Proposition 16

If two numbers multiplied by one another make certain numbers, then the numbers so produced equal one another.

Let A and B be two numbers, and let A multiplied by B make C, and B multiplied by A make D.

I say that C equals D.

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VII.Def.15

Since A multiplied by B makes C, therefore B measures C according to the units in A.

But the unit E also measures the number A according to the units in it, therefore the unit E measures A the same number of times that B measures C.

VII.15

Therefore, alternately, the unit E measures the number B the same number of times that A measures C.

Again, since B multiplied by A makes D, therefore A measures D according to the units in B. But the unit E also measures B according to the units in it, therefore the unit E measures the number B the same number of times that A measures D.

But the unit E measures the number B the same number of times that A measures C, therefore A measures each of the numbers C and D the same number of times.

Therefore C equals D.

Therefore, if two numbers multiplied by one another make certain numbers, then the numbers so produced equal one another.

Q.E.D.

Guide

 This proposition states the commutativity of multiplication of formal numbers, ab = ba.

Outline of the proof

Let a = nu and b = mu. Then by the definition of multiplication of numbers, ab = nb, and ba = ma. By the preceding proposition, n(mu) = m(nu). Therefore, ab = ba.

Use of Proposition 16

This proposition is used in VII.18 and a few others in Book VII.