three times the pyramid with the same base as the prism and equal height.
Therefore that pyramid is greater than V, the volume of the cone: which is impossible, since the cone encloses the pyramid.
Therefore C is not greater than 3V.
Next (2) suppose that C < 3V, so that, inversely,
V > 1⁄3 C.
This time we inscribe successive pyramids in the cone until we arrive at a pyramid such that the portions of the cone left over outside it are together less than the excess of V over 1⁄3 C. It follows that the pyramid is greater than 1⁄3 C. Hence the prism on the same base as the pyramid and inscribed in the cylinder (which prism is three times the pyramid) is greater than C: which is impossible, since the prism is enclosed by the cylinder, and is therefore less than it.
Therefore V is not greater than 1⁄3 C, or C is not less than 3V.
Accordingly C, being neither greater nor less than 3V, must