The philosophy of mathematics · Auguste Comte
METHOD OF NEWTON.
Newton has successively presented his own method of conceiving the transcendental analysis under several different forms. That which is at present the most commonly adopted was designated by Newton, sometimes under the name of the _Method of prime and ultimate Ratios_, sometimes under that of the _Method of Limits_. _Method of Limits._ The general spirit of the transcendental analysis, from this point of view, consists in introducing as auxiliaries, in the place of the primitive quantities, or concurrently with them, in order to facilitate the establishment of equations, the _limits of the ratios_ of the simultaneous increments of these quantities; or, in other words, the _final ratios_ of these increments; limits or final ratios which can be easily shown to have a determinate and finite value. A special calculus, which is the equivalent of the infinitesimal calculus, is then employed to pass from the equations between these limits to the corresponding equations between the primitive quantities themselves. 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, according as the curve is plane or of double curvature. It will now be necessary, for each particular curve, to pass from this equation to that between the arc and the abscissa, which depends on the transcendental calculus properly so called. We could take up, with the same facility, by the method of limits, all the other general questions, the solution of which has been already indicated according to the infinitesimal method. Such is, in substance, the conception which Newton formed for the transcendental analysis, or, more precisely, that which Maclaurin and D'Alembert have presented as the most rational basis of that analysis, in seeking to fix and to arrange the ideas of Newton upon that subject. _Fluxions and Fluents._ Another distinct form under which Newton has presented this same method should be here noticed, and deserves particularly to fix our attention, as much by its ingenious clearness in some cases as by its having furnished the notation best suited to this manner of viewing the transcendental analysis, and, moreover, as having been till lately the special form of the calculus of indirect functions commonly adopted by the English geometers. I refer to the calculus of _fluxions_ and of _fluents_, founded on the general idea of _velocities_.
To facilitate the conception of the fundamental idea, let us consider every curve as generated by a point impressed with a motion varying according to any law whatever. The different quantities which the curve can present, the abscissa, the ordinate, the arc, the area, &c., will be regarded as simultaneously produced by successive degrees during this motion. The _velocity_ with which each shall have been described will be called the _fluxion_ of that quantity, which will be inversely named its _fluent_. Henceforth the transcendental analysis will consist, according to this conception, in forming directly the equations between the fluxions of the proposed quantities, in order to deduce therefrom, by a special calculus, the equations between the fluents themselves. What has been stated respecting curves may, moreover, evidently be applied to any magnitudes whatever, regarded, by the aid of suitable images, as produced by motion.
It is easy to understand the general and necessary identity of this method with that of limits complicated with the foreign idea of motion. In fact, resuming the case of the curve, if we suppose, as we evidently always may, that the motion of the describing point is uniform in a certain direction, that of the abscissa, for example, then the fluxion of the abscissa will be constant, like the element of the time; for all the other quantities generated, the motion cannot be conceived to be uniform, except for an infinitely small time. Now the velocity being in general according to its mechanical conception, the ratio of each space to the time employed in traversing it, and this time being here proportional to the increment of the abscissa, it follows that the fluxions of the ordinate, of the arc, of the area, &c., are really nothing else (rejecting the intermediate consideration of time) than the final ratios of the increments of these different quantities to the increment of the abscissa. This method of fluxions and fluents is, then, in reality, only a manner of representing, by a comparison borrowed from mechanics, the method of prime and ultimate ratios, which alone can be reduced to a calculus. It evidently, then, offers the same general advantages in the various principal applications of the transcendental analysis, without its being necessary to present special proofs of this.
Text from Project Gutenberg, public domain in the United States.