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Proposition 22

If three numbers are in continued proportion, and the first is square, then the third is also square.
Let A, B, and C be three numbers in continued proportion, and let A the first be square. I say that C the third is also square.
| Since between A and C there is one mean proportional number, B, therefore A and C are similar plane numbers. But A is square, therefore C is also square. | VIII.20
Therefore, if three numbers are in continued proportion, and the first is square, then the third is also square.
Q.E.D.

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This proposition is used in a few propositions in this and the next book starting with VIII.24.

Book VIII Introduction - Proposition VIII.21 - Proposition VIII.23.

© 1996
D.E.Joyce
Clark University