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Proposition 13

If two magnitudes are commensurable, and one of them is incommensurable with any magnitude, then the remaining one is also incommensurable with the same.
Let A and B be two commensurable magnitudes, and let one of them, A, be incommensurable with some other magnitude C. I say that the remaining one, B, is also incommensurable with C. |
If B is commensurable with C, while A is also commensurable with B, then A is also commensurable with C. | X.12
But it is also incommensurable with it, which is impossible. Therefore B is not commensurable with C. Therefore it is incommensurable with it.
Therefore, if two magnitudes are commensurable, and one of them is incommensurable with any magnitude, then the remaining one is also incommensurable with the same.
Q.E.D.

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The proposition is a logical variant of the previous. It is used in very frequently in Book X starting with X.18.

Book X Introduction - Proposition X.12 - Proposition X.14.

© 1996
D.E.Joyce
Clark University