(related to Proposition: Prop. 10.020: Quotient of Rational Numbers is Rational)

- For let the rational (area) $AC$ have been applied to the rational (straight line) $AB$, producing the (straight line) $BC$ as breadth.
- I say that $BC$ is rational, and commensurable in length with $BA$.

- For let the square $AD$ have been described on $AB$.
- $AD$ is thus rational [Def. 10.4] .
- And $AC$ (is) also rational.
- $DA$ is thus commensurable with $AC$.
- And as $DA$ is to $AC$, so $DB$ (is) to $BC$ [Prop. 6.1].
- Thus, $DB$ is also commensurable (in length) with $BC$ [Prop. 10.11].
- And $DB$ (is) equal to $BA$.
- Thus, $AB$ (is) also commensurable (in length) with $BC$.
- And $AB$ is rational.
- Thus, $BC$ is also rational, and commensurable in length with $AB$ [Def. 10.3] .
- Thus, if a rational (area) is applied to a rational (straight line), and so on ....∎

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