CH. Ill] 



GEOMETKICAL RECREATIONS 



53 



together form a figure containing 65 such small squares*. This 

 is effected by cutting the original square into four pieces in the 

 manner indicated by the thick lines in the first figure. If these 

 four pieces are put together in the shape of a rectangle in the 

 way shown in the second figure it will appear as if this rectangle 

 contains 65 of the small squares. 



This phenomenon, which in my experience non-mathema- 

 ticians find perplexing, is due to the fact that the edges of 

 the four pieces of paper, which in the second figure lie along 



the diagonal AB, do not coincide exactly in direction. In 

 reality they include a small lozenge or diamond-shaped figure, 

 whose area is equal to that of one of the 64 small squares in 

 the original square, but whose length AB is much greater than 

 its breadth. The diagrams show that the angle between the 

 two sides of this lozenge which meet at A is tan _I | — tan -1 |, 

 that is, is tan -1 ^-, which is less than \\°. To enable the eye 

 to distinguish so small an angle as this the dividing lines in the 

 first figure would have to be cut with extreme accuracy and the 

 pieces placed together with great care. 



This paradox depends upon the relation 5x13 — 8 2 =1. 

 Similar results can be obtained from the formulae 



13 x 34 - 21 a = 1, 34x89-55 a =l,...; 

 or from the formulae 



5 a -3x8 = l, 13 a -8x21 = l, 34 a -21 x55 = l 



* I do not know who discovered this paiadoz. It is given in various modern 

 books, but I cannot find an earlier reference to it than one in the Zeitschri/t 

 filr Mathematik und Physik, Leipzig, 1868, vol. xin, p. 162. Some similar 

 paradoxes were given by Ozanam, 1803 edition, vol. x, p. 299. 



