DEVELOPMENT OF MATHEMATICAL THOUGHT. 721 



theory of forms and 'f unctions : there remains the science 



of numbers of number in the abstract and also of the 58. 



Theory of 



named numbers of ordinary arithmetic. Gauss's earliest numbers. 

 labours were connected with this branch. Superseding 

 the work of Fermat, Euler, and Legendre, he produced 

 that great book with seven seals, the ' Disquisitiones 

 Arithrneticse.' The seals were only gradually broken. 

 Lejeune Dirichlet did much in this way : others followed, 

 notably Prof. Dedekind, who published the lectures of 

 Dirichlet and added much of his own. The question 

 may be asked, Have we gained any new ideas about 

 numbers ? 



In this abstract inquiry we can again facilitate our 

 survey by distinguishing between the practical and the 

 purely theoretical interests which stimulated it. Look- 

 ing at the matter as well as the formal treatment by 

 which it was rendered accessible, we may say Gauss not 

 only taught us some very remarkable new properties of 

 numbers he also invented a new instrument or calculus 

 for their investigation. Let us consider his work and 

 that of his followers from these different points of view. 



First, then, there were certain definite problems con- 

 nected with the properties of numbers which had been 

 handed down from antiquity. Such were the division of 

 the circle into equal parts by a ready geometrical con- 

 struction, the duplication of the cube, and the quad- 

 rature of the circle or the geometrical construction 

 of the number ir. 1 To the latter may be attached the 



1 See above, vol. i. p. 181, note. 

 The student will find much in- 

 teresting matter referring to these 

 problems in Prof. Klein's little 



volume entitled 'Famous Problems 

 in Elementary Geometry,' transl. 

 by Beman and Smith, Boston and 

 London, 1879. In it is also given 



VOL. II. 2 Z 



