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, whe­ther it be possible to frame any clear Con­ception of Fluxions, you express yourself in the following manner, "Pray, Sir, who are those thinking Readers you ap­peal [Page 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 Ma­thematicians. And for those who are not Mathematicians, I shall endeavour so to unveil this Mystery, and put the Contro­versy 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 no­blest 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 Repu­tation and Authority of Sir Isaac New­ton 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 [Page 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 Mathema­tical Learning is to Mankind." Hence you take occasion to declaim on the Usefulness of Mathematics in the several Branches, and then to redouble your Sur­prize 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 unpreju­diced Reader will think with me, that things obscure are not therefore sacred; and that it is no more a Crime to canvass and de­tect 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 Ana­lyst. But for your further Satisfaction shall here tell you, it is very well known, that [Page 6] several Persons who deride Faith and Mys­teries in Religion, admit the Doctrine of Fluxions for true and certain. Now if it be shewn that Fluxions are really most incom­prehensible 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 Mathe­matics, 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 [Page 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 o­thers, who are not Mathematicians, who are influenced by a Regard for their Au­thority. 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 en­deavour to shew, that such Men are defec­tive in Point of Reason and Judgment, and that they do the very Thing they would seem to despise.

[Page 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 never­theless a profound Veneration for Fluxions; and I was willing to set forth the Incon­sistence of such Men. If there be no such Thing as Infidels, who pretend to Knowledge in the modern Analysis, I own my self misin­formed, and shall gladly be found in a Mis­take; but even in that Case, my Remarks upon Fluxions are not the less true; nor will it follow, that I have no Right to exa­mine them on the Foot of humane Science, even though Religion were quite uncon­cerned, 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 a­gainst 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 Com­pliments as you do to our Church, and are just as angry, as you can be, at any who un­dertake to defend Religion by Reason. If [Page 9] we resolve all into Faith, they laugh at us and our Faith: And if we attempt to Rea­son, they are angry at us: They pretend we go out of our Province, and they re­commend 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 repre­sent 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 incon­sistent Part, in pretending to reject the Christian Religion, because he cannot be­lieve what he doth not comprehend; or be­cause 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 sen­sibly touched, and denying it with so much Passion. You, indeed, do not stick to affirm, [Page 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 Ad­dison is one of the persons, whom you are pleased to characterize in those modest and mannerly terms. He assured me that the In­fidelity of a certain noted Mathematician, still living, was one principal reason assign­ed by a witty man of those times for his being an Infidel. Not, that I imagine Geo­metry disposeth Men to Infidelity; but that from other causes, such as Presumption, Ignorance, or Vanity, like other Men, Geo­metricians also become Infidels, and that the supposed light and evidence of their Science gains credit to their Infidelity.

VIII. "YOU reproach me with Ca­lumny, detraction and artifice ( P. 15.) You recommend such means as are in­nocent and just, rather than the crimi­nal method of lessening or detracting from my opponents ( ibid.) You accuse me of the Odium Theologicum, the in­temperate 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, [Page 11] but read and judge for himself. In which case he will be able to discern (though he should be no Mathematician) how passio­nate 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 para­doxes, which render it inaccessible to most Men, will be thought a criminal under­taking 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 af­ter Truth, will at once enter into your pas­sions, and degenerate into a nest of Bigots.

IX. I OBSERVE upon the Incon­sistency of certain Infidel Analysts. I re­mark some defects in the principles of the modern Analysis. I take the liberty de­cently to dissent from Sir Isaac Newton. I propose some helps to abridge the trouble of Mathematical Studies, and render them [Page 12] more useful. What is there in all this, that should make you declaim on the use­fulness of practical Mathematics? that should move you to cry out Spain, Inqui­sition, 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 con­futing 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 Bi­gotry what it will. A very remarkable in­stance 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 Fami­liar 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 [Page 13] or Clergymen, &c. Let us dig up the bodies of Dr. Barrow and Sir Isaac Newton, and burn them under the Gal­lows."

X. THE Reader need not be a Ma­thematician, to see how vain all this Tra­gedy of yours is. And if he be as tho­roughly satisfied as I am, that the cause of Fluxions cannot be defended by rea­son, 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 cal­ling 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 Geo­metry) 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 Wri­ters, 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 wri­ting your defence, you seem to have been [Page 14] overcome with Passion: But now you may be supposed cool, I desire you to re­flect whether it be not wrote in the true spirit of an Inquisitor? Whether this be­comes a Person so exceeding delicate him­self upon that Point? And whether your Brethren the Analysts will think them­selves 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 un­derstanding, suspect he could hardly be quite satisfy'd with his own demonstra­tion: 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 rash­ness [Page 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. Ano­ther 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 sur­prise, 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.

