| If a cubic number multiplied by itself makes some number, then the product is a cube. | ||
| Let the cubic number A multiplied by itself make B.
I say that B is cubic. Take C, the side of A. Multiply C by itself make D. It is then manifest that C multiplied by D makes A. | ||
| Now, since C multiplied by itself makes D, therefore C measures D according to the units in itself. But further the unit also measures C according to the units in it, therefore the unit is to C as C is to D. | VII.Def.20 | |
| Again, since C multiplied by D makes A, therefore D measures A according to the units in C. But the unit also measures C according to the units in it, therefore the unit is to C as D is to A. But the unit is to C as C is to D, therefore the unit is to C as C is to D, and as D is to A.
Therefore between the unit and the number A two mean proportional numbers C and D have fallen in continued proportion. | ||
| Again, since A multiplied by itself makes B, therefore A measures B according to the units in itself. But the unit also measures A according to the units in it, therefore the unit is to A as A is to B. | VII.Def.20 | |
| But between the unit and A two mean proportional numbers have fallen, therefore two mean proportional numbers also fall between A and B. | VIII.8 | |
| But, if two mean proportional numbers fall between two numbers, and the first is a cube, then the second is also a cube. And A is a cube, therefore B is also a cube. | VIII.23 | |
| Therefore, if a cubic number multiplied by itself makes some number, then the product is a cube. | ||
| Q.E.D. | ||
