INTRODUCTION.

THE following Compilation was drawn up at a time when the difficulties, which usually present themselves on a first perusal of the Principia, were fresh in the recollection of its Author. Upon a late accidental revision of it he was induced to think that it might, if printed in a convenient form, prove an useful guide to those, who not enjoying the benefits of Academical or other instruction, are yet desirous of becoming acquainted with so much at least of the Principia, as is necessary to a clear comprehension of the more prominent and obvious laws of the Planetary System. Perhaps even to the regularly educated Student it may not be wholly unacceptable as a book of occasional reference; inasmuch as, besides the Commentary properly so called, it will be found to contain, carefully arranged under proper heads, all or most of those Problems and Deductions from the Text, which, after having been collected by the Student at the expence of much time and trouble, are usually entered without any great regard to order or connexion, in the pages of his Manuscript.

The following is the plan and arrangement of this Treatise.

I. Newtons text entire, with the exception of Props. 3, 5, and 17; Lemmas 12, 13, and 14, relating to well-known properties of the Conic Sections; a few of the Scholia; and the

ulitcr proofs in the 2d and 3d Sections; all of which, as being of less general use and application, might, it was conceived, be omitted without injury to the work.

II. A general Introduction to the three Sections, comprising a concise account, with Examples, of the Methods of Exhaustions and Indivisibles, and the doctrine of Limits.

III. Notes explanatory of Newtons text. In this part, which forms the main body of the Treatise, the following method has been invariably adhered to. (j) Each Lemma and Proposition is prefaced, wherever the subject appeared to require it, with such introductory remarks as were thought necessary to prepare the reader for Newtons demonstration, (jj) The Lemma or Proposition itself, where any difficulty occurs, h explained in as distinct and familiar a way as the subject would admit of. (jjj) At the end of each will be found subjoined, under the appellation of Notes, such further remarks, deductions, and problems as the Proposition under consideration seemed naturally to suggest.

IV. A collection of Miscellaneous Problems, with their solutions.

The reader will observe that the short account given of the doctrine of Exhaustions and Indivisibles, and also Arts. 52, 53, and 54, on curvature, have been extracted almost wholly from Maclaurin j and as utility has been his sole object, the Compiler of the following sheets has throughout unreservedly borrowed from every valuable source within his reach.

Should this attempt be favourably received by those forwhose use it is exclusively designed, and the Authors leisure permit, the 7th and 8th Sectfons may probably follow, upon precisely the same plan. ....