Joseph Louis Lagrange

“History of science”

“History of mathematics”

“Lagrange... was the first to draw sharply the line of demarcation between physics and metaphysics. The mechanical ideas of Descartes, Leibnitz, Maupertius, and even of Euler, had proved to be more or less hazy and unfruitful from a failure to separate those two distinct regions of thought. Lagrange put an end to this confusion, for no serious attempt has since been made to derive the laws of mechanics from a metaphysical basis.”

“The treatment of the kinetics of a material system by the method of generalised coordinates was first introduced by Lagrange, and has since his time been greatly developed by the investigations of different mathematicians.Independently of the highly interesting, although purely abstract science of theoretical dynamics which has resulted from these investigations, they have proved of great and continually increasing value in the application of mechanics to thermal, electrical and chemical theories, and the whole range of {{w|molecular physics}}.”

“Full use of Lagrange's own {{w|calculus of variations}} made the unification of the varied principles of statistics and dynamics possible—in statistics by the use of the principle of virtual velocities, in dynamics by the use of {{w|D'Alembert's principle}}. This led... to generalized coordinates and to the equation of motion in their "Lagrangian" form... Newton's geometrical approach was now fully discarded; Lagrange's book was a triumph of pure analysis.”

“Analytic mechanics... was brought to the highest degree of perfection by Lagrange. Lagrange's aim is ({{w|Mécanique analytique}}... 1788) to dispose, once and for all, of the reasoning necessary to resolve mechanical problems, by embodying as much as possible of it in a single formula. This he did. Every case... can now be dealt with by a very simple, highly symmetrical and perspicuous schema; and whatever reasoning is left is performed by purely mechanical methods. The mechanics of Lagrange is a stupendous contribution to the economy of thought.”

“The questions here dealt with have occupied me since my earliest youth, when my interest for them was powerfully stimulated by the beautiful introductions of Lagrange to the chapters of his Analytic Mechanics...”

“Who has studied the works of such men as Euler, Lagrange, Cauchy, Riemann, {{w|Sophus Lie}}, and Weierstrass, can doubt that a great mathematician is a great artist? The faculties possessed by such men, varying greatly in kind and degree with the individual, are analogous with those requisite for constructive art. Not every mathematician possesses in a specially high degree that critical faculty which finds its employment in the perfection of form, hi conformity with the ideal of logical completeness; but every great mathematician possesses the rarer faculty of constructive imagination.”

“The mathematician is perfect only in so far as he is a perfect being, in so far as he perceives the beauty of truth; only then will his work be thorough, transparent, comprehensive, pure, clear, attractive and even elegant. All this is necessary to resemble Lagrange.”

“Lagrange, in one of the later years of his life, imagined that he had overcome the difficulty [of the parallel axiom]. He went so far as to write a paper, which he took with him to the Institute, and began to read it. But in the first paragraph something struck him which he had not observed : he muttered II faut que fy songe encore, and put the paper in his pocket.”

“It took them only an instant to cut off that head, but France may not produce another like it in a century.”

“Newton was the greatest genius that ever existed, and the most fortunate, for we cannot find more than once a system of the world to establish.”

“An ancient writer said that arithmetic and geometry are the wings of mathematics; I believe one can say without speaking metaphorically that these two sciences are the foundation and essence of all the sciences which deal with quantity. Not only are they the foundation, they are also, as it were, the capstones; for, whenever a result has been arrived at, in order to use that result, it is necessary to translate it into numbers or into lines; to translate it into numbers requires the aid of arithmetic, to translate it into lines necessitates the use of geometry.”

“As long as algebra and geometry proceeded along separate paths, their advance was slow and their applications limited. But when these sciences joined company, they drew from each other fresh vitality and thenceforward marched on at a rapid pace toward perfection.”

“I cannot say whether I will still be doing geometry ten years from now. It also seems to me that the mine has maybe already become too deep and unless one finds new veins it might have to be abandoned. Physics and chemistry now offer a much more glowing richness and much easier exploitation. Also, the general taste has turned entirely in this direction, and it is not impossible that the place of Geometry in the Academies will someday become what the role of the Chairs of Arabic at the universities is now.”