Search Results for Legendre


Biographies

  1. Adrien-Marie Legendre (1752-1833)
    • Adrien-Marie Legendre .
    • Adrien-Marie Legendre would perhaps have disliked the fact that this article contains details of his life for Poisson wrote of him in [',' D Poisson, Discours prononce aux funerailles de M Legendre, Moniteur universel (20 Jan 1833), 162.','12]:- .
    • It is not surprising that, given these views of Legendre, there are few details of his early life.
    • In 1770, at the age of 18, Legendre defended his thesis in mathematics and physics at the College Mazarin but this was not quite as grand an achievement as it sounds to us today, for this consisted more of a plan of research rather than a completed thesis.
    • With no need for employment to support himself, Legendre lived in Paris and concentrated on research.
    • His essay Recherches sur la trajectoire des projectiles dans les milieux resistants Ⓣ won the prize and launched Legendre on his research career.
    • In 1782 Lagrange was Director of Mathematics at the Academy in Berlin and this brought Legendre to his attention.
    • Legendre next studied the attraction of ellipsoids.
    • He then introduced what we call today the Legendre functions and used these to determine, using power series, the attraction of an ellipsoid at any exterior point.
    • Legendre submitted his results to the Academie des Sciences in Paris in January 1783 and these were highly praised by Laplace in his report delivered to the Academie in March.
    • Within a few days, on 30 March, Legendre was appointed an adjoint in the Academie des Sciences filling the place which had become vacant when Laplace was promoted from adjoint to associe earlier that year.
    • Over the next few years Legendre published work in a number of areas.
    • In particular he published on celestial mechanics with papers such as Recherches sur la figure des planetes Ⓣ in 1784 which contains the Legendre polynomials; number theory with, for example, Recherches d'analyse indeterminee Ⓣ in 1785; and the theory of elliptic functions with papers on integrations by elliptic arcs in 1786.
    • This is fair since Legendre's proof of quadratic reciprocity was unsatisfactory, while he offered no proof of the theorem on primes in an arithmetic progression.
    • However, these two results are of great importance and credit should go to Legendre for his work on them, although he was not the first to state the law of quadratic reciprocity since it occurs in Euler's work of 1751 and also of 1783 (see [',' H Pieper, Uber Legendres Versuche, das quadratische Reziprozitatsgesetz zu beweisen, Natur, Mathematik und Geschichte.
    • Legendre's career in the Academie des Sciences progressed in a satisfactory manner.
    • This work resulted in his election to the Royal Society of London in 1787 and also to an important publication Memoire sur les operations trigonometriques dont les resultats dependent de la figure de la terre which contains Legendre's theorem on spherical triangles.
    • On 13 May 1791 Legendre became a member of the committee of the Academie des Sciences with the task to standardise weights and measures.
    • At this time Legendre was also working on his major text Elements de geometrie which he had been encouraged to write by Condorcet.
    • However the Academie des Sciences was closed due to the Revolution in 1793 and Legendre had special difficulties since he lost the capital which provided him with a comfortable income.
    • Following the work of the committee on the decimal system on which Legendre had served, de Prony in 1792 began a major task of producing logarithmic and trigonometric tables, the Cadastre.
    • Legendre and de Prony headed the mathematical section of this project along with Carnot and other mathematicians.
    • In 1794 Legendre published Elements de geometrie Ⓣ which was the leading elementary text on the topic for around 100 years.
    • In his "Elements" Legendre greatly rearranged and simplified many of the propositions from Euclid's "Elements" to create a more effective textbook.
    • Legendre's work replaced Euclid's "Elements" as a textbook in most of Europe and, in succeeding translations, in the United States and became the prototype of later geometry texts.
    • In "Elements" Legendre gave a simple proof that π is irrational, as well as the first proof that π2 is irrational, and conjectured that π is not the root of any algebraic equation of finite degree with rational coefficients.
    • Each section of the Institut contained six places, and Legendre was one of the six in the mathematics section.
    • In 1803 Napoleon reorganised the Institut and a geometry section was created and Legendre was put into this section.
    • Legendre published a book on determining the orbits of comets in 1806.
    • In an appendix Legendre gave the least squares method of fitting a curve to the data available.
    • However, Gauss published his version of the least squares method in 1809 and, while acknowledging that it appeared in Legendre's book, Gauss still claimed priority for himself.
    • This greatly hurt Legendre who fought for many years to have his priority recognised.
    • In 1808 Legendre published a second edition of his Theorie des nombres Ⓣ which was a considerable improvement on the first edition of 1798.
    • For example Gauss had proved the law of quadratic reciprocity in 1801 after making critical remarks about Legendre's proof of 1785 and Legendre's much improved proof of 1798 in the first edition of Theorie des nombres Ⓣ.
    • Gauss was correct, but one could understand how hurtful Legendre must have found an attack on the rigour of his results by such a young man.
    • Of course Gauss did not state that he was improving Legendre's result but rather claimed the result for himself since his was the first completely rigorous proof.
    • Legendre later wrote (see [',' S M Stigler, An attack on Gauss, published by Legendre in 1820, Historia Math.
    • To his credit Legendre used Gauss's proof of quadratic reciprocity in the 1808 edition of Theorie des nombres Ⓣ giving proper credit to Gauss.
    • The 1808 edition of Theorie des nombres Ⓣ also contained Legendre's estimate for π(n) the number of primes ≤ n of π(n) = n/(log(n) - 1.08366).
    • Again Gauss would claim that he had obtained the law for the asymptotic distribution of primes before Legendre, but certainly it was Legendre who first brought these ideas to the attention of mathematicians.
    • Legendre's major work on elliptic functions in Exercices du Calcul Integral Ⓣ appeared in three volumes in 1811, 1817, and 1819.
    • In the first volume Legendre introduced basic properties of elliptic integrals and also of beta and gamma functions.
    • However, despite spending 40 years working on elliptic functions, Legendre never gained the insight of Jacobi and Abel and the independent work of these two mathematicians was making Legendre's new three volume work obsolete almost as soon as it was published.
    • Legendre's attempt to prove the parallel postulate extended over 30 years.
    • In 1832 (the year Bolyai published his work on non-euclidean geometry) Legendre confirmed his absolute belief in Euclidean space when he wrote:- .
    • In 1824 Legendre refused to vote for the government's candidate for the Institut National.
    • Legendre is an extremely amiable man, but unfortunately as old as the stones.
    • As a result of Legendre's refusal to vote for the government's candidate in 1824 his pension was stopped and he died in poverty.
    • A Poster of Adrien-Marie Legendre .
    • Legendre's estimates for the density of primes .
    • Honours awarded to Adrien-Marie Legendre .
    • 2.nLunar featuresnCrater Legendre .
    • 3.nParis street namesnPassage Legendre and Rue Legendre (17th Arrondissement) .
    • http://www-history.mcs.st-andrews.ac.uk/Biographies/Legendre.html .

  2. Carl Jacobi (1804-1851)
    • On 5 August 1827 Jacobi wrote to Legendre who was the leading expert on the topic and this letter, together with 22 others between Jacobi and Legendre, is given in [',' E Knobloch, B Mai (trs.) and H Pieper (ed.), Korrespondenz Adrien-Marie Legendre-Carl Gustav Jacob Jacobi (Stuttgart, 1998).','4].
    • Legendre immediately realised that Jacobi had made fundamental advances in his favourite topic.
    • One would have to say that Legendre reacted extremely well to the realisation that his position as the leading expert on elliptic functions had changed overnight with the new theory being developed not only by Jacobi, but also by Abel.
    • Jacobi's promotion to associate professor on 28 December 1827 was mainly due to the praise heaped on him by Legendre.
    • In a letter, sent to Jacobi on 9 February 1828, Legendre wrote:- .
    • In 1829 Jacobi met Legendre and other French mathematicians such as Fourier and Poisson when he made a visit to Paris in the summer vacation.
    • Jacobi's fundamental work on the theory of elliptic functions, which had so impressed Legendre, was based on four theta functions.
    • Legendre expressed this clearly in a letter he wrote to Jacobi early in 1829:- .
    • A few weeks after Legendre wrote this letter Abel died.

