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 $AB$.

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.

This proposition corresponds to [Prop. 10.43], with minus signs instead of plus signs.

Proofs: 1

Propositions: 1

**Fitzpatrick, Richard**: Euclid's "Elements of Geometry"

**Prime.mover and others**: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016