| If two numbers multiplied by one another make a square number, then they are similar plane numbers. | ||
| Let A and B be two numbers, and let A multiplied by B make the square number C.
I say that A and B are similar plane numbers. Multiply A by itself to make D. Then D is square. | ||
| Now, since A multiplied by itself makes D, and multiplied by B makes C, therefore A is to B as D is to C. | VII.17 | |
| And, since D is square, and C is so also, therefore D and C are similar plane numbers. | ||
| Therefore one mean proportional number falls between D and C. And D is to C as A is to B, therefore one mean proportional number falls between A and B also. | VIII.18
VIII.8 | |
| But, if one mean proportional number falls between two numbers, then they are similar plane numbers, therefore A and B are similar plane numbers. | VIII.20 | |
| Therefore, if two numbers multiplied by one another make a square number, then they are similar plane numbers. | ||
| Q.E.D. | ||
