V.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.

For let A have to C a greater ratio than B has to C; I say that A is greater than B.

For, if not, A is either equal to B or less.

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.

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.

But it was proved not to be equal either;

therefore A is greater than B.

Again, let C have to B a greater ratio than C has to A; I say that B is less than A.

For, if not, it is either equal or greater.

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.

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.

But it was proved that it is not equal either;

therefore B is less than A.

Therefore etc. Q. E. D.