Adrien-Marie Legendre
Encyclopedia
Adrien-Marie Legendre (18 September 1752 – 10 January 1833) was a French
mathematician
.
The Moon
crater Legendre
is named after him.
.
In 1783 he became an adjoint of the Académie des Sciences, and an associé in 1785. 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.
s inspired Galois theory
; Abel's work on elliptic function
s 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
. Today, the term "least squares method" is used as a direct translation from the French
"méthode des moindres carrés".
In 1830 he gave a proof of Fermat's last theorem
for exponent n = 5, which was also proven by Dirichlet
in 1828.
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 1796 conjecture of the Prime number theorem
was rigorously proved by Hadamard
and de la Vallée-Poussin
in 1898.
Legendre did an impressive amount of work on elliptic function
s, including the classification of elliptic integral
s, 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.
(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 famous mathematicians by the French artist Julien-Leopold Boilly
as shown below:
France
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
.
The Moon
Moon
The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...
crater Legendre
Legendre (crater)
Legendre is a lunar impact crater that is located near the eastern limb of the Moon. Just to the southwest is the crater Adams. To the northwest is Palitzsch and the prominent Petavius....
is named after him.
Life
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 LagrangeLagrange
La Grange literally means the barn in French. Lagrange may refer to:- People :* Charles Varlet de La Grange , French actor* Georges Lagrange , translator to and writer in Esperanto...
.
In 1783 he became an adjoint of the Académie des Sciences, and an associé in 1785. 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
Légion d'honneur
The Legion of Honour, or in full the National Order of the Legion of Honour is a French order established by Napoleon Bonaparte, First Consul of the Consulat which succeeded to the First Republic, on 19 May 1802...
.
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 polynomialPolynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
s inspired Galois theory
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
; Abel's work on elliptic function
Elliptic function
In complex analysis, an elliptic function is a function defined on the complex plane that is periodic in two directions and at the same time is meromorphic...
s was built on Legendre's; some of Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
' work in statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
and number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
completed that of Legendre. He developed the least squares
Least squares
The method of least squares is a standard approach to the approximate solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in solving every...
method, which has broad application in linear regression
Linear regression
In statistics, linear regression is an approach to modeling the relationship between a scalar variable y and one or more explanatory variables denoted X. The case of one explanatory variable is called simple regression...
, signal processing
Signal processing
Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...
, statistics, and curve fitting
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 required, or smoothing, in which a "smooth" function...
. Today, the term "least squares method" is used as a direct translation from the French
French language
French is a Romance language spoken as a first language in France, the Romandy region in Switzerland, Wallonia and Brussels in Belgium, Monaco, the regions of Quebec and Acadia in Canada, and by various communities elsewhere. Second-language speakers of French are distributed throughout many parts...
"méthode des moindres carrés".
In 1830 he gave a proof of Fermat's last theorem
Fermat's Last Theorem
In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....
for exponent n = 5, which was also proven by Dirichlet
Johann Peter Gustav Lejeune Dirichlet
Johann Peter Gustav Lejeune Dirichlet was a German mathematician with deep contributions to number theory , as well as to the theory of Fourier series and other topics in mathematical analysis; he is credited with being one of the first mathematicians to give the modern formal definition of a...
in 1828.
In number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, he conjectured the quadratic reciprocity
Quadratic reciprocity
In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic which gives conditions for the solvability of quadratic equations modulo prime numbers...
law, subsequently proved by Gauss; in connection to this, the Legendre symbol
Legendre symbol
In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo a prime number p: its value on a quadratic residue mod p is 1 and on a quadratic non-residue is −1....
is named after him. He also did pioneering work on the distribution of primes
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
, and on the application of analysis to number theory. His 1796 conjecture of the Prime number theorem
Prime number theorem
In number theory, the prime number theorem describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are distributed amongst the positive integers....
was rigorously proved by Hadamard
Jacques Hadamard
Jacques Salomon Hadamard FRS was a French mathematician who made major contributions in number theory, complex function theory, differential geometry and partial differential equations.-Biography:...
and de la Vallée-Poussin
Charles Jean de la Vallée-Poussin
Charles-Jean Étienne Gustave Nicolas de la Vallée Poussin was a Belgian mathematician. He is most well known for proving the Prime number theorem.The king of Belgium ennobled him with the title of baron.-Biography:...
in 1898.
