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Proposition 16

If two incommensurable magnitudes are added together, the sum is also incommensurable with each of them; but, if the sum is incommensurable with one of them, then the original magnitudes are also incommensurable.
Let the two incommensurable magnitudes AB and BC be added together.
I say that the whole AC is also incommensurable with each of the magnitudes AB and BC. For, if CA and AB are not incommensurable, then some magnitude D measures them. Since then D measures CA and AB, therefore it also measures the remainder BC. But it also measures AB, therefore D measures AB and BC. Therefore AB and BC are commensurable, but they were also, by hypothesis, incommensurable, which is impossible. |
Therefore no magnitude measures CA and AB. Therefore CA and AB are incommensurable. | X.Def.1
Similarly we can prove that AC and CB are also incommensurable. Therefore AC is incommensurable with each of the magnitudes AB and BC. Next, let AC be incommensurable with one of the magnitudes AB or BC. First, let it be incommensurable with AB. I say that AB and BC are also incommensurable. For, if they are commensurable, then some magnitude D measures them. Since, then, D measures AB and BC, therefore it also measures the whole AC. But it also measures AB, therefore D measures CA and AB. Therefore CA and AB are commensurable, but they were also, by hypothesis, incommensurable, which is impossible.
Therefore no magnitude measures AB and BC. Therefore AB and BC are incommensurable. | X.Def.1
Therefore, if two incommensurable magnitudes are added together, the sum is also incommensurable with each of them; but, if the sum is incommensurable with one of them, then the original magnitudes are also incommensurable.
Q.E.D.

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This proposition is a logical variant of the previous one, but it is proved afresh. It is used in several others in Book X starting with X.18.

Book X Introduction - Proposition X.15 - Proposition X.17.

© 1996
D.E.Joyce
Clark University