In this connexion, and in the logical order of ideas, the transcendental analysis presents itself as being necessarily the first, since its general object is to facilitate the establishment of equations, an operation which must evidently precede the _resolution_ of those equations, which is the object of the ordinary analysis. But though it is exceedingly important to conceive in this way the true relations of these two systems of analysis, it is none the less proper, in conformity with the regular usage, to study the transcendental analysis after ordinary analysis; for though the former is, at bottom, by itself logically independent of the latter, or, at least, may be essentially disengaged from it, yet it is clear that, since its employment in the solution of questions has always more or less need of being completed by the use of the ordinary analysis, we would be constrained to leave the questions in suspense if this latter had not been previously studied. _Corresponding Divisions of the Calculus of Functions._ It follows from the preceding considerations that the _Calculus of Functions_, or _Algebra_ (taking this word in its most extended meaning), is composed of two distinct fundamental branches, one of which has for its immediate object the _resolution_ of equations, when they are directly established between the magnitudes themselves which are under consideration; and the other, starting from equations (generally much easier to form) between quantities indirectly connected with those of the problem, has for its peculiar and constant destination the deduction, by invariable analytical methods, of the corresponding equations between the direct magnitudes which we are considering; which brings the question within the domain of the preceding calculus. The former calculus bears most frequently the name of _Ordinary Analysis_, or of _Algebra_, properly so called. The second constitutes what is called the _Transcendental Analysis_, which has been designated by the different denominations of _Infinitesimal Calculus_, _Calculus of Fluxions and of Fluents_, _Calculus of Vanishing Quantities_, the _Differential and Integral Calculus_, &c., according to the point of view in which it has been conceived. In order to remove every foreign consideration, I will propose to name it CALCULUS OF INDIRECT FUNCTIONS, giving to ordinary analysis the title of CALCULUS OF DIRECT FUNCTIONS. These expressions, which I form essentially by generalizing and epitomizing the ideas of Lagrange, are simply intended to indicate with precision the true general character belonging to each of these two forms of analysis. Having now established the fundamental division of mathematical analysis, I have next to consider separately each of its two parts, commencing with the _Calculus of Direct Functions_, and reserving more extended developments for the different branches of the _Calculus of Indirect Functions_. CHAPTER II. ORDINARY ANALYSIS, OR ALGEBRA.