| If a number multiplied by itself makes a cubic number, then it itself is also cubic. | ||
| Let the number A multiplied by itself make the cubic number B.
I say that A is also cubic. | ||
| Multiply A by B to make C. Since, then, A multiplied by itself makes B, and multiplied by B makes C, therefore C is a cube. And, 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. And, since A multiplied by B makes C, therefore B measures C according to the units in A. | VII.Def.20 | |
| But the unit also measures A according to the units in it. | ||
| Therefore the unit is to A as B is to C. But the unit is to A as A is to B, therefore A is to B as B is to C. | VII.Def.20 | |
| And, since B and C are cubes, therefore they are similar solid numbers. | ||
| Therefore there are two mean proportional numbers between B and C. And B is to C as A is to B. Therefore there are two mean proportional numbers between A and B also. | VIII.19
VIII.8 | |
| And B is a cube, therefore A is also a cube. | cf. VIII.23 | |
| Therefore, if a number multiplied by itself makes a cubic number, then it itself is also cubic. | ||
| Q.E.D. | ||
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