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-59Reas is for nos repl ing, Uc. Mahon's doctrine. But, no wonder that gentiem an mould not agree with Sir Isaac, since he cannot agree even With himself, hut contradicts whathe se illi et sewhere as the reader may see, even besore he geis to the endos that fame section, wherein he hath told us that the gnomen and the sum of the two rectangles are turned into those two sides by a retro-μ verted motion P. II and Ia).' Whicli proposition is you or any other person mali try to malie sense os, you may possibly be convinced, that this pro und author is as much at variance With common sense, as he is th himself and Sir Ioac Newton. XI. Mr. Mallon in the ninth page of his Vindication, in order to explain the nature os fluxions, aith that to obtain the last ratio of syn- chronat incremenis, the magnitude of those increments must he infi-
he explained the doctrine os fluxions by the ratio os magnitudes infinitely diminis hed. It is an easy matter, sor any author to Write se, asto betray his readers into mistahes about his meaning. But then it is not east to conceive, what right he hath to Dpbraid them with lach theirmistakes. Is I have mistahen his sense, let any one judge is he did notia irly lead me into the mistake. When a man pugZleth his reader, salthand unsaith, useth ambiguous terms and obscure terms, and pulteth themtogether in Q perverse a manner, that it is odds you can mahe out nosense at all, or is a ny, a wrong sense, pray who is in fauit but the writer himselfi Let any one consider Mr. Waltons own words, and then say hether I am not justified in malaing this remarii. XII. In the twentieth page of his fuit ansiser Mr. Giton telis us, that fluxions are measu red by the first or last proportioris of isochronat in- crements generaled or destroyed by motion.' A litile aster he salththese ratios subsist when the isochronat increments have no magnitude.
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ments we are not to understand increments genera ted in equat times twhether there can he an increment where there is no increa se, or in- crease where there is no magnitude t Whether is magnitudes are not generaled in those equat times, what et se is generaled therein, or what et seis it that M r. Walton calis isochronai t I asti the reader these questions. I dare not assi Mr. Walton. For, as I hin ted besere, the subject grows stili more obscure in proportion as this able writer attempis to illuse
nal spaces liath a real existence, forasmuch as it is equat to the ratio of the two motions of two potnis; whicli motions, subsisting when the isochronat si aces a re nothing; preserve the existence of the firsi or last ratio of these spaces, or heep it hom being a ratio of nothings.' inorder to assist your understanding, it must not be omitted that the seidtwo potnis are supposed to exist at the fame time in one potnt, and to bemoved disserent ways without stirring Dom that potnt. Mr. Walton hath the conscience to cali this riddie a fuli and clear answer: to mahe sense oswhich you must suppose it one of his Donies. In the nexi and last arti-cle of his persormance, you stili find him proceed in the same vein ofraillery upon fluxion S. XIV. It will be allowed, that whoe ver serioussy under took to expla in the second, third, and urth fluxions of Sir Isaac Neiston, would havedone it in a way agreeable to that great man's own doctrine. What SirIOae s preci se notion is I will not pretend to say. And yet I will ventu reto say, it is semething that cannot be expla ined by the three dimensitonsos a cube. I frankly own, I do not understand Sir Isaac s domine so faras to frame a positive idea of his fluxions. I have, neverthelest, a nega
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in earnest) that he understands it no more than I do. XU. Sir Ioae telis us that he considers indeterminate quantities asso ing, or in other words, as increasing or decreasing by a perpetuat motion. Whicli quantities he denotes by the lalter letters of the alphabet, and their fluxions or celerities of increasing by the fame letters potntedo ver head, and the fluxions os fluxions or second fluxions, i. e. the mutations more or less Mift of the first celerities by the fame letters potia ted with doubie potnis; and the mutations of those mutations of the first mutations or fluxions or celerities of increasing, which he calis Buxions of fluxions of fluxions or third fluxions, by three potnis; the foui thfuxions by sour potnis; the fifth by five; and so on . Sir I ac, youste, spealis of quantity in generat. And in the Analyst the doctrine igexemplified and the case is piat in lines. Now in lines, where there isonly one dimension, how are we enabled to conceive second, third or urth fluxions by concei ving the generation of three dimensions in acube t Let any one but read what Sir Uaac Newton or what I have se id, and then apply what Mr. Walton hath written about the three dimensions os a cube, and see whether the dissiculties are solved, or the doctrine made one whit the clearer by this explication.
XVI. That you may the better judge of the merit of this part of Mr.
' See his Treatist de quadratura curVarum,
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any sense what ever. One would thinli that men could not speah too exaclly on nice a su est And yet we may osten observe, that the exponents of fluxions or notes representing fluxions are con unded with the fluxions themselves. Is not this the case, When just after the fluxions os flowing quantities, were se id to be the celerities of their in- creasing and the second fluxions to be the mutations of the first fluxions
or celerities, we are told that g. g. g. g. z. g. representS a series of quan-
tities whereos each subsequent quantity is the fluxion of the preceding. and each foregoing is a fluent quantity having the folio ing one sor iis fluxion. Divers series of quantities and expressions geometrical and algebraical may be east ly conceived in lines, in sursaces, in species, to be continued without end or limit. But it Will not be soland so east to
conceive a series, ei ther of mere velocities or os mere nascent incre- menis, distinct thereseom and corresponding thereunto. ' ' Compare
what is here seid with Mr. Waltons genesis of a cube, and you willthen ciearly see how far this answerer is frona explaining the nature offecond, third and Durth fluxions: and hoW justly I might repay that gentieman in hind, and teli him in his own langu age, that ali his stili is Cain and impertinent, Vind. p. 36. XVII. But it doth not hecome me to find Dult with this learned professor, who at bottom militates on my side, and in this very section, mahes it his businest directly to overthrow Sir Ioac Newton s doctrine. For he salth in plain terms, that there can be no fourth fluxion os acube P. et s), that is, there can be no second fluxion os a line, and a fr-tiori, no third, Hurth, fifth, In much, that with one single dash of his pen, Mr. Walton destroys, to the great relies of the learned worid, ata Analyst, Sect. XLIV, XLV, XLVI. indefinite
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Reafons for me replFing, Uc. 263 indesinite rank of fluxions os disserent orders that might have reached nom pote to pote. I had distinctly potnted out the difficulties in severat paris both of my Analyst and Delance, and I leave you to judge whether
he explains Or even attempis to explain one of them. Instead thereos he telis us of the trine dimension os a cube generaled by motion: whencelle takes occasion, as liath heen observed, to explode Sir I ac's own doctrine, whicli is ut terly inconsistent with Mr. Waltons. And can you no doubi the real design of this egregious vindicator.
