(related to Proposition: Prop. 10.074: First Apotome of Medial is Irrational)

- For let the medial (straight line) $BC$, which is commensurable in square only with $AB$, and which makes with $AB$ the rational (rectangle contained) by $AB$ and $BC$, have been subtracted from the medial (straight line) $AB$ [Prop. 10.27].
- I say that the remainder $AC$ is an irrational (straight line).
- Let it be called the first apotome of a medial (straight line).

- For since $AB$ and $BC$ are medial (straight lines), the (sum of the squares) on $AB$ and $BC$ is also medial.
- And twice the (rectangle contained) by $AB$ and $BC$ (is) rational.
- The (sum of the squares) on $AB$ and $BC$ (is) thus incommensurable with twice the (rectangle contained) by $AB$ and $BC$.
- Thus, twice the (rectangle contained) by $AB$ and $BC$ is also incommensurable with the remaining (square) on $AC$ [Prop. 2.7], since if the whole is incommensurable with one of the (constituent magnitudes) then the original magnitudes will also be incommensurable (with one another) [Prop. 10.16].
- And twice the (rectangle contained) by $AB$ and $BC$ (is) rational.
- Thus, the (square) on $AC$ is irrational.
- Thus, $AC$ is an irrational (straight line) [Def. 10.4] .
- Let it be called a *first apotome":bookofproofs$2091 of a medial (straight line).∎

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