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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.
| 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.
Therefore, alternately, the unit E measures the number B the same number of times that A measures C. | VII.15
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.

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This proposition describes the commutativity mentioned in the last proposition more explicitly, ab = ba. It is used in VII.18 and a few others in Book VII.

Book VII Introduction - Proposition VII.15 - Proposition VII.17.

© 1996
D.E.Joyce
Clark University