Legendre did an impressive amount of work on elliptic function
Elliptic function
In complex analysis, an elliptic function is a function defined on the complex plane that is periodic in two directions and at the same time is meromorphic...
s, including the classification of elliptic integral
Elliptic integral
In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse. They were first studied by Giulio Fagnano and Leonhard Euler...
s, but it took Abel
Niels Henrik Abel
Niels Henrik Abel was a Norwegian mathematician who proved the impossibility of solving the quintic equation in radicals.-Early life:...
's stroke of genius to study the inverses of Jacobi
Carl Gustav Jakob Jacobi
Carl Gustav Jacob Jacobi was a German mathematician, widely considered to be the most inspiring teacher of his time and is considered one of the greatest mathematicians of his generation.-Biography:...
's functions
Jacobi's elliptic functions
In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions, and auxiliary theta functions, that have historical importance with also many features that show up important structure, and have direct relevance to some applications...
and solve the problem completely.
He is known for the Legendre transformation
Legendre transformation
In mathematics, the Legendre transformation or Legendre transform, named after Adrien-Marie Legendre, is an operation that transforms one real-valued function of a real variable into another...
, which is used to go from the Lagrangian
Lagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...
to the Hamiltonian
Hamiltonian mechanics
Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...
formulation of classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
. In thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...
it is also used to obtain the enthalpy
Enthalpy
Enthalpy is a measure of the total energy of a thermodynamic system. It includes the internal energy, which is the energy required to create a system, and the amount of energy required to make room for it by displacing its environment and establishing its volume and pressure.Enthalpy is a...
and the Helmholtz
Helmholtz free energy
In thermodynamics, the Helmholtz free energy is a thermodynamic potential that measures the “useful” work obtainable from a closed thermodynamic system at a constant temperature and volume...
and Gibbs
Gibbs free energy
In thermodynamics, the Gibbs free energy is a thermodynamic potential that measures the "useful" or process-initiating work obtainable from a thermodynamic system at a constant temperature and pressure...
(free) energies
Thermodynamic free energy
The thermodynamic free energy is the amount of work that a thermodynamic system can perform. The concept is useful in the thermodynamics of chemical or thermal processes in engineering and science. The free energy is the internal energy of a system less the amount of energy that cannot be used to...
from the internal energy
Internal energy
In thermodynamics, the internal energy is the total energy contained by a thermodynamic system. It is the energy needed to create the system, but excludes the energy to displace the system's surroundings, any energy associated with a move as a whole, or due to external force fields. Internal...
. 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
Electrostatics
Electrostatics is the branch of physics that deals with the phenomena and properties of stationary or slow-moving electric charges....
.
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
Euclid's Elements
Euclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...
to create a more effective textbook.
Portrait debacle
For two centuries, until the recent discovery of the error in 2009, books, paintings and articles have incorrectly shown a side-view portrait of the obscure French politician Louis LegendreLouis Legendre
Louis Legendre 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...
(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 famous mathematicians by the French artist Julien-Leopold Boilly
Julien-Leopold Boilly
Julien-Leopold Boilly , also known as Jules Boilly, was a French artist noted for his 1820 booklet Album de 73 Portraits-Charge Aquarelle’s des Membres de I’Institute containing watercolor caricatures of seventy-three famous mathematicians, in particular French mathematician Adrien-Marie Legendre,...
as shown below:
See also
- Gauss–Legendre algorithm
- Legendre's constantLegendre's constantLegendre's constant is a mathematical constant occurring in a formula conjectured by Adrien-Marie Legendre to capture the asymptotic behavior of the prime-counting function \scriptstyle\pi...
- Legendre's equationLegendre's equationIn mathematics, Legendre's equation is the Diophantine equationax^2+by^2+cz^2=0.The equation is named for Adrien Marie Legendre who proved in 1785 that it is solvable in integers x, y, z, not all zero, if and only if...
- Legendre's conjectureLegendre's conjectureLegendre's conjecture, proposed by Adrien-Marie Legendre, states that there is a prime number between n2 and 2 for every positive integer n. The conjecture is one of Landau's problems and unproven ....
- Legendrian submanifold
- Saccheri–Legendre theoremSaccheri–Legendre theoremIn absolute geometry, the Saccheri–Legendre theorem asserts 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 the axiom that is equivalent to the parallel postulate...
- Legendre transformationLegendre transformationIn mathematics, the Legendre transformation or Legendre transform, named after Adrien-Marie Legendre, is an operation that transforms one real-valued function of a real variable into another...
External links
- The True Face of Adrien-Marie Legendre (Portrait of Legendre)
- Biography at Fermat's Last Theorem Blog
- References for Adrien-Marie Legendre 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)