1 62 POPULAR SCIENCE MONTHLY 



As mathematics enters into questions which are common to all 

 society, from the most primitive to the most civilized, it is clear that 

 some of its elements must always enter into the education of every one. 

 Within these narrow limits there is little freedom of choice. In 

 rising from this low view of mathematics to the one which recognizes 

 its value in developing thought power and thought caution, one soon 

 arrives in regions of the greatest freedom. From such a standpoint 

 one can readily comprehend why so profound a thinker as Simon New- 

 comb should say, ' The mathematics of the twenty-first century may 

 be very different from our own: perhaps the schoolboy will begin 

 algebra with the theory of substitution-groups as he might now but for 

 inherited habits.' 3 



During the last few decades Germany has wielded a predominating 

 influence on the development of mathematics in this country. Hence 

 it is natural that the tendencies of German mathematical development 

 should be strongly felt in this country. Two of these tendencies are 

 especially prominent: viz., the uniting of pure and applied mathe- 

 matics, and the encyclopedic character of pure mathematics. Definite 

 evidence of the former tendency is furnished by the rapid increase in 

 the number of courses on the mathematics of insurance, mathmatics 

 for students of physics and chemistry, and especially by the enactment 

 of 1898 which made applied mathematics a distinct requirement of 

 those who expected to become teachers of mathematics. While the 

 American universities have been more conservative in these directions, 

 yet there are many evidences that these tendencies are strongly reflected 

 in the mathematical courses offered by our higher institutions. 



Jacobi was perhaps the first eminent German mathematician who 

 made special efforts to lead his students to the boundary between the 

 known and unknown as rapidly as possible, and then to make them 

 coworkers with him in investigating new problems. His methods 

 were imitated very largely by others so that German mathematical in- 

 struction became, to an unusually large extent, instruction in research, 

 or, at least, instruction in regions which had been very inadequately 

 explored. These methods have been employed in other countries. In 

 our own country their introduction was hastened by the teaching of 

 Sylvester and Cayley, who employed similar methods while they were 

 connected with Johns Hopkins University. 



While the discovery of new truths gives an interest and charm 



the former standpoint, geometry, as commonly understood, is not mathematics. 

 For a splendid exposition of the definitions of mathematics we refer to Bocher, 

 ' The Fundamental Conceptions and Methods of Mathematics,' Bulletin of the 

 American Mathematical Society, Vol. 11 (1905), p. 115. 



"Newcomb, Bulletin of the American Mathematical Society, Vol. 3 (1894), 

 p. 107. 



