The Teaching of Geometry

by David Eugene Smith

Theorem. All the perpendiculars that can be drawn to a straight line at a given point lie in a plane which is perpendicular to the line at the given point.

Thus the hands of a clock pass through a plane as the hands revolve, if they are, as is usual, perpendicular to the axis; and the same is true of the spokes of a wheel, and of a string with a stone attached, swung as rapidly as possible about a boy's arm as an axis. A clock pendulum too swings in a plane, as does the lever in a pair of scales.

Theorem. Through a given point within or without a plane there can be one perpendicular to a given plane, and only one.

This theorem is better stated to a class as two theorems.

Thus a plumb line hanging from a point in the ceiling, without swinging, determines one definite point in the floor; and, conversely, if it touches a given point in the floor, it must hang from one definite point in the ceiling. It should be noticed that if we cut through this figure, on the perpendicular line, we shall have the figure of the corresponding proposition in plane geometry, namely, that there can be, under similar circumstances, only one perpendicular to a line.

Theorem. Oblique lines drawn from a point to a plane, meeting the plane at equal distances from the foot of the perpendicular, are equal, etc.

There is no objection to speaking of a right circular cone in connection with this proposition, and saying that the slant height is thus proved to be constant. The usual corollary, that if the obliques are equal they meet the plane in a circle, offers a new plan of drawing a circle. A plumb line that is a little too long to reach the floor will, if swung so as just to touch the floor, describe a circle. A 10-foot pole standing in a 9-foot room will, if it moves so as to touch constantly a fixed point on either the floor or the ceiling, describe a circle on the ceiling or floor respectively.

One of the corollaries states that the locus of points in space equidistant from the extremities of a straight line is the plane perpendicular to this line at its middle point. This has been taken by some writers as the definition of a plane, but it is too abstract to be usable. It is advisable to cut through the figure along the given straight line, and see that we come back to the corresponding proposition in plane geometry.

A good many ships have been saved from being wrecked by the principle involved in this proposition.