Maboh 13, 1903.J 



SCIENCE. 



411 



To consider the subject of geometry in 

 all bi-iefness: with the understanding that 

 proper emphasis is laid upon all the con- 

 crete sides of the subject, and that further- 

 more from the beginning exercises in in- 

 formal deduction* are introduced increas- 

 ingly frequently, when it comes to the 

 beginning of the more formal deductive 

 geometry why should not the students be 

 directed each for himself to set forth a 

 body of geometric fundamental principles, 

 on which he would proceed to erect his 

 geometric edifice? This method would be 

 thoroughly practical and at the same time 

 thoroughly scientific. The various students 

 would have different systems of axioms, 

 and the discussions thus arising naturally 

 would make clearer in the minds of all 

 precisely what a!re the functions of the 

 axioms in the theory of geometry. The 

 students would omit very many of the 

 axioms, which to them would go without 

 saying. The teacher would do well not to 

 undertake to make the system of axioms 

 thoroughly complete in the abstract sense. 

 "Sufficient unto the day is the precision 

 thereof." The student would very prob- 

 ably wish to take for granted all the ordi- 

 nary properties of measurement and of mo- 

 tion, and would be ready at once to accept 

 the geometrical implications of coordinate 

 geometry. He could then be brought with 

 extreme ease to the consideration of funda- 

 mental notions of the calculus as treated 

 concretely, and he would find those notions 

 delightfully real and powerful, whether in 

 the domain of mathematics or of physics 

 or of chemistry. 



* In an article shortly to appear in the Educa- 

 tional Review, on ' The Psychological and the 

 Logical in the Teaching of Geometry,' Professor 

 John Dewey, calling attention to the evolutionary 

 character of the education of an individual, in- 

 sists that there should be no abrupt transition 

 from the introductory, intuitional geometry to the 

 systematic, demonstrative geometry. 



To be sure, as Study has well insisted, 

 for a thorough comprehension of even the 

 elementary parts of euclidean geometry 

 the non-euclidean geometries are absolutely 

 essential. But the teacher is teaching the 

 subject for the benefit of the students, and 

 it must be admitted that beginners in the 

 study of demonstrative geometry can not 

 appreciate the very delicate considerations 

 involved in the thoroughly abstract science. 

 Indeed, one may conjecture that, had it not 

 been for the brilliant success of Euclid in 

 his effort to organize into a formally de- 

 ductive system the geometric treasures of 

 his times, the advent of the reign of science 

 in the modern sense might not have been 

 so long deferred. Shall we then hold that 

 in the schools the teaching of demonstrative 

 geometry should be reformed in such a 

 way as to take account of all the wonder- 

 ful discoveries which have been made- 

 many even recently— in the domain of ab- 

 stract geometry? And should similar re- 

 forms be made in the treatment of arith- 

 metic and algebra? To make reforms of 

 this kind, would it not be to repeat more 

 gloriously the error of those followers of 

 Euclid who fixed his 'Elements' as a text- 

 book for elementary instruction in geometry 

 for over two thousand years? Every one 

 agrees that professional mathematicians 

 should certainly take account of these great 

 developments in the technical foundations 

 of mathematics, and that ample provision 

 should be made for instruction in these 

 matters; and on refiection, every one 

 agrees further that this provision should 

 be reserved for the later collegiate and 

 university years. 



The Laboratory Metliod.— This program 

 of reform calls for the development of a 

 thoroughgoing laboratory system of in- 

 struction in mathematics and physics, a 

 principal purpose being as far as possible 

 to develop on the part of every student the 

 true spirit of research, and an appreciation, 



