For let A by multiplying B make C.

Since, then, A by multiplying itself has made B, and by multiplying B has made C, therefore C is cube.

And, since A by multiplying itself has made B, therefore A measures B according to the units in itself.

But the unit also measures A according to the units in it.

Therefore, as the unit is to A, so is A to B. [[VII. Def. 20](elem.7.def.20)]

And, since A by multiplying B has made C, therefore B measures C according to the units in A.

But the unit also measures A according to the units in it.

Therefore, as the unit is to A, so is B to C. [[VII. Def. 20](elem.7.def.20)]

But, as the unit is to A, so is A to B; therefore also, as A is to B, so is B to C.

And, since B, C are cube, they are similar solid numbers.

Therefore there are two mean proportional numbers between B, C. [[VIII. 19](elem.8.19)]

And, as B is to C, so is A to B.

Therefore there are two mean proportional numbers between A, B also. [[VIII. 8](elem.8.8)]

And B is cube; therefore A is also cube. [cf. [VIII. 23](elem.8.23)] Q. E. D.