Proposition: Prop. 10.086: Construction of Second Apotome
(Proposition 86 from Book 10 of Euclid's “Elements”)
To find a second apotome.
 Let the rational (straight line) $A$, and $GC$ (which is) commensurable in length with $A$, be laid down.
 Thus, $GC$ is a rational (straight line).
 And let the two square numbers $DE$ and $EF$ be laid down, and let their difference $DF$ be not square [Prop. 10.28 lem. I] .
 And let it have been contrived that as $FD$ (is) to $DE$, so the square on $CG$ (is) to the square on $GB$ [Prop. 10.6 corr.] .
 Thus, the square on $CG$ is commensurable with the square on $GB$ [Prop. 10.6].
 And the (square) on $CG$ (is) rational.
 Thus, the (square) on $GB$ [is] also rational.
 Thus, $BG$ is a rational (straight line).
 And since the square on $GC$ does not have to the (square) on $GB$ the ratio which (some) square number (has) to (some) square number, $CG$ is incommensurable in length with $GB$ [Prop. 10.9].
 And they are both rational (straight lines).
 Thus, $CG$ and $GB$ 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 second (apotome) .
Modern Formulation
This proposition proves that the second apotome has length
\[\frac{\alpha}{\sqrt{1\beta^{\,2}}}\alpha,\]
where \(\alpha,\beta\) denote positive rational numbers.
Notes
See also [Prop. 10.49].
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