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LEGENDRE, ADRIEN MARIE (1752–1833)
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See also:LEGENDRE, ADRIEN See also:MARIE (1752–1833) , See also:French mathematician, was See also:born at See also:Paris (or, according to some accounts, at See also:Toulouse) in 1752. He was brought up at Paris, where he completed his studies at the See also:College See also:Mazarin. His first published writings consist of articles forming See also:part of the Traite de mecanique (1774) of the See also:Abbe Marie, who was his See also:professor; Legendre's name, however, is not mentioned. Soon afterwards he was appointed professor of See also:mathematics in the Ecole Militaire at Paris, and he was afterwards professor in the Ecole Normale. In 1782 he received the See also:prize from the See also:Berlin See also:Academy for his " Dissertation sur la question de balistique," a memoir See also:relating to the paths of projectiles in resisting See also:media. He also, about this See also:time, wrote his " Recherches sur la figure See also:des planetes," published in the Memoires of the French Academy, of which he was elected a member in See also:succession to J. le Rond d'See also:Alembert in 1783. He was also appointed a See also:commissioner for connecting geodetically Paris and See also:Greenwich, his colleagues being P. F. A. Mechain and C. F. See also:Cassini de Thury; See also:General See also: The French observations were published in 1792 (Expose des operations faitcs en See also:France in 1787 pour la junction des observatoires de Paris et de Greenwich). During the Revolution, he was one of the three members of the See also:council established to introduce the decimal See also:system, and he was also a member of the See also:commission appointed to determine the length of the See also:metre, for which purpose the calculations, &c., connected with the arc of the See also:meridian from See also:Barcelona to See also:Dunkirk were revised. He was also associated with G. C. F. M. See also:Prony (1755–1839) in the formation of the See also:great French tables of logarithms of See also:numbers, sines, and tangents, and natural sines, called the Tables du See also:Cadastre, in which the quadrant was divided centesimally; these tables have never been published (see LOGARITHMS). He was examiner in the Ecole Polytechnique, but held few important See also:state offices. He died at Paris on the loth of See also:January 1833, and the discourse at his See also:grave was pronounced by S. D. See also:Poisson. The last of the three supplements to his Traite des fonctions elliptiques was published in 1832, and Poisson in his funeral oration remarked: " M. Legendre a eu cela de commun avec Ia plupart des geometres qui See also:font precede, que ses travaux n'ont fini qu'avec sa See also:vie. Le dernier See also:volume de nos mmoires renferme encore un memoire de lui, sur une question difficile de la theorie des nombres; et peu de temps avant la maladie qui 1'a conduit au tombeau, ii se procura See also:les observations les plus recentes des cometes a courtes periodes, dont it allait se servir pour appliquer et perfectionner ses methodes." It will be convenient, in giving an See also:account of his writings, to consider them under the different subjects which are especially associated with his name. Elliptic Functions.—This is the subject with which Legendre's name will always be most closely connected, and his researches upon it extend over a See also:period of more than See also:forty years. His first published writings upon the subject consist of two papers in the Memoires de l'Academie Francaise for 1786 upon elliptic arcs. In 1792 he presented to the Academy a memoir on elliptic transcendents. The contents of these See also:memoirs are included in the first volume of his Exercices de calcul integral (1811). The third volume (1816) contains the very elaborate and now well-known tables of the elliptic integrals which were calculated by Legendre himself, with an ac-See also:count of the mode of their construction. In 1827 appeared the Traite des fonctions elliptiques (2 vols., the first dated 1825, the second 1826), a great part of the first volume agrees very closely with the contents of the Exercices; the tables, &c., are given in the second volume. Three supplements, relating to the researches of N. H. See also:Abel and C. G. J. See also:Jacobi, were published in 1828-1832, and See also:form a third volume. Legendre had pursued the subject which would now be called elliptic integrals alone from 1786 to 1827, the results of his labours having been almost entirely neglected by his contemporaries, but his See also:work had scarcely appeared in 1827 when the discoveries which were independently made by the two See also:young and as yet unknown mathematicians Abel and Jacobi placed the subject on a new basis, and revolutionized it completely. The readiness with which Legendre, who was then seventy-six years of See also:age, welcomed these important researches, that quite overshadowed his own, and included them in successive supplements to his work, does the highest See also:honour to him (see See also:FUNCTION). Eulerian Integrals and Integral Calculus.