polygon for base, thereby obtaining successive prisms inscribed in the cylinder, and of the same height with it. Each time we double the number of sides in the base of the prism we take away more than half of the volume by which the cylinder exceeds the prism (since we take away more than half of the excess of the area of the circular base over that of the inscribed polygon, as in Euclid XII., 2). 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