The whole of ordinary algebra, dealing with real or complex numbers a, b, c, is based on three fundamental laws which are usually called:

1. I. The Associative Law,

2. II. The Distributive Law,

3. III. The Commutative Law.

As usual we define the p-multiplied L.T. of f(t) to be (1) and we adopt the abbreviation (2) introduced in 1938. The theorem in § 2 was obtained by the author in 1940. It would have been published, together with other work on L.T., some years ago, but for war-time restrictions.

In this article, we discuss a combinatorial problem arising from the National Lottery. The question considered is the following: Given a works (or any other) National Lottery syndicate of say n persons, how small can n be and still allow to ensure that at least one person in the syndicate has a winning ticket? Of course we will only be able to ensure that the required ticket wins the minimum prize of £10. However, we will show that a system to achieve this end can be constructed using some remarkable combinatorics which are well-known parts of modern mathematics. In fact, the system we propose for this problem has n = 290 and at least two tickets will win £10 prizes. It will be perfectly possible that the system will result in some members of the syndicate winning a fourth, third, second or even first prize. Although the combinatorics are quite interesting, it is not possible to ‘beat’ the National Lottery using our system. We have applied our system to the actual results of the National Lottery since its inception. Our experimental evidence shows that even using our seemingly efficient scheme, payout averages only about 28% of expenditure (showing an average loss of about 72%). However, in a typical draw, our system would actually have produced 5 or so ‘winning’ tickets for the syndicate.

Given two triangles ΔABC and ΔDEF, if each side of ΔDEF contains a vertex of ΔABC, then we call ΔDEF an outscribed triangle of ΔABC. Given ΔABC, let ΦΔABC be the set of all outscribed equilateral triangles of ΔABC. Clearly ΦΔABC is non-empty. In the following we will determine the area of the largest member of ΦΔABC when each angle of ΔABC is smaller than 120° and show that this largest member can be constructed by ruler and compass from ΔABC. The corresponding problem on quadrilaterals has been considered in [1].

We shall try to show that, in some well-defined sense, every right-angled triangle is ‘approximately similar’ to a Pythagorean triangle: and then we’ll ask how large the Pythagorean triangle must be.

1. Let f(x) be a real function of a real variable x. The meanings of when λ is a positive integer, a negative integer and zero, are well known. In the first case, denotes the λth integral of f(x) with respect to x, with an arbitrary lower limit of integration. In the second case, stands for the (−λ)th differential coefficient of f(x) with respect to x. Lastly, when λ = 0, means f(x).

Transformation geometry is now taught in many schools, and complex numbers are frequently used to describe the geometric transformations. For example, a translation is represented by (where b is a complex number), a rotation by a magnification by and a reflection in the x-axis by

The set of all transformations of the form forms a group and consists of precisely those transformations which preserve distance. Among the subgroups of this group there are the groups which give rise to the well known 17 distinct ‘wallpaper patterns’. These can be described in an elementary and geometric way and are (quite rightly) a popular topic in transformation geometry.

When asked to give this lecture I was given to understand that this Society would like a change from the discussion of pedagogical methods and school problems, and could do with something that had nothing to do with teaching at all. In other words, I was to talk about stars and not to mention schools. Well, I promise to do that, though I must confess that the fact that members of the Society are teachers was at the back of my mind when choosing the subject. The schoolboy and schoolgirl and the man in the street know that celestial distances are very large; in fact the adjective “astronomical” has been sadly degraded by being applied to such mundane affairs as the figures of the national debt—and a schoolboy of enquiring mind may ask how the astronomers measure such distances. Such an enquiry is often tinged with a certain scepticism—sometimes politely veiled, sometimes not. When you tell a member of the public that a certain nebula is many million light years away and that a light year is 6 billion miles, he is apt to adopt the attitude of mind indicated by the phrase, “You’re telling me”, and to wonder what meaning such a monstrous distance can have: and how does the astronomer know, anyway?

In this article we shall study the ‘shape’ of a triangle, and its relation to the area formula

Δ = √(s(s - a)(s - b)(s - c))

that occupied the time of sixth formers long ago. In so doing we shall resurrect a little ‘Durrel-type’ geometry, and introduce some ideas from the qualitative theory of differential equations, which are becoming increasingly familiar to undergraduates.

Morley’s theorem is well-known: if we trisect the interior angles of any triangle A0A1A2, then the common point of each pair of trisectors adjacent to a side is a vertex of the Morley triangle, M0M1M2, which is always equilateral (Figure 1). Less well-known is that the two triangles are in perspective, that is, the three lines AiMi, concur at a point M.

The pairs of trisectors further from the sides meet at W0, W1 and W2, the vertices of the anti-Morley (or perhaps the Worley) triangle. It, too, is in perspective with A0A1A2, the centre of perspective being W. Also, the Morley and anti-Morley triangles have a centre of perspective X, the points M, W and X being collinear.

A paradox connected with the idea of a uniform field of force is recalled. The main purpose is to discuss frames of reference in the gravitational fields of systems originally studied by Milne and McCrea in developing their “newtonian cosmology” It is shown that a frame moving with any particle of the material of such a system serves as a newtonian frame for the description of mechanical phenomena throughout a region that may be taken to be arbitrarily large and that a frame of such a sort is the nearest approach possible in the system to the realization of a local newtonian frame ; any two such frames are nevertheless in mutually accelerated motion.