Surfaces, Volumes and Cube Roots; Doubling the Cube

Surface grows as a body departs from the sphere

The surface of a thread wound about its centre is far smaller than that of the same thread stretched straight, so a body extended to greater length has greater surface. A quantity of matter is clothed by the least surface when reduced to a figure most like the sphere, as with a ducat beaten into gold-leaf against a spherical drop.

The 48 surfaces of eight joined cubes reduce to 24

There are 48 surfaces in 8 joined cubes; the containing cube a b is clothed with only half of them, that is 24, because each of the 8 cubes loses half its six sides where they touch two by two inside. Of the 12 sides of the cylinder c e only two are lost, a sixth of the whole.

Equal surfaces need not enclose equal bodies

Just as from the 8 cubes of the cylinder a b one makes the cube c d, so from c d one may remake the cylinder; the science of cubing must go by bodies, not surfaces, since one same quantity has surfaces of infinite magnitudes. Two sacks of four staia each, unstitched into one sack of the same height, hold sixteen staia, because their combined circle is quadruple.

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

Of every four proportional numbers, beginning from one, the second is the cube root of the fourth. The matter of least surface is that most like the sphere, and of greatest surface the most unlike it; since that unlikeness is infinite, the capacity of its surface is infinite, every continuous quantity being divisible to infinity.

A rule for doubling the cube

Make this four times smaller, and you will have made a cube double one of these.

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.