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The MINUTE MATHEMATICIAN: OR, The Free-Thinker no Juſt-Thinker. Set forth in A Second LETTER TO THE Author of the ANALYST; CONTAINING A Defence of Sir ISAAC NEWTON And the Britiſh Mathematicians, Againſt a late Pamphlet, entituled, A Defence of Free-Thinking in Mathematicks.

By PHILALETHES CANTABRIGIENSIS.

It is hard for thee to kick againſt the pricks, Acts ix. 5.
Thou art weighed in the balances, and art found wanting. Dan. v. 27.

LONDON: Printed for T. COOPER, at the Globe in Pater-Noſter-Row. MDCCXXXV Price 1s. 6d.

THE CONTENTS.

[]
  • Sect. I. PHilalethes in a paſſion. Grown cool. His preſumption. Candour of Mathematicians. He depends not on their favour. Takes no advantage of the ignorance of his other readers. His courage and the reaſon of it. Deplorable condition of the Author of the Analyſt.
  • II. Truth not to be hurt by inquiry.
  • III. Some poſitions hard to be underſtood.
  • IV. Motive to writing the Analyſt.
  • V. Faith does not conſiſt in ſeeing.
  • VI. Truth and Religion ſeemingly in oppoſition. Philalethes greatly puzzled. A curious criticiſm. Philalethes grievouſly miſtaken. Tenderly uſed. Miſtake of his Antagoniſt. Philalethes his opinion of the Clergy. Diſlikes a few of them. His Reaſons.
  • VII. Mathematicians plainly proved to be Infidels. Story of a witty man, ſaid to come from Mr. Addiſon.
  • VIII, IX, X. Injuſtice and paſſion of Philalethes. Modeſty of his Opponent. Inquiſition. [] Antient Diſcipline. Britiſh Laity no Fools.
  • XI. Author of the Analyſt abhors an inquiſition. With very good reaſon. Philalethes of the ſame mind. Perſons admirably well qualified for Inquiſitors.
  • XII. An inſtance of ſingular Modeſty.
  • XIII, XIV, XV, XVI. Idolatry of Philalethes. His Ambition. Deſire of knowledge not criminal. Philalethes not very credulous.
  • XVII. Sublime aſſertions of the Author of the Analyſt. Not eaſy to underſtand. Explained by the Author. Hypotheſis of Philalethes confirm'd. An unreaſonable challenge. Uncivil uſage. Attempt to explain the firſt principles of Fluxions.
  • XVIII. The moſt profound Mathematician in the World. What Readers Sir Iſaac Newton wrote for. Definitions not always ſo clear at firſt reading, as after ſome progreſs.
  • XIX. Great pains taken with little ſucceſs.
  • XX. Incomprehenſible Myſteries. Invented by the Author of the Analyſt.
  • XXI. Picture of that Author drawn by himſelf.
  • XXII. Appeal to the Reader.
  • XXIII. Pious arts uſed by the Author of the Analyſt. Attempt to explain ſecond Fluxions.
  • [] XXIV, XXV, XXVI. Great candour. Some perſons big with expectation. Diſappointed. One how undeceived. Diſpute de Lana Caprina.
  • XXVII. An extraordinary piece of Legerdemain.
  • XXVIII. Philalethes diſpoſed to peace. Agrees with his Antagoniſt. Conſequence of this agreement. Mathematical Spectacles. A wonderful aſſertion. A more wonderful Anſwer. A moſt wonderful inference. Even or odd, croſs or pile.
  • XXIX. Singular candour of Philalethes. One of Euclid's Axioms overturned. Author of the Analyſt can conceive no velocity in a horſe that ſtands ſtill. Nor Philalethes neither. A Horſe that is going muſt have ſome velocity. Sir Iſaac Newton quoted againſt Philalethes. The Reader deſired to believe his eyes. Conduct of a Scholar, a Gentleman, and a Chriſtian.
  • XXX. Queſtion for queſtion. Magnitude of Moments cannot be aſſign'd. Their Proportion may. Author of the Analyſt much at a loſs. Aſſiſted by his opponent. Delicacy of that Author's underſtanding. An evaneſcent quantity divided into halves. Strange conceits. Obſtinate Infidelity of a Free-Thinker. [] He loſes his ſight. Conjectures about the cauſe. A ſcience too hard for an Angel. Somebody overheard raving in Miltonick verſe. A dreadful Soliloquy. A Cat without her Membrana nictitans. Project to defeat a Book-ſeller of t'other ſix-pence. Succeeds ill. Strange infirmity of the Author of the Analyſt. Philalethes caſts a figure to remedy it. A moſt unintelligible affair render'd ſurprizingly plain. Reaſon miſtaken for mirth. The Author of the Analyſt catechiſed. Anſwers very badly. Senſe of the word moment finally ſettled between him and Philalethes. This ſenſe ſhown to be Sir Iſaac Newton's ſenſe. A ſecret. A miſtake rectify'd. Airs of Philalethes. Ghoſts and Viſions. Horrible Apparition of a great Apoſtate. His deſperate Speech. A ſecond Apparition. Commiſerates the firſt. Is reproved by him.
  • XXXI. Civility of Philalethes. He is charged with untruth. His generoſity and Chriſtian forgiveneſs. All difficulties vaniſh before him.
  • XXXII. A ſecret. Reſolution of Philalethes. Evaneſcent moment equal to evaneſcent increment. Former Hypotheſis of Philalethes farther confirmed. Candle and lanthorn. How to prove St. Paul an errant Heretick.
  • [] XXXIII, XXXIV. Much wrangling about four Latin words. Evidence to prove a Will. An important Lemma. Ill applied. Conſequence of a ſhower of rain. Philalethes feels himſelf cold and wet. Is not to be perſuaded out of that feeling. Conſequence of a falſe opinion.
  • XXXV. Author of the Analyſt not troubled with too much modeſty. Philalethes obliged to Sir Iſaac Newton. An excellent Eye-water.
  • XXXVI. A requeſt already complied with.
  • XXXVII. Whether will and can have the ſame meaning?
  • XXXVIII. Philalethes miſtaken in thinking Sir Iſaac Newton was accuſed of a double error. He begs pardon. Burns his fine picture. How he came to be miſtaken. Has a ſmall ſcruple ſtill remaining.
  • XXXIX. A challenge that might have been ſpared. Philalethes genteely call'd a matchleſs Lyar.
  • XL. Marquis de l'Hoſpital uſes as much ceremony as Euclid. Acquitted of a double error.
  • XLI. Philalethes acquitted of a charge brought againſt him. Writes for the information of ſome Great Churchmen. Falſely accuſed.
  • XLII. Declamation unanſwered.
  • [] XLIII. Diſcretion of Philalethes. What uniformity among Analyſts.
  • XLIV. A country ſwarming with mathematicians. Philalethes's apprehenſion for Religion in that country.
  • XLV, XLVI, XLVII, XLVIII. Modeſty and ſigns of grace in the Author of the Analyſt. He recants his principal error. Good nature of Philalethes. Difference between a triangle and a round ſquare. Item, between an intellectual Idea and a Picture.
  • XLIX. Arcana of the Boeotian Analyſis.
  • L. Queries uanſwered.

