A DEFENCE OF FREE-THINKING IN MATHEMATICS, &c.

I. WHEN I read your Defence of the Britiſh Mathematicians, I could not, Sir, but admire your Courage in aſſerting with ſuch undoubting Aſſurance things ſo eaſily diſproved. This to me ſeemed unaccountable, till I reflected on what you ſay (p. 32.) when upon my having appealed to every thinking Reader, whe⯑ther it be poſſible to frame any clear Con⯑ception of Fluxions, you expreſs yourſelf in the following manner, "Pray, Sir, who are thoſe thinking Readers you ap⯑peal [4] to? Are they Geometricians, or Perſons wholly ignorant of Geometry? If the former, I leave it to them: If the latter, I ask how well are they qualified to judge of the Method of Fluxions"? It muſt be acknowledged you ſeem by this Dilemma ſecure in the favour of one Part of your Readers, and the ignorance of the other. I am nevertheleſs perſuaded there are fair and candid Men among the Ma⯑thematicians. And for thoſe who are not Mathematicians, I ſhall endeavour ſo to unveil this Myſtery, and put the Contro⯑verſy between us in ſuch a Light, as that every Reader of ordinary Senſe and Reflection may be a competent Judge thereof.

II. "YOU expreſs an extreme Surprize and Concern, that I ſhould take ſo much Pains to depreciate one of the no⯑bleſt Sciences, to diſparage and traduce a Set of learned Men, whoſe Labours ſo greatly conduce to the Honour of this Iſland, (p. 5.) to leſſen the Repu⯑tation and Authority of Sir Iſaac New⯑ton and his Followers, by ſhewing that they are not ſuch Maſters of Reaſon as they are generally preſumed to be; and to depreciate the Science they profeſs, by demonſtrating to the World, that it [5] is not of the Clearneſs and Certainty as is commonly imagined. All which, you inſiſt, appears very ſtrange to you and the reſt of that famous Univerſity, who plainly ſee of how great Uſe Mathema⯑tical Learning is to Mankind." Hence you take occaſion to declaim on the Uſefulneſs of Mathematics in the ſeveral Branches, and then to redouble your Sur⯑prize and Amazement (p. 19. and 20.). To all which Declamation I reply, that it is quite beſide the Purpoſe. For I allow, and always have allowed, its full claim of Merit to whatever is uſeful and true in the Mathematics: But that which is not ſo, the leſs it employs Men's time and thoughts, the better. And after all you have ſaid or can ſay, I believe the unpreju⯑diced Reader will think with me, that things obſcure are not therefore ſacred; and that it is no more a Crime to canvaſs and de⯑tect unſound Principles or falſe Reaſonings in Mathematics, than in any other Part of Learning.

III. YOU are, it ſeems, much at a loſs to underſtand the Uſefulneſs or Tendency or Prudence of my Attempt. I thought I had ſufficiently explained this in the Ana⯑lyſt. But for your further Satisfaction ſhall here tell you, it is very well known, that [6] ſeveral Perſons who deride Faith and Myſ⯑teries in Religion, admit the Doctrine of Fluxions for true and certain. Now if it be ſhewn that Fluxions are really moſt incom⯑prehenſible Myſteries, and that thoſe, who believe them to be clear and ſcientific, do entertain an implicite Faith in the Author of that Method; will not this furniſh a fair Argumentum ad Hominem againſt Men, who reject that very thing in Religion which they admit in human Learning? And is it not a proper Way to abate the Pride, and diſcredit the Pretenſions of thoſe, who inſiſt upon clear Ideas in Points of Faith, if it be ſhewn that they do without them even in Science?

IV. AS to my timeing this Charge; why now and not before, ſince I had publiſhed Hints thereof many Years ago? Surely I am obliged to give no Account of this: If what hath been ſaid in the Analyſt be not ſufficient; ſuppoſe that I had not Leiſure, or that I did not think it expedient, or that I had no Mind to it. When a Man thinks fit to publiſh any Thing, either in Mathe⯑matics, or in any other Part of Learning; what avails it, or indeed what Right hath any one to ask, why at this or that Time; in this or that Manner; upon this or that Motive? Let the Reader judge, if it ſuffice [7] not, that what I publiſh is true, and that I have a Right to publiſh ſuch Truths, when and how I pleaſe, in a free Country.

V. I DO not ſay, that Mathematicians, as ſuch, are Infidels; or that Geometry is a Friend to Infidelity; which you untruly inſinuate, as you do many other Things; whence you raiſe Topics for invective: But I ſay there are certain Mathematicians, who are known to be ſo; and that there are o⯑thers, who are not Mathematicians, who are influenced by a Regard for their Au⯑thority. Some, perhaps, who live in the Univerſity, may not be appriſed of this; but the intelligent and obſerving Reader, who lives in the World, and is acquainted with the Humour of the Times, and the Characters of Men, is well aware, there are too many that deride Myſteries, and yet admire Fluxions; who yield that Faith to a mere Mortal, which they deny to Jeſus Chriſt, whoſe Religion they make it their Study and Buſineſs to diſcredit. The owning this is not to own, that Men who reaſon well, are Enemies to Religion, as you would repreſent it: On the contrary, I en⯑deavour to ſhew, that ſuch Men are defec⯑tive in Point of Reaſon and Judgment, and that they do the very Thing they would ſeem to deſpiſe.

[8] VI. THERE are, I make no doubt, among the Mathematicians many ſincere Believers in Jeſus Chriſt; I know ſeveral ſuch my ſelf; but I addreſſed my Analyſt to an Infidel; and on very good Grounds, I ſuppoſed that beſides him, there were other Deriders of Faith, who had never⯑theleſs a profound Veneration for Fluxions; and I was willing to ſet forth the Incon⯑ſiſtence of ſuch Men. If there be no ſuch Thing as Infidels, who pretend to Knowledge in the modern Analyſis, I own my ſelf miſin⯑formed, and ſhall gladly be found in a Miſ⯑take; but even in that Caſe, my Remarks upon Fluxions are not the leſs true; nor will it follow, that I have no Right to exa⯑mine them on the Foot of humane Science, even though Religion were quite uncon⯑cerned, and though I had no End to ſerve but Truth. But you are very angry (P. 13 and 14.) that I ſhould enter the Liſts with reaſoning Infidels, and attack them upon their Pretenſions to Science: And hence you take Occaſion to ſhew your Spleen a⯑gainſt the Clergy. I will not take upon me to ſay, that I know you to be a Minute Philoſopher your ſelf: But I know, the Minute Philoſophers make juſt ſuch Com⯑pliments as you do to our Church, and are juſt as angry, as you can be, at any who un⯑dertake to defend Religion by Reaſon. If [9] we reſolve all into Faith, they laugh at us and our Faith: And if we attempt to Rea⯑ſon, they are angry at us: They pretend we go out of our Province, and they re⯑commend to us a blind implicite Faith. Such is the Inconſiſtence of our Adverſaries. But it is to be hoped, there will never be wanting Men to deal with them at their own Weapons; and to ſhew, they are by no Means thoſe Maſters of Reaſon, which they would fain paſs for.

VII. I DO not ſay, as you would repre⯑ſent me, that we have no better Reaſon for our Religion, than you have for Fluxions: But I ſay, that an Infidel, who believes the Doctrine of Fluxions, acts a very incon⯑ſiſtent Part, in pretending to reject the Chriſtian Religion, becauſe he cannot be⯑lieve what he doth not comprehend; or be⯑cauſe he cannot aſſent without Evidence; or becauſe he cannot ſubmit his Faith to Authority. Whether there are ſuch Infidels, I ſubmit to the Judgment of the Reader. For my own Part I make no Doubt of it, having ſeen ſome ſhrewd Signs thereof my ſelf, and having been very credibly informed thereof by others. Nor doth this Charge ſeem the leſs credible, for your being ſo ſen⯑ſibly touched, and denying it with ſo much Paſſion. You, indeed, do not ſtick to affirm, [10] that the perſons, who informed me are a pack of baſe, profligate, and impudent liars, (P. 27.) How far the Reader will think fit to adopt your paſſions, I cannot ſay; but I can truly ſay, the late celebrated Mr Ad⯑diſon is one of the perſons, whom you are pleaſed to characterize in thoſe modeſt and mannerly terms. He aſſured me that the In⯑fidelity of a certain noted Mathematician, ſtill living, was one principal reaſon aſſign⯑ed by a witty man of thoſe times for his being an Infidel. Not, that I imagine Geo⯑metry diſpoſeth Men to Infidelity; but that from other cauſes, ſuch as Preſumption, Ig⯑norance, or Vanity, like other Men, Geo⯑metricians alſo become Infidels, and that the ſuppoſed light and evidence of their Science gains credit to their Infidelity.

