The article which follows is the first of a set of papers sponsored by the Modern Trends sub-committee of the Teaching Committee of the Mathematical Association. It is hoped that future papers will be published either in The Gazette or as a supplement. Each paper will represent the personal view of the author rather than being an official policy statement of the Association. It is hoped that they will lead to correspondence in the form of reports of experiments as well as constructive criticism.

An American mathematician, celebrated for a love of epigram tempered by some regard for truth, has asserted that George Boole has the sour distinction of being the most severely under-rated mathematician of the 19th century There is some substance in this judgement. During Boole’s lifetime, his work was recognised and admired in this country, but British mathematics was then very insular, paying small attention to the great progress being made in Europe, and, reciprocally, being little regarded by Continental mathematicians. Boole’s reputation was localised, and, moreover, it rested very much on what we should now consider to be the less valuable of his contributions to mathematics. The title of his greatest work, The Laws of Thought, does not suggest a strictly professional study of pure mathematics but rather a digression into philosophy and metaphysics. Yet Bertrand Russell has declared that Boole, in this work, was the first person to discern clearly the essential characteristics of pure mathematics, and today, when Boolean algebra is a topic for school texts as well as for research journals, we can see that Boole’s work, under-valued at the time of its appearance, was an essential step in the development of pure mathematics as an abstract and axiomatic structure.

Mathematics is relentlessly seeping into the very foundations of the civilisation in which we live and the community of which we are part, but I wonder how many of my readers—teachers of mathematics in grammar schools or in the universities—realise the magnitude and understand the nature of this seepage. It is permeating commerce, sociology, economics, medicine, linguistics, …, as well as more traditional science and technology. In this article I shall confine myself mainly to engineering, although I believe that the principles I shall develop will apply generally to any ancillary mathematics.

In his talk on Decimals at the Annual General Meeting of the Mathematical Association, 1962, (see the Mathematical Gazette, XLVII, 1963, p. 25), Mr. Green observed that the recurring decimal expression for 1/p where p was a prime other than 2 or 5 had the maximum possible length of p – 1 if and only if 10 was a primitive root of the modulus p. In practice there was maximum length for

p = 17, 19, 23, 29, 47, 59, 61, 97

but not for

p = 13, 31, 37, 41, 43, 53, 67, 71, 73, 79, 83, 89,

and he did not see any reason or pattern behind these figures.

It is well known that the cubic equation

can be solved by identifying it with the equation

If, instead of solving (2) by algebraic methods as appears to be customary, we consider the intersections of the circle

with the system of arcs of circles with equation

we quickly obtain information about the nature of the roots of (1) and can also obtain their values.

In their book Mathematical Models (second edition, 1960), Cundy and Rollett describe a dissection model to illustrate the result

It is the purpose of this note to describe similar models for the equalities

and

In addition, a dissection for the sum of the cubes of the first n integers is discussed.

In his article on the flexagon family [1], R. F. Wheeler uses a mapping scheme which illustrates the behaviour of a flexagon. I shall introduce an alternative scheme and use it to expound a method of constructing and folding the net of any flexagon.

In [1], C. D. Langford asked for those plane figures which can be dissected into four “replicas”, congruent to one another and similar to the original figure. (An equivalent formulation is that four identical figures are to be assembled into a scale model, twice as long and twice as high.) In addition to triangles and parallelograms, which always have this property (see Figure 1), he also exhibits three trapezoids (Figure 2) and three hexagons (Figure 3) with this property. Finally, Langford gives an example involving a stellated hexagon (Figure 4). He then asks if the list is complete.

Let pn denote the nth prime number. (p1=2, p2 = 3, etc.)

Let [x] denote the greatest integer which is not greater than x.

From Wilson’s theorem, is an integer for x = 1 and for all prime values of x; but is fractional for all composite values of x.

1. Definition

We define an infinite coprime sequence as an infinite class of positive integers arranged in order of ascending magnitude, in which the highest common divisor of any two members is unity.

2. Note on prime numbers

If an infinite sequence of primes, constructed by some rule which does not invoke a knowledge of the sequence of primes, should ever be found, it will necessarily be an infinite coprime sequence. We thus concern ourselves in this note with the construction of infinite coprime sequences without using a knowledge of the sequence of primes.