| If two numbers have to one another the ratio which a square number has to a square number, and the first is square, then the second is also a square. | ||
| Let the two numbers A and B have to one another the ratio which the square number C has to the square number D, and let A be square.
I say that B is also square. | ||
| Since C and D are square, C and D are similar plane numbers. Therefore one mean proportional number falls between C and D. | VIII.18 | |
| And C is to D as A is to B, therefore one mean proportional number falls between A and B also. And A is square, therefore B is also square. | VIII.18
VIII.22 | |
| Therefore, if two numbers have to one another the ratio which a square number has to a square number, and the first is square, then the second is also a square. | ||
| Q.E.D. | ||
Book VIII Introduction - Proposition VIII.23 - Proposition VIII.25.