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Euclid: Elementa

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Euclid: Elementa
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PROPOSITION 26. 
 
 
Similar plane numbers have to one another the ratio which a square number has to a square number. 
 
 
Let A, B be similar plane numbers;  I say that A has to B the ratio which a square number has to a square number. 
   
   
For, since A, B are similar plane numbers, therefore one mean proportional number falls between A, B. [VIII. 18]  Let it so fall, and let it be C; and let D, E, F, the least numbers of those which have the same ratio with A, C, B, be taken; [VII. 33 or VIII. 2]  therefore the extremes of them D, F are square. [VIII. 2, Por.]  And since, as D is to F, so is A to B, and D, F are square,  therefore A has to B the ratio which a square number has to a square number.  Q. E. D. 
           
           
 
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