You are all mathematicians, but almost all of you are something more than mathematicians; you are astronomers, meteorologists, physicists, or whatever it may be. Few mathematicians who are nothing but mathematicians attend the meetings of our Association regularly, and in this Chair I am not only a fish out of water, but a very small fish at that. you should see the ones that got away! An enquiry into the reasons for this state of affairs might lead to interesting observations on the purpose of conferences in general and of our own meetings in particular, as well as on the differences between one subject and another in respect of semi-vulgarisation, but it would be in the worst of taste to offer you who are here nothing but a disquisition making excuses for those who are not. Let us agree with the French that being absent they are in the wrong, and leave it at that.

The following definition of differential is not uncommon. It is to be found in Goursat’s and Hardy’s classical texts, as well as in a few more recent ones.

It is well known that the cubic equation

can be solved by identifying it with the equation

If, instead of solving (2) by algebraic methods as appears to be customary, we consider the intersections of the circle

with the system of arcs of circles with equation

we quickly obtain information about the nature of the roots of (1) and can also obtain their values.

Gamesmanship is defined as being the art of winning games without actually cheating. I do not intend to define researchmanship similarly but rather as minimum energy research.

One can only start from what one knows and proceed to things unknown in the subject one likes best. For myself, I like the integers and can tolerate fractions but consider series anti-social. As for divergent series, they do not bear thinking about.

In a recent competition (see M.R. 23 (1962), A1482) it was required to show that the set {2, 3, 4} has the property, here called Pv that every number x in (0, 1] (i.e. 0 < x ≤ 1) can be expressed in the form , where m1 is an integer and belongs to the set for all i.

In their book Mathematical Models (second edition, 1960), Cundy and Rollett describe a dissection model to illustrate the result

It is the purpose of this note to describe similar models for the equalities

and

In addition, a dissection for the sum of the cubes of the first n integers is discussed.

Professor Forder’s elegantly simple method in Note 3068 (Math. Gazette 47 (1963), 237-8) can be extended to prove a more general result. Let k be any positive integer, ξ any irrational number, ar the (r + l)-th quotient in the simple continued fraction for ξ and pr/qr the (r+ l)-th convergent, so that

§1. The problem is to evaluate, with an absolute error o(l) or better,

where m and n are large and positive and c is a positive constant.

As a result of the present revolution in Mathematics teaching it is increasingly likely that the word Vector’ may fall into the hands of young people. This word has been used rather carelessly in the past, and I would propose that before it becomes generally available to children we should legislate a few safety regulations. (Polyvalency is one thing, but spreading confusion among the young is another.)

The object of this note is to give a simple geometrical proof of the following well-known

The following is an account of a method of addition and some methods of subtraction which may appeal to teachers who like to lace their teaching with enrichment material, but who sometimes refrain from doing so because of a fear that an “off beat” treatment of a topic may confuse rather than clarify.

In his article on the flexagon family [1], R. F. Wheeler uses a mapping scheme which illustrates the behaviour of a flexagon. I shall introduce an alternative scheme and use it to expound a method of constructing and folding the net of any flexagon.

It has been argued from time to time [1] that the special theory of relativity is suitable for first year undergraduates reading mathematics or physics at British Universities. The pedagogical value of this subject is considerable, since it introduces conceptual problems, whose solutions require clear thought, but only elementary mathematics. The teaching of this subject can therefore bring about a useful broadening in a student’s intellectual equipment.

Readers may be interested in an elementary exposition of the convergence properties of the sequence of iterates obtained by applying the Newton-Raphson process to an arbitrary polynomial. The coefficients of the polynomial may be real or complex. By restricting attention to polynomials we will give some precise results using the least possible analysis, while proofs for more general functions would necessarily require more high-powered methods. The arguments we will use require only an elementary knowledge of limiting processes.

In [1], C. D. Langford asked for those plane figures which can be dissected into four “replicas”, congruent to one another and similar to the original figure. (An equivalent formulation is that four identical figures are to be assembled into a scale model, twice as long and twice as high.) In addition to triangles and parallelograms, which always have this property (see Figure 1), he also exhibits three trapezoids (Figure 2) and three hexagons (Figure 3) with this property. Finally, Langford gives an example involving a stellated hexagon (Figure 4). He then asks if the list is complete.

Now that formal Euclidean geometry is disappearing from the school syllabus the time is ripe for a reappraisal of the background to analytical geometry

Consider the following question. Find the coordinates of the fourth vertex of a parallelogram A BCD if the other vertices are A(2, 4), J5(l, 1) and (7(7, 2).

It has been noticed that heavy wear takes place on the wheel flanges of electric locomotives in countries where mountain railways exist and where curves are numerous and severe. Locomotives have to return to shops after a comparatively small mileage to have their wheels attended to unless steps are taken to reduce flangewear.

Let pn denote the nth prime number. (p1=2, p2 = 3, etc.)

Let [x] denote the greatest integer which is not greater than x.

From Wilson’s theorem, is an integer for x = 1 and for all prime values of x; but is fractional for all composite values of x.