48 UNIVERSITY OF MISSOURI BULLETIN 



them. The only reason is that tradition prescribes the field 

 of mathematical activity. Indeed, I have mentioned already 

 the principal fields which mathematics enters. You will see 

 that practically all of them will be known to any student 

 who has so much as finished the Sophomore course in a 

 university such as this. Past that, remain elaborations of 

 these theories which go perhaps beyond the farthest bounds 

 of your possible imagination. In number work alone the 

 mass of present information is appalling; yet, looking back- 

 ward at it from the standpoint of our present knowledge, 

 it seems as if we were merely on the threshold. In the 

 study of functions we are in perhaps much the same position ; 

 the theory of functions is an enormous accumulation of 

 material which, alone, compares with the entire literature 

 of some of the sciences. In each of the other fields which 

 I have mentioned, this sort of elaboration has proceeded. In 

 the theory of limits, in geometry, in intricate types of reckon- 

 ing, such as the Group Theory, an immense literature exists. 

 The details are of course abstruse, and not fitted for such a 

 brief talk as this, otherwise there would scarcely be any 

 need for extended courses in which to discuss them and for 

 voluminous books in which to record and classify all of this 

 knowledge. 



Applications 



I have spoken repeatedly of the applications of mathematics 

 to other sciences; in fact I have pointed out that the only 

 successful characterization of mathematics is that it demands 

 that the conclusions drawn by its processes shall be useful 

 to the experimental sciences from which its hypotheses are 

 drawn, aside from elaborations of theories which have thus 

 arisen. 



The most common application, the one which is used by 

 every civilized human being, is reckoning, the first of the 

 grand divisions of mathematics which I have mentioned. A 



