Kitap 5
16
Therefore etc. Q. E. D. Let A, B, C, D be four proportional magnitudes, so that, as A is to B, so is C to D. In a number of expressions like this it is absolutely necessary, when translating into English, to interpolate words which are not in the Greek. Thus the Greek here is: Ἕστω τέσσαρα μεγέθη ἀνάλογον τὰ Α, Β, Γ, Δ, ὡς τὸ Α πρὸς τὸ Β, οὕτως τὸ Γ πρὸς τὸ Δ, literally Let A, B, C, D be four proportional magnitudes, as A to B, so C to D. The same remark applies to the corresponding expressions in the next propositions, V. 17, 18, and to other forms of expression in V. 20-23 and later propositions: e.g. in V. 20 we have a phrase meaning literally Let there be magnitudes...which taken two and two are in the same ratio, as A to B, so D to E, etc.: in V. 21 (magnitudes)...which taken two and two are in the same ratio, and let the proportion of them be perturbed, as A to B, so E to F, etc. In all such cases (where the Greek is so terse as to be almost ungrammatical) I shall insert the words necessary in English, without further remark.
PROPOSITION 17.
17
If magnitudes be proportional componendo, they will also be proportional separando.
17
Let AB, BE, CD, DF be magnitudes proportional componendo, so that, as AB is to BE, so is CD to DF; I say that they will also be proportional separando, that is, as AE is to EB, so is CF to DF.
17
For of AE, EB, CF, FD let equimultiples GH, HK, LM, MN be taken, and of EB, FD other, chance, equimultiples, KO, NP.
17
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]
17
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.
17
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]
17
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.
17
Therefore GK, LN are equimultiples of AB, CD.