VIII. "YOU reproach me with Ca⯑lumny, detraction and artifice (P. 15.) You recommend ſuch means as are in⯑nocent and juſt, rather than the crimi⯑nal method of leſſening or detracting from my opponents (ibid.) You accuſe me of the Odium Theologicum, the in⯑temperate Zeal of Divines, that I do ſtare ſuper vias antiquas," (P. 13.) with much more to the ſame effect. For all which charge I depend on the reader's candour, that he will not take your word, [11] but read and judge for himſelf. In which caſe he will be able to diſcern (though he ſhould be no Mathematician) how paſſio⯑nate and unjuſt your reproaches are, and how poſſible it is for a Man to cry out againſt Calumny and practiſe it in the ſame breath. Conſidering how impatient all Mankind are when their prejudices are looked into, I do not wonder to ſee you rail and rage at the rate you do. But if your own Imagination be ſtrongly ſhocked and moved, you cannot therefore conclude, that a ſincere endeavour to free a ſcience, ſo uſeful and ornamental to Humane Life, from thoſe ſubtilties, obſcurities, and para⯑doxes, which render it inacceſſible to moſt Men, will be thought a criminal under⯑taking by ſuch as are in their right Mind. Much leſs can you hope that an illuſtrious ſeminary of Learned Men, which hath produced ſo many free-ſpirited inquirers af⯑ter Truth, will at once enter into your paſ⯑ſions, and degenerate into a neſt of Bigots.

IX. I OBSERVE upon the Incon⯑ſiſtency of certain Infidel Analyſts. I re⯑mark ſome defects in the principles of the modern Analyſis. I take the liberty de⯑cently to diſſent from Sir Iſaac Newton. I propoſe ſome helps to abridge the trouble of Mathematical Studies, and render them [12] more uſeful. What is there in all this, that ſhould make you declaim on the uſe⯑fulneſs of practical Mathematics? that ſhould move you to cry out Spain, Inqui⯑ſition, Odium Theologicum? By what figure of Speech, do you extend, what is ſaid of the modern Analyſis, to Mathematics in general, or what is ſaid of Mathematical Infidels to all Mathematicians, or the con⯑futing an errour in Science to burning or hanging the Authors? But it is nothing new or ſtrange, that Men ſhould chooſe to indulge their paſſions, rather than quit their opinions how abſurd ſoever. Hence the frightful viſions and tragical uproars of Bigotted Men, be the Subject of their Bi⯑gotry what it will. A very remarkable in⯑ſtance of this you give (P. 27.) where, upon my having ſaid that a deference to certain Mathematical Infidels, as I was credibly informed, had been one motive to Infidelity, you ask with no ſmall emotion, "For God's ſake are we in England or in Spain? Is this the language of a Fami⯑liar who is whiſpering an Inquiſitor, &c." And, the page before, you exclaim in the following Words; "Let us burn or hang up all the Mathematicians in Great Britain, or halloo the mob upon them to tear them to pieces every Mother's Son of them, Tros Rutuluſve fuat, Laymen [13] or Clergymen, &c. Let us dig up the bodies of Dr. Barrow and Sir Iſaac Newton, and burn them under the Gal⯑lows."

X. THE Reader need not be a Ma⯑thematician, to ſee how vain all this Tra⯑gedy of yours is. And if he be as tho⯑roughly ſatisfied as I am, that the cauſe of Fluxions cannot be defended by rea⯑ſon, he will be as little ſurpriſed as I am, to ſee you betake your ſelf to the arts of all bigotted men, raiſing terror, and cal⯑ling in the paſſions to your aſſiſtance. Whether thoſe Rhetorical flouriſhes about the Inquiſition and the Gallows are not quite ridiculous, I leave to be determined by the Reader. Who will alſo judge (though he ſhould not be skilled in Geo⯑metry) whether I have given the leaſt grounds for this and a World of ſuch like declamation? and whether I have not conſtantly treated thoſe celebrated Wri⯑ters, with all proper reſpect, though I take the liberty in certain points to differ from them?

XI. AS I heartily abhor an Inquiſition in Faith, ſo I think you have no right to erect one in Science. At the time of wri⯑ting your defence, you ſeem to have been [14] overcome with Paſſion: But now you may be ſuppoſed cool, I deſire you to re⯑flect whether it be not wrote in the true ſpirit of an Inquiſitor? Whether this be⯑comes a Perſon ſo exceeding delicate him⯑ſelf upon that Point? And whether your Brethren the Analyſts will think them⯑ſelves honoured or obliged by you, for having defended their Doctrine, in the ſame manner as any declaiming Bigot would defend Tranſubſtantiation? The ſame falſe colours, the ſame intemperate Sallies, and the ſame Indignation againſt common Senſe!

XII. IN a matter of mere Science, where authority hath nothing to do, you conſtantly endeavour to overbear me with authorities, and load me with envy. If I ſee a Sophiſm in the writings of a great Author, and, in compliment to his un⯑derſtanding, ſuſpect he could hardly be quite ſatisfy'd with his own demonſtra⯑tion: This ſets you on declaiming for ſeveral pages. It is pompouſly ſet forth, as a criminal method of detracting from great men, as a concerted project to leſſen their reputation, as making them paſs for impoſtors. If I publiſh my free thoughts, which I have as much right to publiſh [...] any other man, it is imputed to raſh⯑neſs [15] and vanity and the love of oppoſition. Though perhaps my late publication, of what had been hinted twenty five years ago, may acquit me of this charge in the eyes of an impartial Reader. But when I conſider the perplexities that beſet a man, who undertakes to defend the doctrine of Fluxions, I can eaſily forgive your anger.

XIII. TWO ſorts of learned men there are; one, who candidly ſeek Truth by rational means. Theſe are never averſe to have their principles looked into, and examined by the teſt of Reaſon. Ano⯑ther ſort there is, who learn by route a ſet of principles and a way of thinking which happen to be in vogue. Theſe betray themſelves by their anger and ſur⯑priſe, whenever their principles are freely canvaſſed. But you muſt not expect, that your Reader will make himſelf a party to your paſſions or your prejudices. I freely own that Sir Iſaac Newton hath ſhew'd himſelf an extraordinary Mathematician, a profound Naturaliſt, a Perſon of the greateſt Abilities and Erudition. Thus far I can readily go, but I cannot go the lengths that you do. I ſhall never ſay of him as you do, Veſtigia pronus adoro, (p. 70.) This ſame adoration that you pay to him, I will pay only to Truth.

[16] XIV. YOU may, indeed, your ſelf be an Idolater of whom you pleaſe: But then you have no right to inſult and ex⯑claim at other men, becauſe they do not adore your Idol. Great as Sir Iſaac New⯑ton was, I think he hath, on more occa⯑ſions than one, ſhew'd himſelf not to be infallible. Particularly, his demonſtration of the Doctrine of Fluxions I take to be defective, and I cannot help thinking that he was not quite pleaſed with it himſelf. And yet this doth not hinder but the me⯑thod may be uſeful, conſidered as an art of Invention. You, who are a Mathema⯑tician, muſt acknowledge, there have been divers ſuch methods admitted in Mathe⯑matics, which are not demonſtrative. Such, for inſtance, are the Inductions of Doctor Wallis in his Arithmetic of Infinites, and ſuch, what Harriot and, after him, Deſ⯑cartes have wrote concerning the roots of affected Aequations. It will not, neverthe⯑leſs, thence follow that thoſe methods are uſeleſs; but only, that they are not to be allowed of as Premiſſes in a ſtrict Demonſtration.

XV. NO great Name upon earth ſhall ever make me accept things obſcure for clear, or Sophiſms for Demonſtrations. Nor may you ever hope to deter me from freely ſpeaking what I freely think, by [17] thoſe arguments ab invidia which at every turn you employ againſt me. You re⯑preſent your ſelf (P. 52.) as a man, whoſe higheſt Ambition is in the loweſt degree to imitate Sir Iſaac Newton. It might, per⯑haps, have ſuited better with your appel⯑lation of Philalethes, and been altogether as laudable, if your higheſt ambition had been to diſcover Truth. Very conſiſtently with the character you give of your ſelf, you ſpeak of it as a ſort of crime (P. 70.) to think it poſſible, you ſhould ever ſee further, or go beyond Sir Iſaac Newton. And I am perſuaded you ſpeak the Senti⯑ments of many more beſides your ſelf. But there are others who are not afraid to ſift the Principles of human Science, who think it no honour to imitate the greateſt man in his Defects, who even think it no crime to deſire to know, not only beyond Sir Iſaac Newton, but beyond all Mankind. And whoever thinks otherwiſe, I appeal to the Reader, whether he can properly be called a Philoſopher.

XVI. BECAUSE I am not guilty of your mean Idolatry, you inveigh againſt me as a perſon conceited of my own Abi⯑lities; not conſidering that a perſon of leſs Abilities may know more on a certain point than one of greater; not conſider⯑ing that a purblind eye, in a cloſe and [18] narrow view, may diſcern more of a thing, than a much better eye in a more exten⯑ſive proſpect; not conſidering that this is to fix a ne plus ultra, to put a ſtop to all future inquiries; Laſtly, not conſidering that this is in fact, ſo much as in you lies, converting the Republick of Letters into an abſolute Monarchy, that it is even in⯑troducing a kind of Philoſophic Popery among a free People.

