Adrien-Marie Legendre (18 September 1752 – 10 January 1833) was a
FrenchFrance , officially the French Republic , is a country located in Western Europe, with several overseas islands and territories located on other continents. Metropolitan France extends from the Mediterranean Sea to the English Channel and the North Sea, and from the Rhine to the Atlantic Ocean...
mathematicianA mathematician is a person whose primary area of study and/or research is the field of mathematics. Mathematicians are concerned with particular problems related to logic, space, transformations, numbers and more general ideas which encompass these concepts...
. He made important contributions to
statisticsStatistics is a branch of mathematics concerned with collecting and interpreting data. According to other definitions, it is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. Statisticians improve the quality of data with the...
,
number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
,
abstract algebraAbstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
and
mathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of pure mathematics most explicitly concerned with the notion of a limit, whether the limit of a sequence or the limit of a function...
.
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 a top quality 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 from 1813. 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. During the French Revolution, in 1793, he lost his private fortune, but was able, with the help of his wife, Marguerite-Claudine Couhin, whom he married in the same year, to put his affairs in order. 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'Intérieure 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.
The
MoonThe Moon is Earth's only natural satellite and the fifth largest satellite in the Solar System. The average centre-to-centre distance from the Earth to the Moon is , about thirty times the diameter of the Earth. The common centre of mass of the system is located at about —a quarter the Earth's...
crater
LegendreLegendre 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.
Scientific activity
Most of his work was brought to perfection by others: his work on roots of
polynomialIn mathematics, a polynomial is a finite length expression constructed from variables and constants, by using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents...
s inspired
Galois theoryIn 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 functionIn complex analysis, a mathematical discipline, an elliptic function is a function defined on the complex plane that is periodic in two directions...
s was built on Legendre's; some of
GaussJohann 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...
' work in
statisticsStatistics is a branch of mathematics concerned with collecting and interpreting data. According to other definitions, it is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. Statisticians improve the quality of data with the...
and
number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
completed that of Legendre. He developed the
least squaresThe method of least squares is applied to approximate solutions of overdetermined systems, i.e. systems of equations in which there are more equations than unknowns. Least squares is often applied in statistical contexts, particularly regression analysis....
method, which has broad application in
linear regressionIn statistics, linear regression refers to any approach to modeling the relationship between one or more variables denoted y and one or more variables denoted X, such that the model depends linearly on the unknown parameters to be estimated from the data...
,
signal processingSignal processing is an area of electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time to perform useful operations on those signals...
, statistics, and
curve fittingCurve 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
FrenchFrench is a Romance language globally spoken by about 65 million people as a first language , by 50 million as a second language, and by about another 200 million people as an acquired foreign language, with significant speakers in 57 countries. Most native speakers of the language live in France,...
"méthode des moindres carrés".
In 1830 he gave a proof of
Fermat's last theoremIn 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
DirichletJohann Peter Gustav Lejeune Dirichlet was a German mathematician credited with the modern formal definition of a function.- Biography :...
in 1828.
In number theory, he conjectured the
quadratic reciprocityThe law of quadratic reciprocity is a theorem from modular arithmetic, a branch of number theory, which gives conditions for the solvability of quadratic equations modulo prime numbers. There are a number of equivalent statements of the theorem, which consists of two "supplements" and the...
law, subsequently proved by Gauss; in connection to this, the
Legendre symbolThe Legendre symbol or quadratic character is a function introduced by Adrien-Marie Legendre in 1798 during his partly successful attempt to prove the law of quadratic reciprocity....
is named after him. He also did pioneering work on the distribution of
primesIn mathematics, a prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. The first twenty-six prime numbers are:An infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC. The number 1 is by definition not a prime number...
, and on the application of analysis to number theory. His 1796 conjecture of the
Prime number theoremIn number theory, the prime number theorem describes the asymptotic distribution of the prime numbers. The prime number theorem gives a rough description of how the primes are distributed....
was rigorously proved by
HadamardJacques Salomon Hadamard 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-PoussinCharles-Jean Étienne Gustave Nicolas, Baron de la Vallée Poussin was a Belgian mathematician. He is most well-known for proving the Prime number theorem.- Biography :...
in 1898.