—The Exercices de calcul integral consist of three volumes, a great portion of the first and the whole of the third being devoted to elliptic functions. The See also:remainder of the first volume relates to the Eulerian integrals and to quadratures. The second volume (1817) relates to the Eulerian integrals, and to various integrals and See also:series, developments, See also:mechanical problems, &c., connected with the integral calculus; this volume contains also a numerical table of the values of the See also:gamma function. The latter portion of the second volume of the Traite des fonctions elliptiques (1826) is also devoted to the Eulerian integrals, the table being reproduced. Legendre's researches connected with the " gamma function " are of importance, and are well known; the subject was also treated by K. F. See also:Gauss in his memoir Disquisitiones generales circa series infinitas (1816), but in a very different manner. The results given in the second volume of the Exercices are of too See also:miscellaneous a See also:character to admit of being briefly described. In 1788 Legendre published a memoir on See also:double integrals, and in 1809 one on definite integrals. Theory of Numbers.—Legendre's Theorie des nombres and Gauss's Disquisitiones arithmeticae (18ot) are still See also:standard See also:works upon this subject. The first edition of the former appeared in 1798 under the See also:title Essai sur la theorie des nombres; there was a second edition in 1808; a first supplement was published in 1816, and a second in 1825. The third edition, under the title Theorie des nombres, appeared in 183o in two volumes. The See also:fourth edition appeared in 1900. To Legendre is due the theorem known as the See also:law of quadratic See also:reciprocity, the most important general result in the See also:science of numbers which has been discovered since the time of P. de See also:Fermat, and which was called by Gauss the " See also:gem of See also:arithmetic." It was first given by Legendre in the Memoires of the Academy for 1785, but the demonstration that accompanied it was incomplete. The See also:symbol (See also:alp) which is known as Legendre's symbol, and denotes the See also:positive or negative unit which is the remainder when See also:a4 '> is divided by a See also:prime number p, does not appear in this memoir, but was first used in the Essai sur la theorie des nombres. Legendre's See also:formula x: (See also:log x–I•o8366) for the approximate number of forms inferior to a given number x was first given by him also in this work (2nd ed., p. 394) (see NUMBER). Attractions of Ellipsoids.—Legendre was the author of four important memoirs on this subject. In the first of these, entitled Rechercnes sur 1'attraction des spheroides homogenes," published in the Memoires of the Academy for 1785, but communicated to it at an earlier period, Legendre introduces the celebrated expressions which, though frequently called See also:Laplace's coefficients, are more correctly named after Legendre. The See also:definition of the coefficients is that if (1-211 See also:cos ¢+h2)-1 be See also:expanded in ascending See also:powers of h, and if the general See also:term be denoted by P"h", then P, is of the Legendrian coefficient of the nth See also:order. In this memoir also the function which is now called the potential was, at the See also:suggestion of Laplace, first introduced. Legendre shows that See also:Maclaurin's theorem with respect to confocal ellipsoids is true for any position of the See also:external point when the ellipsoids are solids of revolution. Of this memoir See also:Isaac See also:Todhunter writes: " We may affirm that no single memoir in the See also:history of our subject can See also:rival this in See also:interest and importance. During forty years the resources of See also:analysis, even in the hands of d'Alembert, See also:Lagrange and Laplace, had not carried the theory of the attraction of ellipsoids beyond the point which the See also:geometry of Maclaurin had reached. The introduction of the coefficients now called Laplace's, and their application, commence a new era in mathematical physics." Legendre's second memoir was communicated to the Academie in 1784, and relates to the conditions of See also:equilibrium of a See also:mass of rotating fluid in the form of a figure of revolution which does not deviate much from a See also:sphere. The third memoir relates to Laplace's theorem respecting confocal ellipsoids. Of the fourth memoir Todhunter writes: " It occupies an important position in the history of our subject. The most striking addition which is here made to previous researches consists in the treatment of a See also:planet supposed entirely fluid; the general See also:equation for the form of a stratum is given for the first time and discussed. For the first time we have a correct and convenient expression for Laplace's nth coefficient." (See Todhunter's History of the Mathematical Theories of Attraction and the Figure of the See also:Earth (1873), the twentieth, twenty-second, twenty-fourth, and twenty-fifth chapters of which contain a full and See also:complete account of Legendre's four memoirs. See also SPHERICAL HARIdoNIcs.) See also:Geodesy.