Proposition: Prop. 10.051: Construction of Fourth Binomial Straight Line
(Proposition 51 from Book 10 of Euclid's “Elements”)
To find a fourth binomial (straight line).
 Let the two numbers $AC$ and $CB$ be laid down such that $AB$ does not have to $BC$, or to $AC$ either, the ratio which (some) square number (has) to (some) square number [Prop. 10.28 lem. I] .
 And let the rational (straight line) $D$ be laid down.
 And let $EF$ be commensurable in length with $D$.
 Thus, $EF$ is also a rational (straight line).
 And let it have been contrived that as the number $BA$ (is) to $AC$, so the (square) on $EF$ (is) to the (square) on $FG$ [Prop. 10.6 corr.] .
 Thus, the (square) on $EF$ is commensurable with the (square) on on $FG$ [Prop. 10.6].
 Thus, $FG$ is also a rational (straight line).
 And since $BA$ does not have to $AC$ the ratio which (some) square number (has) to (some) square number, the (square) on $EF$ does not have to the (square) on $FG$ the ratio which (some) square number (has) to (some) square number either.
 Thus, $EF$ is incommensurable in length with $FG$ [Prop. 10.9].
 Thus, $EF$ and $FG$ are rational (straight lines which are) commensurable in square only.
 Hence, $EG$ is a binomial (straight line) [Prop. 10.36].
 So, I say that (it is) also a fourth (binomial straight line).
Modern Formulation
If the rational straight line has unit length then the length of a fourth binomial straight line is \[\alpha+\frac \alpha{\sqrt{1+\beta}},\]
where \(\alpha,\beta\) denote positive rational numbers.
Notes
This, and the fourth apotome, whose length according to [Prop. 10.88] is \[\alpha\frac \alpha{\sqrt{1+\beta}}\] are the roots of the quadratic function \[x^2 2\,\alpha\,x+\frac{\alpha^2\,\beta}{1+\beta}=0,\]
where \(\alpha,\beta\) denote positive rational numbers.
Table of Contents
Proofs: 1
Mentioned in:
Propositions: 1
Thank you to the contributors under CC BYSA 4.0!
 Github:

 nonGithub:
 @Fitzpatrick
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