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Euclid: Elementa
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PROPOSITION 18. 
 
 
Given two numbers, to investigate whether it is possible to find a third proportional to them. 
 
 
Let A, B be the given two numbers, and let it be required to investigate whether it is possible to find a third proportional to them. 
 
 
Now A, B are either prime to one another or not.  And, if they are prime to one another, it has been proved that it is impossible to find a third proportional to them. [IX. 16] 
   
   
Next, let A, B not be prime to one another, and let B by multiplying itself make C.  Then A either measures C or does not measure it.  First, let it measure it according to D;  therefore A by multiplying D has made C.  But, further, B has also by multiplying itself made C;  therefore the product of A, D is equal to the square on B.  Therefore, as A is to B, so is B to D; [VII. 19]  therefore a third proportional number D has been found to A, B. 
               
               
Next, let A not measure C;  I say that it is impossible to find a third proportional number to A, B.  For, if possible, let D, such third proportional, have been found.  Therefore the product of A, D is equal to the square on B.  But the square on B is C;  therefore the product of A, D is equal to C.  Hence A by multiplying D has made C;  therefore A measures C according to D.  But, by hypothesis, it also does not measure it: which is absurd.  Therefore it is not possible to find a third proportional number to A, B when A does not measure C.  Q. E. D. 
                     
                     
 
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