Natural Number Multiplication is Commutative/Euclid's Proof

Theorem

As Euclid defined it:

If two (natural) numbers by multiplying one another make certain numbers, the numbers so produced will be equal to one another.

(The Elements: Book VII: Proposition $16$)

Proof

Let $A, B$ be two (natural) numbers, and let $A$ by multiplying $B$ make $C$, and $B$ by multiplying $A$ make $D$.

We need to show that $C = D$.

We have that $A \times B = C$.

So $B$ measures $C$ according to the units of $A$.

But the unit $E$ also measures $A$ according to the units in it.

So $E$ measures $A$ the same number of times that $B$ measures $C$.

Therefore from Book VII Proposition 15: Alternate Ratios of Multiples‎ $E$ measures $B$ the same number of times that $A$ measures $C$.

We also have that $A$ measures $D$ according to the units of $B$.

But the unit $E$ also measures $B$ according to the units in it.

Therefore from Book VII Proposition 15: Alternate Ratios of Multiples‎ $E$ measures $B$ the same number of times that $A$ measures $D$.

But we also have that $E$ measures $B$ the same number of times that $A$ measures $C$.

So $A$ measures $C$ and $D$ the same number of times.

Therefore $C = D$.

$\blacksquare$

Historical Note

This is Proposition 16 of Book VII of Euclid's The Elements.