| If a square measures a square, then the side also measures the side; and, if the side measures the side, then the square also measures the square. | ||
| Let A and B be square numbers, let C and D be their sides, and let A measure B.
I say that C also measures D. | ||
| Multiply C by D to make E. Then A, E, and B are continuously proportional in the ratio of C to D. | VIII.11 | |
| And, since A, E, and B are continuously proportional, and A measures B, therefore A also measures E. And A is to E as C is to D, therefore C measures D. | VIII.7
VII.Def.20 | |
Next, let C measure D. I say that A also measures B. | ||
| With the same construction, we can in a similar manner prove that A, E, and B are continuously proportional in the ratio of C to D. And since C is to D as A is to E, and C measures D, therefore A also measures E. | VII.Def.20 | |
| And A, E, and B are continuously proportional, therefore A also measures B. | ||
| Therefore, if a square measures a square, then the side also measures the side; and, if the side measures the side, then the square also measures the square. | ||
| Q.E.D. | ||
Book VIII Introduction - Proposition VIII.13 - Proposition VIII.15.