Adrien-Marie Legendre

How to use this course

Adrien-Marie Legendre’s name appears in more corners of mathematics and physics than almost any other: Legendre polynomials in electrostatics, the Legendre transformation in mechanics and thermodynamics, the Legendre symbol in number theory, the Legendre condition in the calculus of variations, Legendre’s relation for elliptic integrals, and the method of least squares in every statistics course. He rarely finished a subject. He opened it, organized it, conjectured its main results and handed it to a younger generation, often to Carl Friedrich Gauss, Niels Henrik Abel or Carl Gustav Jacob Jacobi.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proofs or proof sketches, and exercises with folded solutions. The lessons follow the order of his work: potential theory (1782–1784), variational problems (1786–1787), number theory (1785–1830), elliptic integrals (1786–1832), least squares (1805), and geometry (1794–1833).

Timeline

Paris and the Collège Mazarin, 1752–1775

Adrien-Marie Legendre was born on 18 September 1752, almost certainly in Paris; a little evidence points to Toulouse, with an early move to the capital. His family was wealthy, and very little else is known of his childhood. He left no memoirs, and his surviving letters are almost all scientific.

He was educated at the Collège Mazarin, also called the Collège des Quatre-Nations, one of the best schools in Paris for mathematics and natural philosophy. In 1770, at the age of 18, he defended theses in mathematics and physics there. The thesis was a programme of topics rather than original research, as was usual at the time.

His private income meant he did not need a post. For several years he lived in Paris and studied on his own, reading the mechanics and analysis of Leonhard Euler, d’Alembert and Lagrange. That independence shaped his career. He worked slowly and systematically, returned to the same subjects for decades, and was more interested in complete, well-organized treatises than in quick publication.

The Berlin prize and the Académie, 1775–1787

From 1775 to 1780 Legendre taught mathematics at the École Militaire in Paris, appointed on d’Alembert’s recommendation. Laplace taught at the same school. The posts were teaching posts for young officers, but they placed Legendre at the centre of Parisian science.

In 1782 he won the prize of the Berlin Academy for an essay on the paths of cannonballs and bombs through air, Recherches sur la trajectoire des projectiles dans les milieux résistants. The question was practical: how to correct firing tables for air resistance. Legendre treated it with the new analysis, and the essay brought him to the notice of Lagrange, then director of mathematics in Berlin.

In January 1783 he submitted a memoir on the attraction of ellipsoids to the Paris Académie des Sciences. Laplace reported on it favourably, and on 30 March 1783 Legendre was elected adjoint of the Académie, the junior rank; in 1785 he became associé. Between 1783 and 1787 he wrote the papers that still carry his name in physics and analysis: the attraction of spheroids and the polynomials now called Legendre polynomials (Lesson 04), the test separating maxima from minima in the calculus of variations and the Legendre transformation (Lesson 05), and his first long memoir on number theory (Lesson 06).

This was also the period of a lasting rivalry with Laplace. Both worked on the shape of the Earth and the attraction of spheroids, and Their results overlapped, and questions of credit between them were never fully settled.

Geodesy, the metre and the Revolution, 1787–1799

In 1787 the Académie charged Legendre, with the director of the Paris Observatory and the astronomer Pierre Méchain, with linking the Paris and Greenwich observatories by triangulation, in cooperation with a British team. The French party crossed to Dover and visited London, where they met William Herschel. The survey produced Legendre’s theorem on small spherical triangles (Lesson 10). He was elected a Fellow of the Royal Society of London.

In 1791 the Académie began the work that produced the metric system. Legendre served on the commission on weights and measures and was one of three commissioners for the geodetic measurements on which the metre was to be based; he asked to be relieved of that duty in March 1792. He also worked with Gaspard de Prony on the great tables of logarithms and trigonometric functions for the decimal division of the circle, the tables du cadastre.

In August 1793 the Convention suppressed the Académie. Legendre lost the capital that had given him his independence. In 1793 he married Marguerite-Claudine Couhin, who helped him put his affairs back in order. In the same year he printed at his own expense his memoir on elliptic transcendentals, since the Académie no longer existed to publish it.

