If two numbers have to one another the ratio which a cube number has to a cube number, and the first be cube, the second will also be cube.
For let the two numbers A
have to one another the ratio which the cube number C
has to the cube number D
, and let A
be cube; I say that B
is also cube.
For, since C
are cube, C
are similar solid numbers.
Therefore two mean proportional numbers fall between C
. [VIII. 19
And, as many numbers as fall between C
in continued proportion, so many will also fall between those which have the same ratio with them; [VIII. 8
] so that two mean proportional numbers fall between A
Since, then, the four numbers A
are in continued proportion, and A
is cube, therefore B
is also cube. [VIII. 23
] Q. E. D.