ordinary differentials; the other gives rise to variations, which, there as every where, are only differentials taken under a new point of view. Such is the general acceptation in which we should conceive the Calculus of Variations, in order suitably to appreciate the importance of this admirable logical instrument, the most powerful that the human mind has as yet constructed.
The method of variations being only an immense extension of the general transcendental analysis, I have no need of proving specially that it is susceptible of being considered under the different fundamental points of view which the calculus of indirect functions, considered as a whole, admits of. Lagrange invented the Calculus of Variations in accordance with the infinitesimal conception, and, indeed, long before he undertook the general reconstruction of the transcendental analysis. When he had executed this important reformation, he easily showed how it could also be applied to the Calculus of Variations, which he expounded with all the proper development, according to his theory of derivative functions. But the more that the use of the method of variations is difficult of comprehension, because of the higher degree of abstraction of the ideas considered, the more necessary is it, in its application, to economize the exertions of the mind, by adopting the most direct and rapid analytical conception, namely, that of Leibnitz. Accordingly, Lagrange himself has constantly preferred it in the important use which he has made of the Calculus of Variations in his "Analytical Mechanics." In fact, there does not exist the least hesitation in this respect among geometers.
ITS RELATIONS TO THE ORDINARY CALCULUS.
In order to make as clear as possible the philosophical character of the Calculus of Variations, I think that I should, in conclusion, briefly indicate a consideration which seems to me important, and by which I can approach it to the ordinary transcendental analysis in a higher degree than Lagrange seems to me to have done.
We noticed in the preceding chapter the formation of the calculus of partial differences, created by D'Alembert, as having introduced into the transcendental analysis a new elementary idea; the notion of two kinds of increments, distinct and independent of one another, which a function of two variables may receive by virtue of the change of each variable separately. It is thus that the vertical ordinate of a surface, or any other magnitude which is referred to it, varies in two manners which are quite distinct, and which may follow the most different laws, according as we increase either the one or the other of the two horizontal co-ordinates. Now such a consideration seems to me very nearly allied, by its nature, to that which serves as the general basis of the method of variations. This last, indeed, has in reality done nothing but transfer to the independent variables themselves the peculiar conception which had been already adopted for the functions of these variables; a modification which has remarkably enlarged its use. I think, therefore, that so far as regards merely the fundamental conceptions, we may consider the calculus created by D'Alembert as having established a natural and necessary transition between the ordinary infinitesimal calculus and the calculus of variations; such a derivation of which seems to be adapted to make the general notion more clear and simple.
According to the different considerations indicated in this chapter, the method of variations presents itself as the highest degree of perfection which the analysis of indirect functions has yet attained. In its primitive state, this last analysis presented itself as a powerful general means of facilitating the mathematical study of natural phenomena, by introducing, for the expression of their laws, the consideration of auxiliary magnitudes, chosen in such a manner that their relations are necessarily more simple and more easy to obtain