Let A and B be equal magnitudes, and C any other magnitude: each of the magnitudes A and B shall have the same ratio to C; and C shall have the same ratio to each of the magnitudes A and B.
Take of A and B any equimultiples whatever D and E; and of C any multiple whatever F.
Then, because D is the same multiple of A that E is of B, [Construction.
and that A is equal to B; [Hypothesis.
therefore D is equal to E. [V. Axiom 1. Therefore if D be greater than F, E is greater than F; and if equal, equal; and if less, less.
But D and E are any equimultiples whatever of A and B, and F is any multiple whatever of C; [Construction. therefore A is to C as B is to C. [V. Def. 5.
Also C shall have the same ratio to A that it has to B.
For the same construction being made, it may be shewn, as before, that D is equal to E.
Therefore if F be greater than D, F is greater than E; and if equal, equal; and if less, less.
But F is any multiple whatever of C, and D and E are any equimultiples whatever of A and B; [Construction.
therefore C is to A as C is to B. [V. Definition 5.
Wherefore, equal magnitudes &c. q.e.d.
PROPOSITION 8. THEOREM.
Of unequal magnitudes, the greater has a greater ratio to the same than the less has; and the same magnitude has a greater ratio to the less than it has to the greater.
Let AB and BC be unequal magnitudes, of which AB is the greater; and let D be any other magnitude whatever: AB shall have a greater ratio to D than BC has to D; and D shall have a greater ratio to BC than it has to AB.
