Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/175

Rh some equimultiples of A and B, and some multiple of C, such that the multiple of A is greater than the multiple of C, but the multiple of B is not greater than the multiple of C. Let such multiples be taken; and let D and E be the equimultiples of A and B, and F the multiple of C; so that D is greater than F, but E is not greater than F. Then, because A is to C as B is to C; and of A and B are taken equimultiples D and E, and of C is taken a multiple F; and that D is greater than F; [Construction therefore E is also greater than F.[V. Definition 5.

But E is not greater than F; [Construction which is impossible.

Therefore A and B are not unequal; that is, they are equal.

Next, let C have the same ratio to A and B: 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 is some multiple F of C, and some equimultiples E and D of B and A, such that F is greater than E, but not greater than D.

And, because C is to B as C is to A, [Hypothesis. and that F the multiple of the first is greater than E the multiple of the second, [Construction. therefore F the multiple of the third is greater than D the multiple of the fourth. [V. Definition 5. But F is not greater than D; [Construction. which is impossible.

Therefore A and B are not unequal; that is, they are equal.

Wherefore, magnitudes which &c.