| Between two square numbers there is one mean proportional number, and the square has to the square the duplicate ratio of that which the side has to the side. | ||
| Let A and B be square numbers, and let C be the side of A, and D of B.
I say that between A and B there is one mean proportional number, and A has to B the ratio duplicate of that which C has to D. Multiply C by D to make E. Now, since A is a square and C is its side, therefore C multiplied by itself makes A. For the same reason also, D multiplied by itself makes B. | ||
| Since, then, C multiplied by the numbers C and D makes A and E respectively, therefore C is to D as A is to E. | VII.17 | |
| For the same reason also C is to D as E is to B. Therefore A is to E as E is to B. Therefore between A and B there is one mean proportional number. | VII.18 | |
| I say next that A also has to B the ratio duplicate of that which C has to D. | ||
| Since A, E, and B are three numbers in proportion, therefore A has to B the ratio duplicate of that which A has to E. | V.Def.9 | |
| But A is to E as C is to D, therefore A has to B the ratio duplicate of that which the side C has to D. | ||
| Therefore, between two square numbers there is one mean proportional number, and the square has to the square the duplicate ratio of that which the side has to the side. | ||
| Q.E.D. | ||
Book VIII Introduction - Proposition VIII.10 - Proposition VIII.12.