XVII. I HAVE ſaid (and I venture ſtill to ſay) that a Fluxion is incomprehenſible: That ſecond, third, and fourth Fluxions are yet more incomprehenſible: That it is not poſſible to conceive a ſimple Infiniteſimal, that it is yet leſs poſſible to conceive an In⯑finiteſimal of an Infiniteſimal, and ſo on⯑ward^*. What have you to ſay in anſwer to this? Do you attempt to clear up the noti⯑on of a Fluxion or a Difference? Nothing like it; "you only aſſure me (upon your bare word) from your own experience, and that of ſeveral others whom you could name, that the Doctrine of Fluxions may be clearly conceived and diſtinctly comprehended; and that if I am puz⯑zled about it and do not underſtand it, yet others do". But can you think, Sir, I ſhall take your word when I refuſe to take your Maſter's?

[19] XVIII. UPON this point every Rea⯑der of common ſenſe may judge as well as the moſt profound Mathematician. The ſimple apprehenſion of a thing defined is not made more perfect by any ſubſequent progreſs in Mathematics. What any man evidently knows, he knows as well as you or Sir Iſaac Newton. And every one can know whether the object of this method be (as you would have us think) clearly conceivable. To judge of this, no depth of Science is requiſite, but only a bare at⯑tention to what paſſes in his own mind. And the ſame is to be underſtood of all definitions in all Sciences whatſoever. In none of which can it be ſuppoſed, that a man of Senſe and Spirit will take any definition or principle upon truſt, with⯑out ſifting it to the bottom, and trying how far he can or he cannot conceive it. This is the courſe I have taken and ſhall take, however you and your Brethren may declaim againſt it, and place it in the moſt invidious Light.

XIX. IT is uſual with you to admo⯑niſh me to look over a ſecond time, to conſult, examine, weigh the words of Sir Iſaac. In anſwer to which I will ven⯑ture to ſay, that I have taken as much pains as (I ſincerely believe) any man [20] living, to underſtand that great Author, and to make ſenſe of his principles. No induſtry nor caution nor attention, I aſſure you, have been wanting on my part. So that, if I do not underſtand him, it is not my fault but my misfortune. Upon other ſubjects you are pleaſed to compliment me with depth of thought and uncommon abilities, (P. 5. and 84.) But I freely own, I have no pretence to thoſe things. The only advantage I pretend to, is that I have always thought and judged for my ſelf. And, as I never had a maſter in Mathe⯑matics, ſo I fairly followed the dictates of my own mind in examining, and cenſu⯑ring the authors I read upon that ſubject, with the ſame freedom that I uſed upon any other; taking nothing upon truſt, and believing that no writer was infallible. And a man of moderate parts, who takes this painful courſe in ſtudying the prin⯑ciples of any Science, may be ſuppoſed to walk more ſurely than thoſe of greater abilities, who ſet out with more ſpeed and leſs care.

XX. WHAT I inſiſt on is, that the idea of a Fluxion, ſimply conſidered, is not at all improved or amended by any progreſs, though ever ſo great, in the Ana⯑lyſis: neither are the demonſtrations of the [21] general rules of that method at all cleared up by applying them. The reaſon of which is, becauſe in operating or calculat⯑ing, men do not return to contemplate the original principles of the method, which they conſtantly preſuppoſe, but are em⯑ployed in working, by notes and ſymbols, denoting the Fluxions ſuppoſed to have been at firſt explained, and according to rules ſuppoſed to have been at firſt de⯑monſtrated. This I ſay to encourage thoſe, who are not far gone in theſe Studies, to uſe intrepidly their own judg⯑ment, without a blind or a mean de⯑ference to the beſt of Mathematicians, who are no more qualify'd than they are, to judge of the ſimple apprehenſion, or the evidence of what is delivered in the firſt elements of the method; men by further and frequent uſe or exerciſe becoming only more accuſtomed to the ſymbols and rules, which doth not make either the foregoing notions more clear, or the foregoing proofs more perfect. Every Reader of common ſenſe, that will but uſe his faculties, knows as well as the moſt profound Analyſt what idea he frames or can frame of Velocity without motion, or of motion without ex⯑tenſion, of magnitude which is neither fi⯑nite nor infinite, or of a quantity having no magnitude which is yet diviſible, of a figure where there is no ſpace, of proporti⯑on [22] between nothings, or of a real product from nothing multiplied by ſomething. He need not be far gone in Geometry to know, that obſcure principles are not to be ad⯑mitted in Demonſtration: That if a man deſtroys his own Hypotheſis, he at the ſame time deſtroys what was built upon it: That error in the premiſes, not rectified, muſt produce error in the concluſion.

XXI. IN my opinion the greateſt men have their Prejudices. Men learn the ele⯑ments of Science from others: And every learner hath a deference more or leſs to authority, eſpecially the young learners, few of that kind caring to dwell long up⯑on principles, but inclining rather to take them upon truſt: And things early admit⯑ted by repetition become familiar: And this familiarity at length paſſeth for Evi⯑dence. Now to me it ſeems, there are certain points tacitly admitted by Mathe⯑maticians, which are neither evident nor true. And ſuch points or principles ever mixing with their reaſoning do lead them into paradoxes and perplexities. If the great author of the fluxionary method was early imbued with ſuch notions, it would only ſhew he was a man. And if by vir⯑tue of ſome latent error in his principles a man be drawn into fallacious reaſonings, it is nothing ſtrange that he ſhould take [23] them for true: And, nevertheleſs, if; when urged by perplexities and uncouth conſe⯑quences, and driven to arts and ſhifts, he ſhould entertain ſome doubt thereof, it is no more than, one may naturally ſuppoſe, might befall a great genious grappling with an inſuperable difficulty: Which is the light in which I have placed Sir Iſaac New⯑ton ^*. Herereupon you are pleaſed to re⯑mark, that I repreſent the great author not only as a weak but an ill 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 inſults and outcries, I ſhall paſs them over, on⯑ly deſiring the Reader not to take your word, but read what I have written, and he will want no other anſwer. It hath been often obſerved that the worſt cauſe produceth the greateſt clamour, and in⯑deed you are ſo clamorous throughout your defence that the Reader, although he ſhould be no Mathematician, provided he underſtands common ſenſe and hath obſerved the ways of men, will be apt to ſuſpect you are in the wrong. It ſhould ſeem, therefore, that your Brethren the Analyſts are but little obliged to you, for [24] this new method of declaiming in Mathe⯑matics. Whether they are more obliged by your Reaſoning I ſhall now examine.

XXIII. YOU ask me (p. 32.) where I find Sir Iſaac Newton uſing ſuch expreſſions as the Velocities of Velocities, the ſecond, third, and fourth Velocities, &c. This you ſet forth as a pious fraud and unfair repre⯑ſentation. I anſwer, that if according to Sir Iſaac Newton a Fluxion be the veloci⯑ty of an increment, then according to him I may call the Fluxion of a Fluxion the Ve⯑locity of a Velocity. But for the truth of the antecedent ſee his introduction to the Qua⯑drature of Curves, where his own words are, motuum velincrementorum velocitates nominan⯑do Fluxiones. See alſo the ſecond Lemma of the ſecond Book of his mathematical prin⯑ciples of natural Philoſophy, where he ex⯑preſſeth himſelf in the following manner, velocitates incrementorum ac decrementorum, quas etiam, motus, mutationes & fluxiones quantitatum nominare licet. And that he ad⯑mits Fluxions of Fluxions, or ſecond, third, fourth Fluxions, &c. ſee his Treatiſe of the Quadrature of Curves. I ask now, Is it not plain, that if a Fluxion be a Velocity, then the Fluxion of a Fluxion may agreea⯑bly thereunto be called the Velocity of a Velocity? In like manner if by a Fluxion [25] is meant a naſcent augment, will it not then follow, that the Fluxion of a Fluxion, or ſecond Fluxion is the naſcent augment of a naſcent augment? Can any thing be plain⯑er? Let the Reader now judge who is unfair.

XXIV. I HAD obſerved, that the Great Author had proceeded illegitimately, in ob⯑taining the Fluxion or moment of the Rec⯑tangle of two flowing quantities; and that he did not fairly get rid of the Rectangle of the moments. In anſwer to this you al⯑ledge, that the error ariſing from the o⯑miſſion of ſuch rectangle (allowing it to be an error) is ſo ſmall that it is inſignificant. This you dwell upon and examplify to no other purpoſe, but to amuſe your Reader and miſlead him from the Queſtion; which in truth is not concerning the accuracy of computing or meaſuring in practice, but concerning the accuracy of the reaſoning in ſcience. That this was really the caſe, and that the ſmallneſs of the practical error no wiſe concerns it, muſt be ſo plain to any one who reads the Analyſt, that I won⯑der how you could be ignorant of it.

