(related to Proposition: Prop. 10.115: From Medial Straight Line arises Infinite Number of Irrational Straight Lines)

- Let $A$ be a medial (straight line).
- I say that an infinite (series) of irrational (straight lines) can be created from $A$, and that none of them is the same as any of the preceding (straight lines).

- Let the rational (straight line) $B$ be laid down.
- And let the (square) on $C$ be equal to the (rectangle contained) by $B$ and $A$.
- Thus, $C$ is irrational [Def. 10.4] .
- For an (area contained) by an irrational and a rational (straight line) is irrational [Prop. 10.20].
- And ($C$ is) not the same as any of the preceding (straight lines).
- For the (square) on none of the preceding (straight lines), applied to a rational (straight line), produces a medial (straight line) as breadth.
- So, again, let the (square) on $D$ be equal to the (rectangle contained) by $B$ and $C$.
- Thus, the (square) on $D$ is irrational [Prop. 10.20].
- $D$ is thus irrational [Def. 10.4] .
- And ($D$ is) not the same as any of the preceding (straight lines).
- For the (square) on none of the preceding (straight lines), applied to a rational (straight line), produces $C$ as breadth.
- So, similarly, this arrangement being advanced to infinity, it is clear that an infinite (series) of irrational (straight lines) can be created from a medial (straight line), and that none of them is the same as any of the preceding (straight lines).
- (Which is) the very thing it was required to show.∎

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