§ 87. A method similar to that used in proving Prop. 5 leads to the following results relating to solids bounded by simple curved surfaces:—
Prop. 10. Every cone is the third part of a cylinder which has the same base, and is of an equal altitude with it.
Prop. 11. Cones or cylinders of the same altitude are to one another as their bases.
Prop. 12. Similar cones or cylinders have to one another the triplicate ratio of that which the diameters of their bases have.
Prop. 13. If a cylinder be cut by a plane parallel to its opposite planes or bases, it divides the cylinder into two cylinders, one of which is to the other as the axis of the first to the axis of the other; which may also be stated thus:—
Cylinders on the same base are proportional to their altitudes.
Prop. 14. Cones or cylinders upon equal bases are to one another as their altitudes.
Prop. 15. The bases and altitudes of equal cones or cylinders are reciprocally proportional, and if the bases and altitudes be reciprocally proportional, the cones or cylinders are equal to one another.
These theorems again lead to formulae in mensuration, if we compare a cylinder with a prism having its base and altitude equal to the base and altitude of the cylinder. This may be done by the method of exhaustion. We get, then, the result that their bases are equal, and have, if B denotes the numerical value of the base, and h that of the altitude,
| Volume of cylinder | = Bh, |
| Volume of cone | = 1⁄3Bh. |