SQUARING THE CIRCLE. 295 



diameter of a circle, to determine, by a geometrical 

 construction, in which only straight lines and circles 

 shall be made use of, the side of a square, equal in area 

 to the circle. As I have said, the problem is solved, 

 if, by a construction of the kind described, we can 

 determine the length of the circumference; because 

 then the rectangle under half this length and the 

 radius is equal in area to the circle, and it is a 

 simple problem to describe a square equal to a given 

 rectangle. 



To illustrate the kind of construction required, I 

 give an approximate solution which is remarkably 

 simple, and, so far as I am aware, not generally 

 known. Describe a square about the given circle, 

 touching it at the ends of two diameters, AOB, COB, 

 at right angles to each other, and join CA; let COAE 

 be one of the quarters of the circumscribing square, 

 and from E draw EGr, cutting off from AO a fourth 

 part AGr of its length, and from AC the portion AH. 

 Then three sides of the circumscribing square together 

 with AH are very nearly equal to the circumference 

 of the circle. The difference is so small, that in a 

 circle two feet in diameter, it would be less than the 

 two-hundredth part of an inch. If this construc- 

 tion were exact, the great problem would have been 

 solved. 



One point, however, must be noted ; the circle is 

 of all curved lines the easiest to draw by mechanical 

 means. But there are others which can be so drawn. 

 And if such curves as these be admitted as available, 



