There are other projections than those formed by lines that are perpendicular to the plane. The lines may be oblique to the plane, and this is the case with most projections. A photograph, for example, is not formed by lines perpendicular to a plane, for they all converge in the camera. If the lines of projection are all perpendicular to the plane, the projection is said to be orthographic, from the Greek _ortho-_ (straight) and _graphein_ (to draw). A good example of orthographic projection may be seen in the shadow cast by an object upon a piece of paper that is held perpendicular to the sun's rays. A good example of oblique projection is a shadow on the floor of the schoolroom. THEOREM. _Between two straight lines not in the same plane there can be one common perpendicular, and only one._ The usual corollary states that this perpendicular is the shortest line joining them. It is interesting to compare this with the case of two lines in the same plane. If they are parallel, there may be any number of common perpendiculars. If they intersect, there is still a common perpendicular, but this can hardly be said to be between them, except for its zero segment. There are many simple illustrations of this case. For example, what is the shortest line between any given edge of the ceiling and the various edges of the floor of the schoolroom? If two galleries in a mine are to be connected by an air shaft, how shall it be planned so as to save labor? Make a drawing of the plan. At this point the polyhedral angle is introduced. The word is from the Greek _polys_ (many) and _hedra_ (seat). Students have more difficulty in grasping the meaning of the size of a polyhedral angle than is the case with dihedral and plane angles. For this reason it is not good policy to dwell much upon this subject unless the question arises, since it is better understood when the relation of the polyhedral angle and the spherical polygon is met. Teachers will naturally see that just as we may measure the plane angle by taking the ratio of an arc to the whole circle, and of a dihedral angle by taking the ratio of that part of the cylindric surface that is cut out by the planes to the whole surface, so we may measure a polyhedral angle by taking the ratio of the spherical polygon to the whole spherical surface. It should also be observed that just as we may have cross polygons in a plane, so we may have spherical polygons that are similarly tangled, and that to these will correspond polyhedral angles that are also cross, their representation by drawings being too complicated for class use. The idea of symmetric solids may be illustrated by a pair of gloves, all their parts being mutually equal but arranged in opposite order. Our hands, feet, and ears afford other illustrations of symmetric solids. THEOREM. _The sum of the face angles of any convex polyhedral angle is less than four right angles._