It is known that if a solid is turned right-handedly through an angle θ about a line with direction cosines ni, a particle with co-ordinates xi moves to yi, where

First year mathematics undergraduates entering British universities are often very unprepared for the kind of proof activities that occur throughout undergraduate degree programmes. Their previous experiences of the idea of proof are frequently limited to situations where the ‘proof’ is simply an extended chain of calculations or algebraic manipulations. In fact, this is well-illustrated by their approach to proof by induction, where they can carry out the mechanical details of the inductive step, but have no understanding of why proof by induction works, nor what status the initialisation step and the inductive hypothesis have in the proof. There is no real grasp of what it means to prove that some assertion is true. Much time, in consequence, is spent in the first years of mathematics degree courses up and down the country attempting to bring about this understanding.

Every geometrical truth has an individual beauty which delights the mind: but the most fascinating theorems, as an eminent mathematician rightly asserts, are those which have simplicity, depth, and obvious generality I propose to show that nearly all the important circles of the geometry of the triangle—the in-circle, the nine-points circle, the circum-circle, Taylor’s circle, Lemoine’s circle, and many other useful but nameless circles—are each and all particular cases of one fundamental general circle.

On a récemment signalé des simples constructions qui mènent d’un triangle plan tout à fait arbitraire à un triangle regulier. On peut établir que ce triangle dérivé jouit de cette remarquable qualité en s’inspirant à une profonde maxime d’Abel (” il faut énoncer un problème de manière de le pouvoir toujours résoudre “); on doit alors se proposer en général de déterminer de quelle espèce est le triangle résultant de l’application d’une des constructions citées. Dans plusieurs cas il arrive que l’expression de la longueur d’un quelconque de ses côtés est une fonction symétrique des éléments du triangle primitif ; alors la régularité du triangle d’arrivée est hors de doute. Il est aussi à remarquer qu’à l’expression cherchée on arrive très souvent par un procédé (qui rappelle la “ triangulation géodésique “) qui mérite d’être signalée à tous ceux qui auront à l’avenir à s’occuper de questions analogues à celles étudiées dans le mémoire qui suit.

In recent years we have been teaching mathematical analysis using the computer to give experiences to help students understand the concepts [1]. It has proved to be a valuable aid in many ways, but the method depends on giving a suitably rich environment enabling students to experience the full range of possibilities. A great weakness of the use of regular computer languages is that they hold numbers in memory as a finite expansion in base two, so, effectively, all numbers in such a system are rational. This meant that we were unable to demonstrate any ideas that depend on the difference between rationals and irrationals; for instance, we could not plot graphs of the kind

Much debate about A-level mathematics centres on organisational, assessment and content issues, whilst another major focus of interest concerns the role of the microcomputer in the sixth form classroom. Many of these issues have been discussed at length in a recent article in the Gazette by Bibby [1], but he does also seek to draw attention to the rather limited success there has been in directing attention to the importance of “process” as well as “content” in A-level mathematics.

In ordinary arithmetic a number ab … jk represents the sum of the place values of its digits a, b, …, j, k, these place values being k, 10j, 102i, etc. Also the representation is unique if the digits are restricted to being positive and less than 10. We are going to consider the representation of whole numbers by means of digits with place values, but with a different definition of place value. The digits a, b, …, j, k + 1 of a given number A are to be positive, and the number of them, n, is to be assigned, the units digit k + 1 (in the standard form) is to be greater than zero, and the place value of the digit a in the nth place counting from k + 1 is to be a. a + 1 … a + n - 1 divided by 1.2. …. n, or the number of homogeneous products of a letters taken n at a time; this number or place value is written (a)n.

Many people have had the experience of borrowing a sum of money which is to be repaid with interest by making equal monthly payments over a year or a number of years. Two problems suggest themselves:

1. If the annual rate of interest is known what is the value of each monthly payment?

2. If the monthly payment is known, what is the annual rate of interest?