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LAGRANGE, JOSEPH LOUIS (1736-1813)
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See also:LAGRANGE, See also:JOSEPH See also: Appointed, in 1754, See also:professor of geometry in the royal school of See also:artillery, he formed with some of his pupils—for the most See also:part his seniors—friendships based on community of scientific ardour. With the aid of the See also:marquis de Saluces and the anatomist G. F. Cigna, he founded in 1758 a society which became the Turin See also:Academy of Sciences. The first See also:volume of its See also:memoirs,' published in the following year, contained a See also:paper by Lagrange entitled Recherches sur la nature et la See also:propagation du son, in which the See also:power of his See also:analysis and his address in its application were equally conspicuous. He made his first See also:appearance in public as the critic of See also:Newton, and the arbiter between d'See also:Alembert and Euler. By considering only the particles of See also:air found in a right See also:line, he reduced the problem of the propagation of See also:sound to the See also:solution of the same partial See also:differential equations that include the motions of vibrating strings, and demonstrated the insufficiency of the methods employed by both his great contemporaries in dealing with the latter subject. He further treated in a masterly manner of echoes and the mixture of sounds, and explained the phenomenon of See also:grave harmonics as due to the occurrence of beats so rapid as to generate a musical note. This was followed, in the second volume of the Miscellanea Taurinensia (1762) by his " Essai d'une nouvelle methode pour determiner See also:les See also:maxima et les minima See also:des formules integrales indefinies," together with the application of this important development of analysis to the solution of several dynamical problems, as well as to the demonstration of the See also:mechanical principle of " least See also:action." The essential point in his advance on Euler's mode of investigating curves of maximum or minimum consisted in his purely analytical conception of the subject. He not only freed it from all trammels of geometrical construction, but by the introduction of the See also:symbol b gave it the efficacy of a new calculus. He is thus justly regarded as the inventor of the " method of variations "—a name supplied by Euler in 1766. - By these performances Lagrange found himself, at the age of twenty-six, on the See also:summit of See also:European fame. Such a height had not been reached without cost. Intense application during See also:early youth had weakened a constitution never robust, and led to accesses of feverish exaltation culminating, in the See also:spring of 1761, in an attack of bilious hypochondria, which permanently lowered the See also:tone of his See also:nervous See also:system. See also:Rest and exercise, however, temporarily restored his See also:health, and he gave See also:proof of the undiminished vigour of his See also:powers by carrying off, in 1764, the See also:prize offered by the See also:Paris Academy of Sciences for the best See also:essay on the See also:libration of the See also:moon. His See also:treatise was remark-able, not only as offering a satisfactory explanation of the coincidence between the lunar periods of rotation and revolution, but as containing the first employment of his See also:radical See also:formula of See also:mechanics, obtained by combining with the principle of d'Alembert that of virtual velocities. His success encouraged the Academy to propose, in 1766, as a theme for competition, the hitherto unattempted theory of the See also:Jovian system. The prize was again awarded to Lagrange; and he earned the same distinction with essays on the problem of three bodies in 1772, on the See also:secular See also:equation of the moon in. 1774, and in 1778 on the theory of cometary perturbations. He had in the meantime gratified a See also:long See also:felt See also:desire by a visit to Paris, where he enjoyed the stimulating delight of conversing with such mathematicians as A. C. See also:Clairault, d'Alembert, See also:Condorcet and the See also:Abbe See also:Marie. Illness prevented him from visiting See also:London. The See also:post of director of the mathematical See also:department of the Berlin Academy (of which he had been a member since 1759) becoming vacant by the removal of Euler to St See also:Petersburg, the latter and d'Alembert See also:united to recommend Lagrange as his successor. Euler's eulogium was enhanced by his desire to quit Berlin, d'Alembert's by his dread of a royal command to repair thither; and the result was that an invitation, conveying the wish of the " greatest See also: But before that time Lagrange himself was on the spot. After the See also:death of See also:Frederick the Great, his presence was competed for by the courts of France, See also:Spain and See also:Naples, and a See also:residence in Berlin having ceased to possess any attraction for him, he removed to Paris in 1787. Marie Antoinette warmly patronized him. He was lodged in the Louvre, received the See also: An. i. 166-172, 3rd ed. with See also:miscellaneous studies, especially with that of See also:chemistry, which, in the new form given to it by See also:Lavoisier, he found " aisee comme 1'algebre." The Revolution roused him once more to activity