| If two similar plane numbers multiplied by one another make some number, then the product is square. | ||
| Let A and B be two similar plane numbers, and let A multiplied by B make C.
I say that C is square. Multiply A by itself to make D. Then D is square. | ||
| Since then 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 A and B are similar plane numbers, therefore one mean proportional number falls between A and B. | VIII.18 | |
| Since as many number fall in continued proportion between those which have the same ration, therefore one mean proportional number falls between D and C also. | VIII.8 | |
| And D is square, therefore C is also square. | VIII.22 | |
| Therefore, if two similar plane numbers multiplied by one another make some number, then the product is square. | ||
| Q.E.D. | ||
