension from the École Militaire because he refused to vote for the government candidate at the Institut National—the comte de Corbière, Ministre de L'Intérieur of the ultraroyalist government. His pension was partially reinstated with the change in government in 1828. In 1831 he was made an officer of the Légion d'Honneur.
Legendre is best known as the author of Éléments de géométrie, which was published in 1794 and was the leading elementary text on the topic for around 100 years. This text greatly rearranged and simplified many of the propositions from Euclid's Elements to create a more effective textbook.
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 1798 conjecture of the prime number theorem was rigorously proved by Hadamard and de la Vallée-Poussin in 1896.
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 and firstly communicated it to his contemporaries before Gauss, which has broad application in linear regression, signal processing, statistics, and curve fitting; this was published in 1806 as an appendix to his book on the paths of comets. Today, the term "least squares method" is used as a direct translation from the French "méthode des moindres carrés".
Around 1811 he named the gamma function and introduced the symbol Γ normalizing it to Γ(n+1) = n!.
In 1830 he gave a proof of Fermat's last theorem for exponent n = 5, which was also proven by Lejeune Dirichlet in 1828.
Legendre died in Paris on 10 January 1833, after a long and painful illness, and Legendre's widow carefully preserved his belongings to memorialize him. Upon her death in 1856, she was buried next to her husband in the village of Auteuil, where the couple had lived, and left their last country house to the village. Legendre's name is one of the 72 names inscribed on the Eiffel Tower.
For two centuries, until the recent discovery of the error in 2005, books, paintings and articles have incorrectly shown a side-view portrait of the obscure French Politician Louis Legendre (1752–1797) as that of the Mathematician Legendre. The error arose from the fact that the Sketch was labelled simply "Legendre" and appeared in a book along with contemporary mathematicians such as Lagrange. The only known portrait of Legendre, recently unearthed, is found in the 1820 book Album de 73 portraits-charge aquarellés des membres de I'Institut, a book of caricatures of seventy-three members of the Institut de France in Paris by the French Artist Julien-Leopold Boilly as shown below:
