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Proposition 38

If a number has any part whatever, then it is measured by a number called by the same name as the part.
Let the number A have any part whatever, B, and let C be a number called by the same name as the part B. I say that C measures A.
Since B is a part of A called by the same name as C, and the unit D is also a part of C called by the same name as it, therefore the part B of A is the same part of the unit D of the number C. Therefore the unit D measures the number C the same number of times that B measures A. |
Therefore, alternately, the unit D measures the number B the same number of times that C measures A. Therefore C measures A. | VII.15
Therefore, if a number has any part whatever, then it is measured by a number called by the same name as the part.
Q.E.D.

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This proposition is used in the proof of the next proposition.

Book VII Introduction - Proposition VII.37 - Proposition VII.39.

© 1996
D.E.Joyce
Clark University