Then, since 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;
therefore GK is the same multiple of AB that LM is of CF.
Again, since LM is the same multiple of CF that MN is of FD,
therefore LM is the same multiple of CF that LN is of CD. [V. 1]
But LM was the same multiple of CF that GK is of AB;
therefore GK is the same multiple of AB that LN is of CD.
Therefore GK, LN are equimultiples of AB, CD.
Again, since HK is the same multiple of EB that MN is of FD,
and KO is also the same multiple of EB that NP is of FD,
therefore the sum HO is also the same multiple of EB that MP is of FD. [V. 2]
And, since, as AB is to BE, so is CD to DF,
and of AB, CD equimultiples GK, LN have been taken, and of EB, FD equimultiples HO, MP,
therefore, if GK is in excess of HO, LN is also in excess of MP, if equal, equal, and if less, less.
Let GK be in excess of HO;
then, if HK be subtracted from each, GH is also in excess of KO.
But we saw that, if GK was in excess of HO, LN was also in excess of MP;
therefore LN is also in excess of MP,
and, if MN be subtracted from each, LM is also in excess of NP;
so that, if GH is in excess of KO, LM is also in excess of NP.
Similarly we can prove that, if GH be equal to KO,
LM will also be equal to NP, and if less, less.
And GH, LM are equimultiples of AE, CF, while KO, NP are other, chance, equimultiples of EB, FD;
therefore, as AE is to EB, so is CF to FD.
Complete text
Book I