SQUARING THE CIRCLE. 317 



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. 

 Now, undoubtedly, if any relation of this sort could be 

 established, the problem would be solved ; but, as a 

 matter of fact, no such relation exists, and the solu- 

 tion of the problem does not require that there should 

 be any relation of the sort. For example, we do not 

 look on the determination of the diagonal of a square 

 (whose side is known) as an insoluble, or as otherwise 

 than a very simple problem. Yet in this case no 

 exact relation exists. We cannot possibly express 

 both the side and the diagonal of a square in whole 

 numbers, no matter what unit of measurement we 

 adopt : or, to put the matter in another way, we can- 

 not possibly divide both the side and the diagonal 

 into equal parts (which shall be the same along each), 

 no matter how small we take the parts. If we divide 

 the side into 1,000 parts, there will be 1,414 such 

 parts, and a piece over, in the diagonal ; if we divide 

 the side into 10,000 parts, there will be 14,142, and 

 still a little piece over, in the diagonal ; and so on for 

 ever. Similarly, the mere fact that no exact relation 

 exists between the diameter and the circumference of 



