MONISM IN ARITHMETIC. 25 



first degree led to no new numbers. After the investigation of mul- 

 tiplication and its inverse operations, the positive or negative frac- 

 tional numbers and " infinitely great" were added, and again we 

 could say that the combination of two already defined numbers 

 by operations of the first and second degree in turn also always 

 led to numbers already defined. Now we have reached a point at 

 which we can say that the combination of two complex numbers by 

 all operations of the first, second, and third degree must again 

 always lead to complex numbers ; only that now such a combina- 

 tion does not necessarily always lead to a single number, but may 

 lead to many regularly arranged numbers. For example, the com- 

 bination "logarithm of minus one to a positive base" furnishes a 

 countless number of results which form an arithmetical series of 

 purely imaginary numbers. Still, in no case now do we arrive at new 

 classes of numbers. But just as before the ascent from multiplication 

 to involution brought in its train the definition of new numbers, so 

 it is also possible that some new operation springing out of involution 

 as involution sprang from multiplication might furnish the germ of other 

 new numbers which are not reducible to a -f- ib. As a matter of fact, 

 mathematicians have asked themselves this question and investi- 

 gated the direct operation of the fourth degree, together with its 

 inverse processes. The result of their investigations was, that an 

 operation which springs from involution as involution sprang from 

 multiplication is incapable of performing any real mathematical ser- 

 vice ; the reason of which is, that in involution the laws of commu- 

 tation and association do not hold. It also further appeared that 

 the operations of the fourth degree could not give rise to new num- 

 bers. No more so can operations of still higher degrees. With 

 respect to quaternions, which many might be disposed to regard as 

 new numbers, it will be evident that though quaternions are valu- 

 able means of investigation in geometry and mechanics they are not 

 numbers of arithmetic, because the rules of arithmetic are not un- 

 conditionally applicable to them. 



The building up of arithmetic is thus completed. The exten- 

 sions of the domain of number are ended. It only remains to be 

 asked why the science of arithmetic appears in its structure so logi- 



