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An Appendix, So. 2 0gles vanisheth, whether the rectangles them selves do not ait o vanim ti. e. when a b is nothing, Whether A b φ B a be not also nothingi i. e. whether the momentum of AB be not nothingi Let him then be asked. What his momentum S are good for, When they are thus brought to nothing t Again, I Wissi he were asked to explain the disserence, betweena magnitude infinitoly smali and a magnitude infinitely diminimed. lfhe seith there is no disserence: then let him be further asked, how hedares to explain the method of fluxions, by the ratio os magnitudes infinitely diminimed P. s), When Sir Isaac Newton hath expressty excluded ali consideration os quantities infinitely smali t j I f this able vindicatori hould say that quantities infinitely diminimed a re nothing at all, and
consequently that, according to him, the first and last ratio's a re proportions belween nothings, let him be destred to mahe sense of this, or explain what he means by proportion bremeen nothings. Is he mould say the ultimate proportions are the ratio's of mere limiis, then let him he askedhois the limits of lines can be proportioned or divided y Aster ali, who knowsbiat this gentieman, Who hath already compla ined of me for an uncommon way of treat ing mathematics and mathematicians s P. s), may asweli as the Cantabrigian cry out, S in and the inquisition, when hefinds himself thias closely pursu ed and beset with interrogatori es t That we may not, theresere, stem too hard on an innocent man, Who proba-hly meant nothing, but was be trayed by solio ing another into difficul-ties and stratis that he was not arua re os, I siali propose one single expedient, by whicli his discipies whom it most concern s) may seon satis' thenaseives, whether this vindicator reatly understands what he tahesu pon him to vindicate. It is in mori, that they would assi him to expla in the seconii, third, or Ourth fluxions upon his principies. Be this the touchstone of his vindication. Is he can do it, Ι strali own mystis
u See Vindication, P. I 7.1 See his Introduction to the Quadratures.
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aso An Appendix, Ge. much mistahen: is he cannot, it will he evident that he was much mic talien in himself, when he presumed to defend fluxions without so muchas knowing what they are. So having put the meriis of the cause onthis issue, I leave him to be tried by his scholars.
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There are also certain caretess writers, that in defiance of common sense
ing no further notice of him on the abovementioned considerations, Ileave you and every other reader to judge. But those, sir, are not thereasons I mali assign for not replying to Mr. Walton's fuit ans er. Thetrue reason is, that he stems at bottom a facetious man, who under thecolour of an opponent writes on my side of the question, and reatly belleves no more than I do of Sir Isaac Neistons doctrine a bout fluxions, Which he exposes, contradicts, and confutes With great Mill and humour, under the masque os a grave vindication. II. At first I considered him in another light, as one Who had goodreason sor Leeping to the beaten trach, who had been used to dictate, Who had terms of art at will, but was indeed, at smali trouble about Putting them together, and perfectly east about his reader's understand-ing 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, he a man's assertions e ver se strong in favour of a doctrine, yet is his rea nings are direct ly levelled against it, Whatever question there maybe .about the matter in dispute, there can be non e about the intention
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of the writer. Should a person, so knowing and discreet as Mr. Giton, thwari and contradi et Sir Isaac Neiston under pretence of defending his fluxions, and mould he at e very turn say lach uncouth things of theses a me fluxions, and place them in suta odd lighra, as must set ali men in their wiis against them, could I hope sor a be iter second in this cause tor could there rema in any do ubi of his heing a disgui sed Deeth inher in mathematics, who defended fluxions just as a certain freeth inher in religion did the rights of the christian Church. III. Mr. Walton indeed aster his Dee manner calis my Analyst a libet. But this ingenio us genti eman well knows a bad vindication is the bitterest libet. Had you a mind, sir, to be tray and ridicule any cause under the notion os vindicat ing ii, would you not thin k it the right way to hevery strong and dogmatical in the affirmative, and very weali and pug-gled in the argumentative paris of your performancet To Utter contradictions and paradoxes without remorse, and to be at no patias about
reconcit ing or explaining them l And with great good humour, to be at perpetuat varia iace Mith yourself and the author you preten d to vindicate ' How successislly Μr. Walton hath practised these aris, and howmuch to the honour of the great client he would stem to talie under his protection, I mali particularly examine throughout every article of his fuit an Mer. IV. First then, salth Mr. Walton, I am to be asked, whether I can
conceive veloci ty without motion, or motion without extension, or
extension without magnitudet To which he answereth in positive term S, that he Can conceive veloci ty and motion in a poliat P. ). Andio mahe out this, he undertaris to demonstrate, that is a thing be moved by an agent operat ing continuatly with the fame force, the ve-
loci ty will not be the fame in any two disserent potnts of the described
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had it not been in jest. V. Suppost the center of a salting body to describe a line, divide theti me of iis tali into equat paris, sor instance into minutes. The spaces described in those equat paris of time will be u nequat. That is, Domwhat e ver potnts of the described line you mea re a minute's descent, you wili stili find it a disserent space. This is true. But how or whysrom this plain truth a man staould inser, that motion can be conceived in a potnt, is to me as Oblcure as any the most obscure mysteries that o cur in this pro und aut hor. Let the reader mari the hest of it. Formy pari, I can as easi ly conceive Mr. Walton should walh without stir ring as I can his idea os motion without space. Aster' ait, the question was not whether motion could be proved to exist in a poliat, but onlywhether it could be conceived in a potnt. For, as to the proos of things