  3. Blagoj Popov biography
    • Nowadays, looking back at the list of his publications, it appears that there is no kind of differential equation or special function that has not been studied by him! He had investigated: the equation of ballistics, the hypergeometric, Riccati, Legendre, confluent hypergeometric, Bessel, Weber, Hermite, Darboux, Whittaker, and Laplace differential equations; orthogonal polynomials, Legendre, Gegenbauer, Jacobi, Hermite, Laguerre, Bernoulli, Bessel, and Chebyshev polynomials, the associate spherical Legendre functions, the ultraspherical polynomials, the generalized Legendre and q-Appell polynomials.

  4. John Dougall biography
    • In the following year Dougall published The product of two Legendre polynomials in the same journal.
    • This paper contains a new derivation of the coefficients in the expansion into a series of Legendre polynomials of the product of two Legendre polynomials.

  5. Henri Delannoy biography
    • It is interesting to note that in the 1950s a mysterious connection was spotted between these diagonal Delannoy numbers and Legendre polynomials.
    • Only very recently has Gabor Hetyei explained this connection, which was previously thought to be a coincidence, giving a geometric interpretation of the relation between the diagonal Delannoy numbers and the Legendre polynomials in [',' G Hetyei, Delannony orthants of Legendre polytopes, Discrete Comput.

  6. Bryce McLeod (1929-2014)
    • During the time McLeod was studying for his Oxford B.A., Chaundy taught the courses 'Elementary differential equations and Legendre's functions' and 'Partial differential equations.
    • The article [',' J B McLeod, Review: Solved Problems: Gamma and Beta Functions, Legendre Polynomials, Bessel Functions, by Orin J Farrell and Bertram Ross, The Mathematical Gazette 48 (366) (1964), 482-483.','7] is a review by McLeod of a text on Gamma and Beta Functions, Legendre Polynomials, Bessel Functions.

  7. Augustin-Louis Cauchy biography
    • He submitted his first paper on this topic then, encouraged by Legendre and Malus, he submitted a further paper on polygons and polyhedra in 1812.
    • He failed to obtain this post, Legendre being appointed.
    • When Abel's untimely death occurred on April 6, 1829, Cauchy still had not given a report on the 1826 paper, in spite of several protests from Legendre.

  8. Sophie Germain (1776-1831)
    • Germain wrote to Legendre about problems suggested by his 1798 Essai sur le Theorie des Nombres Ⓣ, and the subsequent Legendre - Germain correspondence became virtually a collaboration.
    • Legendre included some of her discoveries in a supplement to the second edition of the Theorie.

  9. Dominique Cassini biography
    • The Government appointed Cassini as a commissioner, along with Legendre and Mechain, to triangulate the French side.
    • In April 1791 the Academy appointed Cassini, Legendre and Mechain to carry out the task.
    • On 19 June Cassini, Legendre, Mechain, and Borda had an audience with King Louis XVI.

  10. Aurel Angelescu biography
    • The title of his thesis was Sur les polynomes generalisant les polynomes de Legendre et d'Hermite et sur le calcul approche des integrals multiples Ⓣ.
    • He was particularly interested in generating functions for Legendre, Laguerre and Hermite polynomials and one of his main research efforts was put into generalising these polynomials.
    • Examples of his contributions are: Extrait sur les polynomes Ⓣ (1915); Sur le quardatures mechanique Ⓣ (1920); On the linear homogeneous recurrence relations with constant coefficients (Romanian) (1925); The definition of the functions of complex variable (Romanian) (1925); Sur certains developpements des fonctions holomorphes en serie de fractions rationnelles Ⓣ (1925); Sur une formule de M Pompeiu Ⓣ (1929); On some means (Romanian) (1932); Sur le principe de Legendre Ⓣ (1933); Sur une certaine extension des series entieres Ⓣ (1937); and Sur certains polynomes generalisant les polynomes de Laguerre Ⓣ (1938).

  11. Olinde Rodrigues biography
    • Rodrigues was awarded a doctorate in mathematics from the Faculty of Science of the University of Paris in 1816 for a thesis that contains one of the two results for which he is known today, namely the Rodrigues formula for Legendre polynomials: Pn = 1/(2nn!)dn/dxn [(x2 - 1)n].
    • The story of the Rodrigues formula for Legendre polynomials is somewhat more complicated due to the fact that Rodgigues' paper on the subject does not appear to have been noticed at the time, or if it was then it was quickly forgotten.
    • Heine was an expert on Legendre polynomials, Lame functions and Bessel functions and he wrote a book in which he proposed that, since Hermite had shown that Rodrigues had priority in discovering the formula, then it should be known as the Rodrigues formula.

  12. Pierre-Simon Laplace (1749-1827)
    • Laplace served on many of the committees of the Academie des Sciences, for example Lagrange wrote to him in 1782 saying that work on his Traite de mecanique analytique was almost complete and a committee of the Academie des Sciences comprising of Laplace, Cousin, Legendre and Condorcet was set up to decide on publication.
    • The Legendre functions also appear here and were known for many years as the Laplace coefficients.
    • The Mecanique Celeste Ⓣ does not attribute many of the ideas to the work of others but Laplace was heavily influenced by Lagrange and by Legendre and used methods which they had developed with few references to the originators of the ideas.

  13. Jean-Baptiste-Joseph Delambre biography
    • The Academie had already set up a Commission of Weights and Measures in 1790 consisting of Borda, Condorcet, Laplace, Legendre and Lavoisier to advise on a metric system of weights and measures.
    • The Academie des Sciences appointed Mechain, Legendre and Dominique Cassini to carry out this task.

  14. Joseph-Louis Lagrange (1736-1813)
    • It had been approved for publication by a committee of the Academie des Sciences comprising of Laplace, Cousin, Legendre and Condorcet.
    • Legendre acted as an editor for the work doing proof reading and other tasks.

  15. Lejeune Dirichlet biography
    • He had some of the leading mathematicians as teachers and he was able to profit greatly from the experience of coming in contact with Biot, Fourier, Francoeur, Hachette, Laplace, Lacroix, Legendre, and Poisson.
    • Legendre was appointed one of the referees and he was able to prove case 2 thus completing the proof for n = 5.

  16. Pierre Verhulst (1804-1849)
    • This came about since Verhulst bought an edition of the complete works of Legendre in a public sale.
    • He was particularly inspired by Legendre's Traite des fonctions elliptiques Ⓣ and went on to read the works of Niels Abel and Carl Jacobi on elliptic functions.

  17. Pafnuty Chebyshev biography
    • Legendre and Laplace had encountered the Legendre polynomials in their work on celestial mechanics in the late eighteenth century.

  18. Sylvestre Lacroix biography
    • Of the geometry texts of Lacroix and Lagrange, Lamande writes in [',' P Lamande, Trois traites francais de geometrie a l’oree du XIXe siecle : Legendre, Peyrard et Lacroix, Physis Riv.
    • The efforts of Lacroix and Legendre to expose or recast the theory of parallels were notable, as were their additions to the theory.

  19. Siméon-Denis Poisson (1781-1840)
    • A memoir on finite differences, written when Poisson was 18, attracted the attention of Legendre.
    • His first attempt to be elected to the Institute was in 1806 when he was backed by Laplace, Lagrange, Lacroix, Legendre and Biot for a place in the Mathematics Section.

