| 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. |
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| 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. | ||