—Besides the work upon the geodetical operations connecting Paris and Greenwich, of which Legendre was one of the authors, he published in the Memoires de l'Academie for 1787 two papers on trigonometrical operations depending upon the figure of the earth, containing many theorems relating to this subject. The best known of these, which is called Legendre's theorem, is usually given in See also:treatises on spherical See also:trigonometry; by means of it a small spherical triangle may be treated as a See also:plane triangle, certain corrections being applied to the angles. Legendre was also the author of a memoir upon triangles See also:drawn upon a See also:spheroid. Legendre's theorem is a fundamental one in geodesy, and his contributions to the subject are of the greatest importance. Method of Least Squares.—In 18o6 appeared Legendre's Nouvelles Mcthodes pour la determination des orbites des cometes, which is memorable as containing the first published suggestion of the method of least squares (see See also:PROBABILITY). In the See also:preface Legendre re-marks: " La methode qui me paroit la plus See also:simple et la plus generate consiste a rendre minimum la See also:Somme des quarres des erreurs, . et que j'appelle methode des moindres quarres "; and in an appendix in which the application of the method is explained his words are: " Dc tous les principes qu'on peut proposer pour cet objet, je pense qu'il n'en est pas de plus general, de See also:pius exact, ni d'une application plus facile que celui dont nous avons fait usage dans les recherches precedentes, et qui consiste a rendre minimum la sornme des quarres des erreurs." The method was proposed by Legendre only as a convenient See also:process for treating observations, without reference to the theory of probability. It had, however, been applied by Gauss as See also:early as 1795, and the method was fully explained, and the law of facility for the first time given by him in 1809. Laplace also justified the method by means of the principles of the theory of probability; and this led Legendre to republish the part of his Nouvelles Methodes which related to it in the Memoires de l'Academie for 181o. Thus, although the method of least squares was first formally proposed by Legendre, the theory and algorithm and mathematical See also:foundation of the process are due to Gauss and Laplace. Legendre published two supplements to his Nouvelles Mcthodes in 18oe and 1820. The Elements of Geometry.—Legendre's name is most widely known on account of his Elements de geomctrie, the most successful of the numerous attempts that have been made to supersede See also:Euclid as a See also:text-See also:book on geometry. It first appeared in 1794, and went through very many See also:editions, and has been translated into almost all See also:languages. An See also:English See also:translation, by See also:Sir See also:David See also:Brewster, from the See also:eleventh French edition, was published in 1823, and is well known in England. The earlier editions did not contain the trigonometry. In one of the notes Legendre gives a See also:proof of the irrationality, of 71. This had been first proved by J. H. See also:Lambert in the Berlin Memoirs for 1768. Legendre's proof is similar in principle to Lambert's, but much simpler. On account of the objections urged against the treatment of See also:parallels in this work, Legendre was induced to publish in 1803 his Nouvelle Thcorie des paralleles. His Geomitrie gave rise in England also to a lengthened discussion on the difficult question of the treatment of the theory of parallels. It will thus be seen that Legendre's works have placed him in the very foremost See also:rank in the widely distinct subjects of elliptic functions, theory of numbers, attractions, and geodesy, and have given him a conspicuous position in connexion with the integral calculus and other branches of mathematics. He published a memoir on the integration of partial See also:differential equations and a few others which have not been noticed above, but they relate to subjects with which his name is not especially associated. A See also:good account of the See also:principal works of Legendre is given in the Bibliotheque universelle de Geneve for 1833, pp. 45-82. See See also:Elie de See also:Beaumont, Memoir de Legendre," translated by C. A. See also: Additional information and CommentsThere are no comments yet for this article.
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