I. THERE are some Men that can neither give nor take an Answer, but writing mere­ly for the sake of writing multiply words to no pur­pose. There are also cer­tain careless Writers, that in defiance of com­mon sense publish such things as, though they are not asham'd to utter, yet, other men may well be asham'd to answer. Whether there be [Page 4] any thing in Mr. Walton's method of vindi­cating Fluxions, that might justify my taking no further notice of him on the abovemention­ed considerations, I leave you and every other Reader to judge. But those, Sir, are not the reasons I shall assign for not replying to Mr. Walton's full Answer. The true reason is, that he seems at bottom a facetious man, who under the colour of an opponent writes on my side of the Question, and really believes no more than I do of Sir Isaac Newton's Doctrine about Fluxions, which he exposes, contradicts, and confutes with great skill and humour, under the masque of a grave vindication.

II. AT first I consider'd him in another light, as one who had good reason for keeping to the beaten Track, who had been used to dictate, who had terms of art at will, but was indeed, at small trouble about putting them together, and perfectly easy about his Reader's understanding them. It must be owned, in an Age of so much ludicrous humour, it is not e­very one can at first sight discern a Writer's real design. But, be a man's Assertions ever so strong in favour of a Doctrine, yet if his Rea­sonings [Page 5] are directly levelled against it, whate­ver Question there may be about the matter in Dispute, there can be none about the Intenti­on of the Writer. Should a Person, so know­ing and discreet as Mr. Walton, thwart and contradict Sir Isaac Newton under pretence of defending his Fluxions, and should he at eve­ry turn say such uncouth things of these same Fluxions, and place them in such odd lights, as must set all men in their Wits against them, could I hope for a better second in this Cause? or could there remain any doubt of his being a disguised Freethinker in Mathematics, who defended Fluxions just as a certain Freethinker in Religion did the Rights of the Christian Church.

III. MR. Walton indeed after his free man­ner calls my Analyst a Libel. ^* But this inge­nious Gentleman well knows a bad Vindicati­on is the bitterest Libel. Had you a mind, Sir, to betray and ridicule any Cause under the No­tion of vindicating it, would you not think it the right way to be very strong and dogmatical [Page 6] in the Affirmative, and very weak and puzzled in the argumentative Parts of your Perfor­mance? To utter Contradictions and Paradoxes without Remorse, and to be at no pains about reconciling or explaining them? And with great good humour to be at perpetual vari­ance with yourself and the Author you pre­tend to vindicate? How successfully Mr. Wal­ton hath practised these Arts, and how much to the honour of the great Client he would seem to take under his protection, I shall par­ticularly examine throughout every Article of his full Answer.

IV. FIRST then, saith Mr. Walton, ‘I am to be asked, whether I can conceive Velocity without Motion, or Motion without Exten­sion, or Extension without Magnitude?’ To which he answereth in positive Terms, that he can conceive Velocity and Motion in a Point (P. 7). And to make out this, he un­dertakes to demonstrate, ‘that if a thing be moved by an Agent operating continually with the same force, the Velocity will not be the same in any two different Points of the described Space. But that it must va­ry [Page 7] upon the least change of Space.’ Now admitting thus much to be demonstrated, yet I am still at a loss to perceive, how Mr. Walton's Conclusion will follow, to wit, ‘that I am greatly mistaken in imagining there can be no Motion, no Velocity in a Point of Space’ (P. 10). Pray, Sir, consider his Reasoning. The same Velocity cannot be in two Points of Space; therefore Velocity can be in a Point of Space. Would it not be just as good Rea­soning to say, the same man cannot be in two Nutshels; therefore a Man can be in a Nut­shel? Again, Velocity must vary upon the least change of Space; therefore there may be Velocity without Space. Make Sense of this if you can. What have these Consequences to do with their Premises? Who but Mr. Walton could have inferred them? Or how could even he have inferred them, had it not been in jest.

V. SUPPOSE the Center of a falling Bo­dy to describe a Line, divide the time of its Fall into equal Parts, for instance into minutes. The Spaces described in those equal parts of Time will be unequal. That is, from what­soever [Page 8] Points of the described Line you mea­sure a minute's descent, you will still find it a different Space. This is true. But how or why from this plain truth a Man should infer, that Motion can be conceived in a Point, is to me as obscure as any the most obscure myste­ries that occur in this profound Author. Let the Reader make the best of it. For my Part, I can as easily conceive Mr. Walton should walk without stirring, as I can his Idea of Motion without Space. After all, the Questi­on was not whether Motion could be proved to exist in a Point, but only whether it could be conceived in a Point. For, as to the proof of things impossible, some men have a way of proving that may equally prove any thing. But I much question whether any Reader of common Sense will undertake to conceive what this pleasant Man at Inference undertakes to prove.

