Kitap 5
8
And in the same manner, by following the above argument, we complete the demonstration.
8
Therefore etc. Q. E. D.
PROPOSITION 9.
9
Magnitudes which have the same ratio to the same are equal to one another; and magnitudes to which the same has the same ratio are equal.
9
For let each of the magnitudes A, B have the same ratio to C; I say that A is equal to B.
9
For, otherwise, each of the magnitudes A, B would not have had the same ratio to C; [V. 8] but it has; therefore A is equal to B.
9
Again, let C have the same ratio to each of the magnitudes A, B; I say that A is equal to B.
9
For, otherwise, C would not have had the same ratio to each of the magnitudes A, B; [V. 8] but it has; therefore A is equal to B.
9
Therefore etc. Q. E. D.
PROPOSITION 10.
10
Of magnitudes which have a ratio to the same, that which has a greater ratio is greater; and that to which the same has a greater ratio is less.
10
For let A have to C a greater ratio than B has to C; I say that A is greater than B.
10
For, if not, A is either equal to B or less.
10
Now A is not equal to B; for in that case each of the magnitudes A, B would have had the same ratio to C; [V. 7] but they have not; therefore A is not equal to B.
10
Nor again is A less than B; for in that case A would have had to C a less ratio than B has to C; [V. 8] but it has not; therefore A is not less than B.
10
But it was proved not to be equal either; therefore A is greater than B.
10
Again, let C have to B a greater ratio than C has to A; I say that B is less than A.
10
For, if not, it is either equal or greater.
10
Now B is not equal to A; for in that case C would have had the same ratio to each of the magnitudes A, B; [V. 7] but it has not; therefore A is not equal to B.
10
Nor again is B greater than A; for in that case C would have had to B a less ratio than it has to A; [V. 8] but it has not; therefore B is not greater than A.
10
But it was proved that it is not equal either; therefore B is less than A.
10
Therefore etc. Q. E. D.
PROPOSITION 11.
