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SCIENTIFIC THOUGHT. 



62. 

 Kummer's 

 ideal 

 numbers. 



Berlin when dealing with a special problem. This was 

 no other than the celebrated problem of the division of 

 the circle into equal parts, which had been reduced by- 

 Gauss to an arithmetical question. Gauss had shown 

 that the accurate geometrical solution of this problem 

 depended on the solution of certain simple binomial 

 forms or equations. The study of such forms accord- 

 ingly became of special interest : it necessitated the 

 employment of the extended notion of number called by 

 Gauss that of complex numbers. Now it is one of the 

 fundamental laws in the theory of ordinary numbers that 

 every integer can be divided only in one way into prime 

 numbers. This law was found to break down at a 

 certain point if complex numbers were admitted. Rum- 

 mer, however, suggested that the anomaly disappeared if 

 we introduced along with the numbers he was dealing 

 with other numbers, which he termed ideal numbers — 

 i.e., if we considered these complex factors to be divisible 

 into other prime factors. The law of divisibility was 

 thus again restored to its supreme position. These 

 abstract researches led to the introduction of a very 

 useful conception — the conception not only of generalised 

 numbers, but also of a system (body, corpus, or region) 

 of nimibers ; ^ comprising all numbers which, by the 



^ The idea of a clo.sed system or 

 domain of generalised numbers has 

 revolutionised the theory of num- 

 bers. Originally the theory of 

 numbers meant only the theory 

 of the common integers, excluding 

 complex numbers. Gauss, in the 

 introduction to the ' Disquisitiones,' 

 limits the doctrine in this way. 

 He excludes also the arithmetical 

 theories which are implied in 



cyclotomy — i.e., the theorj' of the 

 division of the circle ; stating at 

 the .same time that the principles 

 of the latter depend on theories 

 of higher arithmetic. This con- 

 nection of algebraical problems 

 with the theory of numbers be- 

 came still more evident in the 

 labours of Gauss's successors — 

 Jacobi and Lejeune Dirichlet, and 

 was surprising to them. "The 



