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 Δ_y_ and Δ_x_, we would have at each instant, for the trigonometrical tangent of the angle which the secant makes with the axis of abscissas, _t_ = Δ_y_/Δ_x_; from which, taking the limits, we will obtain, relatively to the tangent itself, this general formula of transcendental analysis, _t_ = _L_(Δ_y_/Δ_x_), the characteristic _L_ being employed to designate the limit. The calculus of indirect functions will show how to deduce from this formula in each particular case, when the equation of the curve is given, the relation between _t_ and _x_, by eliminating the auxiliary quantities which have been introduced. If we suppose, in order to complete the solution, that the equation of the proposed curve is _y_ = _ax²_, we shall evidently have Δ_y_ = 2_ax_Δ_x_ + _a_(Δ_x_)², from which we shall obtain Δ_y_/Δ_x_ = 2_ax_ + _a_Δ_x_. Now it is clear that the _limit_ towards which the second number tends, in proportion as Δ_x_ diminishes, is 2_ax_. We shall therefore find, by this method, _t_ = 2_ax_, as we obtained it for the same case by the method of Leibnitz. 2. _Rectifications._ In like manner, when the rectification of a curve is desired, we must substitute for the increment of the arc s the chord of this increment, which evidently has such a connexion with it that the limit of their ratio is unity; and then we find (pursuing in other respects the same plan as with the method of Leibnitz) this general equation of rectifications: (_LΔs_/Δ_x_)² = 1 + (_LΔy_/Δ_x_)², or (_LΔs_/Δ_x_)² = 1 + (_LΔy_/Δ_x_)² + (_LΔz_/Δ_x_)²,