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Proposition 7

Incommensurable magnitudes do not have to one another the ratio which a number has to a number.
Let A and B be incommensurable magnitudes. I say that A does not have to B the ratio which a number has to a number.
| If A does have to B the ratio which a number has to a number, then A is commensurable with B. | X.6
But it is not, therefore A does not have to B the ratio which a number has to a number.
Therefore, incommensurable magnitudes do not have to one another the ratio which a number has to a number.
Q.E.D.

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This proposition is the contrapositive of the last one. It is used in X.11.

Book X Introduction - Proposition X.6 - Proposition X.8.

© 1996
D.E.Joyce
Clark University