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. q.e.d.
