Many elementary accounts of numerical integration use the trapezium rule and only rarely is the mid-ordinate rule mentioned. This is a great pity for the mid-ordinate rule is often simpler to apply and more accurate.

Fifteen years ago one could purchase a hand operated calculator which, by much lever pulling and handle twirling, performed the four basic arithmetical operations. Such a calculator cost around £40. Today electronic programmable calculators can be bought for prices which are less, in real terms, than that of the old mechanical calculator. As will be illustrated by examples later, such calculators incorporate the major features of a full-scale computer, and by constructing programs for a hand-held electronic calculator one can learn the basic tactics and strategies of computer programming. There are definite advantages in starting programming with a hand-held calculator; it is inexpensive, much of the ‘language’ will be familiar because hand-held calculators are now so commonly used, and one can test and use programs immediately they are written.

In this article we shall study the ‘shape’ of a triangle, and its relation to the area formula

Δ = √(s(s - a)(s - b)(s - c))

that occupied the time of sixth formers long ago. In so doing we shall resurrect a little ‘Durrel-type’ geometry, and introduce some ideas from the qualitative theory of differential equations, which are becoming increasingly familiar to undergraduates.

It follows at once from Pythagoras’ theorem about a right-angled triangle that the sum of the squares of the lengths of the diagonals of a rectangle is equal to the sum of the squares of the lengths of the four sides. Apollonius showed that the assertion holds for a parallelogram and, more recently, Amir-Moez and Hamilton gave a generalization to quadrilaterals by introducing a correction term which depends on the distance between the mid-points of the diagonals.

This is the story of a problem that began in an innocent way and in the course of time spread out in unexpected directions involving some interesting side-issues. It all began with the following problem which I constructed for the Mathematical Challenge competition run by the Scottish Mathematical Council.