I Freely own to you, when I ſat down to write my defenſe of Sir Iſaac Newton and the Bri⯑tiſh Mathematicians, I was not a little moved at the treatment you had been pleaſed to give to one or two of thoſe Great Men, whom I am proud to call my Maſter, and whoſe memories on that ac⯑count I ſhall always reverence and honour. But now, you tell me, I may be ſuppoſed cool. I am ſo: partly through the length of time that has intervened; and partly by conſider⯑ing the ſeverity of the diſcipline you have undergone. It has had, I ſee, a marvellous effect upon you. One may plainly perceive an alteration, notwithſtanding your endea⯑vours to conceal it, not only in your ſenti⯑ments, [2] but in your language, your behaviour, and your very air. You no longer breathe that ſuperiority and contempt of all mankind you were wont to ſhew. This change in you has greatly mitigated the paſſion with which I was before overcome.

In this calm and cool ſtate therefore when I reflect upon what is paſt, I am not a little ſtartled at my own audaciouſneſs and pre⯑ſumption, in entring the liſts againſt ſo re⯑doubtable an adverſary as the Author of the Minute Philoſopher. To you, likewiſe, I find, this preſumption of mine appeared ſo extraordinary, that though you are ſo good as to qualify it by the ſofter name of courage, you could not but admire it, it ſeemed unaccount⯑able to you, till you reflected on my ſeeming ſe⯑cure in the favour of one part of my readers, and the ignorance of the other.

Nevertheleſs you are perſuaded there are fair and candid men among the Mathematicians. I likewiſe am perſuaded not only that there are fair and candid men among the Mathemati⯑cians; but that generally ſpeaking Mathema⯑ticians are fair and candid men. What ſhould make them otherwiſe? Are their opinions di⯑geſted into Creeds and Articles, and eſtabliſh⯑ed by Law? Has the Publick thought fit to [3] beſtow dignities and large poſſeſſions on them, which are not to be obtained without embracing thoſe opinions, nor to be retained without perſevering in them? Were even this the Caſe; yet ſurely there would be found among them fair and candid men. I am ſure I know many ſuch among another ſet of men in theſe very circumſtances.

Ay, but this other ſet of men, you will ſay, have nothing but truth to defend. I grant it. And I take this to be the caſe of the Mathematicians likewiſe. They are at leaſt as good reaſoners as any other ſet of men whatſoever, and conſequently are as likely to know truth, when they meet it, and have nothing to hinder them from embra⯑cing it. It is therefore on their judgment, not their favour, that I depend.

But you ſpeak of the ignorance of the other part of my Readers. Alas! Sir, of what ad⯑vantage can that be to me? To me, Phila⯑lethes, who aim at truth alone, who have no intereſt in deceiving them?

Were I indeed the Author of the Minute Philoſopher: Had I any other end to ſerve than truth: Were I maſter of aſſurance e⯑nough to miſlead my reader at the inſtant that I call out to him to mind his way; to [4] deſire him to examine, while I am miſin⯑forming him; to throw duſt in his eyes, and bid him ſee: then undoubtedly much might be done. But theſe are arts I neither need nor practiſe. I content my ſelf with the plain honeſt way of giving my Reader the beſt light I can, neither miſleading him my ſelf, nor ſuffering him to be miſled by others. With this view it is that I divide my reply into the ſame number of ſections with your defenſe, and confine my ſections to the ſame matter with yours. This will indeed make what I have to ſay ſomewhat leſs methodical, but then it will enable the Reader more ea⯑ſily to compare us together, and to make a more certain deciſion between us.

The taking this method, Sir, will make it plainly appear, that what I aim at is only manifeſting the truth; and conſequently that the reaſon of my courage in encountering you, is my being verily perſuaded that I have truth and juſtice on my ſide. I know and am aware of your ſuperior accompliſhments: But Philalethes is my name: And truth will prevail againſt the pens of men or angels. Your vanity has engaged you in a difficulty, from which all your abilities ſhall never ex⯑tricate you.

Your arms are wedged in the oak you have preſumptuouſly attempted to rend: Your ſtrength is no longer of any uſe to defend you: A woman, a child may be too hard for you.

II. Your ſecond ſection teaches us, 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 Mathe⯑maticks, than in any other part of Learning. I agree with you. I go farther. It can never be a crime, but on the contrary is highly lau⯑dable, to canvaſs and to examine the princi⯑ples and reaſoning made uſe of in any ſcience whatſoever, and that with the utmoſt free⯑dom and impartiality. All ingenuous minds will be pleaſed with ſuch an examination: They will readily conſent that the ſcience they profeſs, be brought to the ſevereſt and ſtricteſt trial. Truth can never be hurt by Inquiry: Truth loves the light: But error, falſhood and impoſture dread and abhor it.

[6] III. I am much at a loſs here. You ſpeak ofmen who reject that VERY THING in Reli⯑gion which they admit in human Learning. Do they admit that Fluxions are to them moſt in⯑comprehenſible Myſteries? Do they, notwith⯑ſtanding this conceſſion, believe them to be clear and ſcientifick? Do they, notwithſtand⯑ing this belief, entertain an implicit faith in the Author of that Method? Theſe things ſeem hard to reconcile.

IV. I do not aſk, Why you choſe to de⯑fame Mathematicians in the month of March, Ann. Dom. 1734, rather than at any time be⯑fore? The only queſtion with me was, Whe⯑ther Vanity or Chriſtianity were the motive to writing the Analyſt. Quae relligio aut quae machina belli? I have fully proved there was no Religion, no Chriſtianity in it. It was partly Vanity, partly Machine.

V. Here I would obſerve, that whoever admires Fluxions, muſt admire them for ſomething of excellence he ſees in the Me⯑thod of Fluxions, and conſequently cannot juſtly be ſaid to yield Faith to the Inventor of that Method. But this whole ſection ſeems [7] to me to be matter of ſecret hiſtory and de⯑clamation of the worſt ſort, namely, the de⯑famatory.

VI. More ſecret hiſtory and declamation, partly about what no body denies, and part⯑ly about what no body believes. You give us to underſtand, you have a right to examine Fluxions, even though Religion were quite un⯑concerned, and though you had no end to ſerve but Truth. No body diſputes your right of examining: but ſurely no good can be ex⯑pected from the examination of a Perſon who has any end to ſerve but Truth, let that end be what it will. But pray what is this other end, this end different from Truth, that you have to ſerve? It looks as if Religion were meant. But I hope better things of you, a Chriſtian, and a Preacher of the Go⯑ſpel. Truth and the Chriſtian Religion are one. I profeſs I am greatly puzzled. I have taken as much pains to underſtand this paſſage as, I ſincerely believe, you have done to make ſenſe of Sir Iſaac Newton's principles. A friend of mine is of opinion the paſſage has been corrupted either by the tranſcribers or the printer, and bids me for Religion read Promotion. Ita legendum cenſeo, ſays he, re⯑clamantibus [8] centum Tonſonis. I am apt to think he is in the right, partly becauſe I take him to be a very able Critick, and partly be⯑cauſe this emendation, though conſiderably differing from the vulgar reading, ſerves to confirm the proof I had before given, that your end was neither Truth nor Religion.

You tell me I am very angry, and refer to page 13 and 14 of my Defenſe. I have looked over thoſe pages to ſee what ſigns of anger I have there ſhown, what injury, what affront I have there offered you. All I can find is, that I have propoſed to you the ex⯑ample of our Saviour and of St. Paul. I beg your pardon, Sir: I took you for one of their followers.

You will not take upon you to ſay you know me to be a Minute Philoſopher. I am much obliged to you for this tenderneſs, and ſhould be more ſo, if it appeared that you knew ſo much as one letter of my name. But it ſeems, you would not be concerned if others ſhould take me to be ſuch a one. You ſpeak of my ſpleen againſt the Clergy; and you tell me the Minute Philoſophers make juſt ſuch com⯑pliments as I do to our Church. Here I appre⯑hend you miſtake the compliments I make to yourſelf, and a few of your credulous [9] friends, for compliments to the body of the Clergy. I aſſure you, I look upon the body of the Clergy as a body of learned and uſe⯑ful men. I know and am known to a great many individuals among them, whom I highly eſteem and honour: I have ſpoke of ſome of them in my Defence with ſingular reſpect. If I laugh at any, it is at ſuch as think you do ſervice to the Church in wri⯑ting the Analyſt: If I diſlike any, they are ſuch as are perpetually graſping at dominion and riches. No wonder. I am a Layman. If the Clergy obtain more power, I ſhall have leſs liberty: If they will have more wealth, I am one of thoſe muſt pay to it.

VII. The chief purport of this ſection ſeems to be to ſtrengthen the proof you had before given of the infidelity of Mathema⯑ticians. You had told us in the Analyſt, you were not a ſtranger to it: It was known: You were credibly informed. Now you go far⯑ther. You make no doubt of it: You have ſeen ſhrewed ſigns: You have been VERY credibly in⯑formed. Can any thing be plainer? I declare myſelf fully ſatisfied with this proof, even without the ſtory told you by Mr. Addiſon, of a witty man who was an Infidel, becauſe [10] of the infidelity of a certain noted Mathema⯑tician. Surely this witty man was in jeſt; at leaſt he was no wiſe man.

VIII, IX, X. In theſe three ſections I meet with nothing but declamation. The ſubject of it is my paſſion and injuſtice, my railing and raging, my rhetorick and writing tragedy; your own ſincerity and laudable en⯑deavours to do ſervice to mathematical learn⯑ing; the proper reſpect you treat Sir Iſaac Newton with, and the decency with which you diſſent from him. For which laſt the reader is deſired to have recourſe to the Ana⯑lyſt, particularly to the thirty-firſt Query, where Sir Iſaac Newton is plainly charged with writing nonſenſe.

As to my frightful viſions and tragical up⯑roars about the Inquiſition and the Gallows, you may laugh at them as much as you pleaſe: But I have heard of perſons hanged and burnt upon as ſlender evidence as that which you bring againſt Mathematicians. And what has been, may be: Eſpecially if the wholſome, ancient diſcipline ſhould ever be reſtored, which ſome perſons ſay is much to be wiſhed. I confeſs I am not of their mind: And I hope the body of the Britiſh [11] Laity ſee too plainly the uſe that would be made of ſuch a power, ever to truſt you Gen⯑tlemen with it.

XI. You ſay, you heartily abhor an Inqui⯑ſition in Faith. Upon my word you have a great deal of reaſon. You have been a grie⯑vous Free-thinker in your time: I do not mean in Mathematicks only. As great a Bi⯑got as I am, poſſeſſed with the true ſpirit of an Inquiſitor, I aſſure you, I ſhould be very ſorry that you and I were at the mercy of ſome men I could name. They ſeem to me to be ſingularly well qualify'd to preſide in the holy Office, and I doubt they would make us confeſs that ſomething elſe exiſted in Nature beſides SPIRIT AND IDEAS.

XII. More declamation about my declaim⯑ing, and your own Modeſty, and the compli⯑ment you pay to Sir Iſaac Newton's Under⯑ſtanding. But, Sir, I don't like that word Sophiſm. It ſeems not very conſiſtent with the decency and proper reſpect you ſo lately talked of.

XIII, XIV, XV, XVI. You tell me, The adoration that I pay to Sir Iſaac Newton, you will pay only to truth: That I may be an Idolater of whom I pleaſe; but I have no right to inſult and exclaim at other men, becauſe they do not adore my Idol: That I inveigh againſt you, becauſe you are not guilty of my mean Ido⯑latry.

