Proposition: Prop. 10.085: Construction of First Apotome

To find a first apotome.

Let the rational (straight line) $A$ be laid down.
And let $BG$ be commensurable in length with $A$.
$BG$ is thus also a rational (straight line).
And let two square numbers $DE$ and $EF$ be laid down, and let their difference $FD$ be not square [Prop. 10.28 lem. I] .
Thus, $ED$ does not have to $DF$ the ratio which (some) square number (has) to (some) square number.
And let it have been contrived that as $ED$ (is) to $DF$, so the square on $BG$ (is) to the square on $GC$ [Prop. 10.6 corr.] .
Thus, the (square) on $BG$ is commensurable with the (square) on on $GC$ [Prop. 10.6].
And the (square) on $BG$ (is) rational.
Thus, the (square) on $GC$ (is) also rational.
Thus, $GC$ is also rational.
And since $ED$ does not have to $DF$ 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).
Thus, $BG$ and $GC$ are 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 first (apotome).