The new republic needed teachers and textbooks. In 1794 Legendre published Éléments de géométrie, which became the most successful geometry textbook of the nineteenth century. In 1795 he taught at the short-lived first École Normale, and on 13 December he was elected a resident member of the mathematics section of the new Institut National. In 1799 he succeeded Laplace as examiner in mathematics for the artillery, a well-paid post he held until 1815.

Attraction of spheroids and Legendre polynomials

For the general reader. The Earth is not a perfect sphere, and the pull of gravity on and off its surface depends on its shape. To compute that pull, Legendre expanded the inverse distance between two points in a series of special polynomials. Those polynomials now appear wherever a problem has spherical symmetry: in gravity, electrostatics, the quantum hydrogen atom and the analysis of the cosmic microwave background.

The generating function

Let a unit point mass sit at a point at distance \(r'\) from the origin, and let a field point lie at distance \(r>r'\), the angle between the two position vectors being \(\theta\). With \(x=\cos\theta\) and \(t=r'/r<1\), the law of cosines gives

Legendre introduced the coefficients \(P_n(x)\) in his work on the attraction of spheroids of revolution, presented in 1783 and developed in Recherches sur la figure des planètes (1784). Expanding gives \(P_0=1\), \(P_1=x\), \(P_2=\tfrac12(3x^2-1)\), \(P_3=\tfrac12(5x^3-3x)\).

Proof sketch of orthogonality

Write \(L y=\big((1-x^2)y'\big)'\). Integration by parts shows \(\int_{-1}^1 (Lf)g\,dx=\int_{-1}^1 f(Lg)\,dx\), because the boundary terms carry the factor \(1-x^2\), which vanishes at \(\pm1\). If \(LP_n=-n(n+1)P_n\) and \(LP_m=-m(m+1)P_m\), then \(\big(n(n+1)-m(m+1)\big)\int P_mP_n=0\), so the integral vanishes for \(m\ne n\). For the normalization, square the generating function and integrate:

while the left side is \(\sum_n t^{2n}\int P_n^2\) by orthogonality. Comparing coefficients gives \(\int P_n^2=2/(2n+1)\).

Potential of an axisymmetric body

For a body with density \(\rho\) symmetric about the \(z\)-axis, and a field point outside a sphere containing it, summing the expansion over the mass gives

the form still used for the Earth’s gravity field (the \(n=1\) term vanishes if the origin is the centre of mass). The coefficient \(J_2\approx1.083\times10^{-3}\) measures the Earth’s flattening. Laplace’s spherical harmonics, introduced soon afterwards, generalize the \(P_n\) to bodies without axial symmetry, and their axisymmetric members are exactly Legendre’s polynomials.

Variations and the Legendre transformation

For the general reader. The calculus of variations looks for curves that make a quantity, such as length or travel time, as small as possible. Euler and Lagrange gave an equation every such curve must satisfy, but a solution of that equation might be a maximum, a minimum or neither. In 1786 Legendre proposed a test to tell them apart. A year later, studying soap-film surfaces, he invented a change of variables that physicists now use every day to move from Lagrangian to Hamiltonian mechanics.

The second variation

Let \(J[y]=\int_a^b F(x,y,y')\,dx\) with fixed endpoints, and let \(y\) satisfy the Euler–Lagrange equation \(F_y-\frac{d}{dx}F_{y'}=0\). For a variation \(y+\varepsilon h\) with \(h(a)=h(b)=0\),

after integrating the cross term \(2F_{yy'}hh'=F_{yy'}(h^2)'\) by parts.

Proof idea. If \(P(x_0)<0\), choose \(h\) to be a narrow tent of half-width \(\delta\), height \(\delta\) and slopes \(\pm1\) centred at \(x_0\). Then \(\int Ph'^2\approx 2\delta P(x_0)\) while \(\int Qh^2=O(\delta^3)\), so the second variation is negative for small \(\delta\). The idea is Legendre’s; the rigorous form is due to later writers.

