SQUARING THE CIRCLE. 291 



difficulty had been propounded, actually set to work 

 to invent an arrangement for measuring the circum- 

 ference 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 

 leno-th of a circle's circumference when the diameter 



e 



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 twenty-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 

 some such sort that many persons have set themselves. 



u 2 



