(related to Proposition: Prop. 10.076: Minor is Irrational)

- For let the straight line $BC$, which is incommensurable in square with the whole, and fulfils the (other) prescribed (conditions), have been subtracted from the straight line $AB$ [Prop. 10.33].
- I say that the remainder $AC$ is that irrational (straight line) called
**minor**.

- For since the sum of the squares on $AB$ and $BC$ is rational, and twice the (rectangle contained) by $AB$ and $BC$ (is) medial, the (sum of the squares) on $AB$ and $BC$ is thus incommensurable with twice the (rectangle contained) by $AB$ and $BC$.
- And, via convertion, the (sum of the squares) on $AB$ and $BC$ is incommensurable with the remaining (square) on $AC$ [Prop. 2.7], [Prop. 10.16].
- And the (sum of the squares) on $AB$ and $BC$ (is) rational.
- The (square) on $AC$ (is) thus irrational.
- Thus, $AC$ (is) an irrational (straight line) [Def. 10.4] .
- Let it be called a
**minor**(straight line). - (Which is) the very thing it was required to show.∎

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