PROPOSITION 11. THEOREM.
Ratios that are the same to the same ratio, are the same to one another.
Let A be to B as C is to D, and let C be to D as E is to F: A shall be to B as E is to F.
Take of A, C, E any equimultiples whatever G, H, K; and of B, D, F any equimultiples whatever L, M, N.
Then, because A is to B as C is to D, [Hypothesis.
and that G and H are equimultiples of A and C, and L and M are equimultiples of B and D; [Construction.
therefore if G be greater than L, H is greater than 'N;
and if equal, equal; and if less, less. [V. Definition 5.
Again, because C is to D as E is to F, [Hypothesis.
and that H and K are equimultiples of C and E, and M and N are equimultiples of D and F; [Construction.
therefore if H be greater than M, K is greater than N; and if equal, equal; and if less, less. [V. Definition 5.
But it has been shewn that if G be greater than L, H is greater than M; and if equal, equal; and if less, less.
Therefore if G be greater than L, K is greater than N;
and if equal, equal; and if less, less.
And G and K are any equimultiples whatever of A and E, and L and N are any equimultiples whatever of B and F. Therefore A is to B as E is to F. [V. Definition 6.
Wherefore, ratios that are the same &c. q.e.d.