Legendre’s trick and its gap

Legendre wanted a sufficient condition too. For any differentiable \(w\), \(\int_a^b (wh^2)'\,dx=0\), so the second variation equals \(\int (Ph'^2+2whh'+(Q+w')h^2)\). If \(w\) solves the Riccati equation \(P(Q+w')=w^2\), the integrand becomes the perfect square \(P\,(h'+wh/P)^2\), which is non-negative when \(P>0\). The flaw, noted by Lagrange, is that the Riccati equation may have no solution on the whole interval: its solutions can blow up. Jacobi repaired the argument in 1836–37 by substituting \(w=-Pu'/u\), which turns the Riccati equation into a linear equation for \(u\); the obstruction is a zero of \(u\), a conjugate point. The strengthened Legendre condition \(P>0\), together with the absence of conjugate points in \((a,b]\), is sufficient for a weak minimum.

The Legendre transformation

For a convex function \(f\), define

If \(f\) is smooth and strictly convex, the supremum is attained where \(p=f'(x)\), and then \(x=(f^*)'(p)\). The transform is an involution on lower semicontinuous convex functions, \(f^{**}=f\), and Young’s inequality \(px\le f(x)+f^*(p)\) holds. In mechanics, with \(p=\partial L/\partial\dot q\), the Hamiltonian \(H(q,p)=p\dot q-L(q,\dot q)\) is the Legendre transform of \(L\) in the velocity variable. In thermodynamics the same operation turns the internal energy \(U(S,V)\) into the Helmholtz energy \(F=U-TS\), the enthalpy and the Gibbs energy.

Legendre’s own use, in his 1787 memoir on the integration of partial differential equations, was for surfaces \(z(x,y)\). With \(p=z_x\), \(q=z_y\) he set \(\omega(p,q)=px+qy-z\). Then \(\omega_p=x\), \(\omega_q=y\), and the second derivatives swap roles:

Quadratic reciprocity and the Legendre symbol

For the general reader. Is 3 a perfect square when we only care about remainders after division by 383? The question sounds arbitrary, but its answers follow a beautiful law. Euler had glimpsed it. Legendre stated it clearly in 1785, gave it a compact notation in 1798, and tried to prove it. Gauss gave the first complete proof in 1801, and the law became the starting point of modern number theory.

The symbol

Let \(p\) be an odd prime and \(p\nmid a\). Legendre’s symbol is \(\left(\frac ap\right)=+1\) if \(a\) is a square modulo \(p\) and \(-1\) otherwise; set \(\left(\frac ap\right)=0\) when \(p\mid a\). He introduced it in the Essai sur la théorie des nombres of 1798.

Proof. The multiplicative group \((\mathbb Z/p)^\times\) is cyclic of even order \(p-1\), generated by some \(g\). The squares are the even powers of \(g\), and \(g^{k(p-1)/2}=(-1)^k\) because \(g^{(p-1)/2}\) is the unique element of order 2.

Legendre’s 1785 argument, revised in 1798, assumed two facts he could not prove: that every arithmetic progression \(a, a+d, a+2d,\dots\) with \(\gcd(a,d)=1\) contains infinitely many primes, and the existence of auxiliary primes with prescribed residue behaviour. The first is Dirichlet’s theorem of 1837. Gauss, who found the law independently as a teenager, criticized the gap in the Disquisitiones Arithmeticae (1801). Legendre adopted Gauss’s proof in his 1808 edition, and in 1830 added a proof by Jacobi.

Two theorems of Legendre

This is the first instance of a local-to-global principle: solvability is decided by congruences and signs. Helmut Hasse’s local-global theorem (1921–24) extends it to quadratic forms in any number of variables, over \(\mathbb Q\) and then over number fields.

Necessity is easy (Exercise 6.2). Legendre’s 1798 proof of sufficiency again used primes in arithmetic progressions; Gauss gave a complete proof in 1801 through his theory of ternary forms.

