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.
Letter tod'Alembert (1781) cited in R. Laubenbacher, D. Pengelly:Mathematical Expeditions: Chronicles by the Explorers (1999) Springer, pp. 233–234.
We have already various treatises on Mechanics, but the plan of this one is entirely new. I intend to reduce the theory of this Science, and the art of solving problems relating to it, to general formulae, the simple development of which provides all the equations necessary for the solution of each problem. I hope that the manner in which I have tried to attain this object will leave nothing to be desired. No diagrams will be found in this work. The methods that I explain require neither geometrical, nor mechanical, constructions or reasoning, but only algebraical operations in accordance with regular and uniform procedure. Those who love Analysis will see with pleasure that Mechanics has become a branch of it, and will be grateful to me for having thus extended its domain.
As long asalgebra andgeometry 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.
Dans Les Leçons Élémentaires sur les Mathématiques (1795) Leçon cinquiéme,Tr. McCormack, cited inRobert Edouard Moritz,Memorabilia mathematica or, The philomath's quotation-book (1914) Ch. V The teaching of mathematics,p. 81.
An ancient writer said thatarithmetic andgeometry are the wings ofmathematics; 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.
Dans Les Leçons Élémentaires sur les Mathématiques (1795) Leçon cinquiéme, Tr. McCormack, cited in Moritz,Memorabilia mathematica or, The philomath's quotation-book (1914) Ch. 15 Arithmetic,p. 261.
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.
As quoted by F. R. Moulton,Introduction to Astronomy (New York, 1906), p. 199.
It took them only an instant to cut off that head, but France may not produce another like it in a century.
As quoted by William Hughes,Annual Editions: Western Civilization (1997) p. 64; about the beheading of his friendAntoine Lavoisier.
The great masters of modernanalysis are Lagrange,Laplace, andGauss, who were contemporaries. It is interesting to note the marked contrast in their styles. Lagrange is perfect both in form and matter, he is careful to explain his procedure, and though his arguments are general they are easy to follow. Laplace on the other hand explains nothing, is indifferent to style, and, if satisfied that his results are correct, is content to leave them either with no proof or with a faulty one. Gauss is as exact and elegant as Lagrange, but even more difficult to follow than Laplace, for he removes every trace of the analysis by which he reached his results, and studies to give a proof which while rigorous shall be as concise and synthetical as possible.
W. W. Rouse Ball,History of Mathematics, (London, 1901), p. 463; as cited in: Moritz (1914, 160); Persons and anecdotes
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 mutteredII faut que fy songe encore, and put the paper in his pocket.
Augustus De Morgan,Budget of Paradoxes, (London, 1872), p. 178; as cited in: Moritz (1914, 161); Persons and anecdotes
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.
Johann Wolfgang von Goethe,Wilhelm Meister's Wanderjahre, Zweites Buck; Spriiche in Prosa; Natur, VI, 950; as cited in: Moritz (1914, 120); About the mathematician
Lagrange, struck with the circumstance that the calculus had never given any inequalities but such as were periodical, applied himself to the investigation of a general question, from which he found by a method peculiar to himself and independent of any approximation, that the inequalities produced by the mutual action of the planets must in effect be all periodical; that the periodical changes are confined within narrow limits; that none of the planets ever has been or ever can be a comet moving in a very eccentric orbit; but that the planetary system oscillates as it were round a medium state from which it never deviates far: that amid all the changes which arise from the mutual actions of the planets, two things remain perpetually the same, viz. the length of the greater axis of the ellipse which the planet describes, and its periodical time round the sun; or, which is the same thing, the mean distance of each planet from the sun and its mean motion remain constant. The plane of the orbit varies, the species of the ellipse and its eccentricity change, but never, by any means whatever, the greater axis of the ellipse, or the time of the entire revolution of the planet. The discovery of this great principle, which we may consider as the bulwark that secures the stability of our system, and excludes all access to confusion and disorder, must render the name of Lagrange for ever memorable in science, and ever revered by those who delight in the contemplation of whatever is excellent and sublime. AfterNewton's discovery of the elliptic orbits of the planets from gravitation, Langrange's discovery of their periodical inequalities is, without doubt, the noblest truth in physical astronomy, and in respect of the doctrine of final causes, it may truly be regarded as the greatest of all.
Who has studied the works of such men asEuler, Lagrange,Cauchy,Riemann,Sophus Lie, andWeierstrass, 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.
E. W. Hobson, "Presidential Address British Association for the Advancement of Science" (1910) in:Nature, Vol. 84, p. 290. Cited in: Moritz (1914, 182); Mathematics as a fine art.
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 hisAnalytic Mechanics...
Analytic mechanics... was brought to the highest degree of perfection by Lagrange. Lagrange's aim is (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.
Ernst Mach,Die Mechanik in Ihrer Entwicklung Historisch-Kritsch Dargestellt (1883, 1888) Tr. Thomas J. McCormick asThe Science of Mechanics (1893). See also, 4th edition (1919)pp. 466-467.
Full use of Lagrange's owncalculus of variations made the unification of the varied principles of statistics and dynamics possible—in statistics by the use of the principle ofvirtual velocities, in dynamics by the use ofD'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.
Dirk Jan Struik,A Concise History of Mathematics (1948) Ch. 8 The Eighteenth Century.
The treatment of thekinetics of a material system by the method ofgeneralised 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 theoreticaldynamics which has resulted from these investigations, they have proved of great and continually increasing value in the application ofmechanics to thermal, electrical and chemical theories, and the whole range ofmolecular physics.
The value of his work [Mécanique Analytique] consists in the exposition of a general method by which every mechanical question may be stated in a single algebraic equation. The entire history of any mechanical system, as for example, the solar system, may thus be condensed into a single sentence; and its detailed interpretation becomes simply a question of algebra. No one who has not tried to cope with the difficulties presented by almost any mechanical problem can form a just appreciation of the great utility of such a labor-saving and thought-saving device. It has been well called 'a stupendous contribution to the economy of thought.'
Lagrange... was the first to draw sharply the line of demarcation between physics and metaphysics. The mechanical ideas ofDescartes,Leibnitz,Maupertius, and even ofEuler, 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.
