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he can or he cannot conceive it. This is the course I have talien and
mali talie, however you and your brethren may declaim against it, and place it in the most invidious light. XIX. It is usual with you to admonim me to look over a second time, to consuli, examine, weigh the words of Sir Isaac. In answer to whicli Iwill venture to say, that I have talien as much pains as I sincerely bclieve)any man living, to understand that great author and to mahe sense of his principies. No industry nor caution nor attention, I asture you, have been wanting on my pari. So that, is I do not understand him, it is not my fauit but my missortune. Upon other subjects you are pleasedio compliment me with depth of thought and uncommon abilities P. sand 8 ). But I Deely own, I have no pretence to those things. Theon ly advantage I preten d to, is that I have always thought and judgediar myself And, as I ne ver had a master in mathematics, so I fairly fol-lowed the dictates of my own mind in examining, and censuring the authors I read upon that subject, with the same Deedom that I used uponany other; taking nothing upon trusi, and belleving that no writer was infalli ble. And a man os moderate paris, Who talies this painful cour in studying the principies of any science, may be supposed to walli more rely than those of greater abilities, who set out with more speed and
XX. What I insist on is, that the idea of a fluxion simply considerodis not at ali improved or amended by any progresis, though ever se great, in the analysis: neither are the demonstrations of the generat rules of
is, hecause in operating or calculating, men do not return to contemplate
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etto A Defence of Free-Ninling, Ge. the original principies of the method, whicli they constantly presuppose,
but are emplo ed in Working, by notes and symbols, denoting the fluxi-ons supposed to have been at first explained, and according to rules sup- posed to have been at first demonstrated. I his I say to encourage those, ho are not far gone in these studies, to u se intrepidly their own judg-ment, Without a blind or a mean deserence to the hest os mathematici- an Who are no more qualified than they are, to judge of the simple apprehension, or the evidence of what is deli vered in the first clements of the method ; men by further and frequent use or exercise hecomingonly more accustomed to the symbols and rules, whicli doth not malle either the foreming notions more clear, or the Dregoing Proosi more perfeci. Every reader os common sense, that will but use his faculties, knows as weli as the most pro und Analyst what idea he frames or canseame of Velocity without motion, or os motion Without extension, of magnitude whicli is nei ther finite nor infinite, or of a quantity having nomagnitude whicli is yet divisibie, of a figure where there is no space, of proportion tWeen nothings, or os a real product stom nothing mul tiplied by something. He need not be far gone in geometry to know, that ob scure principies are not to be admitted in demonstration: that is a mandestroys his own hypothesis, he at the same time destroys what was bulli upon it: that error in the premises, not rectitad, must produce
XXI. In my opinion the greatest men have thein prejudices. Mensearn the elements of science Dom others: and every learner halli a deference more or lese to authority, especialty the young learners, sew of that hind caring to dweli long upon principies, but inclining rathen totalie them Upon trust: and things early admitted by repetition hecomesamiliar: and this. familiari ty at longili passoth sor evidence. to meitiseems, there are certain potnis tacitly admitted by mathematicians, whicli are ncither evident nor true. And sucii potnis or principies ever
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A Defenoe of Free-Ninling, Uc. mixing with their rea nings do lead them into paradoxes and perplexities. I f the great author of the fluxionary method was early imbued with lach notions, it would only shew he was a man. And is by ver- tue of me latent error in his principies a man be drawn into fallacious rea nings, it is nothing strange that he mouid talie them sor true : and ne verthelesi, ic When urged by Perplexities and uncouth consequences, and dri ven to aris and shisis, he mouid entertain seme doubi thereos, itis no more than one may naturalty suppose, might besali a great genius grappling with an insuperable difficulty : whicli is the light in whieli Ihave placed Sir Isaac Newton '. Hereupon you are pleased to rem arti, that I represent the great authot not only as a weali but an ili man, as a deceiver and an impostor. The reader Will judge hoW justly. XXII. As to the rest of your colourings and glosses, your reproaches and insulis and outcries, I mali past them over, only destring the readernot to take your Word, but read What I have writ ten, and he will want no other an Meta It hath been osten observed that the worst cause produceth the greatest clamour, and indeed you are so clamorous throughout your defence that the reader, although he mould be no mathematiaci an, provided he understands common sense and hath observed the ways of men, will be api to suspeet that you a re in the wrong. It mould stem, the refore, that your brethren the Analysts are but litile obliged toyou, for this new method os declaim ing in mathematios. Whether theyare more obliged. by your rea ning I mali now examine.
