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.


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.


  • 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. [Page] 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.
  • [Page] 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. [Page] 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.
  • [Page] 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.
  • [Page] 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.


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.


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, [Page 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 [Page 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 [Page 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.

[Page 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.

[Page 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 [Page 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 [Page 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 [Page 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 [Page 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 [Page 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.

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 [Page 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 [Page 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 [Page 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 [Page 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 [Page 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 [Page 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


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


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.

[Page 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.

The definitions explained by the same example.

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 [Page 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 [Page 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.


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.


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

[Page 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 [Page 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, [Page 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.

[Page 26]


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 [Page 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 [Page 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

[Page 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 [Page 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. [Page 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.

[Page 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 [Page 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 [Page 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 [Page 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 [Page 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.

[Page 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 [Page 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.


[Page 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 [Page 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 [Page 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 [Page 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 [Page 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 [Page 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 [Page 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 [Page 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

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

[Page 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 [Page 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 [Page 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.

[Page 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 [Page 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 [Page 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 [Page 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 [Page 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 [Page 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 [Page 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 [Page 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

[Page 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 [Page 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, [Page 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
[Page 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.

[Page 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 [Page 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.


Let therefore RALB, or RL, represent the flowing rectangle AB in Sir Isaac New­ton's [Page 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.

[Page 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 [Page 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

[Page 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 [Page 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? [Page 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 [Page 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 [Page 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.

[Page 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 [Page 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 [Page 74] them respectively, with an uniform motion, in a given time.

The Decrement.Moment.Increment.

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 [Page 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, [Page 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,
[Page 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
[Page 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 [Page 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 [Page 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, [Page 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 [Page 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 [Page 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: [Page 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.

[Page 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.


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.


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 [Page 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 [Page 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 [Page 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.


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.
  • [Page 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 [Page 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 [Page 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 [Page 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 [Page 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.

[Page 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.

[Page 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, [Page 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 [Page 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 [Page 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.

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, [Page 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: [Page 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 [Page 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.

[Page 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, [Page 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.

[Page 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 [Page 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 [Page 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 [Page 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, [Page 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 [Page 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."

[Page 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 [Page 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,

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