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Euclid: Elementa

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Euclid: Elementa
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PROPOSITION 18. 
 
 
If two numbers by multiplying any number make certain numbers, the numbers so produced will have the same ratio as the multipliers. 
 
 
For let two numbers A, B by multiplying any number C make D, E;  I say that, as A is to B, so is D to E. 
   
   
For, since A by multiplying C has made D,  therefore also C by multiplying A has made D. [VII. 16]  For the same reason also C by multiplying B has made E.  Therefore the number C by multiplying the two numbers A, B has made D, E.  Therefore, as A is to B, so is D to E. [VII. 17]   
           
           
 
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