expressons as the velocities of velocities, the second, third, and fourthvelocities, tae. This you set sorti, as a pious fraud and uniair representation. I an Mer, that is according to Sir Uaac Neiston a fluxion bothe veloci ty of an increment, then according to him I may cali tho fluxion
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of a fluxion the veloci ty of a veloci ty. But for the truth of the antecedent 1ee his introduction to the Quadrature of Curves, where his ownword3 are, motuum Tel incrementorum Telocitates nominando miniones. See
also the second lemma of the second book of his mathematical principies of natural pliilosophy, where he expresseth himself in the following man-ner, Celocitates incrementorum ac decrementorum quas etiam, motus, mintationes ta Iuxiones quantitatum nominare licet. And that he admits fluxions of fluxions, or second, third, seu rili fluxions, tac. see his Treati se of the adrature of Curves. I ask now, is it not plain, that is a fluxion be a velocity, then the fluxion os a fluxion may agreeably thereunto be called the veloci ty of a velocityi In like manner is by a fluxionis meant a nascent augment, will it not then folio , that the fluxion ofa fluxion or second fluxion is the nascent augment of a nascent augment i Can any thing bc plainer. Let the reader now judge who is
XXIV. I had observed, that the great author had proceeded illegiti- mately, in obtaining the fluxion or moment of the rectangle of two Bowing quantities ; and that he did not Dirly get rid of the rectangle of the moments. In ans er to this you alledge, that the error aristing Dom theomission os such rectangle allo ing it to be an error) is se sinali that itis insignificant. This you dweli upon and exempli* to no other pur- pose, but to amuse your reader and missead him Dom the question, whicli in truth is not concerning the accuracy of computing or measum ing in practice, but concerning the accuracy of the reasoning in science. That this was reatly the case, and that the smalinest of the practical error no wise concerns it, must be so pla in to any one who reads the Analysi,1hat I wonder how you could be ignorant of it. XXV. You would se in persuade your reader, that I mahe an absurdquarret against errors of no significancy in practice, and represent mathematicians
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223 A Defence of Free'Thinling. Vothematicians as proceeding blind id in their approximations, in ali whicli I cannot hel p thin king there is on your part ei ther great ignorance orgreat disingenui ty. I f you mean to defend the rea sonabienesi and use of approximations or of the method os indivisibies, I have nothing to say. But theri you must remember this is not the doctrine os fluxions : it is non e of that analysis with whicli I am concern ed. That I am far stomquarret ling at approximations in geometry is manifest frona the thirtythird and fisty third Queries in the Analyst. And that the method offluxions pretends to sonae fiat more than the method of indivisibios is plain; hecause Sir L aae disclaims this method as not geometricat '. Andthat the method os fluxioris is supposed accurate in geometrical rigour is
manifest, to whoe ver considers what the great author writes about it,
especialty in his introduction to the Quadrature os Curves, where he salth, In rebus mathematicis errores quam minimi non sunt contemnendi. Whicli expression you have seen quoted in the Analysi, and yet you stem ignorant thereos, and indeed, of the Very end and design of the great author in this his invention os fluxion S. XXVI. As ost as you talli os finite quantities inconsiderable in pracatice, Sir I ac disowns your apology. Cave, satili he, in esi eris initas. And although quantities test than sensibie may be Os no account in praciatice, yet non e of your masters, nor Will even you yourseis venture to say, they a re of no account in theory and in reasoning. The application ing rosis practice is not the potnt questioned, but the rigour and justnest of the rea ning. And it is evident that, he the subject e ver se litile, ore ver so inconsiderabie, this doth not hinder but that a person treat ingthereos may commit very great errors in logic, Whicli logical errors arein no wi se to be measu red by the sensibie or praelicat inconveniences the nce arising, whicli, perchance may bo none at all. It must be o n-
See the Scholium at the end of the sirst Section. Lib. I. Phil. Nat. Prin. Math.
