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Proposition 8

If two magnitudes do not have to one another the ratio which a number has to a number, then the magnitudes are incommensurable.
Let the two magnitudes A and B not have to one another the ratio which a number has to a number. I say that the magnitudes A and B are incommensurable.
| For, if they are commensurable, then A has to B the ratio which a number has to a number. | X.5
But it does not, therefore the magnitudes A and B are incommensurable.
Therefore, if two magnitudes do not have to one another the ratio which a number has to a number, then the magnitudes are incommensurable.
Q.E.D.

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This proposition is the contrapositive of X.5. It is used in frequently in X.11.

Book X Introduction - Proposition X.7 - Proposition X.9.

© 1996
D.E.Joyce
Clark University