For, since A, B are similar solid numbers, therefore two mean proportional numbers fall between A, B. [[VIII. 19](elem.8.19)]

Let C, D so fall, and let E, F, G, H, the least numbers of those which have the same ratio with A, C, D, B, and equal with them in multitude, be taken; [[VII. 33](elem.7.33) or [VIII. 2](elem.8.2)] therefore the extremes of them E, H are cube. [[VIII. 2, Por.](elem.8.2.p.1)]

And, as E is to H, so is A to B; therefore A also has to B the ratio which a cube number has to a cube number. Q. E. D.