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Proposition 37

If a number is measured by any number, then the number which is measured has a part called by the same name as the measuring number.
Let the number A be measured by any number B. I say that A has a part called by the same name as B.
Let there be as many units in C as the times that B measures A. Since B measures A according to the units in C, and the unit D also measures the number C according to the units in it, therefore the unit D measures the number C the same number of times as B measures A. |
Therefore, alternately, the unit D measures the number B the same number of times as C measures A. Therefore, whatever part the unit D is of the number B, the same part is C of A also. But the unit D is a part of the number B called by the same name as it, therefore C is also a part of A called by the same name as B, so that A has a part C which is called by the same name as B. | VII.15
Therefore, if a number is measured by any number, then the number which is measured has a part called by the same name as the measuring number.
Q.E.D.

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This proposition is used in the proof of proposition VII.39.

Book VII Introduction - Proposition VII.36 - Proposition VII.38.

© 1996
D.E.Joyce
Clark University