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 and

watercolor portrait

#29). Biliotheque de l'Institut de France.

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

Legendre transformation
In mathematics, the Legendre transformation (or Legendre transform), named after Adrien-Marie Legendre, is an involutive transformation on real-valued convex functions of one real variable. In physical problems, it is used to convert functions ...

are named after him.
Life

Adrien-Marie Legendre was born inParis
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. S ...

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 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 mathematic ...

.
He assisted with the Anglo-French Survey (1784–1790) to calculate the precise distance between the Paris Observatory 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 and Pierre Méchain
Pierre François André Méchain (; 16 August 1744 – 20 September 1804) was a French astronomer and surveyor who, with Charles Messier, was a major contributor to the early study of deep-sky objects and comets.
Life
Pierre Méchain was born ...

. The three also visited William Herschel
Frederick William Herschel (; german: Friedrich Wilhelm Herschel; 15 November 1738 – 25 August 1822) was a German-born British astronomer and composer. He frequently collaborated with his younger sister and fellow astronomer Carolin ...

, 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. 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, 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.
Mathematical work

Abel
Abel ''Hábel''; ar, هابيل, Hābīl is a Biblical figure in the Book of Genesis within Abrahamic religions. He was the younger brother of Cain, and the younger son of Adam and Eve, the first couple in Biblical history. He was a shepherd ...

's work on elliptic functions was built on Legendre's, and some of Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refe ...

' 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, "Mathe ...

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 r ...

method which has broad application in linear regression, 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
Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints. Curve fitting can involve either interpolation, where an exact fit to the data is ...

; 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 functions, 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 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, "Mathe ...

, 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
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 elliptic functions, including the classification of elliptic integral
In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising i ...

s, but it took Abel
Abel ''Hábel''; ar, هابيل, Hābīl is a Biblical figure in the Book of Genesis within Abrahamic religions. He was the younger brother of Cain, and the younger son of Adam and Eve, the first couple in Biblical history. He was a shepherd ...

's stroke of genius to study the inverses of Jacobi's functions and solve the problem completely.
He is known for the Legendre transformation
In mathematics, the Legendre transformation (or Legendre transform), named after Adrien-Marie Legendre, is an involutive transformation on real-valued convex functions of one real variable. In physical problems, it is used to convert functions ...

, 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 classica ...

. 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 o ...

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, 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 am ...

.
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 theAmerican 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, ...

(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 ...

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 (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.

See also

* List of things named after Adrien-Marie Legendre * Associated Legendre polynomials * Gauss–Legendre algorithm * Legendre's constant * Legendre's equation in number theory * Legendre's functional relation for elliptic integrals * Legendre's conjecture * Legendre sieve * Legendre symbol * Legendre's theorem on spherical triangles *Saccheri–Legendre theorem
In absolute geometry, the Saccheri–Legendre theorem states that the sum of the angles in a triangle is at most 180°. Absolute geometry is the geometry obtained from assuming all the axioms that lead to Euclidean geometry with the exception of ...

* 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 r ...

* Least-squares spectral analysis
Least-squares spectral analysis (LSSA) is a method of estimating a frequency spectrum, based on a least squares fit of sinusoids to data samples, similar to Fourier analysis. Fourier analysis, the most used spectral method in science, generally ...

* Seconds pendulum
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