THE NOTION OF THE LIMIT
The notion of the limit is one of the most important we have to examine here, for the value of the infinitesimal method, at least insofar as its rigor is concerned, depends entirely upon it; one could even go so far as to say that, ultimately, 'the entire infinitesimal algorithm rests solely on the notion of the limit, for it is precisely this rigorous notion that serves to define and justify all the symbols and formulas of the infinitesimal calculus. [1] Indeed, the object of this calculus 'amounts to calculating the limits of ratios and the limits of sums, that is, to finding the fixed values toward which the ratios or sums of variable quantities converge, inasmuch as these quantities decrease indefinitely according to a given law.'2 To be even more precise, let us say that of the two branches into which the infinitesimal calculus may be divided, the differential calculus consists in calculating the limits of ratios, of which the two terms decrease indefinitely, at the same time following a certain law in such a way that the ratio itself always maintains a finite and determined value; and the integral calculus consists in calculating the limits of sums of elements, of which the multitude increases indefinitely as the value of each element decreases indefinitely, for both of these conditions must be united in order for the sum itself always to remain a finite and determined quantity. This being granted, one can say in a general way that the limit of a variable quantity is another quantity considered to be fixed, which the variable quantity is supposed to approach through the values it successively takes on
in the course of its variation, until it differs from the fixed quantity by as little as one likes, or in other words, until the difference between the two quantities becomes less than any assignable quantity. The point which we must emphasize most particularly, for reasons that will be better understood in what follows, is that the limit is essentially conceived as a fixed and determined quantity; even though it will not be given by the conditions of the problem, one should always begin by supposing it to have a determined value, and continue to regard it as fixed until the end of the calculation.
But the conception of the limit in and of itself is one thing, and the logical justification of the 'passage to the limit' quite another; Leibnitz believed that
what in general justifies this 'passage to the limit' is that the same relations that exist among several variable magnitudes also subsist among their fixed limits when their variations are continuous, for then they will indeed reach their respective limits; this is another way of putting the principle of continuity. [3]
But the entire question is precisely that of knowing whether a variable quantity, which approaches its fixed limit indefinitely and which, consequently, can differ from it by as little as one likes, according to the very definition of a limit, can effectively reach this limit precisely as a consequence of this variability, that is, whether a limit can be conceived as the final term in a continuous variation. We shall see that in reality this solution is unacceptable; but putting aside the question, to return to it later, we will only say for now that the true notion of continuity does not allow infinitesimal quantities to be considered as if they could ever equal zero, for they would then cease to be quantities; now, Leibnitz himself held that they must always preserve the character of true quantities, even when they are considered to be 'vanishing'. An infinitesimal difference can therefore never be strictly null; consequently, a variable, insofar as it is regarded as such, will always really differ from its limit, and could not reach this limit without thereby losing its variable character.
On this point, aside from one slight reservation, we can thus entirely accept the considerations a previously cited mathematician sets forth in these terms:
What characterizes a limit as we have defined it is that the variable can approach it as much as one might wish, while nonetheless never being able to strictly reach it; for in order that the variable in fact reach it, a certain infinity would have to be realized, which is necessarily ruled out.... And one must also keep to the idea of an indefinite, that is to say an even greater, approximation. [4]
Instead of speaking of 'the realization of a certain infinity', which has no meaning for us, we will simply say that a certain indefinitude would have to be exhausted precisely insofar as it is inexhaustible, but that at the same time the possibilities of development contained within this very indefinitude allow the attainment of as close an approximation as might be desired, ut error fiat minor dato [that the error may become smaller than any given error], according to an expression of Leibnitz, for whom 'the method is certain' as soon as this result is attained.
The distinctive feature of the limit, and that which prevents the variable from ever exactly reaching it, is that its definition is different from that of the variable; and the variable, for its part, while approaching the limit more and more closely, never reaches it, because it must never cease to satisfy its original definition, which, as we have said, is different. The necessary distinction between the two definitions of the limit and the variable is met with everywhere.... This fact, that the two definitions, although logically distinct, are nevertheless such that the objects they define can come closer and closer to one another, [5] explains what
might at first seem strange, that is, the impossibility of ever making coincide two quantities over which one has the authority to diminish the difference until it becomes so small as to pass beyond expressibility. [6]
There is hardly any need to say that in virtue of the modern tendency to reduce everything exclusively to the quantitative, some people have not failed to find fault with this conception of the limit for introducing a qualitative difference into the science of quantity itself; but if it must be discarded for this reason, it would likewise be necessary to ban from geometry entirely-among other things-the consideration of similarity, which is also purely qualitative, since it concerns only the form of figures, abstracting them from their magnitudes, and hence from their properly quantitative element, as we have already explained elsewhere. In this connection, it would also be good to note that one of the chief uses of the differential calculus is to determine the directions of the tangents at each point on a curve, the totality of which defines the very form of the curve, and that in the spatial order direction and form are precisely elements of an essentially qualitative character. [7] What is more, it is no solution to claim to purely and simply do away with the 'passage to the limit' on the pretext that the mathematician can dispense with actually passing to it without in any way hindering him from pushing his calculation to its end; this may be true, but what matters is this: under these conditions, up to what point would one have the right to consider this calculus to rest on rigorous reasoning, and even if 'the method is thus certain', will it not be so only as a simple method of approximation? One could object that the conception we just explained also makes the 'passage to the limit' impossible, since the character of this limit is precisely such as to prevent its ever being reached; but this is true only in a certain sense, and only insofar as one considers variable quantities as such, for we did not say that the limit could in no way be reached, but-and it is essential that this be made clear-that it could not be reached within the
variation, and as a term of the latter. The only true impossibility is the notion of a 'passage to the limit' constituting the result of a continuous variation; we must therefore replace it with another notion, and this we shall do more explicitly in what follows.