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Euclid: Elementa

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Euclid: Elementa
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PROPOSITION 32. 
 
 
Each of the numbers which are continually doubled beginning from a dyad is even-times even only. 
 
 
For let as many numbers as we please, B, C, D, have been continually doubled beginning from the dyad A;  I say that B, C, D are eventimes even only. 
   
   
Now that each of the numbers B, C, D is even-times even is manifest;  for it is doubled from a dyad.  I say that it is also even-times even only. 
     
     
For let an unit be set out.  Since then as many numbers as we please beginning from an unit are in continued proportion,  and the number A after the unit is prime,  therefore D, the greatest of the numbers A, B, C, D, will not be measured by any other number except A, B, C. [IX. 13]  And each of the numbers A, B, C is even;  therefore D is even-times even only. [VII. Def. 8]  Similarly we can prove that each of the numbers B, C is even-times even only.  Q. E. D. 
               
               
 
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