---
book: 8
number: 22
id: "VIII.22"
kind: "theorem"
uses: ["[[book-8/proposition-20]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:8.prop.22"
license: "CC-BY-SA-4.0"
---

# VIII.22

*If three numbers be in continued proportion, and the first be square, the third will also be square.*

## Proof

Let *A*, *B*, *C* be three numbers in continued proportion, and let *A* the first be square; I say that *C* the third is also square.

For, since between *A*, *C* there is one mean proportional number, *B*, therefore *A*, *C* are similar plane numbers. [[book-8/proposition-20|VIII. 20]]

But *A* is square; therefore *C* is also square. Q. E. D.