For, since A, B are similar plane numbers, therefore one mean proportional number falls between A, B. [[VIII. 18](elem.8.18)]

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

And since, as D is to F, so is A to B, and D, F are square, therefore A has to B the ratio which a square number has to a square number. Q. E. D.