SCIENCE 



[N. S. Vol. XXVIII. No. 705 



The third period, beginning with 1890, 

 seems to be characterized by a tendency to 

 again associate pure and applied mathe- 

 matics; that is to say, the idea prevails 

 that while a teacher should be thoroughly 

 familiar with pure mathematics, his knowl- 

 edge of its applications in the various 

 fields should also be extensive. This is 

 perhaps one result of the new order of 

 things which puts the real schools on an 

 equality as to privileges with the older 

 gymnasium. There is also a tendency to 

 allow the teacher greater freedom from the 

 dictation of a centralized bureaucracy, and 

 in this freedom lies an opportunity for 

 future development. 



For many decades, under the rule of 

 the new humanism, the value of mathe- 

 matical training was thought to lie in its 

 formal discipline. Before the revival of 

 learning it was the utilitarian factor which 

 received emphasis, but in the last decades 

 the majority have reached a more compre- 

 hensive view. Briefly stated, the modern 

 view is that mathematical thought should 

 be cherished in the schools in its fullest 

 independence, its content being regulated 

 in a measure by the other problems of the 

 school ; that is to say, its content should be 

 such as to establish the liveliest .possible 

 connection with the various parts of the 

 general culture which is typical of the 

 school in question. Here, then, it is not a 

 question of methods of teaching, but rather 

 of the selection of material from the great 

 mass furnished by elementary mathe- 

 matics. 



In the conference of 1900, it was agreed 

 that each type of school should determine 

 what form of culture its particular course 

 should produce. It seems that the Gym- 

 nasium was asserting its claim to be con- 

 sidered preeminently the culture school, not 

 hesitating to stigmatize the others as mere 

 technical schools, while the friends of the 

 Real schools apparently made no efforts at 



defense. Klein emphasized the fact that 

 he considers the three schools of equal im- 

 portance, and whatever he has to say con- 

 cerns all three types. 



Much of the material of instruction, 

 although interesting in itself, lacks con- 

 nection and is partially isolated. In fact, 

 the topics seem for the most part to be the 

 result of chance selection, and afford only 

 a faulty and indirect preparation for a 

 clear understanding of the mathematical 

 element of modern culture. This element 

 clearly rests on the idea of function and 

 its form, both geometrical and analytical, 

 and this idea should, therefore, be made 

 the center of mathematical instruction. 

 Klein's chief thesis is, in fact, that begin- 

 ning with the Untersecunda and proceed- 

 ing in regular, methodical steps, the geo- 

 metrical concept of a function should 

 permeate all mathematical instruction. In 

 this is included a certain consideration of 

 analytic geometry, and the elements of dif- 

 ferential and integral calculus. He refers 

 in this connection to two French publica- 

 tions which to a certain extent carry out 

 his ideas." 



To accomplish this purpose, the graph- 

 ical representation of the simplest ele- 

 ments, such as y = ax-\-'b and y = 1/x, 

 should be begun in the Untersecunda. 

 Trigonometry and the theory of algebraic 

 equations furnish ample material for more 

 complicated work, while in this connection 

 related illustrations can be obtained from 

 applications of mathematics, particularly 

 from the domain of physics. Also the idea 

 should be especially inculcated that a func- 

 tion can be developed empirically, perhaps 

 by means of apparatus. In the Prima the 

 general fundamental principles of both 

 differential and integral calculus should be 

 given, based upon the ideas which the 

 pupil has acquired in the Secunda. 



° " Notions de mathgmatique," Jules Tannery ; 

 "Algebra," E. Borel. 