Now give me leave to ask you a queſtion. Do you really and bona fide believe that I pay idolatrous worſhip to Sir Iſaac Newton, that I make him the object of that adoration which you ſay you will pay only to truth, and which I will pay only to the God of truth? And this becauſe I apply to Sir Iſaac Newton, a Verſe which an inferior Poet appli⯑ed to Virgil? Is adorare veſtigia to be literal⯑ly taken, think you? What can be meant by theſe veſtigia? The mark of his foot in Crane Court? Or the truths diſcovered by him? If the laſt; to what purpoſe all this [13] declamation, and ridiculous rant about Ido⯑latry for four ſections together?

You ſeem to diſlike my profeſſing that the higheſt honour I can ever arrive at, or even deſire, is in any the loweſt degree to imitate Sir Iſaac Newton's example. You think It might have ſuited better with my appellation of Philalethes, and been altogether as laudable, if my higheſt ambition had been to diſcover truth. Why ſo it is. The diſcovering of truth, and his clear, candid, humane way of making it known to Mankind, is the very thing in which I ſhould deſire to imitate Sir Iſaac Newton.

You ſay, I ſpeak of it as a ſort of crime to think it poſſible I ſhould ever ſee farther, or go beyond Sir Iſaac Newton. But there are others who think it no crime to deſire to know not only beyond Sir Iſaac Newton, but beyond all Man⯑kind. You intimate your ſelf to be one of theſe. Now, Sir, I am for ſeeing as much beyond Sir Iſaac Newton as you can be: But firſt let us ſee as far as he has done. I agree with you in this deſire of knowledge; make it as un⯑bounded as you pleaſe. I aſſure you I think [14] it no crime. The only difference between us is this. You ſeem to think you have this knowledge already. I am ſenſible I have it not.

I make no doubt but ſuch a Man as you, or one much inferior to you, may carry a particular point, or many particular points, farther than Sir Iſaac Newton has done. But that ſuch a Man as Sir Iſaac Newton, af⯑ter long conſideration of one thing, after touching and retouching it at different times for above half a Century, after ſetting it in ſeveral various lights, after applying it in an infinite number of examples, after giving ſeveral different demonſtrations of it, ſuch as had ſatisfied all the Mathematicians in Eu⯑rope, ſhould all this while have taken error for truth, and given Sophiſms for demon⯑ſtrations, and thereby deceived all the world except my dear Friend the Author of the Minute Philoſopher, is what muſt be very clearly made out before I believe it.

XVII. You begin this ſection with ad⯑dreſſing your ſelf to me in the following manner. "I have ſaid (and I venture ſtill to [15] 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 infinite⯑ſimal: That it is yet leſs poſſible to con⯑ceive an infiniteſimal of an infiniteſimal, and ſo onward. What have you to ſay to this?" Truly very little. Only I don't well comprehend, how one incomprehenſi⯑ble can be MORE incomprehenſible than another incomprehenſible: How it can poſ⯑ſibly be LESS poſſible to conceive one thing than to conceive another thing, which other thing it is not at all poſſible to conceive.

For clearing up theſe aſſertions I have had recourſe, purſuant to your directions, to the fourth ſection of the Analyſt, which is the only one relating to fluxions, of the three you refer me to. But all the ſatisfaction I there meet with is, That your imagination is very much ſtrained and puzzled with one thing; That it ſeems ſtill more difficult to con⯑ceive another thing; That a third ſeems an obſcure myſtery; That a fourth exceeds, if you miſtake not, all human underſtanding; That take another in what light one pleaſes, the clear conception of it will, if you miſtake not, be found impoſſible. This to me ſeems to [16] amount to thus much. The Author of the Minute Philoſopher cannot comprehend the principles of fluxions: Therefore no man living can comprehend them. He cannot underſtand them: Therefore they exceed all human underſtanding. A notable proof of my Hypotheſis, that that Gentleman has too good an opinion of himſelf, and too mean a one of all other men.

You go on addreſſing your ſelf to me; Do you attempt to clear up the notion of a fluxion? Nothing like it. Very true, nor did I ever undertake it. Sir Iſaac Newton has done it incomparably well to thoſe who are qualified to read his Works, and thither I refer you. May not I expoſe your blunders, without pretending to explain his doctrine better than he has done it himſelf?

But you tell me, I only aſſure you (upon my bare word) from my own experience, and that of ſeveral others whom I could name, that the doctrine of fluxions may be clearly conceived and diſtinctly comprehended. Why pray, Sir, what did you require more? You appealed to the trial of every thinking reader. I am one of your thinking readers. I have made the trial you deſired. I acquaint you with the reſult of that trial, and all the return you make [17] me is, Can you think I will take your word when I refuſe to take your Maſter's? I appeal to all my thinking readers whether this be civil uſage. You ſay, you don't underſtand fluxions. I ſay I do. I believe you: And yet you won't believe me.

This, Sir, my judgment tells me is all the anſwer I ought to make to the invitations you ſo frequently give me upon this head. I am ſenſible it were better to hold ſo ſlippery an adverſary to the points we have already in hand, than before theſe are ſettled, to go up⯑on new matter. Beſides, I am afraid of incur⯑ring the common fate of Sir Iſaac Newton's interpreters, to be leſs intelligible than my Maſter. I apprehend likewiſe that, let me take ever ſo much pains to ſatisfy and oblige you, I ſhall meet with no better uſage than when you appealed to me: That all the re⯑turn I am to expect is, Alas! I find no ſenſe or reaſon in what you ſay. And yet I am ſo deſirous of contributing my aſſiſtance to⯑wards your laudable deſign of putting this controverſy in ſuch a light as that every reader may judge thereof, that I think I muſt run that hazard. But I deſire it may be remem⯑bred that I do not here intend, nor indeed think my ſelf at all qualified to write a com⯑plete [18] treatiſe of Fluxions, that being ex⯑pected from better hands. All that you re⯑quire of me is to ſhew that the principles of Fluxions may be clearly conceived. This therefore is what I ſhall endeavour to do, and in order to render thoſe principles as intelli⯑gible as I can, I ſhall make uſe of the plain⯑eſt and eaſieſt example poſſible, that I may give my Reader no other trouble than only that of comprehending the principles them⯑ſelves.

The foundation of the Method of Fluxions I take to be contained in the following

XVIII. I am not of your opinion, that every reader of common ſenſe may judge as well of the principles of Fluxions as the moſt pro⯑found Mathematician. How well the moſt profound Mathematician can judge, can, I think, be certainly known to the moſt pro⯑found Mathematician only, and I am ſure I am not the man. Conſequently I cannot take upon me to pronounce upon this point with the ſame aſſurance and certitude that you ſeem to do. But this I well know, that Sir Iſaac Newton did not write for every rea⯑der of common ſenſe. He wrote for Mathe⯑maticians.

[32] Nor can I agree with you that the ſimple apprehenſion of a thing defined is not, ſome⯑times at leaſt, made more perfect by any ſub⯑ſequent Progreſs in Mathematicks. It hap⯑pened to me, and I believe it happens to all or moſt other Beginners in Geometry, that the definitions of an angle, of a figure, of parallel lines, and of proportion, all be⯑come clearer upon ſeeing the application of the things defined in different examples, than upon only reading and conſidering thoſe de⯑finitions with what care and attention ſoever.

XIX. I will venture to ſay that you have taken as much pains as (I ſincerely believe) any man living, except a late Philoſopher of our Univerſity, to make nonſenſe of Sir Iſaac Newton's principles. Your ſucceſs indeed has been equally bad with his: But that is not your fault, but your misfortune. I muſt needs ſay, you have done pretty well, con⯑ſidering you never had a Maſter in Mathema⯑ticks.

XX. I find by this Section as well as by the eighteenth, that you are perfectly well [33] acquainted with what may, or may not be done, by any progreſs, though ever ſo great, in the Analyſis, by the beſt of Mathematicians, by the moſt profound Analyſt. Such a man as you, one would think, might give one a little light into ſome very ſtrange things I meet with towards the latter end of this ſection, ſuch as velocity without motion, mo⯑tion without extenſion, magnitude which is nei⯑ther finite nor infinite, a quantity having no magnitude which is yet diviſible, a figure where there is no ſpace, proportion between nothings, and a real product from nothing multiplied by ſomething. To me, I muſt own, theſe ſeem to be Myſteries utterly incom⯑prehenſible; but then I take them to be Myſte⯑ries of your own making: I can find no more ſign of them in Sir Iſaac Newton's wri⯑tings, than of Tranſubſtantiation and ſome other Myſteries in the New Teſtament.

XXI. The Picture you here draw is really a very ingenious portraiture, but it has no manner of reſemblance to Sir Iſaac Newton. I ſhould ſooner have taken it for a picture of Bellarmine, or for a handſome likeneſs of the Author of the Minute Philoſopher drawing up an anſwer to Philalethes. A man driven [34] to arts and ſhifts in order to defend his prin⯑ciples, can hardly take them for true, muſt entertain more than ſome doubt thereof. For inſtance, let any man breathing obſerve the arts and ſhifts you make uſe of throughout your anſwer, and he will plainly ſee you are convinced of your being in an error, but will not own it.

XXII. A new way of paſſing over a thing is never to have done with it. The reader will eaſily judge who colours moſt, is moſt clamorous, reproaches moſt and reaſons leaſt.

XXIII. In the fourth ſection of the A⯑nalyſt, inſtead of fairly giving Sir Iſaac Newton's plain, eaſy, intelligible definition of a ſecond Fluxion, you are pleaſed to lay down three or four definitions of your own, as obſcure, myſterious and abſurd as you can poſſibly deviſe.

After which you appeal to the trial of every thinking reader, whether the clear Conception of them is not impoſſible. This I had taken [35] notice of as a pious art of miſleading and confounding your reader, inſtead of inſtruct⯑ing him, and had put the two following queſtions to you, which I ſhall here tran⯑ſcribe at large; becauſe with another pious art you have thought fit to truncate the one, and to leave out the other, for particular reaſons which I ſhall by and by lay before the reader. Where, ſaid I, do you find Sir Iſaac Newton uſing ſuch expreſſions as the velocities of the velocities, the ſecond, third and fourth velocities, the incipient ce⯑lerity of an incipient celerity, the naſcent aug⯑ment of a naſcent augment? Is this the true and genuine meaning of the words fluxionum mutationes magis aut minus celeres?

To theſe two Queſtions you are ſenſible it is incumbent upon you to ſeem to give an anſwer, and you are likewiſe ſenſible you have none to give. In this perplexity it is wor⯑thy the obſervation of a curious reader to ſee what arts and ſhifts a great Genius may be driven to in grappling with an inſuperable diffi⯑culty.

In the firſt place you curtail my firſt que⯑ſtion, cutting off the latter part of it with an &c. By this means you hide from your rea⯑der one of your definitions, and that the leaſt juſtifiable of them all, namely the incipient [36] celerity of an incipient celerity. There's one difficulty cleverly got over.

In the next place you entirely cut off my ſecond queſtion. In which I find you have two advantages. The firſt is to avoid giving an anſwer to it. The ſecond, not to let your reader ſee Sir Iſaac Newton's definition, which I had inſerted into that queſtion.

But ſetting all this aſide, after you have propoſed my queſtion in your own manner, what anſwer do you give to it? Do you ſhew me where Sir I. N. uſes ſuch expreſ⯑ſions? No. You don't pretend to it. What then? Why truly you endeavour to ſhew, by comparing together two independent Paſ⯑ſages taken from two different treatiſes of Sir I. N. that you may juſtifiably call a ſe⯑cond fluxion ſo and ſo. Be it ſo: Though I think otherwiſe. Yet ſtill this will only ſhew that a definition of your own may be uſed; but will not ſhew it to be Sir Iſaac Newton's definition, nor to be equally clear with Sir Iſaac Newton's definition. There⯑fore the pious art I at firſt mentioned, ſtill ſubſiſts with the addition of two or three more pious arts to ſupport it, as it generally happens when ſuch arts come to be exami⯑ned into by any of our family of Philalethes.

