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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 Encyclopaedia 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.

ELEMENTS

OF

GEOMETRY.

N Grovern, is that branch of natural science which treats of bounded space.

Our knowledge concerning external objects is derived entirely from the information received through the medium of the senses. The science of Physics considers Bodies as they actually exist, invested at once with all their various qualities, and endued with their peculiar affections: Its researches are hence directed by that refined species of observation which is termed Experiment. But Geometry takes a more limited view ; and, selecting only the generic property of Magnitude, it can safely pursue the most lengthened train of investigation, and arrive with perfect certainty at the remotest conclusion. It contemplates mere

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ly the forms which bodies present, and the spaces which they occupy. Geometry is thus founded likewise on external observation ; but such observation is so familiar and obvious, that the primary notions which it furnishes might seem intuitive, and have often been regarded as innate. This science, proceeding from a basis of extreme simplicity, is therefore supereminently distinguished, by the luminous evidence which constantly attends every step of its progress.

PRINCIPLES.

IN contemplating an external object, we can, by successive acts of abstraction, reduce the complex idea which arises in the mind into others that are successively simpler. Body, divested of all its essential characters, presents the mere idea of surjace; a surface, considered apart from its peculiar qualities, exhibits only linear boundaries; and a line, abstracting its continuity, leaves nothing in the imagination, but the points which form its extremities. A solid is bounded by surfaces; a surface is circumscribed by lines ; and a line is terminated by points. A point marks position; a line measures distance ; and a surface presents eatension. A line has only length ; a surface has both length and breadth ; and a solid combines all the three dimensions of length, breadth, and thickness.

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