[Page 16] XIV. YOU may, indeed, your self be an Idolater of whom you please: But then you have no right to insult and ex­claim at other men, because they do not adore your Idol. Great as Sir Isaac New­ton was, I think he hath, on more occa­sions 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 me­thod may be useful, considered as an art of Invention. You, who are a Mathema­tician, must acknowledge, there have been divers such methods admitted in Mathe­matics, 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, Des­cartes have wrote concerning the roots of affected Aequations. It will not, neverthe­less, 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 [Page 17] those arguments ab invidia which at every turn you employ against me. You re­present your self ( P. 52.) as a man, whose highest Ambition is in the lowest degree to imitate Sir Isaac Newton. It might, per­haps, have suited better with your appel­lation 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 Senti­ments 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 Abi­lities; not considering that a person of less Abilities may know more on a certain point than one of greater; not consider­ing that a purblind eye, in a close and [Page 18] narrow view, may discern more of a thing, than a much better eye in a more exten­sive 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 in­troducing 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 In­finitesimal of an Infinitesimal, and so on­ward ^*. What have you to say in answer to this? Do you attempt to clear up the noti­on 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 puz­zled 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?

[Page 19] XVIII. UPON this point every Rea­der 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 at­tention 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, with­out 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 admo­nish me to look over a second time, to consult, examine, weigh the words of Sir Isaac. In answer to which I will ven­ture to say, that I have taken as much pains as (I sincerely believe) any man [Page 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 Mathe­matics, so I fairly followed the dictates of my own mind in examining, and censu­ring 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 prin­ciples 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 Ana­lysis: neither are the demonstrations of the [Page 21] general rules of that method at all cleared up by applying them. The reason of which is, because in operating or calculat­ing, men do not return to contemplate the original principles of the method, which they constantly presuppose, but are em­ployed 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 de­monstrated. This I say to encourage those, who are not far gone in these Studies, to use intrepidly their own judg­ment, without a blind or a mean de­ference 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 ex­tension, of magnitude which is neither fi­nite nor infinite, or of a quantity having no magnitude which is yet divisible, of a figure where there is no space, of proporti­on [Page 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 ad­mitted 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 ele­ments 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 up­on principles, but inclining rather to take them upon trust: And things early admit­ted by repetition become familiar: And this familiarity at length passeth for Evi­dence. Now to me it seems, there are certain points tacitly admitted by Mathe­maticians, 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 vir­tue of some latent error in his principles a man be drawn into fallacious reasonings, it is nothing strange that he should take [Page 23] them for true: And, nevertheless, if; when urged by perplexities and uncouth conse­quences, 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 New­ton ^*. Herereupon you are pleased to re­mark, 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, on­ly 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 in­deed 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 [Page 24] this new method of declaiming in Mathe­matics. 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 repre­sentation. I answer, that if according to Sir Isaac Newton a Fluxion be the veloci­ty of an increment, then according to him I may call the Fluxion of a Fluxion the Ve­locity of a Velocity. But for the truth of the antecedent see his introduction to the Qua­drature of Curves, where his own words are, motuum velincrementorum velocitates nominan­do Fluxiones. See also the second Lemma of the second Book of his mathematical prin­ciples of natural Philosophy, where he ex­presseth himself in the following manner, velocitates incrementorum ac decrementorum, quas etiam, motus, mutationes & fluxiones quantitatum nominare licet. And that he ad­mits 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 agreea­bly thereunto be called the Velocity of a Velocity? In like manner if by a Fluxion [Page 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 plain­er? Let the Reader now judge who is unfair.

XXIV. I HAD observed, that the Great Author had proceeded illegitimately, in ob­taining the Fluxion or moment of the Rec­tangle of two flowing quantities; and that he did not fairly get rid of the Rectangle of the moments. In answer to this you al­ledge, that the error arising from the o­mission 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 won­der how you could be ignorant of it.

XXV. YOU would fain persuade your Reader, that I make an absurd quarrel a­gainst errors of no significancy in practice, and represent Mathematicians as proceeding [Page 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 no­thing to say. But then you must remem­ber this is not the Doctrine of Fluxions: It is none of that Analysis with which I am concerned. That I am far from quar­relling 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 some­what 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 accu­rate in Geometrical rigour is manifest, to whoever considers what the Great Author writes about it; especially in his Intro­duction to the Quadrature of Curves, where he saith In rebus mathematicis errores quam minimi non sunt contemnendi. Which ex­pression you have seen quoted in the Ana­lyst, and yet you seem ignorant thereof, and indeed, of the very End and Design of the Great Author in this his invention of Fluxions.

