Kitap 5
8
For, since AB is greater than C, let BE be made equal to C; then the less of the magnitudes AE, EB, if multiplied, will sometime be greater than D. [V. Def. 4]
8
[Case I.]
8
First, let AE be less than EB; let AE be multiplied, and let FG be a multiple of it which is greater than D; then, whatever multiple FG is of AE, let GH be made the same multiple of EB and K of C; and let L be taken double of D, M triple of it, and successive multiples increasing by one, until what is taken is a multiple of D and the first that is greater than K. Let it be taken, and let it be N which is quadruple of D and the first multiple of it that is greather than K.
8
Then, since K is less than N first, therefore K is not less than M.
8
And, since FG is the same multiple of AE that GH is of EB, therefore FG is the same multiple of AE that FH is of AB. [V. 1]
8
But FG is the same multiple of AE that K is of C; therefore FH is the same multiple of AB that K is of C; therefore FH, K are equimultiples of AB, C.
8
Again, since GH is the same multiple of EB that K is of C, and EB is equal to C, therefore GH is equal to K.
8
But K is not less than M; therefore neither is GH less than M.
8
And FG is greater than D; therefore the whole FH is greater than D, M together.
8
But D, M together are equal to N, inasmuch as M is triple of D, and M, D together are quadruple of D, while N is also quadruple of D; whence M, D together are equal to N.
8
But FH is greater than M, D; therefore FH is in excess of N, while K is not in excess of N.
