THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 31 



series. This sense, which it has been reserved for the modern 

 philosophical mathematicians to discover, must be briefly 

 explained. 



A P Q K S T 



-B 



P 



The straight line AB may be supposed to be the road 

 which an Oriental despot has driven across his empire from 

 one great city to another, while the shorter line ab may be 

 taken as a map or plan of this road showing at the points 

 p, q, r, s, t, the relative positions of the towns and villages, 

 P, Q, E, S, T, to be met with on the way. It will be 

 admitted that though the imperfection of human powers 

 would make it practically impossible to represent every feature 

 of the road AB on the plan ab, yet the difficulty is only a 

 practical one, and that there is no detail of the road that could 

 not conceivably be represented on the plan. The obvious con- 

 sequence of this admission is that every point on the line AB 

 can be correlated with a point on the line ab. According to 

 the idiom which we have employed before, the two series of 

 points are similar. But the line ab could be placed so that its 

 points would become actually a part of the points of the 

 line AB. Thus we should be faced by the disturbing conX 

 elusion that a part of these points may be similar to the whole/ 

 The number of points, then, on the line AB is such that 

 a part of these points is capable of one-one correlation with 

 the whole. But however far we carried the assignment 

 of terms of our compact number series to the points on 

 the line AB, it is perfectly certain that some of these 

 numbers could not be correlated one by one with the whole. 

 We are driven, then, either to the conclusion reached by 

 such philosophers as Mr. Bradley that space is an irrational 

 idea, " riddled with contradictions/' or else to the provisional 

 adoption of the concept of a new type of numbers whose 