impossibie, seme men have a way of proving that may equalty proveany thing. But I much question whether any reader of common sense
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VI. Is ML Walton reatly meant to defend the author of the fluxionarymethod, would he not have done it in a way consistent with this illusetrious author's own principies t Let us now see what may be Sir Isaac's notion abo ut this matter. He distingui meth two foris of motion, absolute and relative. The former he defineth to be a translation Dom absolute place to absolute place, the lalter from One relative place to an other. ' Mr. Mations is pia in ly nei ther of these foris of motion, but semethird hind, which what it is, I am at a tost to comprehend. But I canclearly comprehend that, is we admit motion without space, then Sir Isaac Newtons account of it must be wrong: sor place by which he defines motion is, according to him, a part of space. And is se, then this notabie defender halli cui out new work for himself to defend and explain. But about this, is I mistahe not, he will be very casy. For, as Isaid be re, he stems at bottom a bach frien d to that great man; whichopinion you Will se e further confirmed in the sequel. VII. I mali no more assi Mr. Walton to expla in any thing. For I caci honestly say, the more he explains, the more I am puggled. But I willassi his readers to explain, by What art a man may conceive motion without space. And supposing this to be done, in the second place to explain, how it consist s with Sir Isaac Ne tons account of motion. Is it not evident, that Μr. Walton hath deserted stom his old master, and beenat some palns to expost him, While he defenda one part of his principies by overturning anotheri Let any reader teli me, What Mr. Walton meansis motion, or is he can guess, what this third hind is, whicli is nei ther absolute nor relative, whicli exist s in a potnt, Which may be conceived without space. This learned professor salth, Ι have no clear concep-
Rood and considered the nature os motion.' I belleue I am not alone
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to this intrepid answerer, who is ne ver at a tost, horu osten foe ver his readers may, I entreat you, or any other man of plain sense, to read the folio ing passage cited frona the thirty-first section of the Analyst, and then try to apply Mr. Waltons ans er to it: Whereby you will clearlyperceive What a vein os raillery that gentieman is master of Velocityμ necessarily implies both time and space, and cannot be conceived with- out them. And is the velocities of nascent and evanescent quantities, i. e. abstracted froni time and space, may not be comprehended, horu can we comprehend and demonstrate their proportionst Or consider A their rariones primae G ultimae. For to consider the proportion or ratio of things implieth that such things have magnitude: that such their magnitudes may be mea red, and their relations to each other known. But, as there is no mea re os velocity excepi time and space, the pro-
portiora of velocities heing only compounded of the direct proportion of the spaces and the reciprocat proportion of the times ; doth it notv follow, that to talk of investigating, obtaining, and considering the proportions of velocities, exclusi vely of time and space, is to talli uni n-μ telligibly ρ' Apply now, as I said, Mr. Walton's tuli araswer, and you willsoon sind how sully you are enlightened about the nature os fluxions.lX. In the soliori ing article of Μr. malions fuit an Mer, he stilli divers curious things, whicli, heing derived froin this sanie principie, that
motion may be conceived in a potnt, are altogether as incomprehensi bieas the origin Dorn wheiace they BON. It is obvious and natural to sup-Vo L. II. LI posu
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by increments. Mr. Giton indoed supposeth that when the increments vanissi or become nothing, the velocities remain, which being multiplied by finite lines produce those rectangles P. Ia). But admitting the velocities to remain, yet how can any one conceive a rectangular surisce tobe produced Dom a line multiplied by veloci ty, other ise than by supposing suta line multiplied by a line or increment, whicli mali be exponent of or proportional to such velocity t You may try to conceive itother i . Ι must own I cannot. Is not the increment of a rectangle iistis a rectan glet must not then Ab and B a be rectangi es t And must not the coessicients or sides of rectangles be linest Consequently are noth and a lines or whicli is the fame thing) increments of lines t These increments may indeed be considered as proportional to and exponents of velocity. But exclusive of such exponenis to talk of rectangles underlines and velocities is, I conceive, to talli uia intelligibb. And yet this is what Mr. Multon doth, when he maheth b and a in the rectangles A band B a to denote mere velocities. X. As to the question, whether nothing be not the produci of nothingmultiplied by something, Mr. Waltan is pleased to answer in the assirmative. And nevertheless when a b is nothing, that is, When a and b arenothing, he dentes that A b Φ B a is nothing. This is one of those many inconsistencies whicli I leave the reader is reconcite. But, salth Mr. GLton, the sides of the gi ven rectangle stili rem ain, which two sides accord- ing to him must form the increment of the Boising rectangle. But in this he dilectly contradicis Sir Ioae Newton, who asseris that A b B aand not A -b B is the increment of the rectangle A B. And, indeed, hois is it possibie, a line mould be the increment of a furface t Laterum incrementis totis a et b generatur rectanguli incrementum Ab Φ B a are thewords of Sir I ac ', which words seem ut terly inconsistent With Mr.