BOOK XII.
LEMMA.
If from the greater of two unequal magnitudes there be taken more than its half, and from, the remainder more than its half and so on, there shall at length remain a magnitude less than the smaller of the proposed magnitudes.
Let AB and C be two unequal magnitudes, of which AB is, the greater: if from AB there be taken more than its half, and from the remainder more than its half, and so on, there shall at length remain a magnitude less than C.
For C may be multiplied so as at length to become greater than AB.
Let it be so multiplied, and let DE its multiple be greater than AB, and let DE be divided into DF, FG, GE, each equal to C.
From AB take BH, greater than its half, and from the remainder AH take HK greater than its half, and so on, until there be as many divisions in AB as in DE; and let the divisions in AB be AK, KH, HB, and the divisions in DE be DF, FG, GE.
Then, because DE is greater than AB;
and that EG taken from DE is not greater than its half;
but BH taken from AB is, greater than its half;
therefore the remainder DG is greater than the remainder AH.
