(related to Proposition: Prop. 10.039: Major is Irrational)

- For let the two straight lines, $AB$ and $BC$, incommensurable in square, and fulfilling the prescribed (conditions), be laid down together [Prop. 10.33].
- I say that $AC$ is irrational.

- For since the (rectangle contained) by $AB$ and $BC$ is medial, twice the (rectangle contained) by $AB$ and $BC$ is [thus] also medial [Prop. 10.6], [Prop. 10.23 corr.] .
- And the sum of the (squares) on $AB$ and $BC$ (is) rational.
- Thus, twice the (rectangle contained) by $AB$ and $BC$ is incommensurable with the sum of the (squares) on $AB$ and $BC$ [Def. 10.4] .
- Hence, (the sum of) the squares on $AB$ and $BC$, plus twice the (rectangle contained) by $AB$ and $BC$ - that is, the (square) on $AC$ [Prop. 2.4] - is also incommensurable with the sum of the (squares) on $AB$ and $BC$ [Prop. 10.16] [and the sum of the (squares) on $AB$ and $BC$ (is) rational.
- Thus, the (square) on $AC$ is irrational.
- Hence, $AC$ is also irrational [Def. 10.4] - let it be called a
**major**(straight line). - (Which is) the very thing it was required to show.∎

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