1872.] On the Duplication of the Cube and Quadrature of the Circle. 525 



XXIII. " Approximate Geometrical Solutions of the Problems of the 

 Duplication of the Cube and of the Quadrature of the Circle." 

 By William Hayden. Communicated by Prof. G. G. Stokes, 

 Sec. R.S. Received November 30, 1871. 



I. Duplication of the Cube. 

 Let AB be the given cube root. Erect the perpendicular B K ; bisect 

 A B in C, and with radius C B describe the arcs intersecting at E ; let fall 

 the perpendicular JE D and trisect C D ; then, with radius B F from E as a 

 centre, describe the arc H I cutting B K in L ; join A L, which will be 

 nearly equal to the cube root of double the cube of A B, the amount of 

 error being very small, which is proved as under. 



Let AB = 3. Then 



CB=l-5, 



BF=l-25, 



BD= -75, 



DE=sin60°xCB, 

 <y(BF 2 — BD 2 ) = 1, 

 (sin 60° x CB) + 1=BL, 

 ^/(AB 2 + BL 2 )=AL=3-7796264 . . . 



3/2=1-2599210 

 gAL= 1-2598754 

 Error -0000456 



II. Quadrature of the Circle. 



In and about a given circle, as A fE e, draw the inscribed and tangent 

 squares in the manner indicated by the figure, with their diagonals. Draw 

 the diameter A E bisecting the opposite sides of the said squares ; bisect 

 A B in C, and D E in F ; set off from E and D respectively E G and D I 

 equal to D F, E F and join G I ; from E let fall E H perpendicular to G I, 

 and with radius E H describe the circle H K ; draw the line C L touching 