[Page 27] XXVI. AS oft as you talk of finite quan­tities inconsiderable in practice, Sir Isaac disowns your apology. Cave, saith he, in­tellexeris 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 just­ness 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 com­mit very great errors in Logic, which Lo­gical 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 qua­lified 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 Rectan­gle. 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 rec­tangle of two flowing quantities, nor in [Page 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 in­crementis 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 Incre­menta 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 demonstra­tion understands his incrementum as belong­ing to the Rectangulum quodvis at the begin­ning. Is not the same also evident from the very Lemma it self prefixed to the Demon­stration? The sense whereof is (as the Au­thor there explains it) that if 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 x b A. [Page 29] Either therefore the conclusion of the de­monstration is not the thing which was to be demonstrated, or the rectanguli incremen­tum a B x b A belongs to the rectangle AB.

XXVIII. ALL this is so plain that no­thing can be more so; and yet you would fain perplex this plain case by distinguish­ing between an increment and a moment. But it is evident to every one, who has any notion of Demonstration, that the incre­mentum in the Conclusion must be the mo­mentum 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 mo­mentorum intelligo. And you observe your self that he useth the word moment to sig­nify 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 mo­mentaneous increment or decrement of the [Page 30] flowing quantities, you would have us con­clude, 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 ma­thematical quantities as pleading their rights, as tossing up cross and pile, as dis­puting amicably. You talk of their claim­ing preference, their agreeing, their boy­ishness and their gravity. And after this in­genious digression you address me in the fol­lowing words.—Believe me there is no re­medy, you must acquiesce. But my answer is, that I will neither believe you nor ac­quiesce; 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.

[Page 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 rec­tangle. 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 ve­locity 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 con­ceive 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 mo­tion. 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. In­crementa [Page 32] (saith he) vel decrementa momen­tanea 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 ly­ing at either extremity of the moment, but as extended to the middle of it; as having acquired the one half of the mo­ment, 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 distinguish­ed 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 [Page 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 Rea­der 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 au­thorities, by Ghosts and Visions, and a­bove all by that extreme satisfaction and complacency with which you utter your strange conceits; if words without a mean­ing 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. "Inas­much (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 con­fining it to be either an increment or decrement, is by you inconsiderately and arbitrarily, and without any Shadow of [Page 34] of Reasou given, supposed and deter­mined 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 con­sider 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 pro­ceed to extoll the great Author of the fluxio­nary Method, and to bestow some Brus­queries upon those who unadvisedly dare to differ from him. To all which I shall give no answer.

