The philosophy of mathematics Part 7

But the coincidence of these three princ.i.p.al methods is not limited to the common effect which they produce; it exists, besides, in the very manner of obtaining it. In fact, not only do all three consider, in the place of the primitive magnitudes, certain auxiliary ones, but, still farther, the quant.i.ties thus introduced as subsidiary are exactly identical in the three methods, which consequently differ only in the manner of viewing them. This can be easily shown by taking for the general term of comparison any one of the three conceptions, especially that of Lagrange, which is the most suitable to serve as a type, as being the freest from foreign considerations. Is it not evident, by the very definition of _derived functions_, that they are nothing else than what Leibnitz calls _differential coefficients_, or the ratios of the differential of each function to that of the corresponding variable, since, in determining the first differential, we will be obliged, by the very nature of the infinitesimal method, to limit ourselves to taking the only term of the increment of the function which contains the first power of the infinitely small increment of the variable? In the same way, is not the derived function, by its nature, likewise the necessary _limit_ towards which tends the ratio between the increment of the primitive function and that of its variable, in proportion as this last indefinitely diminishes, since it evidently expresses what that ratio becomes when we suppose the increment of the variable to equal zero?

That which is designated by _dx_/_dy_ in the method of Leibnitz; that which ought to be noted as _L_(?_y_/?_x_) in that of Newton; and that which Lagrange has indicated by _f'_(_x_), is constantly one same function, seen from three different points of view, the considerations of Leibnitz and Newton properly consisting in making known two general necessary properties of the derived function. The transcendental a.n.a.lysis, examined abstractedly and in its principle, is then always the same, whatever may be the conception which is adopted, and the procedures of the calculus of indirect functions are necessarily identical in these different methods, which in like manner must, for any application whatever, lead constantly to rigorously uniform results.

COMPARATIVE VALUE OF THE THREE METHODS.

If now we endeavour to estimate the comparative value of these three equivalent conceptions, we shall find in each advantages and inconveniences which are peculiar to it, and which still prevent geometers from confining themselves to any one of them, considered as final.

_That of Leibnitz._ The conception of Leibnitz presents incontestably, in all its applications, a very marked superiority, by leading in a much more rapid manner, and with much less mental effort, to the formation of equations between the auxiliary magnitudes. It is to its use that we owe the high perfection which has been acquired by all the general theories of geometry and mechanics. Whatever may be the different speculative opinions of geometers with respect to the infinitesimal method, in an abstract point of view, all tacitly agree in employing it by preference, as soon as they have to treat a new question, in order not to complicate the necessary difficulty by this purely artificial obstacle proceeding from a misplaced obstinacy in adopting a less expeditious course. Lagrange himself, after having reconstructed the transcendental a.n.a.lysis on new foundations, has (with that n.o.ble frankness which so well suited his genius) rendered a striking and decisive homage to the characteristic properties of the conception of Leibnitz, by following it exclusively in the entire system of his _Mechanique a.n.a.lytique_. Such a fact renders any comments unnecessary.

But when we consider the conception of Leibnitz in itself and in its logical relations, we cannot escape admitting, with Lagrange, that it is radically vicious in this, that, adopting its own expressions, the notion of infinitely small quant.i.ties is a _false idea_, of which it is in fact impossible to obtain a clear conception, however we may deceive ourselves in that matter. Even if we adopt the ingenious idea of the compensation of errors, as above explained, this involves the radical inconvenience of being obliged to distinguish in mathematics two cla.s.ses of reasonings, those which are perfectly rigorous, and those in which we designedly commit errors which subsequently have to be compensated. A conception which leads to such strange consequences is undoubtedly very unsatisfactory in a logical point of view.

To say, as do some geometers, that it is possible in every case to reduce the infinitesimal method to that of limits, the logical character of which is irreproachable, would evidently be to elude the difficulty rather than to remove it; besides, such a transformation almost entirely strips the conception of Leibnitz of its essential advantages of facility and rapidity.

Finally, even disregarding the preceding important considerations, the infinitesimal method would no less evidently present by its nature the very serious defect of breaking the unity of abstract mathematics, by creating a transcendental a.n.a.lysis founded on principles so different from those which form the basis of the ordinary a.n.a.lysis. This division of a.n.a.lysis into two worlds almost entirely independent of each other, tends to hinder the formation of truly general a.n.a.lytical conceptions.

To fully appreciate the consequences of this, we should have to go back to the state of the science before Lagrange had established a general and complete harmony between these two great sections.

_That of Newton._ Pa.s.sing now to the conception of Newton, it is evident that by its nature it is not exposed to the fundamental logical objections which are called forth by the method of Leibnitz. The notion of _limits_ is, in fact, remarkable for its simplicity and its precision. In the transcendental a.n.a.lysis presented in this manner, the equations are regarded as exact from their very origin, and the general rules of reasoning are as constantly observed as in ordinary a.n.a.lysis.