  20. Pierre Méchain (1744-1804)
    • The Government appointed Mechain as a commissioner, along with Legendre and Dominique Cassini, to triangulate the French side.
    • The Commission of Weights and Measures, which had as its members Condorcet, Lavoisier, Laplace and Legendre, was set up by the Academie des Sciences in 1790 to bring in a uniform system of measurement.

  21. Thomas MacRobert biography
    • Formulae for generalized hypergeometric functions as particular cases of more general formulae (1939) showed how certain known formulae for generalized hypergeometric functions can be derived as particular cases of formulae of more general type involving multiple series; Some formulae for the E-function (1941) showed how special cases of the formulae derived lead to interesting relations between Bessel functions, Legendre functions and confluent hypergeometric functions; and Proofs of some formulae for the hypergeometric function and the E-function (1943) gave alternative proofs for some known theorems on hypergeometric functions, then gives a formula for an integral involving the product of two E-functions.
    • He continued to produce papers on the E-function such as On an identity involving E-functions (1948), Integral of an E-function expressed as a sum of two E-functions (1953), An integral involving an E-function and an associated Legendre functions of the first kind (1953), Integrals involving E-functions (1958), Infinite series of E-functions (1959).

  22. Niels Abel (1802-1829)
    • Two referees, Cauchy and Legendre, were appointed to referee the paper and Abel remained in Paris for a few months [',' D Stander, Makers of modern mathematics : Niels Henrik Abel, Bull.
    • Legendre saw the new ideas in the papers which Abel and Jacobi were writing and said ([',' Biography in Encyclopaedia Britannica.','2]):- .

  23. Athanase Dupré (1808-1869)
    • He won an honourable mention for the 1858 Grand Prix of the Academy of Sciences with his paper Examen d'une proposition de Legendre relative a la Theorie des nombres Ⓣ on Legendre's theory of numbers.

  24. Lorenzo Mascheroni (1750-1800)
    • The French had been working on the introduction of the metric system of weights and measures and on 7 April 1795 the National Convention had passed a law introducing the metric system putting Legendre in charge of the transition to the new system.

  25. François Français (1768-1810)
    • After this Francais did work which was praised by Legendre, Lagrange, Lacroix and Biot but submitted no further memoirs during his lifetime.

  26. Christian Kramp (1760-1826)
    • As Bessel, Legendre and Gauss did, Kramp worked on the generalised factorial or function which applied to non-integers.

  27. Bernard Bolzano biography
    • But he was reading and recording his ideas on a host of other subjects as well, including the problem of how best to approach the proper mathematical understanding of zero; Legendre's work on surfaces, convexity, concavity, and conditions for congruity; analysis of other geometric concepts, including lengths, areas, volumes, and spheres; trigonometric formulas and spherical trigonometry; imaginary and exponential numbers; definition of the differential and discussion of the infinite and various opinions about it, as well as aspects of maxima and minima.

  28. Yulian Vasilievich Sokhotsky (1842-1927)
    • Furthermore, Sokhotskii was the first to apply the calculus of residues to Legendre polynomials.

  29. Niels Nielsen (1865-1931)
    • It was the first major work devoted to the study of the gamma function since a treatise by Legendre.

  30. Leopold Gegenbauer biography
    • The Gegenbauer polynomials are solutions to the Gegenbauer differential equation and are generalizations of the associated Legendre polynomials.

  31. Jean-Charles de Borda biography
    • When Borda was made Chairman of the Commission of Weights and Measures, which had as its members Condorcet, Lavoisier, Laplace and Legendre, he soon put his accurate surveying instrument to good use.

  32. C V Mourey biography
    • He is not remembered as a student at any of the well known educational establishments in the city and has never been referred to in connection with the great mathematicians who were known to have been in Paris during that period; thinking of the young talents of Abel and Galois, but also of Cauchy, Poisson, Legendre, Hachette, Dirichlet, Fourier and Lacroix.

  33. Ivar Bendixson biography
    • These methods had first been used by Legendre to prove that e and π are irrational.

  34. Norman Alling biography
    • The first two parts offer a detailed and scrupulous presentation of the historical development of the theory of elliptic integrals and functions in the 18th and 19th centuries, from Giulio Fagnano and Euler through Legendre, Gauss, Abel and Jacobi to Riemann and Weierstrass.

  35. Leonhard Euler (1707-1783)
    • Legendre called these 'Eulerian integrals of the first and second kind' respectively while they were given the names beta function and gamma function by Binet and Gauss respectively.

  36. Thomas Carlyle biography
    • In mathematics Carlyle is famed for his English translation of Legendre's Elements de geometrie which David Brewster commissioned him to undertake for £50 in 1821.

  37. François-Joseph Servois (1768-1847)
    • Legendre realised that Servois had considerable mathematical talents and he supported a move to have him appointed to the artillery school of Besancon as professor of mathematics.

  38. Christoph Gudermann (1798-1852)
    • In his more extensive work on the theory of special functions Gudermann published several papers beginning in 1830 which extended work which was developed by Euler, Landen, Legendre, Abel and Jacobi.

  39. Tom Cowling biography
    • To give an indication of the topics of Cowling papers we list a few from the first part of his career: On certain expansions involving products of Legendre functions (1940); The non-radial oscillations of polytropic stars (1942); The electrical conductivity of an ionized gas in a magnetic field, with applications to the solar atmosphere and the ionosphere (1945); The oscillations of a rotating star (1949); The condition for turbulence in rotating stars (1951); Magneto-hydrodynamic oscillations of a rotating fluid globe (1955).

  40. Emil Grosswald biography
    • He also wrote two papers which were published in 1950, the first being On a simple property of the derivatives of Legendre's polynomials while the second was Functions of bounded variation.

  41. William Thomson (1824-1907)
    • In particular the works of Lagrange, Laplace, Legendre, Fresnel and Fourier were treated with "reverence" to use a word which Thomson himself would later use to describe the attitude that his lecturers had towards these French mathematicians.

  42. Johann Heinrich Lambert (1728-1777)
    • In [',' R Wallisser, On Lambert’s proof of the irrationality of ¹, in Algebraic number theory and Diophantine analysis, Graz, 1998 (Berlin, 2000), 521-530.','34] there is discussion of the claim that Lambert's proof is incomplete and requires a result by Legendre to complete it.

  43. William Whiston (1667-1752)
    • This is of interest because his study precedes the publication of 'Meyer's method' by thirty-one years and the Legendre-Gauss 'method of least squares' by eighty-six years.

  44. Dunham Jackson biography
    • Many a worker will be inspired by the pages devoted to Fourier and Legendre series and to Chebyshev-Hermite polynomials to try his hand in obtaining similar results for other classes of orthogonal functions, and in particular, for other classes of orthogonal Chebyshev polynomials.

  45. Aleksandr Nikolaevich Korkin biography
    • He had read, and with his wonderful memory could then recall, most works by Abel, Dirichlet, Euler, Fourier, Gauss, Jacobi, Lagrange, Laplace, Legendre, Monge, and Poisson.

  46. William Wallace (1768-1843)
    • A further paper submitted to the Royal Society of Edinburgh in 1802 made it clear that this school teacher of mathematics had an outstanding research talent (although it later transpired that Legendre had discovered the results six years earlier).

  47. André Weil (1906-1998)
    • Weil made a major contribution through his books that include Arithmetique et geometrie sur les varietes algebriques Ⓣ (1935), Sur les espaces a structure uniforme et sur la topologie generale Ⓣ (1937), L'integration dans les groupes topologiques et ses applications Ⓣ (1940), Foundations of Algebraic Geometry (1946), Sur les courbes algebriques et les varietes qui s'en deduisent Ⓣ (1948), Varietes abeliennes et courbes algebriques Ⓣ (1948), Introduction a l'etude des varietes kahleriennes Ⓣ (1958), Discontinuous subgroups of classical groups (1958), Adeles and algebraic groups (1961), Basic number theory (1967), Dirichlet Series and Automorphic Forms (1971), Essais historiques sur la theorie des nombres Ⓣ (1975), Elliptic Functions According to Eisenstein and Kronecker (1976), (with Maxwell Rosenlicht) Number Theory for Beginners (1979), Adeles and Algebraic Groups (1982), Number Theory: An Approach Through History From Hammurapi to Legendre (1984), and Correspondance entre Henri Cartan et Andre Weil Ⓣ (1928-1991) (2011).

