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Euclid: Elementa

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Euclid: Elementa
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PROPOSITION 20. 
 
 
If a rational area be applied to a rational straight line, it produces as breadth a straight line rational and commensurable in length with the straight line to which it is applied. 
 
 
For let the rational area AC be applied to AB, a straight line once more rational in any of the aforesaid ways, producing BC as breadth;  I say that BC is rational and commensurable in length with BA. 
   
   
For on AB let the square AD be described;  therefore AD is rational. [X. Def. 4]  But AC is also rational;  therefore DA is commensurable with AC.  And, as DA is to AC, so is DB to BC. [VI. 1]  Therefore DB is also commensurable with BC; [X. 11]  and DB is equal to BA;  therefore AB is also commensurable with BC.  But AB is rational;  therefore BC is also rational and commensurable in length with AB. 
                   
                   
Therefore etc. 
 
 
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