| If a cubic number multiplied by a cubic number makes some number, then the product is a cube. | ||
| Let the cubic number A multiplied by the cubic number B make C.
I say that C is cubic. | ||
| Multiply A by itself to make D. Then D is a cube. | IX.3 | |
| Since A multiplied by itself C makes D, and multiplied by B makes C, therefore A is to B as D is to C. And, since A and B are cubic numbers, therefore A and B are similar solid numbers. Therefore two mean proportional numbers fall between A and B, so that two mean proportional numbers fall between D and C also. | VII.17
VIII.19 VIII.8 | |
| And D is a cube, therefore C is also a cube. | VIII.23 | |
| Therefore, if a cubic number multiplied by a cubic number makes some number, then the product is cubic. | ||
| Q.E.D. | ||