and cheerfulness. Curiosity impelled him to remain and See also:watch the progress of such a novel phenomenon; but curiosity was changed into dismay as the terrific See also:character of the phenomenon unfolded itself. He now bitterly regretted his temerity in braving the danger. " Tu 1'as voulu" he would repeat self-reproachfully. Even from revolutionary tribunals, however, the name of Lagrange uniformly commanded respect. His See also:pension was continued by the National See also:Assembly, and he was partially indemnified for the depreciation of the currency by remunerative appointments. Nominated See also:president of the Academical See also:commission for the reform of weights and See also:measures, his services were retained when its " See also:purification " by the See also:Jacobins removed his most distinguished colleagues. He again sat on the commission of 1799 for the construction of the metric system, and by his zealous advocacy of the decimal principle largely contributed to its See also:adoption. Meanwhile, on the 31st of May 1792 he married Mademoiselle See also:Lemonnier, daughter of the astronomer of that name, a See also:young and beautiful girl, whose devotion ignored disparity of years, and formed the one tie with See also:life which Lagrange found it hard to break. He had no children by either marriage. Although specially exempted from the operation of the See also:decree of See also:October 1793, imposing banishment on foreign residents, he took alarm at the See also:fate of J. S. See also:Bailly and A. L. Lavoisier, and prepared to resume his former situation in Berlin. His design was frustrated by the See also:establishment of and his See also:official connexion with the Ecole Normale, and the Ecole Polytechnique. The former institution had an ephemeral existence; but amongst the benefits derived from the See also:foundation of the Ecole Polytechnique one of the greatest, it has been observed,4 was the restoration of Lagrange to mathematics. The remembrance of his teachings was long treasured by such of his auditors—amongst whom were J. B. J. See also:Delambre and S. F. See also:Lacroix—as were capable of appreciating them. In expounding the principles of the differential calculus, he started, as it were, from the level of his pupils, and ascended with them by almost insensible gradations from elementary to abstruse conceptions. He seemed, not a professor amongst students, but a learner amongst learners; pauses for thought alternated with luminous exposition; invention accompanied demonstration; and thus originated his Theorie des fonctions analytiques (Paris, 1797). The leading idea of this work was contained in a paper published in the Berlin Memoirs for 1772.5 Its See also:object was the elimination of the, to some minds, unsatisfactory conception of the See also:infinite from the See also:metaphysics of the higher mathematics, and the substitution for the differential and integral calculus of an analogous method depending wholly on the serial development of algebraical functions. By means of this " calculus of derived functions " Lagrange hoped to give to the solution of all analytical problems the utmost " rigour of the demonstrations of the ancients "; 6 but it cannot be said that the See also:attempt was successful. The validity of his fundamental position was impaired by the See also:absence of a well-constituted theory of series; the notation employed was inconvenient, and was abandoned by its inventor in the second edition of his Mecanique; while his scruples as to the See also:admission into analytical investigations of the idea of limits or vanishing ratios have long since been laid aside as idle. Nowhere, however, were the keenness and clearness of his See also:intellect more conspicuous than in this brilliant effort, which, if it failed in its immediate object, was highly effective in secondary results. His purely abstract mode of regarding functions, apart from any mechanical 01 geometrical considerations, led the way to a new and sharply characterized development of the higher analysis in the hands of A. See also:Cauchy, C. G. See also:Jacobi, and others.' The Theorie des fonctions is divided into three parts, of which the first explains the general See also:doctrine of functions, the second deals with its 4 Notice by J. Delambre, cEuvres de Lagrange, i. p. xlii. 5 CEuvres, iii. 441. 5 Theorie des fonctions, p. 6. H. Suter, Geschichte der math. Wiss. ii. 222-223. application to geometry, and the third with its See also:bearings on mechanics. On the establishment of the See also:Institute, Lagrange was placed at the See also:head of the See also:section of geometry; he was one of the first members of the See also:Bureau des Longitudes; and his name appeared in 1791 on the See also:list of foreign members of the Royal Society. On the See also:annexation of See also:Piedmont to France in 1796, a touching compliment was paid to him in the See also:person of his aged father. By direction of Talleyrand, then See also:minister for foreign affairs, the French See also:commissary repaired in See also:state to the old See also:man's residence in Turin, to congratulate him on the merits of his son, whom they declared " to have done See also:honour to mankind by his genius, and whom Piedmont was proud to have produced, and France to possess." See also:Bonaparte, who styled him " la haute pyramide des sciences mathematiques," loaded him with See also:personal