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 a 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 tavola (slab). And the same happens with round bodies.

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

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

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

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

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

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

By lowering or raising the pyramid, the increase or diminution of its base is sought.

The proportion of the cylinder's extension will be the same as the contraction of its base; and the like happens with pyramids. And so the proportion of the cylinder's increase in thickness is the same as the contraction of its length. The same I mean of pyramids as to the increase of their base.

The same for square slabs increased in length, or whose length is increased in breadth; for as much as you take from them in one direction, it grows in the other, which is by their law; and as much as you thicken it, so much the more it becomes narrow, 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.