| If a cubic number multiplied by any number makes a cubic number, then the multiplied number is also cubic. | ||
| Let the cubic number A multiplied by any number B make the cubic number C.
I say that B is cubic. | ||
| Multiply A by itself to make D. Then D is a cube. | IX.3 | |
| 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 and C are cubes, therefore they are similar solid numbers. Therefore two mean proportional numbers fall between D and C. And D is to C as A is to B, therefore two mean proportional numbers fall between A and B, too. | VIII.19
VIII.8 | |
| And A is a cube, therefore B is also a cube. | VIII.23 | |
| Therefore, if a cubic number multiplied by any number makes a cubic number, then the multiplied number is also cubic. | ||
| Q.E.D. | ||