VI. IF Mr. Walton really meant to defend the Author of the Fluxionary Method, would he not have done it in a Way consistent with this illustrious Author's own Principles? Let us now see what may be Sir Isaac's Notion, [Page 9] about this matter. He distinguisheth two sorts of motion, absolute and relative. The former he defineth to be a Translation from absolute place to absolute place, the latter from one re­lative place to another. ^* Mr. Walton's is plainly neither of these sorts of Motion, but some third kind, which what it is, I am at a loss to comprehend. But I can clearly com­prehend that, if we admit Motion without Space, then Sir Isaac Newton's Account of it must be wrong: For place by which he defines Motion is, according to him, a part of Space. And if so, then this notable Defender hath cut out new Work for himself to defend and explain. But about this, if I mistake not, he will be very easy. For, as I said before, he seems at bottom a back Friend to that great Man; which Opinion you will see further con­firmed in the Sequel.

VII. I SHALL no more ask Mr. Walton to explain any thing. For I can honestly say, the more he explains, the more I am puzzled. But I will ask his Readers to explain, by what Art a Man may conceive Motion without [Page 10] Space. And supposing this to be done, in the second place to explain, how it consists with Sir Isaac Newton's Account of Motion. Is it not evident, that Mr. Walton hath deserted from his old Master, and been at some pains to expose him, while he defends one Part of his Principles by overturning another? Let a­ny Reader tell me, what Mr. Walton means by Motion, or if he can guess, what this third kind is, which is neither absolute nor relative, which exists in a Point, which may be con­ceiv'd without Space. This learned Professor saith, ‘I have no clear Conception of the Principles of Motion’ (P. 24). And in a­nother place (P. 7.) he saith, ‘I might have conceived Velocity in a Point, if I had un­derstood and considered the nature of Moti­on.’ I believe I am not alone in not under­standing his Principles. For myself, I freely confess the Case to be desperate. I neither un­derstand them, nor have any hopes of being e­ver able to understand them.

VIII. BEING now satisfied, that Mr. Wal­ton's aim is not to clear up or defend Sir Isaac's Principles, but rather to contradict and expose [Page 11] them, you will not, I suppose, think it strange, if instead of putting Questions to this intrepid An­swerer, who is never at a loss, how often soe­ver his Readers may, I entreat you, or any o­ther Man of plain Sense, to read the following Passage cited from the thirty first Section of the Analyst, and then try to apply Mr. Wal­ton's Answer to it: Whereby you will clearly perceive what a vein of Raillery that Gentle­man is Master of. ‘Velocity necessarily im­plies both Time and Space, and cannot be conceived without them. And if the Velo­cities of nascent and evanescent Quantities, i. e. abstracted from time and space, may not be comprehended, how can we comprehend and demonstrate their Proportions? or con­sider their rationes primae & ultimae. For to consider the Proportion or Ratio of Things implieth that such Things have Magnitude: That such their Magnitudes may be measur­ed, and their Relations to each other known. But, as there is no measure of Velocity ex­cept Time and Space, the proportion of Ve­locities being only compounded of the direct proportion of the Spaces and the reciprocal Proportion of the Times; doth it not follow, [Page 12] that to talk of investigating, obtaining, and considering the proportions of Velocities, ex­clusively of Time and Space, is to talk un­intelligibly?’ Apply now, as I said, Mr. Walton's full Answer and you will soon find how fully you are enlightened about the Na­ture of Fluxions.

IX. IN the following Article of Mr. Walton's full Answer, he saith divers curious things, which, being derived from this same Principle, that motion may be conceived in a point, are altogether as incomprehensible as the Origine from whence they flow. It is obvious and na­tural to suppose Ab and Ba ^* to be Rectan­gles produced from finite lines multiplied by Increments. Mr. Walton indeed supposeth that when the Increments vanish or become no­thing, the Velocities remain, which being mul­tiplied by finite lines produce those Rectangles (P. 13.) But admitting the Velocities to re­main, yet how can any one conceive a Rect­angular [Page 13] Surface to be produced from a line mul­tiplied by Velocity, otherwise than by suppos­ing such line multiplied by a line or Increment, which shall be exponent of or proportional to such Velocity? You may try to conceive it otherwise. I must own I cannot. Is not the Increment of a Rectangle it self a Rectangle? must not then Ab and Ba be Rectangles? and must not the Coefficients or Sides of Rectangles be lines? consequently are not b and a lines or (which is the same thing) Increments of lines? these Increments may indeed be considered as proportional to and exponents of Velocity. But exclusive of such exponents to talk of Rectan­gles under lines and velocities is, I conceive, to talk unintelligibly. And yet this is what Mr. Walton doth, when he maketh b and a in the Rectangles Ab and Ba to denote mere Velo­cities.

