self. And yet this doth not hinder but the method may be useful, considered as an art of invention. You, who are a mathematician, must acknowledge, there have been divers such methods admitted in mathematics, which are not demonstrative. Such, for instance, are the inductions of Doctor Wallis in his Arithmetic of Infinites, and such what Harriott, and after him, Descartes, have wrote concerning the roots of affected equations. It will not, nevertheless, thence follow that those methods are useless; but only, that they are not to be allowed of as premises in a strict demonstration.

XV. No great name upon earth shall ever make me accept things obscure for clear, or sophisms for demonstrations. Nor may you ever hope to deter me from freely speaking what I freely think, by those arguments ab invidia which at every turn you employ against me. You represent yourself (p. 52) as a man whose highest ambition is in the lowest degree to imitate Sir Isaac Newton. It might, perhaps, have suited better with your appellation of Philalethes, and been altogether as laudable, if your highest ambition had been to discover truth. Very consistently with the character you give of yourself, you speak of it as a sort of crime (p. 70) to think it possible, you should ever see farther, or go beyond Sir Isaac Newton. And I am persuaded you speak the sentiments of many more besides yourself. But there are others who are not afraid to sift the principles of human science, who think it no honour to imitate the greatest man in his defects, who even think it no crime to desire to know, not only beyond Sir Isaac Newton, but beyond all mankind. And whoever thinks otherwise, I appeal to the reader, whether he can properly be called a philosopher.

XVI. Because I am not guilty of your mean idolatry, you inveigh against me as a person conceited of my own abilities; not considering that a person of less abilities may know more on a certain point than one of

greater; not considering that a purblind eye, in a close and narrow view, may discern more of a thing, than a much better eye in a more extensive prospect; not considering that this is to fix a ne plus ultra, to put a stop to all future inquiries; lastly, not considering that this is in fact, so much as in you lies, converting the republic of letters into an absolute monarchy, that it is even introducing a kind of philosophic popery among a free people.

XVII. I have said (and I venture still to say) that a fluxion is incomprehensible: that second, third, and fourth fluxions, are yet more imcomprehensible: 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 in answer to this? Do you attempt to clear up the notion of a fluxion or a difference? Nothing like it; " you only assure me (upon your bare word), from your own experience, and that of several others whom you could name, that the doctrine of fluxions may be clearly conceived and distinctly comprehended; and that if I am puzzled about it and do not understand it, yet others do." But can you think, Sir, I shall take your word when I refuse to take your master's?

XVIII. Upon this point every reader of common sense may judge as well as the most profound mathematician. The simple apprehension of a thing defined is not made more perfect by any subsequent progress in mathematics. What any man evidently knows, he knows as well as you or Sir Isaac Newton. And every one can know whether the object of this method be (as you would have us think) clearly conceivable. To judge of this, no depth of science is requisite, but only a bare attention to what passes in his own mind. And the same is to be understood of all definitions in all sciences whatsoever. In none of which can it be supposed, that a man of sense and spirit will take any definition or prin Analyst, sect. iv, v, vi, &c.

ciple upon trust, without sifting it to the bottom, and trying how far he can or he cannot conceive it. This. is the course I have taken and shall take, however you and your brethren may declaim against it, and place it in the most invidious light.

XIX. It is usual with you to admonish me to look over a second time, to consult, examine, weigh the words of Sir Isaac. In answer to which I will venture to say, that I have taken as much pains as (I sincerely believe) any man living, to understand that great author, and to make sense of his principles. No industry nor caution: nor attention, I assure you, have been wanting on my part. So that, if I do not understand him, it is not my fault but my misfortune. Upon other subjects you are pleased to compliment me with depth of thought and uncommon abilities (p. 5 and 84). But I freely own, I have no pretence to those things. The only advantage I pretend to, is, that I have always thought and judged for myself. And, as I never had a master in mathematics, so I fairly followed the dictates of my own mind in examining and censuring the authors I read upon that subject, with the same freedom that I used upon any other; taking nothing upon trust, and believing that no writer was infallible. And a man of moderate parts, who takes this painful course in studying the principles of any science, may be supposed to walk more surely than those of greater abilities, who set out with more speed and less care.

XX. What I insist on is, that the idea of a fluxion simply considered is not at all improved or amended by any progress, though ever so great, in the analysis: neither are the demonstrations of the general rules of that method at all cleared up by applying them. The reason of which is, because in operating or calculating, men do not return to contemplate the original principles: of the method, which they constantly presuppose, but are employed in working, by notes and symbols, denot

ing the fluxions supposed to have been at first explained, and according to rules supposed to have been at first demonstrated. This I say to encourage those, who are not far gone in these studies, to use intrepidly their own judgment, without a blind or a mean deference to the best of mathematicians, who are no more qualified than they are, to judge of the simple apprehension, or the evidence of what is delivered in the first elements of the method; men by farther and frequent use or exercise becoming only more accustomed to the symbols and rules, which doth not make either the foregoing notions more clear, or the foregoing proofs more perfect. Every reader of common sense, that will but use his faculties, knows as well as the most profound analyst what idea he frames or can frame of velocity without motion, or of motion without extension, of magnitude which is neither finite nor infinite, or of a quantity having no magnitude which is yet divisible, of a figure where there is no space, of proportion between nothings, or of a real product from nothing multiplied by something. He need not be far gone in geometry to know, that obscure principles are not to be admitted in demonstration that if a man destroys his own hypothesis, het at the same time destroys what was built upon it: that error in the premises, not rectified, must produce error

in the conclusion.

XXI. In my opinion the greatest men have their prejudices. Men learn the elements of science from others and every learner hath a deference more or less to authority, especially the young learners, few of that kind caring to dwell long upon principles, but inclining rather to take them upon trust: and things early admitted by repetition become familiar: and this familiarity at length passeth for evidence. Now to me it seems, there are certain points tacitly admitted by mathematicians, which are neither evident nor true. And such points or principles ever mixing with their reason

ings do lead them into paradoxes and perplexities. If the great author of the fluxionary method was early imbued with such notions, it would only shew he was a man. And if by virtue of some latent error in his principles a man be drawn into fallacious reasonings, it is nothing strange that he should take them for true: and, nevertheless, if, when urged by perplexities and uncouth consequences, and driven to arts and shifts, he should entertain some doubt thereof, it is no more than one may naturally suppose might befal a great genius grappling with an insuperable difficulty: which is the light in which I have placed Sir Isaac Newton.* Hereupon you are pleased to remark, that I represent the great author not only as a weak but an ill man, as a deceiver and an impostor. The reader will judge how justly.

XXII. As to the rest of your colourings and glosses, your reproaches and insults and outcries, I shall pass them over, only desiring the reader not to take your word, but read what I have written, and he will want no other answer. It hath been often observed, that the worst cause produceth the greatest clamour; and indeed you are so clamorous throughout your defence, that the reader, although he should be no mathematician, provided he understands common sense and hath observed the ways of men, will be apt to suspect that you are in the wrong. It should seem, therefore, that your brethren the analysts are but little obliged to you for this new method of declaiming in mathematics. Whe ther they are more obliged by your reasoning I shall now examine.

XXIII. You ask me (p. 32) where I find Sir Isaac Newton using such expressions as the velocities of velocities, the second, third, and fourth velocities, &c. This you set forth as a pious fraud and unfair representation. I answer, that if according to Sir Isaac Newton a fluxion be the velocity of an increment, then according to him Analyst, sect. xviii.