Proposition: Prop. 10.014: Commensurability of Squares on Proportional Straight Lines

(Proposition 14 from Book 10 of Euclid's “Elements”)

If four straight lines are proportional, and the square on the first is greater than (the square on) the second by the (square) on (some straight line) commensurable [in length] with the first, then the square on the third will also be greater than (the square on) the fourth by the (square) on (some straight line) commensurable [in length] with the third. And if the square on the first is greater than (the square on) the second by the (square) on (some straight line) incommensurable [in length] with the first, then the square on the third will also be greater than (the square on) the fourth by the (square) on (some straight line) incommensurable [in length] with the third.

Let $A$, $B$, $C$, $D$ be four proportional straight lines, (such that) as $A$ (is) to $B$, so $C$ (is) to $D$.
And let the square on $A$ be greater than (the square on) $B$ by the (square) on $E$, and let the square on $C$ be greater than (the square on) $D$ by the (square) on $F$.
I say that $A$ is either commensurable (in length) with $E$, and $C$ is also commensurable with $F$, or $A$ is incommensurable (in length) with $E$, and $C$ is also incommensurable with $F$.