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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 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 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 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 ingenious 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

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viated. The Elements of Trigonometry are much expanded, and now brought to include whatever appears to be most valuable in recent practice. But the greatest additions have been made in the Notes and Illustrations, which will be found to contain a variety of useful and curious information. The more advanced student may peruse with advantage the historical and critical remarks; and some of the disquisitions, with the solutions of certain more difficult problems relative to trigonometry and geodesiacal operations, in which the modern analysis is but sparingly introduced, are of a nature snfficiently interesting to claim the notice of proficients in science. I have simplified, and materially enlarged the formulae connected with trigonometrical computation; explained the art of surveying, in its dif. ferent branches; and given reduced plans, blended with the narrative of the great operations lately carried on both in England and France. I have likewise shown a very simple method of calculating heights from barometrical observations, accompanied by illustrative sections; and I have been thence led to state the law of climate, as it is modified by elevation. On this attractive subject, I should have dwelt with pleasure, had the limits of the volume permitted. My original design was to exhibit, within perhaps the compass of five volumes, the Elements of Mathematical Science in their fullextent, including the principles and application of the Higher Calculus. But, after due reflection, I have abandoned that aspiring project. The publication of abstract works in this country procures neither fame nor emolument; and after having discharged the more pressing obligations which I had contracted, I shall consider my time as more agreeably and perhaps more beneficially employed in pursuing without distraction the labyrinths of physical research. The text of the present volume has, by successive improvements, arrived at such a state of maturity, that I shall hardly be tempted in any future edition to alter it. It will be followed, without delay, by another volume, which is to contain the tract on Geometrical Analysis, enlarged and improved; the Geometry of Lines of the Second Order, expanded to three books, and including the more important of the Higher Curves; and the Geometry of Planes and Solids, embracing Spherical Trigonometry, with Perspective and the Projection of the Sphere. I intend likewise to print, with all convenient speed, a short treatise on the Philosophy of Arithmetic. The substance of it is already before the public, in the Supplement to the Encyclopædia Britannica; but I shall endeavour to abridge, to modify and improve that article. As a sequel, I wish to give a concise and accurate view of the Elements of Algebra, though I will not absolutely pledge myself to the performance of a task so much wanted. It is the nature of genuine science to advance in continual progression. Each step carries it still higher; new relations are descried; and the most distant objects seem gradually to approximate. But, while science thus enlarges its bounds, it likewise tends uniformly to simplicity and concentration. The discoveries of one age are, perhaps in the next, melted down into the mass of elementary truths. What are deemed at first merely objects of enlightened curiosity, become, in due time, subservient to the most important interests. Theory soon descends to guide and assist the operations of practice. To the geometrical speculations of the Greeks, we may distinctly trace whatever progress the moderns have been enabled to achieve in mechanics, navagation, and the various complicated arts of life. A refined analysis has unfolded the harmony of the celestial motions, and conducted the philosopher, through a maze of intricate phenomena, to the great laws appointed for the government of the Universe.

College of EDINBURGH,
March 1. 1817.

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