The power which is given by such an analysis, of expressing with more ease the mathematical laws of phenomena, depends in general on this, that since the calculus applies, not to the increments themselves of the proposed quantities, but to the limits of the ratios of those increments, we can always substitute for each increment any other magnitude more easy to consider, provided that their final ratio is the ratio of equality, or, in other words, that the limit of their ratio is unity. It is clear, indeed, that the calculus of limits would be in no way affected by this substitution. Starting from this principle, we find nearly the equivalent of the facilities offered by the analysis of Leibnitz, which are then merely conceived under another point of view. Thus curves will be regarded as the limits of a series of rectilinear polygons, variable motions as the limits of a collection of uniform motions of constantly diminishing durations, and so on.
Examples. 1. Tangents. Suppose, for example, that we wish to determine the direction of the tangent to a curve; we will regard it as the limit towards which would tend a secant, which should turn about the given point so that its second point of intersection should indefinitely approach the first. Representing the differences of the co-ordinates of the two points by Δy and Δx, we would have at each instant, for the trigonometrical tangent of the angle which the secant makes with the axis of abscissas,
t = Δy/Δx;
from which, taking the limits, we will obtain, relatively to the tangent itself, this general formula of transcendental analysis,
t = L(Δy/Δx),
the characteristic L being employed to designate the limit. The calculus