460 THE POPULAR SCIENCE MONTHLY 



it soon appeared desirable to include all mathematical instruction in 

 the scope of its investigation. 



Sub-commissions were appointed in various countries. The Ameri- 

 can sub-commission is composed of D. E. Smith, Columbia University; 

 W. F. Osgood, Harvard University, and J. W. A. Young, Chicago Uni- 

 versity. Under the general direction of these sub-commissions a vast 

 amount of material relating to the mathematical instruction has been 

 collected and published. In our own country this material was pub- 

 lished by the U. S. Bureau of Education in the form of thirteen reports. 

 Some of the other countries have not yet completed their work, but 

 about one hundred and sixty such reports have already been published 

 in the twenty-six countries which have joined in this vast undertaking. 



In addition to securing these valuable reports the central commis- 

 sion has arranged international meetings for the discussion of some of 

 the fundamental questions relating to mathematical instruction. Such 

 a meeting was held in Paris, France, in April of the present year, and 

 the two subjects under consideration were: (1) The results obtained 

 by the introduction of differential calculus in the advanced classes of 

 the secondary schools, and (2) the place and the role of mathematics in 

 higher technical education. 



Some of the leading French mathematicians (including Appell, 

 Darboux, Borel and d'Ocogne) took an active part in the deliberations. 

 Professor Borel emphasized the fact that mathematics is not composed 

 of a linear sequence of theorems such that each depends upon the pre- 

 ceding one. If this were the case, the only possible changes in methods 

 of instruction would relate to what theorems could be omitted in this 

 sequence or what theorems could be substituted for others. On the 

 contrary, the number of different routes leading from first principles to 

 an advanced mathematical proposition is often exceedingly large, and 

 hence arises the possibility of employing widely different methods to 

 achieve the same general results. 



In other words, mathematics is a network formed by intersecting 

 thought roads and the chief aim of the International Commission on the 

 Teaching of Mathematics is to secure extensive information as regards 

 the choice of roads in various nations. The Italian member of the 

 central committee, G. Castelnouovo of Eome, stated explicitly in his 

 address during the recent conference at Paris, that the commission did 

 not aim to bring about any great reforms, but aimed to gather facts as 

 regards existing conditions in order that the various nations might 

 be enabled to profit by the experiences of other nations in instituting 

 their own reforms. 



In describing mathematics as a network of a certain type of thought- 

 roads, it is not implied that thought is conveyed along these roads as the 

 products of a country are conveyed on a railroad train. On the con- 