XXXII. AFTERWARDS to re­move (as you say) all Scruple and Difficulty about this affair, you observe that the Mo­ment 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 [Page 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 Lem­ma, 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 suf­ficiently considered that Lemma, its De­monstration, and its Consequences. For, however that way of reasoning may do in the Method of exhaustions, where quanti­ties less than assignable are regarded as nothing; yet for a Fluxionist writing a­bout momentums, to argue that quantities must be equal because they have no assign­able difference, seems the most injudici­ous Step that could be taken: it is direct­ly demolishing the very Doctrine you would defend. For it will thence follow, that all homogeneous momentums are e­qual, and consequently the velocities, mutations, or fluxions proportional thereto, are all likewise equal. There is, therefore, only one proportion of equality through­out, which at once overthrows the whole System you undertake to defend. Your [Page 36] moments (I say) not being themselves as­signable 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 ob­jections 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 subver­sive of the doctrine of Fluxions. And, indeed, who sees not that a Demonstra­tion 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 under­take to defend is that method for obtain­ing a rule to find the Fluxion of any Power of a flowing Quantity, which is delivered in the introduction to the Qua­dratures, [Page 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 with­out that supposition you can never bring the quantity or expression [...] &c. down to [...], the very thing aimed at by supposing the evanescence. Say whe­ther 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 affirmati­ons, and yet I believe that even you will not affirm this; it being most evident, that the product of two real quantities is some­thing real; and that nothing real can be [Page 38] rejected either according to the [...] of Geometry, or according to Sir Isaac's own principles; for the truth of which I ap­peal 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 me­thodo 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 an­gry with, to wit, that Sir Isaac's followers are much more eager in applying his me­thod, than accurate in examining his prin­ciples. You raise a dust about evanescent augments which may perhaps amuse and amaze your Reader, but I am much mis­taken 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 in­deed get rid of those quantities in the com­position whereof they are coefficients; but [Page 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 de­sire you to look into his Analysis per aequa­tiones infinitas ( P. 20.) where, in his pre­paration for demonstrating the first rule for the squaring of simple Curves, you will find that on a parallel occasion, speak­ing 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 unaccounta­ble, so horrid, so truly Boeotian a blunder" [Page 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 in­cautiously 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 au­thorities 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 pro­ceed to declaim upon the following pas­sage taken from the seventeenth section of the Analyst. "Considering the various arts and devices used by the great au­thor of the fluxionary method: In how many lights he placeth his Fluxions: and in what different ways he attempts to [Page 41] demonstrate the same point: One would be inclined to think, he was himself sus­picious of the justness of his own de­monstrations." This passage you com­plain of as very hard usage of Sir Isaac Newton. You declaim copiously, and en­deavour 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 per­haps it should have been. Would you like it better if I should say, the various incon­sistent 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 de­monstration? 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 Doc­trine. [Page 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 ac­knowledge 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 recon­cile this with what he saith in the Scho­lium of the second Lemma of the first Section of the first book of his Principles: Cave intelligas quantitates magnitudine de­terminatas, 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 Geome­triam 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 [Page 43] deerant momentorum dimidia, &c. where the moments are supposed to be divided. I should be very glad, a person of such lu­minous intellect would be so good as to explain, whether by Fluxions we are to understand the nascent or evanescent quan­tities themselves, or their motions, or their Velocities, or simply their proportions: and having interpreted them in what sense you will, that you would then conde­scend 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 o­thers 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 ex­perience. 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 diffi­culties, 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 con­ceive your selves, I give you my word [Page 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, dili­gently read and considered the several ac­counts 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 in­clined to think, He was himself suspi­cious of the justness of his own demon­strations: and that he was not enough pleased with any one Notion steadily to adhere to it." After which I ad­ded, "Thus much is plain that he own­ed 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 substitu­ted for others, a Doctrine instead of being illustrated may be explained away. Whe­ther there be not something of this in [Page 45] the present case I appeal to the writings of the Great Author. His methodus ra­tionum primarum et ultimarum, His second Lemma in the second book of his prin­ciples, 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 ob­scure, 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 be­tween saying, I cannot undertake to prove a thing, and I will not under­take it." But in answer to this, I de­sire 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, [Page 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 Ma­thematician to see the vanity of this para­graph? 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 conse­quence [Page 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 ma­ny 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 Demonstra­tion 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 me­dium between being fast asleep and demon­strating? But you will have it, that I re­present Sir Isaac's Conclusions as com­ing out right, because one error is compen­sated 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 misap­plied to Sir Isaac what was intended for the Marquis de l' Hospital and his fol­lowers, [Page 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 ma­nifest 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 unquestiona­ble evidence to convince me of the con­trary, and that all his followers are already apprised, that this very objecti­on 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 up­on 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.

[Page 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 infi­nitely small quantities. If you defend the Marquis, whom you also stile your Ma­ster, 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 sub­divisible. 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 in­finitesimals, 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 Spe­culative Knowledge: in which respect there may be great Logical errors, al­though there should be no sensible mistake in practice: which, it seems, is what you cannot comprehend. He rejects them like­wise in virtue of a Postulatum, which I venture to call rejecting them without ce­remony. And though he inferreth a con­clusion accurately true, yet he doth it, contrary to the rules of Logic, from inac­curate [Page 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 quanti­ty 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 postu­latum 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 ac­curately 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 in­stead of justifying the Reasoning, the Lo­gic or the Theory of the case specified, [Page 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 ex­cept only about your Logic and me­thod: that I consider how you demon­strate; 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, scienti­fic or tentative: That I have on this ve­ry 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 con­clusions 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 opini­on. [Page 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 Mar­quit de l' Hospital. And, be the error in fact or in practice ever so small, it will not thence follow that the error in Rea­soning, which is what I am alone con­cerned about, is one whit the less, it being evident that a man may reason most absurd­ly about the minutest things.

XLII. PRAY answer me fairly, once for all, whether it be your opinion that whatsoever is little and inconsiderable e­nough to be rejected without inconve­nience 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, set­ting forth and insisting on the extreme small­ness 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 im­prudently turn all your sallies and attacks [Page 53] and Veterans to your own overthrow. If the Reader is of my mind, he will de­spair 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 prin­ciples, have not considered them.

