To find a fifth apotome.

- Let the rational (straight line) $A$ be laid down, and let $CG$ be commensurable in length with $A$.
- Thus, $CG$ [is] a rational (straight line).
- And let the two numbers $DF$ and $FE$ be laid down such that $DE$ again does not have to each of $DF$ and $FE$ the ratio which (some) square number (has) to (some) square number.
- And let it have been contrived that as $FE$ (is) to $ED$, so the (square) on $CG$ (is) to the (square) on $GB$.
- Thus, the (square) on $GB$ (is) also rational [Prop. 10.6].
- Thus, $BG$ is also rational.
- And since as $DE$ is to $EF$, so the (square) on $BG$ (is) to the (square) on $GC$.
- And $DE$ does not have to $EF$ the ratio which (some) square number (has) to (some) square number.
- The (square) on $BG$ thus does not have to the (square) on $GC$ the ratio which (some) square number (has) to (some) square number either.
- Thus, $BG$ is incommensurable in length with $GC$ [Prop. 10.9].
- And they are both rational (straight lines).
- $BG$ and $GC$ are thus rational (straight lines which are) commensurable in square only.
- Thus, $BC$ is an apotome [Prop. 10.73].
- So, I say that (it is) also a fifth (apotome) .

This proposition proves that the fifth apotome has length \[\alpha\,(\sqrt{1+\beta}-1),\]

where \(\alpha,\beta\) denote positive rational numbers.

See also [Prop. 10.52].

Proofs: 1

Propositions: 1

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

**Prime.mover and others**: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016