Kitap 10
15
For, since AB, BC are commensurable, some magnitude will measure them.
15
Let it measure them, and let it be D.
15
Since then D measures AB, BC, it will also measure the whole AC.
15
But it measures AB, BC also; therefore D measures AB, BC, AC; therefore AC is commensurable with each of the magnitudes AB, BC. [X. Def. 1]
15
Next, let AC be commensurable with AB; I say that AB, BC are also commensurable.
15
For, since AC, AB are commensurable, some magnitude will measure them.
15
Let it measure them, and let it be D.
15
Since then D measures CA, AB, it will also measure the remainder BC.
15
But it measures AB also; therefore D will measure AB, BC; therefore AB, BC are commensurable. [X. Def. 1]
15
Therefore etc.
PROPOSITION 16.
16
If two incommensurable magnitudes be added together, the whole will also be incommensurable with each of them; and, if the whole be incommensurable with one of them, the original magnitudes will also be incommensurable.
16
For let the two incommensurable magnitudes AB, BC be added together; I say that the whole AC is also incommensurable with each of the magnitudes AB, BC.
16
For, if CA, AB are not incommensurable, some magnitude will measure them.
16
Let it measure them, if possible, and let it be D.
16
Since then D measures CA, AB, therefore it will also measure the remainder BC.
16
But it measures AB also; therefore D measures AB, BC.
16
Therefore AB, BC are commensurable; but they were also, by hypothesis, incommensurable: which is impossible.
16
Therefore no magnitude will measure CA, AB; therefore CA, AB are incommensurable. [X. Def. 1]
16
Similarly we can prove that AC, CB are also incommensurable.
16
Therefore AC is incommensurable with each of the magnitudes AB, BC.
16
Next, let AC be incommensurable with one of the magnitudes AB, BC.
16
First, let it be incommensurable with AB; I say that AB, BC are also incommensurable.
16
For, if they are commensurable, some magnitude will measure them.
16
Let it measure them, and let it be D.
16
Since then D measures AB, BC. therefore it will also measure the whole AC.
