JUI.Y 9, 1909] 



SCIENCE 



35 



preparation for a clear understanding of 

 the mathematical element of modern cul- 

 ture. This element clearly rests on the 

 idea of function, both geometrical and 

 analytical, and this idea should be made the 

 center of mathematical instruction. In 

 fact, Klein's chief thesis is that beginning 

 with the Untersecunda and continuing in 

 regular methodical steps, the geometrical 

 conception of a function should permeate 

 all mathematical instruction. In this is 

 included the consideration of analytic 

 geometry and the elements of differential 

 and integral calculus. 



The ground to be covered depends largely 

 upon the ideal of the school. While the 

 formal side must not be overlooked, and a 

 thorough knowledge of the processes must 

 be obtained, the principal aim should be to 

 give a clear conception of the fundamental 

 ideas and their meaning. 



Mathematical instruction on the level at 

 which it is carried on in the upper classes 

 of the higher schools has existed in Ger- 

 many since about the beginning of the 

 eighteenth century. In this early forma- 

 tive period calculus was not considered ele- 

 mentary, for it was the possession of only 

 a few investigators of the highest type. 

 Cauchy's great work on differential and 

 integral calculus appeared in 1821, but the 

 schools had already been led into certain 

 channels, and it was not possible to divert 

 them toward a subject which was only in 

 the process of formation. In fact, calculus 

 was considered as a sort of witchcraft, and 

 has ever since been regarded with suspicion. 



The official course of study of 1900, how- 

 ever, showed a tendency in the opposite 

 direction, so that Klein believes that ad- 

 vantage should be taken of this favorable 

 attitude to place that which has taken cen- 

 turies for preparation upon a generally 

 recognized basis. 



As a matter of fact, the ideas underlying 



the calculus are actually taught in many 

 schools. In a few Ober Real schools they 

 are regularly taught as calculus, but in the 

 majority of schools they are taught in the 

 most round-about manner. In fact, stu- 

 dents are actually taught to differentiate 

 and integrate as soon as occasion for the 

 same arises, but the terms differential and 

 integral are avoided. 



Klein 's opinion is that instead of making 

 instruction in calculus in those grades 

 whose work demands its employment inci- 

 dental, desultory and generally unsatisfac- 

 tory, it should be made the central idea of 

 instruction, and the other ideas and work 

 grouped around it. At present, calculus is 

 made the beginning of higher mathematics 

 and is accompanied by a revolution in 

 thinking. Klein's suggestions aim to ob- 

 viate this difficulty by gradually accustom- 

 ing the pupil to those methods of thinking 

 which later predominate. 



THE PRUSSIAN SYSTEM 



The Prussian schools are probably the 

 most efficient, in point of actual instruc- 

 tion, of the entire German school system, 

 and for this reason deserve special con- 

 sideration. 



Although but 1.2 years of the nine school 

 years are given to mathematics as com- 

 pared with 2.1 years in this country, the 

 Prussians accomplish fully as much as, if 

 not more than, our American schools, with 

 a saving of .9 of a year, or .1 of the total 

 time of instruction for the nine years. 



The three main causes of the excellence 

 of the Prussian work in mathematics are 

 the central legislation and supervision, the 

 thorough preparation of the teachers and 

 the systematic methods of instruction. 

 Each of these is a result of the preceding; 

 for well-prepared teachers are likely to use 

 good methods of instruction and the thor- 

 ough preparation of teachei-s is sure to be 



