Given two unequal straight lines, to cut off from the greater a straight line equal to the less.

Let $AB$, $C$ be the two given unequal straight lines, and let $AB$ be the greater of them.Thus it is required to cut off from $AB$ the greater a straight line equal to $C$ the less.At the point $A$ let $AD$ be placed equal to the straight line $C$; [Prop. 1.2]and with centre $A$ and distance $AD$ let the circle $DEF$ be described. [Post. 3]Now, since the point $A$ is the centre of the circle $DEF$,$AE$ is equal to $AD$. [Def. 15]But $C$ is also equal to $AD$.Therefore each of the straight lines $AE$, $C$ is equal to $AD$; so that $AE$ is also equal to $C$.Therefore, given the two straight lines $AB$,$C$ from $AB$ the greater $AE$ has been cut of equal to $C$ the less.(Being) what it was required to do.