The philosophy of mathematics Part 5

But most frequently we consider only the cases in which the functions are such as are called _algebraic_, and to which the idea of _degree_ is applicable. In this case we can give more precision to the general proposition by determining the a.n.a.lytical character which must be necessarily presented by the equation, in order that this property may be verified. It is easy to see, then, that, by the modification just explained, all the _terms_ of the first degree, whatever may be their form, rational or irrational, entire or fractional, will become _m_ times greater; all those of the second degree, _m_ times; those of the third, _m_ times, &c. Thus the terms of the same degree, however different may be their composition, varying in the same manner, and the terms of different degrees varying in an unequal proportion, whatever similarity there may be in their composition, it will be necessary, to prevent the equation from being disturbed, that all the terms which it contains should be of the same degree. It is in this that properly consists the ordinary theorem of _h.o.m.ogeneity_, and it is from this circ.u.mstance that the general law has derived its name, which, however, ceases to be exactly proper for all other functions.

In order to treat this subject in its whole extent, it is important to observe an essential condition, to which attention must be paid in applying this property when the phenomenon expressed by the equation presents magnitudes of different natures. Thus it may happen that the respective units are completely independent of each other, and then the theorem of h.o.m.ogeneity will hold good, either with reference to all the corresponding cla.s.ses of quant.i.ties, or with regard to only a single one or more of them. But it will happen on other occasions that the different units will have fixed relations to one another, determined by the nature of the question; then it will be necessary to pay attention to this subordination of the units in verifying the h.o.m.ogeneity, which will not exist any longer in a purely algebraic sense, and the precise form of which will vary according to the nature of the phenomena. Thus, for example, to fix our ideas, when, in the a.n.a.lytical expression of geometrical phenomena, we are considering at once lines, areas, and volumes, it will be necessary to observe that the three corresponding units are necessarily so connected with each other that, according to the subordination generally established in that respect, when the first becomes _m_ times greater, the second becomes _m_ times, and the third _m_ times. It is with such a modification that h.o.m.ogeneity will exist in the equations, in which, if they are _algebraic_, we will have to estimate the degree of each term by doubling the exponents of the factors which correspond to areas, and tripling those of the factors relating to volumes.

Such are the princ.i.p.al general considerations relating to the _Calculus of Direct Functions_. We have now to pa.s.s to the philosophical examination of the _Calculus of Indirect Functions_, the much superior importance and extent of which claim a fuller development.

CHAPTER III.

TRANSCENDENTAL a.n.a.lYSIS:

DIFFERENT MODES OF VIEWING IT.

We determined, in the second chapter, the philosophical character of the transcendental a.n.a.lysis, in whatever manner it may be conceived, considering only the general nature of its actual destination as a part of mathematical science. This a.n.a.lysis has been presented by geometers under several points of view, really distinct, although necessarily equivalent, and leading always to identical results. They may be reduced to three princ.i.p.al ones; those of LEIBNITZ, of NEWTON, and of LAGRANGE, of which all the others are only secondary modifications. In the present state of science, each of these three general conceptions offers essential advantages which pertain to it exclusively, without our having yet succeeded in constructing a single method uniting all these different characteristic qualities. This combination will probably be hereafter effected by some method founded upon the conception of Lagrange when that important philosophical labour shall have been accomplished, the study of the other conceptions will have only a historic interest; but, until then, the science must be considered as in only a provisional state, which requires the simultaneous consideration of all the various modes of viewing this calculus. Illogical as may appear this multiplicity of conceptions of one identical subject, still, without them all, we could form but a very insufficient idea of this a.n.a.lysis, whether in itself, or more especially in relation to its applications. This want of system in the most important part of mathematical a.n.a.lysis will not appear strange if we consider, on the one hand, its great extent and its superior difficulty, and, on the other, its recent formation.

ITS EARLY HISTORY.

