#### PROPOSITION 16.

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

Let A, B be two numbers, and let A by multiplying B make C, and B by multiplying A make D; I say that C is equal to D.

For, since A by multiplying B has made 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 by multiplying A has made 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 measured the number B the same number of times that A measures C;

therefore A measures each of the numbers C, D the same number of times.

Therefore C is equal to D. Q. E. D. 1

1 The Greek has , “the (numbers) produced from them.” By “from them” Euclid means “from the original numbers,” though this is not very clear even in the Greek. I think ambiguity is best avoided by leaving out the words.

This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 United States License.

An XML version of this text is available for download, with the additional restriction that you offer Perseus any modifications you make. Perseus provides credit for all accepted changes, storing new additions in a versioning system.

load focus Greek (J. L. Heiberg, 1883)
hide Display Preferences
 Greek Display: Unicode (precombined) Unicode (combining diacriticals) Beta Code SPIonic SGreek GreekKeys Latin transliteration Arabic Display: Unicode Buckwalter transliteration View by Default: Original Language Translation Browse Bar: Show by default Hide by default