'30 



SCIENTIFIC THOUGHT. 



63. 



Modern 

 algebra. 



64. 



Algebraical 



cende F ntai 

 numbers. 



as we saw above, was vaguely foreshadowed by Peacock, 

 and expressly placed at the head of all mathematical 

 reasoning by Hermann Hankel. In passing it may also 

 be observed how the notion of a system of algebraical 

 numbers, which belong together as generated in certain 

 defined ways, prepares us for the introduction of that 

 general theory of groups which is destined to bring order 

 and unity into a very large section of scattered mathe- 

 matical reasoning. The great importance of this aspect 

 is clearly and comprehensively brought out in Prof. H. 

 Weber's Algebra. Nothing could better convince us of 

 the great change which has come over mathematical 

 thought in the latter half of the nineteenth century 

 than a comparison of Prof. Weber's Algebra with stand- 

 ard works on this subject published a generation earlier. 



I have shown how the definition of algebraical 

 numbers has led to an extension and generalisation of 

 the conception of number. Another question simultane- 

 ously presented itself, Does this extension cover the 

 whole field of numbers as we practically use them in 

 ordinary life ? The reply is in the negative. Practice 

 is richer than theory. Nor is it difficult to assign 

 ^e reason f this. Numbering is a process carried on 

 ' n p rac ti ca l life for two distinct purposes, which we 

 distinguish by the terms counting and measuring. Num- 

 bering must be made subservient to the purpose of 

 measuring. Thus difficulties arising out of this use of 

 numbers for measuring purposes presented themselves 

 early in the development of geometry in what are called 

 the incommensurable quantities : taking the side of a 

 square as ten, what is the number which measures the 



