Commentary on Proposition XII.9

Back to Proposition XII.9.

The volumes of the pyramids are respectively one sixth part of the volumes of the parallelepipeds described, as in the figure, on double the bases and with the same heights as the pyramids.

  1. Thus the parallelepipeds are equal if the pyramids are equal. And, the parallelepipeds being equal, their bases are reciprocally proportional to their heights ; [XI. 34] hence the bases of the equal pyramids (which are the halves of the bases of the parallelepipeds) are proportional to their heights.

  2. If the bases of the pyramids are reciprocally proportional to their heights, so are the bases of the parallelepipeds to their heights (since the bases of the parallelepipeds are double of the bases of the pyramids respectively). Consequently the parallelepipeds are equal. [XI. 34] Therefore their sixth parts, the pyramids, are also equal.