Proposition: Prop. 10.080: Construction of First Apotome of Medial is Unique

(Proposition 80 from Book 10 of Euclid's “Elements”)

Only one medial straight line, which is commensurable in square only with the whole, and contains a rational (area) with the whole, can be attached to a first apotome of a medial (straight line).

For let $AB$ be a first apotome of a medial (straight line), and let $BC$ be (so) attached to $AB$.
Thus, $AC$ and $CB$ are medial (straight lines which are) commensurable in square only, containing a rational (area) - (namely, that contained) by $AC$ and $CB$ [Prop. 10.74].
I say that a(nother) medial (straight line), which is commensurable in square only with the whole, and contains a rational (area) with the whole, cannot be attached to $AB$.

fig079e

Modern Formulation

In other words, the first apotome of a medial straight line is unique, i.e. \[\alpha^{1/4} - \alpha^{3/4} = \beta^{1/4} - \beta^{3/4}\] has only one solution: i.e., \[\beta=\alpha,\] where $\alpha,\beta$ denote positive rational numbers.