| Similar plane numbers have to one another the ratio which a square number has to a square number. | ||
| Let A and B be similar plane numbers.
I say that A has to B the ratio which a square number has to a square number. | ||
| Since A and B are similar plane numbers, therefore one mean proportional number C falls between A and B. | VIII.18 | |
| Take D, E, and F, the least numbers of those which have the same ratio with A, C, and B | VII.33 or VIII.2 | |
| Then the extremes of them D and F are square. And since D is to F as A is to B, and D and F are square, therefore A has to B the ratio which a square number has to a square number. | VIII.2,Cor | |
| Therefore, similar plane numbers have to one another the ratio which a square number has to a square number. | ||
| Q.E.D. | ||
Book VIII Introduction - Proposition VIII.25 - Proposition VIII.27.