  48. Félix Pollaczek biography
    • In this area his main interests were in orthogonal polynomials and as well as working on Legendre polynomials, Hermite polynomials and Laguerre polynomials, he introduced polynomials named by Arthur Erdelyi as the 'Pollaczek polynomials'.

  49. James Pierpont biography
    • On the other hand the author, having in mind the needs of the students of applied mathematics, has dwelt at some length on the theory of linear differential equations, especially as regards the functions of Legendre, Laplace, Bessel, and Lame.

  50. Paul Bernays biography
    • Paul Bernays' extend over the most diverse fields of mathematical science: the representation of positive integers by binary quadratic forms (dissertation supervised by landau), elementary theory of Landau's function of Picard's theorem (habilitation thesis in Zurich), Legendre's condition in the calculus of variations, one-dimensional gas as an example of an ergodic system, axiomatic treatment of Russell's propositional calculus (habilitation thesis, Gottingen, not printed).

  51. Frank Jackson biography
    • He wrote much on basic hypergeometric functions, including the basic functions of Legendre and Bessel.

  52. John William Strutt (1842-1919)
    • Among the publications devoted to mathematics, rather than to its applications, are papers on Bessel functions, the relationship between Laplace functions and Bessel functions, and Legendre functions.

  53. Évariste Galois (1811-1832)
    • He studied Legendre's Geometrie and the treatises of Lagrange.

  54. Marion Gray biography
    • Marion C Gray, Legendre functions of fractional order, Quart.

  55. Derrick Henry Lehmer biography
    • He was a pioneer in the application of mechanical methods, including digital computers, to the solution of problems in number theory and he talked about some of the methods used to factorise numbers including: factor tables, trial division, Legendre's method, factor stencils, the continued fraction method, Fermat's method, methods based on quadratic forms, and Shanks's method.

  56. Mary Fasenmyer biography
    • It examined certain special sets of generalized hypergeometric polynomials containing as special cases Legendre's, Jacobi's, Bateman's polynomials, and others.

  57. Gaspard de Prony biography
    • With the assistance of Legendre, Carnot and other mathematicians, and between 70 to 80 assistants, the work was undertaken over a period of years, being completed in 1801.

  58. Étienne Louis Malus (1775-1812)
    • In 1811 Malus served, along with Lagrange, Legendre, Laplace and Hauy, on the committee to decide on who to award the prize to for the best work on the propagation of heat in solid bodies.

  59. Thomas Hakon Grönwall biography
    • Gronwall's work contains classical analysis (Fourier series, Gibbs phenomenon, summability theory, Laplace and Legendre series), differential and integral equations, analytic number theory (transcendental numbers, divisor function, L-function of Dirichlet), complex function theory (Dirichlet L-series, conformal mappings, univalent functions), differential geometry, mathematical physics (problems of elasticity, ballistics, induction, potential theory, kinetic theory of gases, optics), nomography, atomic physics (wave mechanics of hydrogen and helium atom, lattice theory of crystals) and physical chemistry where he is especially known as a very important contributor.

  60. Mikhail Vasilevich Ostrogradski (1801-1862)
    • These were delivered by Louis Poinsot, Pierre-Simon Laplace, Joseph Fourier, Adrien-Marie Legendre, Simeon-Denis Poisson, Jacques Binet and Augustin-Louis Cauchy.

  61. Harry Bateman biography
    • Two further papers appeared in print in 1904, namely The solution of partial differential equations by means of definite integrals, and Certain definite integrals and expansions connected with the Legendre and Bessel functions.

  62. Paolo Ruffini (1765-1822)
    • Ruffini asked the Institute in Paris to pronounce on the correctness of his proof and Lagrange, Legendre and Lacroix were appointed to examine it.

  63. Francis Murnaghan biography
    • It covers topics such as: vectors and matrices; Fourier series; boundary value problems; Legendre and Bessel functions; integral equations; the calculus of variations and dynamics; and the operational calculus.

  64. Mauro Picone biography
    • Some of his most important books which Picone published during his years in Rome are: Appunti di Analisi superiore Ⓣ (1940), which studies harmonic functions, Fourier, Laplace and Legendre series and the equations of mathematical physics; Lezioni di Analisi funzionale Ⓣ (1946), which concerns the calculus of variations; Teoria moderna dell'integrazione delle funzioni Ⓣ (1946), containing a detailed discussion of the r-dimensional Stieltjes integrals; (with Tullio Viola) Lezioni sulla teoria moderna dell'integrazione Ⓣ (1952), which is basically the previous work by Picone with three extra chapters by Viola; and (with Gaetano Fichera) Trattato di Analisi matematica Ⓣ (Vol 1, 1954, Vol 2, 1955), which puts into a treatise Picone's way of teaching calculus particularly slanted towards the applications studied at the Institute for Applied Calculus.

  65. Jacques Français (1775-1833)
    • This was based on Argand's paper which had been sent, without disclosing the name of the author, by Legendre to Francois Francais.

  66. Stanislaw Knapowski biography
    • On experimental evidence, after extensive calculation, Legendre in 1798 and Gauss in 1793 (according to a letter he wrote 50 years later) suggested that for large n the density of primes behaves like the function 1/log(n).

  67. Karl Weierstrass (1815-1897)
    • when I became aware of [a letter from Abel to Legendre] in Crelle's Journal during my student years, [it] was of the utmost importance.

  68. Eduard Heine (1821-1881)
    • Heine worked on Legendre polynomials, Lame functions and Bessel functions.

  69. Karl Heinrich Weise biography
    • In years soon after the war Weise published a small but concise book Gewohnliche Differentialgleichungen Ⓣ (1948) in which he discusses Legendre, Bessel, and Sturm-Liouville equations.

  70. Jean-Robert Argand biography
    • Legendre was sent a copy of the work and he sent it to Francois Francais although neither knew the identity of the author.

  71. Gilbert Bliss biography
    • Tables show various ways the necessary conditions of Euler, Weierstrass, Legendre and Jacobi may be strengthened to give sufficient conditions.

  72. Ernst Jacobsthal biography
    • He also showed that it is possible to find a solution p = x2 + y2 where x and y can be expressed with simple sums over Legendre symbols.

  73. Bernhard Riemann (1826-1866)
    • On one occasion he lent Bernhard Legendre's book on the theory of numbers and Bernhard read the 900 page book in six days.

  74. Max Dehn biography
    • In this thesis he proved the Saccheri-Legendre theorem which states that in absolute geometry the sum of the angles in a triangle is at most 180°.

  75. Joseph Fourier (1768-1830)
    • Only one other entry was received and the committee set up to decide on the award of the prize, Lagrange, Laplace, Malus, Hauy and Legendre, awarded Fourier the prize.

  76. Ian Sneddon biography
    • It was aimed at students of applied mathematics, physics, chemistry and engineering who needed to work with the 'special' functions of Legendre, Bessel, Hermite and Laguerre.

  77. William McFadden Orr (1866-1934)
    • Orr's first publications were on hypergeometric series, Fourier double integrals involving Bessel functions which he followed up with a similar paper involving Legendre functions.

  78. Henry Fox Talbot (1800-1877)
    • Talbot wrote papers on elliptic integrals, building on work of Euler, Legendre, Jacobi and Abel.

