If a square number do not measure a square number, neither will the side measure the side; and, if the side do not measure the side, neither will the square measure the square.

Let A, B be square numbers, and let C, D be their sides; and let A not measure B; I say that neither does C measure D.

For, if C measures D, A will also measure B. [VIII. 14]

But A does not measure B; therefore neither will C measure D.

Again, let C not measure D; I say that neither will A measure B.

For, if A measures B, C will also measure D. [VIII. 14]

But C does not measure D; therefore neither will A measure B. Q. E. D.