VARIABLE AND FIXED QUANTITIES
Let us now return to the question of the justification of the rigor of the infinitesimal calculus. We have already seen that Leibnitz considers quantities to be equal when their difference, while not strictly null, is nonetheless incomparable with respect to the quantities themselves; in other words, infinitesimal quantities, though not absolute nothingness, are nevertheless nothingness in some respect, and as such must be negligible with respect to ordinary quantities. Unfortunately, the notion of 'incomparability' is still too imprecise for an argument based on it alone to be fully sufficient to establish the rigorous character of the infinitesimal calculus fully; from this point of view, the calculus appears to be in short but a method of indefinite approximation, and we cannot say with Leibnitz that 'once this is affirmed, it follows not only that the error is infinitely small, but that it is nothing at all'; [1] but is there no more rigorous means of arriving at this conclusion? We must at least admit that the error introduced into our calculations can be rendered as small as desired, which is already saying a great deal; but does not precisely this infinitesimal character of the error do away with it completely when one considers, not only the course of the calculation itself, but its final results?
An infinitesimal difference, that is, one decreasing indefinitely, can only be the difference between two variable quantities, for it is obvious that the difference between two fixed quantities can itself only be a fixed quantity; it would thus be meaningless to speak of an infinitesimal difference between two fixed quantities. Hence, we have the right to say that two fixed quantities are 'rigorously equal the moment that their would be difference can be supposed as small as one likes'; [2] now, 'the infinitesimal calculus, like ordinary calculation, really has in view only fixed and determined quantities'; [3] in short, it introduces variable quantities only as auxiliaries having a purely transitory character, and these variables must disappear from the results, which can only express ratios between fixed quantities. Thus, in order to obtain these results, one must pass from a consideration of variable quantities to one of fixed quantities; and this passage has precisely as its result the elimination of infinitesimal quantities, which are essentially variable, and which can appear only as the differences between variable quantities.
It will now be easy to understand why, in the definition we cited earlier, Carnot insisted that infinitesimal quantities as employed in the calculus, are able to be rendered as small as one likes 'without one's being obliged on that account to vary the quantities to which they are compared.' It is because these latter quantities must in reality be fixed quantities; it is true that in the calculus they are considered to be limits of variable quantities, but these latter merely play the role of simple auxiliaries, as do the infinitesimal quantities which they bring with them. In order to justify the rigor of the infinitesimal calculus, the essential point is that only fixed quantities must figure in the results; in terms of the calculus, therefore, it is ultimately necessary to pass from variable quantities to fixed quantities, and this is indeed a 'passage to the limit', but not as conceived by Leibnitz, since there is no result or 'final term' of the variation itself; now-and this is what really matters-the infinitesimal quantities are eliminated of themselves in this passage, and this quite simply by reason of the substitution of fixed quantities for variable quantities. [4]
But must one view their elimination merely as the result of a simple 'compensation of errors', as Carnot would have it? We think not, and it indeed seems that one really can see more in it as soon as one distinguishes between variable and fixed quantities, observing that they constitute as it were two separate domains, between which there doubtless exists a correlation and analogy-which moreover is necessary in order to be able to pass from one to the other, however such a passage is effected-but without their real ratios ever establishing any kind of interpenetration, or even continuity; furthermore, this implies that an essentially qualitative difference exists between the two sorts of quantity, in conformity with what was said earlier concerning the notion of the limit. Leibnitz never made this distinction clearly, and here again, his conception of a universally applicable continuity no doubt prevented him from doing so; he was unable to see that 'passage to the limit' essentially implies a discontinuity, because for him no discontinuity existed. However, it is this distinction alone that allows us to formulate the following proposition: if the difference between two variable quantities can be rendered as small as one likes, then the fixed quantities that correspond to these variables and which are regarded as the respective limits of the latter, are rigorously equal. Thus, an infinitesimal difference can never become nothing; but such a difference can exist only between variables, and between the corresponding fixed quantities, the difference must indeed be nothing; whence it immediately follows that to an error capable of being rendered as small as one likes in the domain of variable quantities (in which there can in fact be no question of anything more than indefinite approximation precisely by reason of the character of these quantities) there necessarily corresponds another error that is rigorously null in the domain of fixed quantities. The true justification for the rigor of the infinitesimal calculus essentially resides in this consideration alone, and not in any others, which, whatever they might be, are always more or less peripheral to the question.