ERRATA.

Pag. 7. l. 25. read, tranſcriber.

Pag. 77. l. 5. read, of thoſe dire arms?

Pag. 80. l. 17. read, Very true, if I take it for increment or decrement of AB. But I will not take it for either, and then

Pag. 91. l. 3. read, XXXIII, XXXIV.

THE Minute Mathematician, &c.

[]
SIR,

I Freely own to you, when I ſat down to write my defenſe of Sir Iſaac Newton and the Britiſh 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 account 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 conſidering 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 endeavours to conceal it, not only in your ſentiments, [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 preſumption, in entring the liſts againſt ſo redoubtable 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 unaccountable to you, till you reflected on my ſeeming ſecure 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 Mathematicians; but that generally ſpeaking Mathematicians are fair and candid men. What ſhould make them otherwiſe? Are their opinions digeſted into Creeds and Articles, and eſtabliſhed 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 embracing 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 advantage can that be to me? To me, Philalethes, 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 enough 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 miſinforming 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 eaſily 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 extricate you.

[5]
Verte omnes tete in facies, & contrahe quicquid
Sive animis, ſive arte vales.

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 Mathematicks, 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 laudable, to canvaſs and to examine the principles and reaſoning made uſe of in any ſcience whatſoever, and that with the utmoſt freedom 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 Religion which they admit in human Learning. Do they admit that Fluxions are to them moſt incomprehenſible Myſteries? Do they, notwithſtanding this conceſſion, believe them to be clear and ſcientifick? Do they, notwithſtanding 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 defame Mathematicians in the month of March, Ann. Dom. 1734, rather than at any time before? The only queſtion with me was, Whether 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 Method 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 declamation of the worſt ſort, namely, the defamatory.

VI. More ſecret hiſtory and declamation, partly about what no body denies, and partly about what no body believes. You give us to underſtand, you have a right to examine Fluxions, even though Religion were quite unconcerned, and though you had no end to ſerve but Truth. No body diſputes your right of examining: but ſurely no good can be expected 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 Goſpel. 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, reclamantibus [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 becauſe 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 example 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 compliments as I do to our Church. Here I apprehend 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 uſeful 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 writing 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 Mathematicians. You had told us in the Analyſt, you were not a ſtranger to it: It was known: You were credibly informed. Now you go farther. You make no doubt of it: You have ſeen ſhrewed ſigns: You have been VERY credibly informed. 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 Mathematician. 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 endeavours to do ſervice to mathematical learning; 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 Analyſt, particularly to the thirty-firſt Query, where Sir Iſaac Newton is plainly charged with writing nonſenſe.

As to my frightful viſions and tragical uproars 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 Gentlemen with it.

Hoc regnum Dea gentibus eſſe,
Si qua fata ſinant, jam jam tendit (que) fovet (que)

XI. You ſay, you heartily abhor an Inquiſition in Faith. Upon my word you have a great deal of reaſon. You have been a grievous Free-thinker in your time: I do not mean in Mathematicks only. As great a Bigot 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 declaiming, and your own Modeſty, and the compliment you pay to Sir Iſaac Newton's Underſtanding. 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.

[12]

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 Idolatry.

—To deify his power,
Who from the terror of this arm ſo late
Doubted his Empire, that were low indeed.

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 applied to Virgil? Is adorare veſtigia to be literally 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 Idolatry for four ſections together?

—Quo numine laeſo?

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 Mankind. 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 unbounded 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, after 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 Europe, ſhould all this while have taken error for truth, and given Sophiſms for demonſtrations, 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.

—Magnis tamen excidis auſis.

XVII. You begin this ſection with addreſſing 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 infiniteſimal: That it is yet leſs poſſible to conceive 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 incomprehenſible can be MORE incomprehenſible than another incomprehenſible: How it can poſſibly 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 conceive 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 upon new matter. Beſides, I am afraid of incurring 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 return 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 towards 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 remembred that I do not here intend, nor indeed think my ſelf at all qualified to write a complete [18] treatiſe of Fluxions, that being expected from better hands. All that you require 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 intelligible as I can, I ſhall make uſe of the plaineſt and eaſieſt example poſſible, that I may give my Reader no other trouble than only that of comprehending the principles themſelves.

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

POSTULATUM.

Mathematical quantities may be deſcribed, and in deſcribing may be generated or deſtroyed, may increaſe or decreaſe, by a continued motion.

DEFINITIONS.

1. A Mathematical quantity increaſing or decreaſing by a continued motion is called a flowing quantity.

2. The velocity with which ſuch flowing quantity increaſes or decreaſes, is called the fluxion of that flowing quantity.

[19] 3. A part of ſuch flowing quantity generated in a very ſmall particle of time is called the augment or increment of the flowing quantity, if the flowing quantity be increaſing; or its decrement, if the flowing quantity be decreaſing.

4. A naſcent increment is an increment juſt beginning to exiſt from nothing, or juſt beginning to be generated, but not yet arrived at any aſſignable magnitude how ſmall ſoever. An evaneſcent increment is the ſame thing as a naſcent increment, but only conſidered in a different manner, as by a continual diminution becoming leſs than any aſſignable quantity, and at laſt vaniſhing into nothing, or ceaſing to exiſt.

Explanation of the Poſtulatum by an example.

If a point as A move in one direction from A to B with a continued motion, it will deſcribe and generate the right line AB. And if the ſame point return from B to A, it will deſcribe, and may thereby be ſuppoſed to deſtroy or annihilate the ſame right Line AB.

[figure]
The definitions explained by the ſame example.
[20]

1. While the generating point is in motion either way, the line deſcribed by it is called a flowing line. This flowing line perpetually increaſes, while the generating point is moving in the direction AB, and perpetually decreaſes, while the generating point is moving in the direction BA.

2. The velocity with which the generating point moves either way, or the velocity with which the flowing line increaſes or decreaſes, is called the fluxion of the flowing line. Ex. gr. The velocity of the generating point in C is called the fluxion of the flowing line AC; and the velocity of the generating point in B is the fluxion of the flowing line AB.

3. If in a very ſmall particle of time the generating point move from C to c, or from c to C, the ſmall line Cc is in the firſt caſe called the increment of the flowing line AC; and in the ſecond caſe is called the decrement of the flowing line Ac.

