PROPOSITION 10. THEOREM.
That magnitude which has a greater ratio than another has to the same magnitude is the greater of the two; and that magnitude to which the same has a greater ratio than it has to another magnitude is the less of the two.
First, let A have to C a greater ratio than B has to C: A shall be greater than B.
For, because A has a greater ratio A to C than B has to C, there are some equimultiples of A and B, and some . multiple of C, such that the multiple C of A is greater than the multiple of C, but the multiple of B is not greater than the multiple of C. [V. Def. 7.
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;
therefore D is greater than E.
And because D and E are equimultiples of A and B, and that D is greater than E,
therefore A is greater than B. [V. Axiom 4.
Next, let C have to B a greater ratio than it has to A: B shall be less than A.
For 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; [V. Definition 7.
therefore E is less than D.
And because E and D are equimultiples of B and A, and that E is less than D, therefore B is less than A. [V. Axiom 4.
Wherefore, that magnitude &c. q.e.d.