[37] In order to get out of this Egyptian dark⯑neſs in which you have ſtudiouſly involved the matter in debate, as well as to complete what I had begun in my ſeventeenth ſection towards clearing up the firſt principles of fluxions, I ſhall now endeavour to give my reader and you too, Sir, if you pleaſe, a clear and intelligible conception not only of ſecond, but of third, fourth and fifth fluxi⯑ons, &c. ad infinitum, upon the foot of Sir Iſaac Newton's definition of ſecond fluxions.

That great Man making uſe of the liberty which has always been allowed to Inventors, of giving new names to new conceptions, and of defining thoſe names as they thought fit, has been pleaſed to call by the name of fluxion, the velocity with which a flowing quantity increaſes or decreaſes. If this velo⯑city do not always continue the ſame, but undergo any change, the velocity of that change is called a fluxion of a fluxion, or a ſe⯑cond fluxion; and as the change is ſwifter or ſlower, the ſecond fluxion is ſaid to be greater [38] or leſs. For inſtance, if the firſt fluxion or velocity of the flowing quantity continually increaſe, the ſecond fluxion is the velocity with which the firſt velocity increaſes, and is proportional to the momentaneous increaſe of that firſt velocity.

In like manner the third fluxion is the ve⯑locity of the change of the ſecond fluxion; the fourth fluxion is the velocity of the change of the third; the fifth the velocity of the change of the fourth, &c. ad infini⯑tum.

Here perhaps it may not be amiſs to aſſiſt the reader's imagination by repreſenting the proportions betwen fluxions of all the ſeveral orders in a ſenſible manner. I ſay their pro⯑portions: for, as I ſaid before, Sir Iſaac Newton makes no enquiry into, nor ever conſiders the abſolute magnitude of fluxions, or moments, or naſcent increments, but on⯑ly the proportion between them. And this I deſire may be carefully remembred.

[39] Let A be a flowing line, and let the velocity with which it flows, be always repreſented by the line 1F. Then if the line A flow uniformly, that is, if the velocity with which it flows, do never change or alter; the line 1F will be a conſtant quantity; and the line A will have only a firſt fluxion and no ſecond fluxion. But if the line A flow with an accelerated velocity, that is, if the velocity with which it flows, do continually increaſe; the line 1F will be a flowing line; and the fluxion of this line 1F, or the ve⯑locity with which that line flows, will be the fluxion of the fluxion 1F, or the ſecond fluxion of the line A.

Now let the velocity with which this line 1F flows, be always repreſented by the line 2F. Then if the line 1F flow uniformly, or the velocity with which it flows, do ne⯑ver change or alter; the line 2F will be a conſtant quantity; and the line 1F will have only a firſt fluxion, and no ſecond fluxion: And the Line A will have a firſt and ſecond fluxion, but no third Fluxion. But if the line 1F flow with an accelerated velocity, or the velocity with which it flows, do conti⯑nually increaſe; the line 2F will be a flowing line; and the fluxion of this Line 2F, or the [40] velocity with which it flows, will be the fluxion of the fluxion 2F, or the ſecond fluxion of the fluxion 1F, or the third fluxion of the line A: And this fluxion or velocity may be repreſented by the line 3F.

In like manner 4F may repreſent the firſt fluxion of 3F, the ſecond fluxion of 2F, the third fluxion of 1F, and the fourth fluxion of the line A. And it is viſible that after this manner we may proceed ad infini⯑tum.

Obſerving the ſame analogy, let 1f repre⯑ſent the firſt fluxion of the flowing line a: 2f, the firſt fluxion of 1f, or the ſecond fluxion of the line a: 3f the firſt fluxion of 2f, the ſecond fluxion of 1f, or the third fluxion of a: 4f the firſt fluxion of 3f, the ſecond fluxion of 2f, the third fluxion of 1f, or the fourth fluxion of a: &c. ad in⯑finitum.

Then is it manifeſt that the proportion between the firſt fluxion of A and the firſt fluxion of a, will be the ſame as that of the two finite lines 1F and 1f: The proportion between the two ſecond fluxions of A and a, will be the ſame as that of the two finite lines 2F and 2f: The propor⯑tion between the third fluxions will be that [41] of the finite lines 3F and 3f: The propor⯑tion of the fourth fluxions that of 4F and 4f, &c. to infinity.

From this methinks it follows, that ſecond, third and fourth fluxions are not more incom⯑prehenſible than a firſt fluxion.

XXIV, XXV, XXVI. I do not remem⯑ber to have met with a greater inſtance of diſtingenuity and wilful miſrepreſentation in any controverſy I have ever looked into, than what the reader will obſerve to run through theſe three ſections. You had in the Analyſt charged the Mathematicians with unjuſtly omitting a certain rectangle in their computation of the increment of the rectan⯑gle of two flowing quantities, and thereupon had thought fit to repreſent them as not pro⯑ceeding ſcientifically, as not ſeeing their way diſtinctly, as proceeding blindfold, as arriving at the truth they know not how nor by what means, with abundance of the like compli⯑ments plentifully diſperſed all over the Analyſt.

To this I had replied, Firſt, that this omiſſion, at the worſt, could not cauſe them to deviate from the truth the leaſt imaginable quantity, in computing the moſt immenſe [42] magnitude: Secondly, that as they clearly ſaw and could plainly demonſtrate this inſig⯑nificancy of the omiſſion, they could not juſtly be ſaid to proceed blindfold: Thirdly, that this pretended error or omiſſion of theirs was only a blunder of your own.

In anſwer to this, you ſpend three ſections in endeavouring to make your reader believe, that the main ſtreſs of my defence of Sir Iſaac Newton and his followers is, That this error of theirs is of no ſignificancy in practice; without taking the leaſt notice of the ſecond part of my reply, and barely mentioning the third. Upon this you declaim very abun⯑dantly in the ſtyle which the Learned call the tautological.

You tell me, and it might have been ſuffi⯑cient once to have told me, that the applica⯑tion in groſs practice is not the point queſtioned. I grant it is not. Why then have I ſaid ſo much about the ſmallneſs of the error? I will even tell you the plain truth. Though, as you take notice, I live in the univerſity, yet I have been in London too, and am a little acquainted with the humour of the times and the characters of men. Now, Sir, I had ob⯑ſerved ſome of thoſe Gentlemen, who are not greatly pleaſed that other Perſons ſhould [43] be poſſeſſed of any learning, which they themſelves have not, to be not a little tick⯑led with the rebuke that you had given to the pride of Mathematicians: I found them curious to know what this diſcovery was, that was like to do ſo much ſervice to the Church: I did my beſt to give them ſatiſ⯑faction, and to let them ſee the greatneſs and importance of it. They ſee it plainly, and apply the old ſaying, Parturiunt montes.

One of them indeed could make nothing of what I had ſaid about the length of a ſubtangent, or the magnitude of the orb of the fixed ſtars; but was fully ſatisfied by the information given him by one of his ac⯑quaintance to the following effect. The Author of the Minute Philoſopher has found out that, if Sir Iſaac Newton were to mea⯑ſure the height of St. Paul's Church by Fluxions, he would be out about three quar⯑ters of a hair's breadth: But yonder is one Philalethes at Cambridge, who pretends that Sir Iſaac would not be out above the tenth part of a hair's breadth. Hearing this, and that two books had been written in this controverſy, the honeſt Gentleman flew in⯑to a great paſſion, and after muttering ſome⯑thing to himſelf about ſome body's being [44] overpaid, he went on making reflections, which I don't care to repeat, as not being much for your honour or mine, any more than for that of another perſon, whom I too highly reverence to name upon this occaſion.

XXVII. Now, gentle Reader, we come to the point. You are to be ſhown the firſt inſtance of my courage in affirming with ſuch undoubting aſſurance things ſo eaſily diſproved. My antagoniſt intreats you to obſerve how fairly I proceed. I deſire you to be upon your guard, to look well about you. After this, if either of us endeavour to throw duſt in your eyes, knock him down: Which⯑ſoever of us ſhall attempt to falſify the words of Sir Iſaac Newton, or thoſe of his opponent, to the Pump, to the Thames, to the Liffy with him, pump him, duck him for a Pickpocket. The diſpute here is about a matter of fact, and I will endeavour to ſtate the caſe ſo plainly, that it ſhall be impoſſible either to miſtake or to evade it.

In Sir Iſaac Newton's demonſtration of the rule for finding the moment of the re⯑ctangle of two flowing quantities, mention is made of three ſeveral rectangles, to each of which the flowing rectangle is equal, at [45] three different times, or in three different ſtates.

The firſt of theſe is the rectangle A−½a × B−½b.

The ſecond is the rectangle AB.

The third is the rectangle A+½a × B+½b.

I had obſerved, Sir, that you were mi⯑ſtaken in taking it for granted, that what Sir Iſaac Newton was endeavouring to find by the ſuppoſitions made in this demonſtra⯑tion, was the increment of the ſecond of theſe rectangles, the rectangle AB. The reaſon I gave for ſuppoſing you in a miſtake, was expreſſed in the following words. "For neither in the demonſtration it ſelf, nor in any thing preceding or following it, is any mention ſo much as once made of the increment of the rectangle AB." This therefore is a matter of fact that you diſpute with me: But how you diſpute it, is worth obſerving. It greatly imports you to con⯑tradict me, and yet you cannot, you dare not contradict what I ſay. Notwithſtanding this you will contradict me. Methinks I ſee my reader ſtare. I ſhall be taken for a Mad⯑man: And yet I ſpeak the words of truth and ſoberneſs. I affirm, ſay you, the direct contrary. Contrary to what? To what I [46] have ſaid? No. You cannot, you dare not do it. Your reader would immediately turn to Sir Iſaac Newton and detect you. But you can firſt alter what I ſay, and then contra⯑dict me. Inſtead of my words alone, you can give the reader other words which are not mine, and yet are ſo intermixed with mine and diſtinguiſhed by inverted comma's, that every reader ſhall take them for mine; and then you can affirm the direct contrary. You cannot ſay Sir Iſaac Newton makes mention of the increment of the rectangle AB: But you can affirm that he makes mention of the rectangle of ſuch flowing quantities: That he makes expreſs mention of the increment of ſuch rectangle: Of the increment of that rectangle whoſe ſides have a and b for their incrementa tota: That he underſtands his in⯑crementum as belonging to the rectangulum quodvis. You go on declaiming about the words, the ſenſe, the context, the concluſion of the demonſtration and the thing to be de⯑monſtrated:

And when the reader has loſt all ſight of the point in queſtion, you refer it to his own eyes.

[47] I refer it to him likewiſe, and reply to all you have here ſaid, that the firſt of the three rectangles mentioned above, namely the rectangle A−½a × B−½b is the re⯑ctangle of two flowing quantities, but is not the rectangle AB; is a rectangle whoſe ſides have a and b for their incrementa tota, but is not the rectangle AB; is the rectan⯑gulum quodvis in its firſt ſtate, but ſtill is not the rectangle AB. The queſtion is, as your ſelf declare, about matter of fact. It is not therefore about what Sir Iſaac New⯑ton means, but what he mentions: Not a⯑bout what he underſtands, but what he declares: Not about his ſenſe, but his words. And in all his words throughout this demonſtration and every thing prece⯑ding and following it, I affirm and aver that he does not ſo much as once mention the in⯑crement of the rectangle AB. Deny it, if you dare.

XXVIII. You tell me, I would fain per⯑plex this plain caſe by diſtinguiſhing between an [48] increment and a moment. But it is evident to every one, who has any notion of demonſtra⯑tion, that the incrementum in the concluſion muſt be the momentum in the Lemma; and to ſuppoſe it otherwiſe is no credit to the Author. Now, Sir, to ſhew you how little I am in⯑clined to perplex the caſe, I hereby declare that I abſolutely and fully agree with you that the incrementum in the concluſion is the momentum in the Lemma. Let us now ſee whither this our agreement will lead us.