But, on the other hand, it is very far from offering such powerful resources for the solution of problems as the infinitesimal method. The obligation which it imposes, of never considering the increments of magnitudes separately and by themselves, nor even in their ratios, but only in the limits of those ratios, r.e.t.a.r.ds considerably the operations of the mind in the formation of auxiliary equations. We may even say that it greatly embarra.s.ses the purely a.n.a.lytical transformations. Thus the transcendental a.n.a.lysis, considered separately from its applications, is far from presenting in this method the extent and the generality which have been imprinted upon it by the conception of Leibnitz. It is very difficult, for example, to extend the theory of Newton to functions of several independent variables. But it is especially with reference to its applications that the relative inferiority of this theory is most strongly marked.

Several Continental geometers, in adopting the method of Newton as the more logical basis of the transcendental a.n.a.lysis, have partially disguised this inferiority by a serious inconsistency, which consists in applying to this method the notation invented by Leibnitz for the infinitesimal method, and which is really appropriate to it alone. In designating by _dy_/_dx_ that which logically ought, in the theory of limits, to be denoted by _L_(?_y_/?_x_), and in extending to all the other a.n.a.lytical conceptions this displacement of signs, they intended, undoubtedly, to combine the special advantages of the two methods; but, in reality, they have only succeeded in causing a vicious confusion between them, a familiarity with which hinders the formation of clear and exact ideas of either. It would certainly be singular, considering this usage in itself, that, by the mere means of signs, it could be possible to effect a veritable combination between two theories so distinct as those under consideration.

Finally, the method of limits presents also, though in a less degree, the greater inconvenience, which I have above noted in reference to the infinitesimal method, of establishing a total separation between the ordinary and the transcendental a.n.a.lysis; for the idea of _limits_, though clear and rigorous, is none the less in itself, as Lagrange has remarked, a foreign idea, upon which a.n.a.lytical theories ought not to be dependent.

_That of Lagrange._ This perfect unity of a.n.a.lysis, and this purely abstract character of its fundamental notions, are found in the highest degree in the conception of Lagrange, and are found there alone; it is, for this reason, the most rational and the most philosophical of all.

Carefully removing every heterogeneous consideration, Lagrange has reduced the transcendental a.n.a.lysis to its true peculiar character, that of presenting a very extensive cla.s.s of a.n.a.lytical transformations, which facilitate in a remarkable degree the expression of the conditions of various problems. At the same time, this a.n.a.lysis is thus necessarily presented as a simple extension of ordinary a.n.a.lysis; it is only a higher algebra. All the different parts of abstract mathematics, previously so incoherent, have from that moment admitted of being conceived as forming a single system.

Unhappily, this conception, which possesses such fundamental properties, independently of its so simple and so lucid notation, and which is undoubtedly destined to become the final theory of transcendental a.n.a.lysis, because of its high philosophical superiority over all the other methods proposed, presents in its present state too many difficulties in its applications, as compared with the conception of Newton, and still more with that of Leibnitz, to be as yet exclusively adopted. Lagrange himself has succeeded only with great difficulty in rediscovering, by his method, the princ.i.p.al results already obtained by the infinitesimal method for the solution of the general questions of geometry and mechanics; we may judge from that what obstacles would be found in treating in the same manner questions which were truly new and important. It is true that Lagrange, on several occasions, has shown that difficulties call forth, from men of genius, superior efforts, capable of leading to the greatest results. It was thus that, in trying to adapt his method to the examination of the curvature of lines, which seemed so far from admitting its application, he arrived at that beautiful theory of contacts which has so greatly perfected that important part of geometry. But, in spite of such happy exceptions, the conception of Lagrange has nevertheless remained, as a whole, essentially unsuited to applications.

The final result of the general comparison which I have too briefly sketched, is, then, as already suggested, that, in order to really understand the transcendental a.n.a.lysis, we should not only consider it in its principles according to the three fundamental conceptions of Leibnitz, of Newton, and of Lagrange, but should besides accustom ourselves to carry out almost indifferently, according to these three princ.i.p.al methods, and especially according to the first and the last, the solution of all important questions, whether of the pure calculus of indirect functions or of its applications. This is a course which I could not too strongly recommend to all those who desire to judge philosophically of this admirable creation of the human mind, as well as to those who wish to learn to make use of this powerful instrument with success and with facility. In all the other parts of mathematical science, the consideration of different methods for a single cla.s.s of questions may be useful, even independently of its historical interest, but it is not indispensable; here, on the contrary, it is strictly necessary.

Having determined with precision, in this chapter, the philosophical character of the calculus of indirect functions, according to the princ.i.p.al fundamental conceptions of which it admits, we have next to consider, in the following chapter, the logical division and the general composition of this calculus.

