I may call the fluxion of a fluxion the velocity of a velocity. But for the truth of the antecedent see his introduction to the Quadrature of Curves, where his own words are, "Motuum vel incrementorum volicitates nominando fluxiones." See also the second lemma of the second book of his Mathematical Principles of Natural Philosophy, where he expresseth himself in the following manner: "Velocitates incrementorum ac decrementorum quas etiam, motus, mutationes et fluxiones quantitatum nominare licet." And that he admits fluxions of fluxions, or second, third, fourth fluxions, &c. see his Treatise of the Quadrature of Curves. I ask now, is it not plain, that if a fluxion be a velocity, then the fluxion of a fluxion may agreeably thereunto be called the velocity of a velocity? In like manner if by a fluxion is meant a nascent augment, will it not then follow, that the fluxion of a fluxion or second fluxion is the nascent augment of a nascent augment? Can any thing be plainer? Let the reader now judge who is unfair.

XXIV. I had observed, that the great author had proceeded illegitimately, in obtaining the fluxion or moment of the rectangle of two flowing quantities; and that he did not fairly get rid of the rectangle of the moments. In answer to this you allege, that the error arising from the omission of such rectangle (allowing it to be an error) is so small that it is insignificant. This you dwell upon and exemplify to no other purpose, but to amuse your reader and mislead him from the ques tion; which in truth is not concerning the accuracy of computing or measuring in practice, but concerning the accuracy of the reasoning in science. That this was really the case, and that the smallness of the practical error no wise concerns it, must be so plain to any one who reads the Analyst, that I wonder how you could be ignorant of it.

XXV. You would fain persuade your reader, that I make an absurd quarrel against errors of no signifi

cancy in practice, and represent mathematicians as proceeding blindfold in their approximations, in all which I cannot help thinking there is on your part either great ignorance or great disingenuity. If you mean to defend the reasonableness and use of approximations or of the method of indivisibles, I have nothing to say. But then you must remember this is not the doctrine of fluxions: it is none of that analysis with which I am concerned, That I am far from quarrelling at approximations in geometry, is manifest from the thirty-third and fiftythird queries in the Analyst. And that the method of fluxions pretends to somewhat more than the method of indivisibles is plain; because Sir Isaac disclaims this method as not geometrical.* And that the method of fluxions is supposed accurate in geometrical rigour is manifest, to whoever considers what the great author writes about it; especially in his introduction to the Quadrature of Curves, where he saith, "In rebus mathematicis errores quam minimi non sunt contemnendi.” Which expression you have seen quoted in the Analyst, and yet you seem ignorant thereof, and indeed of the very end and design of the great author of this his invention of fluxions.

XXVI. As oft as you talk of finite quantities inconsiderable in practice, Sir Isaac disowns your apology. "Cave (saith he) intellexeris finitas." And although quantities less than sensible may be of no account in practice, yet none of your masters, nor will even you yourself, venture to say they are of no account in theory and in reasoning. The application in gross practice is not the point questioned, but the rigour and justness of the reasoning. And it is evident that, be the subject ever so little, or ever so inconsiderable, this doth not hinder but that a person treating thereof may commit very great errors in logic, which logical errors are in no

* See the Scholium at the end of the first section. Lib. i. Phil. Nat. Prin. Math.

wise to be measured by the sensible or practical incon veniences thence arising, which, perchance, may be none at all. It must be owned, 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 yourself to justify Sir Isaac's method of getting rid of the abovementioned rectangle. And here I must entreat the reader to observe how fairly you proceed.