The momentum in the Lemma we both agree to be the momentum of the rectangle AB. The incrementum in the concluſion is manifeſtly the exceſs of the rectangle A+½a × B+½b, above the rectangle A−½a × B−½b, i.e. the increment of the rectangle A−½a × B−½b. Therefore we are agreed that the moment of the rectangle AB is the increment of the rectangle A−½a × B−½b. Conſequently you were miſtaken in ſuppoſing that the moment of the rectangle AB was the increment of the ſame rectangle AB.

You quote Sir Iſaac Newton's words a⯑gainſt me to ſhew that a moment is an in⯑crement or decrement. Why Sir! You make me ſtare. Did not I plainly tell you [49] in my defence that the moment of AB was an increment? Did not I likewiſe tell you what increment it was, namely the in⯑crement of A−½a × B−½b? If you will be pleaſed to put on your mathematical ſpectacles, or rather to put on the ingenuity of a Scholar and a Gentleman, (for your eyes are good enough) you will plainly ſee that the diſtinction I make, is not between a moment in general and an increment in ge⯑neral, but between a particular moment and a particular increment, between the mo⯑ment of the rectangle AB and the increment of the rectangle AB, i.e. the exceſs of the rectangle A+a × B+b above the rectangle AB.

Obſerve me well, Sir, what I have affirm⯑ed, and what I ſtill affirm, and what before I have done, I ſhall prove paſt a poſſibility of being denied, is this. The moment of AB is neither the increment nor the decre⯑ment of AB; neither the exceſs of A+a × B+b above AB; nor the defect of A−a × B−b from AB. This ſeems to you a won⯑derful aſſertion. But one of yours, which you call a very plain and eaſy one, is to me much more wonderful.

[50] I aſked, which of theſe two quantities, the increment of AB, or the decrement of AB, you would be pleaſed to call the mo⯑ment of AB? Your anſwer is, Either of them. This to me is a very wonderful an⯑ſwer for ſo great and ſo accurate a Mathema⯑tician to make, and if I have not quite for⯑got my Logick, I ſhall draw as wonderful an inference from it. The moment of AB is equal to the increment of AB: The ſame moment of AB is equal to the decrement of AB. Ergo, the increment of AB is e⯑qual to the decrement of AB. That is, Ab+Ba+ab=Ab+Ba−ab, i.e. 2ab=o. Therefore the rectangle ab, a⯑bout which the Author of the Minute Phi⯑loſopher has made ſuch a pother, is by his own confeſſion equal to nothing.

Your example in numbers does by no means come up to our caſe. I ſhall beg leave to ſtate it a little more pertinently. It is a⯑greed that all numbers are either odd, or e⯑ven. Upon this you pronounce an unknown number to be even, without giving any rea⯑ſon for it. I repreſent to you that, ſince the number is unknown, it may as well be odd as even; and therefore to pronounce it either the one or the other, without any reaſon for [51] ſo doing, is no better, and no more like an A⯑rithmetician, than to toſs up croſs or pile what you ſhall call it. You may call this mirth, if you pleaſe; but the argument is not the leſs ſtrong againſt you for this ſeem⯑ing levity.

Nor is the accommodation I propoſed in the diſpute between an increment and a de⯑crement for the title of moment, at all the leſs reaſonable for being delivered in a ludi⯑crous manner, under which other perſons can plainly diſcern a ſerious argument, and I perceive you find it much eaſier to rally than to anſwer that argument. To ſay truth, there is no anſwer to be given to it; it is a demonſtration againſt you as ſtrong as any in Euclid, that the moment of the rectangle AB is a middle arithmetical proportional between the increment and decrement of the ſame rectangle AB. If ſo;

XXIX. You are pleaſed to take notice that I very candidly repreſent my caſe to be that of an Aſs between two pottles of hay. I find by this you are duly ſenſible of my candour. Had I been leſs candid, you ſee plainly I had [52] a fair occaſion of repreſenting another per⯑ſon in that perplexity, who might not have had a Ghoſt ſo ready at hand to help him out.

The queſtion with me was, Whether the velocity of the flowing rectangle AB was the velocity with which the increment, or the velocity with which the decrement, of the ſame rectangle AB, might be generated? I could ſee no poſſibility of a reaſon to deter⯑mine me either way. This led me to fix up⯑on a middle arithmetical proportional be⯑tween theſe two velocities, for the velocity of the rectangle AB: As I had before ſhewn its moment to be a middle arithmetical pro⯑portional between the increment and decre⯑ment. But you, who talk ſo much of rea⯑ſoning and logick, and who ſet up for the great and ſole Maſter of the [...] Geome⯑trica, are of opinion that either of theſe veloci⯑ties may be deemed the velocity of the rectangle AB. That is, in your opinion, of two un⯑equal velocities, either the one, or the other, may be deemed equal to a third velocity; or two velocities may be deemed equal and unequal at the ſame time.

You tell me, For your part, in the rectan⯑gle AB conſidered ſimply in itſelf, without ei⯑ther increaſing or diminiſhing, you can conceive [53] no velocity at all. Nor I neither. But in the rectangle AB conſidered as flowing, whether increaſing or diminiſhing, I can conceive ſome velocity or other: And if it flow with an accelerated velocity; I can con⯑ceive that velocity to be different in every point of time: And if we ſuppoſe the incre⯑ment of the rectangle to be generated in a given particle of time, and the decrement of the ſame rectangle to be generated in ano⯑ther equal particle of time; I can conceive the uniform velocity that would generate the increment in the given time, to exceed the uniform velocity that would generate the de⯑crement in the ſame time: And theſe two ve⯑locities being unequal, I can conceive an a⯑rithmetical mean between them; in like manner as I had before ſuppoſed an arithme⯑tical mean between the increment and decre⯑ment of AB, which mean is the moment of AB: And laſtly, while AB flows, I can con⯑ceive that the firſt arithmetical mean is con⯑ſtantly proportional to the laſt, i.e. that the velocity of AB is proportional to the mo⯑ment of AB.

Upon my aſſerting that the moment of the rectangle AB is neither the increment nor decrement of that ſame rectangle AB, you [54] tell me this is in direct oppoſition to what Sir Iſaac himſelf has aſſerted in a paſſage you quote from him, and you bid the reader not believe you, but believe his eyes. Now certain⯑ly would any reaſonable man, that did not thoroughly know the Author of the Minute Philoſopher, conclude that I denied what is expreſſed in the paſſage here quoted againſt me, viz. that moments are either increments or decrements; that increments are affirma⯑tive moments, and decrements are negative moments. Little would any one imagine from the aſſurance with which you here ex⯑preſs yourſelf, that all I maintain is, as I ſaid a while ago, That one particular deter⯑mined moment is not one particular deter⯑mined increment. But your chicaneries are ſo many, ſo groſs, and every way ſo ſhame⯑ful for a Scholar, a Gentleman, and above all for one profeſſing piety and chriſtian zeal, that I grow weary of expoſing and refuting them. I ſolemnly aver, that after I have de⯑tected ſo many, almoſt in every paragraph of your Reply, I have knowingly and volunta⯑rily paſſed by many more, particularly thoſe ſcandalous ones of almoſt perpetually chang⯑ing the Words I uſe, for others that ſeem to make more for your advantage. One would [55] think your aim was to ſhew, that whatever care can be uſed in expreſſion, it ſhall be no fence againſt ſuch an adverſary as you.

XXX. You intreat me, in the name of Truth, to tell what this moment is, which is acquired, which is loſt, which is cut in two, or diſtinguiſh⯑ed into halves. Is it, ſay you, a finite quan⯑tity, or an infiniteſimal, or a mere limit, or nothing at all? You go on to make objections to every one of thoſe ſenſes. If I take it in ei⯑ther of the two former, you ſay, I contradict Sir Iſaac Newton. Very true. If in either of the latter, I contradict common ſenſe. Very true again. But what then? Can I take it in no other ſenſe, but thoſe four you pro⯑poſe? I aſſure you I never had a thought of taking it in any one of thoſe ſenſes.

But, in the name of falſhood, what is the meaning of this queſtion? Would you have me tell you, what a Moment is? Or, what the magnitude of a Moment is? If the for⯑mer; I tell you what Sir Iſaac Newton has told you before, a moment is a momenta⯑neous, or naſcent increment, proportional to the velocity of the flowing quantity. If the latter; I have no buſineſs at all to con⯑ſider [56] the magnitude of a moment.^* Neque enim ſpectatur, ſays Sir Iſaac Newton, mag⯑nitudo momentorum, ſed prima naſcentium pro⯑portio. I may tell you farther, that the magnitude of a moment is nothing fixed nor determinate, is a quantity perpetually fleet⯑ing and altering till it vaniſhes into nothing; in ſhort, that it is utterly unaſſignable.^† Dantur ultimae quantitatum evaneſcentium rationes, non dantur ultimae magnitudines.

You ſeem much at a loſs to conceive how a naſcent increment, a quantity juſt begin⯑ning to exiſt, but not yet arrived to any aſ⯑ſignable magnitude, can be divided or di⯑ſtinguiſhed into two equal parts. Now to me there appears no more difficulty in con⯑ceiving this, than in apprehending how any finite quantity is divided or diſtinguiſhed in⯑to halves. For naſcent quantities may bear all imaginable proportions to one another, as well as finite quantities. One example of this I have already given in ſect. 17. p. 27. where the naſcent increments BD, bd, bear to each other the proportion of 2 to 1; and conſequently the naſcent increment bd is equal to one half of the naſcent increment [57] BD. And by dividing the revolving line AbB into any other aſſignable parts, it is very eaſy to conceive what number one pleaſes of naſcent increments bearing any aſſignable proportions to one another.

It is poſſible you may be ſo exceedingly ſcrupulous as to object that, though a mo⯑ment as bd, is here ſhewn to be equal to half of another moment BD, yet ſtill this does not come up to the caſe of Sir Iſaac Newton's demonſtration, where one mo⯑ment is ſuppoſed not only to be double of another, as in this caſe, but to be actually divided into two equal parts. I am willing to have all poſſible regard for the tenderneſs and delicacy of your underſtanding in con⯑ceiving any thing that makes againſt you, and therefore ſhall readily you give the beſt aſſiſtance I can towards overcoming this dif⯑ficulty likewiſe. And perhaps it may be moſt eaſy to your imagination, if we firſt ſuppoſe our moments to be finite quantities, and afterwards to become evaneſcent, as Sir Iſaac Newton generally does, and ob⯑ſerves to be agreeable to the geometry of the ancients.

Let therefore the line AC be biſected in the point B, and at a given inſtant of time

let a point ſet out from A to deſcribe the line AC with any given velocity. It is plain this point will arrive at C in a given time. Let another point at the ſame given inſtant of time ſet out from B with one half of the former velocity, to deſcribe the line BC. Then will this ſecond point arrive at C in the ſame given time as the firſt point will arrive there. Now let us ſuppoſe the lines AC, BC, to be gradually deſtroyed by this motion of their reſpective deſcribing points A and B. It is manifeſt that theſe lines will be to one another as 2 to 1, not only at the firſt, but all the time they are diminiſhing. And as by the approach of the points A and B to the point C theſe lines will be dimi⯑niſhed ſine fine, and will at laſt vaniſh into nothing by the actual arrival of thoſe points at C; the proportion beforeſaid of 2 to 1 will ſtill ſubſiſt between them to the inſtant of their evaneſcence, and even at that very in⯑ſtant. Here then we have the evaneſcent line AB actually divided into two equal parts, as was above propoſed. For this di⯑viſion does not ceaſe before the line vaniſhes, any more than the line vaniſhes before the [59] diviſion ceaſes. The whole line AC does not vaniſh before its half BC; nor the half BC before the whole AC: But the whole line AC and the half line BC vaniſh at one and the ſame inſtant of time.

