Associative Law of Multiplication/Euclid's Statement

Theorem

In the words of Euclid:

If a first magnitude be the same multiple of a second that a third is of a fourth, and if equimultiples be taken of the first and third, then also ex aequali the magnitudes taken will be equimultiples respectively, the one of the second and the other of the fourth.

(The Elements: Book $\text{V}$: Proposition $3$)

That is, if:

$n a, n b$ are equimultiples of $a, b$

and if:

$m \cdot n a, m \cdot nb$ are equimultiples of $n a, n b$

then:

$m \cdot n a$ is the same multiple of $a$ that $m \cdot n b$ is of $b$

This can also be expressed as:

$m \cdot n a = m n \cdot a$