4. When the generating point, in deſcribing the line AB, is arrived at the point C, and proceeds from thence towards B: At the [21] inſtant of time that it ſets out or departs from the point C, at that very inſtant of time an increment begins to be generated, or begins to exiſt, which therefore is properly called a naſcent increment. And as the generating point at that inſtant of time is ſuppoſed to be juſt ſetting out, and not as yet to have moved to the leaſt imaginable diſtance from the point C, nor conſequently to have generated the leaſt imaginable increment, it is plain that the naſcent increment here conſidered will be leſs than any quantity that can be aſſigned.

In like manner when the generating point returns back from c to C, in order to annihilate the increment cC, that increment will continually grow leſs and leſs, will become leſs than any aſſignable quantity, and will at laſt entirely vaniſh and become nothing by the return of the generating point to the point C. At that inſtant of time therefore that the generating point returns to C, at that very inſtant I ſay the increment vaniſhes, and therefore is then properly called an evaneſcent increment.

Behold good Reader, the difficult, the obſcure, the myſterious, the incomprehenſible principles of Fluxions! I am much miſtaken if a little attention do not enable thee clearly [22] to conceive them. When thou haſt done this, then wilt thou be rightly prepared for underſtanding the following fundamental propoſition, upon which Sir Iſaac Newton has eſtabliſhed his Method of Fluxions: the whole buſineſs of which Method is, from the proportion between the Fluxions, or between the naſcent increments, of flowing quantities, to determine the proportion between the flowing quantities themſelves; & vice verſa.

PROPOSITION.

The Fluxions, or Velocities, of flowing quantities are very nearly as the increments of thoſe flowing quantities, generated in very ſmall equal particles of time: And they are exactly in the firſt proportion of the naſcent increments, or in the laſt proportion of the evaneſcent increments.

DEMONSTRATION.

1. If the velocities are uniform, it is plain that the increments generated in any equal times muſt be as thoſe velocities.

[23] 2. And if the velocities are not uniform, but are perpetually changing, yet in a very ſmall particle of time their change will be very little, and the increments will be very nearly the ſame as if the velocities were uniform, i. e. the increments will be very nearly as the velocities with which they begin to be generated.

3. And as the firſt ratio of the naſcent increments muſt be the ſame, whether the velocities be uniform or variable, it follows that the naſcent increments muſt be exactly as the velocities with which they begin to be generated. Q. E. D.

Here, Sir, I muſt beg leave to obſerve to you, that if Sir Iſaac Newton had proceeded no farther than the firſt part of this propoſition, and had contented himſelf with eſtabliſhing the proportion between the increments and their velocities very nearly, without going to the utmoſt exactneſs, yet his Method had been no leſs ſcientifical and no leſs demonſtrative than it now is. Conſequently you were very much overſeen in charging him and his followers with proceeding blindfold, and not knowing what they were doing, even though you had ſucceeded [24] in proving that his Method did not come up to the rigor of Geometry.

To prevent cavils, I muſt farther obſerve that the third part of this demonſtration might eaſily be put into a more diffuſive form, and might be deduced ſtep by ſtep from the Methodus rationum primarum & ultimarum. But this at preſent is no way neceſſary, eſpecially as you admit the propoſition to be true. All that I have to do therefore is to explain it a little more particularly, and this I ſhall be the more careful in doing; becauſe this proportion of naſcent or evaneſcent increments is what I apprehend, you are ſo often pleaſed to call a proportion between nothings. With what juſtice you do ſo the Reader may eaſily judge, if he gives himſelf the trouble of conſidering what follows.

In the firſt place, and above all, it is here to be diligently attended to, that Sir Iſaac Newton no where ſettles or determines the magnitude of naſcent or evaneſcent increments any farther than to ſay it is leſs than any finite magnitude. On the contrary, he expreſly declares that their magnitude cannot be aſſigned or determined. Nor indeed has he any occaſion to determine their magnitude, [25] but only the proportion between them, this being all that is requiſite in his Method.

Now the proportion between two evaneſcent increments is eaſily to be conceived, though the abſolute magnitude of thoſe increments is utterly imperceptible to the imagination. For thoſe increments may be expounded or repreſented by any two finite quantities bearing the ſame proportion to one another: And as theſe finite quantities may be clearly conceived, the proportion between them may likewiſe be clearly conceived, i. e. the proportion of the evaneſcent increments may be clearly conceived by this means. Of this ſeveral examples may be found in Sir Iſaac's Methodus rationum primarum & ultimarum, and one in imitation of him may be ſeen in this Letter, Sect. XXXII.

This being premiſed, I come now to illuſtrate and explain the Propoſition, to which end I ſhall make uſe of the following eaſy example.

[26]

[figure]

Let the right line AbB, divided into two equal parts in the point b, revolve about the point A, and with any continued motion, even or uneven, remove into the ſituation AcC. Then will the points B and b deſcribe the circular arcs BC, bc: The velocity of the point B will be always double of the velocity of the point b: The arc BC will be double of the arc bc: And the increment BD will be double of the increment bd. In all this there is no difficulty between us.

I ſay farther, The naſcent or evaneſcent increment of the arc BC is double of the naſcent or evaneſcent increment of the arc bc. This you won't underſtand. I explain it thus, beginning firſt with the naſcent increments.

As ſoon as the line AbB begins to revolve upon the point A, and thereby begins to depart from the ſituation AbB; at that inſtant of time do the points B and b begin to generate their ſeveral increments. And as the velocity of the point B is always double of the velocity of the point b, it is manifeſt that [27] the increment of the arc BC begins to be generated with twice the velocity that the increment of the arc bc begins to be generated with; i.e. that the naſcent increment of BC is generated with twice the velocity that the naſcent increment of bc is generated with; and conſequently that the former naſcent increment is by this propoſition double of the latter naſcent increment.

To come now to the evaneſcent increments, let us ſuppoſe the line AbB to have removed into the ſituation AdD very near to AbB, whereby the increments BD, bd have been generated. Next let us imagine the line AbD to return gradually to its firſt ſituation AbB, and thereby let the increments BD, bd grow continually leſs and leſs, and at laſt entirely vaniſh and become nothing. Then as the firſt of theſe two increments is double of the ſecond, and decreaſes twice as faſt as the ſecond, it muſt perpetually bear the ſame proportion to the ſecond; and conſequently the laſt proportion of theſe two increments, their proportion at the inſtant of evaneſcence will be the ſame as at firſt, namely that of 2 to 1. You tell me, when they vaniſh, they become nothing. I allow it. You ſay, to talk of a proportion [28] between nothings is to talk nonſenſe. I agree with you. But their laſt proportion is not their proportion after they are vaniſhed and are become nothing: It is their proportion when they vaniſh: It is the proportion with which they vaniſh.

