# Proposition 15

If two commensurable magnitudes are added together, then the whole is also commensurable with each of them; and, if the whole is commensurable with one of them, then the original magnitudes are also commensurable.

Let the two commensurable magnitudes AB and BC be added together.

I say that the whole AC is also commensurable with each of the magnitudes AB and BC.

Since AB and BC are commensurable, some magnitude D measures them.

X.Def.I.1

Since then D measures AB and BC, therefore it also measures the whole AC. But it measures AB and BC also, therefore D measures AB, BC, and AC. Therefore AC is commensurable with each of the magnitudes AB and BC.

Next, let AC be commensurable with AB.

I say that AB and BC are also commensurable.

Since AC and AB are commensurable, some magnitude D measures them.

Since then D measures CA and AB, therefore it also measures the remainder BC.

X.Def.I.1

But it measures AB also, therefore D measures AB and BC. Therefore AB and BC are commensurable.

Therefore, if two commensurable magnitudes are added together, then the whole is also commensurable with each of them; and, if the whole is commensurable with one of them, then the original magnitudes are also commensurable.

Q.E.D.

## Guide

This fundamental proposition on commensurability of sums and differences is used in very frequently in Book X starting with X.17. It is also used in XIII.11.