The problem I am going to discuss is the generation of the class of all mappings of the space S×S×.. × S, with any number of factors, into S itself, from a single mapping of the class, where S is a finite set. The problem has its origin in the study of truth tables and it is in this setting that the problem may be most simply presented.

The Eulerian numbers are not strangers to readers of the Gazette but they are not normally associated with the subject of this article, despite the similarity of their names. This article seeks to use Eulerian numbers in generalised telescoping sums, a role that is a powerful extension of an established technique – the Euler summation formula.

Twenty years ago it was anticipated that “Standard Form” would prove the ideal method for decimal processes. It was designed to abolish the mechanical teaching of a mechanical rule and to base arithmetical instruction on a conscious grasp of principles. A wide and extended trial has not realised the anticipation, indeed the method appears to-day to have many followers but few champions.

The purpose of this article is to show how elementary properties of the cubic curve can give rise to an abstract structure with interesting properties and to investigate some of the simpler of these.

In Routh’s Elementary Rigid Dynamics (Sixth edition, 1897, § 300) a section is given to the application of the principle of linear momentum to determine the gradual changes produced by alterations of mass, with examples; but I cannot recall that any modern textbook explicitly points out that the formulae

M dV/dt = F .....(1)

and d(MV)/dt = F, .....(2)

which serve to determine the motion of a body of constant mass, are not necessarily true when the mass is changing.

This problem refers to an observation made by George Gamow and Marvin Stern [1]. They found that concerning multi-storey buildings whenever one requires the lift on a particular floor, the probability of the lift coming from above or below varies according to the height of the floor concerned. It was observed that on the second floor of a seven-storey building, the probability of the lift coming from below is very small whereas it is much greater when the sixth floor is concerned. This is very similar to the case of finding the probability of a bus coming from a particular direction at a stop on a bus route.

It is perhaps somewhat remarkable that in such frequently recurring and comparatively simple processes as those of multiplication and division of decimals there should still be no final settling down by teachers of Arithmetic into one standardized method and arrangement of work, shown by experience to be the simplest, safest, least laborious, and most easily understood and assimilated by the pupil. Indeed there seems a tendency in some quarters, after trying in succession all the various modifications of method and set-out which have been suggested, each as an improvement on its predecessor, to come round full circle to the older methods in use, say, fifty years ago.

It has now been possible to study the many comments received since the publication of the “interim statement” in Gazette No. 407 (March 1975). The recommendations given below are based on the original suggestions, taking these comments into account. Sidelining indicates a change or modification of the original suggestion.

It is proposed also to publish these recommendations as a pamphlet, so that they may be widely available to authors and others.

There are four methods which are normally taught in schools for solving quadratic equations:

(i) By factorisation.

(ii) By the use of graphs.

(iii) By completing the square.

(iv) By the use of the formula.

This note describes a possible approach to the direct calculation of both derivatives and integrals of power functions using rather simple algebra and limits that could be easily accessible to students beginning a calculus course. The calculations provide a powerful example of the fundamental theorem, and could be used to allow students to discover the fundamental theorem from the example. The main calculation, which is not found in many calculus textbooks and is due to Descartes, is described by Maor [2]. While this approach may seem like the popular American ‘reform’ calculus, one may find a somewhat similar approach (presenting integral calculus from first principles) in Tom Apostol’s classic and lovely text [1].