THE SQUARING OF THE CIRCLE. 



I J7 



icable mania for solving the quadrature of the circle has also its 

 serious side. Circle-squarers are not always so self- satisfied as 

 the author of the book above mentioned. They often see, or at 

 least divine, the insuperable difficulties that tower up before them, 

 and the conflict between their aspirations and their performances, 

 the consciousness that the problem they long to solve they are un- 

 able to solve, darkens their soul and, lost to the world, they be- 

 come interesting subjects for the science of psychiatry. 



n. 



NATURE OF THE PROBLEM. 



It is easy to determine the length of the radius of a circle, or 

 the length of its diameter, which must be double that of the radius; 

 and the question next arises, what is the number that tells how many 

 times larger the circumference of the circle, that is the length of the 

 circular line, is than its radius or its diameter. From the fact that 

 all circles have the same shape it follows that this proportion will 

 be the same for all circles both large and small. Now, since the 

 time of Archimedes, all civilised nations that have cultivated math 

 ematics have denoted the number that tells how many times larger 

 the circumference of a circle is than the diameter by the symbol n, 

 the Greek initial letter of the word periphery.* To compute n, 

 therefore, means to calculate how many times larger the circumfer- 

 ence of a circle is than its diameter. This calculation is called 

 "the numerical rectification of the circle." 



Next to the calculation of the circumference, the calculation of 

 the superficial contents of a circle by means of its radius or diam- 

 eter is perhaps most important ; that is, the computation of how 

 great an area that part of a plane which lies within a circle meas- 

 ures. This calculation is called the "numerical quadrature." It 

 depends, however, upon the problem of numerical rectification ; 

 that is, upon the calculation of the magnitude of n. For it is de- 

 monstrated in elementary geometry, that the area of a circle is 



*The Greek symbol TT was first employed by W. Jones in 1706 and did not 

 Dome into general use until about the middle of the eighteenth century through 

 thf works of Euler. Trans 



