Proposition: Prop. 10.085: Construction of First Apotome
(Proposition 85 from Book 10 of Euclid's “Elements”)
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).
Modern Formulation
This proposition proves that the first apotome has length
\[\alpha\alpha\,\sqrt{1\beta^{\,2}},\]
where \(\alpha,\beta\) denote positive rational numbers.
Notes
See also [Prop. 10.48].
Table of Contents
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References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"
Adapted from CC BYSA 3.0 Sources:
 Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016