| If two commensurable magnitudes are added together, then the whole is also commensurable with each of them; and, if the whole is commensurable with one of them, then the original magnitudes are also commensurable. | ||
| Let the two commensurable magnitudes AB and BC be added together.
I say that the whole AC is also commensurable with each of the magnitudes AB and BC. Since AB and BC are commensurable, some magnitude D measures them. | ||
| Since then D measures AB and BC, therefore it also measures the whole AC. But it measures AB and BC also, therefore D measures AB, BC, and AC. Therefore AC is commensurable with each of the magnitudes AB and BC. | X.Def.1 | |
| Next, let AC be commensurable with AB.
I say that AB and BC are also commensurable. Since AC and AB are commensurable, some magnitude D measures them. Since then D measures CA and AB, therefore it also measures the remainder BC. | ||
| But it measures AB also, therefore D measures AB and BC. Therefore AB and BC are commensurable. | X.Def.1 | |
| Therefore, if two commensurable magnitudes are added together, then the whole is also commensurable with each of them; and, if the whole is commensurable with one of them, then the original magnitudes are also commensurable. | ||
| Q.E.D. | ||
