The pyramid extended to a given length.

The pyramid shortened to a given lowness.

From the pyramid let the cube be made.

From the cube let its pyramid be made.

From the cube let the pyramid be made to a given height.

From a given height of pyramid let the cube be made.

From a pyramid let the cylinder be made.

That square body which exceeds the height of the cube is called a cylinder; that which is less in height than the cube is called a slab (tavola). And so it happens with the round ones.

From a pyramid let a slab of a given thickness be made.

From a pyramid let the slab of given breadth be made.

From a pyramid let the slab of given breadth and length be made.

From a slab of given breadth and length let a pyramid of given height be made.

Let the base of a pyramid be narrowed to a given narrowness; and the extension of such a pyramid is asked.

Let the base of the pyramid be widened, and let its shortening be given.

Lowering or raising the pyramid, one seeks the increase or diminution of its base.

The proportion of the extension of the cylinder will be the same as the contraction of its base; the like happens in the pyramids. And likewise the proportion of the increase of the cylinder in thickness is the same as the contraction of its length. The like I understand of the pyramids, as to the increase of their base.

The like of the square slabs increased in length, or of the length increased in breadth, because as much as you take from it one way it increases it the other, which is thus by its law; and as much as you thicken it, so much it becomes narrower, and as much as you narrow it, the more it thickens. And because these proportions are surd and irrational, I have therefore taken care to demonstrate them geometrically in every difficult case.