TRANSITION FROM ELEMENTARY ALGEBRA TO 

 THE CALCULUS WITHOUT INFINITE SERIES.* 



By Prof. W. N. Roseveare, M.A. 



This paper divides itself into three parts :— 



I. The details of the transition. 

 II. The introduction of the (natural) logarithm. 

 III. The method of approach to infinite series. 



I frankly confess that I have no doubts left as to the wisdom 

 of developing the idea of limits early. The student of mathe- 

 matics has to face it in the geometry of a tangent and of the 

 circle, in the definition of velocity (when not uniform), and, 

 indeed, in the logic of everyday afifairs. At tennis he uses 

 his sense of sight and his conscious reasoning powers up to the 

 last moment. The process by which he finally decides how 

 and when to hit is almost exactly the mathematical " going to 

 the limit." If we see two motor-cars approaching one another 

 at 60 miles an hour, and within a yard of colliding, we need no 

 further evidence of an actual collision. 



This " limit "' process may be simply described as following 

 a train of reasoning applied to an increasing number of things as 

 far as is necessary for conviction, and then giving full rein to the 

 imagination as to the result to which this conviction will lead 

 when the finite number of things becomes an indefinitely great 

 number. We may add, as another apt illustration, the phy- 

 sical process of ' generalization' ; for instance, Newton's Law of 

 (Gravitation. 



Teaching experience leads me to the conclusion that 

 the average student masters the idea readily enough ; 

 but the ordinary text-book does its best to nullify one's efforts. 

 In the case of the tangent the books are quite sound — are with 

 us. When we reach the regular polygon and the circle, the 

 books refuse the word ' limit.' and thereby increase the teacher's 

 difficulties. With some diffidence (because I think our Cape 

 syllabuses in mathematics are generally very good), I venture 

 to call " logically deplorable " the note in our Intermediate 

 Syllabus : " It is to be assumed that it is impossible to distin- 

 guish between a circle and an inscribed regular polygon of a 

 sufficiently large number of sides." The iniar/iiiatioii, to which 

 at this point we are specially appealing — the main difficulty is 

 to make the student put down his pen, stop counting, and let 

 his imagination work freely — has, of course, no difficulty what- 

 ever in making the distinction between a circle and a polygon, 

 however large the number of sides. Why not " A circle may be 

 regarded as the liinit of an inscribed or circumscribed regular 

 polygon when the number of sides is increased indefinitely "? 



* In this paper no knowledge of Algebra is assumed beyond indices — - 

 no Progressions, no Binomial. 



