PROPOSITION 16.

If two numbers by multiplying one another make certain numbers, the numbers so produced[1] 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.

A B C D E
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.
Therefore etc. Q. E. D.

Dependency Graph

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Required for

VII.18, VII.19, VII.21, VII.24, VII.25, VII.26, VII.27, VII.30, VII.33, VII.34, VII.36, VII.39, VIII.1, VIII.2, VIII.3, VIII.4, VIII.5, VIII.6, VIII.7, VIII.8, VIII.9, VIII.10, VIII.11, VIII.12, VIII.14, VIII.15, VIII.16, VIII.17, VIII.19, VIII.20, VIII.21, VIII.22, VIII.23, VIII.24, VIII.25, VIII.26, VIII.27, IX.1, IX.2, IX.3, IX.4, IX.5, IX.6, IX.8, IX.9, IX.10, IX.12, IX.13, IX.14, IX.15, IX.16, IX.17, IX.18, IX.19, IX.32, IX.36, X.9, X.10, X.12, X.13, X.17, X.18, X.22, X.23, X.24, X.25, X.26, X.27, X.28, X.29, X.30, X.31, X.32, X.33, X.34, X.35, X.36, X.37, X.38, X.39, X.40, X.41, X.42, X.43, X.44, X.45, X.46, X.47, X.48, X.49, X.50, X.51, X.52, X.53, X.54, X.55, X.56, X.57, X.58, X.59, X.60, X.61, X.62, X.63, X.64, X.65, X.66, X.67, X.68, X.69, X.70, X.71, X.72, X.73, X.75, X.76, X.77, X.78, X.79, X.80, X.81, X.82, X.83, X.84, X.85, X.86, X.87, X.88, X.89, X.90, X.91, X.92, X.93, X.94, X.95, X.96, X.97, X.98, X.99, X.100, X.101, X.102, X.103, X.104, X.105, X.106, X.107, X.108, X.109, X.110, X.111, X.112, X.113, X.114, XIII.6, XIII.11, XIII.16, XIII.17, XIII.18

See Commentary on Proposition VII.16.