184 



SCIENTIFIC THOUGHT. 



philosophical and classical spirit. During these twenty- 

 five years Gauss lived and soared in solitary height a 

 name only to the German student, as Euler had been 

 before him. Probably he was better known to the 

 younger astronomers whom he trained, and the elder ones 

 with whom he corresponded. But astronomy was not 

 then within the pale of the universities. To what extent 

 the character of Gauss's own genius was the cause of this 

 it is difficult to say.-^ He himself had not come under 

 the influence of any great teachers such as Paris then 

 possessed ; he was self-taught, and had early imbibed 

 a great admiration for the methods of Euclid, Archimedes, 

 and Newton ; he wrote in the classical style fitted for all 

 times, but not for uninitiated beginners." It is certain, 



^ Bjerknes, in his most interesting 

 memoir on Abel, refers frequently 

 to the awe in which Gauss was held 

 by younger mathematicians. 



" In this Gauss resembled New- 

 ton. He was therefore, like Newton, 

 frequently forestalled by others, 

 who published his new methods 

 and ideas in an unfinished and frag- 

 mentary form ; whereby it is not 

 suggested that these simultaneous 

 discoveries or inventions vt'ere not 

 quite indej^endent. Two examples 

 of this may be added to those given 

 above. When Gauss published the 

 'Disquis. Arith.' in 1801, he left 

 out the last or eighth section, which 

 was to treat of the residues of the 

 higher orders. He had already 

 nearly completed the theory of 

 biquadratic residues. In dealing 

 with this subject he had found it 

 necessary to extend the conception 

 of number beyond the limits then 

 in use. If we confine ourselves to 

 integers, the only extension which 

 then existed of the notion of number 

 was in the use of negative numbers. 



These were counted on a straight 

 line backward, as positive (or or- 

 dinary) numbers were counted for- 

 ward. Gauss conceived the idea of 

 counting numbers laterally from the 

 straight line which represented the 

 ordinary positive and negative 

 numbers. He called numbers which 

 were thus located in the plane 

 " complex numbers," as they had to 

 be counted bj' the use of two units, 

 the ordinary unit 1 and a new unit 

 i. He also showed that this new 

 unit i stood in such relations to the 

 ordinarj' unit 1 as were algebraically 

 defined by the mysterious imagin- 

 ary symbol ^'-1. The complete 

 exposition of this new or complex 

 system of counting was not ex- 

 plained by Gauss till the year 

 1831, when he published the 

 ' Theoria residuorum biquadrati- 

 corum.' In the meantime the 

 geometrical representation of im- 

 aginary quantities had been devised 

 and published by Argand (1806), 

 but not being employed for such 

 important researches, it had re- 



