| If a cubic number does not measure a cubic number, then neither does the side measure the side; and, if the side does not measure the side, then neither does the cube measure the cube. | ||
| Let the cubic number A not measure the cubic number B, and let C be the side of A, and D of B.
I say that C does not measure D. | ||
| For if C measures D, then A also measures B. But A does not measure B, therefore neither does C measure D. | VIII.15 | |
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.15 | |
| Therefore, if a cubic number does not measure a cubic number, then neither does the side measure the side; and, if the side does not measure the side, then neither does the cube measure the cube. | ||
| Q.E.D. | ||
Book VIII Introduction - Proposition VIII.16 - Proposition VIII.18.