I am ſatisfied that what I have here laid before you, in order to aſſiſt you in conceiv⯑ing an evaneſcent quantity diſtinguiſhed into two equal parts, will be of little uſe, unleſs I clear up the reſt of thoſe ſtrange conceits, if words without a meaning may be called ſo, which, you ſay, I utter with that extreme ſatisfaction and complacency, that unintelli⯑gible account, in which you find no ſenſe or reaſon, and bid the reader find it if he can.

And here, I own, you have fairly gra⯑velled me. I am at a ſtand, at a loſs, in as great a perplexity, as when my hunger was equally divided between the two bottles of hay, without ſeeing any poſſibility of its being ſatisfied. Oh for a whiſper from ano⯑ther Ghoſt! But alas! What would even that avail me againſt a Freethinker in Mathe⯑maticks, againſt a man ſo hardened in in⯑fidelity, that he will not believe, though one ſhould ariſe from the dead, not upon the word of a Ghoſt, how venerable ſoever? What then can be done? I had, I thought, [60] rendered that account as clear as words could make it. I had ſhown not only what a moment was, but to prevent, as far as poſſible, all miſtakes about it, I had moſt carefully and circumſpectly ſhown what it was not. Since that account was publiſhed, I had obſerved ſeveral perſons to be greatly ſatisfied with that paragraph, and ſome to have rectified their notions by it. What then can be the reaſon of this phaenomenon, that the perſpicacious Author of the Minute Philoſopher cannot comprehend what every body elſe ſo eaſily underſtands, cannot ſee what to others appears as clear as the day? Is it that he has hurt his ſight by poring ſo long upon objects too ſmall to be diſcerned, as a triangle in a point? Or has he blinded himſelf by gazing upon a light too ſtrong for his eyes, with endeavouring to find ſpots in the Sun? Or has he crack'd his brain by his meditations upon a ſcience too hard for an ^‡ Angel? Hark! Is not that he, ex⯑claiming yonder?

Bleſs us! How the poor Gentleman raves! Huſh! He begins again.

[62] But hold! Theſe circumſtances ſurely can never ſuit my correſpondent; and be⯑ſides, I remember, he abominates the very ſound of Miltonick verſe. I muſt certainly be miſtaken.

Is it then, that by having been long in the dark, and fixing his attention upon dim and obſcure objects which he had not light enough to perceive diſtinctly, his pupil is ſo dilated as not to be able to diſtinguiſh things in open day? I was going to ſay, like a Cat that had loſt her Membrana nicti⯑tans. But perhaps this compariſon, though with ſo ſagacious an Animal, may give him offence. What then ſhall I ſay? I have it. I beg his pardon for theſe offenſive gueſſes. It was my own fault I was not underſtood by him. This comes of ſaving ſixpence to one's reader. Had I put a figure in that place, all had been right. But I was reſol⯑ved to have none. For, I knew, my Book-ſeller, who underſtands his buſineſs as well as Jacob Tonſon, would not have failed of clapping on the other ſixpence to the price of my performance, which would have diſ⯑appointed me in my deſign of making Truth come cheaper than error. But it will be [63] asked, why the want of a figure to that ac⯑count ſhould be of greater diſadvantage to him than to other readers. I anſwer, this proceeds from an infirmity that I have long obſerved in him, though every body may not have taken notice of it, and though it is, as I believe, unknown to himſelf. It is, that his Ideas are almoſt all ſenſible. He has few or none of thoſe Ideas which are pure⯑ly, or partly, intellectual, and which have no ſenſible images to repreſent them. But of this diſeaſe I may perhaps ſpeak more large⯑ly another time; at preſent I ſhall endea⯑vour to obviate this defect in him by the following figure.

Let therefore RALB, or RL, repreſent the flowing rectangle AB in Sir Iſaac New⯑ton's [64] demonſtration; RA the ſide A; and RB the ſide B; ei, iA, Ao, and ou, each, one half of a; and bc, cB, Bd, and df, each, one half of b; and compleat the re⯑ctangles eRbq, iRcr, oRdſ, uRft.

Then will the rectangle A−½a × B−½b be repreſented by the rectangle Rr; the rectangle A+½a × B+½b by the rectangle Rs; and the difference between theſe two rectangles, or the moment of the rectangle AB or RL, will be repreſented by the gno⯑mon rſ; lying partly within and partly with⯑out the rectangle RL.

The rectangle A−a × B−b will be re⯑preſented by Rq; and the difference be⯑tween this rectangle and the rectangle AB, or the decrement of AB, will be repreſent⯑ed by the gnomon Lq lying within the re⯑ctangle RL.

Likewiſe the rectangle A+a × B+b will be repreſented by Rt; and the difference between this rectangle and the rectangle AB, or the increment of AB, will be re⯑preſented by the gnomon Lt lying without the rectangle RL.

Let us now ſee if by the help of this figure my unintelligible account of a moment can be cleared up.

Firſt then, the moment of the rectangle AB, or RL, is neither the increment from AB to A+a × B+b; nor the decrement from AB to A−a × B−b: i.e. rſ; is nei⯑ther Lt nor Lq.

It is not a moment common to AB and A+a × B+b, which may be conſidered as the increment of the former, or as the de⯑crement of the latter: i. e. rſ; is not Lt, common to RL and Rt, which may be conſidered as the increment of RL, or as the decrement of Rt.

Nor is it a moment common to AB and A−a × B−b, which may be conſidered as the decrement of the firſt, or as the incre⯑ment of the laſt: i.e. rſ; is not Lq com⯑mon to RL and Rq, which may be conſi⯑dered [66] as the decrement of RL, or as the in⯑crement of Rq.

But it is the moment of the very indivi⯑dual rectangle AB itſelf, and peculiar to that only; and ſuch as being conſidered in⯑differently either as an increment or decre⯑ment, ſhall be exactly and perfectly the ſame: i.e. rſ; is the moment of RL, and peculiar only to RL; and if RL be conſi⯑dered as an increaſing quantity, rſ; may be conſidered as an increment; if RL be look⯑ed upon as decreaſing, rſ; may be conſidered as a decrement. But whether rſ; be conſi⯑dered as increment or decrement of RL, it is one and the ſame quantity.

And the way to obtain ſuch a moment, (viz. ſuch as being conſidered either as an in⯑crement or decrement of the rectangle RL, ſhall be exactly the ſame, ſuch as is not com⯑mon to RL and ſome other rectangle, but peculiar to RL only) is not to look for a moment lying between AB and A+a × B+b, i.e. between RL and Rt; nor to look for one lying between AB and A−a × B−b i.e. between RL and Rq: Not to ſuppoſe AB as lying at either extremity of the moment, but as extended to the middle

of it, i.e. not to ſuppoſe Lt to be the mo⯑ment and RL lying at the inner extremity of it, nor to ſuppoſe Lq to be the moment and RL lying at the outer extremity of it, but to ſuppoſe rſ; to be the moment, and RL ex⯑tended to the middle of it; as having acqui⯑red rL the one half of the moment, and be⯑ing about to acquire the other half Lſ; or as having loſt Lſ; the one half of the mo⯑ment, and being about to loſe the other half Lr.

I hope, by this time, Sir, you may have diſcovered ſome ſenſe and reaſon in what I ſay in my account of a moment: but if you can⯑not or will not diſcover any, I flatter myſelf the reader both will and can. And having now a figure before me, I ſhall take the op⯑portunity [68] of ſhewing you, that there is ſome reaſoning couched under what you are plea⯑ſed to take for mirth and humour, in the proof that I have given, pag. 45, 46. of my Defence, that the moment of the rect⯑angle AB is not the increment or decrement of AB, but a middle arithmetical propor⯑tional between them.

After propoſing to you what by your own confeſſion is the increment, and what the de⯑crement of the rectangle AB, I aſk, you ſay, with an intention to puzzle you, which of theſe you will call the moment of AB. I ſuppoſed it impoſſible for you to give any anſwer to that queſtion, and therefore I de⯑cided it my own way. You now ſay, Either of them: And you call this a plain and eaſy anſwer. My queſtion was, What is the moment of the rectangle RL? You anſwer, EITHER Lt, or Lq. I aſk again, How can I take EITHER Lt, or Lq, for the moment of RL, when Lt and Lq are une⯑qual? If the moment be equal to Lt, then muſt Lq be leſs than the moment: And if the moment be equal to Lq, then muſt Lt be greater than the moment. Which then muſt I take for the moment, ſince each of them can never be equal to the moment? [69] All the Anſwer I can get out of you is, EI⯑THER of them.

Things ſtanding thus, I offer this argu⯑ment to your conſideration. Since, accord⯑ing to you, I may take Lt for the moment of RL; and ſince, according to you I may likewiſe take Lq for the moment of RL; it is manifeſt that, according to you, I may take Lt and Lq added together for twice the moment of RL. Conſequently, accord⯑ing to you, I may take the half of Lt and the half of Lq added together for the moment of RL, i.e. I may take rſ; for the moment of RL. I hope I may now be al⯑lowed to ſay, "Believe me there is no re⯑medy, you muſt acquieſce."

I ſuppoſe, Sir, you may now compre⯑hend my meaning, when I ſay, that the mo⯑ment of AB is not the increment of AB, tho' I allow the moment of AB to be an in⯑crement, agreeably to Sir Iſaac Newton's definition of the word moment. But ſtill it is poſſible, you may doubt whether the ſenſe I aſſign to the word moment, be Sir Iſaac Newton's ſenſe of that term, or a new one that [70] I have affixed to it in oppoſition to you. This is the next point to be cleared up.

And here I beg leave to obſerve in the firſt place, the preſumption is ſtrong in my fa⯑vour, that by the moment of AB Sir Iſaac means ſomething different from the incre⯑ment of AB. For if theſe two words ſigni⯑fied preciſely the ſame thing, it is probable he would have uſed them indifferently, ſometimes the one and ſometimes the other. Whereas the fact is, that, after he has done with defining his terms, he never mentions the word increment but in one place, and then he does not ſpeak of the increment of the rectangle AB, but only of the increment of the rectangle, i.e. of the flowing rectan⯑gle taken at large. But where he names his rectangle, or other flowing quantity, as AB, ABC, A2, A3, &c. He never mentions the increment of AB, of ABC, of A2, &c. but always the moment of AB, the moment of ABC, the moment of A2, of A3, &c. And when ſuch a writer as Sir Iſaac Newton chuſes conſtantly to uſe one term, rather than another ſeemingly of the ſame ſignification, it is to be preſmed he has ſome reaſon for ſo doing.

But farther, we are to take notice that, ac⯑cording to Sir Iſaac Newton, the moment of [71] a flowing quantity is ever proportional to the velocity of the ſame flowing quantity. Let the velocity and the moment of a flowing quantity vary as they will, yet if any inſtant of time be taken, theſe three things will be given, ſuch as they are at that ſame inſtant, namely, the rectangle itſelf, its velocity, and its moment. And this velocity and moment are always proportional. If therefore it ſhall be ſhown, that the moment of a flowing quantity, ſuch as I ſuppoſe it, is proportio⯑nal to the velocity of that ſame flowing quan⯑tity; it will follow that what I ſuppoſe to be the moment, is the ſame with the mo⯑ment intended by Sir Iſaac Newton.

In order therefore to render the concep⯑tion of this point as eaſy and as clear as poſſible, I ſhall once more have recourſe to that well known and familiar inſtance of a flowing quantity I have ſo often made uſe of, viz. that of a line deſcribed by the motion of a point.

