The mathematical sciences are two, which extend over the two quantities, that is the continuous and the discrete, and no more than these are found in nature. From this is born perspective, which extends into the office of the eye, the first sense, with all its pleasures. The second is music, given to the second sense, less worthy. The third is taste, since astronomy too lies within perspective.

By pyramidal multiplication, 3 surface diagonals of the cube times its 3 sides yield as much as 4 sides of the same cube times its 4 half-axes. What has been said is proved by the foregoing, which shows a b c d to be 1/6 of a cube, and likewise e f g h o to be 1/6 of the same cube, because its 4 lower semidiameters divide it thus, that is into 6 equal and similar pyramids.

And likewise, as to what is said above, that a b c d is 1/6 of its cube, I prove it again, because its base a b c is the base of another similar pyramid a b c e. And it is equal to that one, because its greater height a e is equal to the greater height of the other, at d e. And the triangle b e c is the base of 2 pyramids of equal height, e b c a and e b c f. And they are of equal heights, because the greater height of each is as tall as the lower semidiameter a f.

b, a d, f, e c

This mode of multiplying is purely geometric, because the parts are named by continuous proportion. And otherwise they cannot be named except by surd (irrational) roots.

a c, b f, e, d, g, o

a b c d is 1/6 of its cube. And b c d e f g is 1/3 of that cube.