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