Suppose now that V is the volume of the cone, C that of the cylinder. We have to prove that C = 3V. If C is not equal to 3V, it is either greater or less than 3V.
Suppose (1) that C > 3V, and that C = 3V + E. Continue the construction of prisms inscribed in the cylinder until the parts of the cylinder left over outside the final prism (of volume P) are together less than E.
| Then | C − P < E. |
|---|---|
| But | C − 3V = E; |
| Therefore | P > 3V. |
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