Legendre did an impressive amount of work on
elliptic functionIn complex analysis, a mathematical discipline, an elliptic function is a function defined on the complex plane that is periodic in two directions...
s, including the classification of
elliptic integralIn 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
AbelNiels Henrik Abel was a noted Norwegian mathematician who proved the impossibility of solving the quintic equation in radicals.-Early life:...
's stroke of genius to study the inverses of
JacobiCarl Gustav Jacob Jacobi was a Prussian mathematician, widely considered to be the most inspiring teacher of his time and one of the greatest mathematicians of all time.-Biography:...
's
functionsIn 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 transformIn 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
LagrangianThe 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 known as Lagrangian mechanics. In classical mechanics, the...
to the
HamiltonianHamiltonian 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 mechanicsIn the fields of physics, classical mechanics is one of the two major sub-fields of study in the science of mechanics, which is concerned with the set of physical laws governing and mathematically describing the motions of bodies and aggregates of bodies geometrically distributed within a certain...
. In
thermodynamicsIn physics, thermodynamics is the study of the conversion of energy into work and heat and its relation to macroscopic variables such as temperature, volume and pressure...
it is also used to obtain the
enthalpyIn thermodynamics and molecular chemistry, the enthalpy is a thermodynamic property of a thermodynamic system. It can be used to calculate the heat transfer during a quasistatic process taking place in a closed thermodynamic system under constant pressure...
and the
HelmholtzIn thermodynamics, the Helmholtz free energy is a thermodynamic potential which measures the “useful” work obtainable from a closed thermodynamic system at a constant temperature and volume...
and
GibbsIn thermodynamics, the Gibbs free energy is a thermodynamic potential that measures the "useful" or process-initiating work obtainable from an isothermal, isobaric thermodynamic system...
(free) energiesIn thermodynamics, the term thermodynamic free energy refers to the amount of work that can be extracted from a system, and is helpful in engineering applications...
from the
internal energyIn thermodynamics, the internal energy of a thermodynamic system, or a body with well-defined boundaries, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules and the potential energy associated with the vibrational and electric energy of atoms...
. He is also the namegiver of the
Legendre polynomialsIn mathematics, Legendre functions are solutions to Legendre's differential equation:They are named after Adrien-Marie Legendre. This ordinary differential equation is frequently encountered in physics and other technical fields...
, solutions to Legendre's differential equation, which occur frequently in physics and engineering applications,
e.g. electrostaticsElectrostatics is the branch of science that deals with the phenomena arising from stationary or slow-moving electric charges.Since classical antiquity it was known that some materials such as amber attract light particles after rubbing. The Greek word for amber, ήλεκτρον , was the source of the...
.
He also wrote the influential
Éléments de géométrie in 1794.
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.
See also
- Gauss-Legendre algorithm
The Gauss–Legendre algorithm is an algorithm to compute the digits of π. It is notable for being rapidly convergent, with only 25 iterations producing 45 million correct digits of π...
- Legendre's constant
Legendre'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 . Its value is now known to be exactly 1....
- Legendre's equation
In mathematics, Legendre's equation is the Diophantine equationThe 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 polynomials
In mathematics, Legendre functions are solutions to Legendre's differential equation:They are named after Adrien-Marie Legendre. This ordinary differential equation is frequently encountered in physics and other technical fields...
- Legendre's conjecture
Legendre's conjecture, proposed by Adrien-Marie Legendre, states that there is a prime number between n2 and 2 for every positive integer n...
- 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...
- Legendre symbol
The Legendre symbol or quadratic character is a function introduced by Adrien-Marie Legendre in 1798 during his partly successful attempt to prove the law of quadratic reciprocity....
- Saccheri–Legendre theorem
In absolute geometry, the Saccheri–Legendre theorem asserts that the sum of the angles in a triangle is at most 180° in any geometry satisfying the first four Euclidean postulates.The theorem is named after Giovanni Girolamo Saccheri and Adrien-Marie Legendre....
External links