장음표시 사용
241쪽
229 A Defence of μα-Tbin ling, Scionly one proportion os equality throughoui, whicli at once overthrowsthe whole system you underlahe to defend. Your moments I say) notbeing themselves assignable quantities, their disserences cannot he assign- able: and is this be true, by that way of rea ning it wili sellow, theyare ali equal, upon whicli supposition you cannot mahe one flep in themethod os fluxions. It appears stom hence, how unjustly you blameme P 32 for omitting to give any account of that first section of the first book of the Principia, wherein you say) the Mundation of the me thod os fluxions is geometricatly demonstrated and largely explained, and difficulties and objections against it are clearly solved. Alt whicli is solar Dom being true, that the very sirst and fundamentat lemma of that section is incompati ble with, and subversive of the doctrine os fluxions.
And, indeed, who stes not that a demonstration ad absurdum more veterum Proceed ing on a supposition, that e very disserence must be seme gi venquantity, cannot be admitted ita, or consist With, a method, where quantities, test than any given, are supposed reatly to exist, and be ca-pable os division tXXXIII. The nexi potnt you underia he to defend is that method forobtaining a rule to find the fluxion of any po Ner of a fio ing quantity, whicli is deli vered in his introduction to the Quadratures, and consider-ed in the Analyst '. And here the question belween us is, whether Ihave rightly represented the sense of those words, manescant jam aurmenta illa, in rendering them, let the increments vanish, i. e. let the in- Crements be nothing, or let there be no incrementst This you deny, but, as your manner is, instead os giving a rea n you declaim. I, on thecontrary affirm, the increments must be understood to be qui te gone and
absolutely nothing at all. My reason is, because without that supposition
you can ne ver bring the quantity or expression nX ox q.
242쪽
G. down to v x , the very thing ai med at by supposing the evanes cence. Say whether this he not the truth of the case t Whether the sor-mer expression is not to be reduced to the lalter i And whether this canpossibly be done se long as o is supposed a real quantity t I cannot in-deed say you are scrupulous about your assirmations, and yet I belle vethat even you will not affirm this; it being most evident, that the produci of two reat quantities is semething real; and that nothing real can
Sir Isaacs own principies; for the truth of whicli I appeal to ait whohnow any thing of these matters. Further by evanescant must ei ther bemeant let them the increments) vanissi and become nothing, in the obviolas sense, or else let them hecome infinitely smali. But that this latteris not Sir IDac's sense is evident Dom his own words in the very sanae page, that is, in the last os his introduction to the Quadratures, where he expressy faith Uolui sendere quod in methodo suxionum non opus si Aguras in ite parvas in geometriam introducere. Upon the whole, you seemto have considered this affair so very superficiat ly, as greatly to confirmme in the opinion, you are so angry with, to wit, that Sir Ibaac's followers a re much more eager in applying his method, than accurate in examining his principies. You raise a dust about evanescent augments whichmay perhaps amuse and amaze your reader, but I am much mistahenis it ever instructs or enlightens him. For, to come to the potnt, tho evanescent augments either are real quantities, or they are not. lf you
ties in the composition whereof they are coessicients; but then yota areos the sume opinion with me, whicli opinion you are pleased to cali
243쪽
A Defence of ne Thin ing, Uc. 23 IXXX lU. Nothing I say can be plainer to any impartiat reader, than that by the evanescence of augmenis, in the above cited passage, Sir I ac means their being actualty reduced to nothitag. But to put it out os alidoubi, that this is the truth, and to convince everi you, who me htile disposition to he convinced, I destre you to look into his Anahisis per uationes insultas P. eto) where, in his preparation sor demonstratingthe first rule for the squaring of simple curves, you Will find that on a parallel occasion, speaking of an augment whicli is supposed to vanish, he interpreis the word manescere by es e nihil. Nothing can be plainerthan this, Whicli at orace destroys your defence. And yet, plain as it is, I despair of mahing you acknowledge it, though I am sure you kel it,
244쪽
serent ways he attempis to demonstrate the fame potnt: one Would he
inclined to thinti, he was himself suspicious of the justnesis of his o n