CHAPTER IV.

THE DIFFERENTIAL AND INTEGRAL CALCULUS.

ITS TWO FUNDAMENTAL DIVISIONS.

The _calculus of indirect functions_, in accordance with the considerations explained in the preceding chapter, is necessarily divided into two parts (or, more properly, is decomposed into two different _calculi_ entirely distinct, although intimately connected by their nature), according as it is proposed to find the relations between the auxiliary magnitudes (the introduction of which const.i.tutes the general spirit of this calculus) by means of the relations between the corresponding primitive magnitudes; or, conversely, to try to discover these direct equations by means of the indirect equations originally established. Such is, in fact, constantly the double object of the transcendental a.n.a.lysis.

These two systems have received different names, according to the point of view under which this a.n.a.lysis has been regarded. The infinitesimal method, properly so called, having been the most generally employed for the reasons which have been given, almost all geometers employ habitually the denominations of _Differential Calculus_ and of _Integral Calculus_, established by Leibnitz, and which are, in fact, very rational consequences of his conception. Newton, in accordance with his method, named the first the _Calculus of Fluxions_, and the second the _Calculus of Fluents_, expressions which were commonly employed in England. Finally, following the eminently philosophical theory founded by Lagrange, one would be called the _Calculus of Derived Functions_, and the other the _Calculus of Primitive Functions_. I will continue to make use of the terms of Leibnitz, as being more convenient for the formation of secondary expressions, although I ought, in accordance with the suggestions made in the preceding chapter, to employ concurrently all the different conceptions, approaching as nearly as possible to that of Lagrange.

THEIR RELATIONS TO EACH OTHER.

The differential calculus is evidently the logical basis of the integral calculus; for we do not and cannot know how to integrate directly any other differential expressions than those produced by the differentiation of the ten simple functions which const.i.tute the general elements of our a.n.a.lysis. The art of integration consists, then, essentially in bringing all the other cases, as far as is possible, to finally depend on only this small number of fundamental integrations.

In considering the whole body of the transcendental a.n.a.lysis, as I have characterized it in the preceding chapter, it is not at first apparent what can be the peculiar utility of the differential calculus, independently of this necessary relation with the integral calculus, which seems as if it must be, by itself, the only one directly indispensable. In fact, the elimination of the _infinitesimals_ or of the _derivatives_, introduced as auxiliaries to facilitate the establishment of equations, const.i.tuting, as we have seen, the final and invariable object of the calculus of indirect functions, it is natural to think that the calculus which teaches how to deduce from the equations between these auxiliary magnitudes, those which exist between the primitive magnitudes themselves, ought strictly to suffice for the general wants of the transcendental a.n.a.lysis without our perceiving, at the first glance, what special and constant part the solution of the inverse question can have in such an a.n.a.lysis. It would be a real error, though a common one, to a.s.sign to the differential calculus, in order to explain its peculiar, direct, and necessary influence, the destination of forming the differential equations, from which the integral calculus then enables us to arrive at the finite equations; for the primitive formation of differential equations is not and cannot be, properly speaking, the object of any calculus, since, on the contrary, it forms by its nature the indispensable starting point of any calculus whatever.

How, in particular, could the differential calculus, which in itself is reduced to teaching the means of _differentiating_ the different equations, be a general procedure for establishing them? That which in every application of the transcendental a.n.a.lysis really facilitates the formation of equations, is the infinitesimal _method_, and not the infinitesimal _calculus_, which is perfectly distinct from it, although it is its indispensable complement. Such a consideration would, then, give a false idea of the special destination which characterizes the differential calculus in the general system of the transcendental a.n.a.lysis.

But we should nevertheless very imperfectly conceive the real peculiar importance of this first branch of the calculus of indirect functions, if we saw in it only a simple preliminary labour, having no other general and essential object than to prepare indispensable foundations for the integral calculus. As the ideas on this matter are generally confused, I think that I ought here to explain in a summary manner this important relation as I view it, and to show that in every application of the transcendental a.n.a.lysis a primary, direct, and necessary part is constantly a.s.signed to the differential calculus.

1. _Use of the Differential Calculus as preparatory to that of the Integral._ In forming the differential equations of any phenomenon whatever, it is very seldom that we limit ourselves to introduce differentially only those magnitudes whose relations are sought. To impose that condition would be to uselessly diminish the resources presented by the transcendental a.n.a.lysis for the expression of the mathematical laws of phenomena. Most frequently we introduce into the primitive equations, through their differentials, other magnitudes whose relations are already known or supposed to be so, and without the consideration of which it would be frequently impossible to establish equations. Thus, for example, in the general problem of the rectification of curves, the differential equation,

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

is not only established between the desired function s and the independent variable _x_, to which it is referred, but, at the same time, there have been introduced, as indispensable intermediaries, the differentials of one or two other functions, _y_ and _z_, which are among the data of the problem; it would not have been possible to form directly the equation between _ds_ and _dx_, which would, besides, be peculiar to each curve considered. It is the same for most questions.

