---
book: 7
number: 37
id: "VII.37"
kind: "theorem"
uses: ["[[book-7/proposition-15]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:7.prop.37"
license: "CC-BY-SA-4.0"
---

# VII.37

*If a number be measured by any number, the number which is measured will have a part called by the same name as the measuring number.*

## Proof

For let the number *A* be measured by any number *B*; I say that *A* has a part called by the same name as *B*.

For, as many times as *B* measures *A*, so many units let there be in *C*.

Since *B* measures *A* according to the units in *C*, and the unit *D* also measures the number *C* according to the units in it, therefore the unit *D* measures the number *C* the same number of times as *B* measures *A*.

Therefore, alternately, the unit *D* measures the number *B* the same number of times as *C* measures *A*; [[book-7/proposition-15|VII. 15]] therefore, whatever part the unit *D* is of the number *B*, the same part is *C* of *A* also.

But the unit *D* is a part of the number *B* called by the same name as it; therefore *C* is also a part of *A* called by the same name as *B*, so that *A* has a part *C* which is called by the same name as *B*. Q. E. D.