You will tell me, perhaps, this is unintelligible. I expect it. I ask you therefore, which vaniſhes firſt? The increments themſelves? Or the proportion between them? I think, Sir, even you will not venture to ſay, that the increments vaniſh before their proportion vaniſhes; or that the proportion vaniſhes before the increments vaniſh. If ſo; we are agreed thus far, that the increments vaniſh and their proportion vaniſhes at one and the ſame inſtant of time. This proportion therefore which vaniſhes at the ſame inſtant of time that the increments vaniſh, is the proportion with which the increments vaniſh, or, in other words, is the laſt proportion of the evaneſcent increments.

This, I hope, will appear ſufficiently clear to an attentive Reader: But for his farther ſatisfaction I ſhall beg leave to lay before him another example.

Let the point A with a given uniform velocity deſcribe or generate the flowing line

[29]

AB: And let the point D with a velocity continually increaſing deſcribe or generate the flowing line DE: Alſo let both points arrive at the line CF, (cutting the two flowing lines) at the ſame inſtant of time, and with velocities exactly equal.

Then it is plain, that if we take two increments Ff, Cc, generated in the ſame particle of time, Ff will a little exceed Cc. But if we ſuppoſe the generating points to return towards the line CF, and their reſpective velocities in every point of the increments Ff, Cc, to be the very ſame in returning as they had been before in proceeding from the line CF; it is manifeſt that the more the increments are diminiſhed by the gradual return of the generating points towards the line CF, the nearer will the proportion between them approach to that of a perfect equality. This is eaſily conceived, and admits of no diſpute.

Farther, if the generating points be ſuppoſed to return exactly to the line CF, and thereby [30] by the increments vaniſh and become nothing; the ratio with which the increments vaniſh into nothing, or the laſt proportion of the evaneſcent increments, will be that of a perfect equality. For, as during the time that the generating points are returning towards the line CF, the increments Ff, Cc are continually more and more diminiſhed, and the velocities with which the increments decreaſe, approach more and more to the ratio of equality; ſo at the inſtant of time that thoſe points actually arrive at the line CF, at that ſame inſtant the increments entirely vaniſh, and at the very ſame inſtant the velocities with which they decreaſe and in decreaſing vaniſh, arrive at the ratio of perfect equality: which therefore is the ratio of the velocities with which the increments vaniſh, and conſequently, by this propoſition, is the ratio of the evaneſcent increments.

It is to be carefully attended to, that the proportion here given as the proportion of the evaneſcent increments, is not their proportion before they vaniſh. For then Ff will exceed Cc. Nor is it their proportion after they have vaniſhed. For then they are become nothing and [...] proportion. [31] But it is their proportion at the inſtant that they vaniſh, or the proportion with which they vaniſh.

I might obſerve farther, that as the increments do not come to this proportion before they vaniſh, ſo neither do they vaniſh before they come to this proportion: but at one and the ſame inſtant of time they come to this proportion and vaniſh, they vaniſh and come to this proportion. But I am now afraid I have taken up too much of my reader's time in explaining a point ſufficiently clear before.

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 profound Mathematician. How well the moſt profound Mathematician can judge, can, I think, be certainly known to the moſt profound 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 reader of common ſenſe. He wrote for Mathematicians.

[32] Nor can I agree with you that the ſimple apprehenſion of a thing defined is not, ſometimes at leaſt, made more perfect by any ſubſequent Progreſs in Mathematicks. It happened 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 become clearer upon ſeeing the application of the things defined in different examples, than upon only reading and conſidering thoſe definitions 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, conſidering you never had a Maſter in Mathematicks.

—Neque ego tibi detrahere auſim
Haerentem capiti multa cum laude coronam.

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, motion without extenſion, magnitude which is neither 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 incomprehenſible; but then I take them to be Myſteries of your own making: I can find no more ſign of them in Sir Iſaac Newton's writings, 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 principles, 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 Analyſt, 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.

Eripiunt ſubito nubes coelumque diemque
Lectorum ex oculis.

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 inſtructing him, and had put the two following queſtions to you, which I ſhall here tranſcribe 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 celerity of an incipient celerity, the naſcent augment 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 worthy the obſervation of a curious reader to ſee what arts and ſhifts a great Genius may be driven to in grappling with an inſuperable difficulty.

In the firſt place you curtail my firſt queſtion, cutting off the latter part of it with an &c. By this means you hide from your reader 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 expreſſions? No. You don't pretend to it. What then? Why truly you endeavour to ſhew, by comparing together two independent Paſſages taken from two different treatiſes of Sir I. N. that you may juſtifiably call a ſecond 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. Therefore 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 examined into by any of our family of Philalethes.

[37] In order to get out of this Egyptian darkneſs 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 fluxions, &c. ad infinitum, upon the foot of Sir Iſaac Newton's definition of ſecond fluxions.

Adſpice, nam (que) omnem, quae nunc obducta tuenti
Mortales viſus hebetat tibi, & humida circum
Caligat, nubem eripiam.

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 velocity do not always continue the ſame, but undergo any change, the velocity of that change is called a fluxion of a fluxion, or a ſecond 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 velocity 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 infinitum.

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 proportions: 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 only the proportion between them. And this I deſire may be carefully remembred.

A—a
1F—1f
2F—2f
3F—3f
4F—4f

[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 velocity 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 never 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 continually 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 infinitum.

Obſerving the ſame analogy, let 1f repreſent 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 infinitum.

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 proportion between the third fluxions will be that [41] of the finite lines 3F and 3f: The proportion 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 incomprehenſible than a firſt fluxion.

XXIV, XXV, XXVI. I do not remember 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 rectangle of two flowing quantities, and thereupon had thought fit to repreſent them as not proceeding 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 compliments 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 inſignificancy 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 abundantly in the ſtyle which the Learned call the tautological.

You tell me, and it might have been ſufficient once to have told me, that the application 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 obſerved 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 tickled 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 ſatiſfaction, 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 acquaintance to the following effect. The Author of the Minute Philoſopher has found out that, if Sir Iſaac Newton were to meaſure the height of St. Paul's Church by Fluxions, he would be out about three quarters 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 into a great paſſion, and after muttering ſomething 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: Whichſoever 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 rectangle 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 Aa × Bb.

I had obſerved, Sir, that you were miſtaken in taking it for granted, that what Sir Iſaac Newton was endeavouring to find by the ſuppoſitions made in this demonſtration, 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 contradict 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 Madman: 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 contradict 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 incrementum 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 demonſtrated:

Involvere diem nubes, nox humida coelum
Abſtulit.—

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 rectangle 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 rectangulum 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 Newton means, but what he mentions: Not about 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 preceding and following it, I affirm and aver that he does not ſo much as once mention the increment of the rectangle AB. Deny it, if you dare.

—Vim duram & vincula capto
Tende. Doli circa haec demum frangenturinanes.

