Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a

French French may refer to: * Something of, from, or related to France ** French language, which originated in France ** French people, a nation and ethnic group ** French cuisine, cooking traditions and practices Arts and media * The French (band), ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

who made numerous contributions to

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

. Well-known and important concepts such as the

Legendre polynomials In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of mathematical properties and numerous applications. They can be defined in many ways, and t ...

and

Legendre transformation In mathematics, the Legendre transformation (or Legendre transform), first introduced by Adrien-Marie Legendre in 1787 when studying the minimal surface problem, is an involutive transformation on real-valued functions that are convex on a rea ...

are named after him. He is also known for his contributions to the

method of least squares The method of least squares is a mathematical optimization technique that aims to determine the best fit function by minimizing the sum of the squares of the differences between the observed values and the predicted values of the model. The me ...

, and was the first to officially publish on it, though

Carl Friedrich Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and ...

had discovered it before him.

Life

Adrien-Marie Legendre was born in

Paris Paris () is the Capital city, capital and List of communes in France with over 20,000 inhabitants, largest city of France. With an estimated population of 2,048,472 residents in January 2025 in an area of more than , Paris is the List of ci ...

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 École or Ecole 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 * Éco ...

in Paris from 1775 to 1780 and at the

École Normale École or Ecole 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 * Éco ...

from 1795. At the same time, he was associated with the

Bureau des Longitudes __NOTOC__ The ''Bureau des Longitudes'' () is a French scientific institution, founded by decree of 25 June 1795 and charged with the improvement of nautical navigation, standardisation of time-keeping, geodesy and astronomical observation. Durin ...

. 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 (, ) 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 method, scientific research. It was at the forefron ...

'' 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 Fellows of the Royal Society of London to individuals who have made a "substantial contribution to the improvement of natural science, natural knowledge, incl ...

. 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 the Royal Observatory, Greenwich, Royal Greenwich Observatory and the Paris Observatory via triangulation (surveying), triangulation. The English ...

to calculate the precise distance between the

Paris Observatory The Paris Observatory (, ), a research institution of the Paris Sciences et Lettres University, is the foremost astronomical observatory of France, and one of the largest astronomical centres in the world. Its historic building is on the Left Ban ...

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 G ...

by means of

trigonometry Trigonometry () is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The fiel ...

. 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), also called Cassini IV, was a French astronomer, son of César-François Cassini de Thury and great-grandson of Giovanni Domenico Cassini. Cassini was born at the Paris Observato ...

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 bo ...

. The three also visited

William Herschel Frederick William Herschel ( ; ; 15 November 1738 – 25 August 1822) was a German-British astronomer and composer. He frequently collaborated with his younger sister and fellow astronomer Caroline Herschel. Born in the Electorate of Hanover ...

, the discoverer of the planet

Uranus Uranus is the seventh planet from the Sun. It is a gaseous cyan-coloured ice giant. Most of the planet is made of water, ammonia, and methane in a Supercritical fluid, supercritical phase of matter, which astronomy calls "ice" or Volatile ( ...

. 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 (, ; also known as Polytechnique or l'X ) is a ''grande école'' located in Palaiseau, France. It specializes in science and engineering and is a founding member of the Polytechnic Institute of Paris. The school was founded in 1794 by mat ...

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

Légion d'Honneur The National Order of the Legion of Honour ( ), formerly the Imperial Order of the Legion of Honour (), is the highest and most prestigious French national order of merit, both military and Civil society, civil. Currently consisting of five cl ...

. 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 ( ''Hébel'', in pausa ''Hā́ḇel''; ''Hábel''; , ''Hābēl'') is a biblical figure in the Book of Genesis within the Abrahamic religions. Born as the second son of Adam and Eve, the first two humans created by God in Judaism, God, he ...

's work on

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

s was built on Legendre's, and some of

Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, Geodesy, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observat ...

