THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 33 



existing compact series of " rational " numbers.* It appears to 

 be necessary to have recourse to a totally new series of concepts 

 which, though derived from the rational numbers, are not 

 identical with them. These are the " real " numbers of the 

 mathematician. Their nature may be indicated^ follows : 



123456 7 8 9 10 11 12 13 



In the line above are printed the symbols of the integers 

 from 1 to 13, irregularly spaced so as to emphasise the fact 

 that we are required to attend only to their order, not to any 

 notion of distance of a spatial kind. Hitherto such a number 

 as nine has been regarded as the common property of an 

 indefinite number of " similar classes," each of which would, in 

 ordinary parlance, be said to contain liine terms. But, given 

 the definite ordinal series of integers, thus defined, it becomes 

 possible to regard nine as defining a class of integers, namely, 

 the class containing the integers 1 to 8 which precede it. In 

 this way each of the integers can be correlated with a class of 

 integers, so that " counting," say, of the persons round a dinner 

 table, may be conceived as correlating them, one by one, with 

 the classes of integers with which the symbols 1, 2, 3, &c., are 

 also correlated. 



It is clear that nothing in this will become invalid if we 

 include in the classes of numbers, which are henceforward to be 

 our instruments of correlation, the 'whole of the rational numbers 

 which have received rigorous definition. Thus we can suppose 

 our dining guests to be correlated, one by one, with the 

 successive classes of rationals less than one, less than two, less 

 than three, &c. ; and there will be no danger lest the correlation 

 should be incomplete or ambiguous. It is true that our 

 ingenuity might run the risk of comparison with that shown 

 by the White Knight if we deliberately employed so far-fetched 

 a method in so simple a case as the counting of guests round 



* Kussell, op. cit.j p. 270. 



