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 was 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 modern 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 rela

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tions 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 of 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 the propositions, indeed, might be reached by a sort of calculation; but such an artificial mode of procedure gives only 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 seem loose and defective. In adapting it to the actual state of the science, I have therefore endeavoured carefully

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 hope to have considerably smoothed the toil of the student. The numerous additions which are incorporated in the text, so far from retarding, will rather facilitate his progress, by rendering more continuous the chain of demonstration. To multiply the steps of ascent, is in general the most expeditious mode of gaining a summit.

The view which I have given of the nature of Proportion, in the fifth Book, will, I flatter myself, be found 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 that pave the way to the higher branches of Geometry, or lead immediately to valuable practical results. Yet the Appendix, without claiming the same degree of utility, will not be deemed the least interesting

portion of the volume, since the ingenious resources which it discovers are calculated to afford a very pleasing and instructive exercise.

The part which has cost me the greatest pains, is that devoted to Geometrical Analysis. The first Book consists of a series of the choicest problems, rising above each other in gradual succession. The second and third Books are almost wholly occupied with the researches of the Ancient Analysis. In framing them, I have consulted a great variety of authors, some of whom are of difficult access. The labour of condensing the scattered materials, will be duly estimated by those, who, taking delight in such fine speculations, are thus admitted at once to a rich and varied repast. The analytical investigations of the Greek geometers are indeed models of simplicity, clearness, and unrivalled elegance; and though miserably defaced by the riot of time and barbarism, they will yet be regarded by every person capable of appreciating their beauties, as some of the noblest monuments of human genius. It is matter of deep regret, that Algebra,