164 SCIENTIFIC METHOD IN PHILOSOPHY 



The problem first raised by the discovery of incom- 

 mensurables proved, as time went on, to be one of the 

 most severe and at the same time most far-reaching 

 problems that have confronted the human intellect in its 

 endeavour to understand the world. It showed at once 

 that numerical measurement of lengths, if it was to be 

 made accurate, must require an arithmetic more advanced 

 and more difficult than any that the ancients possessed. 

 They therefore set to work to reconstruct geometry on 

 a basis which did not assume the universal possibility of 

 numerical measurement a reconstruction which, as may 

 be seen in Euclid, they effected with extraordinary skill 

 and with great logical acumen. The moderns, under 

 the influence of Cartesian geometry, have reasserted the 

 universal possibility of numerical measurement, extending 

 arithmetic, partly for that purpose, so as to include what 

 are called " irrational " numbers, which give the ratios 

 of incommensurable lengths. But although irrational 

 numbers have long been used without a qualm, it is only 

 in quite recent years that logically satisfactory definitions 

 of them have been given. With these definitions, the 

 first and most obvious form of the difficulty which con- 

 fronted the Pythagoreans has been solved ; but other 

 forms of the difficulty remain to be considered, and it is 

 these that introduce us to the problem of infinity in its 

 pure form. 



We saw that, accepting the view that a length is com- 

 posed of points, the existence of incommensurables proves 

 that every finite length must contain an infinite number 

 of points. In other words, if we were to take away points 

 one by one, we should never have taken away all the 

 points, however long we continued the process. The 

 number of points, therefore, cannot be counted^ for counting 

 is a process which enumerates things one by one. The 



