---
book: 10
number: 3
id: "X.3"
kind: "construction"
uses: ["[[book-10/proposition-2]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_1.3"
license: "CC-BY-SA-4.0"
---

# X.3

*Given two commensurable magnitudes, to find their greatest common measure*.

## Proof

Let the two given commensurable magnitudes be *AB*, *CD* of which *AB* is the less; thus it is required to find the greatest common measure of *AB*, *CD*.

Now the magnitude *AB* either measures *CD* or it does not.

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

And it is manifest that it is also the greatest; for a greater magnitude than the magnitude *AB* will not measure *AB*.

Next, let *AB* not measure *CD*.

Then, if the less be continually subtracted in turn from the greater, that which is left over will sometime measure the one before it, because *AB*, *CD* are not incommensurable; cf. [[book-10/proposition-2|X. 2]] let *AB*, measuring *ED*, leave *EC* less than itself, let *EC*, measuring *FB*, leave *AF* less than itself, and let *AF* measure *CE*.

Since, then, *AF* measures *CE*, while *CE* measures *FB*, therefore *AF* will also measure *FB*.

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

But *AB* measures *DE*; therefore *AF* will also measure *ED*.

But it measures *CE* also; therefore it also measures the whole *CD*.

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

I say next that it is also the greatest.

For, if not, there will be some magnitude greater than *AF* which will measure *AB*, *CD*.

Let it be *G*.

Since then *G* measures *AB*, while *AB* measures *ED*, therefore *G* will also measure *ED*.

But it measures the whole *CD* also; therefore *G* will also measure the remainder *CE*.

But *CE* measures *FB*; therefore *G* will also measure *FB*.

But it measures the whole *AB* also, and it will therefore measure the remainder *AF*, the greater the less: which is impossible.

Therefore no magnitude greater than *AF* will measure *AB*, *CD*; therefore *AF* is the greatest common measure of *AB*, *CD*.

Therefore the greatest common measure of the two given commensurable magnitudes *AB*, *CD* has been found. Q. E. D.

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