---
book: 7
number: 16
id: "VII.16"
kind: "theorem"
uses: ["[[book-7/proposition-15]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:7.prop.16"
license: "CC-BY-SA-4.0"
---

# VII.16

*If two numbers by multiplying one another make certain numbers, the numbers so produced will be equal to one another.*

## Proof

Let *A*, *B* be two numbers, and let *A* by multiplying *B* make *C*, and *B* by multiplying *A* make *D*; I say that *C* is equal to *D*.

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

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

Therefore, alternately, the unit *E* measures the number *B* the same number of times that *A* measures *C*. [[book-7/proposition-15|VII. 15]]

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

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

But the unit *E* measured the number *B* the same number of times that *A* measures *C*; therefore *A* measures each of the numbers *C*, *D* the same number of times.

Therefore *C* is equal to *D*. Q. E. D.