taken the defence of what you do not understand. To mend the matter, you say, "you do not consider A B as lying at either extremity of the moment, but as extended to the middle of it; as having acquired the one half of the moment, and as being about to acquire the other; or, as having lost one half of it, and being about to lose the other." Now, in the name of truth, I entreat you to tell what this moment is, to the middle whereof the rectangle is extended? This moment, I say, which is acquired, which is lost, which is cut in two, or distinguished into halves? Is it a finite quantity, or an infinitesimal, or a mere limit, or nothing at all? Take it in what sense you will, I cannot make your defencé either consistent or intelligible. For if you take it in either of the two former senses, you contradict Sir Isaac Newton. And if you take it in either of the latter, you contradict common sense; it being plain, that what hath no magnitude, or is no quantity, cannot be divided. And here I must entreat the reader to preserve his full freedom of mind entire, and not weakly suffer his judgment to be overborne by your imagination and your prejudices, by great names and authorities, by ghosts and visions, and above all by that extreme satisfaction and complacency with which you utter your strange conceits; if words without a meaning may be called so. After hav

ing given this unintellegible account, you ask with your 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, alas! I acknowledge no such thing. I find no sense or reason in what you say. Let the reader find it if he can.

XXXI. In the next place (p. 50) you charge me with want of caution," inasmuch (say you) as that quantity which Sir Isaac Newton, through his whole lemma, and all the several cases of it, constantly calls a moment, without confining it to be either an increment or decrement, is by you inconsiderately and arbitra

rily, and without any shadow of reason given, supposed and determined to be an increment." To which charge I reply, that it is as untrue as it is peremptory. For that, in the foregoing citation from the first case of Sir Isaac's lemma, he expressly determines it to be an increment. And as this particular instance or passage was that which I objected to, it was reasonable and proper for me to consider the moment in the same light. But take it increment or decrement as you will, the objections still lie and the difficulties are equally insuperable. You then proceed to extol the great author of the fluxionary method, and to bestow some brusqueries upon those who unadvisedly dare to differ from him. To all which I shall give no answer.

XXXII. Afterwards to remove (as you say) all scruple and difficulty about this affair, you observe that the moment of the rectangle determined by Sir Isaac Newton, and the increment of the rectangle determined by me, are perfectly and exactly equal, supposing a b to be diminished ad infinitum: and for proof of this, you refer to the first lemma of the first section of the first book of Sir Isaac's Principles. I answer that if a and b are real quantities, then a b is somethings, 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. As for the abovementioned lemma, which you refer to, and which you wish I had consulted sooner, both for my own sake and for yours; I tell you I had long since consulted and considered it. But I very much doubt whether you have sufficiently considered that lemma, its demonstration, and its consequences. For, however that way of reasoning may do in the method of exhaustions, where quantities less than assignable are regarded as nothing; yet for a fluxionist writing about momenturns, to argue that quantities must be

equal because they have no assignable difference, seems the most injudicious step that could be taken: it is directly demolishing the very doctrine you would defend. For it will thence follow, that all homogeneous momentums are equal, and consequently the velocities, mutations, or fluxions, proportional thereto, are all likewise equal. There is, therefore, only one proportion of equality throughout, which at once overthrows the whole system you undertake to defend. Your moments (I say) not being themselves assignable quantities, their differences cannot be assignable: and if this be true, by that way of reasoning it will follow, they are all equal, upon which supposition you cannot make one step in the method of fluxions. It appears from hence, how unjustly you blame me (p. 32) for omitting to give any account of that first section of the first book of the Principia, wherein you say the foundation of the method of fluxions is geometrically demonstrated and largely explained, and difficulties and objections against it are clearly solved. All which is so far from being true, that the very first and fundamental lemma of that section is incompatible with and subversive of the doctrine of fluxions. And, indeed, who sees not that a demonstration ad absurdam more veterum proceeding on a supposition, that every difference must be some given quantity, cannot be admitted in, or consist with, a method, wherein quantities, less than any given, are supposed really to exist, and be capable of division?

XXXIII. The next point you undertake to defend is that method for obtaining a rule to find the fluxion of any power of a flowing quantity, which is delivered in his introduction to the Quadratures, and considered in the Analyst. And here 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 no* Sect. xiii, xiv, &c.

thing, or let there be no increments? This you deny, but, as your manner is, instead of giving a reason you declaim. I, on the contrary affirm, the increments must be understood to be quite gone and absolutely nothing at all. My reason is, because without that supposition you can never bring the quantity or expres

the very thing aimed at by supposing the evanescence. Say whether this be not the truth of the case? Whether the former expression is not to be reduced to the latter? And whether this can possibly be done so long as o is supposed a real quantity? I cannot indeed say you are scrupulous about your affirmations, and yet I believe that even you will not affirm this; it being most evident, that the product of two real quantities is something real; and that nothing real can be rejected either according to the άkeißea of geometry, or according to Sir Isaac's own Principles; for the truth of which I appeal to all who know any thing of these matters. Further by evanescent must either be meant, let them (the increments) vanish and become nothing, in the obvious sense, or else let them become infinitely small. But that this latter is not Sir Isaac's sense is evident from his own words in the very same page, that is, in the last of his introduction to the Quadratures, where he expressly saith" volui ostendere quod in methodo fluxionum non opus sit figuras in finite parvas in geometriam introducere." Upon the whole, you seem to have considered this affair so very superficially, as greatly to confirm me in the opinion you are so angry with, to wit, that Sir Isaac's followers are much more eager in applying his method, than accurate in examining his principles. You raise a dust about evanescent augments, which may perhaps amuse and amaze your reader, but I am much mistaken if it ever instructs or enlightens him. For, to come to the

point, those evanescent augments either are real quantities, or they are not. If you say they are; I desire to know, how you get rid of the rejectaneous quantity? If you say they are not; you indeed get rid of those quantities in the composition whereof they are coefficients; but then you are of the same opinion with me," which opinion you are pleased to call (p. 58) a most palpable, inexcusable, and unpardonable blunder, although it be a truth most palpably evident."

XXXIV. Nothing, I say, can be plainer to any impartial reader, than that by the evanescence of augments, in the abovecited passage, Sir Isaac means their being actually reduced to nothing. But to put it out of all doubt, that this is the truth, and to convince even you, who shew so little disposition to be convinced, I desire you to look into his Analysis per Æquationes Infinitas (p. 20), where, in his preparation for demonstrating the first rule for the squaring of simple curves, you will find that on a parallel occasion, speaking of an augment which is supposed to vanish, he interprets the word evanescere by esse nihil. Nothing can be plainer than this, which at once destroys your defence. And yet, plain as it is, I despair of making you acknowledge it; though I am sure you feel it, and the reader if he useth his eyes must see it. The words evanescere sive esse nihil do (to use your own expression) stare us in the face. Lo! "This is what you call (p. 56) so great, so unaccountable, so horrid, so truly Boeotian a blunder," that, according to you, it was not possible Sir Isaac Newton could be guilty of it. For the future, I advise you to be more sparing of hard words; since, as you incautiously deal them about, they may chance to light on your friends as well as your adversaries. As for my part, I shall not retaliate. It is sufficient to say you are mistaken. But I can easily pardon your mistakes. Though, indeed, you tell me, on this very occasion, that I must

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