XXV. YOU would fain perſuade your Reader, that I make an abſurd quarrel a⯑gainſt errors of no ſignificancy in practice, and repreſent Mathematicians as proceeding [26] blindfold in their approximations; in all which I cannot help thinking there is on your part either great ignorance or great diſingenuity. If you mean to defend the reaſonableneſs and uſe of approximations, or of the method of Indiviſibles, I have no⯑thing to ſay. But then you muſt remem⯑ber this is not the Doctrine of Fluxions: It is none of that Analyſis with which I am concerned. That I am far from quar⯑relling at approximations in Geometry is manifeſt from the thirty third and fifty third Queries in the Analyſt. And that the method of Fluxions pretends to ſome⯑what more than the method of Indiviſibles is plain; becauſe Sir Iſaac diſclaims this method as not Geometrical^* And that the method of Fluxions is ſuppoſed accu⯑rate in Geometrical rigour is manifeſt, to whoever conſiders what the Great Author writes about it; eſpecially in his Intro⯑duction to the Quadrature of Curves, where he ſaith In rebus mathematicis errores quam minimi non ſunt contemnendi. Which ex⯑preſſion you have ſeen quoted in the Ana⯑lyſt, and yet you ſeem ignorant thereof, and indeed, of the very End and Deſign of the Great Author in this his invention of Fluxions.

[27] XXVI. AS oft as you talk of finite quan⯑tities inconſiderable in practice, Sir Iſaac diſowns your apology. Cave, ſaith he, in⯑tellexeris finitas. And although Quantities leſs than ſenſible may be of no account in practice, yet none of your maſters, nor will even you yourſelf venture to ſay, they are of no account in Theory and in Reaſoning. The application in groſs practice is not the point queſtioned, but the rigour and juſt⯑neſs of the reaſoning. And it is evident that, be the ſubject ever ſo little, or ever ſo inconſiderable, this doth not hinder but that a perſon treating thereof may com⯑mit very great errors in Logic, which Lo⯑gical errors are in no wiſe to be meaſured by the ſenſible or practical inconveniences thence ariſing, which, perchance, may be none at all. It muſt be owned, that after you have miſlead and amuſed your leſs qua⯑lified Reader (as you call him) you return to the real point in controverſy, and ſet your ſelf to juſtifie Sir Iſaac's method of getting rid of the abovementioned Rectan⯑gle. And here I muſt intreat the Reader to obſerve how fairly you proceed.

XXVII. FIRST then you affirm (P. 44.) "that, neither in the Demonſtration of the Rule for finding the Fluxion of the rec⯑tangle of two flowing quantities, nor in [28] any thing preceding or following it, is any mention ſo much as once made of the increment of the rectangle of ſuch flowing quantities." Now I affirm the direct contrary. For in the very paſſage by you quoted in this ſame page, from the firſt caſe of the ſecond lemma of the ſecond Book of Sir Iſaac's Principles, beginning with Rectangulum quodvis motu perpetuo auctum, and ending with igitur laterum in⯑crementis totis a et b generatur rectanguli incrementum a B x b A Q. E. D. In this very paſſage, I ſay, is expreſs mention made of the increment of ſuch rectangle. As this is matter of fact, I refer it to the Reader's own eyes. Of what rectangle have we here the Increment? Is it not plainly of that whoſe ſides have a and b for their Incre⯑menta tota, that is, of AB? Let any Reader judge whether it be not plain from the words, the ſenſe, and the context, that the Great Author in the end of his demonſtra⯑tion underſtands his incrementum as belong⯑ing to the Rectangulum quodvis at the begin⯑ning. Is not the ſame alſo evident from the very Lemma it ſelf prefixed to the Demon⯑ſtration? The ſenſe whereof is (as the Au⯑thor there explains it) that if the moments of the flowing quantities A and B are cal⯑led a and b, then the momentum vel mutatio geniti rectanguli AB will be a B x b A. [29] Either therefore the concluſion of the de⯑monſtration is not the thing which was to be demonſtrated, or the rectanguli incremen⯑tum a B x b A belongs to the rectangle AB.

XXVIII. ALL this is ſo plain that no⯑thing can be more ſo; and yet you would fain perplex this plain caſe by diſtinguiſh⯑ing between an increment and a moment. But it is evident to every one, who has any notion of Demonſtration, that the incre⯑mentum in the Concluſion muſt be the mo⯑mentum in the Lemma; and to ſuppoſe it otherwiſe is no credit to the Author. It is in effect ſuppoſing him to be one who did not know what he would demonſtrate. But let us hear Sir Iſaac's own words: Earum (quantitatum ſcilicet fluentium) incrementa vel decrementa momentanea ſub nomine mo⯑mentorum intelligo. And you obſerve your ſelf that he uſeth the word moment to ſig⯑nify either an increment or decrement. Hence, with an intention to puzzle me, you propoſe the increment and decrement of AB, and ask which of theſe I would call the moment? The caſe, you ſay, is difficult. My anſwer is very plain and eaſy, to wit, Either of them. You, indeed, make a different anſwer, and from the Author's ſaying that, by a moment he underſtands either the mo⯑mentaneous increment or decrement of the [30] flowing quantities, you would have us con⯑clude, by a very wonderful inference, that his moment is neither the increment nor decrement thereof. Would it not be as good an inference, Becauſe a number is either odd or even, to conclude it is neither? Can any one make ſenſe of this? Or can even your ſelf hope that this will go down with the Reader, how little ſoever qualified? It muſt be owned, you endeavour to obtrude this inference on him, rather by mirth and humour than by reaſoning. You are merry, I ſay, and (P. 46.) repreſent the two ma⯑thematical quantities as pleading their rights, as toſſing up croſs and pile, as diſ⯑puting amicably. You talk of their claim⯑ing preference, their agreeing, their boy⯑iſhneſs and their gravity. And after this in⯑genious digreſſion you addreſs me in the fol⯑lowing words.—Believe me there is no re⯑medy, you muſt acquieſce. But my anſwer is, that I will neither believe you nor ac⯑quieſce; there is a plain remedy in common ſenſe; and, to prevent ſurpriſe, I deſire the Reader always to keep the controverted point in view, to examine your reaſons, and be cautious how he takes your word, but moſt of all when you are poſitive or eloquent or merry.

[31] XXIX. A PAGE or two after, you very candidly repreſent your caſe to be that of an Aſs between two bottles of hay; it is your own expreſſion. The cauſe of your perplexity is, that you know not whether the velocity of AB increaſing or of AB decreaſing is to be eſteemed the Fluxion, or proportional to the moment of the rec⯑tangle. My opinion, agreeably to what hath been premiſed, is that either may be deemed the Fluxion. But you tell us "(P. 49.) that you think, the venerable ghoſt of Sir Iſaac Newton whiſpers you, The Velocity you ſeek for is neither the one nor the other of theſe, but is the ve⯑locity which the flowing rectangle hath, not while it is greater or leſs than AB, but at that very inſtant of time that it is AB." For my part, in the rectangle AB conſidered ſimply in it ſelf, without either increaſing or diminiſhing, I can con⯑ceive no velocity at all. And if the Reader is of my mind, he will not take either your word, or even the word of a Ghoſt, how venerable ſoever, for velocity without mo⯑tion. You proceed and tell us that, in like manner, the moment of the rectangle is neither its increment or decrement. This you would have us believe on the authority of his Ghoſt, in direct oppoſition to what Sir Iſaac himſelf aſſerted when alive. In⯑crementa [32] (ſaith he) vel decrementa momen⯑tanea ſub nomine momentorum intelligo: ita ut incrementa pro momentis addititiis ſeu affirmativis, ac decrementa proſubductitiis ſeu negativis habeantur ^*. I will not in your ſtyle bid the Reader believe me, but believe his eyes.

XXX. TO me it verily ſeems, that you have undertaken the defence of what you do not underſtand. To mend the matter, you ſay, "you do not conſider AB as ly⯑ing at either extremity of the moment, but as extended to the middle of it; as having acquired the one half of the mo⯑ment, and as being about to acquire the other; or, as having loſt one half of it, and being about to loſe the other." Now, in the name of Truth, I intreat you to tell what this moment is, to the middle whereof the rectangle is extended? This moment, I ſay, which is acquired, which is loſt, which is cut in two, or diſtinguiſh⯑ed into halfs? Is it a finite quantity, or an infiniteſimal, or a mere limit, or nothing at all? Take it in what ſenſe you will, I cannot make your defence either conſiſtent or intelligible. For if you take it in either of the two former ſenſes, you contradict Sir Iſaac Newton. And if you take it in [33] either of the latter, you contradict common ſenſe; it being plain, that what hath no magnitude, or is no quantity, cannot be divided. And here I muſt intreat the Rea⯑der to preſerve his full freedom of mind intire, and not weakly ſuffer his judgment to be overborn by your imagination and your prejudices, by great names and au⯑thorities, by Ghoſts and Viſions, and a⯑bove all by that extreme ſatisfaction and complacency with which you utter your ſtrange conceits; if words without a mean⯑ing may be called ſo. After having given this unintelligible account, you ask with your accuſtomed air, "What ſay you, Sir? Is this a juſt and legitimate reaſon for Sir Iſaac's proceeding as he did? I think you muſt acknowledge it to be ſo." But alas! I acknowledge no ſuch thing. I find no ſenſe or reaſon in what you ſay. Let the Reader find it if he can.

