Ladies and Gentlemen,

There are many different forms of Presidential Address. To glance back at those delivered to this Association is at once an exhilarating and a humbling procedure. Indeed, at one stage during it I toyed with the notion of filling this session with an amalgam of abstracts. You might have been better served had I done so; as it is I shall be quoting from three only and the reasons for that particular selection should become apparent before long.

One of the most striking results in the theory of uniform convergence asserts that for a monotonic sequence of continuous functions, continuity of the limit function is a necessary and sufficient condition for uniform convergence. Much has been written about this theorem, but the fact that the proofs given in such works as Bromwich’s Infinite Series and De La Vallée Poussin’s Cours d’Analyse are incomplete, seems to justify the hope that this account may be of some assistance to students and teachers of convergence.

This note was suggested by Dr. Maxwell’s query in the May “Gazette” for a simple proof of the fact that the rectangular hyperbola whose centre is the point of contact of the nine-point circle and the inscribed circle of a triangle and which passes through the vertices, also passes through the incentre. This fact appears as a special case, together with a number of other interesting results, which include a new proof of Feuerbach’s Theorem.

Consider a rational (unicursal) plane curve of degree n given by

x : y : z = f(t) : Φ(t) :ψ(t)

In this paper we will discuss the lemniscate curve and show that its arc length can be bisected and trisected using classical ruler and compasses construction. The method dates back to 1718 when Count Giulio Fagnano (1682-1766) first published these constructions [1]. Fagnano was self-educated in mathematics and treated the subject as a hobby. Euler was impressed by his work on this topic and recommended his admission to the Berlin Academy of Science. Euler then generalised Fagnano's work on integrals. In addition, Fagnano was employed to assist in reinforcing the dome of Saint Peter's which was in danger of collapse. Pope Benedict IV rewarded him for his work by publishing his mathematical papers. Fagnano achieved considerable international fame as a mathematician, and rightly so given the outstanding contributions which he made on a number of different topics.

Which is your favourite formula in triangle geometry? Mine is definitely the formula for the area of triangle OIH . It is well known that the perpendicular bisectors to the sides of any triangle are concurrent at a point O (centre of the circumcircle), that the angle bisectors to the vertex angles are concurrent at a point I (centre of the incircle), and that the altitudes are concurrent at a point H. If the triangle is not isosceles, then these three points are all different and uniquely determine a new triangle OIH (see Figure 1), whose area can be expressed in terms of the sides a, b, c of the original triangle. I derived such a formula 20 years ago, and later found out that it had been studied a century earlier.