The Teaching of Geometry

by David Eugene Smith

The problem becomes still more difficult if we have two roads of different widths that we wish to join on a curve. Here the two centers of curvature are not the same, and the one road narrows to the other on the curve. The solutions will be understood from a study of the figures.

The number of problems of this kind that can easily be made is limitless, and it is well to avoid the danger of hobby riding on this or any similar topic. Therefore a single one will suffice to close this group.

If a road AB on an arc described about O, is to be joined to road CD, described about O', the arc BC should evidently be internally tangent to AB and externally tangent to CD. Hence the center is on BOX and O'CY, and is therefore at P. The problem becomes more real if we give some width to the roads in making the drawing, and imagine them in a park that is being laid out with drives.

It will be noticed that the above problems require the erecting of perpendiculars, the bisecting of angles, and the application of the propositions on tangents.

A somewhat different line of problems is that relating to the passing of a circle through three given points. It is very easy to manufacture problems of this kind that have a semblance of reality.

For example, let it be required to plan a driveway from the gate G to the porch P so as to avoid a mass of rocks R, an arc of a circle to be taken. Of course, if we allow pupils to use the Pythagorean Theorem at this time (and for metrical purposes this is entirely proper, because they have long been familiar with it), then we may ask not only for the drawing, but we may, for example, give the length from G to the point on R (which we may also call R), and the angle RGO as 60°, to find the radius.

A second general line of exercises adapted to Book II is a continuation of the geometric drawing recommended as a preliminary to the work in demonstrative geometry. The copying or the making of designs requiring the describing of circles, their inscription in or circumscription about triangles, and their construction in various positions of tangency, has some value as applying the various problems studied in this book. For a number of years past, several enthusiastic teachers have made much of the designs found in Gothic windows, having their pupils make the outline drawings by the help of compasses and straightedge. While such work has its value, it is liable soon to degenerate into purposeless formalism, and hence to lose interest by taking the vigorous mind of youth from the strong study of geometry to the weak manipulation of instruments. Nevertheless its value should be appreciated and conserved, and a few illustrations of these forms are given in order that the teacher may have examples from which to select. The best way of using this material is to offer it as supplementary work, using much or little, as may seem best, thus giving to it a freshness and interest that some have trouble in imparting to the regular book work.

The best plan is to sketch rapidly the outline of a window on the blackboard, asking the pupils to make a rough drawing, and to bring in a mathematical drawing on the following day.