I.3

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; [I.2] and with centre A and distance AD let the circle DEF be described. [I.post.3] Now, since the point A is the centre of the circle DEF, AE is equal to AD. [I.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. [I.cn.1]

Therefore, given the two straight lines AB, C, from AB the greater AE has been cut off equal to C the less.

Q. E. D.

(Being) what it was required to do.