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Proposition 35

If two numbers measure any number, then the least number measured by them also measures the same.
Let the two numbers A and B measure any number CD, and let E be the least that they measure. I say that E also measures CD.
If E does not measure CD, let E, measuring DF, leave CF less than itself. Now, since A and B measure E, and E measures DF, therefore A and B also measure DF. But they also measure the whole CD, therefore they measure the remainder CF which is less than E, which is impossible. |
Therefore E cannot fail to measure CD. Therefore it measures it.
Therefore, if two numbers measure any number, then the least number measured by them also measures the same.
Q.E.D.

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This proposition is used in the next one and in VIII.4.

Book VII Introduction - Proposition VII.34 - Proposition VII.36.

© 1996
D.E.Joyce
Clark University