X. As to the Question, whether nothing be not the Product of nothing multiplied by some­thing, Mr. Walton is pleased to answer in the affirmative. And nevertheless when ab is no­thing, that is, when a and b are nothing, he denies that Ab + Ba is nothing. This is one [Page 14] of these many Inconsistencies which I leave the Reader to reconcile. But, saith Mr. Walton, the Sides of the given Rectangle still remain, which two Sides according to him must form the Increment of the flowing Rectangle. But in this he directly contradicts Sir Isaac Newton, who asserts that Ab + Ba and not A + B is the Increment of the Rectangle AB. And, indeed, how is it possible, a line should be the Increment of a Surface? Laterum Incrementis totis a et b generatur Rectanguli incrementum Ab + Ba are the Words of Sir Isaac ^*. which words seem utterly inconsistent with Mr. Wal­ton's Doctrine. But, no wonder that Gentle­man should not agree with Sir Isaac, since he cannot agree even with himself; but contradicts what he saith elsewhere as the Reader may see, even before he gets to the End of that same Section, wherein he hath told us that ‘the Gnomon and the Sum of the two Rectan­gles are turned into those two Sides by a re­troverted Motion’ (P. 11 & 12). which [Page 15] proposition if you or any other Person shall try to make Sense of, you may possibly be convin­ced, that this profound Author is as much at variance with common Sense, as he is with himself and Sir Isaac Newton.

XI. MR. Walton in the ninth Page of his Vindication, in order to explain the Nature of Fluxions, saith that ‘to obtain the last ratio of synchronal Increments, the magnitude of those Increments must be infinitely diminish­ed.’ Notwithstanding which, in the twenty third Page of his full Answer he chargeth me as greatly mistaken, in supposing that he ex­plained the Doctrine of Fluxions by the ratio of Magnitudes infinitely diminished. It is an easy matter, for any Author to write so, as to betray his Readers into Mistakes about his meaning. But then it is not easy to conceive, what right he hath to upbraid them with such their Mistakes. If I have mistaken his Sense, let any one judge if he did not fairly lead me in­to the Mistake. When a Man puzzleth his Reader, saith and unsaith, useth ambiguous Terms and obscure Terms, and putteth them together in so perverse a Manner, that it is [Page 16] odds you can make out no sense at all, or if a­ny, a wrong sense, pray who is in fault but the Writer himself? let any one consider Mr. Walton's own words, and then say whether I am not justified in making this Remark.

XII. IN the twentieth Page of his full An­swer Mr. Walton tells us, that ‘Fluxions are measured by the first or last proportions of isochronal Increments generated or destroy­ed by motion.’ A little after he saith these Ratios subsist when isochronal Increments have no Magnitude. Now, I would fain know whether the isochronal Increments themselves subsist when they have no Magnitude? whe­ther by isochronal Increments we are not to understand Increments generated in equal times? whether there can be an Increment where there is no increase, or increase where there is no Magnitude? whether if Magnitudes are not ge­nerated in those equal times, what else is ge­nerated therein, or what else is it that Mr. Walton calls isochronal? I ask the Reader these Questions. I dare not ask Mr. Walton. For, as I hinted before, the Subject grows still [Page 17] more obscure in proportion as this able Wri­ter attempts to illustrate it.

XIII. We are told (P. 22.) ‘that the first or last ratio of the isochronal Spaces hath a real existence, forasmuch as it is e­qual to the ratio of the two motions of two points; which motions, subsisting when the isochronal Spaces are nothing; preserve the existence of the first or last ratio of these Spaces, or keep it from being a ratio of no­things.’ In order to assist your understand­ing, it must not be omitted that the said two points are supposed to exist at the same time in one point, and to be moved different ways without stirring from that point. Mr. Walton hath the Conscience to call this Riddle a full and clear Answer: to make sense of which you must suppose it one of his Ironies. In the next and last Article of his performance, you still find him proceed in the same Vein of Raillery upon Fluxions.

