258 



SCIENCE 



[N. S. Vol. XXVIII. Xo. 713 



Hamilton written seventy-sis years ago, in 

 ■which he maintains that there is no one 

 of the subjects in the curriculum which 

 develops a smaller number of mental facul- 

 ties or develops them in a more imperfect 

 and inadequate manner than mathematics. 

 I have never seen what has seemed to me a 

 conclusive refutation of Sir William 

 Hamilton's main arguments, and for my 

 part I am disposed to agree with him in 

 general, and to assign a comparatively low 

 value to mathematics simply as a training, 

 aside from its applications. I have not 

 observed that students trained in this sub- 

 ject are able to reason any better than stu- 

 dents who have ignored mathematics; in- 

 deed, I believe that many non-mathe- 

 matical subjects afford a better training in 

 reasoning than the study of mathematics. 

 This view may perhaps be justified by re- 

 membering that mathematics, aside from 

 geometry, deals with questions of quantity 

 and number, but not with questions of 

 quality. The student puts certain fixed 

 data into his mathematical machine and 

 grinds out the result. He does not learn 

 to observe and to discover the finer and 

 more elusive, but equally important, 

 sources of error likely to occur in the ordi- 

 nary questions of daily life, because he is 

 dealing with a rigid, unyielding, logical 

 machine. In this way his mind may be- 

 come hardened— he deals with rigid demon- 

 strations and is unwilling or unable to ap- 

 preciate or submit to a less rigid method, 

 which is often the only possible one. The 

 best student of mathematics is frequently 

 one of the poorest of engineers. Give him 

 fixed data and he will get the proper result, 

 but he may be entirely incapable of attack- 

 ing a practical problem, or of deciding 

 what the proper data are. 



I have not observed that students of 

 mathematics are, as a rule, more accuraie 

 than other students, or that a training in 

 the branches of mathematics above arith- 



metic leads to accuracy. Indeed, it more 

 often appears to pervert the sense of per- 

 spective, and to lead students to work out a 

 result to several figures in cases where a 

 smaller number only may be significant. 

 j\Iathematics does not train the obseiTation, 

 neither does it train the imagination, except 

 in the geometrical branches, which are now 

 comparatively neglected since the powerful 

 modern methods in analysis have been in- 

 troduced. 



Hamilton only allowed, as I remember, 

 that mathematics adequately trained one 

 faculty, namely, that of continuous atten- 

 tion: biit I fail to see that this is trained 

 any better by the study of mathematics 

 than by that of language, chemistry or by 

 other natural sciences. Unfortunately, as 

 at present taught it does train the memory, 

 in a way that it ought not to do. Th& 

 ordinary student of mathematics subor- 

 dinates perception to a memorization of 

 formiilffi and rules. 



I believe, therefore, that from the point 

 of view of the engineer, mathematics should 

 be taught with the object of giving the 

 student power to use it as a tool. With 

 reference to this I think it is fair to say 

 that the consensus of opinion among engi- 

 neering teachei-s and practitioners is that 

 the results of the present mathematical 

 training are very poor. The average stu- 

 dent who has completed his mathematical 

 course is frequently quite helpless when 

 called upon to attack a concrete engineer- 

 ing problem, and it is a common remark 

 by civil engineering students that they did 

 not really learn any mathematics until the;^ 

 studied mechanics or the theory of struc- 

 tures. The resiilts seem to be almost 

 equally poor no matter what iustitution the 

 student comes fi"om, for in my classes there 

 have been stiidents from most of the prin- 

 cipal universities and technical schools in 

 the country and I have failed to notice any- 

 great difference in them in that respect. 



