Proof: By Euclid

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.

fig019e

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 ....
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