The Teaching of Geometry

by David Eugene Smith

Theorem. If two planes are perpendicular to each other, a straight line drawn in one of them perpendicular to their intersection is perpendicular to the other.

This and the related propositions allow of numerous illustrations taken from the schoolroom, as of door edges being perpendicular to the floor. The pretended applications of these propositions are usually fictitious, and the propositions are of value chiefly for their own interest and because they are needed in subsequent proofs.

Theorem. The locus of a point equidistant from the faces of a dihedral angle is the plane bisecting the angle.

By changing "plane" to "line," and by making other obvious changes to correspond, this reduces to the analogous proposition of plane geometry. The figure formed by the plane perpendicular to the edge is also the figure of that analogous proposition. This at once suggests that there are two planes in the locus, provided the planes of the dihedral angle are taken as indefinite in extent, and that these planes are perpendicular to each other. It may interest some of the pupils to draw this general figure, analogous to the one in plane geometry.

Theorem. The projection of a straight line not perpendicular to a plane upon that plane is a straight line.

In higher mathematics it would simply be said that the projection is a straight line, the special case of the projection of a perpendicular being considered as a line-segment of zero length. There is no advantage, however, of bringing in zero and infinity in the course in elementary geometry. The legitimate reason for the modern use of these terms is seldom understood by beginners.

This subject of projection (Latin pro-, "forth," and jacere, "to throw") is extensively used in modern mathematics and also in the elementary work of the draftsman, and it will be referred to a little later. At this time, however, it is well to call attention to the fact that the projection of a straight line on a plane is a straight line or a point; the projection of a curve may be a curve or it may be straight; the projection of a point is a point; and the projection of a plane (which is easily understood without defining it) may be a surface or it may be a straight line. An artisan represents a solid by drawing its projection upon two planes at right angles to each other, and a map maker (cartographer) represents the surface of the earth by projecting it upon a plane. A photograph of the class is merely the projection of the class upon a photographic plate (plane), and when we draw a figure in solid geometry, we merely project the solid upon the plane of the paper.

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.