INTRODUCTION. 1. ALL bodies occupy, in the indefinite space which embraces the material universe, a determinate, or finite place which we properly call a space. This finite space occupied by the body, has limits or boundaries, which separate it from the rest of space. These boundaries constitute what we call the Surface of the body. Surface being then the separation of a body from the rest of space, belongs as well to the one as the other; and as an infinite number of bodies may exist, in indefinite space, each having its proper surface, it follows that In space we may conceive of an infinite number of surfaces. When one surface is cut by another surface, the place of their mutual intersection is called a Line. This line belongs to each of the surfaces. Since the intersection of two surfaces gives a line, and any surface may be intersected by an infinite number of other distinct surfaces, it follows that On any surface whatever, we may conceive an infinite number of lines. The place of meeting, or of intersection of two lines, is called a Point. This point is common to the two lines. Since a point results from the meeting of two lines, and any line may be met. by an infinite number of other distinct lines, it follows that Any line may be regarded as having an infinite number of points. 2. Although we acquire the notion of a point by the consideration of lines, the notion of a line by the consideration of surfaces, and that of the surface by the consideration of a body; that is to say, of material things, we must not conclude from this, that points, lines, and surfaces are themselves material objects. In virtue of an inherent faculty of the mind, we may readily conceive of a point without the lines which determine it, of a line without the surfaces of which it is the intersection, of a surface independently of the body, or of the space of which it is the limit; in short, we may conceive of space itself, as absolutely immaterial. It is the result of these different abstractions which we call a point, a line, a surface, or a space. It is the same when we speak of the points of a line, of a surface, of a space; the lines of a surface, &c. 3. A space, a surface, and a line, may be considered in two distinct ways. We may consider them in regard to their different forms, which we name in general their Figures. Or, we may consider them in regard to their relative magnitudes, which is comprehended under the name of Extension. Extension takes the particular name of Volume, of Area, or of Length, according as it is applied to a space, a surface, or a line. Thus, the length of a line, or its linear extension, is the magnitude of this line, estimated or measured in units of a line. In the same way the area of a surface, or its superficial extension, is the magnitude of this surface, estimated or measured in units of a surface. In short, the volume or the extension of space is the magnitude of this space, estimated or measured in units of space. GEOMETRY is the science which treats of these two principal objects: The properties of different kinds of figures, and the measure of extension, considered under the different circumstances, as already noticed. THE POINT. 4. A point has neither figure nor extension; it is this, above all, which distinguishes it from all other objects of Geometry, which are all capable of being described and measured. Nevertheless as it is often necessary to consider one or more isolated points, we represent their position by a dot or distinct mark made on a surface with the pen, pencil, or crayon, and we distinguish them from each other, usually, by letters. Thus we say the point A, the point B, the point C, &c. .A .B .C .D The position of a point in reference to any other point is determined by its direction and distance from that point. This distance, which is the length of the shortest line which can unite these points, is obviously mutual to the two points; that is, the distance of the point B from the point A is the same as the distance of the point A from the point B. The direction of the point A from the point B is opposite to the direction of the point B from the point A. THE STRAIGHT LINE. 5. Of all geometrical lines, the simplest is the straight line. Although the idea of a straight line, is the first to which we are conducted by our experience, and the use of our senses, still it is very difficult to define it. The definition usually given is as follows: A straight line is the shortest distance between two points. A more general definition is as follows: A straight line is an indefinite line, such, that any limited portion whatever, is the shortest distance between the points which fix this limit. In any line the direction of one of its points, B, from another point, A, as has already been noticed, is opposite the direction of A from B; so that a line has two different directions exactly opposite, either of which may be regarded as the direction of the line. A Straight line is one which has the same direction throughout its whole extent. A Curved line is one which changes its direction at every point. Two straight lines are evidently capable of superposition, that is, of being placed the one on the other, so as to coincide. Hence, two straight lines coincide throughout their whole extent when they have two points common. Or, in other words, two points determine the position of a straight line. We also infer that two distinct straight lines can intersect or meet each other in only one point. THE PLANE. 6. The plane surface, or, as usually expressed, the Plane, is the simplest of all surfaces. It may be defined as follows: A plane is an indefinite surface, on which we may conceive that through each of its points a straight line may be made exactly to coincide with it throughout its whole extent. It immediately results from this definition, and the nature of |