Early in the classical period of Greek mathematics there arose a crisis over the meaning of “proportion”, or alternatively of “area”. This crisis, which was resolved by Eudoxus, the pupil of Plato, about the middle of the fourth century B.C., led to the introduction of axioms, or postulates, as “self-evident truths”, necessary for the proof of theorems, equally true but presumably less self-evident. Among these axioms was included the famous Fifth Postulate of Euclid, equivalent to saying (see below for its rather complicated wording) that “through a given point (the point T in Fig. 1) there is at most one (straight) line parallel to a given line (the line MK)”. But this postulate had all the earmarks of a theorem; it looks true yet not altogether self-evident, and its converse, to the effect that there is at least one such line, is an easily provable theorem.

The most important properties of any distribution are its moments. The first four moments, for example, can be used describe any data set fairly well. Moments can also be used for estimation.

Most direct proofs of Morley’s theorem are very involved. Dick Hogg’s neat geometrical proof (Note 67.3) circumvents the problem with its slightly indirect approach, but is very pleasing, none the less. It may however not be generally known that there is a direct straightforward proof using complex numbers which is simple to those familiar with the technique involved; indeed the proof is almost inevitable, once the right start has been made.

I am continually surprised by the mathematical structure which surfaces in unexpected situations, and even more so when arranging a tenth birthday party for my elder son. The invasion of the home by some eight rather energetic boys on a cold February afternoon was viewed with some trepidation. The instinct of self preservation was uppermost in the search for activities which would allow the afternoon to pass with the minimum of discomfort and the maximum enjoyment. A family friend is very good at organising team games at parties, but with a space problem (our house does not have a large living room as so many modern houses have these days) it was important to split the children up for their activities, but yet devise a programme which was coherent.