Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the

Legendre polynomials In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applica ...

and Legendre transformation are named after him.

Life

Adrien-Marie Legendre was born in

Paris Paris () is the capital and most populous city of France, with an estimated population of 2,165,423 residents in 2019 in an area of more than 105 km² (41 sq mi), making it the 30th most densely populated city in the world in 2020. Si ...

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 Bureau ( ) may refer to: Agencies and organizations * Government agency *Public administration * News bureau, an office for gathering or distributing news, generally for a given geographical location * Bureau (European Parliament), the administ ...

. 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 Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia The ''

Académie des sciences The French Academy of Sciences (French: ''Académie des sciences'') is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research. It was at ...

'' made Legendre an adjoint member in 1783 and an associate in 1785. In 1789, he was elected a

Fellow of the Royal Society Fellowship of the Royal Society (FRS, ForMemRS and HonFRS) is an award granted by the judges of the Royal Society of London to individuals who have made a "substantial contribution to the improvement of natural knowledge, including mathemati ...

. He assisted with the

Anglo-French Survey (1784–1790) The Anglo-French Survey (1784–1790) was the geodetic survey to measure the relative position of Greenwich Observatory and the Paris Observatory via triangulation. The English operations, executed by William Roy, consisted of the measurement ...

to calculate the precise distance between the

Paris Observatory The Paris Observatory (french: Observatoire de Paris ), a research institution of the Paris Sciences et Lettres University, is the foremost astronomical observatory of France, and one of the largest astronomical centers in the world. Its histo ...

and the

Royal Greenwich Observatory The Royal Observatory, Greenwich (ROG; known as the Old Royal Observatory from 1957 to 1998, when the working Royal Greenwich Observatory, RGO, temporarily moved south from Greenwich to Herstmonceux) is an observatory situated on a hill in ...

by means of

trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...

. To this end in 1787 he visited Dover and London together with

Dominique, comte de Cassini Jean-Dominique, comte de Cassini (30 June 174818 October 1845) was a French astronomer, son of César-François Cassini de Thury and great-grandson of Giovanni Domenico Cassini. Cassini was born at the Paris Observatory. He succeeded his fath ...

and Pierre Méchain. The three also visited William Herschel, the discoverer of the planet

Uranus Uranus is the seventh planet from the Sun. Its name is a reference to the Greek god of the sky, Uranus ( Caelus), who, according to Greek mythology, was the great-grandfather of Ares (Mars), grandfather of Zeus (Jupiter) and father of ...

. 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 École may refer to: * an elementary school in the French educational stages normally followed by secondary education establishments (collège and lycée) * École (river), a tributary of the Seine flowing in région Île-de-France * École, Savoi ...

. In 1824, Legendre's pension from the École Militaire was stopped because he refused to vote for the government candidate at the Institut National. His pension was partially reinstated with the change in government in 1828. In 1831, he was made an officer of the

Légion d'Honneur The National Order of the Legion of Honour (french: Ordre national de la Légion d'honneur), formerly the Royal Order of the Legion of Honour ('), is the highest French order of merit, both military and civil. Established in 1802 by Napoleon ...

. Legendre died in Paris on 9 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 Auteuil may refer to: Places * Auteuil, Oise, a commune in France * Auteuil, Paris, a neighborhood of Paris ** Auteuil, Seine, the former commune which was on the outskirts of Paris * Auteuil, Quebec, a former city that is now a district within ...

, 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. Tombe Adrien-Marie Legendre, Cimetière d'Auteuil, Paris

Tombe Adrien-Marie Legendre, Cimetière d'Auteuil, Paris

Mathematical work

Abel's work on

elliptic function In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those ...

s was built on Legendre's, and some of Gauss' work in statistics and

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...

completed that of Legendre. He developed, and first communicated to his contemporaries before Gauss, the

least squares The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the re ...

method which has broad application in

linear regression In statistics, linear regression is a linear approach for modelling the relationship between a scalar response and one or more explanatory variables (also known as dependent and independent variables). The case of one explanatory variable is cal ...

signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...

