For two given unequal straight lines, to find by (the square on) which (straight line) the square on the greater (straight line is) larger than (the square on) the lesser. * Let $AB$ and $C$ be the two given unequal straight lines, and let $AB$ be the greater of them. * So it is required to find by (the square on) which (straight line) the square on $AB$ (is) greater than (the square on) $C$.
That is, if $\alpha$ and $\beta$ are the positive real numbers denoting the lengths of two given straight lines, with $\alpha > \beta$, to find a straight line of length $\gamma$ such that $\alpha^2=\beta^2+\gamma^2$. Similarly, we can also find $\gamma$ such that $\gamma^2=\alpha^2+\beta^2$.
Proofs: 1
