And it was shewn that FG is not less than K and EF is greater than D; [Construction.
therefore the whole EG is greater than K and D together.
But K and D together are equal to L; [Construction
therefore EG is greater than L.
But FG is not greater than L.
And EG and FG were shewn to be equimultiples of AB and BC;
and L is a multiple of D. [Construction.
Therefore AB has to D a greater ratio than BC has to D. [V. Definition 7.
Also, D shall have to BC a greater ratio than it has to AB.
For, the same construction being made, it may be shewn, that L is greater than FG but not greater than EG.
And L is a multiple of D, [Construction.
and EG and FG were shewn to be equimultiples of AB and CB.
Therefore D has to C a greater ratio than it has to AB. [V. Definition 7.
Wherefore, of unequal magnitudes &c. q.e.d.
PROPOSITION 9. THEOREM.
Magnitudes which have the same ratio to the same magnitude, are equal to one another; and those to which the same magnitude has the same ratio, are equal to one another.
First, let A and B have the same ratio to C: A shall be equal to B.
For, if A is not equal to B, one of them must be greater than the other; let A be the greater.
Then, by what was shewn in Proposition 8, there are
