Therefore the momentum must be an infinitesimal. And, indeed, though much artifice hath been employed to escape or avoid the admission of quantities infinitely small, yet it seems ineffectual. For aught I see, you can admit no quantity as a medium between a finite quantity and nothing, without admitting infinitesimals. An increment generated in a finite particle of time is itself a finite particle; and cannot therefore be a momentum. You must therefore take an infinitesimal part of time wherein to generate your momentum. It is said, the magnitude of moments is not considered; and yet these same moments are supposed to be divided into parts. This is not easy to conceive, no more than it is why we should take quantities less than A and B in order to obtain the increment of AB , of which proceeding it must be owned the final cause or motive is obvious;but it is not so obvious or easy to explain a just and legitimate reason for it, or show it to be geometrical.
12. From the foregoing principle, so demonstrated, the general rule for finding the fluxion of any power of a flowing quantity is derived. [`Philosophiae Naturalis Principia Mathematica,' lib. ii., lem. 2.] But, as there seems to have been some inward scruple or consciousness of defect in the foregoing demonstration, and as this finding the fluxion of a given power is a point of primary importance, it hath therefore been judged proper to demonstrate the same in a different manner, independent of the foregoing demonstration. But whether this other method be more legitimate and conclusive than the former, I proceed now to examine; and in order thereto shall premise the following lemma:- `If, with a view to demonstrate any proposition, a certain point is supposed, by virtue of which certain other points are attained; and such supposed point be itself afterwards destroyed or rejected by a contrary supposition; in that case, all the other points attained thereby, and consequent thereupon, must also be destroyed and rejected, so as from thenceforward to be no more supposed or applied in the demonstration.' This is so plain as to need no proof.
13. Now, the other method of obtaining a rule to find the fluxion of any power is as follows. Let the quantity x flow uniformly, and be it proposed to find the fluxion of .
In the same time that x by flowing becomes x + o , the power becomes , i.e. by the method of infinite series and the increments are to one another as Let now the increments vanish, and their last proportion will be 1 to .
But it should seem that this reasoning is not fair or conclusive. For when it is said, let the increments vanish, i.e. let the increments be nothing, or let there be no increments, the former supposition that the increments were something, or that there were increments, is destroyed, and yet a consequence of that supposition, i.e. an expression got by virtue thereof, is retained. Which, by the foregoing lemma, is a false way of reasoning. Certainly when we suppose the increments to vanish, we must suppose their proportions, their expressions, and everything else derived from the supposition of their existence to vanish with them.
14. To make this point plainer, I shall unfold the reasoning, and propose it in a fuller light to your view. It amounts therefore to this, or may in other words be thus expressed. I suppose that the quantity x flows, and by flowing is increased, and its increment I call o , so that by flowing it becomes x + o . And as x increaseth, it follows that every power of x is likewise increased in a due proportion. Therefore as x becomes x + o , will become , that is, according to the method of infinite series, And if from the two augmented quantities we subduct the root and the power respectively, we shall have remaining the two increments, to wit, which increments, being both divided by the common divisor o , yield the quotients which are therefore exponents of the ratio of the increments. Hitherto I have supposed that x flows, that x hath a real increment, that o is something. And I have proceeded all along on that supposition, without which I should not have been able to have made so much as one single step. From that supposition it is that I get at the increment of , that I am able to compare it with the increment of x , and that Ifind the proportion between the two increments. I now beg leave to make a new supposition contrary to the first, i.e. I will suppose that there is no increment of x , or that o is nothing; which second supposition destroys my first, and is inconsistent with it, and therefore with everything that supposeth it. I do nevertheless beg leave to retain , which is an expression obtained in virtue of my first supposition, which necessarily presupposed such supposition, and which could not be obtained without it. All which seems a most inconsistent way of arguing, and such as would not be allowed of in Divinity.
15. Nothing is plainer than that no just conclusion can be directly drawn from two inconsistent suppositions. You may indeed suppose anything possible; but afterwards you may not suppose anything that destroys what you first supposed: or, if you do, you must begin de novo . If therefore you suppose that the augments vanish, i.e. that there are no augments, you are to begin again and see what follows from such supposition. But nothing will follow to your purpose. You cannot by that means ever arrive at your conclusion, or succeed in what is called by the celebrated author, the investigation of the first or last proportions of nascent and evanescent quantities, by instituting the analysis in finite ones. I repeat it again: you are at liberty to make any possible supposition:
