effects exactly alike. Such is a sketch of the general manner in which the method of variation is applied to all the different questions which compose what is called the _Theory of Isoperimeters_. It will undoubtedly have been remarked in this summary exposition how much use has been made in this new analysis of the second fundamental property of the transcendental analysis noticed in the third chapter, namely, the generality of the infinitesimal expressions for the representation of the same geometrical or mechanical phenomenon, in whatever body it may be considered. Upon this generality, indeed, are founded, by their nature, all the solutions due to the method of variations. If a single formula could not express the length or the area of any curve whatever; if another fixed formula could not designate the time of the fall of a heavy body, according to whatever line it may descend, &c., how would it have been possible to resolve questions which unavoidably require, by their nature, the simultaneous consideration of all the cases which can be determined in each phenomenon by the different subjects which exhibit it. _Other Applications of this Method._ Notwithstanding the extreme importance of the theory of isoperimeters, and though the method of variations had at first no other object than the logical and general solution of this order of problems, we should still have but an incomplete idea of this beautiful analysis if we limited its destination to this. In fact, the abstract conception of two distinct natures of differentiation is evidently applicable not only to the cases for which it was created, but also to all those which present, for any reason whatever, two different manners of making the same magnitudes vary. It is in this way that Lagrange himself has made, in his "_Méchanique Analytique_," an extensive and important application of his calculus of variations, by employing it to distinguish the two sorts of changes which are naturally presented by the questions of rational mechanics for the different points which are considered, according as we compare the successive positions which are occupied, in virtue of its motion, by the same point of each body in two consecutive instants, or as we pass from one point of the body to another in the same instant. One of these comparisons produces 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.