's work in statistics and

number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

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

least squares The method of least squares is a mathematical optimization technique that aims to determine the best fit function by minimizing the sum of the squares of the differences between the observed values and the predicted values of the model. The me ...

method which has broad application in

linear regression In statistics, linear regression is a statistical model, model that estimates the relationship between a Scalar (mathematics), scalar response (dependent variable) and one or more explanatory variables (regressor or independent variable). A mode ...

signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, Scalar potential, potential fields, Seismic tomograph ...

, 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 function In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral : \Beta(z_1,z_2) = \int_0^1 t^ ...

s and

gamma function In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...

s, introducing the symbol Γ and 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 In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive number, positive integers , , and satisfy the equation for any integer value of greater than . The cases ...

for exponent ''n'' = 5, which was also proven by Lejeune Dirichlet in 1828. In

, he conjectured the

quadratic reciprocity In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard st ...

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 of an odd prime number ''p'': its value at a (nonzero) quadratic residue mod ''p'' is 1 and at a non-quadratic re ...

is named after him. He also did pioneering work on the distribution of

primes A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

, 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 analysis, 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 p ...

was rigorously proved by

Hadamard Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry, and partial differential equations. Biography The son of a tea ...

and de la Vallée-Poussin in 1896. Legendre did an impressive amount of work on

s, 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

's study of the inverses of

Jacobi Jacobi may refer to: People * Jacobi (surname), a list of people with the surname * Jacobi Boykins (born 1995), American basketball player * Jacobi Francis (born 1998), American football player * Jacobi Mitchell (born 1986), Bahamian sprinter ...

's functions to solve the problem completely. He is known for the

, which is used to go from the

Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...

to the

Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...

formulation of

classical mechanics Classical mechanics is a Theoretical physics, physical theory describing the motion of objects such as projectiles, parts of Machine (mechanical), machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics inv ...

. In

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

it is also used to obtain the

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

and the

Helmholtz Hermann Ludwig Ferdinand von Helmholtz (; ; 31 August 1821 – 8 September 1894; "von" since 1883) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The ...

and

Gibbs Gibbs or GIBBS is a surname and acronym. It may refer to: People * Gibbs (surname) Places * Gibbs (crater), on the Moon * Gibbs, Missouri, US * Gibbs, Tennessee, US * Gibbs Island (South Shetland Islands), Antarctica * 2937 Gibbs, an asteroid ...

(free) energies from the

internal energy The internal energy of a thermodynamic system is the energy of the system as a state function, measured as the quantity of energy necessary to bring the system from its standard internal state to its present internal state of interest, accoun ...

. He is also the namesake of the

, solutions to Legendre's differential equation, which occur frequently in physics and engineering applications, such as

electrostatics Electrostatics is a branch of physics that studies slow-moving or stationary electric charges. Since classical antiquity, classical times, it has been known that some materials, such as amber, attract lightweight particles after triboelectric e ...

. 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 (The Academy) 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, and other ...

(1832) *The

Moon The Moon is Earth's only natural satellite. It Orbit of the Moon, orbits around Earth at Lunar distance, an average distance of (; about 30 times Earth diameter, Earth's diameter). The Moon rotation, rotates, with a rotation period (lunar ...

crater Legendre is named after him. *Main-belt asteroid

26950 Legendre __NOTOC__ Year 695 ( DCXCV) was a common year starting on Friday of the Julian calendar. The denomination 695 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe f ...

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 ( ; ) 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 from 1887 to 1889. Locally nicknamed "''La dame de fe ...

when it first opened. *A

street A street is a public thoroughfare in a city, town or village, typically lined with Building, buildings on one or both sides. Streets often include pavements (sidewalks), pedestrian crossings, and sometimes amenities like Street light, streetligh ...

in Paris' 17th Arrondissement is named after him.

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 ide ...

(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. One of only two known portraits 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 Julien-Léopold Boilly (; 30 August 1796 – 14 June 1874), also known as Jules Boilly, was a French artist noted for his album of lithographs ''Iconographie de l'Institut Royal de France'' (1820–1821) and his booklet ''Album de 73 portraits-c ...

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. The other portrait is from the book ''Le Panthéon scientifique de la tour Eiffel''.

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 French number theorists French textbook writers Officers of the Legion of Honour 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