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.

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.

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.

The purpose of this note is to show at a glance the significance of successive steps in the solutions to some of the quadratic equations that have come down in the cuneiform texts as examples of the mathematical instruction given to Babylonian students c. 1600 B.C. It is due to the translations made by O. Neugebauer in Germany and later, with A. Sachs, in the U.S.A. and to those published by Thureau-Dangin in France that this ancient material is available for general study, and, of course, it is from these sources that the following examples are taken. The texts are rhetorical, they instruct the pupil to perform arithmetical operations with specific numbers derived from the problems; in the translations these numbers retain their sexagesimal form f but here they are in our customary notation although expressed in a somewhat unusual way to facilitate, in particular, ready recognition of the re-entry of a number that has been temporarily left behind; in the text the pupil is reminded that this is the number that “your head held”.