If two incommensurable magnitudes are added together then the whole will also be incommensurable with each of them. And if the whole is incommensurable with one of them then the original magnitudes will also be incommensurable (with one another). * For let the two incommensurable magnitudes $AB$ and $BC$ be laid down together. * I say that the whole $AC$ is also incommensurable with each of $AB$ and $BC$. * And so let $AC$ be incommensurable with one of $AB$ and $BC$. * I say that $AB$ and $BC$ are also incommensurable.
(not yet contributed)