XXXI. IN the next Place (P. 50.) you charge me with want of caution. "Inaſ⯑much (ſay you) as that quantity which Sir Iſaac Newton through his whole Lemma, and all the ſeveral Caſes of it, conſtantly calls a Moment, without con⯑fining it to be either an increment or decrement, is by you inconſiderately and arbitrarily, and without any Shadow of [34] of Reaſou given, ſuppoſed and deter⯑mined to be an increment." To which Charge I reply that it is as untrue as it is peremptory. For that, in the foregoing citation from the firſt caſe of Sir Iſaac's Lemma, he expreſly determines it to be an Increment. And as this particular Inſtance or Paſſage was that which I objected to, it was reaſonable and proper for me to con⯑ſider the Moment in that ſame Light. But take it increment or decrement as you will, the Objections ſtill lie, and the Difficulties are equally inſuperable. You then pro⯑ceed to extoll the great Author of the fluxio⯑nary Method, and to beſtow ſome Bruſ⯑queries upon thoſe who unadviſedly dare to differ from him. To all which I ſhall give no anſwer.

XXXII. AFTERWARDS to re⯑move (as you ſay) all Scruple and Difficulty about this affair, you obſerve that the Mo⯑ment of the Rectangle determined by Sir Iſaac Newton, and the Increment of the Rectangle determined by me, are perfectly and exactly equal, ſuppoſing a and b to be diminiſhed ad infinitum: and for proof of this, you refer to the firſt Lemma of the firſt Section of the firſt Book of Sir Iſaac's Principles. I anſwer, that if a and b are real quantities, then a b is ſomething, and [35] conſequently makes a real difference: but if they are nothing, then the Rectangles whereof they are coefficients become nothing likewiſe: and conſequently the momentum or incrementum, whether Sir Iſaac's or mine, are in that Caſe nothing at all. As for the abovementioned Lem⯑ma, which you refer to, and which you wiſh I had conſulted ſooner, both for my own ſake and for yours; I tell you I had long ſince conſulted and conſidered it. But I very much doubt whether you have ſuf⯑ficiently conſidered that Lemma, its De⯑monſtration, and its Conſequences. For, however that way of reaſoning may do in the Method of exhauſtions, where quanti⯑ties leſs than aſſignable are regarded as nothing; yet for a Fluxioniſt writing a⯑bout momentums, to argue that quantities muſt be equal becauſe they have no aſſign⯑able difference, ſeems the moſt injudici⯑ous Step that could be taken: it is direct⯑ly demoliſhing the very Doctrine you would defend. For it will thence follow, that all homogeneous momentums are e⯑qual, and conſequently the velocities, mutations, or fluxions proportional thereto, are all likewiſe equal. There is, therefore, only one proportion of equality through⯑out, which at once overthrows the whole Syſtem you undertake to defend. Your [36] moments (I ſay) not being themſelves aſ⯑ſignable quantities, their differences cannot be aſſignable: and if this be true, by that way of reaſoning it will follow, they are all equal, upon which Suppoſition you cannot make one Step in the Method of Fluxiors. It appears from hence, how unjuſtly you blame me (P. 32.) for omitting to give any Account of that firſt Section of the firſt Book of the Principia, wherein (you ſay) the Foundation of the Method of Fluxions is geometrically demonſtrated and largely explained, and difficulties and ob⯑jections againſt it are clearly ſolved. All which is ſo far from being true, that the very firſt and fundamental Lemma of that Section is incompatible with, and ſubver⯑ſive of the doctrine of Fluxions. And, indeed, who ſees not that a Demonſtra⯑tion ad abſurdum more veterum proceeding on a Suppoſition, that every difference muſt be ſome given quantity, cannot be admitted in, or conſiſt with, a method, wherein Quantities, leſs than any given, are ſuppoſed really to exiſt, and be capable of diviſion?

XXXIII. THE next point you under⯑take to defend is that method for obtain⯑ing a rule to find the Fluxion of any Power of a flowing Quantity, which is delivered in the introduction to the Qua⯑dratures, [37] and conſidered in the Analyſt^*. And here the queſtion between us is, whether I have rightly repreſented the ſenſe of thoſe words, evaneſcant jam augmenta illa, in rendering them, let the increments vaniſh, i. e. let the increments be nothing, or let there be no increments? This you deny, but, as your manner is, inſtead of giving a reaſon you declaim. I, on the contrary affirm, the increments muſt be underſtood to be quite gone and abſolutely nothing at all. My reaſon is, becauſe with⯑out that ſuppoſition you can never bring the quantity or expreſſion [...] &c. down to [...], the very thing aimed at by ſuppoſing the evaneſcence. Say whe⯑ther this be not the truth of the caſe? Whether the former expreſſion is not to be reduced to the latter? And whether this can poſſibly be done ſo long as o is ſuppoſed a real Quantity? I cannot indeed ſay you are ſcrupulous about your affirmati⯑ons, and yet I believe that even you will not affirm this; it being moſt evident, that the product of two real quantities is ſome⯑thing real; and that nothing real can be [38] rejected either according to the [...] of Geometry, or according to Sir Iſaac's own principles; for the truth of which I ap⯑peal to all who know any thing of theſe matters. Further by evaneſcent muſt either be meant, let them (the increments) vaniſh and become nothing, in the obvious ſenſe, or elſe let them become infinitely ſmall. But that this latter is not Sir Iſaac's ſenſe is evident from his own words in the very ſame page, that is, in the laſt of the Introduction to his Quadratures, where he expreſsly ſaith volui oſtendere quod in me⯑thodo Fluxionum non opus fit figuras infinitè parvas in Geometriam introducere. Upon the whole, you ſeem to have conſidered this affair ſo very ſuperficially, as greatly to confirm me in the opinion, you are ſo an⯑gry with, to wit, that Sir Iſaac's followers are much more eager in applying his me⯑thod, than accurate in examining his prin⯑ciples. You raiſe a duſt about evaneſcent augments which may perhaps amuſe and amaze your Reader, but I am much miſ⯑taken if it ever inſtructs or enlightens him. For, to come to the point, thoſe evaneſcent augments either are real quantities, or they are not. If you ſay they are; I deſire to know, how you get rid of the rejectaneous quantity? If you ſay they are not; you in⯑deed get rid of thoſe quantities in the com⯑poſition whereof they are coefficients; but [39] then you are of the ſame opinion with me, "which opinion you are pleaſed to call (P. 58.) a moſt palpable, inexcuſable, and unpardonable blunder, although it be a Truth moſt palpably evident".

XXXIV. NOTHING, I ſay, can be plainer to any impartial Reader, than that by the Evaneſcence of augments, in the above cited paſſage, Sir Iſaac means their being actually reduced to nothing. But to put it out of all doubt, that this is the truth, and to convince even you, who ſhew ſo little diſpoſition to be convinced, I de⯑ſire you to look into his Analyſis per aequa⯑tiones infinitas (P. 20.) where, in his pre⯑paration for demonſtrating the firſt rule for the ſquaring of ſimple Curves, you will find that on a parallel occaſion, ſpeak⯑ing of an augment which is ſuppoſed to vaniſh, he interprets the word evaneſcere by eſſe nihil. Nothing can be plainer than this, which at once deſtroys your defence. And yet, plain as it is, I deſpair of making you acknowledge it; though I am ſure you feel it, and the Reader if he uſeth his eyes muſt ſee it. The words Evaneſcere five eſſe nihil do (to uſe your own expreſſion) ſtare us in the face. Lo! "This is what you call (P. 56.) ſo great, ſo unaccounta⯑ble, ſo horrid, ſo truly Boeotian a blunder" [40] that, according to you, it was not poſſible Sir Iſaac Newton could be guilty of it. For the future, I adviſe you to be more ſparing of hard words: Since, as you in⯑cautiouſly deal them about, they may chance to light on your friends as well as your adverſaries. As for my part, I ſhall not retaliate. It is ſufficient to ſay you are miſtaken, But I can eaſily pardon your miſtakes. Though, indeed, you tell me on this very occaſion, that I muſt expect no quarter from Sir Iſaac's followers. And I tell you that I neither expect nor deſire any. My aim is truth. My reaſons I have given. Confute them, if you can. But think not to overbear me either with au⯑thorities or harſh words. The latter will recoil upon your ſelves: The former in a matter of ſcience are of no weight with indifferent Readers; and as for Bigots, I am not concerned about what they ſay or think.

