Let A, B, C, and D be four proportional magnitudes, so that A is to B as C is to D.
I say that they are also so alternately, that is A is to C as B is to D.
Take equimultiples E and F of A and B, and take other, arbitrary, equimultiples G and H of C and D.
Then, since E is the same multiple of A that F is of B, and parts have the same ratio as their equimultiples, therefore A is to B as E is to F.
But A is to B as C is to D, therefore C is to D also as E is to F.
Again, since G and H are equimultiples of C and D, therefore C is to D as G is to H.
But C is to D as E is to F, therefore as E is to F also as G is to H.
But, if four magnitudes are proportional, and the first is greater than the third, then the second is also greater than the fourth; if equal, equal; and if less, less.
Therefore, if E is in excess of G, F is also in excess of H; if equal, equal; and if less, less.
Now E and F are equimultiples of A and B, and G and H other, arbitrary, equimultiples of C and D, therefore A is to C as B is to D.
Therefore, if four magnitudes are proportional, then they are also proportional alternately.
so by the definition of proportion V.Def.5, a : c = b : d. Q.E.D.
This proposition requires using V.Def.4 as an axiom of comparability.