XXVIII. You tell me, I would fain perplex this plain caſe by diſtinguiſhing between an [48] increment and a moment. But it is evident to every one, who has any notion of demonſtration, that the incrementum in the concluſion muſt be the momentum in the Lemma; and to ſuppoſe it otherwiſe is no credit to the Author. Now, Sir, to ſhew you how little I am inclined 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 Aa × Bb, 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 againſt me to ſhew that a moment is an increment 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 increment 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 general, but between a particular moment and a particular increment, between the moment 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 affirmed, 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 decrement of AB; neither the exceſs of A+a × B+b above AB; nor the defect of Aa × Bb from AB. This ſeems to you a wonderful 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 moment of AB? Your anſwer is, Either of them. This to me is a very wonderful anſwer for ſo great and ſo accurate a Mathematician to make, and if I have not quite forgot 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 equal to the decrement of AB. That is, Ab+Ba+ab=Ab+Baab, i.e. 2ab=o. Therefore the rectangle ab, about which the Author of the Minute Philoſopher 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 agreed that all numbers are either odd, or even. Upon this you pronounce an unknown number to be even, without giving any reaſon 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 Arithmetician, 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 ſeeming levity.

Nor is the accommodation I propoſed in the diſpute between an increment and a decrement for the title of moment, at all the leſs reaſonable for being delivered in a ludicrous 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;

Ridentem dicere verum quid vetat?

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 perſon 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 determine me either way. This led me to fix upon a middle arithmetical proportional between theſe two velocities, for the velocity of the rectangle AB: As I had before ſhewn its moment to be a middle arithmetical proportional between the increment and decrement. But you, who talk ſo much of reaſoning and logick, and who ſet up for the great and ſole Maſter of the [...] Geometrica, are of opinion that either of theſe velocities may be deemed the velocity of the rectangle AB. That is, in your opinion, of two unequal 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 rectangle AB conſidered ſimply in itſelf, without either 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 conceive that velocity to be different in every point of time: And if we ſuppoſe the increment of the rectangle to be generated in a given particle of time, and the decrement of the ſame rectangle to be generated in another 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 decrement in the ſame time: And theſe two velocities being unequal, I can conceive an arithmetical mean between them; in like manner as I had before ſuppoſed an arithmetical mean between the increment and decrement of AB, which mean is the moment of AB: And laſtly, while AB flows, I can conceive that the firſt arithmetical mean is conſtantly proportional to the laſt, i.e. that the velocity of AB is proportional to the moment 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 certainly 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 affirmative moments, and decrements are negative moments. Little would any one imagine from the aſſurance with which you here expreſs yourſelf, that all I maintain is, as I ſaid a while ago, That one particular determined moment is not one particular determined increment. But your chicaneries are ſo many, ſo groſs, and every way ſo ſhameful 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 detected ſo many, almoſt in every paragraph of your Reply, I have knowingly and voluntarily paſſed by many more, particularly thoſe ſcandalous ones of almoſt perpetually changing 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 diſtinguiſhed into halves. Is it, ſay you, a finite quantity, 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 either 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 propoſe? 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 former; I tell you what Sir Iſaac Newton has told you before, a moment is a momentaneous, or naſcent increment, proportional to the velocity of the flowing quantity. If the latter; I have no buſineſs at all to conſider [56] the magnitude of a moment.* Neque enim ſpectatur, ſays Sir Iſaac Newton, magnitudo momentorum, ſed prima naſcentium proportio. I may tell you farther, that the magnitude of a moment is nothing fixed nor determinate, is a quantity perpetually fleeting 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 beginning to exiſt, but not yet arrived to any aſſignable magnitude, can be divided or diſtinguiſhed into two equal parts. Now to me there appears no more difficulty in conceiving this, than in apprehending how any finite quantity is divided or diſtinguiſhed into 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 moment 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 moment 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 conceiving any thing that makes againſt you, and therefore ſhall readily you give the beſt aſſiſtance I can towards overcoming this difficulty 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 obſerves 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

[58]

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 diminiſhed 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 inſtant. Here then we have the evaneſcent line AB actually divided into two equal parts, as was above propoſed. For this diviſion 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 conceiving 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 unintelligible 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 gravelled 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 another Ghoſt! But alas! What would even that avail me againſt a Freethinker in Mathematicks, againſt a man ſo hardened in infidelity, 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, exclaiming yonder?

O thou, that with ſurpaſſing glory crown'd
Look'ſt from thy ſole dominion like the Author
[61] Of this new Method; at whoſe ſight the Sages
Hang their unfurniſh'd heads! To thee I call,
But with no friendly voice, and add thy name,
Iſaac! to tell thee how I hate thy wreaths,
That bring to my remembrance from what ſtate
I fell, elated far above thy ſphere;
Till pride and luſt of M [...]e threw me down,
Warring in vain againſt thee, matchleſs Knight!
Ah wherefore! He deſerv'd no ſuch return
From me. 'Twas he that taught me all I knew
Of fluents, moments, and of increments
Naſcent or evaneſcent, with his ſcience
Upbraiding none, nor were his fluxions hard.

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

O then at laſt relent! Is there no place
Left for repentance, none for pardon left?
None left but recantation, and that word
Diſdain forbids me, and my dread of ſhame
'Mongſt Aaron's Lordly Sons, whom I deluded
With other promiſes, and other vaunts,
Than to recant; boaſting I could ſubdue
The Analyſts. Ay me! They little know
How dearly I abide that boaſt ſo vain,
Under what torments inwardly I groan,
While Br [...]s adore me on the Thr [...]e of Cl [...]
With M [...]e and with Cr [...]r high advanc'd.

[62] But hold! Theſe circumſtances ſurely can never ſuit my correſpondent; and beſides, 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 nictitans. 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 reſolved 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 diſappointed 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 account 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 purely, or partly, intellectual, and which have no ſenſible images to repreſent them. But of this diſeaſe I may perhaps ſpeak more largely another time; at preſent I ſhall endeavour to obviate this defect in him by the following figure.

[figure]

Let therefore RALB, or RL, repreſent the flowing rectangle AB in Sir Iſaac Newton'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 rectangles eRbq, iRcr, oRdſ, uRft.

Then will the rectangle A−½a × B−½b be repreſented by the rectangle Rr; the rectangle Aa × Bb 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 gnomon rſ; lying partly within and partly without the rectangle RL.

The rectangle Aa × Bb will be repreſented by Rq; and the difference between this rectangle and the rectangle AB, or the decrement of AB, will be repreſented by the gnomon Lq lying within the rectangle 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 repreſented 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.

[65]

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 Aa × Bb: i.e. rſ; is neither 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 decrement 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 Aa × Bb, which may be conſidered as the decrement of the firſt, or as the increment of the laſt: i.e. rſ; is not Lq common to RL and Rq, which may be conſidered [66] as the decrement of RL, or as the increment of Rq.

