If two numbers have to one another the ratio which a square number has to a square number, and the first be square, the second will also be square.

For let the two numbers A, 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.

For, since C, D are square, C, D are similar plane numbers.

Therefore one mean proportional number falls between C, D. [VIII. 18]

And, as C is to D, so is A to B; therefore one mean proportional number falls between A, B also. [VIII. 8]

And A is square; therefore B is also square. [VIII. 22] Q. E. D.