*THE POPULAR SCIENCE MONTHLY.*

ings of each other and following the explanations. There continually come up at these times incidents of this sort: Fig. 13 is given as a solution of "Can you raise a perpendicular to a line, and from the end of it?"

In his explanation the pupil points out the given line, the end from which he is to erect the perpendicular, the point from which as a center he sweeps each circle, why he may take that point, and

why he sweeps the circle. Another pupil discovers, before the explanation is finished, that the problem can be solved with one less circle, and there is the keenest interest while he draws and explains his way (Fig. 14).

The original and independent power acquired is shown in Figs.

15, 16, 17, and 18, solutions of the problem, "Can you make an octagon with one side given?"

Spencer's "Inventional Geometry" is one of the most original