method to use the argumentum ad hominem? And being so, whether it ought to surprise either Christians or scholars? Whether in an age wherein so many pretenders to science attack the Christian religion, we may not be allowed to make reprisals, in order to show that the irreligion of those men is not to be presumed an effect of deep and just thinking? Whether an attempt to detect false reasonings, and remedy defects in mathematics, ought to be ill received by mathematicians? Whether the introducing more easy methods and more intelligible principles in any science should be discountenanced? Whether there may not be fair objections as well as cavils? And whether to inquire diligently into the meaning of terms and the proof of propositions, not excepting against any thing without assigning a reason, nor affecting to mistake the signification of words, or stick at an expression where the sense was clear, but considering the subject in all lights, sincerely endeavouring to find out any sense or meaning whatsoever, candidly setting forth what seems obscure and what fallacious, and calling upon those who profess the knowledge of such matters to explain them; whether, I say, such a proceeding can be justly called cavilling? Whether there be an ipse dixit erected? And if so, when, where, by whom, and upon what authority? Whether even where authority was to take place, one might not hope the mathematics, at least, would be excepted? Whether the chief end, in making mathematics so considerable a part of academical education, be not to form in the minds of young students habits of just and exact reasoning? And whether the study of abstruse and subtile matters can conduce to this end, unless they are well understood, examined, and sifted to the bottom? Whether, therefore, the bringing geometrical demonstrations to the severest test of reason should be reckoned a discouragement to the studies of any learned society? Whether to separate the clear parts of things from the obscure, to distinguish the real principles whereon truths rest, and whence they are derived, and to proportion the just measures of assent according to the various degrees of evidence, be a useless or unworthy undertaking? Whether the making more of an argument than it will bear, and placing it in an undue rank of evidence, be not the likely way to disparage it? Whether it may not be of some use, to provoke and stir up the learned professors to explain a part of mathematical learning which is acknowledged to be most profound, difficult, and obscure, and at the same time set forth by Philalethes and many others, as the greatest instance that has ever been given of the extent of human abilities? Whether for the sake of a great man's discoveries, we must adopt his errors? Lastly, whether in an age wherein all other principles are canvassed with the utmost freedom, the principles of fluxions are to be alone excepted?

171

AN APPENDIX

CONCERNING MR. WALTON'S VINDICATION OF SIR ISAAC NEWTON'S PRINCIPLES

OF FLUXIONS.

I. 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 section concerning the rationes primæ et ultima. 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

Philalethes, p. 32.

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 showeth itself. This vindicator, indeed, by his dissembling nine parts in ten of the difficulties proposed in the Analyst, showeth 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 gnomon and the sum of the rectangles* vanisheth, whether the rectangles themselves do not also vanish?

See Vindication, p. 17.

i. e. when ab is nothing, whether Ab + Ba be not also nothing? i. e. whether the momentum of AB 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 further 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 say 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 say, 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 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 by his scholars.

See his Introduction to the Quadratures.

REASONS FOR NOT REPLYING

ΤΟ

MR. WALTON'S FULL ANSWER.

IN A LETTER TO P. T. P.

I. THERE are some men that can neither give nor take an answer, but, writing merely for the sake of writing, multiply words to no purpose. There are also certain careless writers, that in defiance of common sense publish such things as, though they are not ashamed to utter, yet other men may well be ashamed to answer. Whether there be any thing in Mr. Walton's method of vindicating fluxions, that might justify my taking no futher notice of him on the above-mentioned considerations, I leave you and every other reader to judge. But those, sir, are not the reasons I shall assign for not replying to Mr. Walton's full answer. The true reason is, that he seems at bottom a facetious man, who, under the colour of an opponent, writes on my side of the question, and really believes no more than I do of Sir Isaac Newton's doctrine about fluxions, which he exposes, contradicts, and confutes, with great skill and humour, under the mask of a grave vindication.

II. At first I considered him in another light, as one who had good reason for keeping to the beaten track, who had been used to dictate, who had terms of art at will, but was, indeed, at small trouble about putting them together, and perfectly easy about his reader's understanding them. It must be owned, in an age of so much ludicrous humour, it is not every one can, at first sight, discern a writer's real design. But, be a man's assertions ever so strong in favour of a doctrine, yet if his reasonings are directly levelled against it, whatever question there may be about the matter in dispute, there can be none about the intention of the writer. Should a person, so knowing and discreet as Mr. Walton, thwart and contradict Sir Isaac Newton under pretence of defending his fluxions, and should he at every turn say such uncouth things of these same fluxions, and place them in such odd lights, as must set all men in their wits against them, could I hope for a better second in this cause? Or could there remain any doubt of his being a disguised free-thinker in mathematics, who defended fluxions just as a certain free-thinker in religion did the rights of the Christian church.