| If four magnitudes are proportional, and the first is commensurable with the second, then the third also is commensurable with the fourth; but, if the first is incommensurable with the second, then the third also is incommensurable with the fourth. | ||
| Let A, B, C, and D be four magnitudes in proportion, so that A is to B as C is to D, and let A be commensurable with B.
I say that C is also commensurable with D. | ||
| Since A is commensurable with B, therefore A has to B the ratio which a number has to a number. | X.5 | |
| And A is to B as C is to D, therefore C also has to D the ratio which a number has to a number. Therefore C is commensurable with D. | V.11
X.6 | |
| Next, let A be incommensurable with B.
I say that C is also incommensurable with D. | ||
| Since A is incommensurable with B, therefore A does not have to B the ratio which a number has to a number. | X.7 | |
| And A is to B as C is to D, therefore neither has C to D the ratio which a number has to a number. Therefore C is incommensurable with D. | V.11
X.8 | |
| Therefore, if four magnitudes are proportional, and the first is commensurable with the second, then the third also is commensurable with the fourth; but, if the first is incommensurable with the second, then the third also is incommensurable with the fourth. | ||
| Q.E.D. | ||
This proposition is used in repeatedly in Book X starting with X.14. It is also used in the previous proposition which was, no doubt, not in the original Elements.