Primes, factorials and Fermat for n = 5

For the general reader. Primes look random up close but regular from far away. Legendre was among the first to guess a precise law for how many primes there are below a given number, by fitting a formula to tables. He also found a neat rule for how many times a prime divides a factorial, and he helped settle a case of Fermat’s famous equation.

Proof. \(v_p(n!)=\sum_{m\le n}v_p(m)=\sum_{m\le n}\#\{k\ge1:p^k\mid m\}=\sum_{k\ge1}\#\{m\le n:p^k\mid m\}\), and the last count is \(\lfloor n/p^k\rfloor\). For the digit form, write \(n=\sum d_ip^i\); then \(\lfloor n/p^k\rfloor=\sum_{i\ge k}d_ip^{i-k}\), and summing over \(k\) gives \(\sum_i d_i(p^i-1)/(p-1)\).

Counting primes

In the 1798 Essai Legendre proposed that the number \(\pi(x)\) of primes up to \(x\) is approximately \(x/(A\ln x+B)\). In the 1808 edition he fixed the constants from tables up to about 400 000:

Gauss, who had made a similar empirical study, preferred the logarithmic integral \(\mathrm{Li}(x)=\int_2^x dt/\ln t\). Both forms imply \(\pi(x)\sim x/\ln x\), the prime number theorem, proved independently by Jacques Hadamard and Charles de la Vallée Poussin in 1896. Around 1850 Pafnuty Chebyshev showed that if \(x/\pi(x)-\ln x\) has a limit, the limit must be \(1\), not \(1.08366\). Legendre’s constant fits the range of his tables well and is asymptotically wrong.

Fermat’s equation for \(n=5\)

Fermat’s claim that \(x^n+y^n=z^n\) has no positive integer solutions for \(n\ge3\) had been proved only for \(n=3\) (Euler) and \(n=4\) (Fermat). In 1825 the twenty-year-old Dirichlet presented a proof for \(n=5\) to the Académie that covered one of the two cases, and Legendre completed the other shortly afterwards. The method, an infinite descent using numbers of the form \(a+b\sqrt5\), anticipates the arithmetic of the field \(\mathbb Q(\sqrt5)\) and so the ideal theory that Ernst Kummer and Richard Dedekind built for this problem.

Elliptic integrals and Eulerian integrals

For the general reader. The length of a circle’s arc is easy; the length of an ellipse’s arc is not. It leads to integrals that cannot be written with ordinary functions. Legendre spent forty years classifying such elliptic integrals, reducing them to three standard forms, finding their relations and computing tables. Just as his treatise appeared, Abel and Jacobi turned the subject inside out by inverting these integrals.

Normal forms

An elliptic integral is \(\int R\big(x,\sqrt{P(x)}\big)\,dx\) with \(R\) rational and \(P\) a cubic or quartic without repeated roots. Legendre showed, in memoirs from 1786 and 1793 and in the Exercices (1811–1817), that every such integral is a combination of elementary functions and three standard integrals, with modulus \(0<k<1\):

These are the integrals of the first, second and third kind. The complete integrals are \(K(k)=F(\tfrac\pi2,k)\) and \(E(k)=E(\tfrac\pi2,k)\), and \(k'=\sqrt{1-k^2}\) is the complementary modulus. The arc of an ellipse \(x=a\sin\theta\), \(y=b\cos\theta\) with eccentricity \(e\) is \(aE(\varphi,e)\), which gave the subject its name.

Proof sketch. Call the integral \(I(a,b)\). The substitution \(\sin\theta=\dfrac{2a\sin\psi}{a+b+(a-b)\sin^2\psi}\) shows \(I(a,b)=I\big(\frac{a+b}2,\sqrt{ab}\big)\). Iterating, \(I(a,b)=I(M,M)=\pi/(2M)\). The invariance is a form of the transformation of John Landen (1775), which Legendre used systematically to compute his tables; the AGM form is Gauss’s. Convergence is quadratic, so a few steps give many digits.

