PRINTED FOR ARCIII BALD CONSTABLE & Co. 1817. PREFACE. THE volume now laid before the public, is the first of a projected Course of Mathematical Science. Many compendiums or elementary treatises have appeared—at different times, and of various merit; but there seemed still wanting, in our language, a work that should embrace the subject in its full extent, that should unite theory with practice, and connect the ancient with the mo dern discoveries. The magnitude and difficulty of such a task might deter an individual from the attempt, if he were not deeply impressed with the importance of the undertaking, and felt his exertions to accomplish it animated by zeal, and supported by active perseverance. The study of Mathematics holds forth two capital objects:—While it traces the beautiful relations of figure and quantity, it likewise accustoms the mind to the invaluable exercise of patient attention and accurate reasoning. Of these distinct objects, the last is perhaps the most important in a course of liberal education. For this purpose, the Geometry . the Greeks is the most powerfully recommended, as bearing the stamp of that acute people, and displaying the finest specimens of logical deduction. Some of its conclusions, indeed, might be reached by a sort of calculation; but such an artificial mode of procedure gives merely an apparent facility, and leaves no clear or permanent impression on the mind. We should form a wrong estimate, however, did we consider the Elements of Euclid, with all its merits, as a finished production. That admirable work was composed at the period when Geometry was making its most rapid advances, and new prospects were opening on every side. No wonder that its structure should now appear loose and defective. In adapting it to the actual state of the science, I have therefore endeavoured care. fully to retain the spirit of the original, but have sought to enlarge the basis, and to dispose the accumulated materials into a regular and more compact system. By simplifying the order of arrangement, I presume to have materially abridged the labour of the student. The numerous additions that are incorporated in the text, so far from retarding, will rather facilitate his progress, by rendering more continuous the chain of demonstration. The view which I have given of the nature of Proportion, in the Fifth Book, will contribute, I hope, to remove the chief difficulties attending that important subject. The Sixth Book, which exhibits the application of the Doctrine of Ratios, contains a copious selection of propositions, not only beautiful in themselves, but which pave the way to the higher branches of Geometry, or lead immediately to valuable practical results. The Appendix, without claiming the same degree of utility, will not perhaps be deemed the least interesting portion of the volume, since the ingemious resources which it discloses for the construction of certain problems are calculated to afford a very pleasing and instructive exercise. The Elements of Trigonometry are as ample as my plan would allow. I have explained fully the properties of the lines about the circle, and the calculation of the trigonometrical tables; nor have I omitted any proposition which has a distinct reference to practice. Some of the problems annexed are of essential consequence in marine surveying. In the improvement of this edition, I have spared no trouble or expence. The text has been simplified and reduced to a shorter compass, by throwing such propositions as were less elementary to the Notes. Other Notes of a simpler kind are intended chiefly to engage the attention of the young student. In various parts of the work, the demonstrations are occasionally abbre |