But it is the moment of the very individual rectangle AB itſelf, and peculiar to that only; and ſuch as being conſidered indifferently either as an increment or decrement, ſhall be exactly and perfectly the ſame: i.e. rſ; is the moment of RL, and peculiar only to RL; and if RL be conſidered as an increaſing quantity, rſ; may be conſidered as an increment; if RL be looked upon as decreaſing, rſ; may be conſidered as a decrement. But whether rſ; be conſidered 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 increment or decrement of the rectangle RL, ſhall be exactly the ſame, ſuch as is not common 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 Aa × Bb i.e. between RL and Rq: Not to ſuppoſe AB as lying at either extremity of the moment, but as extended to the middle

[67]

of it, i.e. not to ſuppoſe Lt to be the moment 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 extended to the middle of it; as having acquired rL the one half of the moment, and being about to acquire the other half Lſ; or as having loſt Lſ; the one half of the moment, 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 cannot 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 opportunity [68] of ſhewing you, that there is ſome reaſoning couched under what you are pleaſed to take for mirth and humour, in the proof that I have given, pag. 45, 46. of my Defence, that the moment of the rectangle AB is not the increment or decrement of AB, but a middle arithmetical proportional between them.

After propoſing to you what by your own confeſſion is the increment, and what the decrement 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 decided 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 unequal? 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, EITHER of them.

Things ſtanding thus, I offer this argument to your conſideration. Since, according 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, according 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 allowed to ſay, "Believe me there is no remedy, you muſt acquieſce."

—Fruſtra cerno te tendere contra.

I ſuppoſe, Sir, you may now comprehend my meaning, when I ſay, that the moment of AB is not the increment of AB, tho' I allow the moment of AB to be an increment, 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 favour, that by the moment of AB Sir Iſaac means ſomething different from the increment of AB. For if theſe two words ſignified 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 rectangle 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, according 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 proportional to the velocity of that ſame flowing quantity; it will follow that what I ſuppoſe to be the moment, is the ſame with the moment intended by Sir Iſaac Newton.

In order therefore to render the conception 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 flowing line is generated, in all the following caſes, and ſince time flows uniformly, let the conſtant quantity repreſent the moment, or increment, (for in this particular caſe they are both one) of the time x.

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

Caſe 2. Let the velocity be equably accelerated, as in the caſe of a falling body according 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 repreſented by x2. Now the ſuppoſed moment of this line x2, is 2xẋ, and I ſay, 2xẋ is proportional to x the velocity of the flowing line. For ſince 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 velocity 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 velocity 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 conſequently is the moment ſuppoſed by Sir Iſaac Newton.

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

The Decrement.Moment.Increment.
2xẋẋẋ2xẋ2xẋ+ẋẋ
Velocity.Velocity.Velocity.
x/2xx+/2

Here it appears that as the moment is a middle arithmetical proportional between the decrement and increment; ſo is the velocity of the flowing line, or the velocity that would generate the moment in the given time 2ẋ, a middle arithmetical proportional between the velocities that would reſpectively generate the decrement and increment in the ſame given time. And this proportion equally holds, whether the moments 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 velocity 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 moment. But the quantity that Sir Iſaac Newton 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 queſtion, and ask with my accuſtomed air, "What ſay you, Sir? Is this a juſt and legitimate reaſon for Sir Iſaac's proceeding as he did? I think you muſt acknowledge it to be ſo."

But hark you! Why all this outcry about Ghoſts and Viſions? Pray who firſt introduced them? If I brought in one, you might conſider it was to a very good purpoſe, [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.

See where the Phantom comes, a ſable wand
Before his decent ſteps! Of regal port,
But faded ſplendour wan: His flowing hair
Circled with golden Tiar: A gorgeous veſt,
Dyed Meliboean, from his ſhoulders broad
Hangs graceful down: In ſable armour clad,
Sable his body, but in whiteſt mail
His ſinewy arms refulgent: Such the bird
Majeſtick treads the albent cliffs, or wings
The air Royſtonian. Paſſion dims his face
Thrice chang'd with pale, ire, envy and deſpair.
His geſtures fierce, and mad demeanour mark!
His form disfigur'd more than can befal
Spirit of happy ſort: For heavenly minds
From ſuch diſtempers foul are ever clear.
The thought both of loſt fame and laſting ſcorn
Torments him; round he throws his baleful eyes,
That witneſs huge affliction and diſmay,
[77] Mixt with obdurate pride and ſteadfaſt hate;
And breaking ſilence, horrid, thus begins.
Fall'n from what height! So much the ſtronger prov'd
He with his Moments: And till then who knew
The force of theſe dire arms? Yet not for thoſe,
Nor what the potent Victor in his rage
Can elſe inflict, do I repent, or change,
Though chang'd in outward luſtre, that fixt mind,
And high diſdain from ſenſe of ſelf-weigh'd merit,
That with proud Newton rais'd me to contend,
And ſhook his Throne. What though the field be loſt?
All is not loſt. Th'unconquerable will,
And ſtudy of revenge, immortal hate,
And courage never to ſubmit, or yield,
As at the head of battle ſtill defies him,
Undaunted, ſince by Fate the wings of Ganders,
And Sepia ſable-blooded cannot fail.
So ſpake th'Apoſtate Analyſt, though in pain,
Vaunting aloud, but rack'd with deep deſpair.
Frowning he ended, and his look denounc'd
Deſperate revenge, and battel dangerous
To leſs than Philalethes; when upſtood
One next himſelf in crime, in ſtrength ſuperior;
Niſroc, of principalities the prime.
And to that eminence by merit rais'd;
Niſroc, the ſtrongeſt and the fierceſt Spirit,
That fought in this bad cauſe, the ſtrongeſt far,
The fierceſt once, now broken with deſpair.
His truſt was with great Iſaac to be deem'd
[78] A match in ſtrength, and rather than be leſs,
Car'd not to be at all. Grown humbler now,
As one, he ſtood, eſcap'd from cruel fight,
Sore toil'd, his riven arms to havock hewn,
Mangled with ghaſtly wounds through plate and mail.
Clouded his brow, deep on his front engraven
Sat meditation ſilent, in his eye
Shone piercing contemplation, thought profound,
And princely counſel in his face yet ſhone,
Majeſtick, though in ruin. Sage he ſtood,
With Atlantean ſhoulders fit to bear
The weight of loftieſt Theories: His look
Drew audience and attention ſtill as night,
Or ſummer's noontide air, while thus he ſpake.
O Prince, O Chief of many wronghead Powers,
That led th' imbattled Increments to War
Under thy conduct, and with dreadful blunders,
Brainleſs, endanger'd Newton's deathleſs Fame;
And put to proof his high Supremacy,
By chance upheld, or ſcience; and that ſtrife
Was not inglorious, though th'event was dire:
The dire event too well I ſee and rue,
That with ſad overthrow and foul defeat
Hath loſt thy fame, and all this muddy Hoſt
In horrible deſtruction laid thus low,
As far as Ghoſts and ſhadowy Entities
Can periſh.