One proof differentiates the left side in \(k\), using \(\frac{dE}{dk}=\frac{E-K}{k}\) and \(\frac{dK}{dk}=\frac{E-k'^2K}{kk'^2}\), to show it is constant, then evaluates the limit \(k\to0\). In Jacobi’s theory the relation becomes a statement about the periods of an elliptic curve; it is the elliptic analogue of \(\sin^2+\cos^2=1\) for periods, and a special case of the period relations of Bernhard Riemann.

Eulerian integrals

Legendre gave the name Eulerian integrals to Euler’s beta and gamma integrals, and introduced the notation \(\Gamma\), with

He tabulated \(\log\Gamma(x)\) for \(1\le x\le2\) to twelve decimal places in the second volume of the Traité des fonctions elliptiques (1826).

The method of least squares

For the general reader. Surveyors and astronomers always had more measurements than unknowns, and the measurements never quite agreed. How should they be combined? Legendre’s answer, published in 1805, was to choose the values that make the sum of the squared errors as small as possible. The rule is simple, leads to linear equations, and is now the most used method in data analysis.

The method appeared in an appendix, Sur la méthode des moindres quarrés, to Legendre’s Nouvelles méthodes pour la détermination des orbites des comètes (Paris, 1805; the appendix is often dated 1806). His worked example used the meridian measurements of Jean-Baptiste Delambre and Méchain, the data behind the metre.

Proof. \(A\mathbf x\) ranges over the column space \(W\) of \(A\). The closest point of \(W\) to \(\mathbf y\) is its orthogonal projection, characterized by \(\mathbf y-A\mathbf x\perp W\), that is \(A^{\mathsf T}(\mathbf y-A\mathbf x)=0\). The projection exists and is unique; \(\mathbf x\) is unique iff \(A\) is injective, iff \(A^{\mathsf T}A\) is invertible. Legendre obtained the same equations by setting the partial derivatives of the sum of squares to zero.

Priority and justification

In the Theoria motus corporum coelestium (1809), Gauss wrote that he had used the method since 1795, called it “our principle”, and gave a probabilistic justification: if errors are independent and normally distributed, least squares gives the most probable values. Legendre protested in 1811 and again in 1820. Historians agree that Gauss had used the method earlier, notably to recover the asteroid Ceres in 1801, and that Legendre was the first to publish it and to present it as a general method. Laplace soon justified it by the central limit theorem, and in 1821–23 Gauss proved that it gives the best linear unbiased estimates under much weaker assumptions; Andrey Markov later revisited this result, now the Gauss–Markov theorem.

Geometry: Éléments, spherical triangles, parallels

For the general reader. For almost a century, students in France and the United States learned geometry from Legendre, not from Euclid. His textbook rearranged Euclid for clarity and added appendices on new results, including a proof that \(\pi\) is irrational. Legendre also spent thirty years trying to prove Euclid’s parallel postulate. He failed, as everyone must, but proved along the way a theorem that holds in every geometry.

Spherical triangles (1787)

In geodesy, triangles are measured on the curved Earth. On a sphere of radius \(R\), the angles of a geodesic triangle of area \(S\) sum to \(\pi+\varepsilon\), where \(\varepsilon=S/R^2\) is the spherical excess.

Surveyors could therefore solve their triangles by plane trigonometry after subtracting one third of the excess from each measured angle. In 1806 Legendre extended the result to triangles on a spheroid.

Angle sums and the parallel postulate

Proof. Suppose triangle \(ABC\) has angle sum \(\pi+\delta\), \(\delta>0\). Let \(M\) be the midpoint of \(BC\) and extend \(AM\) to \(D\) with \(MD=AM\). Triangles \(AMC\) and \(DMB\) are congruent, so triangle \(ABD\) has the same angle sum as \(ABC\), and the angle at \(A\) splits into two parts, one at \(A\) and one at \(D\), at least one of which is at most half the original angle at \(A\). After \(n\) steps we have a triangle with angle sum \(\pi+\delta\) and one angle at most \(\angle A/2^n<\delta\). Its other two angles then sum to more than \(\pi\), contradicting Euclid I.17, which does not depend on the parallel postulate.

