Proposition: Prop. 10.090: Construction of Sixth Apotome

(Proposition 90 from Book 10 of Euclid's “Elements”)

To find a sixth apotome. * Let the rational (straight line) $A$, and the three numbers $E$, $BC$, and $CD$, not having to one another the ratio which (some) square number (has) to (some) square number, be laid down. * Furthermore, let $CB$ also not have to $BD$ the ratio which (some) square number (has) to (some) square number. * And let it have been contrived that as $E$ (is) to $BC$, so the (square) on $A$ (is) to the (square) on $FG$, and as $BC$ (is) to $CD$, so the (square) on $FG$ (is) to the (square) on $GH$ [Prop. 10.6 corr.] . * Therefore, since as $E$ is to $BC$, so the (square) on $A$ (is) to the (square) on $FG$, the (square) on $A$ (is) thus commensurable with the (square) on on $FG$ [Prop. 10.6]. * And the (square) on $A$ (is) rational. * Thus, the (square) on $FG$ (is) also rational. * Thus, $FG$ is also a rational (straight line). * And since $E$ does not have to $BC$ the ratio which (some) square number (has) to (some) square number, the (square) on $A$ thus does not have to the (square) on $FG$ the ratio which (some) square number (has) to (some) square number either. * Thus, $A$ is incommensurable in length with $FG$ [Prop. 10.9]. * Again, since as $BC$ is to $CD$, so the (square) on $FG$ (is) to the (square) on $GH$, the (square) on $FG$ (is) thus commensurable with the (square) on on $GH$ [Prop. 10.6]. * And the (square) on $FG$ (is) rational. * Thus, the (square) on $GH$ (is) also rational. * Thus, $GH$ (is) also rational. * And since $BC$ does not have to $CD$ the ratio which (some) square number (has) to (some) square number, the (square) on $FG$ thus 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]. * And both are rational (straight lines). * Thus, $FG$ and $GH$ are rational (straight lines which are) commensurable in square only. * Thus, $FH$ is an apotome [Prop. 10.73]. * So, I say that (it is) also a sixth (apotome) .