ANALYSIS

The historical development of the Abstract portion of Mathematical science has, since the time of Descartes, been for the most part determined by that of the Concrete. Yet the Calculus in all its principal branches must be understood before passing on to Geometry and Mechanics. The Concrete portions of the science depend on the Abstract, which are wholly independent of them. We will now therefore proceed to a rapid review of the leading conceptions of the Analysis.

True idea of an equation.

First, however, we must take some notice of the general idea of an equation, and see how far it is from being the true one on which geometers proceed in practice; for without settling this point we cannot determine, with any precision, the real aim and extent of abstract mathematics.

The business of concrete mathematics is to discover the equations which express the mathematical laws of the phenomenon under consideration; and these equations are the starting-point of the calculus, which must obtain from them certain quantities by means of others. It is only by forming a true idea of an equation that we can lay down the real line of separation between the concrete and the abstract part of mathematics.