and clearly and fully removed by Sir Isaac Newton in the first section of the first book of his Principia." All which I do as strongly deny as you affirm. And I do aver, that this is an unquestionable proof of the matchless contempt which you, Philalethes, have for truth. And I do here publicly call upon you, to produce that evidence which you pretend to have, and to make good that fact which you so confidently affirm. And, at the same time, I do assure the reader, that you never will,

nor can.

XL. If you defend Sir Isaac's notions, as delivered in his Principia, it must be on the rigorous foot of rejecting nothing, neither admitting nor casting away infinitely small quantities. If you defend the Marquis, whom you also style your master, it must be on the foot of admitting, that there are infinitesimals, that they may be rejected, that they are nevertheless real quantities, and themselves infinitely subdivisible. But you seem to have grown giddy with passion, and in the heat of controversy to have mistaken and forgot your part. I beseech you, Sir, to consider, that the Marquis (whom alone, and not Sir Isaac, this double error in finding the subtangent doth concern) rejects indeed infinitesimals, but not on the foot that you do, to wit, their being inconsiderable in practical geometry or mixed mathematics. But he rejects them in the accuracy of speculative knowledge: in which respect there may be great logical errors, although there should be no sensible mistake in practice: which, it seems, is what you cannot comprehend. He rejects them likewise in virtue of a postulatum, which I venture to call rejecting them without ceremony. And though he inferreth a conclusion accurately true, yet he doth it, contrary to the rules of logic, from inaccurate and false premises. And how this comes about, I have at large explained in the Analyst, and shewed in that particular case of tangents, that the rejectaneous quantity might have been a finite

quantity of any given magnitude, and yet the conclusion have come out exactly the same way; and, consequently, that the truth of this method doth not depend on the reason assigned by the Marquis, to wit, the postulatum for throwing away infinitesimals, and therefore, that he and his followers acted blindfold, as not knowing the true reason for the conclusions coming out accurately right, which I shew to have been the effect of a double error.

XLI. This is the truth of the matter, which you shamefully misrepresent and declaim upon, to no sort of purpose but to amuse and mislead your reader. For which conduct of yours throughout your remarks, you will pardon me if I cannot otherwise account, than from a secret hope that the reader of your defence would never read the Analyst. If he doth, he cannot but see what an admirable method you take to defend your cause: how, instead of justifying the reasoning, the logic or the theory of the case specified, which is the real point, you discourse of sensible and practical errors: and how all this is a manifest imposition upon the reader. He must needs see that I have expressly said, "I have no controversy except only about your logic and method: that I consider how you demonstrate; what objects you are conversant about; and whether you conceive them clearly? That I have often expressed myself to the same effect, desiring the reader to remember, that I am only concerned about the way of coming at your theorems, whether it be legitimate or illegitimate, clear or obscure, scientific or tentative: that I have, on this very occasion, to prevent all possibility of mistake, repeated and insisted, that I consider the geometrical analyst as a logician, i. e. so far forth as he reasons and argues; and his mathematical conclusions not in themselves but in their premises; not as true or false, useful or insignificant, but as derived from such principles, and by such inferences."* You affirm

* Analyst, sect. xx.

(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. I profess myself of a contrary opinion. My reason is, because nothing cannot be divided into parts. But this difference is capable of being divided into any, or into more than any, given number of parts; for the truth of which consult the Marquis de l'Hospital. And, be the error in fact or in practice ever so small, it will not thence follow, that the error in reasoning, which is what I am alone concerned about, is one whit the less, it being evident, that a man may reason most absurdly about the minutest things.

XLII. Pray answer me fairly, once for all, whether it be your opinion that whatsoever is little and inconsiderable enough to be rejected without inconvenience in practice, the same may in like manner be safely rejected and overlooked in theory and demonstration. If you say no, it will then follow, that all you have been saying here and elsewhere, about yards and inches and decimal fractions, setting forth and insisting on the extreme smallness of the rejectaneous quantity, is quite foreign to the argument, and only a piece of skill to impose upon your reader. If you say yes, it follows that you then give up at once all the orders of fluxions and infinitesimal differences; and so most imprudently turn all your sallies and attacks and veterans to your own overthrow. If the reader is of my mind, he will despair of ever seeing you get clear of this dilemma. The points in controversy have been so often and so distinctly noted in the Analyst, that I very much wonder how you could mistake if you had no mind to mistake. It is very plain, if you are in earnest, that you neither understand me nor your masters. And what shall we think of other ordinary analysts, when it shall be found that even you, who, like a champion step forth to defend their principles, have not considered them?

1

XLIII. The impartial reader is entreated to remark throughout your whole performance how confident you are in asserting and withal how modest in proving or explaining how frequent it is with you to employ figures and tropes instead of reasons: how many difficulties proposed in the Analyst are discreetly overlooked by you, and what and what strange work you make with the rest: how grossly you mistake and misrepresent and how little you practise the advice which you so liberally bestow. Believe me, sir, I had long and maturely considered the principles of the modern analysis, before I ventured to publish my thoughts thereupon in the Analyst. And, since the publication thereof, I have myself freely conversed with mathematicians of all ranks, and some of the ablest professors, as well as made it my business to be informed of the opinions of others, being very desirous to hear what could be said towards clearing my difficulties or answering my objections. But though you are not afraid or ashamed, to represent the analysts as very clear and uniform in their conception of these matters, yet I do solemnly affirm (and several of themselves know it to be true) that I found no harmony or agreement among them, but the reverse thereof, the greatest dissonance and even contrariety of opinions, employed to explain what after all seemed inplicable.

XLIV. Some fly to proportions between nothings. Some reject quantities because infinitesimal. Others allow only finite quantities, and reject them because inconsiderable. Others place the method of fluxions on a foot with that of exhaustions, and admit nothing new therein. Some maintain the clear conception of fluxions. Others hold they can demonstrate about things incomprehensible. Some would prove the algorism of fluxions by reductio ad absurdum; others a priori. Some hold the evanescent increments to be real quantities, some to be nothings, some to be limits. As many men,

so many minds: each differing one from another, and all from Sir Isaac Newton. Some plead inaccurate expressions in the great author, whereby they would draw him to speak their sense, not considering that if he meant as they do, he could not want words to express his meaning. Others are magisterial and positive, say they are satisfied, and that is all, not considering that we, who deny Sir Isaac Newton's authority, shall not submit to that of his disciples. Some insist, that the conclusions are true, and therefore the principles, not considering what hath been largely said in the Analyst * on that head. Lastly, several (and those none of the meanest) frankly owned the objections to be unanswerable. All which I mention by way of antidote to your false colours and that the unprejudiced inquirer after truth may see, it is not without foundation, that I call on the celebrated mathematicians of the present age to clear up these obscure analytics, and concur in giving to the public some consistent and intelligible account of their great master: which if they do not, I believe the world will take it for granted that they cannot.

XLV. Having gone through your defence of the British mathematicians, I find, in the next place, that you attack me on a point of metaphysics, with what success the reader will determine. I had upon another occasion many years ago wrote against abstract general ideas. In opposition to which, you declare yourself to adhere to the vulgar opinion, that neither geometry nor any other general science can subsist without general ideas (p. 74). 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. To me it is plain, there is no consistent idea, the likeness

* Sect. xix, xx, &c.

Introduction to the Treatise concerning the Principles of Human Knowledge.