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 Newcomb 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.'[1]
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 mathematics, 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, mathematics 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 instruction 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
- ↑ Newcomb, Bulletin of the American Mathematical Society, Vol. 3 (1894), p. 107.
the former standpoint, geometry, as commonly understood, is not mathematics. For a splendid exposition of the definitions of mathematics we refer to Bôcher, 'The Fundamental Conceptions and Methods of Mathematics,' Bulletin of the American Mathematical Society, Vol. 11 (1905), p. 115.
