SQUARING THE CIRCLE. 293 



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 dia- 

 meter of the circle. If this were so, scientific so- 

 cieties might well hold out a reward to any one who 

 could enlighten them ; for the determination of this 

 relation (Avith satisfactory exactitude) may be held to 

 lie at the foundation of the whole of our modern 

 system of mathematics. We need hardly say that no 

 doubt whatever rests on the matter. A hundred dif- 

 ferent methods are known to mathematicians by which 

 the circumference may be calculated from the dia- 

 meter 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. 

 AVe 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 



