The Teaching of Geometry

by David Eugene Smith

supplied with draftsmen's materials, and is not compelled to use them in a foolish manner, so much the better. They will not hurt his geometry if the teacher does not interfere, and they will help his practical drawing; but for obvious reasons we cannot demand that the pupil purchase what is not really essential to his study of the subject. The most valuable single instrument of the three just mentioned is the protractor, and since a paper one costs only a few cents and is often helpful in the drawing of figures, it should be recommended to pupils.

There is also another line of work that often arouses a good deal of interest, namely, the simple field measures that can easily be made about the school grounds. Guarding against the ever-present danger of doing too much of such work, of doing work that has no interest for the pupil, of requiring it done in a way that seems unreal to a class, and of neglecting the essence of geometry by a line of work that involves no new principles,—such outdoor exercises in measurement have a positive value, and a plentiful supply of suggestions in this line is given in the subsequent chapters. The object is chiefly to furnish a motive for geometry, and for many pupils this is quite unnecessary. For some, however, and particularly for the energetic, restless boy, such work has been successfully offered by various teachers as an alternative to some of the book work. Because of this value a considerable amount of such work will be suggested for teachers who may care to use it, the textbook being manifestly not the place for occasional topics of this nature.

For the purposes of an introduction only a tape line need be purchased. Wooden pins and a plumb line can easily be made. Even before he comes to the propositions in mensuration in geometry the pupil knows, from his arithmetic, how to find ordinary areas and volumes, and he may therefore be set at work to find the area of the school ground, or of a field, or of a city block. The following are among the simple exercises for a beginner:

1. Drive stakes at two corners, A and B, of the school grounds, putting a cross on top of each; or make the crosses on the sidewalk, so as to get two points between which to measure. Measure from A to B by holding the tape taut and level, dropping perpendiculars when necessary by means of the plumb line, as shown in the figure. Check the work by measuring from B back to A in the same way. Pupils will find that their work should always be checked, and they will be surprised to see how the results will vary in such a simple measurement as this, unless very great care is taken. If they learn the lesson of accuracy thus early, they will have gained much.

2. Take two stakes, X, Y, in a field, preferably two or three hundred feet apart, always marked on top with crosses so as to have exact points from which to work. Let it then be required to stake out or "range" the line from X to Y by placing stakes at specified distances. One boy stands at Y and another at X, each with a plumb line. A third one takes a plumb line and stands at P, the observer at X motioning to him to move his plumb line to the right or the left until it is exactly in line with X and Y. A stake is then driven at P, and the pupil at X moves on to the stake P. Then Q is located in the same way, and then R, and so on. The work is checked by ranging back from Y to X. In some of the simple exercises suggested later it is necessary to range a line so that this work is useful in making measurements. The geometric principle involved is that two points determine a straight line.

3. To test a perpendicular or to draw one line perpendicular to another in a field, we may take a stout cord twelve feet long, having a knot at the end of every foot. If this is laid along four feet, the ends of this part being fixed, and it is stretched as here shown, so that the next vertex is five feet from one of these ends and three feet from the other end, a right angle will be formed. A right angle can also be run by making a simple instrument, such as is described in Chapter XV. Still another plan of drawing a line perpendicular to another line AB, from a point P, consists in swinging a tape from P, cutting AB at X and Y, and then bisecting XY by doubling the tape. This fixes the foot of the perpendicular.