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) . fig086e

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.


See also [Prop. 10.49].

Proofs: 1

Propositions: 1

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Adapted from (subject to copyright, with kind permission)

  1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"

Adapted from CC BY-SA 3.0 Sources:

  1. Prime.mover and others: "Pr∞fWiki",, 2016