1820 watercolor caricature of Adrien-Marie Legendre by French artist Julien-Leopold Boilly (see portrait debacle), the only existing portrait known
18 September 1752|
|Died||10 January 1833
|Alma mater||Collège Mazarin|
|Known for||Legendre transformation and elliptic functions|
Adrien-Marie Legendre (French pronunciation: [adʁiɛ̃ maʁi ləʒɑ̃ːdʁ]) (18 September 1752 – 10 January 1833) was a French mathematician. Legendre made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are named after him.
Adrien-Marie Legendre was born in Paris (or possibly, in Toulouse, depending on sources) on 18 September 1752 to a wealthy family. He was given an excellent education at the Collège Mazarin in Paris, defending his thesis in physics and mathematics in 1770. From 1775 to 1780 he taught at the École Militaire in Paris, and from 1795 at the École Normale, and was associated with the Bureau des longitudes. In 1782, he won the prize offered by the Berlin Academy for his treatise on projectiles in resistant media, which brought him to the attention of Lagrange.
In 1783 he became an adjoint of the Académie des Sciences, and an associé in 1785. In 1789 he was elected a Fellow of the Royal Society. During the French Revolution, in 1793, he lost his private fortune, but was able to put his affairs in order with the help of his wife, Marguerite-Claudine Couhin, whom he married in the same year. In 1795 he became one of the six members of the mathematics section of the reconstituted Académie des Sciences, named the Institut National des Sciences et des Arts, and later, in 1803, of the Geometry section as reorganized under Napoleon. In 1824, as a result of refusing to vote for the government candidate at the Institut National, Legendre was deprived by the Ministre de L'Intérieur of the ultraroyalist government, the comte de Corbière, of his pension from the École Militaire, where he had served from 1799 to 1815 as mathematics examiner for graduating artillery students. This was partially reinstated with the change in government in 1828 and in 1831 he was made an officer of the Légion d'Honneur.
He died in Paris on 9 January 1833, after a long and painful illness. Legendre's widow made a cult of his memory, carefully preserving his belongings. Upon her death in 1856, she left their last country house to the village of Auteuil where the couple had lived and are buried.
His name is one of the 72 names inscribed on the Eiffel Tower.
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; 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".
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.
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 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 (free) energies from the internal energy. He is also the namegiver of the Legendre polynomials, solutions to Legendre's differential equation, which occur frequently in physics and engineering applications, e.g. electrostatics.
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.
Portrait debacle 
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". 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:
- 1782 Recherches sur la trajectoire des projectiles dans les milieux résistants (prize on projectiles offered by the Berlin Academy)
- Eléments de géométrie, textbook 1794
- Essai sur la Théorie des Nombres 1797-8 ("An VI"), 2nd ed. in two vol. 1808, 3rd ed. in 2 vol. 1830
- Nouvelles Méthodes pour la Détermination des Orbites des Comètes, 1806
- Exercices de Calcul Intégral, book in three volumes 1811, 1817, and 1819
- Traité des Fonctions Elliptiques, book in three volumes 1825, 1826, and 1830
Memoires in Histoire de l'Académie Royale des Sciences 
- 1783 Sur l'attraction des Sphéroïdes homogénes (said to contain Legendre polys)
- 1784 Recherches sur la figure des Planètes p. 370
- 1785 Recherches d'analyse indéterminée p. 465 (number theory)
- 1786 Mémoire sur la manière de distinguer les Maxima des Minima dans le Calcul des Variations p. 7 (as Legendre)
- 1786 Mémoire sur les Intégrations par arcs d'ellipse p. 616 (as le Gendre)
- 1786 Second Mémoire sur les Intégrations par arcs d'ellipse p. 644
- 1787 L’intégration de quelques équations aux différences Partielles (Legendre transform)
In Memoires présentés par divers Savants à la l'Académie des Sciences de l'Institut de France 
- 1806 Nouvelle formula pour réduire en distances vraies les distances apparentes de la Lune au Soleil ou à une étoile (30-54)
- 1807 Analyse des triangles tracés sur la surface d'un sphéroide (130-161)
- Tome 10 Recherches sur diverses sortes d'intégrales défines (416-509)
- 1819 Méthode des moindres carrés pour trouver le milieu le plus probable entre les résultats de différentes observations (149-154), Mémoire sur l'attraction des ellipsoïdes homogènes (155-183)
- 1823 Recherches sur quelques objets d'Analyse indéterminée et particulièrement sur le théorème de Fermat (1-60)
- 1828 Mémoire sur la détermination des fonctions Y et Z que satisfont à l'équation 4(X^n-1) = (X-1)(Y^2+-nZ^2), n étant un nombre premier 4i-+1 (81-100)
- 1833 Réflexions sur différentes manières de démontrer la théorie des paralèles ou le théorème sur la somme des trois angles du triangle, avec 1 planche (367-412)
See also 
- List of things named after Adrien-Marie Legendre
- Associated Legendre polynomials
- Gauss–Legendre algorithm
- Legendre's constant
- Legendre duplication formula
- Legendre's equation in number theory
- Legendre functions
- Legendre's functional relation for elliptic integrals
- Legendre's differential equation
- Legendre's conjecture
- Legendre sieve
- Legendrian submanifold
- Legendre symbol
- Saccheri–Legendre theorem
- Duren, Peter (December 2009). "Changing Faces: The Mistaken Portrait of Legendre". Notices of the AMS 56 (11): 1440–1443, 1455.
- "Library and Archive". Royal Society. Retrieved 2012-08-06.
- Boilly, Julien-Leopold. (1820). Album de 73 portraits-charge aquarellés des membres de I’Institut (watercolor portrait #29). Biliotheque de l’Institut de France.
- The True Face of Adrien-Marie Legendre (Portrait of Legendre)
- O'Connor, John J.; Robertson, Edmund F., "Adrien-Marie Legendre", MacTutor History of Mathematics archive, University of St Andrews.
- Biography at Fermat's Last Theorem Blog
- References for Adrien-Marie Legendre
- (French) Eléments de géométrie (Paris : F. Didot, 1817)
- Elements of geometry and trigonometry, from the works of A. M. Legendre. Revised and adapted to the course of mathematical instruction in the United States, by Charles Davies. (New York: A. S. Barnes & co., 1858) : English translation of the above text
- Mémoires sur la méthode des moindres quarrés, et sur l'attraction des ellipsoïdes homogènes (1830)
- Théorie des nombres (Paris : Firmin-Didot, 1830)
- Traité des fonctions elliptiques et des intégrales eulériennes (Paris : Huzard-Courcier, 1825–1828)
- Nouvelles Méthodes pour la Détermination des Orbites des Comètes (Paris : Courcier, 1806)
- Essai sur la Théorie des Nombres (Paris : Duprat, 1798)
- Exercices de Calcul Intégral V.3 (Paris : Courcier, 1816)
- Correspondance mathématique avec Legendre in C. G. J. Jacobis gesammelte Werke (Berlin: 1852)