Let x repreſent the time, in which a flow⯑ing line is generated, in all the following caſes, and ſince time flows uniformly, let the conſtant quantity ẋ repreſent the mo⯑ment, or increment, (for in this particular caſe they are both one) of the time x.

[72] Caſe 1. If the velocity of the genera⯑ting point be uniform, the flowing line will be as the time in which it is generated, and conſequently the line may alſo be repreſent⯑ed by x, and its moment or increment may be repreſented by ẋ In this caſe therefore the moment ẋ, being conſtant, muſt be pro⯑portional to the velocity, which is likewiſe conſtant.

Caſe 2. Let the velocity be equably acce⯑lerated, as in the caſe of a falling body ac⯑cording to Galileo's Theory. Then will the velocity be as the time, and conſequently the velocity likewiſe may be repreſented by x. And the flowing line being as the time and velocity jointly, that line may be repre⯑ſented by x2. Now the ſuppoſed moment of this line x2, is 2xẋ, and I ſay, 2xẋ is proportional to x the velocity of the flow⯑ing line. For ſince ẋ is a conſtant quantity, it is evident that 2xẋ is as 2x; and 2x is as x. Therefore 2xẋ is as x. In this caſe therefore the ſuppoſed moment is as the ve⯑locity of the flowing quantity.

Caſe 3. Let the velocity be as the ſquare of the time, and be repreſented by x2. Then will the flowing line ſtill be as the velocity [73] and the time jointly, and conſequently may be repreſented by x3, the ſuppoſed moment of which, viz. 3x2 ẋ, is evidently as x2, or as the velocity.

Caſe 4. In general, let the velocity be as any power of the time, and conſequently be repreſented by xn−1. Then may the flowing line be repreſented by the time and the velo⯑city jointly, or by x × xn−1, i.e. by x^n. And the ſuppoſed moment of this line will be nxn−1 ẋ, which is manifeſtly as xn−1, that is, as the velocity.

The moment therefore ſuppoſed by me is ever proportional to the velocity, and con⯑ſequently is the moment ſuppoſed by Sir Iſaac Newton.

While I am upon this conſideration, it may not be amiſs for a more compleat illu⯑ſtration of what we have been talking of, to conſider a little more particularly the ſe⯑cond caſe, or that of the flowing quantity x2, anſwering to the rectangle AB of Sir Iſaac Newton. I ſhall therefore take the li⯑berty of laying before my reader in one view, the decrement, the moment, and the increment of the flowing quantity x2, toge⯑ther with the ſeveral velocities that would ge⯑nerate [74] them reſpectively, with an uniform motion, in a given time.

Here it appears that as the moment is a middle arithmetical proportional between the decrement and increment; ſo is the ve⯑locity of the flowing line, or the velocity that would generate the moment in the given time 2ẋ, a middle arithmetical propor⯑tional between the velocities that would re⯑ſpectively generate the decrement and incre⯑ment in the ſame given time. And this pro⯑portion equally holds, whether the mo⯑ments be evaneſcent, or finite quantities of whatſoever magnitude.

Whence I infer, that although it were not poſſible to conceive an evaneſcent moment divided into two equal parts, yet as finite ones may be conceived to be ſo divided, that demonſtration of Sir Iſaac Newton's which you object againſt, will ſtill hold firm and entire, by ſubſtituting finite moments in the [75] room of evaneſcent moments. Which is a ſecret you were not aware of.

One more obſervation, and I have done. You would have us take the increment of AB, or in this caſe the increment of x2, for the moment of x2; that is, you would have us take 2xẋ+ẋẋ, and not 2xẋ, for the moment of x2: And yet you allow that the moment of x2 is proportional to the velo⯑city of x2. But the velocity of x2 is x; and the quantity you give us as the moment, namely 2xẋ+ẋẋ, is not proportional to this velocity x. Therefore by your own conceſſion, that quantity is not the true mo⯑ment. But the quantity that Sir Iſaac New⯑ton aſſigns, namely 2xẋ, has juſt now been ſhown to be proportional to x, the velocity of x2, and therefore is the true moment. Now therefore I may ſafely repeat my que⯑ſtion, and ask with my accuſtomed air, "What ſay you, Sir? Is this a juſt and le⯑gitimate reaſon for Sir Iſaac's proceeding as he did? I think you muſt acknowledge it to be ſo."

But hark you! Why all this outcry about Ghoſts and Viſions? Pray who firſt intro⯑duced them? If I brought in one, you might conſider it was to a very good pur⯑poſe, [76] to help my ſelf out, or rather to help you out, at a dead lift. Whereas you had before needleſly introduced an innumerable multitude of Ghoſts of departed quantities, for no other intent or purpoſe in the world but your own diverſion.

In conſideration of which I hope I may be pardoned for bringing in one more, though I can give no better reaſon for it, than that the Apparition runs ſtrongly in my fancy.

The Viſion would lead me a great deal farther, and I might proceed to relate in [79] heroick verſe the rebuke given by the fallen Chief to this his Aſſociate, for puſillanimity in abandoning the noble undertaking.

But I am afraid, Sir, you begin to be tired. Poſſibly this viſion of mine may give you as little pleaſure, as the Ghoſts you in⯑troduced ſome time ago afforded to any of your readers. I ſhall therefore ſtop here, and hope from your known candour, that if you chance to ſpy any inconſiſtencies, or any little marks of vanity in this my viſion, you will be ſo juſt as to conſider there are but few viſions, apparitions, dreams, or caſtles built in the air, that are not liable to ſome objection.

XXXI. It is now ſo evident even to your ſelf, that the moment of the rectangle AB is not the increment of the rectangle AB, that I expect to be complimented upon my [80] civility in charging you with want of cau⯑tion only, in putting the one for the other. You have indeed replièd, that this charge is as untrue as it is peremptory. But that was in your ſtate of blindneſs, and I forgive you without your asking pardon. You ſay in your juſtification, Sir Iſaac Newton, in the firſt caſe of this Lemma, expreſly determines it to be an increment. Yes, he determines it to be an increment. But an increment of what? An increment of AB? Methinks I ſee the good old Knight hold up his fin⯑ger and cry Cave. It is the increment of A−½a × B−½b.

You ſay, take it increment or decrement as you will, the objections ſtill lie, and the dif⯑ficulties are equally inſuperable. Very true. But I will not take it for either increment or decrement, and then all difficulties and ob⯑jections vaniſh before me, they become no⯑thing, there are no difficulties, no obje⯑ctions, I meet with nothing in my way but the Ghoſts of departed difficulties and obje⯑ctions.

XXXII. Before I proceed to vindicate that aſſertion of mine which makes the ſub⯑ject of this ſection, I crave leave to obſerve, [81] that this aſſertion, true or falſe, is no way material to the point in debate between us. You were fully anſwered before I laid down that aſſertion: And all the ſubterfuges you have ſince made uſe of, are clearly removed before I vindicate it. Why therefore did I make that aſſertion? Dear Sir, the true rea⯑ſon is a ſecret. I ſee plainly it never entered your Pericranium, any more than that of ſome other perſons much ſuperior to you in this part of ſcience. In due time it may come out. In the mean while all I ſhall ſay is, it was made to guard, not againſt pre⯑ſent, but future objections.

Do not miſtake me, Sir, I am not going to excuſe that aſſertion, much leſs to give it up. I intend to vindicate it to the laſt drop of my pen. Like Mackbeth in blood,

My aſſertion was, That the moment of the rectangle AB, determined by Sir Iſaac Newton, namely aB+bA, and the incre⯑ment of the ſame rectangle determined by your ſelf, namely aB+bA+ab, are per⯑fectly and exactly equal, ſuppoſing a and b [82] to be diminiſhed ad infinitum; and this by Lemma 1. Sect. 1. Libr. 1. Princip.

You anſwer, If a and b are real quanti⯑ties, then ab is ſomething, and 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.

By giving this for an anſwer to my aſſer⯑tion, it is plain you have no notion of what Sir Iſaac Newton means by a quantity being infinitely diminiſhed, though he has ſo fully and clearly explained himſelf in the ſcholium of that ſection of the Principia, which I ſo often refer you to.

Suppoſe a given line to be gradually dimi⯑niſhed, during a given time, by the conti⯑nued motion of a point, ſo that at the end of the given time the line would entirely vaniſh and become nothing. Then if the motion of the point, and the gradual diminution of the line conſequent thereupon, be ſup⯑poſed to ſtop before the expiration of the given time, it is plain that the line will not as yet have been diminiſhed ad infinitum; it will ſtill be ſomething, it will be a real [83] quantity, it will be a finite quantity. But if the motion go on, without ſtop or ſtay, to the end of the given time, it is manifeſt that the line muſt be diminiſhed ſine fine, ſine limite, it muſt be diminiſhed ad infinitum, it muſt vaniſh, it muſt become nothing. The end of this diminution ad infinitum, the vaniſhing of the line, and its becoming nothing, theſe three muſt all happen at one and the ſame inſtant of time, namely at the expiration of the given time. So that an in⯑ſtant before the expiration of the given time, or before the quantity becomes nothing, it cannot truly be ſaid to be actually diminiſhed ad infinitum. Therefore while a and b are real quantities, they are not yet diminiſhed ad infinitum, they may be farther dimi⯑niſhed. And conſequently the firſt part of your anſwer is quite beſide the purpoſe: It tends only to ſhew that there is a real diffe⯑rence between the moment and increment, before the inſtant of time when I ſuppoſe them to become equal; that while they are unequal, there is a difference between them. A great diſcovery, and undoubtedly true!

You proceed in your anſwer, If they, i.e. a and b, are nothing, then the rectangles where⯑of they are coefficients, become nothing likewiſe: [84] and conſequently the momentum or incremen⯑tum, whether Sir Iſaac's or mine, are in that caſe nothing at all.

This likewiſe is undoubtedly true. But it is ſo far from contradicting Sir Iſaac New⯑ton's doctrine, that it is perfectly agreeable to it. What he ſays, and what I contend for, is this.

Though ſo long as a and b are real quanti⯑ties, their rectangle ab is a real quantity, and there is a real difference between the two quantities aB+bA and aB+bA+ab: Yet, when by a continual diminution ad infinitum a and b vaniſh, their rectangle ab, or the difference between the two quan⯑tities aB+bA and aB+bA+ab, vani⯑ſhes likewiſe, and there is no longer any dif⯑ference left between thoſe quantities, i.e. thoſe quantities are equal. But you ſay, when ab, when the difference between theſe two quantities vaniſhes, the quantities them⯑ſelves do likewiſe vaniſh. I agree with you. Their difference therefore vaniſhes when they vaniſh: And they vaniſh when their difference vaniſhes: Or, The quantities themſelves, and the difference between them, vaniſh at one and the ſame inſtant of time.

[85] You ſee I agree perfectly with you, that the moment and increment vaniſh at the ſame inſtant that their difference vaniſhes. All I contend for is this, That the moment and increment vaniſh with a ratio of equali⯑ty, and that they do ſo, I am going to de⯑monſtrate after Sir Iſaac Newton's manner.

Let the rectangle AE repreſent 2xẋ, the moment, and let the rectangle AF repreſent 2xẋ+ẋẋ, the increment of the flowing ſquare x2. I ſay when ẋ vaniſhes, the mo⯑ment 2xẋ, and the increment 2xẋ+ẋẋ, will vaniſh with a ratio of equality.

XXXIII. We come now to the method for obtaining a rule to find the fluxion of any power of a flowing quantity, which is deli⯑vered in the introduction to the Quadratures, and conſidered in the Analyſt. And here, ſay you, the queſtion between us is, whether I have rightly repreſented the ſenſe of thoſe words, evaneſcant jam augmenta illa, in ren⯑dering them, let the increments vaniſh, i.e. let the increments be nothing, or let there be no increments? And ſo, Sir, you would have the Reader believe that this is the whole of the queſtion between us: That we have each of us ſpent four or five pages, and may poſ⯑ſibly ſpend twice as many more before we have done, in wrangling about the tranſla⯑tion of four Latin words. If ſo, methinks his beſt way will be to let us wrangle by our ſelves, and to tranſlate thoſe four words himſelf as he thinks fit, without ever trou⯑bling his head about us.

