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Euclid: Elementa

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Euclid: Elementa
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PROPOSITION 83. 
 
 
To a straight line which produces with a rational area a medial whole only one straight line can be annexed which is incommensurable in square with the whole straight line and which with the whole straight line makes the sum of the squares on them medial, but twice the rectangle contained by them rational. 
 
 
Let AB be the straight line which produces with a rational area a medial whole, and let BC be an annex to AB; therefore AC, CB are straight lines incommensurable in square which fulfil the given conditions. [X. 77]  I say that no other straight line can be annexed to AB which fulfils the same conditions. 
   
   
For, if possible, let BD be so annexed;  therefore AD, DB are also straight lines incommensurable in square which fulfil the given conditions. [X. 77]  Since then, as in the preceding cases, the excess of the squares on AD, DB over the squares on AC, CB is also the excess of twice the rectangle AD, DB over twice the rectangle AC, CB, while twice the rectangle AD, DB exceeds twice the rectangle AC, CB by a rational area,  for both are rational, therefore the squares on AD, DB also exceed the squares on AC, CB by a rational area: which is impossible,  for both are medial. [X. 26] 
         
         
Therefore no other straight line can be annexed to AB which is incommensurable in square with the whole and which with the whole fulfils the aforesaid conditions; therefore only one straight line can be so annexed.  Q. E. D. 
   
   
 
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