July 24, 1008] 



SCIENCE 



111 



far as it is continually changing, develop- 

 ing, rising. The above comparisons are, 

 therefore, given merely as a basis for better 

 understanding the efforts that are now 

 made in Germany for the improvement of 

 mathematical teaching in aU its phases. To 

 these efforts I wish to call your special 

 attention. 



The German movement for the reform 

 of the teaching of mathematics is of a 

 somewhat complex nature; at least three 

 different movements may be distinguished. 

 One of these, originating with the German 

 association of engineers (Verein deutscher 

 Ingenieure) had as its direct object the im- 

 provement of the mathematical instruction 

 in the engineering colleges, with a view to 

 making the instruction less abstract and 

 theoretical and more practically useful to 

 the engineer. To a certain extent, this 

 object has been attained. Practical exer- 

 cises for acquiring a working knowledge 

 of mathematics have been introduced every- 

 where, and the lectures on pure mathe- 

 matics have become less theoretical. Some 

 of the originators of this movement, espe- 

 cially Professor Riedler, of the Charlotten- 

 burg College, went so far as to demand that 

 in engineering colleges mathematics should 

 be taught by engineers. Whether or not 

 this was meant as more than a threat I do 

 not undertake to say; certainly, as far as 

 my knowledge goes, no attempt has ever 

 been made in a German engineering college 

 to put the teaching of mathematics in the 

 hands of any one but a trained mathe- 

 matician. But I believe that in the selec- 

 tion of men for such positions more atten- 

 tion has been paid in recent years to the 

 qualifications of the aspirants; mathe- 

 maticians with a bent towards applied sci- 

 ence being given the preference for posi- 

 tions in engineering colleges. 



The second of the three movements re- 

 ferred to above has for its object the re- 

 form of the teaching of mathematics in the 



universities. It is the oldest of these move- 

 ments, and has borne fruit in a variety of 

 ways. But I can here only advert to it 

 very briefly. The tremendous creative 

 mathematical activity that characterized 

 the last three quarters of the nineteenth 

 century in Germany led to a condition in 

 the universities that was injurious to the 

 preparation of teachers for the secondary 

 schools (Gymnasium, Realgymnasium, 

 Oberrealschule) . Too much stress was laid 

 on leading the student as fast as possible 

 to original research in some special line. 

 The system has been described as a system, 

 not of double entry, but of double for- 

 getting; upon entering the university the 

 students, most of whom are fitting for 

 teaching in the secondary schools, are made 

 to forget and almost despise the more ele- 

 mentary mathematics, and when beginning 

 their professional teaching career they are 

 again compelled to forget as fast as possible 

 all the higher and highest mathematics to 

 which they had devoted most of their time 

 at the university. The remarkable devel- 

 opment of mathematical activity in our 

 country during the last fifteen or twenty 

 years may bring about a similar situation. 

 Fortunately, the leaders of American math- 

 ematics are well aware of the danger of 

 losing the healthy contact with the more 

 elementary mathematics and with applied 

 science. Of course, it is, and always will 

 be, the chief object of a real university to 

 foster original research and productive 

 scholarship. But it is well even for the 

 most advanced specialist not to burn the 

 bridges behind him, but to keep in mind 

 the connection of his specialty with the 

 foundations of knowledge, on the one hand, 

 and with kindred branches of science on 

 the other. As Sir Isaac Newton expressed 

 it in his quaint way in a letter to Dr. Lord : 

 "He that in ye mine of knowledge deepest 

 diggeth, hath, like every other miner, ye 



