---
book: 9
number: 2
id: "IX.2"
kind: "theorem"
uses: ["[[book-7/proposition-17]]", "[[book-8/proposition-18]]", "[[book-8/proposition-20]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:9.prop.2"
license: "CC-BY-SA-4.0"
---

# IX.2

*If two numbers by multiplying one another make a square number, they are similar plane numbers.*

## Proof

Let *A*, *B* be two numbers, and let *A* by multiplying *B* make the square number *C*; I say that *A*, *B* are similar plane numbers.

For let *A* by multiplying itself make *D*; therefore *D* is square.

Now, since *A* by multiplying itself has made *D*, and by multiplying *B* has made *C*, therefore, as *A* is to *B*, so is *D* to *C*. [[book-7/proposition-17|VII. 17]]

And, since *D* is square, and *C* is so also, therefore *D*, *C* are similar plane numbers.

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

And, as *D* is to *C*, so is *A* to *B*; therefore one mean proportional number falls between *A*, *B* also. [[book-8/proposition-18|VIII. 8]]

But, if one mean proportional number fall between two numbers, they are similar plane numbers; [[book-8/proposition-20|VIII. 20]] therefore *A*, *B* are similar plane numbers. Q. E. D.