726 



SCIENTIFIC THOUGHT. 



co. same time a metaphysical question presented itself viz., 



Generalised 



conception Can such an extension into more than two dimensions be 



or number. 



consistently and profitably carried out ? Gauss had satis- 

 fied himself that it could not; 1 hut the proof of this 

 was only given in more recent times by Weierstrass, who 

 definitely founded the whole discussion of the subject on 

 the logical principle " that the legitimacy of introducing 

 a number into arithmetic depends solely on the definition 

 of such number." And this leads me to another ex- 

 tension in the region of number suggested by Gauss's 

 treatment, which has also become fundamental, and, in 

 the hands of Dirichlet, Kummer, Liouville, Dedekind, 

 and others, has remodelled the entire science of higher 

 arithmetic. It is based on the logical process of the 



published by Kossak, has been 

 adopted. This proof is based upon 

 the condition that the product of 

 several factors cannot disappear 

 except one of its factors is equal to 

 zero. "We must, therefore, ex- 

 clude from general arithmetic com- 

 plex numbers consisting of three 

 fundamental elements. This is, 

 however, not necessary if the use of 

 them be limited " by some special 

 conditions (Kossak, loc. cit., p. 27). 

 In the course of the further de- 

 velopment of this matter Weier- 

 strass arrives at the fundamental 

 thesis " that the domain of the 

 elementary operations in arithmetic 

 is exhausted by addition and multi- 

 plication, including the inverse 

 operations of subtraction and 

 division/' "There are," says 

 Weierstrass, " no other funda- 

 mental operations at least it is 

 certain that no example is known 

 in analysis where, if an analytical 

 connection exists at all, this cannot 

 be analysed into and reduced to 

 those elementary operations" (p. 

 29). 



1 A concise history of this sub- 

 ject is given by Kossak in the 

 Program referred to above, p. 

 712, note. Gauss had promised 

 to answer the question, "Why 

 the relations between things which 

 have a manifoldness of more than 

 two dimensions would not admit 

 of other " (than the ordinary com- 

 plex numbers introduced by him) 

 "fundamental quantities being in- 

 troduced into general arithmetic ? " 

 He never redeemed his promise. 

 In consequence of this, several 

 eminent mathematicians, notably 

 Hankel, Weierstrass, and Prof. 

 Dedekind, have attempted to reply 

 to this question, and to estab- 

 lish the correctness of the im- 

 plied thesis according to which 

 any system of higher complex 

 numbers becomes superfluous and 

 useless. Prof. Stolz, in the first 

 chapter of the second volume of his 

 'Allgemeine Arithmetik.' gives an 

 account of these several views, 

 which do not exactly coincide. 

 In general, however, the proof 

 given by Weierstrass, and first 