, 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". His major work is ''Exercices de Calcul Intégral'', published in three volumes in 1811, 1817 and 1819. In the first volume he introduced the basic properties of elliptic integrals, beta functions and

gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers excep ...

s, introducing the symbol Γ normalizing it to Γ(n+1) = n!. Further results on the beta and gamma functions along with their applications to mechanics – such as the rotation of the earth, and the attraction of ellipsoids – appeared in the second volume. In 1830, he gave a proof of Fermat's Last Theorem for exponent ''n'' = 5, which was also proven by Lejeune Dirichlet in 1828. In

, he conjectured the quadratic reciprocity law, subsequently proved by Gauss; in connection to this, the

Legendre symbol In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo an odd prime number ''p'': its value at a (nonzero) quadratic residue mod ''p'' is 1 and at a non-quadratic residu ...

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 In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying t ...

was rigorously proved by Hadamard and de la Vallée-Poussin in 1896. Legendre did an impressive amount of work on

s, including the classification of elliptic integrals, but it took Abel's stroke of genius to study the inverses of

Jacobi Jacobi may refer to: * People with the surname Jacobi Mathematics: * Jacobi sum, a type of character sum * Jacobi method, a method for determining the solutions of a diagonally dominant system of linear equations * Jacobi eigenvalue algorithm, ...

'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 Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...

. In

thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws ...

it is also used to obtain the

enthalpy Enthalpy , a property of a thermodynamic system, is the sum of the system's internal energy and the product of its pressure and volume. It is a state function used in many measurements in chemical, biological, and physical systems at a constant ...

and the

Helmholtz Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Associatio ...

and Gibbs (free) energies from the

internal energy The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...

. He is also the namesake of the

, solutions to Legendre's differential equation, which occur frequently in physics and engineering applications, ''e.g.''

electrostatics Electrostatics is a branch of physics that studies electric charges at rest ( static electricity). Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for a ...

. 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.

Honors

*Foreign Honorary Member of the

American Academy of Arts and Sciences The American Academy of Arts and Sciences (abbreviation: AAA&S) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, a ...

(1832) *The

Moon The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...

crater Legendre is named after him. *Main-belt asteroid 26950 Legendre is named after him. *Legendre is one of the 72 prominent French scientists who were commemorated on plaques at the first stage of the

Eiffel Tower The Eiffel Tower ( ; french: links=yes, tour Eiffel ) is a wrought-iron lattice tower on the Champ de Mars in Paris, France. It is named after the engineer Gustave Eiffel, whose company designed and built the tower. Locally nicknamed ...

when it first opened.

Publications

;Essays * 1782 ''Recherches sur la trajectoire des projectiles dans les milieux résistants'' (prize on projectiles offered by the Berlin Academy) ;Books * ''Eléments de géométrie'', textbook 1794 * ''Essai sur la Théorie des Nombres'' 1797-8 ("An VI"), 2nd ed. 1808, 3rd ed. in 2 vol. 1830 * ''Nouvelles Méthodes pour la Détermination des Orbites des Comètes'', 1805 * ''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 homogènes'' (work on Legendre polynomials) * 1784 ''Recherches sur la figure des Planètes'' p. 370 * 1785 ''Recherches d'analyse indéterminée'' p. 465 (work on 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 parallèles ou le théorème sur la somme des trois angles du triangle, avec 1 planche'' (367–412)

Mistaken portrait

For two centuries, until the recent discovery of the error in 2005, books, paintings and articles have incorrectly shown a profile portrait of the obscure French politician

Louis Legendre Louis Legendre (22 May 1752 – 13 December 1797) was a French politician of the Revolution period. Early activities Born at Versailles, he was keeping a butcher's shop in Saint Germain, Paris, by 1789. He was an ardent supporter of the ideas ...

(1752–1797) as a portrait of the mathematician. 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, rediscovered in 2008, 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-Léopold Boilly as shown below:Boilly, Julien-Léopold. (1820). ''Album de 73 portraits-charge aquarellés des membres de I'Institut''
watercolor portrait
#29). Biliotheque de l'Institut de France.

Notes

External links

* *
The True Face of Adrien-Marie Legendre
(Portrait of Legendre)

a
Fermat's Last Theorem Blog

*
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) {{DEFAULTSORT:Legendre, Adrien Marie 1752 births 1833 deaths University of Paris alumni 18th-century French mathematicians 19th-century French mathematicians Number theorists Officiers of the Légion d'honneur Fellows of the American Academy of Arts and Sciences Members of the French Academy of Sciences Fellows of the Royal Society Fellows of the Royal Society of Edinburgh Scientists from Paris