XXXV. IN the next place you pro⯑ceed to declaim upon the following paſ⯑ſage taken from the ſeventeenth ſection of the Analyſt. "Conſidering the various arts and devices uſed by the great au⯑thor of the fluxionary method: In how many lights he placeth his Fluxions: and in what different ways he attempts to [41] demonſtrate the ſame point: One would be inclined to think, he was himſelf ſuſ⯑picious of the juſtneſs of his own de⯑monſtrations." This paſſage you com⯑plain of as very hard uſage of Sir Iſaac Newton. You declaim copiouſly, and en⯑deavour to ſhew that placing the ſame point in various lights is of great uſe to explain it; which you illuſtrate with much Rhetoric. But the fault of that paſſage is not the hard uſage it contains: But on the contrary, that it is too modeſt, and not ſo full and expreſſive of my ſenſe, as per⯑haps it ſhould have been. Would you like it better if I ſhould ſay, the various incon⯑ſiſtent accounts, which 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 there can be no de⯑monſtration? I own frankly that I ſee no clearneſs or conſiſtence in them. You tell me indeed, in Miltonic verſe, that the fault is in my own eyes,

At the ſame time you acknowledge your ſelf obliged for thoſe various lights, which have enabled you to underſtand his Doc⯑trine. [42] But as for me who do not underſtand it, you inſult me, ſaying: "For God's ſake what is it you are offended at, who do not ſtill underſtand him"? May not I anſwer, that I am offended for this very reaſon; becauſe I cannot underſtand him or make ſenſe of what he ſays? You ſay to me, that I am all in the dark. I ac⯑knowledge it, and intreat you who ſee ſo clearly, to help me out.

XXXVI. YOU, Sir, with the bright eyes, be pleaſed to tell me, whether Sir Iſaac's momentum be a finite quantity, or an infiniteſimal, or a mere limit? If you ſay, a finite quantity: Be pleaſed to recon⯑cile this with what he ſaith in the Scho⯑lium of the ſecond Lemma of the firſt Section of the firſt book of his Principles: Cave intelligas quantitates magnitudine de⯑terminatas, ſed cogita ſemper diminuendas ſine limite. If you ſay, an infiniteſimal: reconcile this with what is ſaid in the Introduction to his Quadratures: Volui oftendere quod in methodo Flaxionum non opus fit figuras infinitè parvas in Geome⯑triam introducere. If you ſhould ſay, it is a mere limit, be pleaſed to reconcile this with what we find in the firſt caſe of the ſecond Lemma in the ſecond book of his principles: Ubi de lateribus A et B [43] deerant momentorum dimidia, &c. where the moments are ſuppoſed to be divided. I ſhould be very glad, a perſon of ſuch lu⯑minous intellect would be ſo good as to explain, whether by Fluxions we are to underſtand the naſcent or evaneſcent quan⯑tities themſelves, or their motions, or their Velocities, or ſimply their proportions: and having interpreted them in what ſenſe you will, that you would then conde⯑ſcend to explain the Doctrine of ſecond, third, and fourth Fluxions, and ſhew it to be conſiſtent with common ſenſe if you can. You ſeem to be very ſanguine when you expreſs your ſelf in the following terms. "I do aſſure you, Sir, from my own Experience, and that of many o⯑thers whom I could name, that the Doctrine may be clearly conceived and diſtinctly comprehended" (p. 31.) And it may be uncivil not to believe what you ſo ſolemnly affirm, from your own ex⯑perience. But I muſt needs own, I ſhould be better fatisfied of this, if, inſtead of entertaining us with your Rhetoric, you would vouchſafe to reconcile thoſe diffi⯑culties, and explain thoſe obſcure points abovementioned. If either you, or any one of thoſe many whom you could name, will but explain to others what you ſo clearly con⯑ceive your ſelves, I give you my word [44] that ſeveral will be obliged to you who, I may venture to ſay, underſtand thoſe matters no more than my ſelf. But, if I am not much miſtaken, you and your friends will modeſtly decline this task.

XXXVII. I HAVE long ago done what you ſo often exhort me to do, dili⯑gently read and conſidered the ſeveral ac⯑counts of this Doctrine given by the great Author in different parts of his writings: and upon the whole I could never make it out to be conſiſtent and intelligible. I was even led to ſay, "that one would be in⯑clined to think, He was himſelf ſuſpi⯑cious of the juſtneſs of his own demon⯑ſtrations: and that he was not enough pleaſed with any one Notion ſteadily to adhere to it." After which I ad⯑ded, "Thus much is plain that he own⯑ed himſelf ſatisfied concerning certain points, which nevertheleſs he could not undertake to demonſtrate to others." See the ſeventeenth ſection of the Analyſt. It is one thing when a Doctrine is placed in various lights: and another, when the principles and notions are ſhifted. When new devices are introduced and ſubſtitu⯑ted for others, a Doctrine inſtead of being illuſtrated may be explained away. Whe⯑ther there be not ſomething of this in [45] the preſent caſe I appeal to the writings of the Great Author. His methodus ra⯑tionum primarum et ultimarum, His ſecond Lemma in the ſecond book of his prin⯑ciples, his Introduction and Treatiſe of the Quadrature of Curves. In all which it appears to me, there is not one uniform doctrine explained and carried throughout the whole, but rather ſundry inconſiſtent accounts of this new Method, which ſtill grows more dark and confuſed the more it is handled: I could not help thinking, the greateſt genius might lie under the influence of falſe principles; and where the object and notions were exceeding ob⯑ſcure, he might poſſibly diſtruſt even his own demonſtrations. "At leaſt thus much ſeemed plain, that Sir Iſaac had ſometime owned himſelf ſatisfied, where he could not demonſtrate to others. In proof whereof I mentioned his letter to Mr. Collins; Hereupon you tell me: there is a great deal of difference be⯑tween ſaying, I cannot undertake to prove a thing, and I will not under⯑take it." But in anſwer to this, I de⯑ſire you will be pleaſed to conſider, that I was not making a preciſe extract out of that letter, in which the very words of Sir Iſaac ſhould alone be inſerted. But I made my own remark and inference, [46] from what I remembred to have read in that letter; where, ſpeaking of a certain Mathematical matter, Sir Iſaac expreſſeth himſelf, in the following terms. "It is plain to me by the fountain I draw it from; though I will not undertake to prove it to others." Now whether my inference may not be fairly drawn from thoſe words of Sir Iſaac Newton; and whether the difference as to the ſenſe be ſo great between will and can in that particular caſe, I leave to be determined by the Reader.

XXXVIII. IN the next paragraph you talk big, but prove nothing. You ſpeak of driving out of intrenchments, of ſallying and attacking and carrying by aſſault; of ſlight and untenable works, of a new-raiſed and undiſciplined militia, and of veteran regular troops. Need the Reader be a Ma⯑thematician to ſee the vanity of this para⯑graph? After this you employ (p. 65) your uſual colouring, and repreſent the great Author of the method of Fluxions "as a Good old Gentleman faſt aſleep, and ſnoring in his eaſy chair; while dame Fortune is bringing him her apron full of beautiful theorems and problems, which he never knows or thinks of." This you would have paſs for a conſe⯑quence [47] of my notions. But I appeal to all thoſe who are ever ſo little knowing in ſuch matters, whether there are not divers fountains of Experiment, Induction, and Analogy, whence a man may derive and ſatisfy himſelf concerning the truth of ma⯑ny points in Mathematics and Mechanical Philoſophy, although the proofs thereof afforded by the modern Analyſis ſhould not amount to demonſtration? I further appeal to the conſcience of all the moſt profound Mathematicians, whether they can, with perfect acquieſcence of mind free from all ſcruple, apply any propoſition merely upon the ſtrength of a Demonſtra⯑tion involving ſecond or third Fluxions, without the aid of any ſuch experiment or analogy or collateral proof whatſoever? Laſtly, I appeal to the Reader's own heart, whether he cannot clearly conceive a me⯑dium between being faſt aſleep and demon⯑ſtrating? But you will have it, that I re⯑preſent Sir Iſaac's Concluſions as com⯑ing out right, becauſe one error is compen⯑ſated by another contrary and equal error, which perhaps he never knew himſelf nor thought of: that by a twofold miſtake he arrives though not at ſcience yet at Truth: that he proceeds blindfold, &c. All which is untruly ſaid by you, who have miſap⯑plied to Sir Iſaac what was intended for the Marquis de l' Hoſpital and his fol⯑lowers, [48] for no other end (as I can ſee) but that you may have an opportunity to draw that ingenious portraiture of Sir Iſaac Newton and Dame Fortune, as will be ma⯑nifeſt to whoever reads the Analyſt.

XXXIX. YOU tell me (p. 70), if I think fit to perſiſt in aſſerting, "that this affair of a double error is entirely a new diſcovery of my own, which Sir Iſaac and his followers never knew nor thought of, that you have unqueſtiona⯑ble evidence to convince me of the con⯑trary, and that all his followers are already appriſed, that this very objecti⯑on of mine was long ſince foreſeen, and clearly and fully removed by Sir Iſaac Newton in the firſt ſection of the firſt book of his Principia". All which I do as ſtrongly deny as you affirm. And I do aver, that this is an unqueſtionable proof of the matchleſs contempt which you, Philalethes, have for Truth. And I do here publickly call up⯑on you, to produce that evidence which you pretend to have, and to make good that fact which you ſo confidently affirm. And, at the ſame time, I do aſſure the Reader that you never will, nor can.

