---
book: 7
number: 36
id: "VII.36"
kind: "construction"
uses: ["[[book-7/proposition-34]]", "[[book-7/proposition-35]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:7.prop.36"
license: "CC-BY-SA-4.0"
---

# VII.36

*Given three numbers, to find the least number which they measure.*

## Proof

Let *A*, *B*, *C* be the three given numbers; thus it is required to find the least number which they measure.

Let *D*, the least number measured by the two numbers *A*, *B*, be taken. [[book-7/proposition-34|VII. 34]]

Then *C* either measures, or does not measure, *D*.

First, let it measure it.

But *A*, *B* also measure *D*; therefore *A*, *B*, *C* measure *D*.

I say next that it is also the least that they measure.

For, if not, *A*, *B*, *C* will measure some number which is less than *D*.

Let them measure *E*.

Since *A*, *B*, *C* measure *E*, therefore also *A*, *B* measure *E*.

Therefore the least number measured by *A*, *B* will also measure *E*. [[book-7/proposition-35|VII. 35]]

But *D* is the least number measured by *A*, *B*; therefore *D* will measure *E*, the greater the less: which is impossible.

Therefore *A*, *B*, *C* will not measure any number which is less than *D*; therefore *D* is the least that *A*, *B*, *C* measure.

Again, let *C* not measure *D*, and let *E*, the least number measured by *C*, *D*, be taken. [[book-7/proposition-34|VII. 34]]

Since *A*, *B* measure *D*, and *D* measures *E*, therefore also *A*, *B* measure *E*.

But *C* also measures *E*; therefore also *A*, *B*, *C* measure *E*.

I say next that it is also the least that they measure.

For, if not, *A*, *B*, *C* will measure some number which is less than *E*.

Let them measure *F*.

Since *A*, *B*, *C* measure *F*, therefore also *A*, *B* measure *F*; therefore the least number measured by *A*, *B* will also measure *F*. [[book-7/proposition-35|VII. 35]]

But *D* is the least number measured by *A*, *B*; therefore *D* measures *F*.

But *C* also measures *F*; therefore *D*, *C* measure *F*, so that the least number measured by *D*, *C* will also measure *F*.

But *E* is the least number measured by *C*, *D*; therefore *E* measures *F*, the greater the less: which is impossible.

Therefore *A*, *B*, *C* will not measure any number which is less than *E*.

Therefore *E* is the least that is measured by *A*, *B*, *C*. Q. E. D.