SQUARING THE CIRCLE. 289 



its nature is. One ingenious workman, to whom the 

 difficulty had been propounded, actually set to wcrk to 

 invent an arrangement for measuring the circumference 

 of the circle ; and was perfectly satisfied that he had 

 thus solved a problem which had mastered all the 

 mathematicians of ancient and modern times. That 

 we may not fall into a similar error, let us clearly 

 understand what it is that is required for the solution 

 of the problem of ' squaring the circle.' 



To begin with, we must note that the term ' squar- 

 ing the circle ' is rather a misnomer ; because the true 

 problem to be solved is the determination of the 

 length of a circle's circumference when the diameter 

 is known. Of course, the solution of this problem, 

 or, as it is termed, the rectification of the circle, in- 

 volves the solution of the other, or the quadrature of 

 the circle. But it is well to keep the simpler issue 

 before us. 



Many have supposed that there exists some exact 

 relation between the circumference and the diameter of 

 the circle, and that the problem to be solved is the 

 determination of this relation. Suppose, for example, 

 that the approximate relation discovered by Archi- 

 medes (who found, that if a circle's diameter is repre- 

 sented by seven, the circumference may be almost 

 exactly represented by tiuenty-two) were strictly cor- 

 rect, and that Archimedes had proved it to be so; 

 then, according to this view, he would have solved the 

 great problem ; and it is to determine a relation of 

 seme such sort that many persons have set themselves. 



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