---
book: 9
number: 30
id: "IX.30"
kind: "theorem"
uses: ["[[book-9/proposition-23]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:9.prop.30"
license: "CC-BY-SA-4.0"
---

# IX.30

*If an odd number measure an even number, it will also measure the half of it.*

## Proof

For let the odd number *A* measure the even number *B*; I say that it will also measure the half of it.

For, since *A* measures *B*, let it measure it according to *C*; I say that *C* is not odd.

For, if possible, let it be so.

Then, since *A* measures *B* according to *C*, therefore *A* by multiplying *C* has made *B*.

Therefore *B* is made up of odd numbers the multitude of which is odd.

Therefore *B* is odd: [[book-9/proposition-23|IX. 23]] which is absurd, for by hypothesis it is even.

Therefore *C* is not odd; therefore *C* is even.

Thus *A* measures *B* an even number of times.

For this reason then it also measures the half of it. Q. E. D.