---
book: 7
number: 17
id: "VII.17"
kind: "theorem"
uses: ["[[book-7/proposition-13]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:7.prop.17"
license: "CC-BY-SA-4.0"
---

# VII.17

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

## Proof

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

For, since *A* by multiplying *B* has made *D*, therefore *B* measures *D* according to the units in *A*.

But the unit *F* also measures the number *A* according to the units in it; therefore the unit *F* measures the number *A* the same number of times that *B* measures *D*.

Therefore, as the unit *F* is to the number *A*, so is *B* to *D*. [[book-7/definitions#Definition 20|VII. Def. 20]]

For the same reason, as the unit *F* is to the number *A*, so also is *C* to *E*; therefore also, as *B* is to *D*, so is *C* to *E*.

Therefore, alternately, as *B* is to *C*, so is *D* to *E*. [[book-7/proposition-13|VII. 13]] Q. E. D.