Part 5 (2/2)

3. _Quadrature of a Curve._ It would be the same with the quadrature of curvilinear areas. If the curve is a plane one, and referred to rectilinear co-ordinates, we will conceive the area A comprised between this curve, the axis of the abscissas, and two extreme co-ordinates, to increase by an infinitely small quant.i.ty _d_A, as the result of a corresponding increment of the abscissa. The relation between these two differentials can be immediately obtained with the greatest facility by subst.i.tuting for the curvilinear element of the proposed area the rectangle formed by the extreme ordinate and the element of the abscissa, from which it evidently differs only by an infinitely small quant.i.ty of the second order. This will at once give, whatever may be the curve, the very simple differential equation

_d_A = _ydx_,

from which, when the curve is defined, the calculus of indirect functions will show how to deduce the finite equation, which is the immediate object of the problem.

4. _Velocity in Variable Motion._ In like manner, in Dynamics, when we desire to know the expression for the velocity acquired at each instant by a body impressed with a motion varying according to any law, we will consider the motion as being uniform during an infinitely small element of the time _t_, and we will thus immediately form the differential equation _de_ = _vdt_, in which _v_ designates the velocity acquired when the body has pa.s.sed over the s.p.a.ce _e_; and thence it will be easy to deduce, by simple and invariable a.n.a.lytical procedures, the formula which would give the velocity in each particular motion, in accordance with the corresponding relation between the time and the s.p.a.ce; or, reciprocally, what this relation would be if the mode of variation of the velocity was supposed to be known, whether with respect to the s.p.a.ce or to the time.

5. _Distribution of Heat._ Lastly, to indicate another kind of questions, it is by similar steps that we are able, in the study of thermological phenomena, according to the happy conception of M.

Fourier, to form in a very simple manner the general differential equation which expresses the variable distribution of heat in any body whatever, subjected to any influences, by means of the single and easily-obtained relation, which represents the uniform distribution of heat in a right-angled parallelopipedon, considering (geometrically) every other body as decomposed into infinitely small elements of a similar form, and (thermologically) the flow of heat as constant during an infinitely small element of time. Henceforth, all the questions which can be presented by abstract thermology will be reduced, as in geometry and mechanics, to mere difficulties of a.n.a.lysis, which will always consist in the elimination of the differentials introduced as auxiliaries to facilitate the establishment of the equations.

Examples of such different natures are more than sufficient to give a clear general idea of the immense scope of the fundamental conception of the transcendental a.n.a.lysis as formed by Leibnitz, const.i.tuting, as it undoubtedly does, the most lofty thought to which the human mind has as yet attained.

It is evident that this conception was indispensable to complete the foundation of mathematical science, by enabling us to establish, in a broad and fruitful manner, the relation of the concrete to the abstract.

In this respect it must be regarded as the necessary complement of the great fundamental idea of Descartes on the general a.n.a.lytical representation of natural phenomena: an idea which did not begin to be worthily appreciated and suitably employed till after the formation of the infinitesimal a.n.a.lysis, without which it could not produce, even in geometry, very important results.

_Generality of the Formulas._ Besides the admirable facility which is given by the transcendental a.n.a.lysis for the investigation of the mathematical laws of all phenomena, a second fundamental and inherent property, perhaps as important as the first, is the extreme generality of the differential formulas, which express in a single equation each determinate phenomenon, however varied the subjects in relation to which it is considered. Thus we see, in the preceding examples, that a single differential equation gives the tangents of all curves, another their rectifications, a third their quadratures; and in the same way, one invariable formula expresses the mathematical law of every variable motion; and, finally, a single equation constantly represents the distribution of heat in any body and for any case. This generality, which is so exceedingly remarkable, and which is for geometers the basis of the most elevated considerations, is a fortunate and necessary consequence of the very spirit of the transcendental a.n.a.lysis, especially in the conception of Leibnitz. Thus the infinitesimal a.n.a.lysis has not only furnished a general method for indirectly forming equations which it would have been impossible to discover in a direct manner, but it has also permitted us to consider, for the mathematical study of natural phenomena, a new order of more general laws, which nevertheless present a clear and precise signification to every mind habituated to their interpretation. By virtue of this second characteristic property, the entire system of an immense science, such as geometry or mechanics, has been condensed into a small number of a.n.a.lytical formulas, from which the human mind can deduce, by certain and invariable rules, the solution of all particular problems.

_Demonstration of the Method._ To complete the general exposition of the conception of Leibnitz, there remains to be considered the demonstration of the logical procedure to which it leads, and this, unfortunately, is the most imperfect part of this beautiful method.

In the beginning of the infinitesimal a.n.a.lysis, the most celebrated geometers rightly attached more importance to extending the immortal discovery of Leibnitz and multiplying its applications than to rigorously establis.h.i.+ng the logical bases of its operations. They contented themselves for a long time by answering the objections of second-rate geometers by the unhoped-for solution of the most difficult problems; doubtless persuaded that in mathematical science, much more than in any other, we may boldly welcome new methods, even when their rational explanation is imperfect, provided they are fruitful in results, inasmuch as its much easier and more numerous verifications would not permit any error to remain long undiscovered. But this state of things could not long exist, and it was necessary to go back to the very foundations of the a.n.a.lysis of Leibnitz in order to prove, in a perfectly general manner, the rigorous exact.i.tude of the procedures employed in this method, in spite of the apparent infractions of the ordinary rules of reasoning which it permitted.