But I take the queſtion between us to be of a little more extent, and of ſomewhat more importance. What I have endeavour⯑ed to eſtabliſh the ſenſe of, is not thoſe four [92] words alone, but the whole paſſage taken together, i.e. in the ſtyle of divines, the text and the context. The whole paſſage is, Evaneſcant jam augmenta illa & eorum ratio ultima erit, and I have endeavoured to ſettle the meaning of this whole paſſage taken toge⯑ther, by comparing it with an equivalent paſ⯑ſage, but expreſſed in ſuch terms as not to be liable to any ſophiſtication, Naſcantur jam augmenta illa & eorum ratio prima erit.

The queſtion therefore between us is not barely how thoſe four words may be tran⯑ſlated: If they ſtood alone, they might be tran⯑ſlated twenty different ways: But the queſtion is, how theſe four words ought to be tranſlated in conjunction with the other words that follow; how the whole paſſage ought to be tranſlated, ſo as to let the Reader underſtand the meaning and deſign of Sir Iſaac Newton in that paſſage. His deſign is manifeſtly to conſider the proportion between the evane⯑ſcent augments, or to conſider the pro⯑portion with which the augments vaniſh. He plainly makes two ſuppoſitions in this paſſage. The firſt is, that the aug⯑ments vaniſh, or become nothing. The ſe⯑cond, that the augments have a laſt ratio. And his buſineſs is to determine what this laſt ratio is: Now the queſtion between you [93] and me is, when, at what inſtant of time Sir Iſaac Newton ſuppoſes the augments to have this laſt ratio? You will needs have it, that he ſuppoſes the augments firſt to vaniſh, to become nothing, and then conſiders the proportion between thoſe nothings. I maintain, that he conſiders the proportion between the augments, not after they are va⯑niſhed, but at the inſtant that they vaniſh, in the very point of evaneſcence. And I am juſtified by his own words, where he more fully explains himſelf,^* intelligendam eſſe ra⯑tionem quantitatum non antequam evaneſcutn NON POSTEA, ſed quacum evaneſcunt. You ſee therefore, Sir, the hard words, you ſay, I have uſed, do not fall upon my Friends, but fall where I intended them. The blun⯑der of making the quantities firſt become no⯑thing, and then ſettling the proportion be⯑tween thoſe nothings, ſtands juſt as it did. It puts me in mind of an Evidence, who was inſtructed to ſwear that a certain will was made juſt as the Teſtator was dying, and was therefore ſubſequent to another will made ſome time before his death. This per⯑ſon reſolved to make ſure work, and ſwore poſitively that this was the laſt will, for it was made after the Teſtator was dead.

[94] You ſee likewiſe you had no reaſon to de⯑ſpair of making me acknowledge, that va⯑niſhing and becoming nothing were equiva⯑lent terms with Sir Iſaac Newton. Indeed, how was it poſſible to think otherwiſe? A naſcent augment muſt have been nothing before it began to exiſt, and an evaneſcent augment muſt be nothing after it ceaſes to exiſt.

As it is my buſineſs chiefly to keep upon the defenſive, and I have hitherto had very little occaſion to act offenſively, I did in my firſt Letter conſider your important Lemma and reaſoning upon it, no farther than was neceſſary to juſtify Sir Iſaac Newton againſt the conſequences you draw from that Lem⯑ma. But now, as you are pleaſed to ſhew more than ordinary arrogance in this and the following ſection, I hope the reader will ex⯑cuſe me, if I ſtep out of my way to call you a little to account. A vigilant General, who is aſſaulted in his entrenchments by an overbearing and inſolent Enemy, may ſome⯑times obſerve that Enemy in the heat of his attack, to lay himſelf ſo open, as to give a fair opportunity of ſallying out and chaſtiſing him.

[95] And it may not be amiſs to ſhew, that Mathematicians are not the only perſons, who falſely imagine their rational faculties to be more improved than thoſe of other men, which have been exerciſed in a different manner, and on different ſubjects. That there are other perſons, who erect themſelves into judges and oracles, concerning matters, which they have never ſufficiently conſidered nor comprehended. And if this appear, it will ſurely furniſh a fair argumentum ad hominem againſt men, who reject that very thing in Geometry which they admit in Logick. It will be a proper way to abate the pride, and diſcredit the pretenſions of theſe Logicians and Metaphyſicians, who inſiſt upon clear Ideas in points of Mathe⯑maticks, if it be ſhewn that they do without them in their own ſcience.

The ſubſtance of your Lemma is this. If one ſuppoſition be made, and be afterwards deſtroy'd by a CONTRARY ſuppoſition; then every thing that followed from the firſt ſuppoſition, is deſtroyed with it. This being laid down, you proceed thus. Sir Iſaac Newton ſuppoſes certain increments to exiſt, or that there are certain increments. In conſequence of their ſuppoſed exiſtence, he forms certain expreſſions of thoſe incre⯑ments, [96] with intent to deduce the proportion of the increments from thoſe expreſſions. He afterwards ſuppoſes that thoſe incre⯑ments vaniſh, i.e. ſay you, that the incre⯑ments are nothing, that there are no incre⯑ments.

I forbear making any remarks upon your interpretation of the word vaniſh. I admit it to be as you are pleaſed to make it, that the firſt ſuppoſition is, there are increments; and that the ſecond ſuppoſition is, there are no increments. What do you infer from this? The ſecond ſuppoſition, ſay you, is contrary to the former, and deſtroys the for⯑mer, and in deſtroying the former it de⯑ſtroys the expreſſions, the proportions, and every thing elſe derived from the former ſuppoſition. Not too faſt, good Mr. Logi⯑cian. If I ſay, the increments now exiſt, and, the increments do not now exiſt; the latter aſſertion will be contrary to the for⯑mer, ſuppoſing now to mean the ſame in⯑ſtant of time in both aſſertions. But if I ſay at one time, the increments now exiſt; and ſay an hour after, the increments do not now exiſt; the latter aſſertion will neither be contrary, nor contradictory to the for⯑mer, becauſe the firſt now ſignifies one [97] time, and the ſecond now ſignifies another time, ſo that both aſſertions may be true. The caſe therefore in your argument does not come up to your Lemma, unleſs you will ſay Sir Iſaac Newton ſuppoſes that there are increments, and that there are no increments, at the ſame inſtant of time. Which is what you have not ſaid, and what, I hope, you will not dare to ſay.

But perhaps you will ſtill maintain, that whether the ſecond ſuppoſition be eſteemed contrary, or not contrary, to the firſt, yet as the increments, which were ſuppoſed at firſt to exiſt, are now ſuppoſed not to exiſt, but to be vaniſhed and gone, all the conſequen⯑ces of their ſuppoſed exiſtence, as their ex⯑preſſions, proportions, &c. muſt now be ſup⯑poſed to be vaniſhed and gone with them. I cannot allow of this neither.

Let us imagine your ſelf and me to be debating this matter, in an open field, at a di⯑ſtance from any ſhelter, and in the middle of a large company of Mathematicians and Logicians. A ſudden violent rain falls. The conſequence is, we are all wet to the ſkin. Before we can get to covert, it clears up, and the Sun ſhines. You are for going on with the diſpute. I deſire to be excuſed, I muſt go home and ſhift my cloaths, and [98] adviſe you to do the ſame. You endeavour to perſuade me I am not wet. The ſhower, ſay you, is vaniſhed and gone, and conſe⯑quently your coldneſs, and wetneſs, and every thing derived from the exiſtence of the ſhower, muſt have vaniſhed with it. I tell you I feel my ſelf cold and wet. I take my leave, and make haſte home. I am perſuaded the Mathematicians would all take the ſame courſe, and ſhould think them but very in⯑different Logicians, that were moved by your arguments to ſtay behind.

Another example may make all clear. I know a certain Gentleman, who about the firſt day of April 1734, was verily perſua⯑ded he ſaw more clearly into the principles of fluxions, than Sir Iſaac Newton had ever done. The conſequence of this perſuaſion was, that he publiſhed a book, which imme⯑diately convinced all mankind of the con⯑trary. He has ſince had ſuch reaſons given him, as have entirely altered his opinion. His former perſuaſion is vaniſhed and gone; but the book that was the conſequence of that perſuaſion, is not vaniſhed and gone with it. It would have been much for his credit, and for the quiet of the poor Genle⯑man's mind, if it had.

XXXV. You miſtake me, Sir: What I diſlike in you is not your modeſty, but your arrogance. 'Tis your unparallel'd and ama⯑zing inſolence, to the greateſt diſcoverer of truth, of a mere mortal, that ever appeared in the world.

I am of opinion, that placing the ſame point in various lights is of great uſe to ex⯑plain it.

You have not ſhown Sir Iſaac Newton's various accounts of fluxions to be inconſiſtent. I find them perfectly conſiſtent, and do again profeſs my ſelf greatly obliged to him for his condeſcenſion, in ſetting his doctrine in ſeveral different lights, without which, I ſtill doubt, I ſhould never have underſtood it.

But you ſeem to think it great vanity in me, to talk as if I underſtood the doctrine of Fluxions. Why, Sir! I hope Sir Iſaac Newton wrote ſo as to be underſtood by ſomebody. I have taken pains to under⯑ſtand him, and I ſuppoſe many others to underſtand him likewiſe: I prefer my ſelf to no body, and I never compare my ſelf with any body but one. It is where I ſpeak of ſuch ordinary Proficients in Mathema⯑ticks, as you and me. Even there, you ſee, [100] I have the good manners to place you firſt. Had I ſaid, no body underſtands him, but I: Or, I don't underſtand him, and therefore no body can underſtand him, it were unpar⯑donable vanity.

You ſay, I inſult you, in asking what it is you are offended at, who do not ſtill under⯑ſtand him? I neither inſult you, nor blame you, for not underſtanding him: But it is, I think, pretty extraordinary for a man, who ſo often profeſſes not to underſtand Sir Iſaac Newton, to complain that Sir Iſaac takes too much pains to explain and illuſtrate his do⯑ctrine, by ſetting it in ſeveral different lights. As to your requeſt to help you out of the dark, I have done my beſt, and hope you ſee much better than you did. The eye-water I have applied, might poſſibly give you ſome pain; but it will do you a power of good. E coelo deſcendit [...].

XXXVI. I flatter my ſelf, I have already done to your mind what you here requeſt.

XXXVII. If I were to ſay, there are a hundred mean and low artifices in a certain pamphlet, ſcarce a ſection without one or more too ſcandalous and too trifling to men⯑tion: [101] This is plain to me; but I will not undertake to demonſtrate it to others: Is this the ſame as to ſay, I cannot demonſtrate it to others? No. But it would take up too much of my time, it would ſwell my letter to too great a bulk to demonſtrate it. You ſay below, I neither will, nor can. You make therefore a difference between the meaning of theſe two words.

