Consider the hyperbola y = 1/x. (It will be convenient to restrict attention to that part for which x > 1.) Everyone will agree that the x-axis is an “asymptote” to this curve; however, different people are likely to advance different reasons for this fact.

When the authors met at a geometry conference at Easter 1999, they found that each had discovered and investigated an elegant result (Theorem 2.1) about an octagon inscribed in a conic, Larry Evans algebraically and John Rigby synmetically. This turned out to be a rediscovery, as the result had been discussed by E. H. Lockwood in the Gazette in 1967, in an article that complements the present article. Also, almost a century earlier in 1872, the result had been posed somewhat imprecisely by T. T. Wilkinson as a problem in the Educational Times. There is a surprising connection between a special case of the octagon theorem and the golden cross-ratio.

Illustrative examples which draw on everyday experience are particularly useful in the classroom. For this reason, sport can be a good source of examples. We present four illustrations of the ways that mathematics arises in different sports. The ideas should be understandable to those studying A-level mathematics.

The celebrated Steiner-Lehmus theorem states that if two internal bisectors of a triangle are equal then the triangle is isosceles. It beautifully illustrates the point that the converse of a theorem may be far more difficult to prove than the direct statement. The literature of the Steiner-Lehmus theorem is considerable with many contributions in the Mathematical Gazette.

An interesting variation on this theme may be found by examining the triangle formed by the second intersections of the medians with the circumcircle. Let the median point of a triangle ABC be G and let the medians meet the opposite sides at DEF and the circumcircle again at LMN.

A Little more than three hundred years ago Napier announced to the world his discovery of Logarithms in the Mirifici Logarithmorum Canonis Description “ I have always endeavoured,” we read in one of his works, “ according to my strength and the measure of my ability to do away with the difficulty and tediousness of calculations, the irksomeness of which is wont to deter very many from the study of mathematics. With this aim before me, I undertook the publication of the Canon of Logarithms which I had worked at for a long time in former years; this canon rejected the natural numbers and the more difficult operations performed by them, substituting others which bring out the same results by easy additions, subtractions and divisions by two and by three.” Soon after the publication of the Canon, what he calls “another and better kind of logarithms “occurred to him, practically logarithms to the base 10, as we now know them, and a similar change in his system also suggested itself to Briggs, then Gresham Professor of Geometry in London, one of the first to recognise the immense value of Napier’s discovery In the interval between 1614 and Napier’s death in 1617, they had devised several methods for the construction of the new Tables; but owing to Napier’s failing health he had decided to leave the actual computation to Briggs. As early as 1617 Briggs was able to publish the logarithms of the numbers from 1 to 1000 to eight places. In 1624 he followed this with the numbers from 1 to 20,000 and 90,000 to 100,000 to fourteen places. In 1628 Vlacq, a Dutchman, published the logarithms of all numbers from 1 to 100,000 to ten places, along with a Table of the Logarithms of the Trigonometrical Ratios for every minute, to the same number of places. These Tables of Briggs and Vlacq have never been superseded—their work was done for all time. They contain errors which have gradually been discovered and corrected, but all the Tables that have been published since their time, with one single exception, have been copied, directly or indirectly, from their work.

I recently came across a very appealing result about the product of the lengths of the chords of a regular polygon with n sides inscribed in a unit circle. If one vertex is selected and the product of the chord lengths from that vertex to each of the others is evaluated, the result is just n. There is a simple proof. Consider such a polygon and denote its vertices by the points A0, A1, … , An. Suppose that O is the centre of the circle and the origin of the complex plane. The roots of the equation

are just where ω is a primitive nth root of unity. Each of these roots represents one of the vertices. Suppose for convenience that the real number 1 is represented by A0.