XVIII. Before ever Sir Isaae N ton thought of his fluxions, every hody knew there Nere three dimensions in a cube, and that a solid might be generaled by the motion os a surisce, a sursace by the motion os aline, and a line by the motion os a potnt. And this in effect is ali weknow Dom Mr. Waltons explication. As sor his dNelling se minutely orithe genesis of the solid paris of a cube, a thing so foretgn stom the pur- pose, the only rational account I can give of it is, that Μr. Walton, bypugetling the imagination of his vulgar readers, hoped the better to dis guise his betraying the doctrine of his great client, Which to a discerningeye he manifestly gives up , and instead thereos humouroussy substitutes what ali the world kncis besere Sir Isaac was horn, to wit, the three dimensions os a cube and the genesis thereos by motion. XIX. Upon the whole I appeal to you and every intelligent reader, whether this thing, which Mr. Wallon is pleased ironicatly to cali a fuit, an sever, doth not carry throughout a sty insinuation, that the pro undscience of fluxions cannot be maintained but by the hel p of most unintelligibie paradoxes and inconsistencies. So far, indeed, as affirmations gohe naeweth himself an able lapport of Sir Isaac Newton. But then in his rea nings he drops that great man upon the most important potnis, towit, his doctrine os motion and his doctrine os fluxions, not regardinghow far the demonstration of his samous Principia is interessed therein.
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Τo convince you stili more and more of the truth hereos, do but reflect alitile on Mr. Waltons conduci. Can you thinli it probabie, that solearned and clear-headed a writer would have laid down such a direct repugnancy.to common sense, as his idea os motion in a potnt, sor theground work of his explanation, had it been his real intention to explain t Or can you suppose, he would have been absolutoly silent, on somany potnis urged home, both in the Analyst and Defence, whicli it con- cerned a vindicator of Sir Ioac not to have overtooked i Can you imagine. that is he meant seriousty to desend the doctrine os fluxions, hewould have contented himself with barely asserting that Sir L ac μυ- ton in the introduction to his Quadrature of Curves, in the second lem- ma of the second book, and in the scholium to the first section of the
first book of his principies of philosophy, hath deli vered his doctrine of
fluxions in Q clear and distinct a manner, without the least inconsist- ency in term S or arguments, that one would have thought it impossibie
mouid talie his hare Word, as much more credi ble than Sir L ac's, and not rather have endeavoured to answer the questions and reconcile the difficulties set fortii in my de nce of Dee-thinking for instance, in Sect. XXX vi. Wherein I entreat my antagonist to expla in v whether Siro Isaac's momentum he a finite quantity or an infinitesimal or a mere li-' Init, adding, is you say a finite quantity, he pleased to reconcile this Mith what he salth in the scholium of the second lemma of the first section' of the firR book of his Principies : Cave intelligas quantitates magnitudine determinatas, sed cogita semper diminuendas e limite. lf you say an in- finitesimat: reconcile this with what is seid in his introduction to the Quadratures: Volui ostendere quod in methodo furionum non opus si Aura; in ite parvas in geometriam inducere. Is you should say it is a mere limit, be pleased to reconcile this With what we find in the fit si casu of
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Reafons for nor repl ing, GC. the second lemma in the second book of his Principies: Ubi do lateribus M A G B deerant momentorum dimidia, &c. where the moments are sup- posed to be divided. I mali scarce thin k it worth my while to besto a serious thought on any writer who stati preten d to maintain Sir Isaac's doctrine, and yet leave this passage without a rei ly. And thereader, I belle ve, will thin k with me that, in an swer to dissi culties disitinctly proposed and insisted on, to offer nothing but a magisteriai assertion is a mere gri mace of ono who made merry with fluxions, under thenotion os defending them. And he will he further confirmed in this wayof thin hing, when he observes that Mr. Walton hath not se id one syllabie, in reply to those severat sections of my desence, whicli I had particularly reserred to, as containing a suli an Mer to his Vindication. But it is nowonder is, with Sir Isaae's doctrine, he mould drop also his own argummenis in favour thereos XXI. I have been at the pa ins Once sor ali to write this mori commenton Mr. Giton, as the only way I could think of sor mahing him intelligibie, which will also serve as a key to his suture writings on this subjeci. An d Ι was the rather inclined to talae this trouble, hecause it seemeth to me, there is no part os learning that wanis to be cleared up morethan this fame doctrine os fluxions, which hath hitherio walhed a bout in a mist to the stupefaction of the literari of the present age. To con clude, I accept this professor's recantation, nor a m at ali displeased at theingenio us method he talies to dilui se it. Some gealous fluxionisl may Perhaps ans er him. Vo L. II.