XIV. It will be allowed, that who ever se­riously undertook to explain the second, third, and fourth Fluxions of Sir Isaac Newton, would [Page 18] have done it in a way agreeable to that great Man's own Doctrine. What Sir Isaac's precise notion is I will not petend to say. And yet I will venture to say, it is something that cannot be explained by the three dimensions of a Cube I frankly own, I do not understand Sir Isaac's Doctrine so far as to frame a positive Idea of his Fluxions. I have, nevertheless, a nega­tive conception thereof, so far as to see that Mr. Walton is in jest, or (if in earnest) that he understands it no more than I do.

XV. Sir Isaac tells us that he considers in­determinate quantities as flowing, or in other words, as increasing or decreasing by a perpetu­al motion. Which quantities he denotes by the latter Letters of the Alphabet, and their Fluxions or Celerities of increasing by the same Letters pointed over head, and the Fluxions of Fluxions or second Fluxions, i. e. the Muta­tions more or less swift of the first Celerities by the same Letters pointed with double points; and the Mutations of those Mutations of the first Mutations or Fluxions or Celerities of in­creasing, which he calls Fluxions of Fluxions of Fluxions or third Fluxions, by three [Page 19] points; the fourth Fluxions by four points; the fifth by five; and so on ^*. Sir Isaac, you see, speaks of quantity in general. And in the Analyst the Doctrine is exemplified and the Case is put in lines. Now in lines, where there is only one Dimension, how are we enabled to conceive second, third or fourth Fluxions by conceiving the generation of three dimensions in a Cube? Let any one but read what Sir I­saac Newton or what I have said, and then ap­ply what Mr. Walton hath written about the three dimensions of a Cube, and see whether the difficulties are solved or the Doctrine made one whit the clearer by this Explication.

XVI. That you may the better judge of the merit of this Part of Mr. Walton's performance, I shall beg leave to set down a Passage or two from the Analyst. ‘As it is impossible to conceive Velocity without time or space, without either finite length or finite duration, it must seem above the Power of Man to comprehend even the first Fluxions. And if the first are incomprehensible, what shall [Page 20] we say of the second and third Fluxions, &c. He who can conceive the beginning of a be­ginning or the end of an end, somewhat be­fore the first or after the last, may perhaps be sharpsighted enough to conceive these things. But most Men, I believe, will find it impossible to understand them in any sense whatsoever. One would think that Men could not speak too exactly on so nice a sub­ject. And yet we may often observe, that the exponents of Fluxions or notes represent­ing Fluxions are confounded with the Flux­ions themselves. Is not this the Case, when just after the Fluxions of flowing quantities, were said to be celerities of their increasing and the second Fluxions to be the Mutati­ons of the first Fluxions or celerities, we are told that [...] represents a series of quantities whereof each subsequent quantity is the Fluxion of the preceding; and each foregoing is a fluent quantity having the following one for it's Fluxion. Divers seri­es of quantities and expressions Geometrical and Algebraical may be easily conceived in lines, in surfaces, in species, to be continu­ed without end or limit. But it will not be [Page 21] found so easy to conceive a series, either of mere Velocities or of mere nascent Incre­ments, distinct therefrom and corresponding thereunto.’ ^* Compare what is here said with Mr. Walton's Genesis of a Cube, and you will then clearly see how far this answerer is from explaining the nature of second, third and fourth Fluxions: And how justly I might repay that Gentleman in kind, and tell him in his own language, that all his Skill is vain and impertinent, (vind. p. 36).

XVII. BUT it doth not become me to find fault with this learned Professor, who at bot­tom militates on my Side, and in this very Section, makes it his business directly to over­throw Sir Isaac Newton's Doctrine. For he saith in plain Terms, that there can be no fourth Fluxion of a Cube (P. 25.) that is, there can be no second Fluxion of a line, and a fortiori, no third, fourth, fifth, &c. Inso­much that with one single dash of his Pen Mr. Walton destroys, to the great relief of the learned World, an indefinite rank of Fluxions [Page 22] of different Orders that might have reached from Pole to Pole. I had distinctly pointed out the difficulties in several Parts both of my Analyst and Defence, and I leave you to judge whether he explains or even attempts to explain one of them. Instead thereof he tells us of the true Dimension of a Cube generated by Moti­on: Whence he takes occasion, as hath been observed, to explode Sir Isaac's own Doctrine, which is utterly inconsistent with Mr. Walton's And can you now doubt the real design of this egregious Vindicator.

