32 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD, 



distinctive property is to be that they are the numbers to be 

 assigned to the terms of a series when the part is similar to the- 

 whole. Fqr such numbers we may reserve the term infinite. 

 Fortunately we are not compelled at this stage in the progress- 

 of science to accept the former of the alternatives just offered. 

 The consequences of the admission of a type of " infinite " 

 numbers having the property explained have been carefully 

 traced out by competent thinkers, and the contradictions which 

 would, on Mr. Bradley's principles, condemn the concept have- 

 not appeared ; while, on the other hand, large provinces of fact 

 have fallen into order under the application of the, at first 

 sight, startling concept which we have reached. 



But before the ideas which we have just been considering 

 had shaken off their scarcely reputable association with 

 Eleatic paradox, and had entered into the sober service of the 

 modern mathematician, it had been found that the compact 

 number series was inadequate to the demands that geometrical 

 theory put upon it. Thus, if the two sides of. a right-angled 

 triangle are equal in length, then the number which geometrical 

 theory compels us to correlate with the length of the hypo- 

 thenuse is not one of the compact series. If the number one 

 is correlated with the magnitude of the side, the number 

 which should be correlated with the magnitude of the hypo- 

 thenuse is one whose " square " is two. It is easy to satisfy 

 oneself that no such number is included in the compact series. 

 Two alternatives arise. Either we must conclude that 

 numbers are inadequate to the concept of space which our 

 experience forces upon us, or else we must arrive at some 

 means of supplementing the number . series still further by 

 means of clearly definable " Objective subsistences " which will 

 be available for correlation with the newly discovered magni- 

 tudes. 



As a matter of fact, it seems doubtful whether we can define 

 with logical rigidity numbers with the required characteristics 

 that can be regarded as interpolated between the terms of the 



