Second case: Where the sides are as the proportion of 3:4.
In this figure the three squares are filled with small squares of three different colors, arranged as follows: in the square on the shorter side, 32 = 9; in that on the larger side, 42 = 16; in that on the hypotenuse, 52 = 25.
Second Case
The substitution game suggests itself. The two squares formed on the sides can be entirely filled by the small squares composing the square on the hypotenuse, so that they are both of the same color; while the square formed on the hypotenuse can be filled with varied designs by various combinations of the small squares of the sides which are in two different colors.
Third case: This is the general case.
The large frame is somewhat complicated and difficult to describe. It develops a considerable intellectual exercise. The entire frame measures 44 × 24 cm. and may be likened to a chess-board, where the movable pieces are susceptible of various combinations. The principles already proved or inductively suggested which lead to the demonstration of the theorem are:
(1) That two quadrilaterals having an equal base and equal altitude are equivalent.
(2) That two figures equivalent to a third figure are equivalent to each other.