Euclid, Elements
Thomas L. Heath, Sir Thomas Little Heath, Ed.

Therefore the cone ABCDL has not to any solid greater than the cone EFGHN the ratio triplicate of that which BD has to FH.

But it was proved that neither has it this ratio to a less solid than the cone EFGHN.

Therefore the cone ABCDL has to the cone EFGHN the ratio triplicate of that which BD has to FH.

But, as the cone is to the cone, so is the cylinder to the cylinder, for the cylinder which is on the same base as the cone and of equal height with it is triple of the cone; [XII. 10] therefore the cylinder also has to the cylinder the ratio triplicate of that which BD has to FH.

Therefore etc. Q. E. D.

PROPOSITION 13.

For let the cylinder AD be cut by the plane GH which is parallel to the opposite planes AB, CD, and let the plane GH meet the axis at the point K; I say that, as the cylinder BG is to the cylinder GD, so is the axis EK to the axis KF.

For let the axis EF be produced in both directions to the points L, M, and let there be set out any number whatever of axes EN, NL equal to the axis EK, and any number whatever FO, OM equal to FK; and let the cylinder PW on the axis LM be conceived of which the circles PQ, VW are the bases.

Let planes be carried through the points N, O parallel to AB, CD and to the bases of the cylinder PW, and let them produce the circles RS, TU about the centres N, O.

Then, since the axes LN, NE, EK are equal to one another,