  79. Carl Friedrich Gauss (1777-1855)
    • Comparison with Legendre's estimate .

  80. Daniel da Silva biography
    • In this area the author knew the works of Euler, Lagrange, Legendre, Gauss and Poinsot.

  81. Georg Simon Ohm (1789-1854)
    • As he had done for so much of his life, Ohm continued his private studies reading the texts of the leading French mathematicians Lagrange, Legendre, Laplace, Biot and Poisson.

  82. George Hill biography
    • These texts included Lacroix' Traite du calcul differentiel et integral, Lagrange's Mechanique analytique, Laplace's Mechanique celeste and Legendre's Fonctions elliptiques.

  83. Marshall Stone biography
    • For example he published An unusual type of expansion problem (1924), A comparison of the series of Fourier and Birkhoff (1926), Developments in Legendre polynomials (1926), and Developments in Hermite polynomials (1927).

  84. Guglielmo Libri biography
    • Indeed he became a French citizen three years later and was, in that same year of 1833, elected to the Academie des Sciences to succeed Legendre.

  85. Joseph Pérès (1890-1962)
    • The next chapter considers the period from Newton to Euler, followed by a chapter covering the period from 1780 to 1860 where Peres looks at the contributions of Lagrange, Laplace, Legendre, Cauchy, Galois, Gauss, Jacobi, Riemann, and Weierstrass.

  86. Edmund Whittaker (1873-1956)
    • He found expressions for the Bessel functions as integrals involving Legendre functions.

  87. Giorgio Bidone biography
    • His research at this time was on the solution of transcendental equations and also on definite integrals with papers such as Sur diverses integrals definies Ⓣ (1813), in which he used the method of Mascheroni series to reduce various integrals to known cases, and Sur les transcendantes elliptiques Ⓣ (1818) in which he extended the work of Legendre on the numerical values of elliptic functions of the first and second kind.

  88. Vito Volterra (1860-1940)
    • His interest in mathematics started at the age of 11 when he began to study Legendre's Geometry.

  89. Ernst Meissel biography
    • Meissel must be judged as a classical mathematician, continuing a tradition from an earlier epoch associated with names like Euler, Laplace, Legendre, Gauss, Jacobi, and Dirichlet.


History Topics

  1. Prime numbers
    • Legendre and Gauss both did extensive calculations of the density of primes.
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    • Both Legendre and Gauss came to the conclusion that for large n the density of primes near n is about 1/log(n).
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    • Legendre gave an estimate for π(n) the number of primes ≤ n of .
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    • You can see the Legendre estimate at THIS LINK and the Gauss estimate at THIS LINK and can compare them at THIS LINK.
    • Legendre's estimate for π(n) .
    • Legendre and Gauss estimates compared .

  2. Non-Euclidean geometry
    • Legendre spent 40 years of his life working on the parallel postulate and the work appears in appendices to various editions of his highly successful geometry book Elements de Geometrie Ⓣ.
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    • Legendre proved that Euclid's fifth postulate is equivalent to:- .
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    • Legendre showed, as Saccheri had over 100 years earlier, that the sum of the angles of a triangle cannot be greater than two right angles.
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    • In trying to show that the angle sum cannot be less than 180° Legendre assumed that through any point in the interior of an angle it is always possible to draw a line which meets both sides of the angle.
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    • This turns out to be another equivalent form of the fifth postulate, but Legendre never realised his error himself.
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  3. References for Prime numbers
    • J Pintz, On Legendre's prime number formula, Amer.
    • A Weil, Number Theory: An Approach Through History from Hammurapi to Legendre (1984).

  4. Ring Theory
    • For example Legendre and Gauss investigated integer congruences in 1801.
    • n = 5Legendre and Dirichlet1825 .

  5. Fermat's last theorem
    • Sophie Germain proved Case 1 of Fermat's Last Theorem for all n less than 100 and Legendre extended her methods to all numbers less than 197.
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    • Legendre was able to prove Case 2(ii) and the complete proof for n = 5 was published in September 1825.
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  6. References for Elliptic functions
    • L A Sorokina, Legendre's works on the theory of elliptic integrals (Russian), Istor.-Mat.

  7. Weather forecasting
    • The term Pnm(φ) is the associated Legendre function used in spherical harmonics (but will not be explained here).


Societies etc

  1. Paris Academy of Sciences
    • The prize was awarded to Kummer, even although he had not entered! The 1858 Grand Prix was awarded half to Dupre with a paper on Legendre's theory of numbers.


Honours

  1. Passage Legendre
    • Passage Legendre .

  2. Rue Legendre
    • Rue Legendre .

  3. Legendre
    • Adrien Marie Legendre .

  4. Paris street names
    • Passage Legendre (17th Arrondissement) WnMn .
    • Rue Legendre (17th Arrondissement) WnMn .

  5. Lunar features
    • (W) (L) Legendre .

  6. Eiffel Tower
    • Legendre .

  7. Fellows of the RSE
    • Adrien Marie Legendre1820More infoMacTutor biography .

  8. Fellows of the RSE
    • Adrien Marie Legendre1820More infoMacTutor biography .

  9. Eiffel scientists
    • Legendre (Geometer) .

  10. Lunar features
    • Legendre .


References

  1. References for Adrien-Marie Legendre
    • References for Adrien-Marie Legendre .
    • http://www.britannica.com/biography/Adrien-Marie-Legendre .
    • A Aubry, Sur les travaux arithmetiques de Lagrange, de Legendre et de Gauss, Enseignement mathematique 11 (1909), 430-450.
    • E de Beaumont, Eloge historique d'Adrien-Marie Legendre, Memoires de l'Academie des sciences 32 (1864), XXXVII-LXXXVII.
    • C C Gillispie, Memoires inedits ou anonymes de Laplace sur la theorie des erreurs, les polynomes de Legendre, et la philosophie des probabilites, Rev.
    • C D Hellman, Legendre and the French reform of weights and measures, Osiris 1 (1936), 314-340.
    • P Lamande, Trois traites francais de geometrie a l'oree du XIXe siecle : Legendre, Peyrard et Lacroix, Physis Riv.
    • L Maurice, Memoire sur les travaux et ecrits de M Legendre, Bibliotheque universelle des sciences, belles-lettres et arts.
    • S Maracchia, Legendre e l'incommensurabilita tra lato e diagonale di uno stesso quadrato, Archimede 29 (2) (1977), 123-125.
    • S Maracchia, 'Dimostrazione' del V postulato secondo Legendre, Archimede 25 (1973), 98-102.
    • D Poisson, Discours prononce aux funerailles de M Legendre, Moniteur universel (20 Jan 1833), 162.
    • J-B Pecot, Le probleme de l'ellipsoide et l'analyse harmonique : la controverse entre Legendre et Laplace, in Analyse diophantienne et geometrie algebrique (Paris, 1993), 113-157.
    • J M Querard, Legendre, Adrien-Marie, France litteraire 5 (1833), 94-95.
    • J Pintz, On Legendre's prime number formula, Amer.
    • O B Sheinin, Previous publication of Legendre's attack on Gauss: 'An attack on Gauss, published by Legendre in 1820' by S M Stigler, Historia Math.
    • D E Smith, Legendre on least squares, in A source book of mathematics (New York, 1929), 576-579.
    • L Sorokina, A Legendre's works on the theory of elliptic integrals (Russian), Istor.-Mat.
    • S M Stigler, An attack on Gauss, published by Legendre in 1820, Historia Math.

  2. References for Michel Plancherel
    • De la sommation des series de Legendre .
    • Les problemes de Cantor et de Dubois-Reymond dans la theorie des series de polynomes de Legendre.
    • Sur la sommation des series de Laplace de de Legendre.
    • Unicite du developpement d'une fonction en serie de polynomes de Legendre et expression anaytique des coefficients de ce developpement.
    • Les problemes de Cantor et de Du Bois-Reymond dans la theorie des series de polynomes de Legendre.
    • Sur l'unicite du developpement d'une fonction en serie de polynomes de Legendre.

