“The ambitious learner should carefully study a new fact; he should turn it over and over, consider it under various aspects, scrutinize it from all sides … Moreover, he should try to expand and enlarge any newly acquired knowledge by application, generalization, specialization, analogy, and in all other ways.” [1]

Vincenzo Viviani, a 17th century mathematician, proved that the sum of the (perpendicular) distances from a point within an equilateral triangle to its sides is constant as shown in Figure 1. The theorem, named after him, generalises to polygons that are equilateral or equi-angled, or to 2n-gons with opposite sides parallel. Viviani's theorem is easily proved by summing the areas of triangles APB, BPC and CPA, equating to the area of the triangle ABC, and then simplifying to obtain h1 + h2 + h3 = H, where H is the altitude of ABC.

The ‘midpoint’ approximation to the integral $$\int_0^1 f $$ is

$${M_n}\left( f \right) = {1 \over n}\sum\limits_{r = 1}^n f \left( {{{2r - 1} \over {2n}}} \right)$$

In an earlier communication to the Gazette [1], the authors in effect showed, in a somewhat complicated manner, how to evaluate the integral One can show in a simpler manner, however, how to evaluate, for integers n and m, a more general integral of the form where n ≥ m, provided that if m = 1, then n is odd.

In addition, the final section to this article shows how to extend the procedure to include integrals for which m does not even have to be an integer, and also how to integrate where such an integral converges.

In May 2014, soon after the MA Conference in Nottingham, and soon after the passing of Roger Wheeler, a staunch member of the MA, who had been my teacher at school, I had the idea that my presidential address in April 2016 might be entitled ‘Inspiring Teachers’. The concept was a talk that might trace my mathematical experiences from grammar school sixth form, through my development in retirement as a contributor to masterclasses for the UK Mathematics Trust and the Royal Institution, learning from the students and from the inspiring teachers at whose masterclasses I assist, to the small understanding of Key Stage 2 Mathematics that I am gradually acquiring through an hour a week with some Year 6 students and their remarkable teachers in a local primary school.

A month or so later, when the organising committee was giving shape to Conference 2016, my words were taken as the title of the conference itself. It is deliberately ambiguous and it was gratifying to me to find this last April that its ambiguity had struck melodious chords with other speakers. My presidential address, however, was conceived as a lecture. Like all my lectures it was designed to be an oral presentation. It was not designed to be written down and published as an article. Please bear that in mind gentle reader, and judge accordingly. If you find something of value here I shall be delighted; if not, I shall not be surprised.

In 1877, the English mathematician J J. Sylvester (1) announced the discovery of a rather complete formal analogy between the mathematical theory of the invariants and covariants of binary quantics and the chemical theory of valency as it then stood.

It is widely, almost universally, believed that the Hebrew Bible gives the value of π as the crude approximation 3, a value much less accurate than those adopted by other ancient civilisations, such as the Babylonian, the Egyptian and the Chinese. In the English of the Authorised Version, the biblical source goes:

‘And he made a molten sea, ten cubits from one brim to the
other: it was round all about, and his [i.e. its] height was five
cubits: and a line of thirty cubits did compass it round about.’