ON THE NATURE OF MATHEMATICAL KNOWLEDGE. 35 



and in the enunciation of new problems, nor solely in deductions 

 and solutions, but culminates rather in the discovery of new meth- 

 ods and points of view in which the old disconnected atid isolated 

 results appear suddenly in a new connexion or as different interpre- 

 tations of a common fundamental truth, or finally, as a single or- 

 ganic whole. 



Thus, for example, the idea of representing imaginary and com- 

 plex numbers in a plane, two apparently different branches, the theory 

 of dividing the circumference of a circle into any given number of 

 equal parts, and the theory of the solutions of the equation x" = i t 

 have been made to exhibit an extremely simple connexion with one 

 another which enables us to express many a truth of algebra in a 

 corresponding truth of geometry and vice versa. Another example is 

 afforded by the discovery which we chiefly owe to Alfred Clebsch, 

 of the relation which subsists between the higher theory of func- 

 tions and the theory of algebraic curves, a relation which led to the 

 discovery of the condition under which two curves can be co-ordi- 

 nated to each other, point for point, and hence also adequately rep- 

 resented on each other. Of course such combinations and exten- 

 sions of view possess a much greater charm for the mathematician 

 than the mere accumulation of truths and solutions, whose fascina- 

 tion consists entirely in their truth or correctness. 



From these three cardinal characteristics, now, which distin- 

 guish mathematical research from research in other fields, we may 

 gather at once the three positive characteristics that distinguish 

 mathematical knowledge from other knowledge. They may be briefly 

 expressed as follows; first, mathematical knowledge bears more 

 distinctly the imprint of truth on all its results than any other 

 kind of knowledge ; secondly, it is always a sure preliminary step to 

 the attainment of other correct knowledge ; thirdly, it has no need 

 of other knowledge. Naturally, however, there are associated with 

 these characteristics which place mathematical knowledge high 

 above all other knowledge, other characteristics which somewhat 

 counterbalance the great superiority which mathematics thus ap- 

 pears to have over the other sciences. In order to show more dis- 

 tinctly the nature of these characteristics, which we prefer to call 



