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 [later] 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 [later] 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 [later] 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.*

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.*

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.

Now *F* and *H* are relatively prime. And *F* is to *H* as *A* is to *C,* therefore neither does *A* measure *C.*

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.

Suppose that some number in the sequence divides a later number. We may call that the former number *a* since it divides the next number in the sequence, and call the number it divides *c.* Take the continued proportion *a, b,* ..., *c* and, using
VII.33, reduce it a continued proportion *f, g,* ..., *h* in lowest terms. Since that’s in lowest terms, *f* and *h* are relatively prime. Since *a* : *b* = *f* : *g*, and *a* does not divide *b,* therefore *f* does not divide *g.* Since *f* does not divide *g,* in particular *f* does not equal 1, but *f* and *h* are relatively prime by VIII.3, therefore *f* does not divide *h.* Finally, since *a* : *c* = *f* : *h,* therefore *a* does not divide *c* either.