Proposition: Prop. 10.053: Construction of Sixth Binomial Straight Line
(Proposition 53 from Book 10 of Euclid's “Elements”)
To find a sixth binomial (straight line).
 Let the two numbers $AC$ and $CB$ be laid down such that $AB$ does not have to each of them the ratio which (some) square number (has) to (some) square number.
 And let $D$ also be another number, which is not square, and does not have to each of $BA$ and $AC$ the ratio which (some) square number (has) to (some) square number either [Prop. 10.28 lem. I] .
 And let some rational straight line $E$ be laid down.
 And let it have been contrived that as $D$ (is) to $AB$, so the (square) on $E$ (is) to the (square) on $FG$ [Prop. 10.6 corr.] .
 Thus, the (square) on $E$ (is) commensurable with the (square) on on $FG$ [Prop. 10.6].
 And $E$ is rational.
 Thus, $FG$ (is) also rational.
 And since $D$ does not have to $AB$ the ratio which (some) square number (has) to (some) square number, the (square) on $E$ thus does not have to the (square) on $FG$ the ratio which (some) square number (has) to (some) square number either.
 Thus, $E$ (is) incommensurable in length with $FG$ [Prop. 10.9].
 So, again, let it have be contrived that as $BA$ (is) to $AC$, so the (square) on $FG$ (is) to the (square) on $GH$ [Prop. 10.6 corr.] .
 The (square) on $FG$ (is) thus commensurable with the (square) on on $HG$ [Prop. 10.6].
 The (square) on $HG$ (is) thus rational.
 Thus, $HG$ (is) rational.
 And since $BA$ does not have to $AC$ the ratio which (some) square number (has) to (some) square number, the (square) on $FG$ does not have to the (square) on $GH$ the ratio which (some) square number (has) to (some) square number either.
 Thus, $FG$ is incommensurable in length with $GH$ [Prop. 10.9].
 Thus, $FG$ and $GH$ are rational (straight lines which are) commensurable in square only.
 Thus, $FH$ is a binomial (straight line) [Prop. 10.36].
 So, we must show that (it is) also a sixth (binomial straight line).
Modern Formulation
If the rational straight line has unit length then the length of a sixth binomial straight line is \[\sqrt{\alpha}+\sqrt{\beta},\]
where \(\alpha,\beta\) denote positive rational numbers.
Notes
This, and the sixth apotome, whose length according to [Prop. 10.90] is \[\sqrt{\alpha}\sqrt{\beta},\] are the roots of the quadratic function \[x^2 2\,\sqrt{\alpha}\,x+(\alpha\beta)=0,\]
where \(\alpha,\beta\) denote positive rational numbers.
Table of Contents
Proofs: 1
 Lemma: Lem. 10.053: Construction of Rectangle with Area in Mean Proportion to two Square Areas
Mentioned in:
Propositions: 1
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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