  3. References for André Weil
    • An approach through history: From Hammurapi to Legendre, by Andre Weil, Isis 77 (1) (1986), 153-154.
    • An approach through history: From Hammurapi to Legendre, by Andre Weil, Science.
    • An approach through history: From Hammurapi to Legendre, by Andre Weil, Revue d'histoire des sciences 41 (2), Algebre, Analyse, Topologie.
    • An approach through history: From Hammurapi to Legendre, by Andre Weil, American Scientist 73 (5) (1985), 489.

  4. References for Pierre-Simon Laplace
    • C C Gillispie, Memoires inedits ou anonymes de Laplace sur la theorie des erreurs, les polynomes de Legendre, et la philosophie des probabilites, Rev.
    • J-B Pecot, Le probleme de l'ellipsoide et l'analyse harmonique : la controverse entre Legendre et Laplace, in Analyse diophantienne et geometrie algebrique (Paris, 1993), 113-157.

  5. References for Sylvestre Lacroix
    • P Lamande, Trois traites francais de geometrie a l'oree du XIXe siecle : Legendre, Peyrard et Lacroix, Physis Riv.

  6. References for Carl Jacobi
    • E Knobloch, B Mai (trs.) and H Pieper (ed.), Korrespondenz Adrien-Marie Legendre-Carl Gustav Jacob Jacobi (Stuttgart, 1998).

  7. References for Henri Delannoy
    • G Hetyei, Delannony orthants of Legendre polytopes, Discrete Comput.

  8. References for Bryce McLeod
    • J B McLeod, Review: Solved Problems: Gamma and Beta Functions, Legendre Polynomials, Bessel Functions, by Orin J Farrell and Bertram Ross, The Mathematical Gazette 48 (366) (1964), 482-483.

  9. References for Carl Friedrich Gauss
    • S M Stigler, An attack on Gauss, published by Legendre in 1820, Historia Math.


Additional material

  1. Sansone publications
    • Giovanni Sansone, Un criterio sufficiente di convergenza in medio per le serie di polinomi di Legendre, Bollettino U.
    • Giovanni Sansone, Sulla chiusura dei polinomi di Legendre, Bollettino U.
    • Giovanni Sansone, Sulle serie lacunari di polinomi di Legendre di funzioni sommabili, Annali Pisa (2) 2 (1933), 289-296.
    • Giovanni Sansone, La chiusura dei sistemi ortogonali di Legendre, di Laguerre e di Hermite rispetto alle funzione di quadrato sommabile, Giorn.
    • Giovanni Sansone, La chiusura dei sistemi ortogonali di Legendre, di Laguerre e di Hermite rispetto alle funzioni di quadrato sommabile, Atti Soc.
    • Giovanni Sansone, Sulle serie lacunari di polinomi di Legendre di funzioni sommabili, Atti Soc.
    • Giovanni Sansone, Sulla convergenza delle serie di Legendre, Ann.
    • Giovanni Sansone, Condizioni sufficienti per il problema dei momenti rispetto al sistema ortogonale di Legendre, Atti primo Congr.
    • Giovanni Sansone, Condizioni sufficienti per il problema dei momenti rispetto al sistema ortogonale di Legendre, Rend.
    • Giovanni Sansone, Su una immediata limitazione delle derivate dei polinomi di Legendre, Boll.
    • Giovanni Sansone, Su una disuguaglianza di P Turan relativa ai polinomi di Legendre, Boll.
    • Giovanni Sansone, Su una disuguaglianza relativa ai polinomi di Legendre, Boll.

  2. Weil reviews
    • These theories were developed by Legendre, Gauss, Jacobi, Eisenstein, Kronecker and many others in the last century.
    • An approach through history: From Hammurapi to Legendre (1983), by Andre Weil.
    • The chapters are (i) a quick trip through the ancient world, with particular attention to the contributions of the Mesopotamians, Greeks and Indians; (ii) a visit with Fermat (1601-1665); (iii) an extended stay with Euler (1707-1783); and (iv) brief stops to see Lagrange (1736) and Legendre (1752-1833).
    • But in my opinion, the main strength of the book lies in the penetrating analysis of the thoughts and achievements in number theory of Fermat, Euler, Lagrange, and Legendre.
    • This book is a study of various number-theoretic texts ranging from the Babylonian tablet known as Plimpton 322 to A M Legendre's Essay on the theory of numbers, and covering a time span from ca.

  3. T M MacRobert: 'Spherical Harmonics' Contents
    • The Legendre polynomials.
    • The Legendre functions.
    • The associated Legendre functions of integral order.
    • Applications of Legendre coefficients to potential theory.
    • Associated Legendre functions of general order.

  4. Byrne: Doctrine of Proportion
    • Why it should be considered so, will readily be conceived when such men as Legendre, Leslie, Keith, Bonnycastle, Austin, Brewster, Young, and in fact every one who has attempted to treat the doctrine of Geometrical Proportion on any plan differing from Euclid's, have committed errors, overlooked mistakes, retrenched the generality of Euclid's reasonings, fallen into logical absurdities, or confined the general application of a subject which pervades a whole course of mathematics; while there is not one mistake, oversight, or logical objection in the whole of Euclid's Fifth Book.
    • Legendre, the great French geometer, does not give proportion a place in his Elements of Geometry, for he was of opinion that the subject belonged to arithmetic, and algebra, not to geometry at all.
    • Sir David Brewster, in his Translation of Legendre s Geometry, falls into a notable error, insomuch that he makes assertions which are not at all true.
    • In speaking of Legendre, he says, "the author has provided for the application of proportion to incommensurable quantities, and demonstrated every case of this kind as it occurred, by means of the reductio ad absurdum." This assertion Professor Young very justly questioned, and has given examples from Brewster's translation, where the inference does not hold good.

  5. MacRobert: 'Spherical Harmonics' Preface
    • Chapters V, VI, and VII are devoted respectively to the Legendre Coefficients, the Legendre Functions, and the Associated Legendre Functions.

  6. Weil on history
    • An approach through history: From Hammurapi to Legendre (1984).
    • An approach through history: From Hammurapi to Legendre (1984).
    • After that day, Euler never lost sight of this topic and of number theory in general; eventually Lagrange followed suit, then Legendre, then Gauss with whom number theory reached full maturity.

  7. Gauss: 'Disquisitiones Arithmeticae
    • The really profound discoveries are due to more recent authors like those men of immortal glory P de Fermat, L Euler, L Lagrange, A M Legendre (and a few others).
    • I will not recount here the individual discoveries of these geometers since they can be found in the Preface to the appendix which Lagrange added to Euler's Algebra and in the recent volume of Legendre (which I shall soon cite).
    • Meanwhile there appeared the outstanding work of that man who was already an expert in Higher Arithmetic, Legendre's "Essai d'une theorie des nombres." Here he collected together and systematized not only all that had been discovered up to that time but added many new results of his own.

  8. Sansone books
    • The improvements and additions of Sansone are very pleasing to the harmonious structure of the author; they contribute to making the work a reliable and easy guide for one who wants to know the most beautiful and the highest theories of the modern analysis of the real variable, as well as for a suitable introduction to the subject matter of which Sansone gives a concise and clear representation in the second volume: the theory of the developments in series of orthogonal functions, and especially in the series of Legendre, of Chebyshev- Laguerre, and from Chebyshev-Hermite polynomials.
    • The topics discussed are, by chapters: (1) Developments in orthogonal functions, introduction to Hilbert space; (2) Fourier series; (3) Series of Legendre polynomials and of spherical harmonics; (4) Laguerre and Hermite series; (5) Approximation and interpolation; (6) Stieltjes integrals (including a discussion of Fourier transforms of distribution functions; this chapter has little connection with the rest of the book).
    • This is a translation of the first four chapters of the third edition (1952) of Sansone's second volume of Vitali's 'Moderna teoria delle funzioni di variabile reale', dealing with general theorems, Fourier series, series of Legendre polynomials and spherical harmonics, and series of Laguerre and Hermite functions.

