Conrad Lagrange

“ The Ranger, to tease the artist, had remarked casually,--after complimenting them upon the location of their camp,--"And you've got some mighty nice neighbors, less than a mile above too."
"Neighbors!" ejaculated Conrad Lagrange--in a tone that left no doubt as to his sentiment in the matter.
The others laughed; while the officer said, "Oh, I know how you feel! You think you don't want anybody poaching on your preserves. You're up here in the hills to get away from people, and all that. But you don't need to be uneasy. You won't even see these folks--unless you sneak up on them." ”

“ 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 ”

“ Lagrange?"
"Let it be exhibited as 'The Spirit of Nature--A Portrait'," answered Conrad Lagrange.
As the novelist finished speaking, Yee Kee appeared in the doorway. "They come--big automobile. Whole lot people. Misse Taine, Miste' Lutlidge, sick man, whole lot--I come tell you."
The artist spoke quickly,--"Stop them in the house, Kee; I'll be right in,"--and the Chinaman vanished.
At Yee Kee's announcement, Myra Willard's face went white, and she gave a low cry.
"Never mind, dear," said the girl, soothingly. "We can slip away through the garden--come." ”

“ His emoluments as treasurer at war, together with his wife’s fortune, provided him with ample means, which he lost by rash speculations, a circumstance regarded by his son as the prelude to his own good fortune; for had he been rich, he used to say, he might never have known mathematics.
The genius of Lagrange did not at once take its true bent. His earliest tastes were literary rather than scientific, and he learned the rudiments of geometry during his first year at the college of Turin, without difficulty, but without distinction. ”

“ 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. ”

“ 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. ”

“ He was part author of a Histoire de Périgord left unfinished, and made a further contribution to history, or perhaps, more exactly, to romance, in a letter to Élie Fréron on the identity of the Man with the Iron Mask. Lagrange’s family life was embittered by a long lawsuit against his son. He died at Périgueux at the end of December 1758. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ Lagrange ceased to smile. He replied indifferently that such objects concerned one of his colleagues.
“Ah!” said Madame Martin, “then they are not in your showcase.”
She observed that learned men were not curious, and that it is indiscreet to question them on things that are not in their own showcases. It is true that Lagrange had made a scientific fortune in studying meteors. This had led him to study comets. But he was wise. For twenty years he had been preoccupied by nothing except dining out.
When he had left, Countess Martin told Madame Marmet what she expected of her. ”

“ 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. ”

“ 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 ”

“ 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. ”

“ 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 ”

“ 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. ”

“ 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." ”

“ London, Published by Charles Knight, Pall Mall East.
The father of Lagrange, luckily perhaps for the fame of his son, was ruined by some unfortunate speculation. The latter used to say, that had he possessed fortune, he should probably never have turned his attention to the science in which he excelled. He was placed at the College of Turin, and applied himself diligently and with enthusiasm to classical literature, showing no taste at first for mathematics. In about a year he began to attend to the geometry of the ancients. ”

“ 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 king in Europe” to have the “greatest mathematician” at his court, was sent to Turin. On the 6th of November 1766, Lagrange was installed in his new position, with a salary of 6000 francs, ample leisure for scientific research, and royal favour sufficient to secure him respect without exciting envy. The national jealousy of foreigners, was at first a source of annoyance to him ”

“ 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 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. ”

“ The expression of this fact was generalized in a brief statement by Newton in the Principia, and more in detail by the Bernoullis, until, in the analytical guise of the so-called principle of “virtual velocities” or virtual work, it finally became the basis of Lagrange’s general formulation of dynamics. ”

“ 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. ”

“ 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 ”

“ His development of the equation xm + px = q in an infinite series was extended by Leonhard Euler, and particularly by Joseph Louis Lagrange. In 1761 he proved the irrationality of π; a simpler proof was given somewhat later by Legendre. The introduction of hyperbolic functions into trigonometry was also due to him. His geometrical discoveries are of great value, his Die freie Perspective (1759–1774) being a work of great merit. Astronomy was also enriched by his investigations, and he was led to several remarkable theorems on conics which bear his name. ”

“ Lagrange’s own proof will be found under Dynamics, § Analytical. In a conservative system free from extraneous force we have
Σ (X δx + Y δy + Z δz) = −δV,
(12) where V is the potential energy. Hence
(13) and
(14)
If we imagine any given state of motion (q̇1, q̇2 ... q̇n) through the configuration (q1, q2, ... qn) to be generated instantaneously from rest by the action of suitable impulsive forces, we find on integrating (11) with respect to t over the infinitely short duration of the impulse
(15) where Qr′ is the time integral of Qr and so represents a generalized component of impulse. ”

“ 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. ”