Part 5 (1/2)

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 distinguis.h.i.+ng _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.