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this second supposition had been made besore the common division by o, all. had vani med at once, and you must have got nothing by your supposition. Whereas by this artifice of first dividing, and then changingyour supposition, you retain I and n xv-I. But, notis it dianding allthis address to cover it, the fallacy is si ill the fame. For whether it bedone seoner or later, when once the second supposition or assumption is made, in the same instant the former assumption and ali that you got byit is destroyed, and goes out together. And this is universalty true, bethe subject what it wili, throughout ali the branches of human knowledge ; in any other of whicli, I belleve men would hardly admit such area ning as this, whicli in mathematiclis is accepted sor demonstration. XVII. It may not be amisi to observe, that the method sor finding the fluxion os a reclangle os two flowing quantities, as it is set sortii in the Treati se of Quadratures, differs fio m the above mentioned tal en Dom thesecond book of the Principies, and is in effect the fame with that u sed in the calculus di erentialis '. For the supposing a quantity infinitely diminis hed, and there re rejecting it, is in effect the rejecting an infinitesimal, and indeed it requires a mar vel lous smari nesse of discern ment, to be ableto distinguish het ween evanescent increments and infinitesimal disse ences. It may perhaps be said that the quantity being infinitely diminis hed becomes nothing, and so nothing is rejected. But according to the received principies it is evident, that no geometrical quantity cara byany division or subdivision what e ver be exhausted, or reduced to nothing. Considering the various aris and devices used by the great authoros the fluxionary method, in how many lighis he placeth his fluxions, and in What disserent ways he attempis to demonstrate the fame potnt ;one would inclined to thin k, he was himself suspici ous of the justnesis of his own dc monstrations, and that he was not enough pleased with any one notion stea lily to adhere to it. Thus much at least is plain, that
Analyse des infiniment petitS, pari. I. Prop. 2.
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he owned himself satisfied concerning certain potnis, whicli neverthelesthe would not undertat e to demonstrate to others '. Whether this satisfaction arose Bom tentative methods or inductions, which have ostenbeen admitted by mathematicians, Vor instance by Dr. Wallis in his Arithmetic os Infinites) is What I mali not pretend to determine. Butwhatever the case might have been with respect to the author, it appears that his followers have me n themselves more eager in applyingliis method, than accurate in examining his principies. XVIII. It is curious to observe, What subtilty and skill this great genius employs to struggle with an insuperable difficulty, and through what lata byrintlis he en dea uours to escape the doctrine os infinitesimais, whicli asit in trudes Mon him whether he wili or no, so it is admitted and em-braced by others without the least repugnance ; Leibnita and his sollowers in their calculus dimerentialis mahing no manner of scrupte, first tosuppose, and secondiy to reject quantities infinitely smali: with what clearnesi in the apprehension and justnest in the reasoning, any thinhinginan, who is not prejudiced in favour of those things, may easily discern. The notion or idea os an infinitesimal quantity, as it is an object simplyapprehended by the mi ad, hath been atready consideredi. I mali nowonly observe as to the method of getling rid of such quantities, that itis done Without the least ceremony. As in fluxions the potnt of firstimportance, and whicli paves the way to the rest, is to find the fluxionos a product of two indeterminate quantities, se in the calculus disserentialis whicli method is supposed to have been borrowed seom the sormerwith sorne sinali alterations the main potnt is to obtain the difference offuch produα Now the rule sor this is got by rejecting the produci orrectan e of the differences. And in generat it is supposed, that no quan tity is bigger or tesser sor the addition or subduction os iis infinitesimat:
and consequently no error can arise hom sucti rejection os infinitesimais.
