places before the pupil a problem that he sees at once is fictitious, and that has no human interest.

The designs given on page 283 involve chiefly the square as a basis, but it will be seen from one of the

figures that the equilateral triangle and the hexagon also enter. The possibilities of endless variation of a single design are shown in the illustration on page 284, the basis

in this case being the square. The variations in the use of the triangle and hexagon have been the object of study of many designers of Gothic windows, and some

examples of these forms are shown on page 285. In more simple form this ringing of the changes on elementary figures is shown on page 286. Some teachers have used color work with such designs for the purpose of increasing the interest of their pupils, but the danger of thus using the time with no serious end in view will be apparent.

In the matter of the mensuration of the circle the annexed design has some interest. The figure is not uncommon in decoration, and it is interest

ing to show, as a matter of pure geometry, that the area of the circle is divided into three equal portions by means of the four interior semicircles.

An important application of the formula a = r2 is seen in the area of the annulus, or ring, the formula being a = r222 = π (p2 — p12) = π (r + r') (r−r'). It is used in finding the area of the cross section

12 ' —

of pipes, and this is needed when we wish to compute the volume of the iron used.

Another excellent application is that of finding the area of the surface of a cylinder,

there being no reason why such simple cases from solid geometry should not furnish working material for plane geometry, particularly as they have already been met by the pupils in arithmetic.

A little problem that always has some interest for pupils is one that Napoleon is said to have suggested to his staff on his voyage to Egypt: To divide a circle into four equal parts by the use of circles alone.