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mention se much as onee made of the increment of the rectangle of such slowing quantities. Now I assirm the direct contrary. For in the very passage by you quoted in this fame page, Dom the first caseos the second lemma of the second book of Sir Isaars principies, begin- ning with Rectangulum quodvis motu perpetuo auctum, and ending with igiatur laterum incrementis totis a et b generatur rectanguli incrementum a Bxb A d E. D. In this very passage I say is express mentiora made of the increment of such rectangle. As this is matter of Dct, I reser it tothe reader's own Cyes. Os What rectangle have we here the increment tIs it not plainly of that whose fides have a and b sor their incrementa tota, that is, of A B. Let any reader judge whether it be not plain froni thewords, the sense, and the contexi, that the great author in the end of his demonstration understands his incrementum as belonging to the rectangulum quodvis at the begianing. IS Dot the fame also evident froin the very lemma itself prefixed to the demonstration t The sense whereos issas the author there explains it) that is the moments of the flowing quantities A and B are called a and b, then the momentum vel mutatio geniti rectanguli AB will be a B Y b A. Elther there re the conclusion of the demonstration is not the thing which was to be demonstrated, or therectanguli incrementum a B x b A belongs to the rectangle AB.
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ment and a moment. But it is evident to every one, who has any notion os demonstration, that the incrementum in the conclusion must bethe momentum in the lemma; and to suppost it other i se is no credit to the a uilior. It is in essedi suppossing him to be one who did not knowwhat he would demonstrate. But let us hear Sir Uano's own words: Earum quantitatum scilicet fluentium incrementa Cel decrementa momentanea sub nomine momentorum intelligo. And you observe yourself that he useth the word moment to signi' et ther an increment or decrement. Hence with an intention to pugZle me you propose the increment and
decrement of A B, and ask whicli of these I would cali the moment t The case you say is dissiculi. My ans er is very pla in and ea sy, to wit,
ei ther of them. You, indeed, mahe a different ans er, and Dom theauthor's saying that, by a moment he understands either the momentaneous increment or decrement of the fowing quantities, you would have us conclude, by a very wonderisi inference, that his moment isnei ther the increment nor decrement thereos Would it not be as good an in rence, hecause a number is Cither Odd or even, to conclude it isti ei theri Can any one malie sense of thisi Or can even you rself hope
-Belleve me there is no remody, you must acquiesce. But my answeris that Ι will nei ther belleue you nor acquiesce; there is a plain remedyin common sense ; and that to prevent surprise I desii re the reader alwaysto Leep the controverted potnt in vie , to eXamine yOur reasons, and
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XXIX. A page or two after, you very candidly represent your case tobe that os an asi belween two botiles of hay : it is your own expression. The cause of your perplexi ty is that you kno not, Whether the veloci tyof A B increasing or of A B decrea sing is to be esteemed the fluxion, or proportional to the moment of the rectangle. My opinion, agreeablyto what hath been premi sed, is that ei ther may be de emed the fluxion.
the other of these, but is the veloci ty whicli the flowing rectangle hath, not while it is greater or lese than A B, but at that very instant of time that it is A B. For my pari, in the rectangle AB considered simply in iiself, Without ei ther increasing or diminis hing I can concei veno veloci ty at all. And is the reader is of my mind, he will not tahe
ei ther your rd, or even the word of a gliost how venerable Qever, forveloci ty without motion. You proceed and teli us that, in like manner, the moment of the rectangle is neither iis increment or decrement. Thisyou would have us belle ve on the aut hori ty of his gliost, in direct opposition to what Sir Isaar himself asserted when alive. Incrementa saithhe) vel decrementa momentanea sub nomine momentorum intelligo: ita us incrementa pro momentis addititiis seu affirmativis, ac decrementa pro subductitiis seti negativis habeantur '. I will not in your style bid the reader be-lieve me. but belle ve his eyes. XXX. To me it verily seems, that you have underia hen the defence of what you do not understand. To mend the matter, you say, you do not consider A B as lying at ei ther extremi ty of the moment, butv as extended to the middie of it; as having acquired the one half of
Princi p. Phil. Nat. Lib. II. Lem. II.
