Legendre sentence example
Legendre there was a feeling of "more than coldness," owing to his appropriation, with scant acknowledgment, of the fruits of the other's labours; and Dr Thomas Young counted himself, rightly or wrongly, amongst the number of those similarly aggrieved by him.
C. Maclaurin, Legendre and d'Alembert had furnished partial solutions of the problem, confining their 1 Annales de chimie et de physique (1816), torn.
Legendre, in 1783, extended Maclaurin's theorem concerning ellipsoids of revolution to the case of any spheroid of revolution where the attracted point, instead of being limited to the axis or equator, occupied any position in space; and Laplace, in his treatise Theorie du mouvement et de la figure elliptique des planetes (published in 1784), effected a still further generalization by proving, what had been suspected by Legendre, that the theorem was equally true for any confocal ellipsoids.
The device known as the method of least squares, for reducing numerous equations of condition to the number of unknown quantities to be determined, had been adopted as a practically convenient rule by Gauss and Legendre; but Laplace first treated it as a problem in probabilities, and proved by an intricate and difficult course of reasoning that it was also the most advantageous, the mean of the probabilities of error in the determination of the elements being thereby reduced to a minimum.
The early death of this talented mathematician, of whom Legendre said "quelle tete celle du jeune Norvegien!", cut short a career of extraordinary brilliance and promise.Advertisement
His first published writings consist of articles forming part of the Traite de mecanique (1774) of the Abbe Marie, who was his professor; Legendre's name, however, is not mentioned.
This is the subject with which Legendre's name will always be most closely connected, and his researches upon it extend over a period of more than forty years.
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 account of the mode of their construction.
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 work had scarcely appeared in 1827 when the discoveries which were independently made by the two young and as yet unknown mathematicians Abel and Jacobi placed the subject on a new basis, and revolutionized it completely.
In 1788 Legendre published a memoir on double integrals, and in 1809 one on definite integrals.Advertisement
To Legendre is due the theorem known as the law of quadratic reciprocity, the most important general result in the science of numbers which has been discovered since the time of P. de Fermat, and which was called by Gauss the " gem of arithmetic."
It was first given by Legendre in the Memoires of the Academy for 1785, but the demonstration that accompanied it was incomplete.
The symbol (alp) which is known as Legendre's symbol, and denotes the positive or negative unit which is the remainder when au s (-1) is divided by a prime number p, does not appear in this memoir, but was first used in the Essai sur la theorie des nombres.
Legendre was the author of four important memoirs on this subject.
In the first of these, entitled " Recherches sur l'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 Laplace's coefficients, are more correctly named after Legendre.Advertisement
Legendre shows that Maclaurin's theorem with respect to confocal ellipsoids is true for any position of the external point when the ellipsoids are solids of revolution.
Legendre's second memoir was communicated to the Academie in 1784, and relates to the conditions of equilibrium of a mass of rotating fluid in the form of a figure of revolution which does not deviate much from a sphere.
The best known of these, which is called Legendre's theorem, is usually given in treatises on spherical trigonometry; by means of it a small spherical triangle may be treated as a plane triangle, certain corrections being applied to the angles.
Legendre was also the author of a memoir upon triangles drawn upon a spheroid.
Legendre's theorem is a fundamental one in geodesy, and his contributions to the subject are of the greatest importance.Advertisement
The method was proposed by Legendre only as a convenient process for treating observations, without reference to the theory of probability.
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 1810.
Legendre published two supplements to his Nouvelles Methodes in 1806 and 1820.
In one of the notes Legendre gives a proof of the irrationality of 7r.
Legendre's proof is similar in principle to Lambert's, but much simpler.Advertisement
On account of the objections urged against the treatment of parallels in this work, Legendre was induced to publish in 1803 his Nouvelle Theorie des paralleles.
It will thus be seen that Legendre's works have placed him in the very foremost 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.
A good account of the principal works of Legendre is given in the Bibliotheque universelle de Geneve for 18 33, pp. 45-82.
On the French side the work was conducted by Count Cassini, Legendre, and Mechain; on the English side by General Roy.
He received £50 for a translation of Legendre's Geometry; and an introduction, explaining the theory of proportion, is said by De Morgan to show that he could have gained distinction as an expounder of mathematical principles.Advertisement
When Danton was arrested, Legendre at first defended him, but was soon cowed and withdrew his defence.
After the fall of Robespierre, Legendre took part in the reactionary movement, undertook the closing of the Jacobin Club, was.
Legendre, who recommended that it should be published in the Recueil des savants strangers, an unparalleled honour for a youth of eighteen.
This writer stated that he had found the germ of his remarks among the papers of his deceased brother, and that they had come from Legendre, who had himself received them from some one unnamed.
This led to a letter from Argand, in which he stated his communications with Legendre, and gave a résumé of the contents of his pamphlet.Advertisement
The readiness with which Legendre, who was then seventy-six years of age, welcomed these important researches, that quite overshadowed his own, and included them in successive supplements to his work, does the highest honour to him (see Function).
In 1806 appeared Legendre's Nouvelles Methodes pour la determination des orbites des cometes, which is memorable as containing the first published suggestion of the method of least squares (see Probability).
Thus, although the method of least squares was first formally proposed by Legendre, the theory and algorithm and mathematical foundation of the process are due to Gauss and Laplace.
The " method of least squares," by which the most probable result can be educed from a body of observational data, was published by Adrien Marie Legendre in 1806, by Carl Friedrich Gauss in his Theoria Motus (1809), which described also a mode of calculating the orbit of a planet from three complete observations, afterwards turned to important account for the recapture of Ceres, the first discovered asteroid (see Planets, Minor).
Under the general heading "Analysis" occur the subheadings "Foundations of Analysis," with the topics theory of functions of real variables, series and other infinite processes, principles and elements of the differential and of the integral calculus, definite integrals, and calculus of variations; "Theory of Functions of Complex Variables," with the topics functions of one variable and of several variables; "Algebraic Functions and their Integrals," with the topics algebraic functions of one and of several variables, elliptic functions and single theta functions, Abelian integrals; "Other Special Functions," with the topics Euler's, Legendre's, Bessel's and automorphic functions; "Differential Equations," with the topics existence theorems, methods of solution, general theory; "Differential Forms and Differential Invariants," with the topics differential forms, including Pfaffians, transformation of differential forms, including tangential (or contact) transformations, differential invariants; "Analytical Methods connected with Physical Subjects," with the topics harmonic analysis, Fourier's series, the differential equations of applied mathematics, Dirichlet's problem; "Difference Equations and Functional Equations," with the topics recurring series, solution of equations of finite differences and functional equations.Advertisement
The director, Schmalfuss, encouraged him in his mathematical studies by lending him books (among them Leonhard Euler's works and Adrien Marie Legendre's Theory of Numbers), which Riemann read, mastered and returned within a few days.
In 1761 he proved the irrationality of 7r; a simpler proof was given somewhat later by Legendre.