demonstrations.' This passage you complain os as very hard usage of Sir Isaac Ne ton. You declaim copioussy, and e eavour to me that placing the fame poliat in various lighis is os great use to explain it; which you illustrate with much rhetoric. But the fauit of that passage isnot the hard usam it contains: but on the contrary, that it is too modest, and not so fuli and expressive of my sense, as perhaps it mould have been. Would you like it helter is I should say, the various incon- Asent accounts, whici, this great author gives of his momentums and his fluxions, may convince every intelligent reader that he had no clear and steady notions of them, without which whicli there can he no demonstration t I own frankly that I see no clearnesi or consistence in them. You teli me indeed, in Miltonie verse that the fauit is in my own eyes, So thiel a drop serene has quenob'd their orbs
Sir Isaae's momentum be a finite quantity, or an infinitesimal, or a merelimiti Is you say a finite quantity: be pleased to reconcile this withwhat he laith in the scholium os the second lemma of the first section ofthe
245쪽
A Defence of Free Thin Ag, Uc. the fir st book of his principies: Cave intelligas quantitates magnitudine determinatas, sed cogita semper diminuendas sine limite. Is you say, an infinitesimat: reconcile this with what is Oid in his introduction to the Quadratures: Volui sendere quod in methodo sinimum non opus si Auras in ite parvas in geometriam inducere. Is you mould say, it is a merelimit, be pleased to reconcile this with what we find in the first case of the second lemma in the second book of his principies: Ubi de lateribus
I mould be bet ter satisfied of this, ic instead os entertaining us with
246쪽
by the great author in disserent paris of his writings: andupon the wholet could ne ver mahe it out to be conssistent and intelligibie. I was evenled to an that one would be inclined to thini , he was himself suspi- elous of the justia est of his own demonstrations: and that he was not enough plea sed with any one notion Readi ly to adhere to it.' Asterwhicli I added, thus muci, is plain that he o ned himself satisfied con- ' cerning certain potnis, whicis neverthelesi he could not underialie to demonstrate to others.' See the seventeenth section of the Analyst. It is one thing when a doctrine is placed in various lighis: and another, When the principies and notions a re misted. When new devices a re introduced and substitu ted for othera, a doctrine instead of being illustraledmay be explained a y. Whether there be not something of this in thepresent case I appeal to the writings of the great author. His methodus rationum primarum et ultimarum, his seeond lemma in the second book of his principies, his introduction and treatise of the quadrature os curves. In ali whicli it appears to me, there is not one uniform doctrine explainedand carried throughout the whole, hut rather fundry inconsistent accounts
of this new method, whicli stili grows more dark and consu sed the more it is handled: I could not help thinhing, the greatest genius might lye under the influence of false principies; and where the object and notions Mere exceeding obscure, he might possibiy distrust even his own demonstrations. At least thus much seemed plain that Sir Ioac had semeti me
247쪽
of heautiit theorems and problems, which he never knows or thinlis of This you would have past for a consequence of my notions. But I appeal to ali those who are ever so littie knowing in lach matters, whether there are not divers Duntains os experiment, induction, and an logy, Whenee a man may derive and satis0 himself concerning the truthos many potnis in mathematics and mechanical philosophy, although the proose thereos afforded by the modern analysis mouid not amount to demonstration 8 I further appeal to the conscience of ali the most pro undmathematicia nq, whether they can, With perfect acquiescence of mindsree stom ali scrupte, apply any proposition merely upon the strengili ofa demonstration involving second or third fluxions, without the aid ofany such experiment or analogy or collaterat proos Whatseeuer 8 Lastin I appeal to the reader's own heari, whether he cannot clearly conceivea medium belween being fast asseep and demonstratingi But you willhave it, that I represent Sir Isaac's conclusions as Coming out right, bc
cause one error is compensated by another contrary and equat error,
hicli perhaps he ne ver linew himself nor thought os: that by a two id
248쪽
that this assair of a doubie error is enti rely a new disco very of my o Π, which Sir Ione and his followers ne ver knew nor thought of that you - have unquestion able eviden ce to convince me of the contrary, and that
