Only one medial straight line, which is commensurable in square only with the whole, and contains a medial (area) with the whole, can be attached to a second apotome of a medial (straight line). * Let $AB$ be a second apotome of a medial (straight line), with $BC$ (so) attached to $AB$. * Thus, $AC$ and $CB$ are medial (straight lines which are) commensurable in square only, containing a medial (area) - (namely, that contained) by $AC$ and $CB$ [Prop. 10.75]. * I say that a(nother) medial straight line, which is commensurable in square only with the whole, and contains a medial (area) with the whole, cannot be attached to $AB$.
In other words,
\[\alpha^{1/4}-\frac{\beta^{1/2}}{\alpha^{1/4}} = \gamma^{1/4}-\frac{\delta^{1/2}}{\gamma^{1/4}}\] has only one solution: i.e., \[\gamma=\alpha\quad\text{ and }\quad \delta=\beta,\] where \(\alpha,\beta,\gamma,\delta\) denote positive rational numbers.
This proposition corresponds to [Prop. 10.44], with minus signs instead of plus signs.
Proofs: 1
Propositions: 1
