The philosophy of mathematics · Auguste Comte
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 constitute the general elements of our analysis. 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 analysis, 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, constituting, 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 analysis without our perceiving, at the first glance, what special and constant part the solution of the inverse question can have in such an analysis. It would be a real error, though a common one, to assign 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 analysis 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 analysis.
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 analysis; 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 circumstances exist in the problem of _cubatures_, and in some others equally important.
_Three classes of Questions hence resulting._ As a general result of the previous considerations, it is then necessary to divide into three classes the mathematical questions which require the use of the transcendental analysis; the _first_ class 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 constitutes 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.
Ideas discussed in this chapter
Text from Project Gutenberg, public domain in the United States.