If we had to trace here the systematic history of the successive formation of the transcendental a.n.a.lysis, it would be necessary previously to distinguish carefully from the _calculus of indirect functions_, properly so called, the original idea of the _infinitesimal method_, which can be conceived by itself, independently of any _calculus_. We should see that the first germ of this idea is found in the procedure constantly employed by the Greek geometers, under the name of the _Method of Exhaustions_, as a means of pa.s.sing from the properties of straight lines to those of curves, and consisting essentially in subst.i.tuting for the curve the auxiliary consideration of an inscribed or circ.u.mscribed polygon, by means of which they rose to the curve itself, taking in a suitable manner the limits of the primitive ratios. Incontestable as is this filiation of ideas, it would be giving it a greatly exaggerated importance to see in this method of exhaustions the real equivalent of our modern methods, as some geometers have done; for the ancients had no logical and general means for the determination of these limits, and this was commonly the greatest difficulty of the question; so that their solutions were not subjected to abstract and invariable rules, the uniform application of which would lead with certainty to the knowledge sought; which is, on the contrary, the princ.i.p.al characteristic of our transcendental a.n.a.lysis. In a word, there still remained the task of generalizing the conceptions used by the ancients, and, more especially, by considering it in a manner purely abstract, of reducing it to a complete system of calculation, which to them was impossible.

The first idea which was produced in this new direction goes back to the great geometer Fermat, whom Lagrange has justly presented as having blocked out the direct formation of the transcendental a.n.a.lysis by his method for the determination of _maxima_ and _minima_, and for the finding of _tangents_, which consisted essentially in introducing the auxiliary consideration of the correlative increments of the proposed variables, increments afterward suppressed as equal to zero when the equations had undergone certain suitable transformations. But, although Fermat was the first to conceive this a.n.a.lysis in a truly abstract manner, it was yet far from being regularly formed into a general and distinct calculus having its own notation, and especially freed from the superfluous consideration of terms which, in the a.n.a.lysis of Fermat, were finally not taken into the account, after having nevertheless greatly complicated all the operations by their presence. This is what Leibnitz so happily executed, half a century later, after some intermediate modifications of the ideas of Fermat introduced by Wallis, and still more by Barrow; and he has thus been the true creator of the transcendental a.n.a.lysis, such as we now employ it. This admirable discovery was so ripe (like all the great conceptions of the human intellect at the moment of their manifestation), that Newton, on his side, had arrived, at the same time, or a little earlier, at a method exactly equivalent, by considering this a.n.a.lysis under a very different point of view, which, although more logical in itself, is really less adapted to give to the common fundamental method all the extent and the facility which have been imparted to it by the ideas of Leibnitz.

Finally, Lagrange, putting aside the heterogeneous considerations which had guided Leibnitz and Newton, has succeeded in reducing the transcendental a.n.a.lysis, in its greatest perfection, to a purely algebraic system, which only wants more apt.i.tude for its practical applications.

After this summary glance at the general history of the transcendental a.n.a.lysis, we will proceed to the dogmatic exposition of the three princ.i.p.al conceptions, in order to appreciate exactly their characteristic properties, and to show the necessary ident.i.ty of the methods which are thence derived. Let us begin with that of Leibnitz.

METHOD OF LEIBNITZ.

_Infinitely small Elements._ This consists in introducing into the calculus, in order to facilitate the establishment of equations, the infinitely small elements of which all the quant.i.ties, the relations between which are sought, are considered to be composed. These elements or _differentials_ will have certain relations to one another, which are constantly and necessarily more simple and easy to discover than those of the primitive quant.i.ties, and by means of which we will be enabled (by a special calculus having for its peculiar object the elimination of these auxiliary infinitesimals) to go back to the desired equations, which it would have been most frequently impossible to obtain directly.

This indirect a.n.a.lysis may have different degrees of indirectness; for, when there is too much difficulty in forming immediately the equation between the differentials of the magnitudes under consideration, a second application of the same general artifice will have to be made, and these differentials be treated, in their turn, as new primitive quant.i.ties, and a relation be sought between their infinitely small elements (which, with reference to the final objects of the question, will be _second differentials_), and so on; the same transformation admitting of being repeated any number of times, on the condition of finally eliminating the constantly increasing number of infinitesimal quant.i.ties introduced as auxiliaries.

