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G.188 magnitudes ratio greater — Heath¶

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 ; [G.183] 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 ; [G.186] 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 ; [G.183] 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 ; [G.186] 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.

Dependency Graph¶