A Letter published in the year 1734, under the title of The Analyst, first gave occasion to the ensuing Treatise, and several reasons concurred to induce me to write on this Subject at so great length. The Author of that Piece had represented the Method of Fluxions as founded on false Reasoning, and full of Mysteries. His Objections seemed to have been occasioned, in a great measure, by the concise manner in which the Elements of this Method have been usually described; and their having been so much misunderstood by a person of his abilities appeared to me a sufficient proof that a fuller Account of the Grounds of them was requisite.
Though there can be no comparison made betwixt the extent or usefulness of the ancient and modern Discoveries in Geometry, yet it seems to be generally allowed that the Ancients took greater care, and were more successful in preserving the Character of its Evidence entire. This determined me, immediately after that Piece came to my hands, and before I knew any thing of what was intended by others in answer to it, to attempt to deduce those Elements after the manner of the Ancients, from a few unexceptionable principles, by Demonstrations of the strictest form. In my first Essay of this kind, I contented myself with demonstrating the principal Cases of the Propositions of the four first Chapters of the first Book, and of the first Chapter of the second Book of the following Treatise, nearly in the same form in which they now appear. But when it was communicated to some gentlemen, they expressed a desire that the same Method of Demonstration should be extended to other branches of this Theory, and that I should enlarge the Plan. While I proceeded in this Work I perceived that some Rules were defective or inaccurate; that the Resolution of several Problems which had been deduced in a mysterious manner, by second and third Fluxions, could be completed with greater evidence, and less danger of error, by first Fluxions only; and that other Problems had been resolved by Approximations, when an accurate Solution could be obtained with the same or greater facility. These with other observations concerning this method, and its application, led me on gradually to compose a Treatise of a much greater extent than I intended, or would have engaged in, if I had been aware of it when I began this Work, because my attendance in the University could allow me to bestow but a small part of my time in carrying it on. And as this has been the occasion of my delay in publishing it, so I hope it will serve for an apology, if some mistakes have escaped me in treating of such a variety of subjects, in a manner different from that in which they have been usually explained.
In the mean time the Defence of the Method of Fluxions, and of the great Inventor, was not neglected. Besides an answer to The Analyst that appeared very early under the name of Philalethes Cantabrigiensis (for the Author had concealed his real name as the Analyst whom he opposed had done), a second, by the same hand, in Defence of the first, a Discourse by Mr Robins, a Treatise of Sir Isaac Newton's, with a Commentary by Mr Colson, and several other Pieces, were published on this Subject. After I saw that so much had been written upon it to so good a purpose, I was the rather induced to delay the publication of this Treatise, till I could finish my design. I accommodated my Definition of the Variation of Curvature in Chap. xi. to Sir Isaac Newton's, to prevent mistakes, as I have observed in Article 386, but made no material alteration in any thing else. The greatest part of the first Book was printed in 1737: but it could not have been so useful to the Reader without the second; and I would recommend to him (if he is not already acquainted with this method), to peruse the two first Chapters of the second Book, before the five last of the first; there being a few passages in these which I could not well avoid, that will be better understood by one who has some knowledge of the principal Rules of the Method of Computation delivered in the second Book.
In explaining the Notion of a Fluxion, I have followed Sir Isaac Newton in the first Book, imagining that there can be no difficulty in conceiving Velocity wherever there is Motion; nor do I think that I have departed from his Sense in the second Book; and in both I have endeavoured to avoid several expressions, which, though convenient, might be liable to exceptions, and, perhaps, occasion disputes. I have always represented Fluxions of all Orders by finite Quantities, the Supposition of an infinitely little Magnitude being too bold a Postulatum for such a Science as Geometry. But, because the Method of Infinitesimals is much in use, and is valued for its conciseness. I thought it was requisite to account explicitly for the truth, and perfect accuracy, of the conclusions that are derived from it; the rather, that it does not seem to be a very proper reason that is assigned by Authors, when they determine what is called the Difference (but more accurately the Fluxion) of a Quantity, and tell us, That they reject certain Parts of the Element because they become infinitely less than the other parts; not only because a proof of this nature may leave some doubt as to the accuracy of the conclusion; but because it may be demonstrated that those parts ought to be neglected by them at any rate, or that it would be an error to retain them. If an Accountant, that pretends to a scrupulous exactness, should tell us that he had neglected certain Articles, because he found them to be of small importance, and it should appear that they ought not to have been taken into consideration by him on that occasion, but belong to a different account, we should approve his conclusions as accurate, but not his reason. This method, however, may be considered as an easy and ready way of distinguishing what Parts of an Element are to be rejected, and which are to be retained, in determining the precise Fluxion of a Quantity, or the rate according to which it increases or decreases.
After I found that this Treatise could not be conveniently contained in one volume, I was obliged to reprint two leaves (pages 411 &c.) that it might be divided into two. I have reprinted likewise the first sheet, chiefly on account of several errors of the press that had got into it, and one other leaf (page 244), for the sake of a Passage, the omission of which possibly would have been misinterpreted. There are some Demonstrations in the first Chapter of the first Book that might have been abridged, and some, perhaps, will appear unnecessary. I have mentioned the reasons that induced me to insist so fully on those Elementary Parts, in Articles 69, 104, 494, and 697.
Several Treatises have appeared while this was in the press, wherein some of the same Problems have been considered, though generally in a different manner. I have had occasion to mention most of them in the last Chapter of the second Book; but had not there an opportunity to take notice, that the Problem in 480 has been considered by Mr Euler in his Mechanics. In most of the instances wherein my conclusions did not agree with those given by other Authors, I have not mentioned their names.
If, upon the whole, the Evidence of this method he represented to the satisfaction of the Reader, some of the abstruse parts illustrated, or any improvements of this useful Art be proposed, I shall be under no great concern, though exceptions may be made to some modes of Expression, or to such Passages of this Treatise as are not essential to the principal design.