Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/174

150 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.

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