Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/182
If four magnitudes of the same kind be proportionals, they shall also he proportionals when taken alternately.
Let A, B, C, D be four magnitudes of the same kind which are proportionals ; namely, as A is to B so let C be to D: they shall also be proportionals when taken alternately, that is, A shall be to C as B is to D.
Take of A and B any equimultiples whatever E and F, and of C and D any equimultiples whatever G and H.
Then, because E is the same multiple of A that F is of B, and that magnitudes have the same ratio to one another that their equimultiples have ; [V. 15.
therefore A is to B as E is to F.
But A is to B as C is to D. [Hypothesis.
Therefore C is to D as E is to F. [V. 11.
Again, because G and H are equimultiples of C and D, therefore C is to D as G is to H. [V. 15.
But it was shewn that C is to D as E is to F.
Therefore E is to F as G is to H. [V. 11.
But when four magnitudes are proportionals, if the first be greater than the third, the second is greater than the fourth ; and if equal, equal ; and if less, less. [V. 14.
Therefore if E be greater than G, F is greater than H ; and if equal, equal ; and if less, less.
But E and F are any equimultiples whatever of A and B, and G and H are any equimultiples whatever of C and D. [Construction.
Therefore A is to C as B is to D. [V. Definition 5.
Wherefore if four magnitudes &c. q.e.d.