Joseph-Louis Lagrange

“ At the age of nineteen he communicated to Leonhard Euler his idea of a general method of dealing with “isoperimetrical” problems, known later as the Calculus of Variations. It was eagerly welcomed by the Berlin mathematician, who had the generosity to withhold from publication his own further researches on the subject, until his youthful correspondent should have had time to complete and opportunity to claim the invention. This prosperous opening gave the key-note to Lagrange’s career. ”

“ Laplace regarded analytical triumphs as the means of solving the problems of nature. One mind seemed the complement of the other; and both, united in honourable rivalry, formed an instrument of unexampled perfection for the investigation of the celestial machinery. What may be called Lagrange’s first period of research into planetary perturbations extended from 1774 to 1784 (see 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 planets. ”

“ A very well executed bust, which is now in the Library of the Institute, was made from a sketch by a young Italian artist, sent by the Academy of Turin. From this bust our portrait is engraved.
Of the character of Lagrange as a philosopher, no description, in so few words, can be better than that of M. Laplace: “Among the discoverers who have most enlarged the bounds of our knowledge, Newton and Lagrange appear to me to have possessed in the highest degree that happy tact, which leads to the discovery of general principles, and which constitutes true genius for science. ”

“ His determination [9] of maximum and minimum values for the slowly varying planetary eccentricities was the earliest attempt to 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 U. J. J. Leverrier. [10] As a mathematical writer Lagrange has perhaps never been surpassed. ”

“ Lagrange has a grey moustache, a grey beard and long grey hair. He is overflowing with soured generosity, charitable violence and a sort of chivalrous demagogy; there is a love in his heart with which he stirs up hatred; he is tall, thin, young looking at a distance, old when seen nearer, wrinkled, bewildered, hoarse, flurried, wan, has a wild look in his eyes and gesticulates ”

“ There are few great ideas pursued by succeeding analysts which were not suggested by Euler, or of which he did not share the honour of invention. With, perhaps, less exuberance of invention, but with more comprehensive genius and profounder reasoning, Lagrange developed the infinitesimal calculus and placed analytical mechanics into the form in which we now know it. Laplace applied the calculus and mechanics to the elaboration of the theory of universal gravitation, and thus, largely extending and supplementing the labours of Newton, gave a full analytical discussion of the solar system. ”

“ The final achievement of Lagrange in this direction was the 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. [8] “Not 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. ”

“ 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 rival in practical sagacity and the intuition of physical truth. Lagrange saw in the problems of nature so many occasions for analytical triumphs ”

“ That Aaron King had won the approval of the Ranger was evidenced by the mountaineer's inviting himself to supper the camp in the sycamores. The fact that the officer considerately told Conrad Lagrange only that he had met the artist with his creel full of trout, and so had been tempted to accompany him, won the enduring gratitude of the young man. Thus the circumstances of their meeting introduced each to the other, with recommendations of peculiar value, and marked the beginning of a genuine and lasting friendship. ”

“ He had planned a visit to London, but he fell dangerously ill after a dinner in Paris, and was compelled to return to Turin. In 1766 Euler left Berlin for St. Petersburg, and he pointed out Lagrange as the only man capable of filling the place. D'Alembert recommended him at the same time. Frederick the Great thereupon sent a message to Turin, expressing the wish of "the greatest king of Europe" to have "the greatest mathematician" at his court. Lagrange went to Berlin, and staid there twenty years. ”

“ The discussion on partial differential equations of the second order, carried on by D'Alembert, Euler, and Lagrange, has already been referred to in our account of D'Alembert.
While in Berlin, Lagrange wrote the Méchanique Analytique," the greatest of his works (Paris, 1788) . From the principle of virtual velocities he deduced, with aid of the calculus of variations, the whole system of mechanics so elegantly and harmoniously that it may fitly be called, in Sir William Rowan Hamilton's words, "a kind of scientific poem." It is a most consummate example of analytic generality. ”

“ With the aid of the marquis de Saluces and the anatomist G. F. Cigna, he founded in 1758 a society which became the Turin Academy of Sciences. The first volume of its memoirs, published in the following year, contained a paper by Lagrange entitled Recherches sur la nature et la propagation du son, in which the power of his analysis and his address in its application were equally conspicuous. He made his first appearance in public as the critic of Newton, and the arbiter between d’Alembert and Euler. ”

“ The greatest and least values of the bending moments of loaded rods are often at the extremities of the divisions of the rods and not at points given by critical values.
In the case of a function of several variables, X1, x2, ... xn, not independent but connected by m functional relations u1 = 0, u2 = 0, ..., um = 0, we might proceed to eliminate m of the variables; but Lagrange’s “Method of undetermined Multipliers” is more elegant and generally more useful. ”

“ Some years afterwards the names of Daniel Barnoulli, of Euler, and of Fontaine came to be added to so many great names. Finally, Lagrange in his turn embarked in the same career, and at the very commencement of his researches he succeeded in substituting for the imperfect, although very ingenious, essays of his predecessors, a complete method which was free from every objection. From that instant the dignity of science was satisfied; but in such a case it would not be permitted to say with the poet:
"Le temps ne fait rien à l'affaire." ”