XVIII. BEFORE ever Sir Isaac Newton thought of his Fluxions, every body knew there were three Dimensions in a Cube, and that a Solid might be generated by the motion of a Surface, a Surface by the motion of a Line, and a Line by the motion of a Point. And this in effect is all we know from Mr. Walton's Explication. As for his dwelling so minutely on the Genesis of the solid Parts of a Cube, a thing so foreign from the Purpose, the only rational Account I can give of it is, that Mr. Walton, by puzzling the Imagination of his vulgar Readers, hoped the better to dis­guise [Page 23] his betraying the Doctrine of his great Client, which to a discerning eye he manifest­ly gives up; and instead thereof humourously substitutes, what all the World knew before Sir Isaac was born, to wit, the three Dimensi­ons of a Cube and the genesis thereof by Mo­tion.

XIX. UPON the whole I appeal to you and every intelligent Reader, whether this thing, which Walton is pleased ironically to call a full Answer, doth not carry through­out a fly Insinuation, that the profound Sci­ence of Fluxions cannot be maintained but by the help of most unintelligible Paradoxes and Inconsistencies. So far, indeed, as Affirmati­ons go he sheweth himself an able Support of Sir Isaac Newton. But then in his Rea­sonings he drops that great man upon the most important Points, to wit, his Doctrine of Mo­tion and his Doctrine of Fluxions, not regard­ing how far the demonstration of his famous Principia is interested therein. To convince you still more and more of the Truth hereof, do but reflect a little on Mr. Walton's Conduct. Can you think it probable, that so learned and [Page 24] clear-headed a Writer would have laid down such a direct repugnancy to common Sense, as his Idea of Motion in a Point, for the ground work of his Explanation, had it been his real Intention to explain? Or can you suppose, he would have been absolutely silent, on so many Points urged home, both in the Analyst and Defence, which it concerned a Vindicator of Sir Isaac not to have overlooked? Can you ima­gine, that if he meant seriously to defend the Doctrine of Fluxions, he would have content­ed himself with barely asserting that ‘Sir Isaac Newton in the Introduction to his Quadra­ture of Curves, in the second Lemma of the second Book, and in the Scholium to the first Section of the first Book of his Princi­ples of Philosophy, hath delivered his Doc­trine of Fluxions in so clear and distinct a manner, without the least Inconsistency in terms or Arguments, that one would have thought it impossible for any Person not to have understood him’ (P. 30).

XX. IS it possible, I say, that Mr. Walton could in earnest hope we should take his bare Word, as so much more credible that Sir I­saac's, [Page 25] and not rather have endeavoured to an­swer the Questions and reconcile the Difficul­ties set forth in my Defence of Free-thinking, for instance, in Sect xxxvi. Wherein I intreat my Antagonist to explain ‘whether Sir Isaac's Momentum be a finite Quantity or an In­finitesimal or a mere Limit, adding, 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 ostendere quod in methodo Fluxionum non opus sit figuras infinite 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 & B deerant momentorum dimidia, &c. where the Moments are supposed to be divided.’ I shall scarce think it worth my while to bestow a serious thought on any Writer who shall pre­tend [Page 26] to maintain Sir Isaac's Doctrine, and yet leave this Passage without a Reply. And the Reader, I believe, will think with me that, in answer to difficulties distinctly proposed and in­sisted on, to offer nothing but a magisterial As­sertion is a mere grimace of one who made merry with Fluxions, under the Notion of de­fending them. And he will be further con­firmed in this way of thinking, when he ob­serves that Mr. Walton hath not said one Syl­lable, in Reply to those several Sections of my Defence, which I had particularly referred to, as containing a full answer to his Vindication. But it is no wonder if, with Sir Isaac's Doc­trine, he should drop also his own Arguments in favour thereof.

XXI. I HAVE been at the Pains once for all to write this short Comment on Mr. Wal­ton, as the only way I could think of for mak­ing him intelligible, which will also serve as a Key to his future Writings on this Subject. And I was the rather inclined to take this trou­ble, because it seemeth to me, there is no part of Learning that wants to be clear'd up more than this same Doctrine of Fluxions, which hath hi­therto [Page 27] walked about in a mist to the Stupefac­tion of the Literati of the present Age. To conclude, I accept this Professor's Recantation, nor am at all displeased at the ingenious me­thod he takes to disguise it. Some zealous Flu­xionist may perhaps answer him.