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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.

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This proposition is used in several propositions in this and the next book starting with VIII.25.

Book VIII Introduction - Proposition VIII.22 - Proposition VIII.24.

© 1996
D.E.Joyce
Clark University