(related to Proposition: Prop. 10.018: Condition for Incommensurability of Roots of Quadratic Equation)

- Let $A$ and $BC$ be two unequal straight lines, of which (let) $BC$ (be) the greater.
- And let a (rectangle) equal to the fourth [part] of the (square) on the lesser, $A$, falling short by a square figure, have been applied to $BC$.
- And let it be the (rectangle contained) by $BDC$.
- And let $BD$ be incommensurable in length with $DC$.
- I say that the square on $BC$ is greater than the (square on) $A$ by the (square) on (some straight line) incommensurable (in length) with ($BC$).

- For, similarly, by the same construction as before, we can show that the square on $BC$ is greater than the (square on) $A$ by the (square) on $FD$.
- Therefore, it must be shown that $BC$ is incommensurable in length with $DF$.
- For since $BD$ is incommensurable in length with $DC$, $BC$ is thus also incommensurable in length with $CD$ [Prop. 10.16].
- But, $DC$ is commensurable (in length) with the sum of $BF$ and $DC$ [Prop. 10.6].
- And, thus, $BC$ is incommensurable (in length) with the sum of $BF$ and $DC$ [Prop. 10.13].
- Hence, $BC$ is also incommensurable in length with the remainder $FD$ [Prop. 10.16].
- And the square on $BC$ is greater than the (square on) $A$ by the (square) on $FD$.
- Thus, the square on $BC$ is greater than the (square on) $A$ by the (square) on (some straight line) incommensurable (in length) with ($BC$).
- So, again, let the square on $BC$ be greater than the (square on) $A$ by the (square) on (some straight line) incommensurable (in length) with ($BC$).
- And let a (rectangle) equal to the fourth [part] of the (square) on $A$, falling short by a square figure, have been applied to $BC$.
- And let it be the (rectangle contained) by $BD$ and $DC$.
- It must be shown that $BD$ is incommensurable in length with $DC$.
- For, similarly, by the same construction, we can show that the square on $BC$ is greater than the (square) on $A$ by the (square) on $FD$.
- But, the square on $BC$ is greater than the (square) on $A$ by the (square) on (some straight line) incommensurable (in length) with ($BC$).
- Thus, $BC$ is incommensurable in length with $FD$.
- Hence, $BC$ is also incommensurable (in length) with the remaining sum of $BF$ and $DC$ [Prop. 10.16].
- But, the sum of $BF$ and $DC$ is commensurable in length with $DC$ [Prop. 10.6].
- Thus, $BC$ is also incommensurable in length with $DC$ [Prop. 10.13].
- Hence, via separation, $BD$ is also incommensurable in length with $DC$ [Prop. 10.16].
- Thus, if there are two ... straight lines, and so on ....∎

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