Again, because HK is the same multiple of EB that MN is of FD, and that KX is the same multiple of EB that NP is of FD, [Construction.
therefore HX is the same multiple of EB that MP is of FD; [V. 2.
that is, HX and MP are equimultiples of EB and FD.
And because AB is to BE as CD is to DF, [Hypothesis.
and that GK and LN are equimultiples of AB and CD, and HX and MP are equimultiples of EB and FD,
therefore if GK be greater than HX, LN is greater than MP; and if equal, equal; and if less, less. [V. Def. 5.
But if GH be greater than HX, then, by adding the common magnitude HK to both, GK is greater than HX;
therefore also LN is greater than MP; and, by taking away the common magnitude MN from both, LM is greater than NP. Thus if GH be greater than KX, LM is greater than NP.
In like manner it may be shewn that, if GH be equal to KX, LM is equal to NP; and if less, less.
But GH and LM are any equimultiples whatever of AE and CF, and KX and NP are any equimultiples whatever of EB and FD; [Construction.
therefore AE is to EB as CF is to FD. [V. Definition 5,
Wherefore, if four magnitudes &c. q.e.d.
PROPOSITION 18. THEOREM.
If magnitudes, taken separately, he proportionals, they shall also he proportionals when taken jointly; that is, if the first he to the second as the third to the fourth, the first and second together shall be to the second as the third and fourth together to the fourth.
