In Sir Thomas Heath’s entrancing Manual of Greek Mathematics it is stated on p. 52 that the Babylonians discovered “a most perfect or musical proportion” between two numbers, which Pythagoras introduced into Greece. This is the ratio a ½(a + b):: 2ab/(a + b) b. The two middle terms are the arithmetic and harmonic means of the extreme terms. Not knowing of this when considering the fundamental principle of the logarithm, it seemed to me that the use of these two means should assist in evaluating the natural logarithm of a ratio in which the numerator and denominator are the first and last terms. In this way I found the surprisingly exact but approximate expression for the numerical value of such a logarithm which appeared in Nature of 14th March, 1931.

In a September 2005 note in The Mathematics Teacher, Gordon [1] gave a variation on the method of completing the square that fraction-phobic students are likely to find easier, along with the comment: ‘This variation is so obvious it must have been discovered many times over, but I have never seen it in print and thought it would be useful to disseminate it more widely.’ The method Gordon [1] gave is to multiply by 4a, and then add b2, to both sides of ax2 + bx = −c to produce 4a2x2 + 4abx + b2 = b2 −4ac. The left side of this last equation can be factorised as (2ax + b)2, and now we have a pure quadratic whose solution is straightforward.

In part I of this article [1], we described developments in mathematical logic before 1931. Gödel’s famous paper On formally undecidable propositions of Principia Mathematica and related systems, was published in January of that year. In it Gödel established two theorems:

1. (1) Incompleteness: For any consistent theory at least as strong as arithmetic there are sentences which can neither be proved nor disproved in the theory.

2. (2) Unprovability of consistency: The consistency of such a theory cannot be proved within the theory.

We are going to outline, with minor adjustments to notation, Gödel’s actual proof of (1) given, in translation, in [2, pp. 5–38].

What are the units, irreducibles, and primes of the ring ${\mathbb{Z}}\sqrt n $, the set of all numbers $a + b\sqrt n $ where a and b are integers and n is a fixed positive square-free integer? In the ring ${\mathbb{Z}}$, primes and irreducibles are synonymous and its units are ±1. ${\mathbb{Z}}\sqrt n $ is wilder, and our modest goal here is to catalogue all such numbers for ${\mathbb{Z}}\sqrt {{\rm{[10]}}} $, where a and b range from 0 to 10; the result appears in Figure 1. Here are a few teasers that may induce a reader to read on: $3 + \sqrt {10} $ is a unit; 2, 3, 5, and 7 are irreducibles, but not 31; and 7 is the least positive integer that is prime in both ${\mathbb{Z}}$ and ${\mathbb{Z}}\sqrt {{\rm{[10]}}} $.

Recursively defined sequences (in other words difference equations) appear in modelling systems changing in discrete time steps. The asymptotic behaviour of such sequences tells us information about the state of the modelled system after a long time. The simplest difference equations are linear, but the standard solving method of even these simple equations involves eigenvalue and eigenvector computation, complex numbers and linear algebra.

Some three weeks into the academic year I presented, off the cuff, the following morsel to engage some lower-sixth formers who had polished off some routine derivatives all too quickly:

Sketch the curve whose equation is given by x3 + y3 + 3xy – 1 = 0 , and find the gradient at the point (–1 , –1).

Since this was near the end of a double period I thought no more of it. At the end of the next double period one student remarked that he had been having difficulty in sketching it. “It seems to be a straight line”, he said. I rashly suggested that he could well be mistaken, but promised to look at it. If my faith in the student had been greater I could have saved myself some labour searching for simple solutions of the cubic. I had noticed the solution (½ , ½) by the way. I began to have delusions of grandeur, imagining myself to be the natural successor of Diophantus, as I found the solutions (5,–4) and (–4,5).

Probability and expectation are two distinct measures, both of which can be used to indicate the likelihood of certain events. However, expectation values, which are often associated with waiting times for success, may, at times, speak more clearly and poignantly about the uncertainty of an event than a theoretical probability. To illustrate the point, suppose the probability of choosing a winning lottery ticket is 2.5 × 10−8. This information may not communicate the unlikely odds of winning as clearly as a statement like, “If five lottery tickets are purchased per day, the expected waiting time for a first win is about 22000 years.”

Given a parallelogram, construct squares outwardly on its sides; hereafter we will call this the ‘Thébault configuration’. Our name derives from Thébault’s celebrated first problem, which states that the centres of these newly constructed squares also form a square.