  9. L R Ford - Differential Equations
    • The next chapter on certain classical equations gives a good introduction to the hypergeometric, the Legendre, and the Bessel differential equations.
    • Subsequent chapters cover special methods for equations of first order, linear equations of any order with a brief account of the use of the Laplace transform, solution in series of the hypergeometric, Legendre's and Bessel's equations, approximate numerical solutions, and two chapters on partial differential equations.

  10. Barlow Numbers
    • It appears that "Theory of Numbers" in the title of this book is the first occurrence of this phrase in English although Legendre wrote Essai sur la theorie des nombres in 1798.
    • It is, however, but lately that this branch of analysis has been reduced into a regular system, a task that was first performed by Legendre, in his "Essai sur la Theorie des Nombres;" and nearly at the same time Gauss published his "Disquisitiones Arithmeticae:" these two works eminently display the talents of their respective authors, and contain a complete development of this interesting theory.

  11. V Lebesgue publications
    • Victor Amedee Lebesgue, Demonstration nouvelle et elementaire de la loi de reciprocite de Legendre, par M Eisenstein, precedee et suivie de remarques sur d'autres demonstrations qui peuvent etre tirees du meme principe, Journal de Mathematique pures et appliquees.
    • Victor Amedee Lebesgue, Sur un theoreme des nombres (Legendre, Theorie des nombres, II, 144), Nouvelles Annales de Mathematiques (1) XV (1856), 403-407.

  12. Hille publications
    • On the zeros of Legendre functions, Ark.
    • On the complex zeros of the associated Legendre functions, J.

  13. Gábor Szegö's books
    • Now a depth of critical understanding which scarcely went beyond the fundamental cases of Fourier and Legendre series has come to prevail with unifying authority over a wider range of generalization than had been even tentatively surveyed, and the diverse fields into which the applications extend derive clarification from a common body of coordinated knowledge.
    • However, these omissions are compensated by several interesting features: (a) an elaborate treatment of the asymptotic behaviour of orthogonal polynomials, by various methods, with applications, in particular, to the "classical" polynomials of Legendre, Jacobi, Laguerre and Hermite; (b) a detailed study of expansions in series of orthogonal polynomials, regarding convergence and summability; (c) a detailed study of orthogonal polynomials in the complex domain; (d) a study of the zeros of orthogonal polynomials, particularly of the classical ones, based upon an extension of Sturm's theorem for differential equations.

  14. Percy MacMahon addresses the British Association in 1901, Part 2
    • A combination of great intellects Legendre, Gauss, Eisenstein, Stephen Smith, etc.
    • Euler, Legendre, Gauss, Eisenstein, Jacobi, Kronecker, Poincare, and Klein are great names that will be for ever associated with it.

  15. Cayley: 'Elliptic Functions
    • The present treatise is founded upon Legendre's Traite des Fonctions Elliptiques, and upon Jacobi's Fundamenta Nova, and Memoirs by him in Crelle's Journal: comparatively very little use is made of the investigations of Abel or of those of later authors.
    • I show how the transition is made from Legendre's Elliptic Integrals of the three kinds to Jacobi's Amplitude, which is the argument of the Elliptic Functions (the sine, cosine, and delta of the amplitude, or as with Gudermann I write them, sn, cn, dn), and also of Jacobi's functions Z, P, which replace the integrals of the second and third kinds, and of the functions Q, H, which he was thence led to.

  16. V Lebesgue books
    • The work of Legendre is no longer sufficient in spite of its extent, and by this very fact the author has not been willing to confine himself to the simple role of translator.
    • The excellent Recherches arithmetiques of Gauss and the translation of this work being completed, Legendre's Theorie des Nombres having also become rare, it seemed to me that it would be useful to write a new Treatise presenting roughly the present state of the science of numbers.

  17. Catalan retirement
    • Encore gamin de Paris (bon gamin !), j'ai vu Legendre et j'ai connu Bouvart parfois, je fus aide (benevole) de Hachette et d'Ampere.

  18. Andrew Forsyth addresses the British Association in 1905, Part 2
    • It is to be desired that those persons who are best fitted to improve the science of calculation should direct their labours to these important applications.' Abel was soon to pass beyond the range of admonition; but Jacobi, in a private letter to Legendre, protested that the scope of the science was not to be limited to the explanation of natural phenomena.

  19. Catalan retirement
    • Still a child of Paris (a good child!), I saw Legendre and I knew Bouvart: sometimes I was helped (kindly) by Hachette and Ampere.

  20. Who was who 1852
    • J L Lagrange (1736-1813) spent the last twenty years of his wandering life in France, men like J Fourier (1768-1830), P S Laplace (1749-1827), A M Legendre (1752-1833) and S D Poisson (1781-1840) brought French mathematics to new heights.

  21. Craig Differential Equations
    • Previous to this the only class of linear differential equations for which a general method of integration was known was the class of equations with constant coefficients, including of course Legendre's well-known equation which is immediately transformable into one with constant coefficients.

  22. Johnson pre1900 books
    • Chapter VIII is devoted to the general solution of the binomial equation in the notation of the hypergeometric series, and Chapter IX to Riccati's, Bessel's and Legendre's equations.

  23. Teixeira on da Silva
    • In this area the author knew the works of Euler, Lagrange, Legendre, Gauss and Poinsot.

  24. Three Sadleirian Professors
    • Most of Professor Hobson's researches have been connected with the theory of functions of real variables, but he has also dealt with Legendre's and Bessel's functions, integral equations, potential theory, the conduction of heat, and the calculus of variations.

  25. Walk Around Paris
    • After that year, during which he studied algebra and analysis, he studied very little for his rhetoric course, and was continuing his mathematical journey, studying Gauss's algebra, and the calculations taught from Lagrange and Legendre, even the Abel-Ruffini theorem.

  26. Abel Crelle letter
    • Thank you very much for taking the trouble to copy and send me excerpts from the letters you received from Jacobi and Legendre.

  27. Encke Obituary
    • For the method itself we are mainly indebted to Legendre and to Gauss, but for the first exhibition of its vast practical value, we are indebted to the example of Encke.

  28. Milnor's books
    • The theory of quadratic forms and the intimately related theory of symmetric bilinear forms have a long and rich history, highlighted by the work of Legendre, Gauss, Minkowski, and Hasse.

  29. Valdivia Infinity
    • In this connection, I quote from a letter from the German mathematician Jacobi in the last century to the French mathematician Legendre.

  30. Newcomb Elements of Geometry
    • In the present work is developed what is commonly known as the ancient or Euclidian Geometry, the ground covered being nearly the same as in the standard treatises of Euclid, Legendre, and Chauvenet.

  31. Bolzano publications
    • In these entries Bolzano comments on mathematical texts he has read in preparation for writing his Grossenlehre particularly Legendre's Elements de geometrie.

  32. Speiser books
    • The poverty of Hegel's own contribution to the foundations of mathematics was due, Speiser considers, to the lack of interest in mathematics in Germany at the time; France's Lagrange, Laplace, Legendre, Carnot, Dupin, Lame, Monge, Poncelet, Lacroix and Cauchy were challenged in Germany only (but outstandingly) by Gauss.