[49] XL. IF you defend Sir Iſaac's notions as delivered in his Principia, it muſt be on the rigorous foot of rejecting nothing, neither admitting nor caſting away infi⯑nitely ſmall quantities. If you defend the Marquis, whom you alſo ſtile your Ma⯑ſter, it muſt be on the foot of admitting that there are infiniteſimals, that may be rejected, that they are nevertheleſs real quantities, and themſelves infinitely ſub⯑diviſible. But you ſeem to have grown giddy with paſſion, and in the heat of controverſy to have miſtaken and forgot your part. I beſeech you, Sir, to conſider, that the Marquis (whom alone, and not Sir Iſaac this double error in finding the ſubtangent doth concern) rejects indeed in⯑finiteſimals, but not on the foot that you do, to wit, their being inconſiderable in practical Geometry or mixed Mathematics. But he rejects them in the accuracy of Spe⯑culative Knowledge: in which reſpect there may be great Logical errors, al⯑though there ſhould be no ſenſible miſtake in practice: which, it ſeems, is what you cannot comprehend. He rejects them like⯑wiſe in virtue of a Poſtulatum, which I venture to call rejecting them without ce⯑remony. And though he inferreth a con⯑cluſion accurately true, yet he doth it, contrary to the rules of Logic, from inac⯑curate [50] and falſe premiſes, And how this comes about, I have at large explained in the Analyſt, and ſhewed in that particular caſe of Tangents, that the Rejectaneous Quantity might have been a finite quanti⯑ty of any given magnitude, and yet the concluſion have come out exactly the ſame way; and conſequently, that the truth of this method doth not depend on the reaſon aſſigned by the Marquis, to wit, the poſtu⯑latum for throwing away Infiniteſimals; and therefore that he and his followers acted blindfold, as not knowing the true reaſon for the concluſion's coming out ac⯑curately right, which I ſhew to have been the effect of a double error.

XLI. THIS is the truth of the matter, which you ſhamefully miſrepreſent and declaim upon, to no ſort of purpoſe but to amuſe and miſlead your Reader. For which conduct of yours throughout your remarks, you will pardon me if I cannot otherwiſe account, than from a ſecret hope that the reader of your defence would never read the Analyſt. If he doth, He cannot but ſee what an admirable Method you take to defend your cauſe: How in⯑ſtead of juſtifying the Reaſoning, the Lo⯑gic or the Theory of the caſe ſpecified, [51] which is the real point, you diſcourſe of ſenſible and practical errors: And how all this is a manifeſt impoſition upon the Reader. He muſt needs ſee that I have expreſly ſaid, "I have no controverſy ex⯑cept only about your Logic and me⯑thod: that I conſider how you demon⯑ſtrate; what objects you are converſant about; and whether you conceive them clearly? That I have often expreſſed my ſelf to the ſame effect, deſiring the Reader to remember, that I am only concerned about the way of coming at your theorems, whether it be legitimate or illegitimate, clear or obſcure, ſcienti⯑fic or tentative: That I have on this ve⯑ry occaſion, to prevent all poſſibility of miſtake, repeated and inſiſted that I conſider the Geometrical Analyſt as a Logician, i. e. ſo far forth as he reaſons and argues; and his mathematical con⯑cluſions not in themſelves but in their premiſes; not as true or falſe, uſeful or inſignificant, but as derived from ſuch principles, and by ſuch inferences"^*. You affirm (and indeed what can you not affirm?) that the difference between the true ſubtangent and that found without any compenſation is abſolutely nothing at all. I profeſs my ſelf of a contrary opini⯑on. [52] My reaſon is becauſe nothing cannot be divided into parts. But this difference is capable of being divided into any, or into more than any given number of parts; For the truth of which conſult the Mar⯑quit de l' Hoſpital. And, be the error in fact or in practice ever ſo ſmall, it will not thence follow that the error in Rea⯑ſoning, which is what I am alone con⯑cerned about, is one whit the leſs, it being evident that a man may reaſon moſt abſurd⯑ly about the minuteſt things.

XLII. PRAY anſwer me fairly, once for all, whether it be your opinion that whatſoever is little and inconſiderable e⯑nough to be rejected without inconve⯑nience in practice, the ſame may in like manner be ſafely rejected and overlooked in Theory and Demonſtration. If you ſay no, it will then follow, that all you have been ſaying here and elſewhere, about yards and inches and decimal fractions, ſet⯑ting forth and inſiſting on the extreme ſmall⯑neſs of the rejectaneous quantity, is quite foreign to the argument, and only a piece of skill to impoſe upon your Reader. If you ſay yes, it follows that you then give up at once all the orders of Fluxions and Infiniteſimal Differences; and ſo moſt im⯑prudently turn all your ſallies and attacks [53] and Veterans to your own overthrow. If the Reader is of my mind, he will de⯑ſpair of ever ſeeing you get clear of this Dilemma. The points in controverſy have been ſo often and ſo diſtinctly noted in the Analyſt, that I very much wonder how you could miſtake if you had no mind to miſtake. It is very plain, if you are in earneſt, that you neither underſtand me nor your Maſters. And what ſhall we think of other ordinary Analyſts, when it ſhall be found that even you, who, like a Champion ſtep forth to defend their prin⯑ciples, have not conſidered them.

XLIII. THE impartial Reader is in⯑treated to remark throughout your whole performance, how confident you are in aſ⯑ſerting, and withall how modeſt in proving or explaining: How frequent it is with you to employ Figures and Tropes in⯑ſtead of Reaſons: How many difficulties propoſed in the Analyſt are diſcreetly over⯑looked by you, and what ſtrange work you make with the reſt: How groſly you miſtake and miſrepreſent, and how little you practiſe the advice which you ſo libe⯑rally beſtow. Believe me, Sir, I had long and maturely conſidered the principles of the modern Analyſis, before I ventured to publiſh my thoughts thereupon in the [54] Analyſt. And ſince the publication there⯑of, I have my ſelf freely converſed with Mathematicians of all ranks, and ſome of the ableſt Profeſſors, as well as made it my buſineſs to be informed of the Opinions of others, being very deſirous to hear what could be ſaid towards clearing my diffi⯑culties or anſwering my objections. But though you are not afraid or aſhamed to repreſent the Analyſts as very clear and uni⯑form in their Conception of theſe matters, yet I do ſolemnly affirm (and ſeveral of themſelves know it to be true) that I found no harmony or agreement among them, but the reverſe thereof, the greateſt diſſo⯑nance, and even contrariety of Opinions, employed to explain what after all ſeem⯑ed inexplicable.

XLIV. SOME fly to proportions be⯑tween nothings. Some reject quantities be⯑cauſe infiniteſimal. Others allow only finite quantities, and reject them becauſe incon⯑ſiderable. Others place the method of Fluxions on a foot with that of Exhauſt⯑ions, and admit nothing new therein. Some maintain the clear conception of Fluxions. Others hold they can demon⯑ſtrate about things incomprehenſible. Some would prove the Algoriſm of Fluxions by reductio ad abſurdum; others a priori. [55] Some hold the evaneſcent increments to be real quantities, ſome to be nothings, ſome to be limits. As many Men, ſo many minds: Each differing one from another, and all from Sir Iſaac Newton. Some plead inaccurate expreſſions in the great Author, whereby they would draw him to ſpeak their ſenſe, not conſidering that if he meant as they do, he could not want words to expreſs his meaning. Others are magiſterial and poſitive, ſay they are ſatisfied, and that is all, not con⯑ſidering that we, who deny Sir Iſaac Newton's Authority, ſhall not ſubmit to that of his Diſciples. Some inſiſt, that the Concluſions are true, and therefore the principles, not conſidering what hath been largely ſaid in the Analyſt^* on that head. Laſtly ſeveral (and thoſe none of the meaneſt) frankly owned the objections to be unanſwerable. All which I mention by way of Antidote to your falſe Colours: and that the unprejudiced Inquirer after Truth may ſee, it is not without founda⯑tion, that I call on the celebrated Mathema⯑ticians of the preſent Age to clear up theſe obſcure Analytics, and concur in giv⯑ing to the publick ſome conſiſtent and intelligible account of the principles of their great Maſter: which if they do not, [56] I believe the World will take it for gran⯑ted that they cannot.

XLV. HAVING gone through your Defence of the Britiſh Mathematicians, I find in the next place, that you attack me on a point of Metaphyſics, with what ſucceſs the Reader will determine. I had upon another Occaſion many years ago wrote againſt Abſtract general Ideas^*. In oppoſition to which, you declare your ſelf to adhere to the vulgar opinion, that nei⯑ther Geometry nor any other general Sci⯑ence can ſubſiſt without general Ideas. (P. 74.) This implies that I hold there are no general Ideas. But I hold the direct contrary, that there are indeed general Ideas, but not formed by abſtraction in the man⯑ner ſet forth by Mr. Locke. To me it is plain, there is no conſiſtent Idea, the likeneſs whereof may not really exiſt. Whatſoever therefore is ſaid to be ſome⯑what which cannot exiſt, the Idea thereof muſt be inconſiſtent. Mr. Locke acknowledg⯑eth it doth require Pains and Skill to form his general Idea of a Triangle. He further expreſly ſaith, it muſt be neither oblique nor rectangular, neither equilateral, equi⯑crural, nor ſcalenum; but all and none [57] of theſe at once. He alſo ſaith, it is an Idea wherein ſome parts of ſeveral diffe⯑rent and inconſiſtent Ideas are put toge⯑ther^†. All this looks very like a Con⯑tradiction. But to put the Matter paſt diſpute, it muſt be noted, that he affirms it to be ſomewhat imperfect that cannot ex⯑iſt; conſequently the Idea thereof is im⯑poſſible or inconſiſtent.

