There are many named quadrilaterals. In our hierarchical classification in [1, Figure 10] we included 18, and at least 10 more have been named, but the properties of the latter have only scarcely (or not at all) been studied. However, only a few of all these quadrilaterals are defined in terms of properties of the sides alone. Two well-known classes are the rhombi and the kites, defined to be quadrilaterals with four equal sides or two pairs of adjacent equal sides respectively. The orthodiagonal quadrilaterals are defined to have perpendicular diagonals, but an equivalent defining condition is quadrilaterals where the consecutive sides a, b, c, d satisfy a2 + c2 = b2 + d2. Then it is possible to prove that the diagonals are perpendicular and that no other quadrilaterals have perpendicular diagonals (see [2, pp. 13-14]). In the same way tangential quadrilaterals can be defined to be convex quadrilaterals where a + c = b + d. Starting from this equation, it is possible to prove that these and only these quadrilaterals have an incircle (since this equation is a characterisation of tangential quadrilaterals, see [3, pp. 65-67]).

The study of unique factorisation is almost as old as mathematics itself. The so-called Fundamental Theorem of Arithmetic, which states that every integer greater than 1 can be factorised into a product of primes in only one way, was probably the first major theorem proved. It is interesting to note that Euclid (c. 300 B.C.) did not give the result in this form. He proved (IX 14) that “if a number be the least that is measured by prime numbers, it will not be measured by any other prime number except those originally measuring it” (see [1]).

I am not acquainted with any investigation of the problem of dissecting two given rectilineal figures of equal area, by straight lines, so that the parts of either will fit on the other, and the results given here are new to me. The problem is one which suggests a variety of interesting geometrical constructions. It is natural to consider first how, and in how many different ways, a dissection can be made so as to give the smallest possible number of parts ; but types of dissection which do not give the smallest possible number of parts may also be worth attention. I exclude cases of dissections depending on some accidental or unspecified coincidence of dimensions in the given figures.

A very great amount of scientific effort has been expended in the last few years on Rocket research and development. This work has been carried out in many countries besides Great Britain, notably the U.S.A., Germany and Russia. In the main it has been done by government scientists or by firms and universities under government contract. This is understandable since the chief use of rockets so far has been a military one. There is a very long history of the use of rockets in warfare (they were used at Waterloo), culminating in the production and firing of the German V2. Most of the major rocket weapons produced in the last war have been described in general terms in the Press, but there is, at present, remarkably little mathematical or scientific literature on the subject. Various technical journals,e.g. Flight, have published articles on the various aspects of the rocket problem, and there are one or two books. This lack is due to the fact that most of the papers written during the war years are classified as Secret or Confidential and are not yet available for open publication. Two recent papers are worthy of mention : one by Perring (1) on German Long Range Rocket Development, which gives a considerable amount of data on the V2 and one by Malina and Summerfield (2) on Rocket Escape from the Earth which discusses the possibilities of very large rockets. These may be regarded as the forerunners of very many papers which should eventually find their way into scientific literature.

It is nearly a hundred years since Boole published An Introduction to the Laws of Thought (1). An attempt to reduce the art of logical deduction to algebra was a bold venture, and perhaps not all of Boole’s original hopes have been fulfilled. Nevertheless, Boole algebra has become a well-established branch of mathematics which has given rise to theoretical masterpieces such as Principia Mathematica (2), and has found practical applications in electrical circuitry, switching problems, insurance law and statistics.

A continued fraction is an expression of the form

The expression can continue for ever, in which case it is called an infinite continued fraction, or it can stop after some term, when we call it a finite continued fraction. For irrational numbers, a continued fraction expansion often reveals beautiful number patterns which remain obscured in their decimal expansion. The interested reader is referred to [1] for a collection of many interesting continued fractions for famous mathematical constants.

It is more than 91 years since, on January 17th 1871, a group of twenty-six men met in another place and formed the Association for the Improvement of Geometrical Teaching, which twenty-five years later became the Mathematical Association. Most were schoolmasters; some were clergymen; the account of the proceedings in the first annual report adds: “There were also a few gentlemen present who were not members of the Association.” The ladies were allowed to join later.

‘After Jordan came Lebesgue, and we enter on the subject of another Book’, declared Bourbaki. J.C. Burkill remarked in, ‘It cannot be doubted that (Lebesgue's thesis) is one of the finest which any mathematician has ever written.’ Loève in picturesquely sets the scene for us thus, ‘… the Archimedes of the extension (i.e. modern theory of measure) period was Henri Lebesgue. He took the decisive step in his thesis … . In fact contemporary (measure theory) still dances to Lebesgue's tunes.’ Arguably, before 1902, mathematicians had yet to develop a theory of integration; Lebesgue's great thesis of that year changed this state of affairs irrevocably. In it, difficulties which had begun to plague the Riemannn integral were swept away as Lebesgue boldly extended the concept of the integral by what later came to be regarded as a completion process as profound as that leading from the rational numbers to the reals. Lebesgue created no School but his influence on twentieth century mathematics was profound. In this article celebrating the centenary of Lebesgue's thesis, we look first at Lebesgue's life and then in more detail at his seminal work on integration.

Our Association was founded deliberately to “improve” the teaching of geometry, and its work has extended to cover every branch of the mathematics taught in the pre-undergraduate years. It is no surprise that, among the many distinguished mathematicians who have contributed to this development, a few stand apart clearly for the extent of their influence. Judged by any standards, E. H. Neville was one of the greatest of them all, and this tribute, by one of his earliest pupils at University College, Reading, is written with a sense of deep personal affection and gratitude.

The December 1978 issue of the Mathematical Gazette [1] contains an elegant and humorous contribution from Anthony Hughes. His Note gives a recipe for determining the optimal spot from which to make a conversion attempt in rugby. Others then elaborated on Hughes' idea; see [2] and [3] in particular. In 1996 Isaksen [4] rediscovered Hughes' results while investigating the kicking of extra points in American gridiron. There are also a number of popularisations and summaries, in which the above results are presented, and in instances rediscovered. The ones of which we are aware are listed in the references; see [5, 6, 7, 8, 9 and 10].