526 Duplication of the Cube and Quadrature of the Circle, [June 20, 



the circle H K, and cutting the circle A /E e in P ; then through the point 

 P, parallel to A B, draw M N terminating in its intersections with th.3 



diagonals, as shown by the diagram, which will be one side of a square very 

 nearly equal to the circle A fE e. 



[The author has appended a mathematical calculation proving, in a 

 somewhat indirect manner, that his construction gives an exceedingly close 

 approximation. The construction leads directly to the following result : — 



Let the radius of the circle be taken as unity, and let 5 be a side of the 

 square given by the construction ; then 



* — - > 



1+m 2 



where 



m = 2+ g_'v ^ r =| (VI »- V5) 



(r is the radius of the circle H K, and m the tangent of the inclination of 

 CLto AE). 



Mr. Hayden has reduced these expressions to numbers, with the follow- 

 ing result : — 



r= -13893145240028844 . . ., 

 m= -322999477624 . . 



5=1-7724538677 . . 



5 2 = 3-141592713..., 



instead of 3-141592653 . . . 



—September. 1872 G. G. S.] 



