PROPOSITION 17. THEOREM.
If magnitudes, taken jointly, be proportionals, they shall also he proportionals when taken separately; that is, if two magnitudes taken together have to one of them the same ratio which two others have to one of these, the remaining one of the first two shall have to the other the same ratio which the remaining one of the last two has to the other of these.
Let AB, BE, CD, DF be the magnitudes which, taken jointly, are proportionals; that is, let AB be to BE as CD is to DF: they shall also be proportionals when taken separately; that is, AE shall be to EB as CF is to FD.
Take of AE, EB, CF, FD any equimultiples whatever GH, HK, LM,MN;
and, again, of EB, FD take any equimultiples whatever KX, NP.
Then, because GH is the same multiple of AE that HK is of EB;
therefore GH is the same multiple of AE that GK is of AB. [V. 1.
But GH is the same multiple of AE that LM is of CF, [Constr.
therefore GK is the same multiple of AB that LM is of CF.
Again, because LM is the same multiple of CF that MN is of FD, [Construction.
therefore LM is the same multiple of CF that LN is of CD. [V. 1.
But LM was shewn to be the same multiple of CF that GK of AB.
Therefore GK is the same multiple of AB that LN is of CD;
that is, GK and LN are equimultiples of AB and CD.
