Proposition 23

 If four numbers are in continued proportion, and the first is a cube, then the fourth is also a cube. Let A, B, C, and D be four numbers in continued proportion, and let A be a cube. I say that D is also a cube. Since between A and D there are two mean proportional numbers B and C, therefore A and D are similar solid numbers. But A is a cube, therefore D is also a cube. VIII.21 Therefore, if four numbers are in continued proportion, and the first is a cube, then the fourth is also a cube. Q.E.D.
This proposition is used in several propositions in this and the next book starting with VIII.25.

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