552 



NATURE 



[July 31, 1913 



— we fail to see the superiority of Prof. Perry's 

 questions over the worst examples of the old 

 Cambridge school. 



A very few lines of explanation, based on the 

 definition as a limit, would make the student take 

 an intelligent interest in e. But having started 

 on the wrong tack, Prof. Perry, on p. 150, fails 

 to find a satisfactory proof of the differentiation 

 formulae involving e without assuming the ex- 

 ponential series, and by the time he uses the 

 formula?, on p. 189, it is too late to exhibit the 

 significance of this important limit. An intelli- 

 gent boy ought to be able to understand the com- 

 pound interest law and the ordinary differentiation 

 formulas long before he learns how to differentiate 

 the infinite series employed in Prof. Perry's proof. 

 Take next the formula for the belt slipping on 

 the pulley (p. 37). The formula N/'M = e* 8 conveys 

 no meaning to the student of average intelligence, 

 and it is not the method that anyone with common 

 sense would employ in experimental work. What 

 he would do would be to use the formula N/M = 

 c", where c is a constant and n the number of 

 turns, c being found by experiment. 



The same mistake is made with the radian. 

 Prof. Perry (p. 62) expects his students to be as 

 ready to think in radians as in degrees, but he 

 conspicuously fails to impress his readers suffici- 

 ently with the utility of the radian in connection 

 with the relation between angular and linear 

 velocity and differentiation formulae. 



The chapter on algebra is a good feature, if for 

 no other reason than the fact that existing text- 

 books on algebra are still so unsatisfactory. The 

 proper method of introducing algebra is in con- 

 nection with the use of formulae, and the converse 

 use of formulas naturally leads to the problem of 

 solving an equation. In the conventional treat- 

 ment the utility of the subject is completely 

 ignored, and the study is presented in the form of 

 hateful drudgery. But here, again, Prof. Perry 

 lays stress on such problems as, " Divide a number 

 into two parts," or "A father is 3*5 times as old 

 as his son," of which we have had too many 

 already. 



In the chapters on mensuration, squared paper, 

 and important curves, Prof. Perry is working on 

 what is now well-known ground ; at the same 

 time his treatment is in many respects unsatis- 

 factory, particularly in connection with curves. 

 Thus we all know the importance of the cycloid 

 in geometry, mechanics, and physics. But all 

 that Prof. Perry does is to make the student plot 

 this curve on squared paper by means of the 

 equations x = a(4> — sin</>) and y = 0(1 — cos<£). 

 When this is done the student knows nothing 

 whatever about what a cycloid really is. An in- 

 NO. 2283, VOL. 91] 



telligent boy should learn to plot curves not only 

 from their equations, but from their geometrical 

 definitions ; and, further, he should be trained 

 to plot envelopes as well as loci. The mere draw- 

 ing of graphs may easily degenerate into unintelli- 

 gent drudgery quite as objectionable as any of 

 the old algebraical drill of our schooldays. What 

 is the use of asking boys such questions as the 

 following? 



" Find a value of x to satisfy 



.., ^o.io4i s j n a o 8.1 +078-1'' ■■''- COS .1'- 2'126 = 0. 



" The student must remember that o'Sx is in 

 radians, and must be multiplied by 57'2g6 to con- 

 vert it into degrees. Ans. x = C74." 



In the sections on the calculus there is not very 

 much fault to find with the practical illustrations, 

 and, indeed, most of them are based on fairly 

 reasonable views. But when the author comes to 

 establishing differentiation formulae he falls into 

 the error of denning a differential coefficient as 

 the limit of 



/fr+atr)- /.. 

 Sx 



instead of regarding it as the limit of 



when x 2 and .Vj both approach a common limit x, 

 which may or may not be taken to be equal to 

 either x 2 or x v Consequently he introduces 

 higher powers of Sx, which he afterwards has to 

 neglect, and which ought never to have come in. 

 The alternative definition here suggested leads at 

 once to Lagrange's remainder theorem in the form, 



/(* 2 ) =/(*!) +(*2-*l)f(*)> 



where x has some value between x^ and x». 



The result is that in differentiating x n Prof. 

 Perry assumes the binomial theorem plus certain 

 other assumptions not stated, whereas any pupil 

 ought to differentiate x 11 long before he has heard 

 of the binomial theorem. In speaking of limits 

 Prof. Perry says : — 



"The plain man of common sense finds no diffi- 

 culty in catching the idea. Two thousand years 

 ago neither he nor a small boy would have had a 

 difficulty in understanding that a hare would beat 

 a tortoise in a race ; it is the mathematical philo- 

 sopher who makes a difficulty about such matters, 

 and in these days he says that this fundamental 

 idea of the calculus can only be comprehended by 

 a mathematician. This would not matter if these 

 philosophers were not entrusted with the educa- 

 tion of youth, a trust for which all their training 

 has unfitted them. When they come to explain 

 the essential idea of the limiting value of Ss/St, 

 they talk foolishly." 



Readers of " Elementarv Practical Mathematics " 





