Proposition: Prop. 10.081: Construction of Second Apotome of Medial is Unique

(Proposition 81 from Book 10 of Euclid's “Elements”)

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 $AB$.

fig081e

Modern Formulation

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.