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XIX. And yet it mould stem that, whatever errors are admitted in the premi ses, proportional errors ought to be apprehended in the conclusion, be they finite or infinitesimat: and there re the 'ουιριφία of geometry requires nothing mould be neglected or rejected. In answer to this youwili perhaps say, that the conclusions are accurately true, and that there- fore the principies and methods Dom whence they are derived must beso tota But this inverted way of demonstrating youx principies by your conclusions, as it would be peculiar to you gentiemen, so it is contraryto the rules of logic. The truth of the conclusion will not prove either the form or the matter of a syllogism to be true; inasmuch as the illationmight have been wrong or the premises false, and the conclusion never-theleia true, though not in virtve of such illation or of suci, premises. I say that in every other science men prove their conclusions by their principies, and not their principies by the conclusions. But is in yoursyou fhould allow yourselves this tinnaturat way of proceeding, the con sequence would be that you must take up with induction, and bid adleuto demonstration. And is you submit to this, your authori ty will nolonger lead the way in potnts of rea n and science. XX. I have no controversy about your conclusions, but only aboutyour logic and method : hois you demonstratet what objects you are conversant with, and whether you conceive them clearly t What princi
themt It must be remembered that I am not concerned about the truthos four theorems, but only about the way of coming at them; Whetherit be legitimate or illegitimate, clear or obscure, scientific or tentative.
To prevent ali possibili ty of your mistaking me, I beg leave to repeat and insist, that I consider the geometrical Analyst as a logician, i. e. so far
sortii as he reasons and argues, and his mathematical conclusons, not in them selves, but in their premi ses; not as true or false, useful or insignificant, but as derived Dom fuch principies, and by such inferences. AndZ et forasmuch
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sorasmuch as it may perhaps seem an Unaccountable paradox, that mathematicians silould deduce true propositions Dom false principies, beright in the conclusion, and yet err in the premises; I mali endeavourparticularly to explain Why this may come to past, and shew how error may bring sortii truth, though it cannot bring sortii science. XXI. In order there re to clear Dp this potni, we wili suppose for instance that a tangent is to be drawn to a parabola, and examine the progress of this assair, as it is persormed by infinitesimal differences. Let AB be a curve, the abscisse AP α x, the ordinate P B αγ, the disserence os the abscisse Pura , the disserence of the ordinate RN 6. No by suppossing the curve to be a polygon, and consequently B Vthe increment or disserence of the curve, to be a straight line coincident with the tangent, and the differentiat triangle BRN to be similar to the triangle I PB, the ubtangent P T is found a Qurth proportional to RN:
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by the nature of the curve γγ p x, supposing p to be the parameter,
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XXIII. Now I observe in the first place, that the conclusion comes out right, not hecause the rejected square of dγ was infinitely smali; buthecause this error was compensa ted by another contrary and equat error. Ι observe in the second place, that whateuer is rejected, be it ever se sinati, is it be real and consequently mahes a real error in the premises, it will produce a proportional reat error in the conclusion. Your theorems there re Cannot be accurately true, nor your Problems accuratelysolved, in virtve of premi s whicli them selves are not accurate; it beinga rule in logic that concluso sequitur partem debiliorem. There re 1 observe in the third place, that when the conclusion is evident and the premi ses obscure, or the conclusion accurate and the premises inaccurate, we may sasely pronounce that such conclusion is nei ther evident nor accurate, in virtve of those obscure inaccurate premises or principies ;hut in virtve of some other principies whicli perhaps the demonstrator himself never lineis or thought of I observe in the last place, that in case the differences are suppo sed finite quantities ever se great, the Conclusion will ne verthelesi come out the fame; inasmuch as the rejected quantities
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quantities are legitimately thrown oui, not for their smalinest, hut soranother reason, to wit, because of contrary errors, whicli destroyingeach other do upon the whole cause that nothing is reatly, though semething is apparently thrown oui. And this rea n holds equally, with respect to quantities finite as Heli as infinitesimal, great as weli as smali, a Dot or a yard long as weli as the minutest increment.
