Let the two numbers A and B have to one another the ratio which the cubic number C has to the cubic number D, and let A be a cube.
I say that B is also a cube.
Since C and D are cubes, C and D are similar solid numbers, therefore two mean proportional numbers fall between C and D.
Since as many numbers fall in continued proportion between those which have the same ratio with C and D as fall between C and D, therefore two mean proportional numbers E and F fall between A and B.
Since, then, the four numbers A, E, F, and B are in continued proportion, and A is a cube, therefore B is also a cube.
Therefore, if two numbers have to one another the ratio which a cubic number has to a cubic number, and the first is a cube, then the second is also a cube.
This proposition is used in IX.10.