For two given unequal straight-lines, to cut off from the greater a straight-line equal to the lesser.  Let $AB$ and $C$ be the two given unequal straight-lines, of which let the greater be $AB$. So it is required to cut off a straight-line equal to the lesser $C$ from the greater $AB$.  Let the line $AD$, equal to the straight-line $C$, have been placed at  point $A$ [Prop.~1.2]. And let the circle $DEF$ have been drawn with center $A$ and radius $AD$ [Post.~3].  And since  point $A$ is the center of  circle $DEF$, $AE$ is equal to $AD$ [Def.~1.15]. But, $C$ is also equal to $AD$. Thus, $AE$ and $C$ are each equal to $AD$. So $AE$ is also equal to $C$ [C.N.~1]. Thus, for two given unequal straight-lines, $AB$ and $C$, the (straight-line) $AE$, equal to the lesser $C$, has been cut off from the greater $AB$. (Which is) the very thing it was required to do.