---
book: 8
number: 24
id: "VIII.24"
kind: "theorem"
uses: ["[[book-8/proposition-18]]", "[[book-8/proposition-8]]", "[[book-8/proposition-22]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:8.prop.24"
license: "CC-BY-SA-4.0"
---

# VIII.24

*If two numbers have to one another the ratio which a square number has to a square number, and the first be square, the second will also be square.*

## Proof

For let the two numbers *A*, *B* have to one another the ratio which the square number *C* has to the square number *D*, and let *A* be square; I say that *B* is also square.

For, since *C*, *D* are square, *C*, *D* are similar plane numbers.

Therefore one mean proportional number falls between *C*, *D*. [[book-8/proposition-18|VIII. 18]]

And, as *C* is to *D*, so is *A* to *B*; therefore one mean proportional number falls between *A*, *B* also. [[book-8/proposition-8|VIII. 8]]

And *A* is square; therefore *B* is also square. [[book-8/proposition-22|VIII. 22]] Q. E. D.