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Euclid: Elementa
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PROPOSITION 5. 
 
 
Commensurable magnitudes have to one another the ratio which a number has to a number. 
 
 
Let A, B be commensurable magnitudes;  I say that A has to B the ratio which a number has to a number. 
   
   
For, since A, B are commensurable, some magnitude will measure them.  Let it measure them, and let it be C.  And, as many times as C measures A, so many units let there be in D;  and, as many times as C measures B, so many units let there be in E. 
       
       
Since then C measures A according to the units in D, while the unit also measures D according to the units in it,  therefore the unit measures the number D the same number of times as the magnitude C measures A;  therefore, as C is to A, so is the unit to D; [VII. Def. 20]  therefore, inversely, as A is to C, so is D to the unit. [cf. V. 7, Por.]  Again, since C measures B according to the units in E, while the unit also measures E according to the units in it,  therefore the unit measures E the same number of times as C measures B;  therefore, as C is to B, so is the unit to E.  But it was also proved that, as A is to C, so is D to the unit;  therefore, ex aequali, as A is to B, so is the number D to E. [V. 22] 
                 
                 
Therefore the commensurable magnitudes A, B have to one another the ratio which the number D has to the number E.  Q. E. D. 
   
   
 
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