The philosophy of mathematics Part 6

In this formula we shall have to neglect the term _dx_ as an infinitely small quant.i.ty of the second order. Then the combination of the two _imperfect_ equations.

_t_ = _dy/dx_, _dy_ = 2_ax(dx)_,

being sufficient to eliminate entirely the infinitesimals, the finite result, _t_ = 2_ax_, will necessarily be rigorously correct, from the effect of the exact compensation of the two errors committed; since, by its finite nature, it cannot be affected by an infinitely small error, and this is, nevertheless, the only one which it could have, according to the spirit of the operations which have been executed.

It would be easy to reproduce in a uniform manner the same reasoning with reference to all the other general applications of the a.n.a.lysis of Leibnitz.

This ingenious theory is undoubtedly more subtile than solid, when we examine it more profoundly; but it has really no other radical logical fault than that of the infinitesimal method itself, of which it is, it seems to me, the natural development and the general explanation, so that it must be adopted for as long a time as it shall be thought proper to employ this method directly.

I pa.s.s now to the general exposition of the two other fundamental conceptions of the transcendental a.n.a.lysis, limiting myself in each to its princ.i.p.al idea, the philosophical character of the a.n.a.lysis having been sufficiently determined above in the examination of the conception of Leibnitz, which I have specially dwelt upon because it admits of being most easily grasped as a whole, and most rapidly described.

METHOD OF NEWTON.

Newton has successively presented his own method of conceiving the transcendental a.n.a.lysis 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 a.n.a.lysis, from this point of view, consists in introducing as auxiliaries, in the place of the primitive quant.i.ties, or concurrently with them, in order to facilitate the establishment of equations, the _limits of the ratios_ of the simultaneous increments of these quant.i.ties; 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 pa.s.s from the equations between these limits to the corresponding equations between the primitive quant.i.ties themselves.

The power which is given by such an a.n.a.lysis, 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 quant.i.ties, but to the limits of the ratios of those increments, we can always subst.i.tute 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 subst.i.tution. Starting from this principle, we find nearly the equivalent of the facilities offered by the a.n.a.lysis 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 diminis.h.i.+ng 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 ?_y_ and ?_x_, we would have at each instant, for the trigonometrical tangent of the angle which the secant makes with the axis of abscissas,

_t_ = ?_y_/?_x_;

from which, taking the limits, we will obtain, relatively to the tangent itself, this general formula of transcendental a.n.a.lysis,

_t_ = _L_(?_y_/?_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 quant.i.ties which have been introduced. If we suppose, in order to complete the solution, that the equation of the proposed curve is _y_ = _ax_, we shall evidently have

?_y_ = 2_ax_?_x_ + _a_(?_x_),

from which we shall obtain

?_y_/?_x_ = 2_ax_ + _a_?_x_.

Now it is clear that the _limit_ towards which the second number tends, in proportion as ?_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 subst.i.tute 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:

(_L?s_/?_x_) = 1 + (_L?y_/?_x_), or (_L?s_/?_x_) = 1 + (_L?y_/?_x_) + (_L?z_/?_x_),

according as the curve is plane or of double curvature. It will now be necessary, for each particular curve, to pa.s.s 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 a.n.a.lysis, or, more precisely, that which Maclaurin and D'Alembert have presented as the most rational basis of that a.n.a.lysis, 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 a.n.a.lysis, 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 quant.i.ties 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 quant.i.ty, which will be inversely named its _fluent_. Henceforth the transcendental a.n.a.lysis will consist, according to this conception, in forming directly the equations between the fluxions of the proposed quant.i.ties, 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 ident.i.ty 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 quant.i.ties 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 s.p.a.ce 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 quant.i.ties 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 princ.i.p.al applications of the transcendental a.n.a.lysis, without its being necessary to present special proofs of this.

METHOD OF LAGRANGE.

_Derived Functions._ The conception of Lagrange, in its admirable simplicity, consists in representing the transcendental a.n.a.lysis as a great algebraic artifice, by which, in order to facilitate the establishment of equations, we introduce, in the place of the primitive functions, or concurrently with them, their _derived_ functions; that is, according to the definition of Lagrange, the coefficient of the first term of the increment of each function, arranged according to the ascending powers of the increment of its variable. The special calculus of indirect functions has for its constant object, here as well as in the conceptions of Leibnitz and of Newton, to eliminate these _derivatives_ which have been thus employed as auxiliaries, in order to deduce from their relations the corresponding equations between the primitive magnitudes.

_An Extension of ordinary a.n.a.lysis._ The transcendental a.n.a.lysis is, then, nothing but a simple though very considerable extension of ordinary a.n.a.lysis. Geometers have long been accustomed to introduce in a.n.a.lytical investigations, in the place of the magnitudes themselves which they wished to study, their different powers, or their logarithms, or their sines, &c., in order to simplify the equations, and even to obtain them more easily. This successive _derivation_ is an artifice of the same nature, only of greater extent, and procuring, in consequence, much more important resources for this common object.

