Many members of the Association know the Gazette only in its hampered and restricted war-time form. The time is therefore opportune for some remarks on the pre-war Gazette and on what may reasonably be expected from the post-war Gazette. The history of the Gazette, however, cannot be fully understood without some reference to the history of the Association; such reference need be only brief, since a full account is available in the admirable Presidential Address by Mr. A. W Siddons (XX, No. 237, February, 1936).

Welcome to Leicester, our first Annual Conference here since it became the home of the Association in 1975 as the result of the centenary appeal in 1971 to buy a permanent headquarters building. The city is, of course, also home of the Association's library. Welcome to the University and to the Annual Conference, one of the many manifestations in which our Association appears. It is a great pleasure and honour to be here tonight to speak on the opening day of our 1990 Conference.

Charles M. Whish (of the Madras establishment of the East India Company Civil Service) visited Kerala, in southern India, during the early part of the nineteenth century, and collected some antiquities. The information he could gather about these antiquities was summarised and the curious looking items were displayed in his “showroom”, an article entitled “On the Hindu quadrature of the circle and the infinite series of the proportion of the circumference to the diameter exhibited in the four Sastras, Tantra Sangraham, Yukti-Bhasa, Karana-Paddhati and Sadratnamala.” This paper was read at the Royal Asiatic Society in 1832 and published in 1835 in [12]. Sightseers who passed by Whish's “showroom” were unaware of the worth of its exhibits, and the explorative and organisational labours of Whish remained unrecognised for over a century. By the 1940s though, an occasional enthusiast would wander in and the specimens kept mere amazed them and roused their interest and enthusiasm. Gradually more people were drawn to the antiquities, until by the 1980s Whish's small showroom had become die nucleus of a substantial institution. In this article mere will be a tour of the showroom, and some of the information on me exhibits will be brought up to date.

The entertaining book, There’s one born every minute, by Harry Blackstone, presents many situations where most people would intuitively be prepared to wager even money on the outcome, whereas a proper study of the probabilities concerned shows that the bet is heavily biased. One particular problem is interesting because the author claims to have sought evaluation of the probability concerned, without success. It is as follows:

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.

It is a commonplace that pictures leave a deeper impress on the human mind than abstractions and, in spite of its abstract nature, that is also true of many part? of mathematics. A proof of a theorem or the construction of a counter-example is often suggested by pictorial considerations, and that is true of algebra and analysis as well as of obviously geometrical subjects like topology. (For entertaining examples see Professor J. E. Littlewood’s book [1].) There is obviously a connection between a developed geometrical intuition of that kind and the kind of spatial experience which ought to be furnished by a school mathematics course. No one would deny its importance and the writers of textbooks on ‘modern’ mathematics are well aware of the need to stimulate geometrical intuition. What has been called in question is the relation between that kind of geometrical intuition and the formal treatment of the euclidean plane based, albeit insecurely, on Euclid’s Elements. Indeed, the traditional treatment has been abandoned with indecent haste and without serious question. In Part I of this paper we make a plea for a re-consideration of the case for some kind of formal geometry as an indispensable part of the education of the more mathematically able children.

Consider a square array of 25 grid squares, the grid squares being of two types distinguished by the colours black and white. A basic square of four adjacent grid squares can be selected from such a square array in 16 possible ways. If these 16 basic squares are all distinguishably coloured, then the 5x5 square array is said to be pantactic.

(I). The following inequality is a straight generdisation of one of the most important inequalities occurring in elementary analysis. It is consequently of some intrinsic interest, even though it has to do with a determinant.

Let n be a positive integer (≥ 2) and let a, b, . . . , h be n real numbers (unrestricted as to sign) arranged in descending order of magnitude, and no two being equal. Let x be a positive number, which will be regarded as variable.