XXVII. First then you affirm (p. 44), "that neither in the demonstration of the rule for finding the fluxion of the rectangle of two flowing quantities, nor in any thing preceding or following it, is any mention so much as once made of the increment of the rectangle of such flowing quantities." Now I affirm the direct contrary. For in the very passage by you quoted in this same page, from the first case of the second lemma of the second book of Sir Isaac's Principles, beginning with," Rectangulum quodvis motu perpetuo auctum," and ending with, " igitur laterum incrementis totis a et b generatur rectanguli incrementum a B x b AQ. E. D.” In this very passage I say is express mention made of the increment of such rectangle. As this is matter of fact, I refer it to the reader's own eyes. Of what rectangle have we here the increment? Is it not plainly of that whose sides have a and b for their incrementa tota, that is, of AB. Let any reader judge whether it be not plain from the words, the sense, and the context, that the great author in the end of his demonstration understands his incrementum as belonging to the rectangulum quodvis at the beginning. Is not the same also evident from the very lemma itself prefixed to the demonstration? The sense whereof is (as the author there explains it), that if the moments of the flowing quantities A and B are called a and b, then the momentum vel mutatio geniti rectanguli A B will be a Bxb A. Either therefore the conclusion of the demonstration is not the thing which was to be demonstrated, or the

rectanguli incrementum a B x b A belongs to the rectangle AB.

XXVIII. All this is so plain that nothing can be more so; and yet you would fain perplex this plain case by distinguishing between an increment and a moment. But it is evident to every one, who has any notion of demonstration, that the incrementum in the conclusion must be the momentum in the lemma; and to suppose it otherwise is no credit to the author. It is in effect supposing him to be one who did not know what he would demonstrate. But let us hear Sir Isaac's own words: "Earum (quantitatum scilicet fluentium) incrementa vel decrementa momentanea sub nomine momentorum intelligo." And you observe yourself that he useth the word moment to signify either an increment or decrement. Hence with an intention to puzzle me you propose the increment and decrement of AB, and ask which of these I would call the monent? The case you say is difficult. My answer is very plain and easy, to wit, either of them. You, indeed, make a different answer, and from the author's saying, that by a moment he understands either a momentaneous increment or decrement of the flowing quantities, you would have us conclude, by a very wonderful inference, that his moment is neither the increment nor decrement thereof. Would it not be as good an inference, because a number is either odd or even, to conclude it is neither? Can any one make sense of this? Or can even yourself hope that this will go down with the reader, how little soever qualified? It must be owned, you endeavour to intrude this inference on him, rather by mirth and humour than by reasoning. You are merry, I say, and (p. 46) represent the two mathematical quantities as pleading their rights, as tossing up cross and pile, as disputing amicably. You talk of their claiming preference, their agreeing, their boyishness, and their gravity. And after this ingenious digression you address me in

the following words-Believe me, there is no remedy, you must acquiesce. But my answer is, that I will neither believe you nor acquiesce; there is a plain remedy in common sense; and that to prevent surprise I desire the reader always to keep the controverted point in view, to examine your reasons, and be cautious how he takes your word, but most of all when you are positive, or eloquent, or merry.

XXIX. A page or two after, you very candidly represent your case to be that of an ass between two bottles of hay it is your own expression. The cause of your perplexity is that you know not, whether the velocity of A B increasing or of A B decreasing, is to be esteemed the fluxion, or proportional to the moment of the rectangle. My opinion, agreeably to what hath been premised, is that either may be deemed the fluxion. But you tell us (p. 49) " that you think, the venerable ghost of Sir Isaac Newton whispers you, the velocity you seek for is neither the one nor the other of these, but is the velocity which the flowing rectangle hath not while it is greater or less than AB, but at that very instant of time that it is A B." For my part, in the rectangle A B considered simply in itself, without either increasing or diminishing, I can conceive no velocity at all. And if the reader is of my mind, he will not take either your word, or even the word of a ghost, how venerable soever, for velocity without motion. You proceed and tell us that, in like manner, the moment of the rectangle is neither its increment or decrement. This you would have us believe on the authority of his ghost, in direct opposition to what Sir Isaac himself asserted when alive. “Incrementa (saith he) vel decrementa momentanea, sub nomine momentorum intelligo: ita ut incrementa pro momentis addititiis seu affirmativis, ac decrementa pro subductitiis seu negativis habeantur."* I will not in your style bid the reader believe me, but believe his eyes.

XXX. To me it verily seems, that you have under