more readily seen what is understood by plan and elevation, e.J., First,

(B) The elevation of a rectangular prism thus

5. To any one not conversant with the principles of solid geometry, the above drawings convey no idea of a cube, cone, &c., both

plan and elevation being represented as a surface, drawn on the same plane-whereas they really represent objects as covering two planes at right angles to one another.

6. These two "planes of projection," as they are called, are distinguished as the horizontal plane and the vertical plane. They might be conveniently illustrated by the floor and walls of a room; the floor representing the horizontal plane, and the several walls so many vertical planes. The line in which the floor and any given wall intersect each other is called the "line of intersection;" and sometimes the ground line, or base line.

7. The two drawings which represent the plan and elevation of an object are in solid geometry called the projections of that object. Now as every solid is bounded by planes or surfaces, surfaces by lines, and lines by points, we proceed to show what is meant by the projection of a point, and of a line on the two planes of projection.

8. First, the projection of a point is obtained thus

Let bac be the end view of a sheet of paper, folded in such a manner as to form a right angle at a. Then ba may be regarded as an end view of the vertical plane of projection, and ac an end view of the horizontal plane of projection.

Let A be a point in space. It is required to find its projections upon ab, ac.

From A, draw AA' perpendicular to ba, and AA perpendicular to ac. Then the points A, A', where the perpendiculars meet the given planes, are the projections of the point A in space.

NOTE 1.-The projection of a point upon a plane is the foot of a perpendicular let fall from the point upon the given plane.

NOTE 2.-The line which projects a point upon a plane, is termed the projector of that point, e. g., AA', AA are the projectors of the given point A.

NOTE 3.-From this it is evident that, when the projections of a point are given, the point may be found, since it is the point of intersection of the projectors of the point.

9. Secondly, the projections of a line are obtained thus—

Let abay be the horizontal plane of projection, and cdyx the vertical, also let AB be the position of a line in space.

It is required to find the projections of the line AB upon abxy and cdyx.

The projection of B upon the plane abry is the foot of the perpendicular let fall from B upon the given plane, say point B. Similarly, the projection of A upon the plane abxy is the foot of the perpendicular let fall from A, say point A.

Join AB; then AB is the plan or projection of AB upon the horizontal plane.

Next, the projection of B upon the plane cdyx is the foot of the perpendicular drawn from B to the given plane, say point B. Similarly, we obtain the projection of A upon the given plane, say A'.

Join A'B'; then A'B' is the elevation or projection of AB upon the vertical plane.

NOTE 1.-Having found A, B, the projections of A, B, upon the horizontal plane, the elevation of A'B' is thus found-Am, Bn are drawn at right angles to the plane cdyx, meeting xy, the intersecting line of the two planes, in m and n. From m and n lines are drawn parallel to AA, and BB; then the intersections A' and B' of these lines, with the perpendicuJars let fall from A and B, will be the required projections.

NOTE 2. From the elevation A'B', it may be readily seen how we obtain the plan AB-the operation being just the converse of that shown in the preceding note.

10. As soon as the foregoing projections are thoroughly understood, the student will easily comprehend the projection of a solid; e. g.

Let abxy and cdxy be the planes of projection, and ABCD, &c., the position in space of a regular solid. It is required to find the projection of the solid upon the two given planes.

The plan of C will be the foot of a perpendicular let fall from C upon the horizontal plane cdxy. Let C be its plan. In the same manner we find B. Join BC; then BC is the plan of the line BC. In the same manner we find AD, the plan of the line AD. Join AB and CD; then ABCD is the plan or projection of the given solid ABCD, &c., upon the horizontal plane of projection.

The intersections of the perpendiculars from the points C, D, E, F, with the plane abxy will give the elevation, or projection of the solid upon the vertical plane. The plane BBCC, passing through the line BC, projects that line upon cdxy. Also, the plane CC'D'D passing through the line CD, projects that line upon abxy.

11. The line BC, and all lines parallel to it, are parallel to the horizontal plane of projection; and the line CF, and all lines parallel to it, are parallel to the vertical plane of projection. Also, the line BC, and all lines parallel to it, are projected upon the horizontal plane of projection in lines equal and parallel to themselves; and the same

remark applies to the projections of the line CF, and to all lines parallel to it, upon the vertical plane of projection.

The line CF, and all lines parallel to it, are perpendicular to the horizontal plane of projection, and are projected on that plane in points. Also, the line BC, and all lines parallel to it, are perpendicular to the vertical plane of projection, and are projected on that plane in points.

NOTE 1.-When a line is parallel to the horizontal and vertical plane, its projections are lines parallel to xy, the line of intersection, and equal in length to the original line. The projections C', D' and C, D, of the line CD are parallel to xy, and equal in length to CD.

NOTE 2.-When a line is perpendicular to the plane of projection its projection on that plane is a point. The lines CB, CF, respectively perpendicular to the vertical and horizontal plane, are projected on those planes in the points C'C.

NOTE 3.—Just as when the projections of a point are given, the point itgelf may be found, so the preceding solid may be determined from its projection on the two planes. For example, the surface BCFG is the intersection of the projecting plane of BC with the projecting plane of C'F'. The remaining surfaces are the intersections of the projecting planes of the lines which are the projections of those surfaces.

12. Since objects whose surfaces lie in different planes have to be represented upon a sheet of paper which is but one plane, the vertical plane must be supposed to revolve upon the line of intersection of the planes of projection until it coincides with the horizontal plane.

13. Thus in the figure following, the vertical plane cdxy after revolving one-fourth of a revolution, as shown by the arcs ed, fc, will

assume the position efxy, which is a continuation of the horizontal plane abxy. Now after the vertical plane cday has revolved as de