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Proposition 15

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.

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This fundamental proposition on commensurability of sums and differences is used in very frequently in Book X starting with X.17. It is also used in XIII.11.

Book X Introduction - Proposition X.14 - Proposition X.16.

© 1996
D.E.Joyce
Clark University