328 



SCIENCE. 



[N. S. Vol. XIX. No. 478. 



discussions of obscure points should be 

 postponed as long as a due regard for log- 

 ical development will allow. Time is 

 wasted in removing difficulties whose ex- 

 istence and importance the student has not 

 yet recognized. 



These are some of the necessary exten- 

 sions into college work of the reformation 

 now urged upon the secondary schools, and 

 though every one of them seems familiar 

 enough when taken separately ; all together 

 their united application to the mathematical 

 courses in our technical colleges amounts 

 to a departure from our present traditional 

 methods little short of revolutionary. Yet 

 isn't this the thing our engineers are de- 

 manding, and isn't this the logical way to 

 train an engineer in higher mathematics? 

 Isn't it the way to approach the higher 

 mathematics anywhere or in any kind of a 

 school? 



The pure mathematician may object and 

 exclaim, What is to become of our cur- 

 ricula which have been evolved after so 

 many years of intellectual conflict! The 

 rule is so much algebra, so much geometry, 

 so much trigonometry, so much analytical 

 geometry and so much calculus. At the 

 end the student has passed with greater or 

 less success so many formal examinations 

 upon so many formal topics and his ac- 

 quirements are supposed to range some- 

 where between the maximum and minimum 

 grade of passing. But are these the ques- 

 tions which the enlightened educator of to- 

 day is asking? Is it not How much power? 

 A dry, barren, fruitless familiarity with 

 a number of highly specialized and unre- 

 lated things can not be education. The 

 engineer demands that the unity of the 

 mathematical branches should be emphas- 

 ized and that they should accumulate in 

 the soul of the student not as dry and un- 

 related facts, but as a magazine of energy.* 



* Little has been said in this paper about de- 



You may ask for some definite concrete 

 expression upon the way that the study of 

 calculus should be undertaken. This 

 paper will close with an attempt at a brief 

 answer to this question. 



We will suppose that experimentally or 

 otherwise the student is familiar with the 

 equation of falling bodies s = ^gt^. By 

 this time also the student must be some- 

 what skilled in the use of squared paper 

 and acquainted with this curve itself 

 through its application to parabolic mir- 

 rors or otherwise. Perhaps, our parabola 

 had been studied from its geometrical side 

 as a conic section. It now takes on a 

 symbolic meaning, for it gives in a certain 

 sense a picture of the first law of falling 

 bodies. But does the student grasp the 

 full meaning of the picture? Using the 

 approximation g^d2, we have a numerical 

 equation. The abscissas of the curve repre- 

 sent elapsed time; the corresponding ordi- 

 nates represent total space traversed. At 

 some point on the curve proceed geometri- 

 cally and analytically to construct the 

 tangent, at every step making a threefold 

 interpretation, one of the curve, one of the 

 analysis, and one of the fact connected 

 with these in the familiar phenomena of a 

 falling body. Show the limiting position 

 of the secant, deduce the number towards 

 which your successive numerical approxi- 

 mations tend, and connect both of these 

 with the velocity of the body at the point 

 considered. Draw the tangent and show 



scriptive geometry and mechanical drawing as 

 necessary parts of a general mathematical train- 

 ing. Both of these subjects are of the highest 

 value as disciplinary studies. They make definite 

 and effective other mathematical material. Is not 

 one reason for the barrenness of mathematics in 

 university courses the fact that these branches 

 simple though they are, have been so long ne- 

 glected? Do we not find one important explana- 

 tion of the effectiveness of technical college mathe- 

 matics in the fact that these subjects are always 

 a large part of a technical training? 



