(related to Proposition: Prop. 10.080: Construction of First Apotome of Medial is Unique)

- 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 $AB$.

- For, if possible, let $DB$ also be (so) attached to $AB$.
- Thus, $AD$ and $DB$ are medial (straight lines which are) commensurable in square only, containing a rational (area) - (namely, that) contained by $AD$ and $DB$ [Prop. 10.74].
- And since by whatever (area) the (sum of the squares) on $AD$ and $DB$ exceeds twice the (rectangle contained) by $AD$ and $DB$, the (sum of the squares) on $AC$ and $CB$ also exceeds twice the (rectangle contained) by $AC$ and $CB$ by this (same area).
- For [again] both exceed by the same (area) - (namely), the (square) on $AB$ [Prop. 2.7].
- Thus, alternately, by whatever (area) the (sum of the squares) on $AD$ and $DB$ exceeds the (sum of the squares) on $AC$ and $CB$, twice the (rectangle contained) by $AD$ and $DB$ also exceeds twice the (rectangle contained) by $AC$ and $CB$ by this (same area).
- And twice the (rectangle contained) by $AD$ and $DB$ exceeds twice the (rectangle contained) by $AC$ and $CB$ by a rational (area).
- For both (are) rational (areas).
- Thus, the (sum of the squares) on $AD$ and $DB$ also exceeds the (sum of the) [squares] on $AC$ and $CB$ by a rational (area).
- The very thing is impossible.
- For both are medial (areas) [Prop. 10.15], [Prop. 10.23 corr.] , and a medial (area) cannot exceed a(nother) medial (area) by a rational (area) [Prop. 10.26].
- Thus, 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).
- (Which is) the very thing it was required to show.∎

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