(related to Proposition: Prop. 10.040: Side of Rational plus Medial Area is Irrational)

- For let the two straight lines, $AB$ and $BC$, incommensurable in square, (and) fulfilling the prescribed (conditions), be laid down together [Prop. 10.34].
- I say that $AC$ is irrational.

- For since the sum of the (squares) on $AB$ and $BC$ is 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 $BC$.
- Hence, the (square) on $AC$ is also incommensurable with twice the (rectangle contained) by $AB$ and $BC$ [Prop. 10.16].
- And twice the (rectangle contained) by $AB$ and $BC$ (is) rational.
- The (square) on $AC$ (is) thus irrational.
- Thus, $AC$ (is) irrational [Def. 10.4] - let it be called the square root of a rational plus a medial (area) .
- (Which is) the very thing it was required to show.∎

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