But, although we can readily conceive, _a priori_, that the auxiliary consideration of these derivatives _may_ facilitate the establishment of equations, it is not easy to explain why this _must_ necessarily follow from this mode of derivation rather than from any other transformation. Such is the weak point of the great idea of Lagrange.

The precise advantages of this a.n.a.lysis cannot as yet be grasped in an abstract manner, but only shown by considering separately each princ.i.p.al question, so that the verification is often exceedingly laborious.

EXAMPLE. _Tangents._ This manner of conceiving the transcendental a.n.a.lysis may be best ill.u.s.trated by its application to the most simple of the problems above examined--that of tangents.

Instead of conceiving the tangent as the prolongation of the infinitely small element of the curve, according to the notion of Leibnitz--or as the limit of the secants, according to the ideas of Newton--Lagrange considers it, according to its simple geometrical character, a.n.a.logous to the definitions of the ancients, to be a right line such that no other right line can pa.s.s through the point of contact between it and the curve. Then, to determine its direction, we must seek the general expression of its distance from the curve, measured in any direction whatever--in that of the ordinate, for example--and dispose of the arbitrary constant relating to the inclination of the right line, which will necessarily enter into that expression, in such a way as to diminish that separation as much as possible. Now this distance, being evidently equal to the difference of the two ordinates of the curve and of the right line, which correspond to the same new abscissa _x_ + _h_, will be represented by the formula

(_f'_(_x_) - _t_)_h_ + _qh_ + _rh_ + etc.,

in which _t_ designates, as above, the unknown trigonometrical tangent of the angle which the required line makes with the axis of abscissas, and _f'_(_x_) the derived function of the ordinate _f_(_x_). This being understood, it is easy to see that, by disposing of _t_ so as to make the first term of the preceding formula equal to zero, we will render the interval between the two lines the least possible, so that any other line for which _t_ did not have the value thus determined would necessarily depart farther from the proposed curve. We have, then, for the direction of the tangent sought, the general expression _t_ = _f'_(_x_), a result exactly equivalent to those furnished by the Infinitesimal Method and the Method of Limits. We have yet to find _f'_(_x_) in each particular curve, which is a mere question of a.n.a.lysis, quite identical with those which are presented, at this stage of the operations, by the other methods.

After these considerations upon the princ.i.p.al general conceptions, we need not stop to examine some other theories proposed, such as Euler's _Calculus of Vanis.h.i.+ng Quant.i.ties_, which are really modifications--more or less important, and, moreover, no longer used--of the preceding methods.

I have now to establish the comparison and the appreciation of these three fundamental methods. Their _perfect and necessary conformity_ is first to be proven in a general manner.

FUNDAMENTAL IDENt.i.tY OF THE THREE METHODS.

It is, in the first place, evident from what precedes, considering these three methods as to their actual destination, independently of their preliminary ideas, that they all consist in the same general logical artifice, which has been characterized in the first chapter; to wit, the introduction of a certain system of auxiliary magnitudes, having uniform relations to those which are the special objects of the inquiry, and subst.i.tuted for them expressly to facilitate the a.n.a.lytical expression of the mathematical laws of the phenomena, although they have finally to be eliminated by the aid of a special calculus. It is this which has determined me to regularly define the transcendental a.n.a.lysis as _the calculus of indirect functions_, in order to mark its true philosophical character, at the same time avoiding any discussion upon the best manner of conceiving and applying it. The general effect of this a.n.a.lysis, whatever the method employed, is, then, to bring every mathematical question much more promptly within the power of the _calculus_, and thus to diminish considerably the serious difficulty which is usually presented by the pa.s.sage from the concrete to the abstract. Whatever progress we may make, we can never hope that the calculus will ever be able to grasp every question of natural philosophy, geometrical, or mechanical, or thermological, &c., immediately upon its birth, which would evidently involve a contradiction. Every problem will constantly require a certain preliminary labour to be performed, in which the calculus can be of no a.s.sistance, and which, by its nature, cannot be subjected to abstract and invariable rules; it is that which has for its special object the establishment of equations, which form the indispensable starting point of all a.n.a.lytical researches. But this preliminary labour has been remarkably simplified by the creation of the transcendental a.n.a.lysis, which has thus hastened the moment at which the solution admits of the uniform and precise application of general and abstract methods; by reducing, in each case, this special labour to the investigation of equations between the auxiliary magnitudes; from which the calculus then leads to equations directly referring to the proposed magnitudes, which, before this admirable conception, it had been necessary to establish directly and separately. Whether these indirect equations are _differential_ equations, according to the idea of Leibnitz, or equations of _limits_, conformably to the conception of Newton, or, lastly, _derived_ equations, according to the theory of Lagrange, the general procedure is evidently always the same.