Cube roots, solid surfaces, and man compared with the earth

Half of 48 surfaces clothe a cube half the size

Of 48 surfaces, 24 clothe a cube smaller by half than the one that clothes 4, since from those 4 cubes one must make 8 cubes. The exercise sets up the passage between counts of surfaces and the bodies they enclose.

The cube and the sphere have no common surface

Two disjoined cubes a b have as much body as the joined c d, but c d shows five sixths of the surface, so equalizing surfaces varies the bodies and equalizing bodies varies the surfaces. Hence a cube and a sphere cannot have one and the same surface.

Cube root by dividing twice by three

Dividing 27 by 3 gives 9, and 9 by 3 gives 3, so dividing twice by 3 reverses multiplying twice by 3 (3 times 3 is 9, 3 times 9 is 27). Multiplying the root cubically makes the cube, and dividing the cube cubically makes its root.

Unrolling a cube of 64 cells into a parallelepiped

A cube of eight cubes divided into 8 parallels and laid out lengthwise gives 64 little squares; extending the 4 sides (32 squares), adding the 64 of the greater square a b c d gives 96, and a further 64 of the base gives 160, which divided by 6 gives the side count.

The cube of man against the cube of the earth

The cube root of man is one braccio and the root of the earth twenty-one million miles, so the cube of the earth is 441 million of millions of millions. As man passes four harmonic times in breathing and the hour holds 1080 such times, the earth might breathe in a time as much greater as it exceeds man.

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