XXXVIII. In this Section you addreſs yourſelf to me in the following words. "You will have it, that I repreſent Sir Iſaac Newton's concluſions as coming out right, becauſe one error is compenſated by ano⯑ther contrary and equal error, which per⯑haps 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 Newton, what was intended for the Marquis de l'Hoſpital and his fol⯑lowers." If this was untruly ſaid by me, I aſſure you it was not a wilful untruth. You ſee Mr. Walton fell into that miſtake as well as I. And I do not know a ſingle perſon who has read the Analyſt, but is in the ſame [102] miſtake. However, a miſtake it undoubt⯑edly is; no body ought in the leaſt to diſpute it, after a perſon of your character has made the publick declaration juſt now recited, and has farther aſſured us, that this double error doth concern the Marquis alone, and not Sir Iſaac Newton. Far be it from me to call the truth or ſincerity of this declaration in que⯑ſtion. On the contrary, I aſk your pardon for my miſtake; and to make you all the ſa⯑tisfaction in my power, I do hereby retract, recant and abjure my error, and abandon my picture, my ingenious portraiture of Sir Iſaac Newton and Dame Fortune, to the flames. If you are not yet ſatisfied, I beg leave to al⯑ledge the following reaſons in mitigation of my offence.

1. Your diſcourſe ſeemed to me to be di⯑rected to a follower of Sir Iſaac Newton. And as in Sect. XX. of the Analyſt, where this affair of the double error begins, you perpetually addreſs yourſelf to him in the ſecond perſon, as you demonſtrate, you are converſant, you conceive, you proceed, you ap⯑ply, your concluſions, your logick and method, &c. I too haſtily judged that the double er⯑ror related to this follower, and conſequently to his maſter.

[103] 2. As this affair is purſued through eleven Sections, beginning at Sect. XX. and ending with Sect. XXX. I find Sir Iſaac Newton's way of notation to be uſed in three of thoſe Sections, and the Marquis's notation in two. I find Sir Iſaac's language and expreſſion, as increments, moments, fluxions, infinitely diminiſhed, vaniſh, &c. to be uſed in nine of thoſe Sections, and the Marquis's language and expreſſion, as differences, infiniteſimals, &c. in ſeven of thoſe Sections. Whence it ſeemed to me, that Sir Iſaac Newton was as much concerned in this matter, as the Mar⯑quis.

3. In one of thoſe Sections, namely Sect. XXVI. you refer to Sect. XII, and XIII.; in the firſt of which Sections, viz. Sect. XII. I find this Quotation, Philoſophiae Naturalis Principia Mathematica, lib. 2. lemm. 2. and Sect. XIII. contains nothing elſe but your inſtance of falſe reaſoning taken from Sir Iſaac Newton's Book of Quadratures. Like⯑wiſe in another of thoſe Sections, namely Sect. XXVIII. I find the ſame thirteenth Section quoted. From all which it ſeemed to me, that Sir Iſaac Newton was rather more concerned in this affair than the Marquis, [104] whoſe works I do not find to be quoted in any of thoſe Sections, ſo much as once.

4. The arguments uſed in the Analyſt ſeemed to me to bear equally hard againſt Sir Iſaac Newton and the Marquis; ſo that I could not ſee how you could condemn the one, and acquit the other, of either of the two errors.

Theſe conſiderations had ſo fully poſſeſ⯑ſed my mind, that Sir Iſaac Newton was ſup⯑poſed by you to be guilty of this double er⯑ror, that nothing, but my firm perſuaſion of your veracity and integrity, could ever have removed that apprehenſion. I muſt own, I have ſtill one ſcruple upon my thoughts. If you will be ſo good as to remove that, my mind will be perfectly eaſy about this affair. It is this.

The firſt error in giving 2xdx for the dif⯑ference, or 2xẋ for the moment of xx, is common to the Marquis and Sir Iſaac New⯑ton.

The Marquis makes a ſecond error, which perfectly corrects the firſt, whence his con⯑cluſion comes out right.

Sir Iſaac Newton makes no ſecond error to correct his firſt, and therefore his concluſion ought to come out wrong.

[105] And yet Sir Iſaac's concluſion comes out exactly right, and is the ſame with the Mar⯑quis's concluſion. The more I conſider this, the more it puzzles me: Poſſibly, for want of the Philoſophia prima, which you are ſo great a maſter of.

XXXIX. As you do not perſiſt, nay, on the contrary, deſiſt, and entirely diſown your accuſing Sir Iſaac Newton of this double er⯑ror, methinks there is now no occaſion for my producing any evidence to juſtify him. But you are pleaſed to call publickly upon me to produce it, to deny as ſtrongly as I affirm, to aver, that my declaring I have ſuch evidence, is an unqueſtionable proof of the matchleſs con⯑tempt that I, Philalethes, have for truth. Why this indeed is matchleſs—Blindneſs, or aſſurance, ſhall I call it? I beg the Reader will turn to p. 70. of my Defence. There he will ſee I have already produced my e⯑vidence, and have named the paſſages, where theſe very objections of yours appear to have been foreſeen, and to be clearly and fully re⯑moved. I have there named the paſſages, I ſay, though you have ſuppreſſed them, and every Reader, who is qualified to examine thoſe paſſages, will find what I ſay to be [106] true; and that the pretence of your firſt er⯑ror is fully removed by Lemma 7. and that of the ſecond by Lemma 1.

XL. I have nothing to ſay to the princi⯑ples of the Marquis de l'Hoſpital, I defend nothing but his reaſoning. You ſay, he re⯑jects infiniteſimals in virtue of a Poſtulatum, and this you venture to call rejecting them with⯑out ceremony. I know of no greater ceremo⯑ny uſed by Euclid, than to reject a thing in ne⯑ceſſary and unavoidable conſequence of a Po⯑ſtulatum. You tell me, he inferreth a concluſion accurately true, contrary to the rules of Lo⯑gick, from inaccurate and falſe premiſes. This I deny: for if his premiſes be allowed, his concluſion will follow by the ſtricteſt rules of Logick, though thoſe premiſes are falſe. Allow him his firſt poſtulatum, and then 2xdx will be equal to 2xdx+dxdx. Allow him his ſecond Poſtulatum, and then RN in your figure (Analyſt, p. 32.) will be equal to RL. And his concluſion muſt come out right. It ſeems therefore, that the Marquis is acquitted of this double error, as well as Sir Iſaac Newton, and that it is you alone, who have acted blindfold, as not know⯑ing the true reaſon of the concluſion's coming out accurately right, which I ſhew not to have [107] been the effect of a double error, but of his two Poſtulata.

XLI. To all this declamation I ſhall need to give no other reply, than one you furniſh me with, p. 27. of this very anſwer. It muſt be owned, ſay you, that after you have miſled and amuſed your leſs qualified reader▪ (as you call him) you return to the REAL POINT in controverſy, and ſet your ſelf to juſtify Sir Iſaac's method in getting rid of the above-mentioned quantity. I think I have already told you, that I had talked ſo much of the ſmallneſs of the error, only for the information of ſome great Churchmen, to make them ſenſible of the conſequence of your diſcovery, in order to induce them the more readily to join in the hymn to your honour.

You ſay to me, 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. Theſe are not my words. You will perhaps affirm, that they expreſs my ſenſe. I deny it. I neither ſpeak thus, nor mean thus, nor have any meaning like this, but the direct contrary, with regard to the [108] ſubtangent determined without any compen⯑ſation, upon the principles of the Marquis de l'Hoſpital, who alone is here referred to.

XLII. Empty, childiſh declamation.

XLIII. The ſame, or ſomething worſe. I apprehend it was, as you ſay, diſcreetly done, to fix upon two or three of the main points, and to overlook the reſt of the difficul⯑ties propoſed in the Analyſt, particularly the Queries, threeſcore and ſeven in number. You tell me, I am not afraid nor aſhamed to repreſent the Analyſts as very clear and uniform in their conceptions of theſe matters. Where have I ſo repreſented them? I know there is a great diverſity of opinions among Analyſts: Some follow Monſieur Leib⯑nitz, ſome the Marquis de l'Hoſpital, ſome, other writers, and ſome, whom I take to be the better judges, follow Sir Iſaac Newton, and theſe are uniform ſo far as they follow their maſter, and clear ſo far as they underſtand him.

XLIV. If you have met with all theſe dif⯑ferent opinions, in converſation with Ana⯑lyſts, in ten months time, and ſome Analyſts, perhaps 5 or 6, of every one of thoſe opi⯑nions, [109] one would think that Country, where you have reſided for thoſe ten months, muſt be better ſtocked with Mathematicians than all the reſt of Europe. I hope they are not all Infidels. If they are, it is a mercy they are not very able Infidels, at leaſt ſo far as one can judge of them by their mathema⯑tical opinions. Otherwiſe, I ſhould appre⯑hend Religion to be in great danger there, unleſs that Country be well ſtocked with men able to deal with them at their own wea⯑pons, and to ſhew, they are by no means thoſe maſters of reaſon, which they would fain paſs for.

XLV, XLVI, XLVII, XVIII. You come now to the point of Metaphyſicks in diſpute between us, about which you write, con⯑trary to your uſual manner, ſo very inaccu⯑rately and unintelligibly, as plainly con⯑vinces me you have ſome other end to ſerve than truth. And upon reviſing what I had before addreſſed to you upon this ſubject, I think I neither can, nor need, ſet that matter in a clearer light, than I have already done. I perceive likewiſe, my rebukes have had a good effect upon you. You make excuſes. It was not, you ſay, with intent to carp or cavil at [110] a ſingle paſſage. You talk no more of mani⯑feſt, ſtaring contradictions. No, you expreſs your ſelf with ſome modeſty, all this looks very like a contradiction; with ſome other ſigns of grace, that give me hopes, as you are now made to ſee your errors, you may in time be brought to acknowledge them.

It muſt be owned in your favour, you have already recanted the principal of them, and that which led to all the reſt, as amply and fully, as from you could poſſibly be ex⯑pected. You had expreſſed your ſelf in Art. CXXV of your new Theory of Viſion, in the following words. "After reiterated endea⯑vours to apprehend the GENERAL IDEA of a Triangle, I have found it al⯑together incomprehenſible." But now, ‘Ut primum diſcuſſaeumbrae, & lux redditamenti,’ your eyes being opened, (pardon me this va⯑nity) by the arguments I have done my ſelf the honour of laying before you, you are pleaſed to ſay, "This implies that I hold, there are no GENERAL IDEAS. But I hold the direct contrary, that there are INDEED GENERAL IDEAS, but not formed by abſtraction in the man⯑ner ſet forth by Mr. Locke."

[111] I am ſo much pleaſed with this piece of ingenuity and candid proceeding, that for the ſake of it I willingly excuſe all that fol⯑lows, however inconſiſtent with this recanta⯑tion: Particularly your making no difference between a round ſquare, and a ſpace com⯑prehended by three right lines. For the ſame reaſon I willingly paſs by your ſuppo⯑ſing, that the words of my definition have no ideas, or conceptions of the mind, joined with them, and conſequently that the definition has no meaning. For to make the defi⯑nition have a meaning, ſome particular idea, ſimple or compound, muſt be joined to every word uſed in it; and a compound idea, made up of all thoſe particular ideas, muſt be joined to, and always go along with the whole defini⯑tion: And theſe two, the compound idea and the definition are inſeparable, if the de⯑finition be underſtood. Methinks therefore, inſtead of ſeparating theſe, it were better to make a diſtinction between this compound idea anſwering to the definition of a triangle, and the image, or ſenſible repreſentation of a triangle; two things which I have obſerv⯑ed you often to confound, both here and in your other writings. The compound idea is general, but the image, if exactly attend⯑ed [112] to and adequately perceived by the mind, muſt always be particular.

XLIX. You here propoſe ſome points for the Reader to reflect upon and examine by my light, when you well know I never endea⯑voured to give him any light about them. In this ſecond letter indeed I have, at your re⯑queſt, endeavoured to explain ſome part of them. But there are ſome others, which I am ſo far from being able to explain, that I never heard of them before, and cannot ima⯑gine what you mean by them. Poſſibly they may be ſome arcana of the Boeotian Ana⯑lyſis, explicable only by the Philoſophia prima.

L. As theſe Queries are not propoſed to me, I leave it to the conſideration of my learned Friends of this Univerſity, whether they deſerve or need any anſwer.