(related to Proposition: Prop. 10.019: Product of Rational Numbers is Rational)

- For let the rectangle $AC$ have been enclosed by the rational straight lines $AB$ and $BC$ (which are) commensurable in length.
- I say that $AC$ is rational.

- For let the square $AD$ have been described on $AB$.
- $AD$ is thus rational [Def. 10.4] .
- And since $AB$ is commensurable in length with $BC$, and $AB$ is equal to $BD$, $BD$ is thus commensurable in length with $BC$.
- And as $BD$ is to $BC$, so $DA$ (is) to $AC$ [Prop. 6.1].
- Thus, $DA$ is commensurable with $AC$ [Prop. 10.11].
- And $DA$ (is) rational.
- Thus, $AC$ is also rational [Def. 10.4] .
- Thus, the ... by rational straight lines ... commensurable, and so on ....∎

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