V.16

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

Proof

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.