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.

If thou beeſt he: But O how fall'n! How chang'd
From him, from that ſworn Friend, whom mutual league,
United thoughts and counſels, equal hope
And hazard in the glorious Enterprize,
Join'd with me once, now miſery hath join'd
In equal ruin—

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 introduced 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 caution 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 finger 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 difficulties are equally inſuperable. Very true. But I will not take it for either increment or decrement, and then all difficulties and objections vaniſh before me, they become nothing, there are no difficulties, no objections, I meet with nothing in my way but the Ghoſts of departed difficulties and objections.

XXXII. Before I proceed to vindicate that aſſertion of mine which makes the ſubject 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 reaſon 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 preſent, 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,

—I am in ink
Stept in ſo far, that ſhould I wade no more,
Returning were as tedious as go o'er.

My aſſertion was, That the moment of the rectangle AB, determined by Sir Iſaac Newton, namely aB+bA, and the increment of the ſame rectangle determined by your ſelf, namely aB+bA+ab, are perfectly 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 quantities, 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 aſſertion, 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 diminiſhed, during a given time, by the continued 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 ſuppoſed 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 inſtant 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 diminiſhed. And conſequently the firſt part of your anſwer is quite beſide the purpoſe: It tends only to ſhew that there is a real difference 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 whereof they are coefficients, become nothing likewiſe: [84] and conſequently the momentum or incrementum, 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 Newton'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 quantities, 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 quantities aB+bA and aB+bA+ab, vaniſhes likewiſe, and there is no longer any difference 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 themſelves 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 equality, and that they do ſo, I am going to demonſtrate after Sir Iſaac Newton's manner.

[figure]

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 vaniſhes, the moment 2xẋ, and the increment 2xẋ+ẋẋ, will vaniſh with a ratio of equality.

DEMONSTRATION.

Produce the lines AD, BE, CF, to the diſtant points d, e, f, and draw the right line def parallel to DEF. Then will the [86] rectangles AE, AF be proportional to the rectangles Ae, Af. Now let CB be diminiſhed ad infinitum, and vaniſh into nothing by the coincidence of the point C with the point B. At the inſtant that theſe points coincide, the lines CFf, BEe will likewiſe coincide, i.e. the rectangles Ae, Af, will coincide and become perfectly equal, and at the ſame inſtant the rectangles AE, AF, i.e. the moment and increment, will vaniſh. But at the inſtant that the rectangles Ae, Af become equal, the rectangles proportional to theſe, AE and AF muſt likewiſe become equal. Therefore theſe rectangles vaniſh and become equal at one and the ſame inſtant of time, or vaniſh with a ratio of equality. Q.E.D.

I am ſo deſirous of leaving both you, Sir, and my reader without any ſcruple upon this point, that I cannot content myſelf with only demonſtrating, that in fact the thing is as I ſay, unleſs I likewiſe ſhew you by what means it comes to be ſo. The caſe is this.

While is gradually diminiſhed, the ratio between the increment and moment is likewiſe perpetually diminiſhed, and tends to a certain limit which it can never paſs, and can never arrive at till is diminiſhed ad [87] infinitum, and vaniſhes into nothing. That limit is equality.

Likewiſe while is gradually diminiſhed, the increment and moment are perpetually diminiſhed, and tend to a certain limit, which they can never arrive at till is diminiſhed ad infinitum, and vaniſhes into nothing. That limit is nothing.

So that the ratio of the increment and moment, and the increment and moment, do both arrive at their ſeveral limits, i.e. at equality and at nothing, at one and the ſame inſtant of time. That is, the increment and moment become equal and vaniſh, vaniſh and become equal, at the ſame inſtant.

Methinks, Sir, you and I are now ſo far agreed, that it is pity there ſhould be any difference between us about the Lemma I quoted to you. But as it may be of ſome ſervice to you, and may poſſibly ſave trouble to us both another time, I am willing to take a little farther pains for your information; though I greatly fear it will be loſt upon you, and that you will make no better uſe of it, than you did of the friendly advice I gave you in my laſt letter, to weight very well what Sir Iſaac Newton ſays, before you cenſure him. For I ſee my hypotheſis about [88] the cauſe of your errors ſtill holds good: You have too good an opinion of your own underſtanding, to think you can ever be miſtaken. Elſe how was it poſſible for you to ſay, when ſuch a man as Sir Iſaac Newton was laying down the foundation of the Method of Fluxions, That his very firſt and fundamental Lemma was incompatible with and ſubverſive of the Doctrine of Fluxions? That it ſeemed the moſt injudicious ſtep that could be taken? That it was directly demoliſhing the very doctrine I would defend? Pray let us ſee what this Lemma is.

LEMMA. I.

Quantitates, ut & quantitatum rationes, quae ad aequalitatem tempore quovis finito conſtanter tendunt, & ante finem temporis illius propius ad invicem accedunt quam pro data quavis differentia, fiunt ultimo aequales.

In this Lemma are manifeſtly contained the following ſuppoſitions.

  • 1. That the quantities or ratio's of quantities, tend to equality.
  • 2. That this tendency to equality conſtantly holds during a given time.
  • [89] 3. That they come nearer to equality than to have any aſſignable difference between them.
  • 4. That they come thus near to equality before the expiration of the given time.

Upon theſe ſuppoſitions Sir Iſaac affirms and demonſtrates, that the quantities do at laſt become equal, i.e. do become equal at the end of the given time.

We come now to ſee what you object to this; you, I ſay, who have long ſince conſulted and conſidered this Lemma; you, who very much doubt whether I have ſufficiently conſidered this Lemma, its demonſtration, and its conſequences; you, who have taken as much pains as (you ſincerely believe) any man living to underſtand that great Author, and to make ſenſe of his principles; you, on whoſe part, you aſſure me, no induſtry, nor caution, nor attention have been wanting: So that, if you do not underſtand him, it is not your fault but your misfortune. I am going to take my candle and lanthorn, as Harlequin did a while ago at Paris to look for the complete victory at Parma; and ſhall make a diligent ſearch after your induſtry, caution and attention in conſidering this ſhort Lemma. It certainly deſerves all the caution you can uſe, ſince [90] it contains, according to Sir Iſaac Newton, the foundation not only of the method of fluxions, but of the Principia themſelves, of that book which is the admiration and aſtoniſhment of all mankind, except the Author of that greater and more ſtupendous work, The Minute Philoſopher.

