DEVELOPMENT OF MATHEMATICAL THOUGHT. 729 



ordinary operations of arithmetic, can be formed out of 

 the units or elements we start with. Thus all rational 

 integers form a system ; we can compound them, but 

 also resolve them into their elements. Where we intro- 

 duce new elements or units we only arrive at cor- 

 rect laws if we are careful to cover the whole field or 

 system which is measured by the application of the 

 fundamental operations of arithmetic. Throughout all 

 our abstract reasoning it is the fundamental operations 

 which remain permanent and unaltered, a rule which, 



reason for this connection is now 

 completely cleared up. The theory 

 of algebraical numbers and Galois's 

 ' theory of equations ' have their 

 common root in the general theory 

 of algebraical systems ; especi- 

 ally the theory of the system of 

 algebraical numbers has become 

 at the same time the most im- 

 portant province of the theory of 

 numbers. The merit of having laid 

 down the first beginnings of this 

 theory belongs again to Gauss. 

 He introduced complex numbers, 

 he formulated and solved the 

 problem of transferring the 

 theorems of the ordinary theory 

 of numbers, above all, the pro- 

 perties of divisibility and the re- 

 lation of congruence, to these 

 complex numbers. Through the 

 systematic and general develop- 

 ment of this idea, based upon the 

 far-reaching ideas of Kummer, 

 Dedekind and Kronecker suc- 

 ceeded in establishing the modern 

 theory of the system of algebraical 

 numbers" (Prof. Hilbert in the 

 preface to his " Theorie der Alge- 

 braischen Zahlkbrper, " ' Bericht der 

 Math. Ver.,' vol. iv. p. 3). In the 

 further course of his remarks Prof. 

 Hilbert refers to the intimate con- 

 nection in which this general or 

 analytical theory of numbers 

 stands with other regions of 



modern mathematical science, not- 

 ably the theory of functions. " We 

 thus see," he says, " how arith- 

 metic, the queen of mathematical 

 science, has conquered large do- 

 mains and has assumed the leader- 

 ship. That this was not done 

 earlier and more completely, seems 

 to me to depend on the fact 

 that the theory of numbers has 

 only in quite recent times arrived 

 at maturity." He mentions the 

 spasmodic character which even 

 under the hands of Gauss the 

 progress of the science exhibited, 

 and says that this was characteristic 

 of the infancy of the science, which 

 has only in recent times entered 

 on a certain and continuous de- 

 velopment through the systematic 

 construction of the theory in ques- 

 tion. This systematic treatment 

 was given for the first time in the 

 last supplement to Dedekind's edi- 

 tion of Dirichlet's lectures (1894, 

 4th ed., p. 134). A very clear 

 account will also be found in Prof. 

 H. Weber's ' Lehrbuch der Algebra ' 

 (vol. ii., 1896, p. 487, &c.) He 

 refers (p. 494) to the different 

 treatment which the subject has 

 received at the hands of its two 

 principal representatives Prof. 

 Dedekind (1871 onwards) and Kron- 

 ecker (1882) and tries to show 

 the connection of the two methods. 



