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