But A is to B as E is to F, [Hypothesis.
Therefore G is to H as M is to N. [V. 11.
And because B is to C as D is to E, [Hypothesis.
and that H and K are equimultiples of B and D,
and L and M are equimultiples of C and E; [Constr.
therefore H is to L as K is to M. [V. 4.
And it has been shewn that G is to H as M is to N.
Then since there are three magnitudes G, H, L, and other three K, M,N, which have the same ratio, taken two and two in a cross order;
therefore if G be greater than L, K is greater than N; and if equal, equal; and if less, less. [V. 21.
But G and K are any equimultiples whatever of A and D, and L and 'N are any equimultiples whatever of C and F;
therefore A is to C as D is to F. [V. Definition 5.
Next, let there be four magnitudes A, B, C, D, and other four E, F, G, H, which have the same ratio, taken two and two in a cross order; namely, let A be to B as G is to H, and B to C as F is to G, and C to D as E is to F:
A shall be to D as E is to H.
For, because A, B, C are three magnitudes, and F, G, H other three, which have the same ratio, taken two and two in a cross order; [Hypothesis.
therefore, by the first case, A is to C as F is to H.
But C is to D as E is to F; [Hypothesis.
therefore also, by the first case, A is to D as E is to H.
And so on, whatever be the number of magnitudes.
Wherefore, if there he any number &c. q.e.d.
