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]
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]
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]
And, as B is to C, so is A to B.
Therefore there are two mean proportional numbers between A, B also. [VIII. 8]
And B is cube;
therefore A is also cube. [cf. VIII. 23]
Q. E. D.
Complete text
Book I