Book 5
16
Now E, F are equimultiples of A, B, and G, H other, chance, equimultiples of C, D; therefore, as A is to C, so is B to D. [V. Def. 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.
