Proposition: Prop. 10.089: Construction of Fifth Apotome

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) .