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.
3.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 in 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 it 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 catechized 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,'sheweth no inclination to be catechized by me.But his scholars have a right to be informed.Itherefore 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.
4.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 [See'Vindication,'p.17.]vanisheth,whether the rectangles themselves do not vanish?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 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?[See his'Introduction to the Quadratures.']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.
