| If three numbers are in continued proportion, and the first is square, then the third is also square. | ||
| Let A, B, and C be three numbers in continued proportion, and let A the first be square.
I say that C the third is also square. | ||
| Since between A and C there is one mean proportional number, B, therefore A and C are similar plane numbers. But A is square, therefore C is also square. | VIII.20 | |
| Therefore, if three numbers are in continued proportion, and the first is square, then the third is also square. | ||
| Q.E.D. | ||
Book VIII Introduction - Proposition VIII.21 - Proposition VIII.23.