About four years ago after consulting my copy of Cundy and Rollett [1] and then leaving it on my desk, I noticed that it had fallen open at a diagram of an approximation to Von Koch’s ‘snowflake’ curve. Interest in such curves has increased of late but for most mathematics students they make a fleeting appearance in the analysis course as examples of closed, continuous curves of infinite length and differentiable nowhere.

One of the many definite integrals that Euler was the first to evaluate was

(1)

He did this, almost as an afterthought, at the end of his short, seven-page paper catalogued as E675 in [1] and with the matter-of-fact title, On the values of integrals from x = 0 to x = ∞. It is a beautiful Euler miniature which neatly illustrates the unexpected twists and turns in the history of mathematics. For Euler's derivation of (1) emerges as the by-product of a solution to a problem in differential geometry concerning the clothoid curve which he had first encountered nearly forty years earlier in his paper E65, [1]. As highlighted in the recent Gazette article [2], E675 is notable for Euler's use of a complex number substitution to evaluate a real-variable integral. He used this technique in about a dozen of the papers written in the last decade of his life. The rationale for this manoeuvre caused much debate among later mathematicians such as Laplace and Poisson and the technique was only put on a secure footing by the work of Cauchy from 1814 onwards on the foundations of complex function theory, [3, Chapter 1]. Euler's justification was essentially pragmatic (in agreement with numerical evidence) and by what Dunham in [4, p. 68] characterises as his informal credo, ‘Follow the formulas, and they will lead to the truth.’ Smithies, [3, p. 187], contextualises Euler's approach by noting that, at that time, ‘a function was usually thought of as being defined by an analytic expression; by the principle of the generality of analysis, which was widely and often tacitly accepted, such an expression was expected to be valid for all values, real or complex, of the independent variable’. In this article, we examine E675 closely. We have tweaked notation and condensed the working in places to reflect modern usage. At the end, we outline what is, with hindsight, needed to make Euler's arguments watertight: it is worth noting that all of his conclusions survive intact and that the intermediate functions of one and two variables that he introduces in E675 remain the key ingredients for much subsequent work on these integrals.

I read the first page of John Hersee’s article in the June 1984 Gazette and sat down immediately, “stimulated to contribute comments” on his question. “What do we mean by ‘investigationary work’… in the context of A-level?”

One of the questions that I set my students in a recent A level revision lesson was to express (√2 - 1)6 in the form a + b√2, where a and b are integers. While they were calculating the answer, it occurred to me that, as a consequence of the fact that (√2 - 1)n → 0 as n → ∞ (since 0 < √2 - 1 < 1 ), a and b could be used to give us a ‘good’ rational approximation to √2. Within a minute or so everyone agreed that (√2 - 1)6 = 99 - 70√2, allowing us to write √2 ≈ 99/70.

It has been pointed out that Mayer's method of solving the differential equation

is quite general and only requires one integration, whereas the other general methods require two, or even thee, integrations. The reason for this is rather obscure in most text-books ; and the proof of the validity of the method rests on an existence theorem which leads to a solution of an apparently different form. I shall prove here that the two forms are identical.

Without doing the morally unspeakable or the medically infeasible, can a preference for daughters rather than sons increase their relative number? If, to be more precise, the only variable over which you have control is your number of children, can you increase the ratio.

Expected value (no. of daughters) : Expected value (no. of sons) ? Naïvely, you might think so. If for example you adopt the policy ‘stop procreating as soon as a girl is born’ won't you bear more girls compared to boys than you would otherwise? No, in fact.

Pascal's triangle, the Binomial expansion and the recurrence relation between its entries are all inextricably linked. In the normal course of events, the Binomial expansion leads to Pascal's Triangle, and thence to the recurrence relation between its entries. In this article we are going to reverse this process to make it possible to explore a particular type of generalisation of such interlinked structures, by generalising the recurrence relation and then exploring the resulting generalised ‘Pascal Triangle’ and ‘Binomial expansion’. Within the spectrum of generalisations considered, we find exactly four of particular significance: those concerned with the Binomial coefficients, the Stirling numbers of both kinds, and a lesser known set of numbers – the Lah numbers. We also examine the combinatorial properties of the entries in these triangles and a prime number divisibility property that they all share. Thereby, we achieve a remarkable synthesis of these different entities.