Now in these cases it is evident that the differential equation is not immediately suitable for integration. It is previously necessary that the differentials of the functions supposed to be known, which have been employed as intermediaries, should be entirely eliminated, in order that equations may be obtained between the differentials of the functions which alone are sought and those of the really independent variables, after which the question depends on only the integral calculus. Now this preparatory elimination of certain differentials, in order to reduce the infinitesimals to the smallest number possible, belongs simply to the differential calculus; for it must evidently be done by determining, by means of the equations between the functions supposed to be known, taken as intermediaries, the relations of their differentials, which is merely a question of differentiation. Thus, for example, in the case of rectifications, it will be first necessary to calculate _dy_, or _dy_ and _dz_, by differentiating the equation or the equations of each curve proposed; after eliminating these expressions, the general differential formula above enunciated will then contain only _ds_ and _dx_; having arrived at this point, the elimination of the infinitesimals can be completed only by the integral calculus.

Such is, then, the general office necessarily belonging to the differential calculus in the complete solution of the questions which exact the employment of the transcendental a.n.a.lysis; to produce, as far as is possible, the elimination of the infinitesimals, that is, to reduce in each case the primitive differential equations so that they shall contain only the differentials of the really independent variables, and those of the functions sought, by causing to disappear, by elimination, the differentials of all the other known functions which may have been taken as intermediaries at the time of the formation of the differential equations of the problem which is under consideration.

2. _Employment of the Differential Calculus alone._ For certain questions, which, although few in number, have none the less, as we shall see hereafter, a very great importance, the magnitudes which are sought enter directly, and not by their differentials, into the primitive differential equations, which then contain differentially only the different known functions employed as intermediaries, in accordance with the preceding explanation. These cases are the most favourable of all; for it is evident that the differential calculus is then entirely sufficient for the complete elimination of the infinitesimals, without the question giving rise to any integration. This is what occurs, for example, in the problem of _tangents_ in geometry; in that of _velocities_ in mechanics, &c.

3. _Employment of the Integral Calculus alone._ Finally, some other questions, the number of which is also very small, but the importance of which is no less great, present a second exceptional case, which is in its nature exactly the converse of the preceding. They are those in which the differential equations are found to be immediately ready for integration, because they contain, at their first formation, only the infinitesimals which relate to the functions sought, or to the really independent variables, without its being necessary to introduce, differentially, other functions as intermediaries. If in these new cases we introduce these last functions, since, by hypothesis, they will enter directly and not by their differentials, ordinary algebra will suffice to eliminate them, and to bring the question to depend on only the integral calculus. The differential calculus will then have no special part in the complete solution of the problem, which will depend entirely upon the integral calculus. The general question of _quadratures_ offers an important example of this, for the differential equation being then _dA = ydx_, will become immediately fit for integration as soon as we shall have eliminated, by means of the equation of the proposed curve, the intermediary function _y_, which does not enter into it differentially. The same circ.u.mstances exist in the problem of _cubatures_, and in some others equally important.

_Three cla.s.ses of Questions hence resulting._ As a general result of the previous considerations, it is then necessary to divide into three cla.s.ses the mathematical questions which require the use of the transcendental a.n.a.lysis; the _first_ cla.s.s comprises the problems susceptible of being entirely resolved by means of the differential calculus alone, without any need of the integral calculus; the _second_, those which are, on the contrary, entirely dependent upon the integral calculus, without the differential calculus having any part in their solution; lastly, in the _third_ and the most extensive, which const.i.tutes the normal case, the two others being only exceptional, the differential and the integral calculus have each in their turn a distinct and necessary part in the complete solution of the problem, the former making the primitive differential equations undergo a preparation which is indispensable for the application of the latter. Such are exactly their general relations, of which too indefinite and inexact ideas are generally formed.

Let us now take a general survey of the logical composition of each calculus, beginning with the differential.

THE DIFFERENTIAL CALCULUS.

In the exposition of the transcendental a.n.a.lysis, it is customary to intermingle with the purely a.n.a.lytical part (which reduces itself to the treatment of the abstract principles of differentiation and integration) the study of its different princ.i.p.al applications, especially those which concern geometry. This confusion of ideas, which is a consequence of the actual manner in which the science has been developed, presents, in the dogmatic point of view, serious inconveniences in this respect, that it makes it difficult properly to conceive either a.n.a.lysis or geometry. Having to consider here the most rational co-ordination which is possible, I shall include, in the following sketch, only the calculus of indirect functions properly so called, reserving for the portion of this volume which relates to the philosophical study of _concrete_ mathematics the general examination of its great geometrical and mechanical applications.