19 SUCCESSIVE DIFFERENTIATIONS
The preceding still leaves a difficulty regarding the consideration of different orders of infinitesimal quantity: how can one conceive of quantities as infinitesimal not only with respect to ordinary quantities, but with respect to other quantities that are themselves infinitesimal? Here again Leibnitz has recourse to the notion of 'incomparables', but this is much too vague to satisfy us, and it does not sufficiently explain the possibility of successive differentiations. No doubt, this possibility can best be understood by a comparison or example from mechanics: 'As for ddx, it is to dx as the conatus [force] of weight or the centrifugal tendency is to speed. [1] And Leibnitz develops this idea in his response to the objections of the Dutch mathematician Nieuwentijt, who, while admitting differentials of the first order, maintained that those of higher orders could only be null quantities:
Ordinary quantity, the first infinitesimal or differential quantity, and the second infinitesimal or diffentio-differential quantity, are to each other as movement, speed, and solicitation, [2] which is an element of speed. Movement describes a line, speed an element of the line, and solicitation an element of the element. [3]
But here we have only a particular example or case, which can in short serve only as a simple 'illustration', not an argument, and it is necessary to furnish justification of a general order, which this example, moreover, in a certain sense contains implicitly.
Indeed, differentials of the first order represent the increases-or, better, the variations, since depending on the case they could as easily be in the decreasing as in the increasing direction-that are at each instant received by ordinary quantities; such is speed with respect to the space covered in a given movement. In the same way, differentials of a given order represent the instantaneous variations of differentials of the preceding order, which in turn are taken as magnitudes existing within a certain interval; such is acceleration with respect to speed. Thus the distinction between different orders of infinitesimal quantities in fact rests on the consideration of different degrees of variation, much more than on that of incomparable magnitudes.
In order to state precisely the way in which this must be understood, let us simply make the following remark: one can establish among the variables themselves distinctions analogous to those established earlier between fixed and variable quantities; under these conditions, to go back once again to Carnot's definition, a quantity is said to be infinitesimal with respect to others when one can render it as small as one likes 'without one being obliged thereby to vary these other quantities.' Indeed, this is because a quantity that is not absolutely fixed, or even one that is essentially variable-as is the case with infinitesimal quantities, whatever the order in question-can nevertheless be regarded as fixed and determined, that is, as capable of playing the role of fixed quantity with respect to certain other variables. Only under these conditions can a variable quantity be considered the limit of another variable, which, by the very definition of the term limit, presupposes that it be regarded as fixed, at least in a certain respect, namely relative to that which it limits; inversely, a quantity can be variable not only in and of itself or, what amounts to the same, with respect to absolutely fixed quantities, but even with respect to other variables, insofar as the latter are regarded as relatively fixed.
Instead of speaking in this regard of degrees of variation, as we have just done, one could equally well speak of degrees of indeterminacy, which ultimately would be exactly the same thing, only considered from a slightly different point of view: a quantity, though indeterminate by its nature, can nevertheless be determined in a relative sense by the introduction of certain hypotheses, which allow the indeterminacy of other quantities to subsist at the same time; these latter quantities will therefore be more indeterminate, so to speak, than the others, or indeterminate to a greater degree, and they will therefore be related to the others in a manner comparable to that in which the indeterminate quantities are themselves related to quantities that truly are determined. We shall confine ourselves to these remarks on the subject, for however summary they might be, we believe that they are at least sufficient for understanding the possibility of the existence of differentials of various successive orders; but, in connection with this same question, it still remains for us to show more explicitly that there is really no logical difficulty in considering multiple degrees of indefinitude, and this as much in the order of decreasing quantities, to which infinitesimals and differentials belong, as in that of increasing quantities, in which one can likewise envisage integrals of different orders, which are as it were symmetric with respect to the successive differentiations; and this is moreover in conformity with the correlation that exists between the indefinitely increasing and the indefinitely decreasing, as we have explained. Of course, in all this it is only a question of degrees of indefinitude, and not of 'degrees of infinity', such as Jean Bernoulli understood them, which notion Leibnitz dared neither adopt nor reject absolutely in this regard; and here we have yet another case in which the difficulties can be immediately resolved by substituting the notion of the indefinite for that of the so-called infinite.