Legendre also proved that if one triangle has angle sum \(\pi\), all do. His many attempts to exclude a sum less than \(\pi\) all assumed, in some hidden form, an equivalent of the parallel postulate. His last essay on the question appeared in 1833, shortly after János Bolyai and Nikolai Lobachevsky had published non-Euclidean geometry, which he did not take into account. Giovanni Saccheri had proved the angle-sum result in 1733; Legendre found it independently.

The Éléments and \(\pi\)

A note in the 1794 Éléments de géométrie proves that \(\pi\) and \(\pi^2\) are irrational, simplifying Johann Heinrich Lambert’s continued-fraction argument for \(\tan x\); the result for \(\pi^2\) was new. Legendre conjectured that \(\pi\) is not a root of any algebraic equation with rational coefficients. Ferdinand von Lindemann proved this in 1882. The book went through many editions, and its English translations, especially Charles Davies’s American adaptation, dominated geometry teaching in the United States for much of the nineteenth century.

Empire, Restoration and two young rivals, 1800–1830

Under Napoleon, Legendre became a pillar of the scientific establishment. He was examiner for the artillery, a member of the geometry section of the reorganized Institut from 1803, a conseiller titulaire of the Imperial University from 1808, and a chevalier de l’Empire in 1811. In 1813 he succeeded Lagrange at the Bureau des Longitudes. He published steadily: the comet orbits with least squares (1805), the second edition of the number theory (1808), the three volumes of Exercices de calcul intégral (1811–1817).

Two priority quarrels with Gauss marked these years. The first concerned quadratic reciprocity, where Gauss’s Disquisitiones of 1801 pointed out the gaps in Legendre’s proof and claimed the first rigorous one. The second, over least squares, was sharper; Legendre felt that Gauss took credit for a method Legendre had published first.

In 1824 Legendre refused to vote for the government’s candidate in an election at the Institut. His pension was stopped, and he spent his last years in reduced circumstances. He kept working. The Traité des fonctions elliptiques, his life’s main project, appeared from 1825.

Abel and Jacobi

In 1827 two young mathematicians, Abel in Norway and Jacobi in Königsberg, independently transformed Legendre’s subject. They inverted the elliptic integral, treating \(\varphi\) as a function of \(u=F(\varphi,k)\), and discovered that the inverse functions are doubly periodic. Jacobi wrote to Legendre in 1827. Legendre, then 74, responded with generous enthusiasm, presented Jacobi’s results to the Académie, and praised him to Alexander von Humboldt. When Jacobi told him that Abel was working on the same questions, Legendre wrote to Abel as well. Between 1828 and 1832 he published three supplements to his Traité setting out their new theory, which made much of his own approach obsolete. The correspondence with Jacobi, published later, is one of the warmest documents in the history of mathematics. Abel died in April 1829, aged 26.

Last years and legacy, 1830–1833

In 1830 Legendre published the third edition of his number theory, now titled Théorie des nombres, in two volumes, including Jacobi’s proof of reciprocity. He was made an officer of the Légion d’honneur in 1831 and a Foreign Honorary Member of the American Academy of Arts and Sciences in 1832. In June 1832 he sent Jacobi his last reflections on the theory of parallels.

He died in Paris on 9 January 1833, aged 80, after a long and painful illness; some sources give 10 January. He was buried in the cemetery of Auteuil, where the couple had a house. His widow lived until 1856, carefully preserved his memory, and left their country house to the village.

A face that was not his

For about two centuries, the portrait printed in books as Legendre the mathematician showed the revolutionary politician Louis Legendre (1752–1797). The error was noticed in 2005. The only authenticated contemporary likeness is a watercolour caricature by Julien-Léopold Boilly, from his 1820 album of caricatures of members of the Institut, identified a few years later.

