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Euclid: Elementa

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Euclid: Elementa
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PROPOSITION 30. 
 
 
If an odd number measure an even number, it will also measure the half of it. 
 
 
For let the odd number A measure the even number B;  I say that it will also measure the half of it. 
   
   
For, since A measures B, let it measure it according to C;  I say that C is not odd. 
   
   
For, if possible, let it be so.  Then, since A measures B according to C,  therefore A by multiplying C has made B.  Therefore B is made up of odd numbers the multitude of which is odd.  Therefore B is odd: [IX. 23] which is absurd, for by hypothesis it is even.  Therefore C is not odd; therefore C is even.  Thus A measures B an even number of times.  For this reason then it also measures the half of it.  Q. E. D. 
                 
                 
 
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