XLIII. THE impartial Reader is in­treated to remark throughout your whole performance, how confident you are in as­serting, and withall how modest in proving or explaining: How frequent it is with you to employ Figures and Tropes in­stead of Reasons: How many difficulties proposed in the Analyst are discreetly over­looked 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 libe­rally 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 [Page 54] Analyst. And since the publication there­of, 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 diffi­culties or answering my objections. But though you are not afraid or ashamed to represent the Analysts as very clear and uni­form 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 disso­nance, and even contrariety of Opinions, employed to explain what after all seem­ed inexplicable.

XLIV. SOME fly to proportions be­tween nothings. Some reject quantities be­cause infinitesimal. Others allow only finite quantities, and reject them because incon­siderable. Others place the method of Fluxions on a foot with that of Exhaust­ions, and admit nothing new therein. Some maintain the clear conception of Fluxions. Others hold they can demon­strate about things incomprehensible. Some would prove the Algorism of Fluxions by reductio ad absurdum; others a priori. [Page 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 con­sidering 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 founda­tion, that I call on the celebrated Mathema­ticians of the present Age to clear up these obscure Analytics, and concur in giv­ing to the publick some consistent and intelligible account of the principles of their great Master: which if they do not, [Page 56] I believe the World will take it for gran­ted 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 nei­ther Geometry nor any other general Sci­ence 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 man­ner 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 some­what which cannot exist, the Idea thereof must be inconsistent. Mr. Locke acknowledg­eth 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, equi­crural, nor scalenum; but all and none [Page 57] of these at once. He also saith, it is an Idea wherein some parts of several diffe­rent and inconsistent Ideas are put toge­ther ^†. All this looks very like a Con­tradiction. But to put the Matter past dispute, it must be noted, that he affirms it to be somewhat imperfect that cannot ex­ist; consequently the Idea thereof is im­possible 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 above­mentioned 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 [Page 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 Au­thor's meaning. You give me to under­stand ( P. 82.) that this account of a ge­neral 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 ge­neral 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 re­strained to a more limited signification. The same may be said of the word Tri­angle. 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 there­of, [Page 59] or of a triangle which shall answer Mr. Locke's account, prescinding and ab­stracting 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 se­parated or joined as you please, but ideas always cannot. It is as easy to say a round square as an oblong square, though the for­mer 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 illu­strate 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 impossi­ble triangle; upon which trial the issue of [Page 60] this dispute may be fairly put. This do­ctrine of abstract general ideas seemed to me a capital error, productive of number­less difficulties and disputes, that runs not only throughout Mr. Locke's book, but through most parts of Learning. Conse­quently, 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 per­sons. 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 con­sidered and detected by sincere Inquirers after Knowledge.

XLIX. IN the close of your perfor­mance, 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 [Page 61] be owned you have shewn a singular talent therein. But I am comforted under the severity of your reprehensions, when I con­sider 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 re­flect and examine by your light, how clear­ly 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 with­out 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 re­turn for the advice you have so freely im­parted 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 re­present as being equally surprised at the tendency of my Analyst.

L. I desire to know, whether those who can neither demonstrate nor conceive [Page 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 mys­teries? Whether it is impossible to find a­mong the Physicians, mechanical Philoso­phers, Mathematicians, and Philomathe­maticians 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 Scho­lars? Whether in an age wherein so ma­ny pretenders to science attack the Chri­stian 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 think­ing? Whether an attempt to detect false reasonings, and remedy defects in Mathe­matics, ought to be ill received by Ma­thematicians? Whether the introducing more easy methods and more intelligible principles in any science should be dis­countenanced? Whether there may not be fair objections as well as cavils? And whe­ther 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 [Page 63] the signification of words, or stick at an expression where the sense was clear, but considering the subject in all lights, sin­cerely endeavouring to find out any sense or meaning whatsoever, candidly setting forth what seems obscure and what fal­lacious, and calling upon those, who pro­fess the knowledge of such matters, to explain them; whether, I say, such a pro­ceeding 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 conside­rable a part of Academical Education, be not to form in the minds of young Stu­dents 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, examin­ed, and sifted to the bottom? Whether, therefore, the bringing Geometrical demon­strations to the severest test of Reason should be reckoned a discouragement to the studies of any learned Society? Whe­ther to separate the clear parts of things from the obscure, to distinguish the real Principles, whereon Truths rest, and [Page 64] whence they are derived, and to propor­tion the just measures of assent according to the various degrees of evidence, be an useless or unworthy Undertaking? Whe­ther the making more of an argument than it will bear, and placing it in an un­due 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 acknow­ledged 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? Whe­ther for the sake of a Great man's disco­veries, we must adopt his errors? Lastly, whether in an age wherein all other prin­ciples are canvassed with the utmost free­dom, the principles of Fluxions are to be alone excepted?