“ I have wondered at the artist's friendship with him."
"I tell you, Myra, Lagrange is all right," said Brian Oakley, stoutly. "He's odd and eccentric and rough spoken sometimes; but he's not at all what you would think him from the stuff he writes. He's a true man at heart, and you needn't worry about Sibyl getting anything but good from an acquaintance with him. As for King--well--Conrad Lagrange vouches for him. If you knew Lagrange, you'd understand what that means. He and the young fellow's mother grew up together. He swears the lad is right ”

“ The first formal proof of Lagrange’s theorem for the development in a series of an implicit function was furnished by Laplace, who gave to it an extended generality. He also showed that every equation of an even degree must have at least one real quadratic factor, reduced the solution of linear differential equations to definite integrals, and furnished an elegant method by which the linear partial differential equation of the second order might be solved. ”

“ Accordingly, we find Euler and D’Alembert devoting their talent and their patience to the establishment of the laws of the rotation of solid bodies. Lagrange has incorporated his own analysis of the problem with his general treatment of mechanics, and since his time M. Poinsôt has brought the subject under the power of a more searching analysis than that of the calculus, in which ideas take the place of symbols, and intelligible propositions supersede equations. ”

“ The new ideas which came to them in the fecund period of youth and which they introduced into the common domain have given them all they could have expected; they have fulfilled their task and feel the need of turning their mental activity towards wholly new subjects. This need, as we recognize, manifested itself with particular force at the epoch of Lagrange. At this moment, in fact, the program of researches opened to geometers by the discovery of the infinitesimal calculus appeared very nearly finished up. ”

“ The chord and arc were not taken by Newton as equal before vanishing, nor after vanishing, but when they vanish. "That method," said Lagrange, "has the great inconvenience of considering quantities in the state in which they cease, so to speak, to be quantities; for though we can always well conceive the ratios of two quantities, as long as they remain finite, that ratio offers to the mind no clear and precise idea, as soon as its terms become both nothing at the same time." D'Alembert's method of limits was much the same as the method of prime and ultimate ratios. ”

“ Of these the most conspicuous was offered by the opposite inequalities of Jupiter and Saturn, which the emulous efforts of L. Euler and J. L. Lagrange had failed to bring within the bounds of theory. The discordance of their results incited Laplace to a searching examination of the whole subject of planetary perturbations, and his maiden effort was rewarded with a discovery which constituted, when developed and completely demonstrated by his own further labours and those of his illustrious rival Lagrange, the most important advance made in physical astronomy since the time of Newton. ”

“ Poisson went on to show that this result was by no means all that might with advantage be borrowed from the theory of + Attractions. Lagrange, in a memoir on the motion of gravitating bodies, had shown [77] that the components of the attractive force at any point can be simply expressed as the derivates of the function which is obtained by adding together the masses of all the particles of an attracting system, each divided by its distance from the point ”

“ The distinction plays a considerable part in Euler’s work (§ 236) , but it was Lagrange who first recognised its full importance, particularly for planetary theory, and who made a special study of secular inequalities.
When the perturbations of one planet by another are being studied, it becomes necessary to obtain a mathematical expression for the disturbing force which the second planet exerts. ”

“ The results for three or more variables can be written in the same form. The extension of Taylor’s theorem was given by Lagrange (1797) ; the form written above is due to Cauchy (1823) . For the validity of the theorem in this form it is necessary that all the differential coefficients up to the nth should be continuous in a region bounded by x = a ± h, y = b ± k. When all the differential coefficients, no matter how high the order, are continuous in such a region, the theorem leads to an expansion of the function in a multiple power series. ”

“ It was shown by Sir James Cockle and Robert Harley (1858–1859) in connexion with the Jerrardian form, and by Arthur Cayley (1861) , that Lagrange’s resolvent equation of the sixth order can be replaced by a more simple sextic equation occupying a like place in the theory. ”

“ The problem of three bodies has been treated in various ways since the time of Lagrange, but no decided advance towards a more complete algebraic solution has been made, and the problem stands substantially where it was left by him. He had made a reduction in the differential equations to the seventh order. This was elegantly accomplished in a different way by Jacobi in 1843. ”

“ Euler had been under the necessity of resolving an integral into a sum of elements, recording the magnitude of the change produced in each element by a slight change in the unknown function, and thence forming an expression for the total change in the sum under consideration. Lagrange proposed to free the theory from this necessity. Euler had allowed such changes in the position of the curve, along which the integral, to be made a maximum or minimum, is taken, as can be produced by displacement parallel to the axis of ordinates. ”

“ This is, of course, useless for the solution of the quintic equation, which, as already mentioned, does not admit of solution by radicals; but the equation of the sixth order, Lagrange’s resolvent sextic, is very important, and is intimately connected with all the later investigations in the theory. ”

“ He there became acquainted with Lagrange, and was thus led to turn his attention to physical science. After some years spent on political economy and on the principles of the fine arts (in connexion with which he wrote, for the Berlin Memoirs, a remarkable dissertation on poetry) he returned to Geneva and began his work on magnetism and on heat. Interrupted occasionally in his studies by political duties, in which he was often called to the front, he remained professor of philosophy at Geneva till he was called in 1810 to the chair of physics. He died at Geneva on the 8th of April 1839. ”