Leibnitz, urged to answer, had presented an explanation entirely erroneous, saying that he treated infinitely small quant.i.ties as _incomparables_, and that he neglected them in comparison with finite quant.i.ties, ”like grains of sand in comparison with the sea:” a view which would have completely changed the nature of his a.n.a.lysis, by reducing it to a mere approximative calculus, which, under this point of view, would be radically vicious, since it would be impossible to foresee, in general, to what degree the successive operations might increase these first errors, which could thus evidently attain any amount. Leibnitz, then, did not see, except in a very confused manner, the true logical foundations of the a.n.a.lysis which he had created. His earliest successors limited themselves, at first, to verifying its exact.i.tude by showing the conformity of its results, in particular applications, to those obtained by ordinary algebra or the geometry of the ancients; reproducing, according to the ancient methods, so far as they were able, the solutions of some problems after they had been once obtained by the new method, which alone was capable of discovering them in the first place.

When this great question was considered in a more general manner, geometers, instead of directly attacking the difficulty, preferred to elude it in some way, as Euler and D'Alembert, for example, have done, by demonstrating the necessary and constant conformity of the conception of Leibnitz, viewed in all its applications, with other fundamental conceptions of the transcendental a.n.a.lysis, that of Newton especially, the exact.i.tude of which was free from any objection. Such a general verification is undoubtedly strictly sufficient to dissipate any uncertainty as to the legitimate employment of the a.n.a.lysis of Leibnitz.

But the infinitesimal method is so important--it offers still, in almost all its applications, such a practical superiority over the other general conceptions which have been successively proposed--that there would be a real imperfection in the philosophical character of the science if it could not justify itself, and needed to be logically founded on considerations of another order, which would then cease to be employed.

It was, then, of real importance to establish directly and in a general manner the necessary rationality of the infinitesimal method. After various attempts more or less imperfect, a distinguished geometer, Carnot, presented at last the true direct logical explanation of the method of Leibnitz, by showing it to be founded on the principle of the necessary compensation of errors, this being, in fact, the precise and luminous manifestation of what Leibnitz had vaguely and confusedly perceived. Carnot has thus rendered the science an essential service, although, as we shall see towards the end of this chapter, all this logical scaffolding of the infinitesimal method, properly so called, is very probably susceptible of only a provisional existence, inasmuch as it is radically vicious in its nature. Still, we should not fail to notice the general system of reasoning proposed by Carnot, in order to directly legitimate the a.n.a.lysis of Leibnitz. Here is the substance of it:

In establis.h.i.+ng the differential equation of a phenomenon, we subst.i.tute, for the immediate elements of the different quant.i.ties considered, other simpler infinitesimals, which differ from them infinitely little in comparison with them; and this subst.i.tution const.i.tutes the princ.i.p.al artifice of the method of Leibnitz, which without it would possess no real facility for the formation of equations. Carnot regards such an hypothesis as really producing an error in the equation thus obtained, and which for this reason he calls _imperfect_; only, it is clear that this error must be infinitely small.

Now, on the other hand, all the a.n.a.lytical operations, whether of differentiation or of integration, which are performed upon these differential equations, in order to raise them to finite equations by eliminating all the infinitesimals which have been introduced as auxiliaries, produce as constantly, by their nature, as is easily seen, other a.n.a.logous errors, so that an exact compensation takes place, and the final equations, in the words of Carnot, become _perfect_. Carnot views, as a certain and invariable indication of the actual establishment of this necessary compensation, the complete elimination of the various infinitely small quant.i.ties, which is always, in fact, the final object of all the operations of the transcendental a.n.a.lysis; for if we have committed no other infractions of the general rules of reasoning than those thus exacted by the very nature of the infinitesimal method, the infinitely small errors thus produced cannot have engendered other than infinitely small errors in all the equations, and the relations are necessarily of a rigorous exact.i.tude as soon as they exist between finite quant.i.ties alone, since the only errors then possible must be finite ones, while none such can have entered. All this general reasoning is founded on the conception of infinitesimal quant.i.ties, regarded as indefinitely decreasing, while those from which they are derived are regarded as fixed.

_Ill.u.s.tration by Tangents._ Thus, to ill.u.s.trate this abstract exposition by a single example, let us take up again the question of _tangents_, which is the most easy to a.n.a.lyze completely. We will regard the equation _t_ = _dy/dx_, obtained above, as being affected with an infinitely small error, since it would be perfectly rigorous only for the secant. Now let us complete the solution by seeking, according to the equation of each curve, the ratio between the differentials of the co-ordinates. If we suppose this equation to be _y_ = _ax_, we shall evidently have

_dy_ = 2_axdx_ + _adx_.

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