XLVI. I DESIRE to know, whether it is not poſſible for any thing to exiſt, which doth not include a contradiction: And if it is, whether we may not infer, that what cannot poſſibly exiſt, the ſame doth include a contradiction: I further deſire to know, whether the reader can frame a diſtinct idea of any thing that includes a contradiction? For my part, I cannot, nor conſequently of the above⯑mentioned Triangle; Though you (who it ſeems know better than my ſelf what I can do) are pleaſed to aſſure me of the contrary. Again, I ask whether that, which it is above the power of man to form a compleat idea of, may not be called incomprehenſible? And whether the Reader can frame a compleat idea of this imperfect, impoſſible Triangle? And if not, whether it doth not follow that it [58] is incomprehenſible? It ſhould ſeem, that a diſtinct aggregate of a few conſiſtent parts was nothing ſo difficult to conceive, or impoſſible to exiſt; and that, therefore your Comment muſt be wide of the Au⯑thor's meaning. You give me to under⯑ſtand (P. 82.) that this account of a ge⯑neral Triangle was a trap which Mr. Locke ſet to catch fools. Who is caught therein let the Reader judge.

XLVII. IT is Mr. Locke's opinion, that every general name ſtands for a ge⯑neral abſtract idea, which preſcinds from the ſpecies or individuals comprehended under it. Thus, for example, according to him, the general name Colour ſtands for an idea, which is neither Blue, Red, Green nor any other particular colour, but ſomewhat diſtinct and abſtracted from them all. To me it ſeems, the word Colour is only a more general name applicable to all and each of the particular colours; while the other ſpecific names, as Blue, Red, Green, and the like are each re⯑ſtrained to a more limited ſignification. The ſame may be ſaid of the word Tri⯑angle. Let the Reader judge whether this be not the caſe; and whether he can diſtinctly frame ſuch an idea of colour as ſhall preſcind from all the ſpecies there⯑of, [59] or of a triangle which ſhall anſwer Mr. Locke's account, preſcinding and ab⯑ſtracting from all the particular ſorts of triangles, in the manner aforeſaid.

XLVIII. I intreat my Reader to think. For if he doth not, he may be under ſome influence from your confident and poſitive way of talking. But any one who thinks may, if I miſtake not, plainly perceive that you are deluded, as it often happens, by miſtaking the terms for ideas. Nothing is eaſier, than to define in terms or words that which is incomprehenſible in idea, foraſmuch as any words can be either ſe⯑parated or joined as you pleaſe, but ideas always cannot. It is as eaſy to ſay a round ſquare as an oblong ſquare, though the for⯑mer be inconceivable. If the Reader will but take a little care to diſtinguiſh between the Definition and the Idea, between words or expreſſions and the conceptions of the mind, he will judge of the truth of what I now advance, and clearly perceive how far you are miſtaken, in attempting to illu⯑ſtrate Mr. Locke's Doctrine, and where your miſtake lies. Or, if the Reader is minded to make ſhort work, he needs only at once to try whether laying aſide the words he can frame in his mind the idea of an impoſſi⯑ble triangle; upon which trial the iſſue of [60] this diſpute may be fairly put. This do⯑ctrine of abſtract general ideas ſeemed to me a capital error, productive of number⯑leſs difficulties and diſputes, that runs not only throughout Mr. Locke's book, but through moſt parts of Learning. Conſe⯑quently, my animadverſions thereupon were not an effect of being inclined to carp or cavil at a ſingle paſſage, as you would wrongfully inſinuate, but proceeded from a love of Truth and a deſire to baniſh, ſo far as in me lay, falſe principles and wrong ways of thinking, without reſpect of per⯑ſons. And indeed, though you and other Party-men are violently attached to your reſpective Maſters, yet I, who profeſs my ſelf only attached to Truth, ſee no reaſon why I may not as freely animadvert on Mr. Locke or Sit Iſaac Newton, as they would on Ariſtole or Deſcartes. Certainly the more extenſive the influence of any Error, and the greater the authority which ſupports it, the more it deſerves to be con⯑ſidered and detected by ſincere Inquirers after Knowledge.

XLIX. IN the cloſe of your perfor⯑mance, you let me underſtand, that your Zeal for Truth and the reputation of your Maſters hath occaſioned your reprehending me with the utmoſt freedom. And it muſt [61] be owned you have ſhewn a ſingular talent therein. But I am comforted under the ſeverity of your reprehenſions, when I con⯑ſider the weakneſs of your arguments, which, were they as ſtrong as your reproofs, could leave no doubt in the mind of the Reader concerning the matters in diſpute between us. As it is, I leave him to re⯑flect and examine by your light, how clear⯑ly he is enabled to conceive a fluxion, or the fluxion of a fluxion, a part infinitely ſmall ſubdivided into an infinity of parts, a naſcent or evaneſcent increment, that which is neither ſomething nor nothing, a triangle formed in a point, velocity with⯑out motion, and the reſt of thoſe arcana of the modern Analyſis. To conclude, I had ſome thoughts of adviſing you how to conduct your ſelf for the future, in re⯑turn for the advice you have ſo freely im⯑parted to me: but, as you think it becomes me rather to inform my ſelf than inſtruct others, I ſhall, for my further information, take leave to propoſe a few Queries to thoſe learned Gentlemen of Cambridge, whom you aſſociate with your ſelf, and re⯑preſent as being equally ſurpriſed at the tendency of my Analyſt.

L. I deſire to know, whether thoſe who can neither demonſtrate nor conceive [62] the principles of the modern Analyſis, and yet give into it, may not be juſtly ſaid to have Faith, and be ſtyled believers of myſ⯑teries? Whether it is impoſſible to find a⯑mong the Phyſicians, mechanical Philoſo⯑phers, Mathematicians, and Philomathe⯑maticians of the preſent age, ſome ſuch Believers, who yet deride Chriſtians for their belief of Myſteries? Whether with ſuch men it is not a fair, reaſonable, and legitimate method to uſe the Argumentum ad Hominem? And being ſo, whether it ought to ſurpriſe either Chriſtians or Scho⯑lars? Whether in an age wherein ſo ma⯑ny pretenders to ſcience attack the Chri⯑ſtian Religion, we may not be allowed to make repriſals, in order to ſhew that the Irreligion of thoſe men is not to be preſumed an effect of deep and juſt think⯑ing? Whether an attempt to detect falſe reaſonings, and remedy defects in Mathe⯑matics, ought to be ill received by Ma⯑thematicians? Whether the introducing more eaſy methods and more intelligible principles in any ſcience ſhould be diſ⯑countenanced? Whether there may not be fair objections as well as cavils? And whe⯑ther to inquire diligently into the meaning of terms and the proof of propoſitions, not excepting againſt any thing without aſſigning a reaſon, nor affecting to miſtake [63] the ſignification of words, or ſtick at an expreſſion where the ſenſe was clear, but conſidering the ſubject in all lights, ſin⯑cerely endeavouring to find out any ſenſe or meaning whatſoever, candidly ſetting forth what ſeems obſcure and what fal⯑lacious, and calling upon thoſe, who pro⯑feſs the knowledge of ſuch matters, to explain them; whether, I ſay, ſuch a pro⯑ceeding can be juſtly called cavilling? Whether there be an ipſe dixit erected? And if ſo, when, where, by whom, and upon what Authority? Whether even where Authority was to take place, one might not hope the Mathematics, at leaſt, would be excepted? Whether the chief end, in making Mathematics ſo conſide⯑rable a part of Academical Education, be not to form in the minds of young Stu⯑dents habits of juſt and exact Reaſoning? And whether the ſtudy of abſtruſe and ſubtile matters can conduce to this end, unleſs they are well underſtood, examin⯑ed, and ſifted to the bottom? Whether, therefore, the bringing Geometrical demon⯑ſtrations to the ſevereſt teſt of Reaſon ſhould be reckoned a diſcouragement to the ſtudies of any learned Society? Whe⯑ther to ſeparate the clear parts of things from the obſcure, to diſtinguiſh the real Principles, whereon Truths reſt, and [64] whence they are derived, and to propor⯑tion the juſt meaſures of aſſent according to the various degrees of evidence, be an uſeleſs or unworthy Undertaking? Whe⯑ther the making more of an argument than it will bear, and placing it in an un⯑due rank of evidence, be not the likely way to diſparage it? Whether it may not be of ſome uſe, to provoke and ſtir up the learned profeſſors to explain a part of Mathematical Learning, which is acknow⯑ledged to be moſt profound, difficult, and obſcure, and at the ſame time ſet forth by Philalethes and many others, as the greateſt inſtance that has ever been given of the extent of humane abilities? Whe⯑ther for the ſake of a Great man's diſco⯑veries, we muſt adopt his errors? Laſtly, whether in an age wherein all other prin⯑ciples are canvaſſed with the utmoſt free⯑dom, the principles of Fluxions are to be alone excepted?