The total of the fund which was opened on the occasion of the 200th number of the Gazette was £270. Of this amount, £200 was given to Mr. Greenstreet in June of last year, and the balance, with a list of subscribers, in May of this year, when it became all but certain that the last contribution had been made. From the first it was impressed on Mr. Greenstreet that the desire of his friends was to add to his personal comforts. He arranged at once to take a much needed and much enjoyed holiday in the West country, a holiday of which he had previously been afraid to incur the expense.

From the earliest times the movements of the heavenly bodies have excited the interest of the inhabitants of the earth. The ancients believed that the earth was a flat disc floating in a great ocean, and they thought the sun was a god who drove his chariot across the sky every day. During the night he travelled back to the east across the great ocean, but his light was shut off by a range of high mountains which existed in the northern part of the earth. At night the majority of stars were noticed to move across the sky as if they were all fixed in some spherical shell which revolved. Several stars, however, seemed to move irregularly and were called planets or wanderers. Seven of these bodies were known—Moon, Mercury, Venus, Sun, Mars, Jupiter—and it is probably to these seven bodies that we owe our week of seven days.

Interest for mathematicians in general rather than concern with problems intelligible to small groups was the characteristic of most of the papers in mathematics at the centenary meeting of the British Association. The Department had a full programme for four mornings, and Prof. A. R. Forsyth occupied the chair throughout.

During the year 1929, 73 new members of the Association have been admitted. The number of members now on the Roll is 1223 : of these, 7 are Honorary Members, 61 are Life Members by composition, 17 are Life Members under the old rule, and 1138 are ordinary members.

The title of my address is “Mathematics for the Study of Frequency Statistics.”

By frequency statistics I mean statistics which give the result of counting. A statement as to the number of teachers receiving annual salaries in excess of a certain amount would come under the head of frequency statistics : a statement as to the total amount paid annually to these teachers would not. The sort of thing that I have mainly in mind is a table giving the numbers of boys of specified heights, the heights being taken to the nearest inch, or the numbers of girls who, having worked a paper with full marks 100, get under 5 marks, 5 and under 10, and so on. I may mention that the word “frequency” is usually applied to the actual number in a specified category; the proportion of this number to the total is called the “relative frequency.”

The discussions which have taken place on the methods of teaching Pure Geometry have been concerned almost entirely with the elementary stages, and the problem of the more advanced teaching has hardly been touched. We should, however, face the phenomenon which occurs time after time—of the boy of undoubted mathematical ability, proceeding to specialise in mathematics, who makes good progress in other branches but who fails to make any real headway in Pure Geometry. There is no doubt that difficulties in the earliest stages, however ably overcome, must of necessity foreshadow further difficulties in the more advanced ones. We propose, therefore, to touch on some points in the senior course. We limit ourselves to the discussion of general points and omit difficulties of detail that arise in the teaching of particular theorems, though we are aware that the problems which arise in the teaching of any section of Pure Geometry cannot be fully solved by examining that section alone. Moreover we do not attempt to discuss the philosophical aspect of the subject, but take the viewpoint of the teacher, whose first desire is to present an effective and interesting weapon for mathematical work.

I am afraid that the title of my address is not quite explicit. It lacks that precision of statement which ought to characterise the utterances of a mathematician. I have undertaken to speak to you about the End of the World, but I have not told you which end. The world—or space-time—is a four-dimensional continuum, and consequently offers a choice of a great many directions in which we might start off to look for an end; and it is by no means easy to describe “from the standpoint of mathematical physics” the direction in which I intend to lead you. I have therefore to examine at some length the preliminary question, Which end? But that leads us so deeply into the whole problem that by the time I have finished with it my address will be over.

If a pack of playing cards is shuffled systematically and the operation of shuffling repeated exactly, then after a certain number of repetitions of the operation the original order of the pack will be restored.

The method of shuffling in ordinary card-games is a rough and ready inter-leaving of two parts of the pack. If this method were carried out in simple perfection, one half of the pack would be interleaved with the other, so that no two cards adjacent before the shuffling would be together after.

