336 



SCIENCE 



[N. S. Vol. XXVI. No. 663 



gestive instance of the application of the 

 historical method. It is evident, however, 

 that Ziller's interpretation of the dual the- 

 ory of the culture epochs and concentration 

 centers is open to criticism. This is ap- 

 parent in the arbitrary selection of the cul- 

 ture epochs, as they only partly typify the 

 great epochs of history. Furthermore, the 

 concentration material selected by Ziller by 

 no means embodies the total experience of 

 the race in any particular epoch, and for 

 this reason is inconsistent with the prin- 

 ciple by which it Avas selected. In apply- 

 ing the theory this weakness has also made 

 itself felt by reason of the impossibility of 

 reproducing historical environment, and 

 the difficulty of adequately presenting the 

 notable characters of ancient civilization 

 without it. These objections have led to 

 severe criticism of the whole culture-epoch 

 theory, and in some cases to its entire re- 

 jection. 



It should be noted, however, that the 

 difficulties attending the culture-epoch the- 

 ory are inherent in this theory and not in 

 the historical method. In the ease of 

 mathematics the question of environment 

 does not arise, and thus the chief difficulty 

 is at once removed. Moreover, the nature 

 of the subject matter in elementary mathe- 

 matics is such that none of it can be 

 omitted, thus obviating all possibility of 

 error in the selection of proper materials. 



Perhaps the most convincing proof of 

 the applicability of the historical method 

 to mathematics is furnished by the practical 

 methods attained by teachers as the result 

 of long experience. Special aptitude for 

 teaching consists largely in the ability to 

 assume the mental attitude of the pupil, 

 and establish the connection between the 

 ideas already formed and those which it is 

 desired to communicate ; or, more briefly, in 

 the ability to stimulate apperception. By 

 long and earnest efforts of this kind such 



noted teachers as De Morgan, Grube and 

 others have arrived at methods of presenta- 

 tion which in the main follow the historical 

 sequence of development, thus affording a 

 strong inductive proof of the validity of 

 this method. The recognition of the his- 

 torical method as the universal principle 

 underlying experience by means of which 

 these results may be codified and extended, 

 is, then, all that is necessary to furnish a 

 rational basis for mathematical pedagogy. 



In applying the historical method to 

 mathematics, one of the most interesting 

 results is the light which is thrown on the 

 nature of the difficulties encountered in 

 studying the subject. From the fact that 

 mathematics has formed the basis of all 

 civilization, and has developed independ- 

 ently among nations widely separated, it 

 may be assumed that it possesses a certain 

 imiversality akin to that of mind itself. 

 This is by no means true, however, of the 

 special branches of the subject. Thus it is 

 by no means merely fortuitous that the 

 Greeks excelled in geometry but produced 

 no great algebraists, and that the reverse 

 was the case with the Semitic races. The 

 mathematical attainments of any nation 

 are, in fact, an integral part of its national 

 culture, and may, therefore, be expected to 

 differ in direction with the latter. In so 

 far, then, as mathematics satisfy the com- 

 mon needs of humanity they may justly lay 

 claim to universality, but beyond this point 

 are characterized by the spirit and aims of 

 the nation which gave them birth. 



It is not surprising, therefore, that 

 those reared under modern conditions 

 should experience difficulty in assimilating 

 results attained hundreds or thousands of 

 years ago and expressive of a culture en- 

 tirely foreign to our own; or that they 

 should at times fail to recognize the value 

 of certain branches of the subject. For 

 instance, geometry is still taught in prac- 



