---
book: 7
number: 39
id: "VII.39"
kind: "construction"
uses: ["[[book-7/proposition-36]]", "[[book-7/proposition-37]]", "[[book-7/proposition-38]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:7.prop.39"
license: "CC-BY-SA-4.0"
---

# VII.39

*To find the number which is the least that will have given parts.*

## Proof

Let *A*, *B*, *C* be the given parts; thus it is required to find the number which is the least that will have the parts *A*, *B*, *C*.

Let *D*, *E*, *F* be numbers called by the same name as the parts *A*, *B*, *C*, and let *G*, the least number measured by *D*, *E*, *F*, be taken. [[book-7/proposition-36|VII. 36]]

Therefore *G* has parts called by the same name as *D*, *E*, *F*. [[book-7/proposition-37|VII. 37]]

But *A*, *B*, *C* are parts called by the same name as *D*, *E*, *F*; therefore *G* has the parts *A*, *B*, *C*.

I say next that it is also the least number that has.

For, if not, there will be some number less than *G* which will have the parts *A*, *B*, *C*.

Let it be *H*.

Since *H* has the parts *A*, *B*, *C*, therefore *H* will be measured by numbers called by the same name as the parts *A*, *B*, *C*. [[book-7/proposition-38|VII. 38]]

But *D*, *E*, *F* are numbers called by the same name as the parts *A*, *B*, *C*; therefore *H* is measured by *D*, *E*, *F*.

And it is less than *G* : which is impossible.

Therefore there will be no number less than *G* that will have the parts *A*, *B*, *C*. Q. E. D.