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.’

According to Richard Courant and Herbert Robbins writing more than fifty years ago in their classic book What is Mathematics? (which is still in print)

Sadly for many students entering university with A level Mathematics, and for some leaving university with a degree in Mathematics, this is not so. As was noted by John Smith, for some students there appears to be little understanding of the principles.

While reading a book by David Wells on curious and interesting geometry, I came across the following remarkable theorem named after Holditch.

In Figure 1 the point R divides a straight stick ST into lengths p and q, where p, q 0. We restrict the end points of the stick, S and T, to lie on a plane, simple, closed, convex contour, C1, and ST slides around C1. Assuming C1 is such that ST can pass around C1 once, the locus of R is another plane closed contour, C, inside C1.

It is well known that as I was trying to illustrate this to some students by calculating and so on, using the ‘story’ about a sum of £1 invested at 100% interest. After one year this would yield £2 if interest is added annually, £2.25 if added (compounded) at each half-year, £2.37 if compounded every 4 months, and so on.

In a recent Note Nick Lord proved that if a polynomial f(x) of degree n takes prime values for 2n + 1 integer values of x, then it is irreducible. (All polynomials discussed here are assumed to have integer coefficients and, like Lord, I regard numbers of the form –p where p is a positive prime as primes in their own right.) He wondered whether his result is the best possible one of its kind. If we restrict attention to positive primes, a best possible result can be achieved (Theorems 2 and 3 below). When we allow negative primes, Lord's result can be improved (see Theorems 1, 3 and 4) by an observation which still surprises me.

The Fibonacci series 0,1,1,2,3,5,8,13,.... is well known because so many properties have been found for it and because there are many instances of it occurring both in nature and mathematics. It is of course formed from the recurrence relationship

The Tribonacci series 0,1,1,2,4,7,13,24,44,81,149,... is formed in a similar way but from the recurrence

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.

The problem of dissecting a square into squares of unequal sizes has been extensively discussed in the recreational mathematics literature (see for example), but the dissection of triangles appears not to have received so much attention; there are however a number of naturally occurring problems in the dissection of triangles into triangles. The problem we considerhere is the dissection of triangles into five similar triangles; the component triangles must be similar to each other, but may or may not be similar to the original triangle.

In the first part of this paper we traced the early development of quaternions in the hands of Hamilton’s successor, Peter Guthrie Tait, Professor of Natural Philosophy in the University of Edinburgh from 1860 to 1900. Tait had neither the intuitive feel for physical concepts that Maxwell possessed nor the entrepreneurial talent of Thomson (Lord Kelvin) and yet he fulfilled a pivotal role in nineteenth century British physics through his correspondence with the former, with whom he went to school, and his collaboration with the latter. Though ambivalent towards quaternions, Maxwell was chivvied by Tait into drafting his 1873 Treatise on Electricity and Magnetism in both Cartesian and quaternion form.

Being aware of John Branfield’s request and having enjoyed Tom Roper’s response may I offer a contribution to the problem of describing the curve on which the bowls move? The important first thing is to decide upon a model of the system which is hopefully sound, and closely approximates the truth. It must also be amenable to mathematical analysis.