favours and official distinctions. He became a senator, a See also:count of the See also:empire, a See also:grand officer of the See also:legion of honour, and just before his death received the grand See also:cross of the See also:order of See also:reunion. The preparation of a new edition of his Mecanique exhausted his already failing powers. Frequent fainting fits gave presage of a speedy end, and on the 8th of See also:April 1813 he had a final interview with his See also:friends B. Lacepede, G. See also:Monge and J. A. See also:Chaptal. He spoke with the utmost See also:calm of his approaching death; " c'est une derniere fonction," he said, " qui n'est ni penible ni desagreable." He nevertheless looked forward to a future See also:meeting, when he promised to complete the autobiographical details which weakness obliged him to interrupt. They remained untold, for he died two days later on the loth of April, and was buried in the See also:Pantheon, the funeral oration being pronounced by See also:Laplace and Lacepede. Amongst the brilliant See also:group of mathematicians whose magnanimous rivalry contributed to accomplish the task of generalization and deduction reserved for the 18th See also:century, Lagrange occupies an eminent See also:place. It is indeed by no means easy to distinguish and apportion the respective merits of the competitors. This is especially the See also:case between Lagrange and Euler on the one See also:side, and between Lagrange and Laplace on the other. The calculus of variations lay undeveloped in Euler's mode of treating isoperimetrical problems. The fruitful method, again, of the variation of elements was introduced by Euler, but adopted and perfected by Lagrange, who first recognized its supreme importance to the analytical investigation of the planetary movements. Finally, of the grand series of researches by which the stability of the See also:solar system was ascertained, the See also:glory must be almost equally divided between Lagrange and Laplace. In analytical invention, and mastery over the calculus, the Turin mathematician was admittedly unrivalled. Laplace owned that he had despaired of effecting the integration of the differential equations relative to secular inequalities until Lagrange showed him the way. But Laplace unquestionably surpassed his See also:rival in See also:practical sagacity and the See also:intuition of See also:physical truth. Lagrange saw in the problems of nature so many occasions for analytical triumphs; Laplace regarded analytical triumphs as the means of solving the problems of nature. One mind seemed the See also:complement of the other; and both, united in See also:honourable rivalry, formed an See also:instrument of unexampled perfection for the investigation of the See also:celestial machinery. What may be called Lagrange's first period of research into planetary perturbations extended from 1794 to 1784 (see See also:ASTRONOMY: History). The notable group of treatises communicated, 1781-1784, to the Berlin Academy was designed, but did not prove to be his final contribution to the theory of the See also:planets. After an See also:interval of twenty-four years the subject, re-opened by S. D. See also:Poisson in a paper read on the loth of See also:June 18o8, was once more attacked by Lagrange with all his pristine vigour and fertility of invention. Resuming the inquiry into the invariability of mean motions, Poisson carried the approximation, with Lagrange's formulae, as far as the squares of the disturbing forces, hitherto neglected, with the same result as to the stability of the system. He had not attempted to include in his calculations the orbital variations of the disturbing bodies; but Lagrange, by the happy artifice of transferring the origin of co-ordinates from the centre of the See also:sun to the centre of gravity of the sun and planets, obtained a simplification of the formulae, by which the same analysis was rendered equally applicable to each of the planets severally. It deserves to be recorded as one of the numerous coincidences of See also:discovery that Laplace, on being made acquainted by Lagrange with his new method, produced analogous expressions, to which his See also:independent researches had led him. The final achievement of Lagrange in this direction was the See also:extension of the method of the variation of arbitrary constants, successfully used by him in the investigation of periodical as well as of secular inequalities, to any system whatever of mutually interacting bodies.' " Not i fEuvres, vi. 771. without astonishment," even to himself, regard being had to the great generality of the differential equations, he reached a result so wide as to include, as a particular case, the solution of the planetary problem recently obtained by him. He proposed to apply the same principles to the calculation of the disturbances produced in the rotation of the planets by See also:external action on their See also:equatorial protuberances, but was anticipated by Poisson, who gave formulae for the variation of the elements of rotation strictly corresponding with those found by Lagrange for the variation of the elements of revolution. The revision of the Mecanique analytique was undertaken mainly for the purpose of embodying in it these new methods and final results, but was interrupted, when two-thirds completed, by the death of its author. In the See also:advancement of almost every See also:branch of pure