---
book: 10
number: 15
id: "X.15"
kind: "theorem"
uses: []
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_1.15"
license: "CC-BY-SA-4.0"
---

# X.15

*If two commensurable magnitudes be added together, the whole will also be commensurable with each of them; and, if the whole be commensurable with one of them, the original magnitudes will also be commensurable*.

## Proof

For let the two commensurable magnitudes *AB*, *BC* be added together; I say that the whole *AC* is also commensurable with each of the magnitudes *AB*, *BC*.

For, since *AB*, *BC* are commensurable, some magnitude will measure them.

Let it measure them, and let it be *D*.

Since then *D* measures *AB*, *BC*, it will also measure the whole *AC*.

But it measures *AB*, *BC* also; therefore *D* measures *AB*, *BC*, *AC*; therefore *AC* is commensurable with each of the magnitudes *AB*, *BC*. [[book-10/definitions#Definition 1|X. Def. 1]]

Next, let *AC* be commensurable with *AB*; I say that *AB*, *BC* are also commensurable.

For, since *AC*, *AB* are commensurable, some magnitude will measure them.

Let it measure them, and let it be *D*.

Since then *D* measures *CA*, *AB*, it will also measure the remainder *BC*.

But it measures *AB* also; therefore *D* will measure *AB*, *BC*; therefore *AB*, *BC* are commensurable. [[book-10/definitions#Definition 1|X. Def. 1]]

Therefore etc.