The (square) on a rational (straight line), applied to a binomial (straight line), produces as breadth an apotome whose terms are commensurable (in length) with the terms of the binomial, and, furthermore, in the same ratio. Moreover, the created apotome will have the same order as the binomial. * Let $A$ be a rational (straight line), and $BC$ a binomial (straight line), of which let $DC$ be the greater term. * And let the (rectangle contained) by $BC$ and $EF$ be equal to the (square) on $A$. * I say that $EF$ is an apotome whose terms are commensurable (in length) with $CD$ and $DB$, and in the same ratio, and, moreover, that $EF$ will have the same order as $BC$.
(not yet contributed)
Heiberg considers this proposition, and the succeeding ones, to be relatively early interpolations into the original text (translator's note). ↩