The cubic holds a double fascination, since not only is it interesting in its own right, but its solution is also the key to solving quartics. This article describes five fundamental parameters of the cubic (δ, λ, h, xN, and yN), and shows how they lead to a significant modification of the standard method of solving a cubic, generally known as Cardan’s solution.

Let us first consider the 2 x 2 matrices and Their sum and product are given by

Here the entries a, b, c, d, e, f, g, h can come from a field, such as the real numbers, or more generally from a ring, commutative or not.

In a recent paper (1) formulae were given for the numerical integration of a function in terms of its values at a set of arguments at equal intervals. In this companion paper, formulae for numerical differentiation, using the same data, are collected. Their utility in enabling derivatives of a function given numerically at such a set of arguments to be computed is obvious, the need arises in several approximate methods which are coming more and more into use (2), (3). The formulae avoid the labour of preliminary differencing, and are indeed more convenient than the finite difference formulae when the derivative is required at all the points of subdivision of a limited range.

A number of particles are constrained to move freely on the surface of a sphere. Between every pair of them there acts a force which depends only on the distance between them, the law of force being the same for all particles. For a given law of force and a given number of particles, there are a number of configurations of the particles in which they are in equilibrium. In general, these configurations will depend on the law of force: we do not expect the particles to remain in equilibrium if the law of force is changed. But there are certain configurations in which the particles are in equilibrium whatever the law of force may be, for instance if the particles are at the vertices of a regular polyhedron. In this paper these configurations are investigated. It is evident that if a configuration has rotational symmetry about the diameter through a certain particle P of it, i.e. if the configuration can be rotated through a fraction of a revolution about the diameter through P and thereby brought into coincidence with its initial position, then the particle P is in equilibrium whatever the law of force may be. Thus among the configurations with the desired property are those which have this rotational symmetry about every diameter through particles of the system. It is shown that these are in fact the only such configurations. Specifically, the only such configurations are those in which the particles are equally spaced on a great circle, with or without two further particles one at each pole, and those which, in relation to a regular polyhedron, comprise a set of particles at its vertices or a set at the mid-points of its edges or a set at the centres of its faces or any two or all three of these sets taken together. (This classification is redundant but it seems the most convenient.) No discussion of the stability of the configurations is given, as this depends in any given case on the law of force, the present discussion being confined to the property of equilibrium with independence of the law of force. The main proof occupies §§ 2-4. In §5 there is discussed the problem of two or more sets of particles on the sphere which are to be in equilibrium whatever the laws of force may be between particles both of the same set or of different sets. §6 relates the present problem to that of maximising the least distance between any two of an assigned number of particles.

Recall the integral definition of the gamma function: for a > 0. By splitting this integral at a point x ⩾ 0, we obtain the two incomplete gamma functions:

(1)

(2)

Γ(a, x)is sometimes called the complementary incomplete gamma function. These functions were first investigated by Prym in 1877, and Γ(a, x) has also been called Prym's function. Not many books give these functions much space. Massive compilations of results about them can be seen stated without proof in [1, chapter 9] and [2, chapter 8]. Here we offer a small selection of these results, with proofs and some discussion of context. We hope to convince some readers that the functions are interesting enough to merit attention in their own right.

It was conjectured by David Gregory that a sphere can touch thirteen non-overlapping spheres equal to it. In a recent paper proofs have been given that this in fact is impossible. In the present paper I outline an independent proof of this impossibility, certain details which are tedious rather than difficult being omitted.

Suppose there are n towns every pair of which are connected by a single one-way road (roads meet only at towns). Is it possible to choose the direction of the traffic on all the roads so that if any two towns are named there is always a third from which the two named can be reached directly by road?

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.

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.

In [1], C. D. Langford asked for those plane figures which can be dissected into four “replicas”, congruent to one another and similar to the original figure. (An equivalent formulation is that four identical figures are to be assembled into a scale model, twice as long and twice as high.) In addition to triangles and parallelograms, which always have this property (see Figure 1), he also exhibits three trapezoids (Figure 2) and three hexagons (Figure 3) with this property. Finally, Langford gives an example involving a stellated hexagon (Figure 4). He then asks if the list is complete.

Generalization of the ellipse to

is credited to Gabriel Lamé, the 19th-century French physicist and the eponym of this family of curves. Lamé curves have recently been taken out of mathematical limbo due to their appeal to certain designers and architects. Particularly to be mentioned is Piet Hein, the contemporary Danish poet-designer-scientist (and inventor of mathematical games) who rediscovered the curves and has been using “superellipses” (Lamé curves with n > 2, and therefore oval) in objets d’art, furniture, pottery, fabric patterns, and so on. His major achievement to date is a sunken oval shopping plaza, promenade, and pool in the centre of Stockholm; its shape is a superellipse with and . Derivatively, Gerald Robinson, a Toronto architect, has incorporated a superelliptic parking garage into a shopping complex (called a Superblock) in downtown Peterborough, Ontario; the parameters are and n = 2.71828 … (the exponential coefficient). In such urban contexts the two-parameter super-ellipse is efficiently tailorable to the dimensions of the site. It is said to be very elegant.

Lessons on sets have become commonplace in schools today. Venn diagrams proliferate and, even in primary schools, children can be seen sorting and classifying objects by size, colour and shape and placing them in spaces marked out on the floor by chalk outlines or wooden hoops. Older children learn that such diagrams are named after the English logician, John Venn, and that through them we can represent the relations of membership and inclusion and the operations of union, intersection and complementation. A rectangle is drawn to represent the universe U: subsets of U are represented by the interiors of circles, or other closed curves within U, i.e. subspaces of the rectangle. The elements of U are represented by points within the rectangle, the elements of a subset A by points within the corresponding subspace of the rectangle and the elements of A′ by points within the rectangle but outside the region representing A.

The central role of the resolvent cubic in the solution of the quartic was first appreciated by Leonard Euler (1707-1783). Euler's quartic solution first appeared as a brief section (§ 5) in a paper on roots of equations [1, 2], and was later expanded into a chapter entitled ‘Of a new method of resolving equations of the fourth degree’ (§§ 773-783) in his Elements of algebra [3,4].