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Euclid: Elementa

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Euclid: Elementa
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PROPOSITION 39. 
 
 
To find the number which is the least that will have given parts. 
 
 
Let A, B, C be the given parts;  thus it is required to find the number which is the least that will have the parts A, B, C. 
   
   
Let D, E, F be numbers called by the same name as the parts A, B, C, and let G, the least number measured by D, E, F, be taken. [VII. 36] 
 
 
Therefore G has parts called by the same name as D, E, F. [VII. 37]  But A, B, C are parts called by the same name as D, E, F;  therefore G has the parts A, B, C.  I say next that it is also the least number that has. 
       
       
For, if not, there will be some number less than G which will have the parts A, B, C.  Let it be H.  Since H has the parts A, B, C, therefore H will be measured by numbers called by the same name as the parts A, B, C. [VII. 38]  But D, E, F are numbers called by the same name as the parts A, B, C; therefore H is measured by D, E, F.  And it is less than G: which is impossible.  Therefore there will be no number less than G that will have the parts A, B, C.  Q. E. D. 
             
             
BOOK VΙΙI. 
 
 
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