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.