For of A , B let equimultiples E , F be taken, and of C , D other, chance, equimultiples G , H .

Then, since E is the same multiple of A that F is of B , and parts have the same ratio as the same multiples of them, [[V. 15](elem.5.15)] therefore, as A is to B , so is E to F .

But as A is to B , so is C to D ; therefore also, as C is to D , so is E to F . [[V. 11](elem.5.11)]

Again, since G , H are equimultiples of C , D , therefore, as C is to D , so is G to H . [[V. 15](elem.5.15)]

But, as C is to D , so is E to F ; therefore also, as E is to F , so is G to H . [[V. 11](elem.5.11)]

But, if four magnitudes be proportional, and the first be greater than the third, the second will also be greater than the fourth; if equal, equal; and if less, less. [[V. 14](elem.5.14)]

Therefore, if E is in excess of G , F is also in excess of H , if equal, equal, and if less, less.

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](elem.5.def.5)]

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](elem.5.17 elem.5.18), and to other forms of expression in [V. 20-23](elem.5.20 elem.5.21 elem.5.22 elem.5.23) and later propositions: e.g. in [V. 20](elem.5.20) 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](elem.5.21) (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.