SQUARING THE CIRCLE. 297 



meter of a circle, to determine) by a geometrical con- 

 struction, in which only straight lines and circles shall 

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

 the circle. As we 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, we 

 give an approximate solution which is remarkably 

 simple, and, so far as we are aware, not generally 

 known. Describe a square about the given circle, touch- 

 ing it at the ends of two diameters, AOB, COD, at 

 right angles to each other, and join C A ; let COAE be 

 one of the quarters of the circumscribing square, and 

 from E draw EG, cutting off from AO a fourth part 

 AG 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 construct- 

 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, 



