Joseph-Louis de Lagrange - Collected Quotes

"I consider it as one of the most important steps made by Analysis in the last period, that of not being bothered any more by imaginary quantities, and to be able to submit them to calculus, in the same way as the real ones." (Joseph-Louis de Lagrange, [letter to Antonio Lorgna] 1777)

"It seems to me that the mine [of mathematics] has already gone too deep, and that unless someone discovers new veins it will be necessary sooner or later to abandon it. Physics and chemistry now offer more brilliant riches and are more easily exploited; even the taste of the century seems to have entirely turned in this direction, and it is not impossible that the place of geometry in the academies will one day become like that which is currently occupied by the chairs of Arabic in the universities." (Joseph-Louis de Lagrange, [letter to D'Alembert] 1781)

"Statics is the science of the equilibrium of forces. In general, force or power is the cause, whatever it may be, which induces or tends to impart motion to the body to which it is applied. The force or power must be measured by the quantity of motion produced or to be produced. In the state of equilibrium, the force has no apparent action. It produces only a tendency for motion in the body it is applied to. But it must be measured by the effect it would produce if it were not impeded. By taking any force or its effect as unity, the relation of every other force is only a ratio, a mathematical quantity, which can be represented by some numbers or lines. It is in this fashion that forces must be treated in mechanics." (Joseph-Louis de Lagrange, "Mechanique Analytique", 1788)

"The reader will find no figures in this work. The methods which I set forth do not require either constructions or geometrical or mechanical reasonings: but only algebraic operations, subject to a regular and uniform rule of procedure." (Joseph-Louis de Lagrange, "Mechanique Analytique", 1788)

"When we have grasped the spirit of the infinitesimal method, and have verified the exactness of its results either by the geometrical method of prime and ultimate ratios, or by the analytical method of derived functions, we may employ infinitely small quantities as a sure and valuable means of  shortening and simplifying our proofs." (Joseph-Louis de Lagrange, "Mechanique Analytique", 1788)

"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." (Joseph-Louis de Lagrange, "Leçons Élémentaires de Mathématiques", 1795)

"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." (Joseph-Louis de Lagrange, "Leçons Élémentaires de Mathématiques", 1795)

"I regarded as quite useless the reading of large treatises of pure analysis: too large a number of methods pass at once before the eyes. It is in the works of application that one must study them; one judges their utility there and appraises the manner of making use of them." (Joseph-Louis de Lagrange)

"In general, nothing measurable can be measured except by fractions expressing the result of the measurement, unless the measure be contained an exact number of times in the thing to be measured." (Joseph-Louis de Lagrange, "Leçons Élémentaires de Mathématiques", 1795)

"The ordinary operations of algebra suffice to resolve problems in the theory of curves." (Joseph-Louis de Lagrange, "Théorie des fonctions analytiques", 1797)