It is a very great honour and privilege to be addressing you today. It has been a particularly interesting year politically at least as far as mathematics is concerned. Despite what seems in many ways a prosperous and optimistic situation in this country at the moment, mathematics is in a very serious state. Our subject is the key to living successfully in the modern technological world, but we have a grave shortage of mathematicians. In addition there are endless complaints from employers and the general public that young people leave school without the necessary mathematical skills to cope with everyday life.

The use of the Greek letter π to denote the ratio of the circumference of a circle to its diameter appears to have been popularised by Euler following his adoption of the symbol in 1737 and later in his many popular textbooks. The original use of the symbol for this purpose was in fact by the Welsh mathematician William Jones who used it his book Synopsis Palmariorum Matheseos published in 1706. This book was described in 1824, over a hundred years later as follows ‘This work has long been esteemed as a compendious, but comprehensive summary of Mathematical Science’.

Except for the circle, for which the true anomaly v is proportional to the time t, the position of a body in orbit about a central body at a given time is simplest to derive for a parabola. The classical determination of the time of flight on a parabolic trajectory is through the integration of the dynamic equations of motion. (See Appendix.)

A theorem of Van Aubel, which first appeared in concerns the figure obtained by erecting squares on the sides of a quadrilateral, and considering various line segments such as the joins of the centres of opposite squares. In parts of the theorem are extended to the case of (i) four similar rhombi, and (ii) four similar rectangles erected on the sides of a quadrilateral. Further, these two results are described as in some sense ‘dual’ to each other. Here we show that they are in fact both special cases of a further extension of the original theorem (see Proposition 6, and the remarks that follow it), and we also find several extra squares in the original Van Aubel figure (Proposition 11).

In standard textbooks (e.g. [1, p. 877]) as well as in Encylopaedia Britannica [2, Vol. 13, p. 611], one finds variants of the following statement:

This statement is actually false, and can only be understood in the following sense: It is true only up to a linear approximation, and in fact, as soon as a mirror focuses parallel rays, it must be a parabolic mirror.

An appreciation of the geometry underlying algebraic techniques invariably enhances understanding, and this is particularly true with regard to polynomials.

With visualisation as our theme, this article considers the cubic equation and describes how the French mathematicians François Viète (1540–1603) and René Descartes (1596–1650) related the ‘three-real-roots’ case (casus irreducibilis) to circle geometry. In particular, attention is focused on a previously undescribed aspect, namely, how the lengths of the chords constructed by Viète and Descartes in this setting relate geometrically to the curve of the cubic itself.

Suppose the unit circle is divided into n > 1 equal arcs and points are placed at the ends of the arcs. Choose one of these points to be the ‘base point’ and draw chords connecting each of the other points to the base point. (See Figure 1 for the case n = 8.) Then the product of the lengths of these chords equals n (see Sichardt). In a related article, Eisemann considered the product of the lengths of the (nondegenerate) perpendiculars drawn from the centre of the unit circle to these chords.

Family history research has become a very popular hobby during the last few years due to the ready access to records of births and marriages. Many people have rigorously traced their ancestors back to 1700 and earlier, corresponding to some 12 generations. The principal motivation of most amateur genealogists is to learn about the lives of their forebears and they trust these ‘legal’ records without question and do not usually acknowledge that cuckoldry may have occurred and that their true ancestors may not have been the ones recorded.

In the configuration, illustrated in Figure 1, ABC is a triangle with I1; I2, I3 the excentres opposite A, B, C respectively. The triangles I1BC, I2CA, I3AB are denoted by T1, T2, T3 respectively. Ok, Hk, are the circumcentre, orthocentre respectively of triangle Tk, k = 1, 2, 3. Lk is the Euler line of Tk, k = 1, 2, 3. Define the point Ek (t) on Lk by the equation

In [1, Chapter 3, Section 2], we collected together results we had previously obtained relating to the question of which positive integers m were Lucasian, that is, factors of some Lucas number Ln. We pointed out that the behaviors of odd and even numbers m were quite different. Thus, for example, 2 and 4 are both Lucasian but 8 is not; for the sequence of residue classes mod 8 of the Lucas numbers, n ⩾ 0, reads

and thus does not contain the residue class 0*. On the other hand, it is a striking fact that, if the odd number s is Lucasian, then so are all of its positive powers.

In common with, I suspect, many people the author does not have access to the NAG library and so, when I was asked recently to calculate the value of the integral

correct to 10 decimal places my first reaction was to try several different calculators as well as several mathematical software packages. On doing so it was disappointing to find they either gave widely differing values such as 7.9065200767, 4.1317217452 or 0.9174196842 or an error message indicating that the method had not converged.

A topic of current interest is that of finding defining sets (preferably smallest defining sets) for various combinatorial structures such as latin squares and block designs, so it seemed interesting (though of no practical use) to try to solve the same problem for magic squares of small size. (The germ of this idea was suggested in the Gazette Note [1] of 1988 under the sub-heading ‘Squares from clues’.)

One of the bedrock theorems of mathematics is the statement that a real polynomial of degree n has at most n real zeros. Probably the best-known proof is the algebraic one, by factorisation. But there is also a pleasant analytic proof, by deduction from Rolle’s theorem.

A slightly different question is how many positive zeros a polynomial has. Here the basic result is known as ‘Descartes′ rule of signs’. It says that the number of positive zeros is no more than the number of sign changes in the sequence of coefficients. Descartes included it in his treatise La Géométrie which appeared in 1637. It can be proved by a method based on factorisation, but, again, just as easily by deduction from Rolle’s theorem.

Problem 87.I in a recent Gazette concerns cyclic quadrilaterals.

We shall investigate generalisations of the problem by considering cyclic polygons and also points on a sphere, by distinguishing between sides and diagonals, and by looking at minima as well as maxima. The term ‘edge’ in the title is intended to encompass both sides and diagonals. It is interesting to see the wide variety of methods needed to solve the generalised problems.

Smallest defining sets

A standard Sudoku square is a 9 × 9 latin square in which each of the nine 3 × 3 subsquares into which it can be separated contains each of the integers 1 to 9 exactly once.

A current problem is to complete such a square when only some of the cells have been filled. These cells are often called ‘givens’. (Such problems are currently (2005) published daily in British newspapers.) In more mathematical terms, the given filled cells constitute a defining set or uniquely completable set for the square if they lead to a unique completion of the square. If, after deletion of any one of these givens, the square can no longer be completed uniquely, the givens form a critical set. The investigation of critical sets for ‘ordinary’ latin squares is a topic of current mathematical interest. (See [1] for more details.)