Kitap 12
11
But, as the circle ABCD is to the circle EFGH, so is the cone AL to the solid O, and, as the polygon DTAUBVCW is to the polygon HPEQFRGS, so is the pyramid of which the polygon DTAUBVCW is the base and the point L the vertex to the pyramid of which the polygon HPEQFRGS is the base and the point N the vertex. [XII. 6]
11
Therefore also, as the cone AL is to the solid O, so is the pyramid of which the polygon DTAUBVCW is the base and the point L the vertex to the pyramid of which the polygon HPEQFRGS is the base and the point N the vertex; [V. 11] therefore, alternately, as the cone AL is to the pyramid in it, so is the solid O to the pyramid in the cone EN. [V. 16]
11
But the cone AL is greater than the pyramid in it; therefore the solid O is also greater than the pyramid in the cone EN.
11
But it is also less: which is absurd.
11
Therefore the cone AL is not to any solid less than the cone EN as the circle ABCD is to the circle EFGH.
11
Similarly we can prove that neither is the cone EN to any solid less than the cone AL as the circle EFGH is to the circle ABCD.
11
I say next that neither is the cone AL to any solid greater than the cone EN as the circle ABCD is to the circle EFGH.
11
For, if possible, let it be in that ratio to a greater solid O; therefore, inversely, as the circle EFGH is to the circle ABCD, so is the solid O to the cone AL.
11
But, as the solid O is to the cone AL, so is the cone EN to some solid less than the cone AL; therefore also, as the circle EFGH is to the circle ABCD, so is the cone EN to some solid less than the cone AL: which was proved impossible.
11
Therefore the cone AL is not to any solid greater than the cone EN as the circle ABCD is to the circle EFGH.
11
But it was proved that neither is it in this ratio to a less solid; therefore, as the circle ABCD is to the circle EFGH, so is the cone AL to the cone EN.
11
But, as the cone is to the cone, so is the cylinder to the cylinder, for each is triple of each; [XII. 10]
