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Euclid: Elementa

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Euclid: Elementa
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PROPOSITION 17. 
 
 
If a number by multiplying two numbers make certain numbers, the numbers so produced will have the same ratio as the numbers multiplied. 
 
 
For let the number A by multiplying the two numbers B, C make D, E;  I say that, as B is to C, so is D to E. 
   
   
For, since A by multiplying B has made D,  therefore B measures D according to the units in A.  But the unit F also measures the number A according to the units in it;  therefore the unit F measures the number A the same number of times that B measures D.  Therefore, as the unit F is to the number A, so is B to D. [VII. Def. 20]  For the same reason, as the unit F is to the number A, so also is C to E;  therefore also, as B is to D, so is C to E.  Therefore, alternately, as B is to C, so is D to E. [VII. 13]  Q. E. D. 
                 
                 
 
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