I.3

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

Proof

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. 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. C.N. 1

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

(Being) what it was required to do.