Surface versus volume: cube roots and doubling the cube

The more a body departs from the sphere, the greater its surface

A thread wound about its centre has far less surface than the same thread laid straight, so the more a body is extended the greater its surface. A quantity is clothed by the least surface when its figure is nearest the sphere, as with a ducat beaten into gilders' leaf.

Cube root of 8 by a quadrilateral cylinder of cubes

Imagining 8 cubes joined in a line as a quadrilateral cylinder whose sides each show 8 squares, multiply 8 by the 4 sides to get 32, divide by 4 to get 8, whose cube root 2 multiplied cubically makes a cube holding the 8 cubes.

Contact faces lost when cubes are joined

The 8 joined cubes show 48 surfaces; each cube loses half of its six sides in contact, so 24 are lost and the cube a b is clothed by 24 surfaces. The cylinder c e of 12 sides loses only two, a sixth of the whole.

Equal surfaces do not enclose equal bodies

Two sacks each holding four staia present eight staia of surface, but unstitched and sewn into a single sack of the same height they hold sixteen staia, because their circle becomes the semicircle of a circle quadruple the first. So the science of the cube must follow the bodies, not the surfaces.

The second of four proportional numbers is the cube root of the fourth

Of every four proportional numbers beginning from one, the second is the cube root of the fourth. Since every continuous quantity divides to infinity and each division adds two surfaces, a body's surface capacity is potentially infinite.

Nella metà superiore del foglio:

Figure con lettere.

A lato delle figure:

Sotto le figure:

Nella metà inferiore del foglio:

Figura con lettere:

Due figure con lettere.

A lato delle figure:

Figura con numeri:

In the upper half of the sheet:

Figures with letters.

Beside the figures:

Below the figures:

In the lower half of the sheet:

Figure with letters:

Two figures with letters.