Dedicated to the memory of Russell Towle, a remarkable man who contributed so much to geometry and to other aspects of the quality of life.

We introduce an expanded notation where r + s = n, for the binomial coefficient , and then use this expanded notation to develop theorems involving 8 binomial coefficients, analogous to the Star of David Theorem, which. in its original form, involved the 6 neighbours of a given binomial coefficient in the Pascal Triangle (see Section 3), that appeared in [1,2,3,4,5,6,7,8,9].

The December 1978 issue of the Mathematical Gazette [1] contains an elegant and humorous contribution from Anthony Hughes. His Note gives a recipe for determining the optimal spot from which to make a conversion attempt in rugby. Others then elaborated on Hughes' idea; see [2] and [3] in particular. In 1996 Isaksen [4] rediscovered Hughes' results while investigating the kicking of extra points in American gridiron. There are also a number of popularisations and summaries, in which the above results are presented, and in instances rediscovered. The ones of which we are aware are listed in the references; see [5, 6, 7, 8, 9 and 10].

The beautiful infinite product of radicals

due to Vieta [1] in 1592, is one of the oldest non-iterative analytical expressions for π, In a previous paper [2] the author proved the following two Vieta-like products:

for N even, and

for N odd. Here N is a positive integer, FN and LN are the Fibonacci and Lucas numbers, and is the golden section. (The Fibonacci numbers are F1 = 1, F2 = 1, with the recursion relation , while the Lucas numbers are L1 = 1, L2 = 3 with the same recursion relation )

The probability integral theorem, which states that

is on the record for having enticed numerous mathematicians to find alternative proofs for it, over a period of more than two centuries, till recent times (for a few recent references see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]). The most popular demonstration (attributed to the French mathematician Poisson; see [11], in references of [12]), the one found in almost all textbooks, relies on the double integral

(taken over the upper-right quarter of the Cartesian plane) obtained by squaring the probability integral. By resorting to polar coordinates and writing down the above integral as:

the value of the probability integral is obtained by taking the square root of the result on the right-hand side of this latter relation.

This work grew out of a simple problem: how could I introduce the basic concepts of calculus to students who were mathematically under-prepared? It has been pointed out by many authors (see [1] and its references) that limits are a major stumbling block for many students beginning their study of calculus. Part of the difficulty with limits is the conceptual confusion between process and value. Some limits can be evaluated by a simple process of substitution, for example,

while others cannot, for example,

The purpose of this paper is to present some results regarding the limit is of the sequence

In the problem proposed by A. G. Ioachimescu in 1895 [1], it is asked to be shown that the sequence , defined by , for each , is convergent and its limit lies between -2 and -1.

There have been given many generalisations and other results regarding Ioachimescu's problem in the literature (see, for example, [2, problem 3.1, p. 431], [3,4], [5, Theorem 1, parts (a) and (b)], [6, problem P2, parts (i) and (ii)], [7, 8]).

I often recommend to my students that they follow their noses when solving mathematical problems. The role of intuition in creative mathematical work is difficult to explain. Still its value should never be underestimated and its development may be fostered by experience, including even unsuccessful trips down dead ends and meandering forays into thickets of apparently untameable complexity. In this article I invite you to corne with me and explore a simple calculus problem using any tools and ideas that seem to offer some promise of moving us along in the right direction. The scenic route embraces Taylor's formula, geometric series expansions, relations in Pascal's triangle, and matrix diagonalisation. Don't forget to bring along a pencil and some scrap paper… and a computer algebra system too, if you like.

The ellipse is well known to students of mathematics at an advanced high school level and at an undergraduate level through its associations with, say, conic sections and their deployment as hardware in dual reflector antenna systems. Perhaps less well known is its appearance on the software side, so to speak, for example in RF (Radio Frequency) engineering in the context of polarisation, where it is known as the polarisation ellipse. It is the purpose here to discuss this manifestation and dwell particularly on the concept of a circumscribing rectangle for this ellipse because such aspects are of interest to the RF engineer who may be involved with measurements to determine its properties. This exercise affords an interesting insight into a very practical application of tertiary level mathematics and whilst it is of course helpful to appreciate the physical concepts involved, a lack of such knowledge does not detract from an appreciation of the underlying sums, which should be within the grasp of the above, interested student.

In this note we will discuss five topics from triangle geometry and occasionally encounter something new. Areal coordinates (presented as Appendix A in [1]) will be used throughout, with the exception of the final section where it is advantageous to employ Cartesians.

The topics are:

(1) examples of collinearities where the distance ratios for important points vary with triangle shape;

(2) a pencil of lines which includes the Fermat, Napoleon and Vecten axes;

(3) two more circles related to the Lester circle;

(4) the family of rectangular hyperbolae which circumscribe the triangle of reference ABC and whose principal member is the Kiepert hyperbola [2];

(5) Cartesian coordinates leading to a solution of the ‘Euler disc problem’ [3].

In [1], Chris Ormell raised a question about the uncountability of the real numbers. Ormell affirmed that only a countable collection of numbers may be defined by a finite number of words, a notion that stems from a letter of Richard (1905), reproduced in [2, p. 142]. This appears to conflict with Cantor's proof (1874) of the uncountability of the reals [3, p. 839]. Poincaré, whom Ormell claims as a witness in his defence, said ‘Now as is well known, Cantor proved that the continuum is not denumerable; this contradicts the proof of Richard. The question therefore arises which of the two proofs is correct. I maintain that they are both correct and that the contradiction is only apparent.’ [3, p. 1072] Poincaré then supplied a new proof that the points on an interval are not countable.