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1. I had no sooner considered the performance of Philalethes, but Mr. Walton's Vindication of Fluxions was put into my hands. As this Dublin professor gleans after the Cantabrigian, only endeavouring to translate a few passages from Sir Isaac Newton's Principia, and enlarge on a hint or two of Philalethes, he deserves no particular notice. It may suffice to advertise the reader, that the foregoing Defence contains a full and explicit answer to Mr. Walton as he will find, if he thinks it worth his pains to read what this gentleman hath written, and compare it therewith: particularly with sect. 18. 20. 30 32–36. 43. It is not, I am sure, worth mine to repeat the same things, or confute the same notions twice over, in mere regard to a writer who hath copied even the manners of Philalethes, and whom in answering the other I have, if I am not much mistaken, sufficiently answered.

II. Mr. Walton touches on the same points that the other had touched upon before him. He pursues a hint which the other had given, * about Sir Isaac's first

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section concerning the rationes primæ et ultimæ. He discreetly avoids, like the other, to say one syllable of second, third, or fourth fluxions, and of divers other points mentioned in the Analyst, about all which I observe in him a most prudent and profound silence. And yet he very modestly gives his reader to understand, that he is able to clear up all difficulties and objections that have ever been made (p.5). Mr. Walton in the beginning, like Philalethes, from a particular case makes a general inference, supposing that infidelity to be imputed to mathematicians in general, which I suppose only in the person to whom the Analyst was addressed, and certain other persons of the same mind with him. Whether this extraordinary way of reasoning be the cause or effect of his passion, I know not: but before I had got to the end of his Vindication, I ceased to be surprised at his logic and his temper in the beginning. The double error, which in the Analyst was plainly meant to belong to others, he with Philalethes (whose very oversights he adopts) supposeth to have been ascribed to Sir Isaac Newton (p. 36). And this writer also, as well as the Cantabrigian, must needs take upon him to explain the motive of my writing against fluxions: which he gives out, with great assurance, to have been, because Sir Isaac Newton had presumed to interpose in prophecies and revelations, and to decide in religious affairs (p. 4), which is so far from being true, that, on the contrary, I have a high value for those learned remains of that great man, whose original and free genius is an eternal reproach to that tribe of followers, who are always imitating but never resemble him. This specimen of Mr. Walton's truth will be a warning to the reader to use his own eyes, and in obscure points never to trust the gentleman's candour, who dares to misrepresent the plainest.

III. I was thinking to have said no more concerning this author's performance, but lest he should imagine himself too much neglected, I entreat the reader to have the patience to peruse it ; and if he finds any one point of the doctrine of fluxions cleared up, or any one objection in the Analyst answered, or so much as fairly stated, let him then make his compliments to the author. But, if he can no more make sense of what this gentleman has written than I can, he will need no answer to it. Nothing is easier than for a man to translate or copy, or compose a plausible discourse of some pages in technical terms, whereby he shall make a show of saying somewhat, although neither the reader nor himself understand one tittle of it. Whether this be the case of Mr. Walton, and whether he understands either Sir Isaac Newton, or me, or himself (whatever I may think), I shall not take upon me to say. But one thing I know, that many an unmeaning speech passeth for significant by the mere assurance of the speaker, till he cometh to be catechised upon it ; and then the truth sheweth itself. This vindicator, indeed, by his dissembling nine parts in ten of the difficulties proposed in the Analyst, sheweth no inclination to be catechised by me. But his scholars have a right to be informed. I therefore recommend it to them, not to be imposed on by hard words and magisterial assertions, but carefully to pry into his sense, and sift his meaning, and particularly to insist on a distinct answer to the following questions.

IV. Let them ask him, whether he can conceive velocity without motion, or motion without extension, or extension without magnitude? If he answers that he can, let him teach them to do the same.

If he cannot, let him be asked, how he reconciles the idea of a fluxion which he gives (p. 13), with common sense ? Again, let him be asked, whether nothing be not the product of nothing multiplied by something; and if so, when the difference between the gnomen and the sum of the rectangles * vanisheth, whether the rectangles themselves do not also vanish? i. e. when a b is nothing, whether Ab + B a be not also nothing ? i. e. whether the momentum of A B be not nothing? Let him then be asked, what his momentums are good for, when they are thus brought to nothing? Again, I wish he were asked to explain the difference, between a magnitude infinitely small and a magnitude infinitely diminished. If he saith, there is no difference; then let him be farther asked, how he dares to explain the method of fluxions, by the ratio of magnitudes infinitely diminished (p. 9), when Sir Isaac Newton hath expressly excluded all consideration of quantities infinitely small ? If this able vindicator should


that quantities infinitely diminished are nothing at all, and consequently that, according to him, the first and last ratios are proportions between nothings, let him be desired to make sense of this, or explain what he means by proportion between nothings. If he should

the ultimate proportions are the ratios of mere limits, then let him be asked how the limits of lines can be proportioned or divided ? After all, who knows but this gentleman, who hath already complained of me for an uncommon way of treating mathematics and mathematicians (p. 5), may (as well as the Cantabrigian) cry out, Spain and the inquisition ! when he finds himself thus closely pursued and beset with interrogatories ? That we may not, therefore, seem too hard on an innocent man, who probably meant nothing, but was betrayed by following another into difficulties and straits that he was not aware of, I shall propose one single expedient, by which his disciples (whom it most concerns) may soon satisfy themselves, whether this vindicator really understands what he takes upon him to vindicate. It


• See Vindication, p. 17.
+ See his Introduction to the Quadratures.

is, in short, that they would ask him to explain the second, third, or fourth fluxions upon his principles. Be this the touchstone of his vindication. If he can do it, I shall own myself much mistaken: if he cannot, it will be evident that he was much mistaken in himself, when he presumed to defend fluxions without so much as knowing what they are. So having put the merits of the cause on this issue, I leave him to be tried his scholars.

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