| If there are as many numbers as we please in continued proportion, and the first does not measure the second, then neither does any other measure any other. | ||
| Let there be as many numbers as we please, A, B, C, D, and E, in continued proportion, and let A not measure B.
I say that neither does any other measure any other. Now it is manifest that A, B, C, D, and E do not measure one another in order, for A does not even measure B. I say, then, that neither does any other measure any other. | ||
| If possible, let A measure C. And, however many A, B, and C are, take as many numbers F, G, and H, the least of those which have the same ratio with A, B, and C. | VII.33 | |
| Now, since F, G, and H are in the same ratio with A, B, and C, and the multitude of the numbers A, B, and C equals the multitude of the numbers F, G, and H, therefore, ex aequali A is to C as F is to H. | VII.14 | |
| And since A is to B as F is to G, while A does not measure B, therefore neither does F measure G. Therefore F is not a unit, for the unit measures any number. | VII.Def.20 | |
| Now F and H are relatively prime. And F is to H as A is to C, therefore neither does A measure C. | VIII.3 | |
| Similarly we can prove that neither does any other measure any other. | ||
| Therefore, if there are as many numbers as we please in continued proportion, and the first does not measure the second, then neither does any other measure any other. | ||
| Q.E.D. | ||
Book VIII Introduction - Proposition VIII.5 - Proposition VIII.7.