---
book: 7
number: 2
id: "VII.2"
kind: "construction"
uses: ["[[book-7/proposition-1]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:7.prop.2"
license: "CC-BY-SA-4.0"
---

# VII.2

*Given two numbers not prime to one another, to find their greatest common measure.*

## Proof

Let *AB*, *CD* be the two given numbers not prime to one another.

Thus it is required to find the greatest common measure of *AB*, *CD*.

If now *CD* measures *AB*—and it also measures itself—*CD* is a common measure of *CD*, *AB*.

And it is manifest that it is also the greatest; for no greater number than *CD* will measure *CD*.

But, if *CD* does not measure *AB*, then, the less of the numbers *AB*, *CD* being continually subtracted from the greater, some number will be left which will measure the one before it.

For an unit will not be left; otherwise *AB*, *CD* will be prime to one another [[book-7/proposition-1|VII. 1]], which is contrary to the hypothesis.

Therefore some number will be left which will measure the one before it.

Now let *CD*, measuring *BE*, leave *EA* less than itself, let *EA*, measuring *DF*, leave *FC* less than itself, and let *CF* measure *AE*.

Since then, *CF* measures *AE*, and *AE* measures *DF*, therefore *CF* will also measure *DF*.

But it also measures itself; therefore it will also measure the whole *CD*.

But *CD* measures *BE*; therefore *CF* also measures *BE*.

But it also measures *EA*; therefore it will also measure the whole *BA*.

But it also measures *CD*; therefore *CF* measures *AB*, *CD*.

Therefore *CF* is a common measure of *AB*, *CD*.

I say next that it is also the greatest.

For, if *CF* is not the greatest common measure of *AB*, *CD*, some number which is greater than *CF* will measure the numbers *AB*, *CD*.

Let such a number measure them, and let it be *G*.

Now, since *G* measures *CD*, while *CD* measures *BE*, *G* also measures *BE*.

But it also measures the whole *BA*; therefore it will also measure the remainder *AE*.

But *AE* measures *DF*; therefore *G* will also measure *DF*.

But it also measures the whole *DC*; therefore it will also measure the remainder *CF*, that is, the greater will measure the less: which is impossible.

Therefore no number which is greater than *CF* will measure the numbers *AB*, *CD*; therefore *CF* is the greatest common measure of *AB*, *CD*.

Porism. From this it is manifest that, if a number measure two numbers, it will also measure their greatest common measure.

Q. E. D.