Kitap 12
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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.
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Therefore the cone AL is not to any solid greater than the cone EN as the circle ABCD is to the circle EFGH.
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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.
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But, as the cone is to the cone, so is the cylinder to the cylinder, for each is triple of each; [XII. 10]
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Therefore also, as the circle ABCD is to the circle EFGH, so are the cylinders on them which are of equal height.
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Therefore etc. Q. E. D.
PROPOSITION 12.
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Similar cones and cylinders are to one another in the triplicate ratio of the diameters in their bases.
12
Let there be similar cones and cylinders, let the circles ABCD, EFGH be their bases, BD, FH the diameters of the bases, and KL, MN the axes of the cones and cylinders; I say that the cone of which the circle ABCD is the base and the point L the vertex has to the cone of which the circle EFGH is the base and the point N the vertex the ratio triplicate of that which BD has to FH.
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For, if the cone ABCDL has not to the cone EFGHN the ratio triplicate of that which BD has to FH, the cone ABCDL will have that triplicate ratio either to some solid less than the cone EFGHN or to a greater.