XXlV. For the fuller illustration of this piant, I mali consider it in another light, and proceeding in finite quantities to the conclusion, Ι mallonly then malle use of one infinitesimal. Supposse the straight linecuis the curve AT in the potnis R and S. Suppost L Je a tangent at thepoint R, AN the abscisie, NR and OS ordinates. Let AN be produced to O, and R P be drawn parallel to N O. Suppost A N - MNR F, NO m v, P S m z, the subsecant MN S. Let the equation
α - , Wherein is sor I and et we substitute their values, we get
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- α - . whicli is the true value of the sub tangent. And since this was
obtained by one only error, i. e. by once rejecting one only infinitesimal, it mould stem, contrary to what hath been seid, that an infinitesimal quantity or disserenoe may be neglected or thrown away, and the conclusion neverthelest be accurately true, although there was no doublem istahe or recti'ing of one error by another, as in the first case. But is this potnt be throughly considered, we smali find there is even here adolabie misi ake, and that one compensates or rectifies the other. For in the fir9 place, it was supposed, that when No is infinitely dimini medor becomes an infinites mal, then the subsequent NM becomes equat tothe subtangent N L. But this is a plain mistahe ; sor it is evident, thatas a secant cannot be a tangent, se a subsecant cannot be a subtangent. Be the disserenoe ever small, yet stili there is a difference. And is O be infinitely smali, there will even then he an infinitely smali disser- ence bet ween N M and N L. Theresere N M or S was too litile soryour supposition, when you supposed it equat to N L) and this error was compensated by a second error in throwing out v, which last error made s bigger than iis triae value, and in lieu thereos gave the value of the subtangent. This is the true state of the case, howe ver it may bedis uised. And to this in reali ty it amounts, and is at bottom these me thing, is we mouid pretend to find the subtangent by havingsirsi Mund, froin the equation of the curve and similar triangles, a generat expression for ali subsecanis, and then reducing the subtangent under this generat rute, by considering it as the subsecant when v vanis hes
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XXV. Upon the whole I observe, First, that v can ne ver be nothing so long as there is a secant. Secondiy, that the fame line cannot heboth tangent and secant. Thirdly, that when v or NO ' vanisheth, P Sand S R do also vanish, and with them the proportionality of the similartriangles. Consequently the whole eX pression, which was obtained by means thereos and grounded thereupon, vanisheth when O vanisheth. Fourthly, that the method for finding secanis or the expression os secanis, be it e ver so generat, cannot in common sense extend any further thanto ali secants what ever: and, as it necessari ly supposed similar trian- gles, it cannot he supposed to talie place where there are not similar triangles. Fifthly, that the subsecant will always be lesis than the subtangent, and can never coincide With it; whicli coincidence to suppose would be absurd; sor it would be supposing the fame line at the fame time to cui and not to cui another gi ven line, whicli is a manifest contradiction, such as subveris the hypothesis and gives a demonstration ofit; salmood. Sixthly, is this be not admitted, Ι demand a rea n whyany other apagogical demonstration, or demonstration ad absurdum moti ldhe admitted in geometry rather than this; or that seme real differendo beassigned belween this and others as Rch. Seventhly, I observe that it is sophistical to suppost No or RP, PS, and SR to be finite real lines inorder to forna the triangle E P S, in order to obtain proportions by similar trian es, and after ards to suppose there are no such lines, nor conta sequently similar trian es, and ne vertheless to retain the consequence of the sirsi supposition, after such supposition hath been destroyed by a contrary one. Eighthly, That although, in the present case, by inconsistent suppositions truth may be obtained, yet such truth is not de monstrated: that sucii method is not conformabie to the rules of logicand right reason; that, however usefui it may be, it must be considere lonly as a presumption, as a linach, an ari rather an artifice, but not ascientific demonstration.
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XXVI. The doctrine premised may be sarther illustra ted by the follo ing simple and easy case, wherein I mali proceed by evanescent increments. Suppose A B BC m γ, B D m o, and that xx is equat tothe area ABC: it is proposed to find the ordinate γ or BC. When x
been at ready remarhed ', that it is not legitimate or logical to suppose oto vanish, i. e. to he nothing, i. e. that there is no increment, uniess wer edi at the fame time With the increment itself e very consequence of such increment, i. e. What ever could not be obta ined but by supposingsuch increment. It must ne vertheleia be achnowledged, that the problem is rightly solved, and the conclusion true, to which we are led by this method. It will there re bo asked, how comes it to pasi that thethrowing out o is attended with no error in the conclusion Z I ans er, the