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Euclid: Elementa

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Euclid: Elementa
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PROPOSITION 33. 
 
 
If a number have its half odd, it is even-times odd only. 
 
 
For let the number A have its half odd;  I say that A is even-times odd only. 
   
   
Now that it is even-times odd is manifest;  for the half of it, being odd, measures it an even number of times. [VII. Def. 9]  I say next that it is also even-times odd only. 
     
     
For, if A is even-times even also, it will be measured by an even number according to an even number; [VII. Def. 8]  so that the half of it will also be measured by an even number though it is odd: which is absurd.  Therefore A is even-times odd only.  Q. E. D. 
       
       
 
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