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Proposition 26

Similar plane numbers have to one another the ratio which a square number has to a square number.
Let A and B be similar plane numbers. I say that A has to B the ratio which a square number has to a square number.
| Since A and B are similar plane numbers, therefore one mean proportional number C falls between A and B. | VIII.18
Take D, E, and F, the least numbers of those which have the same ratio with A, C, and B | VII.33 or VIII.2
Then the extremes of them D and F are square. And since D is to F as A is to B, and D and F are square, therefore A has to B the ratio which a square number has to a square number. | VIII.2,Cor
Therefore, similar plane numbers have to one another the ratio which a square number has to a square number.
Q.E.D.

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This proposition is used in propositions IX.10 and X.9.

Book VIII Introduction - Proposition VIII.25 - Proposition VIII.27.

© 1996
D.E.Joyce
Clark University