The first table of logarithms was published by John Napier in 1614 under the title of Mirifici Logarithmorum Canonis Descriptio…, at Edinburgh, but the explanation of the construction was not then given. The author, however, promised to publish this later, when he had been assured that the leading mathematicians of the day approved of his work. Any doubt on this point was very soon removed for, within a short time, Edward Wright, who, after studying at Cambridge, had devoted himself to navigation, was engaged in translating the Descriptio into English. Henry Briggs, Gresham Professor in London since 1596, lectured to his students on the work of Napier as soon as he had grasped its significance, and he took it up with great enthusiasm. Soon he was in communication with the inventor and even travelled twice to Scotland in the summers of 1615 and 1616, before Napier’s death, which took place in 1617. The Descriptio seems to have taken some time to reach the Continent, for it was not till 1617 that Kepler first saw a copy. He apparently did not realise its importance at that time, but in the following year he became so enthusiastic that he wrote a letter to Napier (i.e. after the latter’s death) expressing his admiration and hoping for the publication of the method of construction of the Canon. In 1619 the posthumous work of Napier appeared with the title, Mirifici Logarithmorum Canonis Constructio. The construction of the table of logarithms is fully given and the exposition of the procedure, step by step, is exceedingly simple and beautiful. This work was admirably translated by W. R. Macdonald in 1889.

It is perhaps somewhat remarkable that in such frequently recurring and comparatively simple processes as those of multiplication and division of decimals there should still be no final settling down by teachers of Arithmetic into one standardized method and arrangement of work, shown by experience to be the simplest, safest, least laborious, and most easily understood and assimilated by the pupil. Indeed there seems a tendency in some quarters, after trying in succession all the various modifications of method and set-out which have been suggested, each as an improvement on its predecessor, to come round full circle to the older methods in use, say, fifty years ago.

In selecting a subject for submission to your attention, I have avoided topics belonging to the customary ranges of the school room, the lecture theatre, and the examination arena. My wish is to discuss the notion of dimensions in abstract geometry. In recent years, the “fourth dimension” has been obtruded upon an unsophisticated world in phrases which occasionally defy comprehension. Little more than a year ago, our Sabbath peace was perturbed by a newspaper announcement that a fifth dimension had been discovered, though the explanation which introduced the discovery was little more than incoherent verbiage. Novelists have ventured upon the intangible theme of multiple space : did not one of them, in an attenuated remembrance of Cambridge training, malign a great mathematician by assigning to his genius a fatuous dogma that ghosts are possible by means of the fourth dimension? Theologians of a sort, on both sides of the Atlantic, have not disdained to make a new departure by placing heaven in the fourth dimension : one of them, indeed, has postulated an unlimited number of dimensions as constituting the heaven of heavens, the dwelling-place of The Most High. Nay, coming nearer to our mathematical science, some popular expositions of the successive theories of relativity have stated, almost pontifically as an article in a creed, that Time is the fourth dimension. Even more bewildering was the rhetoric launched at us some six months ago when we were told, in a metaphorical summary, that the four-dimensional space-time continuum had

It is well known that a curve has only one equation in Cartesian co-ordinates When referred to a given pair of lines as axes, but it does not appear to be so well known that a curve can have many equations in polar co-ordinates, all referred to the same pole and initial line. This fact, although of considerable importance in the study of curves by means of their polar equations, appears to be consistently ignored by most text-books.

One of the oldest problems in the theory of Fourier series is that of looking for a criterion that a Fourier series shall converge. No one, however, has been able to find a simple, necessary and sufficient condition for this. Thus, for instance, bounded variation of the function is sufficient but not necessary. Continuity is neither necessary nor sufficient. That is to say, there are functions whose Fourier series converge at points of discontinuity, and others whose Fourier series diverge at points of continuity. If we consider the same problem for Cesàro summability of any particular order, similar difficulties arise.