THE DIFFERENTIAL CALCULUS.
In the exposition of the transcendental analysis, it is customary to intermingle with the purely analytical part (which reduces itself to the treatment of the abstract principles of differentiation and integration) the study of its different principal applications, especially those which concern geometry. This confusion of ideas, which is a consequence of the actual manner in which the science has been developed, presents, in the dogmatic point of view, serious inconveniences in this respect, that it makes it difficult properly to conceive either analysis or geometry. Having to consider here the most rational co-ordination which is possible, I shall include, in the following sketch, only the calculus of indirect functions properly so called, reserving for the portion of this volume which relates to the philosophical study of concrete mathematics the general examination of its great geometrical and mechanical applications.
Two Cases: explicit and implicit Functions. The fundamental division of the differential calculus, or of the general subject of differentiation, consists in distinguishing two cases, according as the analytical functions which are to be differentiated are explicit or implicit; from which flow two parts ordinarily designated by the names of differentiation of formulas and differentiation of equations. It is easy to understand, à priori, the importance of this classification. In fact, such a distinction would be illusory if the ordinary analysis was perfect; that is, if we knew how to resolve all equations algebraically, for then it would be possible to render every implicit function explicit; and, by differentiating it in that state alone, the second part of the differential calculus would be immediately comprised in the first, without giving rise to any new difficulty. But the algebraical resolution of equations being, as we have seen, still almost in its infancy, and as yet impossible for most cases, it is plain that the case is very different, since we have, properly speaking, to differentiate a function without knowing it, although it is determinate. The differentiation of implicit functions constitutes then, by its nature, a question truly distinct from that presented by explicit functions, and necessarily more complicated. It is thus evident that we must commence with the differentiation of formulas, and reduce the differentiation of equations to this primary case by certain invariable analytical considerations, which need not be here mentioned.
These two general cases of differentiation are also distinct in another point of view equally necessary, and too important to be left unnoticed. The relation which is obtained between the differentials is constantly more indirect, in comparison with that of the finite quantities, in the differentiation of implicit functions than in that of explicit functions. We know, in fact, from the considerations presented by Lagrange on the general formation of differential equations, that, on the one hand, the same primitive equation may give rise to a greater or less number of derived equations of very different forms, although at bottom equivalent, depending upon which of the arbitrary constants is eliminated, which is not the case in the differentiation of explicit formulas; and that, on the other hand, the unlimited system of the different primitive equations, which correspond to the same derived equation, presents a much more profound analytical variety than that of the different functions, which admit of one same explicit differential, and which are distinguished from each other only by a constant term. Implicit functions must therefore be regarded as being in reality still more modified by differentiation than explicit functions. We shall again meet with this consideration relatively to the integral calculus, where it acquires a preponderant importance.
Two Sub-cases: A single Variable or several Variables. Each of the two fundamental parts of the Differential Calculus is subdivided into two very distinct theories, according as we are required to differentiate functions of a single variable or functions of several independent variables. This second case is, by its nature, quite