Now it has been proved that C, the fourth from the unit, is cube, as also are all those which leave out two; [IX. 8]
I say that all the rest are also cube.
For, since, as the unit is to A, so is A to B,
therefore the unit measures A the same number of times as A measures B.
But the unit measures A according to the units in it;
therefore A also measures B according to the units in itself;
therefore A by multiplying itself has made B.
And A is cube.
But, if a cube number by multiplying itself make some number, the product is cube. [IX. 3]
Therefore B is also cube.
And, since the four numbers A, B, C, D are in continued proportion, and A is cube,
D also is cube. [VIII. 23]
For the same reason E is also cube, and similarly all the rest are cube.
Q. E. D.