Similar plane numbers have to one another the ratio which a square number has to a square number.

Let A, B be similar plane numbers; I say that A has to B the ratio which a square number has to a square number.

For, since A, B are similar plane numbers, therefore one mean proportional number falls between A, B. [VIII. 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 or VIII. 2] therefore the extremes of them D, F are square. [VIII. 2, Por.]

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.