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