The two great monuments of Western science are Euclid’s Elements and Newton’s Principia. Both are severely mathematical works; both set forth cosmological theories; in their day both were considered daring and speculative; both achieved unprecedented accuracy in application, so much so that at one time they were thought to be based upon a priori truths of reason, and even today are preferred to experiment over a wide range of phenomena. These two theories remain indispensable as elements of modern scientific thought despite the fact that they were simultaneously superseded in 1915 when Einstein published his theory of gravitation, a theory even more mathematical and boldly speculative than its predecessors. (One must, of course, remember that the nineteenth century discovery of non-euclidean geometry, whilst destroying the belief that Euclid’s geometry was true a priori, did not shake the belief that it was true as a matter of fact.)

The rank size rule is an empirical law which describes part of the size distributions of towns within individual countries and regions. It is of particular interest to geographers; and it is one of many applications of mathematics which are now being introduced into school science and mathematics courses. Since a collection of topics does not necessarily add up to a satisfactory mathematics course, it is useful to consider their integration into more general courses. The rank size rule could be taught as part of a course in descriptive statistics and linked to a more general discussion of frequency distributions.

One card up the sleeve of many a teacher of mathematics involves the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, 34,…, in which each number is the sum of the preceding two. These numbers and the closely related golden ratio (√5 − 1):2 have intriguing geometric and algebraic properties and appear mysteriously in nature ([1], 160-172). The main purpose of this article, which stems from a recent talk to prizewinners in the Scottish schools’ problem competition, Mathematical Challenge, is to advertise some less widely known occurrences. A reader interested in further references to any part of this article is invited to contact the author, who himself is grateful to several colleagues.

The method of least squares is a well known technique of curve fitting, giving estimates for the values of the coefficients in equations such as y = ax, y = ax + b, etc, in terms of observed values of the variables x, y. It is less widely realised that the technique can be extended to estimating matrix coefficients.

The Fibonacci sequence

0, 1, 1, 2, 3, 5, 8, 13, …

is defined by u0 = 0, u1 = 1, and un = un−1 + un−2 for . There are many elementary formulae relating the various un, most of which, since the sequence is defined inductively, are themselves usually proved by induction.