Two prominent themes in the Presidential Address upon which this article is based were Enthusiasm and Challenge.

Many, if not most, practitioners of Mathematics can trace their interest in the subject to an enthusiastic teacher at some stage of their education. In this day and age, it is all too easy to get bogged down in a morass of assessment and quality assurance. Yet it is vital that teachers of Mathematics at all levels retain their enthusiasm and pass this on to their students.

How refreshing to read C. D. Collinson’s [1] article in the March 1986 Gazette on the importance of stressing approximations in teaching mechanics. Once students get over their initial scepticism that mechanics has anything to do with the real world, I too find an enthusiastic response from classes invited to contribute their ideas on the sort of approximations and assumptions which have to be made to model a physical situation. Approximations and assumptions are fundamental to the art of mathematical modelling in mechanics and cause more difficulties to students than is often realised; class discussions reveal that many of the problems which students have with mechanics are related to approximations, and such discussions can also bring out some unsuspected fundamental misconceptions. One such misconception, related to approximations, is discussed here and the proposed method of clarification includes developing a little more Collinson’s use of “the linear approximation”.

The sums of powers of the first n positive integers Sp(n) = 1p + 2p + …+np, (p = 0, 1, 2, … )satisfy the fundamental identity

(1)

from which we can successively compute S0 (n), S1 (n), S2 (n), etc. Identity (1) can easily be proved by using the binomial theorem; see e.g. [1, 2]. Several variations of (1) are also well known [3, 4, 5].

In this note, we derive the following lesser-known recursive formula for Sp (n):

(2)

where denote the (unsigned) Stirling numbers of the first kind, also known as the Stirling cycle numbers (see e.g. [6, Chapter 6]). Table 1 shows the first few rows of the Stirling number triangle. Although the recursive formula (2) is by no means new, our purpose in dealing with recurrence (2) in this note is two-fold. On one hand, we aim to provide a new algebraic proof of (2) by making use of two related identities involving the harmonic numbers.

It is proposed to discuss in this paper some properties of the general Frégier-point and point out its connections with well known conics like the Apollonian hyperbola and the nine-point conic of a quadrangle. Incidentally, we discover a pedal conic of a quadrangle which has very intimate relations with the nine-point conic.

An elementary algebraic problem attributed to Leonardo of Pisa is analysed and some illogical elements in its formulation and solution are exposed. The natural context in which the problem was formulated is then proposed, and some consequences are discussed.

How many times have you heard that somebody is a wizard? And how many times did you take it literally? Most likely, the answer to the second question is ‘never’. And yet, there are reasons to believe that some people among us are real wizards, of the kind described with so much charm in the recently published series of books on Harry Potter [ 1, 2, 3, 4 ]. These books have provided us with a wealth of details on wizards and their society.