DEVELOPMENT OF MATHEMATICAL THOUGHT. 727 



inversion of operations in the most general manner. In 

 the direct process we build up algebraical formula; 

 called equations or forms by a combination of addition 

 and multiplication. We can omit subtraction and 

 division, as through the use of negative quantities and 

 fractions these are reduced to the former. Now, given 

 the most general algebraical equation or form, we can 

 search out and define the simple factors or forms into 

 which it can be split up, and these factors and their pro- 

 ducts we can take to serve as the definition of numbers. 

 The question then arises, What are the properties of ei. 

 numbers thus inversely defined ? and, secondly, Do these inversion. 

 numbers exhaust or cover the whole extent of number as 

 it is defined by the uses of practical life ? The answer 

 to the former question led to the introduction of complex 

 and subsequently of ideal numbers ; the discovery by 

 liouville that the latter is not the case has led 

 to the conception of transcendental, i.e., non- algebraic, 

 numbers. 



The idea of generalising the conception of number, by 

 arguing backward from the most general forms into 

 which ordinary numbers can be cast by the processes of 

 addition and multiplication, has led to a generalised 

 theory of numbers. Here, again, the principal object 

 is the question of the divisibility of such generalised 

 algebraical numbers and the generalised notion of prime 

 numbers i.e., of prime factors into which such num- 

 bers can be divided. Before the general theory was 

 attempted by Prof. Dedekind, Kronecker, and others, 

 the necessity of some extension in this direction had 

 already been discovered by the late Prof. Kummer of 



