Archimedes

But it has been proved in earlier propositions that P is equal to 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 be equal to it; that is, V = 1⁄3 C.

It only remains to add that Archimedes is fully acquainted with the main properties of the conic sections. These had already been proved in earlier treatises, which Archimedes refers to as the “Elements of Conics”. We know of two such treatises, (1) Euclid’s four Books on Conics, (2) a work by one Aristæus called “Solid Loci,” probably a treatise on conics regarded as loci. Both these treatises are lost; the former