and you may destroy one supposition by another: but then you may not retain the consequences, or any part of the consequences, of your first supposition so destroyed. I admit that signs may be made to denote either anything or nothing: and consequently that in the original notation x + o , o might have signified either an increment or nothing. But then, which of these soever you make it signify, you must argue consistently with such its signification, and not proceed upon a double meaning: which to do were a manifest sophism. Whether you argue in symbols or in words the rules of right reason are still the same. Nor can it be supposed you will plead a privilege in mathematics to be exempt from them.
16. If you assume at first a quantity increased by nothing, and in the expression x + o , o stands for nothing, upon this supposition, as there is no increment of the root, so there will be no increment of the power; and consequently there will be none except the first of all those members of the series constituting the power of the binomial; you will therefore never come at your expression of a fluxion legitimately by such method. Hence you are driven into the fallacious way of proceeding to a certain point on the supposition of an increment, and then at once shifting your supposition to that of no increment.
There may seem great skill in doing this at a certain point or period.
Since, if this second supposition had been made before the common division by o , all had vanished at once, and you must have got nothing by your supposition. Whereas, by this artifice of first dividing and then changing your supposition, you retain 1 and .
But, notwithstanding all this address to cover it, the fallacy is still the same. For, whether it be done sooner or later, when once the second supposition or assumption is made, in the same instant the former assumption and all that you got by it is destroyed, and goes out together. And this is universally true, be the subject what it will, throughout all the branches of human knowledge; in any other of which, I believe, men would hardly admit such a reasoning as this, which in mathematics is accepted for demonstration.
17. It may not be amiss to observe that the method for finding the fluxion of a rectangle of two flowing quantities, as it is set forth in the Treatise of Quadratures, differs from the above-mentioned taken from the second book of the Principles, and is in effect the same with that used in the calculus differentialis . [`Analyse des Infiniment Petits,' Part I., prop. 2.] For the supposing a quantity infinitely diminished, and therefore rejecting it, is in effect the rejecting an infinitesimal;and indeed it requires a marvellous sharpness of discernment to be able to distinguish between evanescent increments and infinitesimal differences.
It may perhaps be said that the quantity being infinitely diminished becomes nothing, and so nothing is rejected. But, according to the received principles, it is evident that no geometrical quantity can by any division or subdivision whatsoever be exhausted, or reduced to nothing. Considering the various arts and devices used by the great author of the fluxionary method; in how many lights he placeth his fluxions; and in what different ways he attempts to demonstrate the same point; one would be inclined to think, he was himself suspicious of the justness of his own demonstrations, and that he was not enough pleased with any notion steadily to adhere to it.
Thus much at least is plain, that he owned himself satisfied concerning certain points which nevertheless he would not undertake to demonstrate to others. [See `Letter to John Collins,' Nov. 8, 1676.] Whether this satisfaction arose from tentative methods or inductions, which have often been admitted by mathematicians (for instance, by Dr. Wallis, in his Arithmetic of Infinites), is what I shall not pretend to determine. But, whatever the case might have been with respect to the author, it appears that his followers have shown themselves more eager in applying his method, than accurate in examining his principles.
18. It is curious to observe what subtlety and skill this great genius employs to struggle with an insuperable difficulty;and through what labyrinths he endeavours to escape the doctrine of infinitesimals;which as it intrudes upon him whether he will or no, so it is admitted and embraced by others without the least repugnance; Leibnitz and his followers in their calculus differentialis ****** no manner of scruple, first to suppose, and secondly to reject, quantities infinitely small; with what clearness in the apprehension and justness in the reasoning, any thinking man, who is not prejudiced in favour of those things, may easily discern.
The notion or idea of an infinitesimal quantity , as it is an object simply apprehended by the mind, hath already been considered. [Sect. 5and 6.] I shall now only observe as to the method of getting rid of such quantities, that it is done without the least ceremony. As in fluxions the point of first importance, and which paves the way to the rest, is to find the fluxion of a product of two indeterminate quantities, so in the calculus differentialis (which method is supposed to have been borrowed from the former with some small alterations) the main point is to obtain the difference of such product. Now the rule for this is got by rejecting the product or rectangle of the differences. And in general it is supposed that no quantity is bigger or lesser for the addition or subduction of its infinitesimal: and that consequently no error can arise from such rejection of infinitesimals.
