---
book: 8
number: 6
id: "VIII.6"
kind: "theorem"
uses: ["[[book-7/proposition-33]]", "[[book-7/proposition-14]]", "[[book-8/proposition-3]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:8.prop.6"
license: "CC-BY-SA-4.0"
---

# VIII.6

*If there be as many numbers as we please in continued proportion, and the first do not measure the second, neither will any other measure any other.*

## Proof

Let there be as many numbers as we please, *A*, *B*, *C*, *D*, *E*, in continued proportion, and let *A* not measure *B*; I say that neither will any other measure any other.

Now it is manifest that *A*, *B*, *C*, *D*, *E* do not measure one another in order; for *A* does not even measure *B*.

I say, then, that neither will any other measure any other.

For, if possible, let *A* measure *C*.

And, however many *A*, *B*, *C* are, let as many numbers *F*, *G*, *H*, the least of those which have the same ratio with *A*, *B*, *C*, be taken. [[book-7/proposition-33|VII. 33]]

Now, since *F*, *G*, *H* are in the same ratio with *A*, *B*, *C*, and the multitude of the numbers *A*, *B*, *C* is equal to the multitude of the numbers *F*, *G*, *H*, therefore, ex aequali, as *A* is to *C*, so is *F* to *H*. [[book-7/proposition-14|VII. 14]]

And since, as *A* is to *B*, so is *F* to *G*, while *A* does not measure *B*, therefore neither does *F* measure *G*; [[book-7/definitions#Definition 20|VII. Def. 20]] therefore *F* is not an unit, for the unit measures any number.

Now *F*, *H* are prime to one another. [[book-8/proposition-3|VIII. 3]]

And, as *F* is to *H*, so is *A* to *C*; therefore neither does *A* measure *C*.

Similarly we can prove that neither will any other measure any other. Q. E. D.