*INVENTIONAL GEOMETRY.*

into four equal angles without using more than four circles?" (Fig. 5).

"Can you construct a square on a line without using any other radius than the length of the line?" (Figs. 6, 7, 8, and 9).

Such problems as that solved in Fig. 10, "Can you place four octagons to meet in one point and to overlap each other to an equal extent?" delight the eye by beauty of form, and teach the pupil the basis of geometrical design.

Figs. 11 and 12, solutions to "Can you fit a square inside a circle, and another outside, in such positions with regard to each other as shall show the ratio the inner one has to the outer?"

and "Place a hexagon inside a circle, and another outside, in such positions with regard to each other as to show the ratio the inner one has to the outer," illustrate one way in which comparative area of figures is treated.

We have spoken of the pleasure a class experiences in putting their solutions upon the blackboard, and in examining the draw-