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the moment, and as heing about to acquire the other; or, as having tost one half of it, and being about to lose the other. No , in thena me of truth, I intreat you to teli what this moment is, to the middie hereos the rectangle is extended i This moment, I say, whicli is acquired, whicli is tost, whicli is cui in two, or distinguis hed into halia tIs it a finite quantity, or an infinitesimal, or a mere limit, or nothing at alit Tahe it in what sense you will, I cannot malle your defence either consistent or intelligibie. For is you talie it in cither of the two formersenses, you contradidi Sir Isaac Neiston. And is you talae it in ei ther of the lalter, you contradiet common sense; it heing plain, that what hathno magnitude, or is no quantity, cannot be divided. And here I mustini reat the reader to preserve his fuit freedom os mirad intire, and notweahly suffer his judgment to be overborn by your imagination and your Prejudices, by great names and authorities, by ghosts and visions, andabove ali by that extreme satisfaction and complacency with which youulter your strange concelis; is Words without a mea ning may be called . Aster having gi ven this unintelligibie account, you ask with your accustomed air, ' What say you, siri Is this a just and legitimate rea n for Sir Isaac s proceeding as he did i I thin k you must achnowledge it to be Q.V But alas i l acknowledge no suci, thing. I find no sense or reason in What you say. Let the reader find it is he can. XXXI. In the nexi place P. SO) you charge me with want of caution. Inasmuch say you as that quantity whicli Sir I ac Newtonse through his whole lemma, and ali the severat cases of it, constantly
calis a moment, without confining it to be ei ther an increment or decre-μ ment, is by you inconsiderately and arbitrarily, and without any sha- dow of rea n given, supposed and determined to he an increment. To whicli charge I rei ly that it is as untriae as it is peremptory. Forthat, in the foregoing citation Dom the first case of Sir Isaac's lemma, heexpressy determines it to be an increment. And as this particular in-
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228 A Defende V Free-r inling, Uc. stance or passage was that whicli I objected to, it was rea nable and proper for me to consider the moment in that fame light. But talie itincrement or decrement as you wili, the objections stili lie and the dissi- eulties are equalty insuperabie. You then proceed to cxtol the great author of the fluxionary method, and to beslow me Brinqueries uponthose who unad visedly dare to differ from him. To ait which Ι mall
about this affair, you observe that the moment of the rectangle dete
mined by Sir Isaac M.ton, and the increment os the rectangle determined by me, are perfectly and exactly equat, supposing a and b to bedimini med ad infinitum and sor proos of this, you refer to the fi ritiem ma of the first section of the first book of Sir I ac's principies. Ian Mer, that is a and b are real quantities, then a b is somethings, and
consequently mahes a real difference : but is they are noth in g, then therectangles whereos they are coessicients become nothing lihewise: and consequently the momentum or incrementum, whether Sir Isaac's or mine, are in that case nothing at all. As for the above mentioned lemma,
considered that lemma, iis demonstration and iis consequences. For,hovi ever that way of rea ning may do in the method of exhaustons, Where quantities test than assignable are regarded as nothing; yet for afluxionist writing about momentums, to argue that quantities must becqual because they have no assignable difference ems the mosi injudi-ctous step that could be talient it is directly demoliming the very doctrineyou Would defend. For it will thence follow, that ali homogeneoUS momentUmS are eqUal, and consequently the velocities, mutations, or fluxi-cns Proportionat thereio, a re ali likewise equat. There is, there re,