Geometry is gradually coming back into the school syllabus, but so far only 2-dimensional geometry. I would like to make a case for including some 3-dimensional geometry as well, because the latter is vital for describing the world throughout science, engineering and architecture. Higher-dimensional geometry also comprises a major part of modern research within mathematics itself. Also 3-dimensional geometry fosters both our intuitive understanding and our geometric imagination. It teaches us to see things in the round. It also trains us to see all sides of an argument simultaneously, as opposed to algebra and computing which emphasise thinking sequentially.

In [1] there is a rather nice story regarding the coming into being of the hook-length formula. The year was 1953, and the Canadian mathematician Gilbert Robinson was visiting a fellow mathematician, James Frame, at Michigan State University. One of their discussions concerned the work of Ralph Staal [2], an ex-student of Robinson, and this led to Frame conjecturing the formula. Apparently, Robinson was not at all convinced initially that the formula could be as simple as the one Frame was proposing. He was, however, eventually won over, and the combined efforts of these two mathematicians soon elicited a proof.

Ask anyone who has studied mathematics to a moderate level how many trigonometric functions there are and one is likely to be presented with a range of answers depending on what the person being asked is most likely to remember. Perhaps the ‘calculator button’ three of sine, cosine, and tangent will come to mind as these are the three trigonometric functions found on any standard scientific calculator. At a stretch, perhaps the names for their respective reciprocals, cosecant, secant and cotangent, will be recalled. Beyond the modern standard six, looking at calculus or trigonometric texts published prior to 1900 one soon discovers others going by strange names such as versine, haversine, or coversine (see, for example, [1, pp. 53, 63]). There are at least six others with as many as perhaps ten to twelve having received a name at one time or another. Today all these additional trigonometric functions considered important enough to grace the pages of texts in centuries past have fallen by the wayside, to be largely forgotten in favour of the modern standard six. Of course the pedant amongst us would say there is only one trigonometric function, the sine function, which currently stands as the preferred fundamental trigonometric entity, with all others being simple variations of this function, and they would not be incorrect in asserting this. But having the current standard six seems about the right balance between the minimalistic on the one hand and convenience on the other hand.

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.

In this note we will discuss five topics from triangle geometry and occasionally encounter something new. Areal coordinates (presented as Appendix A in [1]) will be used throughout, with the exception of the final section where it is advantageous to employ Cartesians.

The topics are:

(1) examples of collinearities where the distance ratios for important points vary with triangle shape;

(2) a pencil of lines which includes the Fermat, Napoleon and Vecten axes;

(3) two more circles related to the Lester circle;

(4) the family of rectangular hyperbolae which circumscribe the triangle of reference ABC and whose principal member is the Kiepert hyperbola [2];

(5) Cartesian coordinates leading to a solution of the ‘Euler disc problem’ [3].

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.

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]}}} $.

Three familiar properties of a parabola are that it is the locus of points that are equidistant from the focus and the directrix, that it can be created by an intersection of a plane and a cone, and that incoming rays parallel to the axis are reflected to a single point. The first two are often used as definitions, and the third may be used as an alternative definition or characterisation.

In the July 2011 Gazette Paseau [1] considers a population in which each birth is an independent event with probability g of being a girl and b of being a boy (b + g = 1). He supposes that, due to a preference for girls, all couples stop after producing k girls, where k is an integer greater or equal to 1. Each new birth will, even after some couples have stopped producing, still have the probability g of being a girl and b of being a boy. Paseau correctly states and shows that this will mean that the ratio

Expected number of girl births : Expected number of boy births is g : b.

So it is the same as if all couples stopped randomly. Paseau was showing that this result was a consequence of the ratio g : b always being the same for all families.

However in humans the ratio g : b is not always the same. The ratio g : b can vary significantly with many variables between couples, which influence either the ratio at conception or the subsequent implantation and survival of male or female foetuses to birth. For example, mothers whose menstrual cycles have a short follicular phase (normally days 1-14 of the 28-day cycle) tend to produce boys [2], resulting in a smaller g : b ratio. Also the timing of insemination within the menstrual cycle is associated with the gender outcome such that insemination early or late in the fertile period means that the offspring is more likely to be male; if insemination occurs in the middle it is more likely to be female [3]. It is therefore likely that changes in parental hormone concentrations during the fertile period can affect the sexes of the children [4].