A person not yet familiar with these considerations does not perceive at once how the employment of these auxiliary quant.i.ties can facilitate the discovery of the a.n.a.lytical laws of phenomena; for the infinitely small increments of the proposed magnitudes being of the same species with them, it would seem that their relations should not be obtained with more ease, inasmuch as the greater or less value of a quant.i.ty cannot, in fact, exercise any influence on an inquiry which is necessarily independent, by its nature, of every idea of value. But it is easy, nevertheless, to explain very clearly, and in a quite general manner, how far the question must be simplified by such an artifice. For this purpose, it is necessary to begin by distinguishing _different orders_ of infinitely small quant.i.ties, a very precise idea of which may be obtained by considering them as being either the successive powers of the same primitive infinitely small quant.i.ty, or as being quant.i.ties which may be regarded as having finite ratios with these powers; so that, to take an example, the second, third, &c., differentials of any one variable are cla.s.sed as infinitely small quant.i.ties of the second order, the third, &c., because it is easy to discover in them finite multiples of the second, third, &c., powers of a certain first differential. These preliminary ideas being established, the spirit of the infinitesimal a.n.a.lysis consists in constantly neglecting the infinitely small quant.i.ties in comparison with finite quant.i.ties, and generally the infinitely small quant.i.ties of any order whatever in comparison with all those of an inferior order. It is at once apparent how much such a liberty must facilitate the formation of equations between the differentials of quant.i.ties, since, in the place of these differentials, we can subst.i.tute such other elements as we may choose, and as will be more simple to consider, only taking care to conform to this single condition, that the new elements differ from the preceding ones only by quant.i.ties infinitely small in comparison with them. It is thus that it will be possible, in geometry, to treat curved lines as composed of an infinity of rectilinear elements, curved surfaces as formed of plane elements, and, in mechanics, variable motions as an infinite series of uniform motions, succeeding one another at infinitely small intervals of time.

EXAMPLES. Considering the importance of this admirable conception, I think that I ought here to complete the ill.u.s.tration of its fundamental character by the summary indication of some leading examples.

1. _Tangents._ Let it be required to determine, for each point of a plane curve, the equation of which is given, the direction of its tangent; a question whose general solution was the primitive object of the inventors of the transcendental a.n.a.lysis. We will consider the tangent as a secant joining two points infinitely near to each other; and then, designating by _dy_ and _dx_ the infinitely small differences of the co-ordinates of those two points, the elementary principles of geometry will immediately give the equation _t_ = _dy_/_dx_ for the trigonometrical tangent of the angle which is made with the axis of the abscissas by the desired tangent, this being the most simple way of fixing its position in a system of rectilinear co-ordinates. This equation, common to all curves, being established, the question is reduced to a simple a.n.a.lytical problem, which will consist in eliminating the infinitesimals _dx_ and _dy_, which were introduced as auxiliaries, by determining in each particular case, by means of the equation of the proposed curve, the ratio of _dy_ to _dx_, which will be constantly done by uniform and very simple methods.

2. _Rectification of an Arc._ In the second place, suppose that we wish to know the length of the arc of any curve, considered as a function of the co-ordinates of its extremities. It would be impossible to establish directly the equation between this arc s and these co-ordinates, while it is easy to find the corresponding relation between the differentials of these different magnitudes. The most simple theorems of elementary geometry will in fact give at once, considering the infinitely small arc _ds_ as a right line, the equations

_ds_ = _dy_ + _dx_, or _ds_ = _dx_ + _dy_ + _dz_,

according as the curve is of single or double curvature. In either case, the question is now entirely within the domain of a.n.a.lysis, which, by the elimination of the differentials (which is the peculiar object of the calculus of indirect functions), will carry us back from this relation to that which exists between the finite quant.i.ties themselves under examination.

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 establishing 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 establishing 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_.