Now that the proofs of the theorems in the first book are clearly discussed by us, the next thing is the same kind of study with the theorems of the second book.

First he says in the 1st theorem:

“Let a cylinder be taken, half as large again as the given cone or Arch. 188 cylinder.” This can be done in two ways, either keeping in both the same base, or the same height. And to make what I said clearer, let a cone or a cylinder be imagined, whose base is the circle A, and its height AΓ, and let the requirement be to find a cylinder half as large again as it.

(a) Let the cylinder AΓ be laid down, (b) and let the height of the cylinder, AΓ, be produced, (c) and let ΓΔ be set out <as> half AΓ; (1) therefore ΓΔ is half as large again as AΓ. (d) So if we imagine a cylinder having, <as> base, the circle A, and, <as> height, the line AΔ, (2) it shall be half as large again as the <cylinder> set forth, AΓ; (3) for the cones and cylinders which are on the same base are to each other as the height.