If a square measure a square, the side will also measure the side; and, if the side measure the side, the square will also measure the square.

Let A, B be square numbers, let C, D be their sides, and let A measure B; I say that C also measures D.

For let C by multiplying D make E; therefore A, E, B are continuously proportional in the ratio of C to D. [VIII. 11]

And, since A, E, B are continuously proportional, and A measures B, therefore A also measures E. [VIII. 7]

And, as A is to E, so is C to D; therefore also C measures D. [VII. Def. 20]

Again, let C measure D; I say that A also measures B.

For, with the same construction, we can in a similar manner prove that A, E, B are continuously proportional in the ratio of C to D.

And since, as C is to D, so is A to E, and C measures D, therefore A also measures E. [VII. Def. 20]

And A, E, B are continuously proportional; therefore A also measures B.

Therefore etc. Q. E. D.