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. |
V.15 |
But A is to B as C is to D, therefore C is to D also as E is to F. |
V.11 |
Again, since G and H are equimultiples of C and D, therefore C is to D as G is to H. |
V.15 |
But C is to D as E is to F, therefore as E is to F also as G is to H. |
V.11 |
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. |
V.14 |
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. |
V.Def.5 |
Therefore, if four magnitudes are proportional, then they are also proportional alternately.
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Q.E.D. |