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Euclid: Elementa

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Euclid: Elementa
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PROPOSITION 16. 
 
 
If two numbers be prime to one another, the second will not be to any other number as the first is to the second. 
 
 
For let the two numbers A, B be prime to one another;  I say that B is not to any other number as A is to B. 
   
   
For, if possible, as A is to B, so let B be to C.  Now A, B are prime, primes are also least, [VII. 21]  and the least numbers measure those which have the same ratio the same number of times, the antecedent the antecedent and the consequent the consequent; [VII. 20]  therefore A measures B as antecedent antecedent.  But it also measures itself;  therefore A measures A, B which are prime to one another: which is absurd.  Therefore B will not be to C, as A is to B.  Q. E. D. 
               
               
 
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