4fi Colorado- College Studies. 



unit. Thus, the connotation of ratio (embracing both the 

 rational and irrational) as number enables us to make num- 

 ber conti)iHons, so that, starting with a ratio as small as we 

 please, we can conceive number to increase continuously, 

 passing through iall the stages of primary number and all 

 the intersi^ersed stages of irrational number, to a ratio as 

 large as we please. 



But the notion of continuity is not as simple as it, at first, 

 appears to be. Few notions in mathematics are more subtle. 

 It is in considerations of this sort that the logical superiority 

 of the reasoning based on numbers is asserting itself. In 

 advanced mathematical research, the greatest rigor of treat- 

 ment is secured, not by relying on intuition, not by depend- 

 ing on geometrical figures, but by an entire separation from 

 the world of our senses, and making mathematical demon- 

 strations wholly arithmetical. Through reliance on their in- 

 tuitions, mathematicians have been led to some erroneous 

 results; for instance, that every continuous function must 

 have a derivative at all points in a given interval. 



The tendency at the present time is to arUhmetise mafhe- 

 inctiics. The earlier explanation of irrational numbers, like 

 that of fractions, involved the idea of measurement. Formerly 

 an irrational number was defined as the expression of the 

 incommensurable ratio of two geometrical quantities — that 

 is, as the ratio between two quantities having no common 

 measure. For the purpose of removing certain logical diffi- 

 culties, G. Cantor, K. Weierstrass, R. Dedekind and others 

 have treated irrational number in a manner free of ratio and 

 measurement and of all geometrical considerations. This 

 arithmetical theory of the irrational is now about one quarter 

 of a century old; but our college text-books contain nothing 

 of these new ideas; the opinion strongly prevails among 

 teachers everywhere that the arithmetical theories of the 

 irrational are not suited for elementary instruction in the 

 differential and integral calculus, or in analysis in general. 



To the teacher of elementary arithmetic the chief point of 

 interest of these remarks on higher mathematics lies in the fact 

 that the use of the number concept, which is free of ratio and 

 measurement, is assuming a more and more central position in 

 the rigorously logical exposition of the advanced branches. 



