---
book: 9
number: 16
id: "IX.16"
kind: "theorem"
uses: ["[[book-7/proposition-21]]", "[[book-7/proposition-20]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:9.prop.16"
license: "CC-BY-SA-4.0"
---

# IX.16

*If two numbers be prime to one another, the second will not be to any other number as the first is to the second.*

## Proof

For let the two numbers *A*, *B* be prime to one another; I say that *B* is not to any other number as *A* is to *B*.

For, if possible, as *A* is to *B*, so let *B* be to *C*.

Now *A*, *B* are prime, primes are also least, [[book-7/proposition-21|VII. 21]] and the least numbers measure those which have the same ratio the same number of times, the antecedent the antecedent and the consequent the consequent; [[book-7/proposition-20|VII. 20]] therefore *A* measures *B* as antecedent antecedent.

But it also measures itself; therefore *A* measures *A*, *B* which are prime to one another: which is absurd.

Therefore *B* will not be to *C*, as *A* is to *B*. Q. E. D.