For if the cylinder is not triple of the cone, the cylinder will be either greater than triple or less than triple of the cone.
First let it be greater than triple, and let the square ABCD be inscribed in the circle ABCD; [IV. 6]
then the square ABCD is greater than the half of the circle ABCD.
From the square ABCD let there be set up a prism of equal height with the cylinder.
Then the prism so set up is greater than the half of the cylinder,
inasmuch as, if we also circumscribe a square about the circle ABCD, [IV. 7]
the square inscribed in the circle ABCD is half of that circumscribed about it,
and the solids set up from them are parallelepipedal prisms of equal height,
while parallelepipedal solids which are of the same height are to one another as their bases; [XI. 32]
therefore also the prism set up on the square ABCD is half of the prism set up from the square circumscribed about the circle ABCD; [cf. XI. 28, or XII. 6 and 7, Por.]
and the cylinder is less than the prism set up from the square circumscribed about the circle ABCD;
therefore the prism set up from the square ABCD and of equal height with the cylinder is greater than the half of the cylinder.
Let the circumferences AB, BC, CD, DA be bisected at the points E, F, G, H, and let AE, EB, BF, FC, CG, GD, DH, HA be joined;
then each of the triangles AEB, BFC, CGD, DHA is greater than the half of that segment of the circle ABCD which is about it, as we proved before. [XII. 2]
On each of the triangles AEB, BFC, CGD, DHA let prisms be set up of equal height with the cylinder;
then each of the prisms so set up is greater than the half part of that segment of the cylinder which is about it,
inasmuch as, if we draw through the points E, F, G, H parallels to AB, BC, CD, DA,
complete the parallelograms on AB, BC, CD, DA,
and set up from them parallelepipedal solids of equal height with the cylinder, the prisms on the triangles AEB, BFC, CGD, DHA are halves of the several solids set up;
and the segments of the cylinder are less than the parallelepipedal solids set up;
hence also the prisms on the triangles AEB, BFC, CGD, DHA are greater than the half of the segments of the cylinder about them.
Thus, bisecting the circumferences that are left, joining straight lines, setting up on each of the triangles prisms of equal height with the cylinder, and doing this continually, we shall leave some segments of the cylinder which will be less than the excess by which the cylinder exceeds the triple of the cone. [X. 1]
Let such segments be left, and let them be AE, EB, BF, FC, CG, GD, DH, HA;
therefore the remainder, the prism of which the polygon AEBFCGDH is the base and the height is the same as that of the cylinder, is greater than triple of the cone.
But the prism of which the polygon AEBFCGDH is the base and the height the same as that of the cylinder is triple of the pyramid of which the polygon AEBFCGDH is the base and the vertex is the same as that of the cone; [XII. 7, Por.]
therefore also the pyramid of which the polygon AEBFCGDH is the base and the vertex is the same as that of the cone is greater than the cone which has the circle ABCD as base.
But it is also less, for it is enclosed by it: which is impossible.
Therefore the cylinder is not greater than triple of the cone.