Legacy

• Physics: Legendre polynomials and associated functions are the angular part of every problem with spherical symmetry; the Legendre transformation links Lagrangian and Hamiltonian mechanics and the thermodynamic potentials.
• Analysis: the normal forms of elliptic integrals are still used in numerical libraries; Legendre’s relation, the gamma notation and the duplication formula are standard.
• Number theory: the Legendre symbol, its generalizations by Jacobi and Leopold Kronecker, and the local-to-global idea of his ternary theorem run through the subject.
• Statistics: least squares is the foundation of regression and of most data fitting.
• Teaching: the Éléments de géométrie shaped school geometry in Europe and the Americas for a century.
• Honours: his name is one of the 72 engraved on the Eiffel Tower; a lunar crater, the asteroid 26950 Legendre and a street in the 17th arrondissement of Paris bear his name.

Connections

People linked to Adrien-Marie Legendre across this collection. A name without a link has no page here yet. See Adrien-Marie Legendre in the network of all 337 lives →

Influenced by

• Gaspard Monge
• Giovanni Girolamo Saccheri
• Jean le Rond d'Alembert
• Johann Heinrich Lambert
• Leonhard Euler

Influenced

• Augustin-Jean Fresnel
• Niels Henrik Abel 1828-1829
• Nikolai Lobachevsky
• Évariste Galois

Colleagues and collaborators

• Augustin-Louis Cauchy
• Joseph Fourier
• Joseph-Louis Lagrange
• Peter Gustav Lejeune Dirichlet 1825
• Pierre-Simon Laplace
• Siméon Denis Poisson
• Sophie Germain 1798-1831

Rivals

• Carl Friedrich Gauss
• Pierre-Simon Laplace 1775-1827

Friends

• Carl Gustav Jacob Jacobi 1827-1832

Also mentioned in this life

• André Weil
• János Bolyai
• Pafnuty Chebyshev
• Richard Dedekind
• Euclid
• Pierre de Fermat
• Israel Gelfand
• Josiah Willard Gibbs
• Jacques Hadamard
• Helmut Hasse
• Hermann von Helmholtz
• Alexander von Humboldt
• Leopold Kronecker
• Ernst Eduard Kummer
• Andrey Markov
• Bernhard Riemann
• William Herschel

Fields

• number theory 45 lives
• calculus of variations 9 lives
• potential theory 9 lives
• statistics 7 lives
• geometry 32 lives
• geodesy 12 lives

On CoursesHub

• Variational Calculus · Mathematics for Physics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Adrien-Marie Legendre, MacTutor History of Mathematics.
2. J. Itard, Legendre, Adrien-Marie, Dictionary of Scientific Biography.
3. Adrien-Marie Legendre, Wikipedia; French edition.
4. P. Duren, Changing faces: the mistaken portrait of Legendre, Notices of the AMS 56 (2009).
5. S. M. Stigler, The History of Statistics, Harvard University Press, 1986, chapter 1.

Mathematics

1. A.-M. Legendre, Recherches d’analyse indéterminée, Histoire de l’Académie royale des sciences (1785).
2. A.-M. Legendre, Mémoire sur la manière de distinguer les maxima des minima dans le calcul des variations, Mémoires de l’Académie (1786).
3. A.-M. Legendre, Essai sur la théorie des nombres, Paris, 1798; 2nd ed. 1808; Théorie des nombres, 3rd ed., 1830.
4. A.-M. Legendre, Nouvelles méthodes pour la détermination des orbites des comètes, Paris, 1805, with the appendix on least squares.
5. A.-M. Legendre, Traité des fonctions elliptiques et des intégrales eulériennes, 3 vols. and supplements, Paris, 1825–1832.
6. A.-M. Legendre, Éléments de géométrie, Paris, 1794 and many later editions.
7. Legendre polynomials and Legendre transformation, Wikipedia.
8. J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987.
9. I. M. Gelfand and S. V. Fomin, Calculus of Variations, Prentice-Hall, 1963, chapter 5.
10. A. Weil, Number Theory: An Approach through History from Hammurapi to Legendre, Birkhäuser, 1984.