  33. Gyula König Prize
    • This contains, as a special case, both Legendre and power series.

  34. Finkel's Solution Book
    • This reaction, it may be said, started as early as 1832, the time when Benjamin Peirce, the first American worthy to be ranked with Legendre, Wallis, Abel and the Bernouillis, became professor of mathematics and natural philosophy at Harvard University.

  35. Weyl on Hilbert
    • Paul Bernays's publications cover a variety of areas of mathematical science: the representation of positive integers by binary quadratic forms (dissertation supervised by Edmund Landau), elementary theory of landau's function of Picard's theorem (habilitation thesis in Zurich), Legendre's condition in the calculus of variations, one-dimensional gas as an example of an ergodic system, axiomatic treatment of Russell's propositional calculus (habilitation thesis, Gottingen, not printed).

  36. Percy MacMahon addresses the British Association in 1901
    • Whereas in 1801 on the Continent there were the leaders Lagrange, Laplace and Legendre, and of rising men, Fourier, Ampere, Poisson and Gauss, we could only claim Thomas Young and Ivory as men who were doing notable work in research.

  37. Peres books
    • Nowhere, perhaps, would it be more legitimate to speak of a Greek miracle." The following chapters deal successively with the progress made up to Newton exclusively; from the period of Newton to Euler, which represents the science of the eighteenth century, and then of the period from 1780 to 1860, which was that of the great French geometers: Lagrange, Laplace, Legendre, Gauchy, Galois, and which is also brilliantly illustrated by the German school with Gauss, Jacobi, Riemann, Weierstrass, the latter being connected, however, rather with the contemporary movement.

  38. Henry Baker addresses the British Association in 1913, Part 2
    • The attempt to extend the possibilities of integration to the case when the function to be integrated involves the square root of a polynomial of the fourth order, led first, after many efforts, among which Legendre's devotion of forty years was part, to the theory of doubly-periodic functions.

  39. Smith's Teaching Books
    • The present work is a compromise between the traditional treatment of Euclid and Legendre, and the more natural and heuristic methods of the modern geometers.

  40. NAS Memoir of Chauvenet
    • Yet, following the system of Legendre, he wrote the whole in his own clear, precise style, improving wherever improvements could be made, and occasionally introducing a new proposition or a new mode of solution, and enlarging the limits of this fundamental branch.

  41. Magnus books
    • The gamma function and related functions; The hypergeometric function; Bessel functions; Legendre functions; Orthogonal polynomials; Rummer's function; Whittaker functions; Parabolic cylinder functions and parabolic functions; The incomplete gamma function and special cases; Elliptic integrals, theta functions and elliptic functions; Integral transforms; Transformation of systems of coordinates.

  42. Sneddon: 'Special functions
    • This book is intended primarily for the student of applied mathematics, physics, chemistry or engineering who wishes to use the "special" functions associated with the names of Legendre, Bessel, Hermite and Laguerre.

  43. Dubreil-Jacotin on Sophie Germain
    • Attacking the proof of Fermat's last theorem with the help of Legendre's formulas, she supplied an important theorem and its application to the proof of Fermat's theorem up to the hundredth degree.

  44. Ahrens book reviews
    • An effective impression of the progress of mathematical instruction is made by a brief account of mathematical instruction in the "good old times." The account of the self-taught Arago's successes when examined on one occasion by Louis Monge and on another by Legendre is inspiring; any one who does not know these stories will be repaid if he looks up the booklet for them alone.

  45. Smith's Obituaries and Biographies
    • The mathematician is Adrien Marie Legendre, - great in the theory of numbers (including the principle of least squares), in the field of elliptic functions, and in the applications of the calculus; a prolific writer upon a variety of minor mathematical subjects, and the one who, with better right than any other man, save Euclid, can be called the father of American geometry as taught in our schools today.


Quotations

  1. Quotations by Legendre
    • Quotations by Adrien-Marie Legendre .

  2. Quotations by Riemann
    • From Euclid to Legendre, to name the most renowned of modern writers on geometry, this darkness has been lifted neither by the mathematicians nor the philosophers who have laboured upon it.

  3. Quotations by Bell
    • [As a young teenager] Galois read [Legendre's] geometry from cover to cover as easily as other boys read a pirate yarn.


Famous Curves

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Chronology

  1. Mathematical Chronology
    • Legendre introduces his "Legendre polynomials" in his work Recherches sur la figure des planetes on celestial mechanics.
    • Legendre states the law of quadratic reciprocity but his proof is incorrect.
    • Legendre publishes Elements de geometrie, an account of geometry which would be a leading text for 100 years.
    • Legendre develops the method of least squares to find best approximations to a set of observed data.
    • This is named "Germain's theorem" by Legendre.
    • Jacobi writes a letter to Legendre detailing his discoveries on elliptic functions.
    • Legendre points out the flaws in 12 "proofs" of the parallel postulate.

  2. Chronology for 1780 to 1800
    • Legendre introduces his "Legendre polynomials" in his work Recherches sur la figure des planetes on celestial mechanics.
    • Legendre states the law of quadratic reciprocity but his proof is incorrect.
    • Legendre publishes Elements de geometrie, an account of geometry which would be a leading text for 100 years.

  3. Chronology for 1800 to 1810
    • Legendre develops the method of least squares to find best approximations to a set of observed data.
    • This is named "Germain's theorem" by Legendre.

  4. Chronology for 1820 to 1830
    • Jacobi writes a letter to Legendre detailing his discoveries on elliptic functions.

  5. Chronology for 1830 to 1840
    • Legendre points out the flaws in 12 "proofs" of the parallel postulate.


EMS Archive

  1. Edinburgh Mathematical Society Lecturers 1883-2016
    • (School House, Cowbridge) Generalised forms of the series of Bessel and Legendre .
    • (StnAndrews) Note on Legendre's and Bertrand's proof of the parallel postulate by infinite areas .
    • (Edinburgh) On an integral-equation whose solutions are the Legendre polynomials .
    • (Glasgow) The addition theorem for the Legendre functions of the second kind .
    • (Glasgow) On some Legendre function formulae .
    • (Glasgow) On some relations between the Bessel and Legendre functions .
    • (Durham) Some properties of Legendre polynomials .

  2. 1925-26 Dec meeting
    • MacRobert, Thomas Murray: "On some Legendre function formulae", [Proceedings, Vol.

  3. 1902-03 Mar meeting
    • Jackson, Frank Hilton: "Generalised forms of the series of Bessel and Legendre" .

  4. 1911-12 Dec meeting
    • Sommerville, Duncan M Y: "Note on Legendre's and Bertrand's proof of the parallel postulate by infinite areas" .

  5. 1929-30 Mar meeting
    • MacRobert, Thomas Murray: "On some relations between the Bessel and Legendre functions", [Not printed in an EMS publication] .

  6. 1914-15 Jan meeting
    • Whittaker, Edmund Taylor: "On an integral-equation whose solutions are the Legendre polynomials", [Title] .

  7. EMS Proceedings papers
    • A proof of the addition theorem for the Legendre functions .

  8. EMS Proceedings papers
    • Note on Legendre's and Bertrand's proof of the parallel postulate by infinite areas .

  9. 1924-25 Mar meeting
    • MacRobert, Thomas M: "The addition theorem for the Legendre functions of the second kind", [Title] .


BMC Archive

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Gazetteer of the British Isles

  1. London individuals A-C
    • Thomas Carlyle (1795-1881) is best known as a historian, but started life as a teacher of mathematics and made an influential translation of Legendre's Elements de Geometrie which went through 33 editions in America.

  2. Kirkcaldy, Fife
    • Here he translated Legendre's Elements de Geometrie in 1822 for David Brewster (whose name appeared in the book as editor, without mentioning Carlyle).


Astronomy section

  1. List of astronomers

  2. List of astronomers
    • Legendre, Adrien-Marie .


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JOC/BS August 2001