| If a square does not measure a square, then neither does the side measure the side; and, if the side does not measure the side, then neither does the square measure the square. | ||
| Let A and B be square numbers, and let C and D be their sides, and let A not measure B.
I say that neither does C measure D. | ||
| If C measures D, A also measures B. But A does not measure B, therefore neither does C measure D. | VIII.14 | |
Next, let C not measure D. I say that neither does A measure B. | ||
| If A measures B, then C also measures D. But C does not measure D, therefore neither does A measure B. | VIII.14 | |
| Therefore, if a square does not measure a square, then neither does the side measure the side; and, if the side does not measure the side, then neither does the square measure the square. | ||
| Q.E.D. | ||
Book VIII Introduction - Proposition VIII.15 - Proposition VIII.17.