## Proposition 25

 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. 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. VIII.19 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. VIII.18 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. VIII.23 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. Q.E.D.
This proposition is analogous to the previous one about squares. Its proof is straightforward.

This proposition is used in IX.10.

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