| If there are three magnitudes, and others equal to them in multitude, which taken two and two together are in the same ratio, and the proportion of them be perturbed, then they are also in the same ratio ex aequali. | |
| Let there be three magnitudes A, B, and C, and others D, E, and F, equal to them in multitude, which, taken two and two together, are in the same proportion, and let the proportion of them be perturbed, so that A is to B as E is to F, and B is to C as D is to E. | V.Def.18 |
| I say that A is to C as D is to F.
Take equimultiples G, H, and K of A, B, and D, and take other, arbitrary, equimultiples L, M, and N of C, E, and F. | |
| Then, since G and H are equimultiples of A and B, and parts have the same ratio as their multiples, therefore A is to B as G is to H. | V.15 |
| For the same reason E is to F as M is to N. And A is to B as E is to F, therefore G is also to H as M is to N. | V.11 |
| Next, since B is to C as D is to E, alternately, also, B is to D as C is to E. | (V.16) |
| And, since H and K are equimultiples of B and D, and parts have the same ratio as their equimultiples, therefore B is to D as H is to K. | V.15 |
| But B is to D as C is to E, therefore also, H is to K as C is to E. | V.11 |
| Again, since L and M are equimultiples of C and E, therefore C is to E as L is to M. | V.15 |
| But C is to E as H is to K, therefore also, H is to K as L is to M, and, alternately, H is to L as K is to M. | V.11
(V.16) |
| But it was also proved that G is to H as M is to N. | |
| Since, then, there are three magnitudes G, H, and L, and others equal to them in multitude K, M, and N, which taken two and two together are in the same ratio, and the proportion of them is perturbed, therefore, ex aequali, if G is in excess of L, K is also in excess of N; if equal, equal; and if less, less. | V.21 |
| And G and K are equimultiples of A and D, and L and N of C and F. | |
| Therefore A is to C as D is to F. | V.Def.5 |
| Therefore, if there are three magnitudes, and others equal to them in multitude, which taken two and two together are in the same ratio, and the proportion of them be perturbed, then they are also in the same ratio ex aequali. | |
| Q.E.D. | |
The proof given here uses proposition V.16 and alternate ratios, and that means it only applies when all six magnitudes are of the same kind. There is a shorter variation of the proof that uses V.4 instead of V.16 and applies in the general situation.
It is not used in the rest of the Elements.