ELEMENTS OF GEOMETRY. Groverty 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 B 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. The uniform tracing of a line which through its whole extent stretches in the same direction, gives the idea of a straight line. No more than one straight line can thereforejoin two points; and if a straight line be conceived to turn as an axis about both extremities, none of its intermediate points will change their position. From our idea of the straight line is derived that of a plane surface, which, though more complex, has a like uniformity of character. A straight line connecting any two points situate in a plane, lies wholly on the surface; and consequently planes must admit, in every way, a mutual and perfect application. ... Two points ascertain the position of a straight line; for the line may continue to turn about one of the points till it falls upon the other. But to determine the position of a plane, it requires three points; because a plane touching the straight line which joins two of the points, may be made to revolve, till it meets the third point. The separation or opening of two straight lines at their point of intersection, constitutes an angle. If we obtain the idea of distance, or linear extent, from contemplating progressive motion, we derive that of divergence, or angular magnitude, from the consideration of revolving motion. ! GEOMETRY is divided into Plane and Solid; the former confining its views to the properties of space figured on the same plane; the latter embracing the relations of different planes or surfaces, and of the solids which these describe or terminate. In the following definitions, therefore, the points and lines are all considered as existing in the same plane. BOOK I. DEFINITIONS. 1. A crooked line is that which con sists of straight lines not continued in the TS_2^_ same direction. 2. A curved line is that of which no ~0 S 5 portion is a straight line. 3. The straight lines which contain an angle are termed its sides, and their point of origin or intersection, its vertear. To abridge the reference, it is usual to denote an angle by tracing over its sides; the letter at the vertex, which is common to them both, being - placed in the middle. Thus, the angle contained by the straight lines AB and BC, or A. E. the opening formed by turning BA about the point B into the position BC, is named ABC or CBA. 4. A right angle is the fourth part of an entire circuit or revolution of a straight line. It is manifest that all right angles, being derived from the same measure, must be equal to each other. If a straight line CB stand at equal angles CBA and CBD on another straight line AD, and if the surface ACD be conceived laid over towards the opposite side, the point B and |