VIII.23

If four numbers be in continued proportion, and the first be cube, the fourth will also be cube.

Proof

Let A, B, C, D be four numbers in continued proportion, and let A be cube; I say that D is also cube.

For, since between A, D there are two mean proportional numbers B, C, therefore A, D are similar solid numbers. VIII. 21

But A is cube; therefore D is also cube. Q. E. D.