SQUARING THE CIRCLE. 2pl 



and proceed with the division thus indicated. The 

 result, 3-1415929 . . . ., expresses the circumfer- 

 ence of a circle whose diameter is 1, correctly to 

 the sixth decimal place, the true relation being 

 3-14159265. 



Again, many people imagine that mathematicians 

 are still in a state of uncertainty as to the relation 

 which exists between the circumference and the 

 diameter of the circle. If this were so, scientific 

 societies might well hold out a reward to anyone who 

 could enlighten them; for the determination of this 

 relation (with satisfactory exactitude) may be held 

 to lie at the foundation of the whole of our modern 

 system of mathematics. I need hardly say that no 

 doubt whatever rests on the matter. A hundred 

 different methods are known to mathematicians by 

 which the circumference may be calculated from the 

 diameter with any required degree of exactness. Here 

 is a simple one, for example : Take any number of 

 the fractions formed by putting one as a numerator 

 over the successive odd numbers. Add together the 

 alternate ones beginning with the first, which, of 

 course, is unity. Add together the remainder. Sub- 

 tract the second sum from the first. The remainder 

 will express the circumference (the diameter being 

 taken as unity) to any required degree of exactness. 

 We have merely to take enough fractions. The 

 process would, of course, be a very laborious one, 

 if great exactness were required, and as a matter of 

 fact mathematicians have made use of much more 



