The contents of this article result from the perplexities and questions which assail one in teaching or expounding, in talking and writing about, mathematics. What do we mean by “thinking mathematically”? Whence comes the “certainty” attached to the result of a mathematical argument—a certainty not shared by theology, artistic criticism, politics, history, or even by biology, chemistry or physics? How comes it that the abstractions of mathematics enable us to deal better with the problems of the real world around us? How does mathematics grow—is it created, evolved, or discovered? And so on! I cannot deal completely or comprehensively with all these questions, and I doubt whether final answers to most of them are yet known. I will therefore confine myself to four topics: proof, classification (including recognition), contradiction (including consistency), and discovery

The election of this year’s President has in some respects made history. There have been three previous women Presidents in the 95 years of the Association’s life and I recall with admiration the names of such eminent predecessors as Dr. Busbridge and Dr Cartwright, and with affectionate sadness at this time of loss, Miss L. D. Adams. But this is the first occasion on which one woman has succeeded another in the Presidency That there should be two women Presidents in sequence must be a significant event in the somewhat slow development of the recognition of women’s interest in mathematics. Moreover it is the first time that a woman who has had the ordinary responsibility of bringing up a family of children has also had the wonderful opportunity of continuing her mathematical work to the point of serving this Association as its President. This indeed makes me realise that mathematics is taking on a new role with a universal appeal.

J. Cable [1] suggests that if the word “vector” is going to be introduced into secondary school mathematics then it should at least be done properly Probably nobody would explicitly disagree with this, although there is a fair amount of evidence that implicitly many have and still do. But why must “vector” be introduced properly? Why the special fuss about vectors? The basic mathematics necessary to introduce vectors properly is common to many other mathematical ideas. (We wish to make it clear at the outset that our remarks in this article only refer to the printed text of the paper mentioned above. We do not wish to imply, for instance, that Mr. Cable does make a special fuss about vectors, but that the implication of his article is that vectors deserve special attention.)

The purpose of this article is to show how elementary properties of the cubic curve can give rise to an abstract structure with interesting properties and to investigate some of the simpler of these.

Miss L. D. Adams had a personality of such warmth and generosity that her loss will be deeply felt by very many people associated with education. Some will have known her personally through her visits to schools as H.M.I., or her lively contributions at conferences and refresher courses. We in this Association mourn her as one of our most distinguished members. She had an unbroken record of fifty-one years of membership and during that period had served on the Teaching Committee for some twenty years, exerting a strong formative influence which helped the Committee to broaden its enquiries until it can now claim to have produced reports on the whole range of school mathematics. The quality of thought that she brought to the Association’s discussions was recognized in 1959 when she was elected President, only the second woman to hold that office in the long years from 1871. To Miss Adams this tribute from the Association brought surprise and much pleasure, for in her natural modesty she had not thought of herself as possessing any gifts to merit her election.

It seems likely that the teaching of statistics in schools will become more common and will be the responsibility of teachers of mathematics. Many of the pupils will not be mathematical specialists.

A sheet of graph paper, marked in inches and tenths or (preferably) smaller subdivisions, is placed on a horizontal surface.

A halfpenny is placed flat near its edge, and flicked along the paper.

The collection of papers and books of the late Dr. F. W. Lanchester, F.R.S., held in the library of the Lanchester College of Technology, Coventry, contains a paper cover booklet entitled Lanchester’s “Potted Logs”, Parts I and II, printed and published by Taylor and Francis Ltd., Red Lion Court, Fleet Street, E.C.4, in 1939, priced 2s. The booklet measures 4ξ × 7¼ and consists of 27 pages devoted to “a concise tabulation (slide rule auxiliary)” of common and natural logarithms for engineers. In Part I (14 pages) “a new system is disclosed by which the Logarithms of Numbers running to seven or eight figures may be evaluated to seven places of decimals and vice versa” Of these 14 pages, 5 only are required for the tables, the remainder being devoted to preface, introduction and worked examples. In Part II (8 pages, 4 of which are tables) the system is extended to cover natural logarithms. The remaining pages contain a note on the musical scale (“of interest to the musician, it is a table in antilog form to the base 1-059463 . . and is the foundation of the system of equal temperament”) and lists of papers, lectures, reprints, reports, memoranda and books by Lanchester.

In The Gazette for February 1962 (Vol. XLVI, pp. 1–5), Mr. Roger North gave a very interesting method for constructing a group of any even order whose elements are functions of x, the rule of combination being substitution of one function in another. He also showed that the group of order 8 so formed is isomorphic to the octic group, the group of rotations of a square. (See also the article by Miss Joan Holland in The Gazette for February 1964, pp. 47-57.)

George Boole gave (in his books, listed under [1] and [2] in the references at the end of this article) a method for solving any equation in his algebra. This part of his work is now largely ignored in the many modern books on boolean algebra which are making their appearance (I can think of several reasons why). Nevertheless, the method can be adapted to conform with our standards of rigour.

The Report was prepared at the request of the Sub-committee on Instruction of the British National Committee for Mathematics. It formed the basis of the presentation of the United Kingdom point of view in a discussion on the topic at the International Congress of Mathematicians held in Moscow in August, 1966.

This is a very beautiful natural phenomenon, seen on occasions when the primary bow is a strong one. As will appear later, the occurrence necessitates raindrops of a homogeneous size, and of course bright sunshine at a low angle. The diffraction rainbow is therefore most commonly seen in tropical countries, in a thunderstorm that has already passed over and is moving away from the setting (or rising) sun.