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PROPOSITION 16.

If four magnitudes be proportional, they will also be proportional alternately.

Let A, B, C, D be four proportional magnitudes, so that, as A is to B, so is C to D; I say that they will also be so alternately, that is, as A is to C, so is B to D.

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] 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]

Again, since G, H are equimultiples of C, D, therefore, as C is to D, so is G to H. [V. 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]

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]

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]

Therefore etc. Q. E. D. 1

1 “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.

Euclid. Euclid's Elements. Sir Thomas Little Heath. New York. Dover. 1956.

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