mathematics Lagrange took a conspicuous part. The calculus of variations is indissolubly associated with his name. In the theory of See also:numbers he furnished solutions of many of P. See also:Fermat's theorems, and added some of his own. In See also:algebra he discovered the method of approximating to the real roots of an equation by means of continued fractions, and imagined a general See also:process of solving algebraical equations of every degree. The method indeed fails for equations of an order above the See also:fourth, because it then involves the solution of an equation of higher dimensions than they proposed. Yet it possesses the great and characteristic merit of generalizing the solutions of his predecessors, exhibiting them all as modifications of one principle. To Lagrange, perhaps more than to any other, the theory of differential equations is indebted for its position as a science, rather than a collection of ingenious artifices for the solution of particular problems. To the calculus of finite See also:differences he contributed the beautiful formula of See also:interpolation which bears his name; although substantially the same result seems to have been previously obtained by Euler. But it was in the application to mechanical questions of the instrument which he thus helped to form that his singular merit lay. It was his just boast to have transformed mechanics (defined by him as a " geometry of four dimensions ") into a branch of analysis, and to have exhibited the so-called mechanical " principles " as simple results of the calculus. The method of " generalized co-ordinates," as it is now called, by which he attained this result, is the most brilliant achievement of the analytical method. Instead of following the motion of each individual part of a material system, he showed that, if we determine its configuration by a sufficient number of variables, whose number is that of the degrees of freedom to move (there being as many equations as the system has degrees of freedom), the kinetic and potential energies of the system can be expressed in terms of these, and the differential equations of motion thence deduced by simple differentiation. Besides this most important contribution to the general fabric of dynamical science, we owe to Lagrange several See also:minor theorems of great elegance,—among which may be mentioned his theorem that the kinetic See also:energy imparted by given impulses to a material system under given constraints is a maximum. To this entire branch of knowledge, in See also:short, he successfully imparted that character of generality and completeness towards which his labours invariably tended. His See also:share in the gigantic task of verifying the Newtonian theory would alone suffice to immortalize his name. His co-operation was indeed more indispensable than at first sight appears. Much as was done by him, what was done through him was still more import-See also:ant. Some of his brilliant rival's most conspicuous discoveries were implicitly contained in his writings, and wanted but one step for completion. But that one step, from the abstract to the See also:concrete, was precisely that which the character of Lagrange's mind indisposed him to make. As notable instances may be mentioned Laplace's discoveries See also:relating to the velocity of sound and the secular See also:acceleration of the moon, both of which were led See also:close up to by Lagrange's analytical demonstrations. In the Berlin Memoirs for 1778 and 1783 Lagrange gave the first See also:direct and theoretically perfect method of determining cometary orbits. It has not indeed proved practically available; but his system of calculating cometary perturbations by means of " mechanical quadratures " has formed the starting-point of all subsequent researches on the subject. His determination2 of maximum and minimum values for the slowly varying planetary eccentricities was the earliest attempt to See also:deal with the problem. Without a more accurate knowledge of the masses of the planets than was then possessed a satisfactory solution was impossible; but the upper limits assigned by him agreed closely with those obtained later by Ti. J. J. See also:Leverrier.3 As a mathematical writer Lagrange has perhaps never been surpassed. His treatises are not only storehouses of ingenious methods, but See also:models of symmetrical form. The clearness, elegance and originality of his mode of presentation give lucidity to what is obscure; novelty to what is See also:familiar, and simplicity to what is abstruse. His genius was one of generalization and See also:abstraction; and the aspirations of the time towards unity and perfection received, by his serene labours, an embodiment denied to them in the troubled See also:world of politics.
B1sLIoGRAPrY.-Lagrange's numerous scattered memoirs have been collected and published in seven 4to volumes, under the title
2 tEuvres, v. 211 seq.
3 Grant, History of Physical Astronomy, p. 117.
tEuvres de Lagrange, publiees sous les soins de M. J. A. Serret (Paris, 1477 the See also:chapel was presented by See also: The See also:original granja (i.e. See also:grange or See also:farm), established by the monks, was See also:purchased in 1719 by See also: The See also:glass factory of See also:San Ildefonso was founded by See also: Additional information and CommentsThere are no comments yet for this article.
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