Incommensurability of Sum of Incommensurable Magnitudes

Proof
Let $AB$ and $BC$ be incommensurable magnitudes which are added together to make $AC$.

Suppose $AC$ and $AB$ are not incommensurable.

Then some magnitude $D$ will measure them both.

Since $D$ measures both $AC$ and $AB$, $D$ also measures the remainder $BC$.

That is, $D$ measures $AB$ and $BC$.

Therefore from, $AB$ and $BC$ are commensurable.

But by hypothesis $AB$ and $BC$ are incommensurable.

Therefore no magnitude $D$ will measure both $AC$ and $AB$.

Therefore $AC$ and $AB$ are incommensurable.

In the same way it is shown that $AC$ and $BC$ are also incommensurable.

Now let $AC$ be incommensurable with one of either $AB$ and $BC$.

, let $AC$ be incommensurable $AB$.

Suppose $AB$ and $BC$ are not incommensurable.

Then some magnitude $D$ will measure them both.

Since $D$ measures both $AB$ and $BC$, $D$ also measures the whole $AC$.

But $D$ measures $AB$.

Therefore $D$ measures both $AC$ and $AB$.

Therefore $AC$ and $AB$ are commensurable.

But by hypothesis $AB$ and $AC$ are incommensurable.

Therefore no magnitude will measure both $AB$ and $BC$.

Therefore $AB$ and $BC$ are incommensurable.