294 



SCIENCE 



[N. S. Vol. XXVIII. No. 714 



the use of that power discover methods of 

 solving on paper synthetic problems of 

 much difficulty. The German schools 

 teach descriptive geometry as a mathe- 

 matical subject, the American schools as a 

 body of problems to be solved by rule on 

 the drawing board. The former method 

 makes descriptive geometry the finest dis- 

 cipline of the four yeare' course; from 

 the other method little educational benefit 

 arises. Some years ago at the Rose Poly- 

 technic, where for a time we taught de- 

 scriptive geometry in the German way, it 

 was not unusual to meet students who 

 declared enthusiastically that they got 

 more real good from this subject than from 

 anything else in their entire course. 



I would ask the new committee to inquire 

 how and by whom descriptive geometry 

 should be taught 1 



By C. B. Williams, Professor of Mathe- 

 matics, Kalamazoo College. 

 The teachers of mathematics in the 

 small colleges of the middle west are pre- 

 paring many men for work in the better 

 technical schools. From our standpoint 

 there is substantial agreement between the 

 two representatives of the Massachusetts 

 Institute of Technology (Professors Wood 

 and Swain). They expressed themselves 

 so differently that one might easily fail to 

 see how closely they agree. Both want 

 longer and stronger courses in mathe- 

 matics in the secondary schools. I would 

 like to know the coUege teacher of mathe- 

 matics who does not agree with them. 

 They want more mathematics taught and 

 to have it taught better, to have longer and 

 more consecutive mathematical courses in 

 the secondary and primary schools. In 

 other words, the faculties of the technical 

 schools and colleges are working toward 

 the same end, that is, to have more effective 

 courses in primary and secondary mathe- 

 matics so that college students can do more 



and better mathematical work. If we 

 could have properly prepared students, we 

 could turn out the kind of men the better 

 technical schools should have. 



The engineers and teachers of engineer- 

 ing have insisted that the most necessary 

 qualification for a real engineer is that he 

 should be able to realize his mathematics, 

 to ' ' think mathematically, ' ' as they express 

 it. The mathematicians want the same 

 thing. We are trying to make use of and 

 to train the faculty of geometric intuition, 

 to emphasize the functional notion and to 

 develop functional thinking. There is 

 substantial agreement that the best way 

 to do this is through geometry, with per- 

 haps some help from elementary me- 

 chanics. It is true that sometimes we are 

 tempted to use too big and complicated 

 machines for little problems, but this is 

 only because we are attempting to develop 

 methods powerful enough to solve big 

 problems. 



By J. B. Webb, Professor of Mathematics 

 and Mechanics, Stevens Institute. 

 Every practical problem requiring 

 mathematics for its solution consists of 

 three parts: 



(a) An Analysis, which resolves the 

 problem into its elements, examines these 

 in the light of natural laws, rejects unim- 

 portant ones and defines the relations exist- 

 ing between those upon which the solution 

 depends. This involves the adoption or 

 discovery of methods of measuring the 

 elements, so that they may be expressed 

 quantitatively by symbols, and of the re- 

 duction of the relations between them to 

 the standard mathematical forms of ex- 

 pression. The result is a mathematical 

 statement of the proUem by one or more 

 equations. 



(&) A solution of the equations by which 

 the relations sought for between the quan- 

 tities are clearly expressed or the quanti- 



