For let C, the side of A, be taken, and let C by multiplying itself make D.
It is then manifest that C by multiplying D has made A.
Now, since C by multiplying itself has made 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, as the unit is to C, so is C to D. [VII. Def. 20]
Again, since C by multiplying D has made A,
therefore D measures A according to the units in C.
But the unit also measures C according to the units in it;
therefore, as the unit is to C, so is D to A.
But, as the unit is to C, so is C to D;
therefore also, as the unit is to C, so is C to D, and D to A.
Therefore between the unit and the number A two mean proportional numbers C, D have fallen in continued proportion.
Again, 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]
But between the unit and A two mean proportional numbers have fallen;
therefore two mean proportional numbers will also fall between A, B. [VIII. 8]
But, if two mean proportional numbers fall between two numbers, and the first be cube, the second will also be cube. [VIII. 23]
And A is cube;
therefore B is also cube.
Q. E. D.
Complete text
Book I