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.

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.

The practice of long jumping a motor cycle from an inclined ramp over obstacles such as parked buses has become popular with stunt-men. There have been serious accidents, some with fatal results. Some crashes have been caused by the faulty technique of keeping the throttle wide open after leaving the take-off ramp. In some cases this can lead to a backward rotation of the cycle by up to 90° during the jump, with inevitable disaster on landing.

In this article I shall take a problem, the quadrature of the cissoid, and compare a hard (but brilliant!) pre-calculus solution of 1658 with the easy (and routine) calculus solution of only 13 years later. Afterwards we shall perhaps see the calculus through different eyes and maybe even teach it differently. Calculus is mechanical, but that it is mechanical is a marvel.

This article provides solutions to three problems. The solutions are interesting in themselves; but the problems, which are very different in nature, have a common characteristic. Each can be solved relatively quickly—some would say, elegantly—by a leap of intuition. On the other hand, all three can be tackled in a more exploratory, pedestrian fashion. Some people may want to argue that one approach is “better” than another; but a more useful discussion concerns the appropriateness of a method in particular circumstances, whether it provides insight into the underlying structure and principles involved, and indeed whether this is important.

Most of us, I suppose, can quote a few phrases read or heard, which have greatly affected our outlook on life in general and our own work in particular. Of about half a dozen such, two came vividly into my mind as I recently attended, at the University of London Institute of Education, a course of lectures in which a closer link between the sixth form and the University courses in mathematics was sought. The first goes back to my youth, and comes from Ruskin—“Know what you have to do—and do it!” Ruskin goes on to say that insufficiency of means is far less frequently a source of failure than an inadequate understanding of the thing to be done. The second is more recent, and came as a gentle rebuke from more than one of the nurses at St. Bartholomew’s Hospital, during my stay there after becoming blind, when I apologetically asked for some small non-routine service—“What are we here for?” Surely Ruskin’s precept and the nurses’ question are very relevant to our work.

The solid which is the three-dimensional analogue of the plane quadrilateral appears to have been neglected by geometers, possibly on account of the difficulty of visualising it adequately from a plane figure. A model, however, makes an interesting object and is readily constructed from thin stiff wire and thread. The extraordinary number of collinearities, which are its principal feature, then become at once obvious. For the benefit of any one who wishes to construct such a model I would suggest the desirability of commencing with the external diagonal triangle PQR and the line P′Q′R′ (see Figs. 1 and 2), as otherwise it is difficult to get a compact model.