(Only) one rational straight line, which is commensurable in square only with the whole, can be attached to an apotome. * Let $AB$ be an apotome, with $BC$ (so) attached to it. * $AC$ and $CB$ are thus rational (straight lines which are) commensurable in square only [Prop. 10.73]. * I say that another rational (straight line), which is commensurable in square only with the whole, cannot be attached to $AB$.
In other words,
\[\alpha - \sqrt{\beta} = \gamma - \sqrt{\delta}\] has only one solution: i.e., \[\gamma=\alpha\quad\text{ and }\quad\delta=\beta,\] where \(\alpha,\beta,\gamma,\delta\) denote positive rational numbers. Likewise, \[\sqrt{\alpha} - \sqrt{\beta} =\sqrt{\gamma}-\sqrt{\delta}\] has only one solution: i.e., \[\gamma=\alpha\quad\text{ and }\quad\delta=\beta.\]
This proposition corresponds to [Prop. 10.42], with minus signs instead of plus signs.
Proofs: 1