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Euclid: Elementa

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Euclid: Elementa
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PROPOSITION 38. 
 
 
If a number have any part whatever, it will be measured by a number called by the same name as the part. 
 
 
For 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. 
   
   
For, 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, whatever part the unit D is of the number C, the same part is B of A also;  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. [VII. 15]  Therefore C measures A.  Q. E. D. 
           
           
 
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