Similar Triangles in Quadruple Proportion; the Cube in Six Pyramids
A triangle of base 1 against one of base 2, plus cubes divided by their lower diameters
Nested triangle diagrams show that triangles of like figure stand in the same ratio as the squares of their bases: the small triangle d c e counts as one, while a b e, with double the base, is four. Below, cubes crossed by their four lower diameters are used to prove that the cube divides into six equal pyramids and that a pyramid of double height is one-third of the cube. The two demonstrations connect plane proportion with the dissection of solids. Letter-labels a through r key the figures to the text.
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Similar triangles scale as the square of the base
Among triangles of like figure, the same proportion holds in their magnitude as that of the multiplication of their bases by themselves. Triangle d c e with base 1 gives one times one, one; the larger a b e with base 2 gives two times two, four. So the ratio of the one to the other is quadruple.
The four lower diameters divide the cube into six pyramids
If the four lower diameters a h, b g, d e and e f of the cube are drawn, they meet at the lower centre n and the cube is divided into six equal pyramids, so pyramid e f g h n is one-sixth of the cube. The pyramid e f g h m, being twice as tall, is therefore one-third of the cube.