You ſuppoſe Sir Iſaac Newton to argue, that quantities muſt be equal, becauſe they have no aſſignable difference. Is this then the only ſuppoſition he makes, that quantities have no aſſignable difference? Does he not plainly make the firſt, the ſecond and fourth ſuppoſition above-mentioned, as well as the third? Are the following words, ad aequalitatem tempore quovis finito conſtanter tendunt, & ante finem temporis illius, left out of your copy? If not, where were your eyes, that you overlooked them? Or your integrity, that you ſuppreſſed them? Might not the moſt orthodox Father of the Church, or the great Apoſtle St. Paul himſelf, be proved an errant Heretick by ſuch a proceeding? For ſhame go and look over that Lemma again, read it diligently, conſider it throughly, underſtand it if you can, and till you have done ſo, never dare to take the venerable [91] name of Sir Iſaac Newton within your lips, much leſs to condemn him.

XXXIII. We come now to the method for obtaining a rule to find the fluxion of any power of a flowing quantity, which is delivered in the introduction to the Quadratures, 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 rendering 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 poſſibly ſpend twice as many more before we have done, in wrangling about the tranſlation 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 troubling 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 endeavoured 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 together, by comparing it with an equivalent paſſage, 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 tranſlated: If they ſtood alone, they might be tranſlated 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 evaneſcent augments, or to conſider the proportion with which the augments vaniſh. He plainly makes two ſuppoſitions in this paſſage. The firſt is, that the augments vaniſh, or become nothing. The ſecond, 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 vaniſhed, 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 rationem 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 blunder of making the quantities firſt become nothing, and then ſettling the proportion between 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 perſon 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 deſpair of making me acknowledge, that vaniſhing and becoming nothing were equivalent 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 Lemma. But now, as you are pleaſed to ſhew more than ordinary arrogance in this and the following ſection, I hope the reader will excuſe 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 ſometimes 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 Mathematicks, 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 increments, [96] with intent to deduce the proportion of the increments from thoſe expreſſions. He afterwards ſuppoſes that thoſe increments vaniſh, i.e. ſay you, that the increments are nothing, that there are no increments.

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 former, and in deſtroying the former it deſtroys the expreſſions, the proportions, and every thing elſe derived from the former ſuppoſition. Not too faſt, good Mr. Logician. If I ſay, the increments now exiſt, and, the increments do not now exiſt; the latter aſſertion will be contrary to the former, ſuppoſing now to mean the ſame inſtant 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 former, 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 conſequences of their ſuppoſed exiſtence, as their expreſſions, proportions, &c. muſt now be ſuppoſed 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 diſtance 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 conſequently 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 indifferent 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 perſuaded 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 immediately convinced all mankind of the contrary. 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 Genleman's mind, if it had.

[99]

XXXV. You miſtake me, Sir: What I diſlike in you is not your modeſty, but your arrogance. 'Tis your unparallel'd and amazing 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 explain 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 underſtand 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 Mathematicks, 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 unpardonable vanity.

You ſay, I inſult you, in asking what it is you are offended at, who do not ſtill underſtand 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 doctrine, 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 mention: [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 another contrary and equal error, which perhaps he never knew himſelf nor thought of: that by a twofold miſtake he arrives, though not at Science yet at Truth: that he proceeds blindfold, &c. All which is untruly ſaid by you, who have miſapplied to Sir Iſaac Newton, what was intended for the Marquis de l'Hoſpital and his followers." 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 undoubtedly 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 queſtion. On the contrary, I aſk your pardon for my miſtake; and to make you all the ſatisfaction 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 alledge the following reaſons in mitigation of my offence.

1. Your diſcourſe ſeemed to me to be directed 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 apply, your concluſions, your logick and method, &c. I too haſtily judged that the double error 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 Marquis.

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. Likewiſe 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 poſſeſſed my mind, that Sir Iſaac Newton was ſuppoſed by you to be guilty of this double error, 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 difference, or 2xẋ for the moment of xx, is common to the Marquis and Sir Iſaac Newton.

The Marquis makes a ſecond error, which perfectly corrects the firſt, whence his concluſion 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 Marquis'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 error, 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 contempt 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 evidence, and have named the paſſages, where theſe very objections of yours appear to have been foreſeen, and to be clearly and fully removed. 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 error is fully removed by Lemma 7. and that of the ſecond by Lemma 1.

XL. I have nothing to ſay to the principles of the Marquis de l'Hoſpital, I defend nothing but his reaſoning. You ſay, he rejects infiniteſimals in virtue of a Poſtulatum, and this you venture to call rejecting them without ceremony. I know of no greater ceremony uſed by Euclid, than to reject a thing in neceſſary and unavoidable conſequence of a Poſtulatum. You tell me, he inferreth a concluſion accurately true, contrary to the rules of Logick, 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 knowing 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 compenſation, 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 difficulties 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 Leibnitz, 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 different opinions, in converſation with Analyſts, in ten months time, and ſome Analyſts, perhaps 5 or 6, of every one of thoſe opinions, [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 mathematical opinions. Otherwiſe, I ſhould apprehend Religion to be in great danger there, unleſs that Country be well ſtocked with men able to deal with them at their own weapons, and to ſhew, they are by no means thoſe maſters of reaſon, which they would fain paſs for.

XLV, XLVI, XLVII, XVIII. You come now to the point of Metaphyſicks in diſpute between us, about which you write, contrary to your uſual manner, ſo very inaccurately and unintelligibly, as plainly convinces 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 manifeſt, 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 expected. You had expreſſed your ſelf in Art. CXXV of your new Theory of Viſion, in the following words. "After reiterated endeavours to apprehend the GENERAL IDEA of a Triangle, I have found it altogether incomprehenſible." But now, ‘Ut primum diſcuſſaeumbrae, & lux redditamenti,’ your eyes being opened, (pardon me this vanity) 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 manner 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 follows, however inconſiſtent with this recantation: Particularly your making no difference between a round ſquare, and a ſpace comprehended by three right lines. For the ſame reaſon I willingly paſs by your ſuppoſing, 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 definition 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 definition: And theſe two, the compound idea and the definition are inſeparable, if the definition 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 obſerved you often to confound, both here and in your other writings. The compound idea is general, but the image, if exactly attended [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 endeavoured to give him any light about them. In this ſecond letter indeed I have, at your requeſt, 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 imagine what you mean by them. Poſſibly they may be ſome arcana of the Boeotian Analyſis, 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.

I am, Sir,
Your moſt Obedient Servant, PHILALETHES CANTABRIGIENSIS.
FINIS.
Notes
*
Princ. Lib. 11. Lemm. 2.
Princ. Lib. 1. Sect. 1. Schol.
Eſſay on Viſion. Sect. CLIII to CLIX.
*
Schol. Sect. 1. Libr. 1. Princip.
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TextGrid Repository (2020). TEI. 4647 The minute mathematician or the free thinker no just thinker Set forth in a second letter to the author of The analyst containing a defence of Sir Isaac Newton and the British mathematicians. University of Oxford Text Archive. . https://hdl.handle.net/21.T11991/0000-001A-5F5E-2