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].

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.

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 every textbook which deals with second order linear finite-difference equations, methods will be found for solving such equations when the coefficients are constants. In the author’s experience it is unknown for textbooks to treat second order linear difference equations with variable coefficients. Yet such equations certainly occur in practical applications. It is the present intention to remedy this defect in the literature. It will be seen that, by the methods discussed here, we can always reduce the solution of a linear second order difference equation to that of a first order equation.

In his New Pathways in Science (p. 121) Sir Arthur Eddington propounds the following problem

“If A, B, C, D each speak the truth once in three times (independently), and A affirms that B denies that C declares that D is a liar, what is the probability that D is speaking the truth ?” and he has dealt with it further in the Mathematical Gazette for October 1935.

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.

Just how many matrices have inverses? In elementary linear algebra courses, many of the matrices encountered are singular, but perhaps the reason is that such matrices provide rich and interesting examples. How many of them occur naturally? Many beginning students observe that a singular matrix can be made nonsingular by very minor tweaking – changing just one entry, for example, will make a matrix of rank n – 1 into a full rank (nonsingular) matrix. In fact, changing one entry just a tiny bit will do it. Looking at the question from that point of view, with a little experimentation students begin to discover that singular matrices are quite rare. Entire rows (or columns) have to be rigged exactly right in order to get one, while minor changes in individual entries undo all the work and give us another nonsingular matrix. If we were to reach into a hat full of all the numbers – each equally likely to be chosen – and draw enough to fill in a square matrix randomly, we would certainly expect to get a nonsingular one.

“First books” on mathematics usually begin with numbers. Some histories of mathematics say in so many words. Mathematics begins with numeration. I believe this is true, literally, if mathematics is considered as a set of signs, with rules and relations between them. That is to say: the history of mathematical notation does begin with the numerals. The earliest known mathematical signs are no doubt number-signs, and history of mathematical notation has in fact little else to discuss but numerals up to the 14th century of our era. Of present-day mathematical signs, + and - appeared first in the 15th century, = in the 16th, <and> in the 17th.

I was interested to see in a recent issue of the Gazette a note from Hoare and Wright [1] about the distribution of leading significant digits, l.s.d., and “Benford’s Law” that this distribution is logarithmic. This distribution has fascinated many people since Benford’s original observation [2] in 1938 of the dirty pages at the beginning of a book of log tables.

The Report on “The Teaching of English in England” lays great stress on the important truth that the use of English concerns all teachers, but when it deals with the matter more closely it enumerates teachers of history, languages, geography and science, and says nothing about mathematics (e.g. § 110).