Decembeb 13, 1912] 



SCIENCE 



the volumes, being assisted by Mr. Lietzmann, 

 the secretary of the German subcommittee. 

 Furthermore, Mr. Klein gave special atten- 

 tion to the volumes I., III. and V., while the 

 second volume was due in large measure to 

 Mr. Treutlein, one of the most prominent of 

 our secondary teachers, whose death, three 

 weeks before this congress, is a great loss to 

 our country. Volume IV., which relates to 

 the mathematics at the technical schools, is 

 largely in the hands of Mr. Staeckel. 



At first sight it might be expected that the 

 report would begin with the elementary 

 schools, proceeding then to the secondary 

 schools and finishing with the institutions of 

 university rank. It is not possible, however, 

 to give such a systematical description of the 

 German schools. The variety is too great, the 

 development of the diilerent types of schools 

 too peculiar, the mathematical instruction 

 too varied, to make it possible to arrange our 

 school-system in a straight line. 



The points of view that have been set forth 

 in the several papers can not be completely 

 given in this short report. I may say, how- 

 ever, that in general there is given in every 

 case a sketch of the historical development 

 and of the organization of the special types 

 of schools. Perhaps you will allow me to call 

 especial attention to the fact that the reports 

 of the German delegation not only present a 

 fairly adequate picture of the mathematical 

 instruction, but also of the whole German 

 system of schools. 



In order to set forth a general summary of 

 the German reports, without entering into de- 

 tails, I beg to call attention to the third vol- 

 ume, which contains the discussion of certain 

 general questions of the secondary mathe- 

 matical instruction, and to mention in some 

 detail the several papers. 



During the last 10 or 20 years the reform 

 of the teaching of mathematics has often been 

 discussed not only in Germany, but in all cul- 

 tivated countries. Therefore it is of interest 

 that the first paper of the third volume is es- 

 pecially devoted to the development of these 

 reform tendencies in Germany. 



The second paper treats of the relation be- 



tween mathematics and physics in the second- 

 ary schools, showing by numerous examples 

 the great value of physics when founded on a 

 high grade of modern mathematical instruc- 

 tion. We may assume that this paper wiU 

 have a good reception in the native country 

 and in the university of Newton. 



The following three papers treat of applied 

 mathematics, and especially of descriptive 

 geometry, astronomy and practical arith- 

 metic. These are followed by an essay on 

 the history of mathematics as a means for 

 raising the interest of the student in the sub- 

 jects of the secondary school. 



The last essay of the third volume that just 

 appeared sets forth the relation between math- 

 ematics and philosophy. It shows us how the 

 higher classes in mathematics in the second- 

 ary school receive a valuable training in phi- 

 losophy as well. I am of the opinion that this 

 paper will be found to contain much that will 

 prove to be of value and of general interest to 

 all readers. 



Though all of the volumes of the German 

 report treat more or less at length of the 

 training of teachers, nevertheless it has been 

 thought desirable to prepare a special paper 

 on the study of mathematics at the German 

 universities since 1870. I am sorry to say that 

 this report is not yet printed, but it is just 

 going to press. I need not urge the great im- 

 portance of such a report, for it is evident 

 that the education of teachers is the center of 

 any substantial educational reform. 



The reform of mathematical instruction is 

 extending itself everywhere in Germany, and 

 this tendency naturally leads, little by little, 

 to a standardization of the instruction in the 

 different parts of the country. But in spite 

 of this tendency it must be said that, in mat- 

 ters of public education, Germany enjoys very 

 great freedom. I dare say that this freedom 

 is a notable characteristic of our country, and 

 that there exists scarcely an analogy in any 

 other of the leading countries of the world. 

 I may illustrate this liberty of teachers and 

 of the educational system by two examples: 

 First, in Germany the teachers are merely 

 obliged to follow certain general outlines 