XL. Is you defend Sir Uaae s notions as deli vered in his Principia, it must be on the rigorous ot of rejecting nothing, netther admitting nor casting away infinitely smali quantities. Is you defend the Marquis, whom you also style your master, it must be on the ot os admittingthat there are infinitesimais, that they may be rejected, that they arene verthelesse real quantities, and themselves infinitely subdivisibie. Butyou seem to have grown giddy with passion, and in the heat os controversy to have mistaken and Drgot your pari. I beseech yoU, Sir, to consider, that the Marquis whom alone, and not Sir Isaac this doubie error in finding the subtangent doth concern) rejecis in deed infinitesimais, but
249쪽
not on the laot that you do, to wit, their being inconsiderable in practical geometry or mixed mathematics. But he rejecis them in the accuracy of speculative linowledge: in Whicli respect there may be great logical errors, although there should be no sensibie mistahe in practice: hicli, it stems, is What you cannot comprehend. He rejects them like-wi se in veriue os a postulatum, Whicli I venture to cali rejecting them ithout ceremony. And though he inserreth a conclusion accuratelytrue, yet he doth it, contrary to the rules of logic, frona inaccurate and false premi ses. And how this comes about, I have at large explained in the Analyst, and shewed in that particular case of tangenis, that the rejectaneous quantity might have been a finite quantity of any gi ven magnitude, and yet the conclusion have come out exactly the fame wayiand consequently, that the truth of this method doth not depend oti thereason assigned by the Mamuis, to wit, the psulatum ser throwing awayinfinitesimais, and there re that he and his followers acted blinit id, asnot knowing the true reason sor the conclusion S coming out accuratelyr hi, whicli I meis to have been the effect of a doubie error. XLI. This is the truth of the matter, Which you mam esully mi re present and declaim Upon, to no sort of Purpost but to anause and mis lead your reader. For whicli conduct of yOUrs throughout your remarlis, you wili pardon me is I cannot other iste account, than Do m a secrethope that the reader of your defence Would ne ver read the Analyst. Is he doth, he cannot but see what an admirabie method you take to de send your cause: how instead os justisting the rea ning, the logic or thetheory of the case specified, whicli is the real potnt, you discourse of sentasii ble and practical errors: and how ait this is a manifest imposition ut onthe readcr. He must Meds see that I have expressty said, v I have no controversy excepi only about your logic and method: that I consider how you demonstrate; What objects you are conversant aboui; and whether you conceive them clearlyt That. I have osten expressed my-μ seli.
250쪽
self to the fame effect destring the reader to remember, that I am on ly concerned about the way of coming at your theorems, Whether it be legitimate or illegitimate, clear or obscure, scientific or tentative: that I have on this very occasion, to prevent ali possibili ty of mista ke, re peated and insit sted that I consider the geometrical analyst asia logician, i. e. so far sortii as he reasons and argues; and his mathematical co clusiotas not in themselves but in their premises; not as true or false, usesul or insignificant, but as derived from sucti principies, and by such inferen ces. You assirm and indeed what can you not assirmi) that the differcnce belween the true subtangent and that found without any compensation is absolutely nothing at all. I prosesi mystis of a contraryOpinion. My reason is because nothing cannot be divided into paris. Butthis differetice is capable of being divided into any, or into more than any gi ven number of paris; sor the truth of whicli consuli the Marquis de rHospital. And, be the error in sact or in practice e ver so sinati, it ill not theiace sol low that the error in rea ning, whicli is What I amatone concerned abolit, is one whit the lest, it heing evident that a manmay rea n most absurdly abo ut the minutest thin P. XLII. Pray an Mer me Dirly, once sor ali, whether it be your opinionthat what ever is litile and inconsiderable enough to be rejected without inconvenience in practice, the same may in like manner be sesely rejected and overtooked in theory and demonstration. I f you say no, itwill then sol low, that ali you have been saying here and else here, a boutyards and luches and decimal fractions, setting sortii and insisting on theextreme smalinest of the rejectaneous quantity, is qui te fore ign to the argument, and only a plece of skill to impose upon your reader. Is yousay res, it sollows that you then give up at once ali the orders of fluxions and infinitesimal differences ; and se most imprudently turn ali your sal