(related to Proposition: Prop. 10.109: Two Irrational Straight Lines arising from Medial Area from which Rational Area Subtracted)

- For let the rational (area) $BD$ have been subtracted from the medial (area) $BC$.
- I say that one of two irrational (straight lines) arise (as) the square root of the remaining (area), $EC$ - either a first apotome of a medial (straight line), or that (straight line) which with a rational (area) makes a medial whole.

- For let the rational (straight line) $FG$ be laid down, and let similar areas (to the preceding proposition) have been applied (to it).
- So, accordingly, $FH$ is rational, and incommensurable in length with $FG$, and $KF$ (is) also rational, and commensurable in length with $FG$.
- Thus, $FH$ and $FK$ are rational (straight lines which are) commensurable in square only [Prop. 10.13].
- $KH$ is thus an apotome [Prop. 10.73], and $FK$ an attachment to it.
- So, the square on $HF$ is greater than (the square on) $FK$ either by the (square) on (some straight line) commensurable (in length) with ($HF$), or by the (square) on (some straight line) incommensurable (in length with $HF$).
- Therefore, if the square on $HF$ is greater than (the square on) $FK$ by the (square) on (some straight line) commensurable (in length) with ($HF$), and (since) the attachment $FK$ is commensurable in length with the (previously) laid down rational (straight line) $FG$, $KH$ is a second apotome [Def. 10.12] .
- And $FG$ (is) rational.
- Hence, the square root of $LH$ - that is to say, (of) $EC$ - is a first apotome of a medial (straight line) [Prop. 10.92].
- And if the square on $HF$ is greater than (the square on) $FK$ by the (square) on (some straight line) incommensurable (in length with $HF$), and (since) the attachment $FK$ is commensurable in length with the (previously) laid down rational (straight line) $FG$, $KH$ is a fifth apotome [Def. 10.15] .
- Hence, the square root of $EC$ is that (straight line) which with a rational (area) makes a medial whole [Prop. 10.95].
- (Which is) the very thing it was required to show.∎

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