SQUARING THE CIRCLE. 319 



the successive odd numbers. Add together the alter- 

 nate ones beginning with the first, which, of course, 

 is unity. Add together the remainder. Subtract 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 convenient methods for de- 

 termining the required relation ; but the method is 

 strictly exact. 



The largest circle we have much to do with in 

 scientific questions is the earth's equator. As a matter 

 of. curiosity, we may inquire what the circumference 

 of the earth's orbit is ; but as we are far from being 

 sure of the exact length of the radius of that orbit 

 (that is, of the earth's distance from the sun), it is 

 clear that we do not need a very exact relation 

 between the circumference and the diameter in deal- 

 ing with that enormous circle. Confining ourselves, 

 therefore, to the circle of the earth's equator, let us 

 see what exactness we seem to require. We will 

 suppose for a moment that it is possible to measure 

 round the earth's equator without losing count of a 

 single yard, and that we want to gather from our 

 estimate what the diameter of this great circle may 

 be. This seems, indeed, the only .use to which, in 



