---
book: 9
number: 1
id: "IX.1"
kind: "theorem"
uses: ["[[book-7/proposition-17]]", "[[book-8/proposition-18]]", "[[book-8/proposition-8]]", "[[book-8/proposition-22]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:9.prop.1"
license: "CC-BY-SA-4.0"
---

# IX.1

*If two similar plane numbers by multiplying one another make some number, the product will be square.*

## Proof

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

For let *A* by multiplying itself make *D*.

Therefore *D* is square.

Since then *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 *A*, *B* are similar plane numbers, therefore one mean proportional number falls between *A*, *B*. [[book-8/proposition-18|VIII. 18]]

But, if numbers fall between two numbers in continued proportion, as many as fall between them, so many also fall between those which have the same ratio; [[book-8/proposition-8|VIII. 8]] so that one mean proportional number falls between *D*, *C* also.

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