Commentary on Proposition XII.15¶
Back to Proposition XII.15.
I. If the heights of the two cylinders are equal, and their volumes are equal, the bases are equal, since the latter are proportional to the volumes. [XII. 11] If the heights are not equal, cut off from the higher cylinder a cylinder of the same height as the lower. Then, if LK, QN be the equal heights, we have, by XII. 11, (base ABCD) : (base EFGH) = (cylinder AO) : (cylinder ES) = (cylinder EP) : (cylinder ES), by hypothesis, = MN : QN [XII. 13] = MN : KL.
II. In the converse part of the proposition, Euclid omits the case where the cylinders have equal heights. In this case of course the reciprocal ratios are both ratios of equality; the bases are therefore equal, and consequently the cylinders. If the heights are not equal, we have, with the same construction as before, (base ABCD) : (base EFGH) = MN : KL. But [XII. 11] (base ABCD) : (base EFGH) = (cylinder AO) : (cylinder ES), and MN : KL = MN : QN = (cylinder EP) : (cylinder ES). [XII. 13] Therefore (cylinder AO) : (cylinder ES) = (cylinder EP) : (cylinder ES), and consequently (cylinder AO) = (cylinder EP).
Similarly for the cones, which are equal to one-third of the cylinders respectively.
Legendre deduces these propositions about cones and cylinders from two others which he establishes by a method similar to that adopted by him for the theorem of XII. 2 (see note on that proposition). The first (for the cylinder) is as follows. The volume of a cylinder is equal to the product of its base by its height. Suppose CA to be the radius of the base of the given cylinder, $h$ its height. For brevity let us denote by (surf. CA) the area of the circle of which CA is the radius. If (surf. CA) $\times h$ is not the measure of the given cylinder, it will be the measure of a cylinder greater or less than it.
I. First let it be the measure of a less cylinder, that, for example, of which the circle with radius CD is the base, and $h$ is the height.
Circumscribe about the circle with radius CD a regular polygon GHI… such that its sides do not anywhere meet the circle with radius CA. [See note on XII. 2, p. 393 above, for Legendre’s lemma relating to this construction.] Imagine a prism erected on the polygon as base and with height $h$. Then (volume of prism) = (polygon GHI…) $\times h$. [Legendre has previously proved this proposition, first for a parallelepiped (by transforming it into a rectangular one), then for a triangular prism (half of a parallelepiped of the same height), and lastly for a prism with a polygonal base.] But (polygon GHI…) $<$ (surf. CA). Therefore (volume of prism) $<$ (surf. CA) $\times h$ $<$ (cylinder on circle of rad. CD), by hypothesis. But the prism is greater than the latter cylinder, since it includes it: which is impossible.
II. In order not to multiply figures let us, in this second case, suppose that CD is the radius of the base of the given cylinder, and that (surf. CD) $\times h$ is the measure of a cylinder greater than it, e.g. a cylinder on the circle with radius CA as base and of height $h$. Then, with the same construction, (volume of prism) = (polygon GHI…) $\times h$. And (polygon GHI…) $>$ (surf. CD). Therefore (volume of prism) $>$ (surf. CD) $\times h$ $>$ (cylinder on surf. CA), by hypothesis. But the volume of the prism is also less than that cylinder, being included by it: which is impossible.
Therefore (volume of cylinder) = (its base) $\times$ (its height).
It follows as a corollary that Cylinders of the same height are to one another as their bases [XII. 13], and cylinders on the same base are to one another as their heights [XII. 14].
Also Similar cylinders are as the cubes of their heights, or as the cubes of the diameters of their bases [Eucl. XII. 12].
For the bases are as the squares on their diameters; and, since the cylinders are similar, the diameters of the bases are as their heights. Therefore the bases are as the squares on the heights, and the bases multiplied by the heights, or the cylinders themselves, are as the cubes of the heights.
I need not reproduce Legendre’s proofs of the corresponding propositions for the cone.