Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/173

Rh If the magnitude which is not the greater of the two AC, CB, be not less than D, take EF, FG the doubles of AC, CB (Figure 1).

But if that which is not the greater of the two AC, CB, be less than D (Figures 2 and 3), this magnitude can be multiplied, so as to become greater than D, whether it be AC or CB. Let it be multiplied until it becomes greater than D, and let the other be multiplied as often. Let EF be the multiple thus taken of AC, and FG the same multiple of CB; therefore EF and FG are each of them greater than D.

And in all the cases, take H the double of D, K its, triple, and so on, until the multiple of D taken is the first which is greater than FG. Let L be that multiple of D, namely, the first which is greater than FG; and let K be the multiple of D which is next less than L. Then, because L is the first multiple of D which is greater than FG, [Construction. the next preceding multiple K is not greater than FG; that is, FG is not less than K. And because EF is the same multiple of AC that FG is of CB, [Construction. therefore EG is the same multiple of AB that FG is of CB; [V.l. that is, EG and FG are equimultiples of AB and CB.