---
book: 7
number: 18
id: "VII.18"
kind: "theorem"
uses: ["[[book-7/proposition-16]]", "[[book-7/proposition-17]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:7.prop.18"
license: "CC-BY-SA-4.0"
---

# VII.18

*If two numbers by multiplying any number make certain numbers, the numbers so produced will have the same ratio as the multipliers.*

## Proof

For let two numbers *A*, *B* by multiplying any number *C* make *D*, *E*; I say that, as *A* is to *B*, so is *D* to *E*.

For, since *A* by multiplying *C* has made *D*, therefore also *C* by multiplying *A* has made *D*. [[book-7/proposition-16|VII. 16]] For the same reason also *C* by multiplying *B* has made *E*.

Therefore the number *C* by multiplying the two numbers *A*, *B* has made *D*, *E*.

Therefore, as *A* is to *B*, so is *D* to *E*. [[book-7/proposition-17|VII. 17]]