August 7, 1908] 



SCIENCE 



163 



also be the chief aim: one does not know 

 a foreign language unless one is able to 

 think in that language ; one does not know 

 mathematics unless one is able to think 

 mathematically. It is not necessary for 

 that to go up into the highest mathematics, 

 but it is necessary to be thoroughly drilled 

 in elementary principles of each subject. 

 These elementary principles should be- 

 come a second nature to the student, just 

 as a language becomes a second nature 

 when it is thoroughly acquired. Problems 

 arise every day in the practise of an engi- 

 neer, which a mathematical mind can solve 

 without going into calculations, such prin- 

 ciples as those of maxima and minima, 

 those of least work, of cumulative effect of 

 forces and others are invaluable in assist- 

 ing to arrive at a logical solution of many 

 problems without the use of a scrap of 

 paper; but in order that they may be ap- 

 plied, one has to be able to think mathe- 

 matically. With a proper foundation, the 

 engineer's mind becomes so trained that he 

 applies those fundamental principles un- 

 consciously ; they direct his line of thought 

 automatically, so to speak. How to secure 

 such a foundation in a student must be left 

 to those who make a life-study of teach- 

 ing. 



Ealph Modjeski 

 Chicago, III. 



The methods of teaching mathematics to 

 engineering students in vogue twenty years 

 or more ago, while often sufficiently 

 strenuous, were invariably far from satis- 

 factory, in that they failed to show the ap- 

 plication of the subjects to engineering 

 practise and to explain that mathematical 

 quantities represent something real and 

 tangible, not merely abstractions. Possibly 

 methods have changed of late years; but 

 nothing that the writer has seen or heard 

 indicates to him that any fundamental im- 



provement has been effected. Most people 

 continue to believe that mathematical sub- 

 jects are taught mainly for the purpose of 

 training the mind, and that the manipula- 

 tions involved in this branch of science are 

 simply mental gymnastics. Moreover, even 

 among engineers and professors, only a few 

 recognize adequately the great importance 

 of mathematics in engineering and that it 

 is something real and substantial instead of 

 fictitious and imaginary. It is true that 

 higher powers than the third are not con- 

 ceivable entities; but the mathematician 

 recognizes them as temporary multiples for 

 future reduction to entities. 



The engineering student in his pure- 

 mathematical classes is not taught what 

 equations really mean, nor what are their 

 denominations or those of their component 

 parts. All that he learns is how to juggle 

 with quantities in order to produce certain 

 results. It is left to the professor of 

 rational mechanics to teach engineering 

 students the reality of mathematics; and 

 too often he fails to do so, sometimes, per- 

 haps, because his own conception thereof is 

 rather vague. 



Concerning the teaching of pure mathe- 

 matics by the professor of rational me- 

 chanics the writer speaks from personal ex- 

 perience ; for more than a quarter of a cen- 

 tury ago he taught that branch of engi- 

 neering education in one of America 's lead- 

 ing technical schools. Notwitlistanding 

 the fact that the courses in pure mathe- 

 matics then given there were rigid and even 

 severe, the students, as a rule, had no idea 

 of how properly to apply the knowledge 

 they had accumulated ; nor did they know 

 what the mathematical terms employed 

 really meant. It was necessary for the 

 writer not only to teach his own branch, 

 but also to supplement the students' knowl- 

 edge of pure mathematics by explaining 

 such things as limits, differential coeffi- 



