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

# IX.21

*If as many even numbers as we please be added together, the whole is even.*

## Proof

For let as many even numbers as we please, *AB*, *BC*, *CD*, *DE*, be added together; I say that the whole *AE* is even.

For, since each of the numbers *AB*, *BC*, *CD*, *DE* is even, it has a half part; [[book-7/definitions#Definition 6|VII. Def. 6]] so that the whole *AE* also has a half part.

But an even number is that which is divisible into two equal parts; [*id*.] therefore *AE* is even. Q. E. D.