.'\ 34 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 



I a dinner table ; but we should reap the reward of it when we 



^ came to deal with such questions as the correlation of the 



\ members of some convenient series with such spatial magni- 



, tudes as the sides of the triangle recently under consideration. 



For it will refdily be .seen that if we correlate the magnitude 



of the sides with the -class of rationals less than one, we may 



quite unambiguously correlate the magnitude of the hypo- 



"~fchenuse with the class of ratibnals whose squares are less than 



two since no doubt can ever arise as to whether a given 



rational number falls in this class, nor confusion of limits with 



another class be feared. In this way the " irrational" numbers 



which are necessary to enable us to correlate the terms of the 



number series with all the points of space that anyone has 



- ever found himself led to postulate can be supplied. 



It need hardly be added that by an appropriate symbolism 

 Jwhich is strictly a non-logical matter these new members can 

 x be made to perform the services in which we have seen the 

 ^ great practical importance of numbers to lie. 



The difficulties that beset the measurement of Space were 

 imported into Time when attempts were made to consider the 

 question of Motion philosophically. Hence arose the famous 

 paradoxes of Zeno. But there is no reason why Time should not 

 be conceived as having the same kind of " continuity " as Space, so 

 that for the correlation of the moments of Time, as for the con-e- 

 lation of points of Space, we must have recourse to the " real " 

 as distinguished from the "rational" numbers. Motion then 

 loses its "contradictory" character; for we can regard a 

 " material particle " as simply a means by which correlation is 

 set up between (1) a single point of Space and a number of 

 successive moments of Time, if the particle is " at rest " ; and 

 i, (2) a series of points and moments, one by one, if the particle is 

 "in motion."* 



For details see Euasell, Ch. LIV. 



