A method of calculating real roots by a simple division process when an integral approximation is known is described. It is very powerful and economical in labour as well as being of extreme simplicity It surpasses Horner’s method in these respects and in many applications it is preferable to the Newton-Raphson method and others. A root is expressed directly in any scale of notation or, alternatively, as a continued fraction, and the processes involved would appear to be suitable for use in an automatic computer.

The inclusion of statistics in the school curriculum is generally advocated for two reasons. All citizens need to develop “the habit of disciplined thinking about ordinary affairs in terms of quantities” and many scientists need to learn techniques essential to their special studies. Perhaps a third reason should be added; statistics can be regarded as part of the training of mathematicians.

About 1901 I saw something of a member of the firm which was laying out at South Harrow what is now known as the Piccadilly Line.

He explained to me how they marked out the central line and how on curves they used sighting posts well away from the central line.

The last examination of the Cambridge Mathematical Tripos, when the candidates were placed in each class in Order of Merit and not in alphabetical order, was held in 1909. Last year was therefore the fiftieth anniversary of the end of a famous epoch. The University Calendar used annually to reproduce all the lists from 1747, the year when they were first printed, and they made fascinating reading, well known to mathematicians. Of course candidates were examined before that date, but full details have not been preserved. It was earlier called ‘The Senate House Examination’, and not before the start of the Classical Tripos in 1824 did the name gradually change.

The theorem to be discussed in this note depends only on the plane incidence axioms (with Desargues), the Pappus axiom, and the property that a point and its harmonic conjugate are in general distinct. When algebra is applied to the geometry it may be taken over any commutative field of characteristic other than 2.

Some children’s remarks are funny and some are sad, most people realise that, but what interests me most is their instructive value in showing us how to improve our teaching.

I remember once telling a class that I thought they ought to have learned one item in arithmetic before. Tom E., a bright little fellow, promptly replied—“Oh yes Sir, we did it before with Mr. H. last year, but as we didn’t understand it, we forgot it.” I asked him if he really meant what he had said! He looked at me puzzled, so I pointed out that he had said that if he didn’t understand anything, then he forgot it. The whole class laughed and agreed that generally speaking this was true.

The present paper is a sequel to two others which have appeared in the Gazette (July, 1941 and December, 1957) and in which the possibility of attacking a number of familiar triangle theorems by cartesian methods was successfully investigated. The same method is here applied to Morley’s and Pascal’s Theorems.

The late Mr. A. S. Gosset Tanner conducted the Bureau for the Solution of Problems from 1928 almost up to the day of his death in 1953 and many members of the Association will remember with gratitude the help he gave so freely, and with undiminished vigour as the years went on. Mr Gosset Tanner kept files of solutions to the papers for scholarships in Mathematics at Cambridge colleges. These files have been kept up to date, and the solutions are available to any member of the Association who, after a genuine attempt to solve a problem, has failed or has a solution with which he is not satisfied.

I am convinced that many difficulties that arise in the Primary and Secondary schools would never do so if from the very beginning division were taught differently

What I wish to urge is a complete break away from the traditional way of setting out division.

There are two types of problem on impulsive motion which receive a somewhat cursory treatment in most of the standard text books on dynamics. The first, and more elementary, is as follows: Two rods AB and BC are jointed at B and an impulse J is applied to one of them at a certain point. Find the motion. As one of the steps in the solution of this problem we are usually recommended to take moments about the point B. But, we may remark, B is neither a fixed point nor the centre of gravity of the system. How then are we justified in taking moments about this point? This question will be considered in paragraph 2 below.

In the autumn of 1693 the Writing Master of Christ’s Hospital, one John Smith, asked Samuel Pepys the diarist, who was a Governor of Christ’s Hospital and eventually became its Vice-President, for the answer to the following question:

Recent articles on equations characterising the trigonometric functions ([1], [2]) prompt a consideration of still another equation, which might be classified under the heading of “Students’ Mathematical Mythology” In [3], Mr. Heafford multiplies the “obvious” sin (x + y) = sin x + sin y by the equally “clear” sin (x − y) = sin x − sin y to obtain the (correct!) result, sin (x+y) sin(x − y) = sin2x − sin2y.

The subject matter of my talk is geometry in the sense that I shall deal with such entities as lines, circles, and squares. But the results I aim at bear little resemblance to the theorems of traditional geometry, which describe properties of given configurations. Our present object is quite different; we shall begin by specifying not configurations but processes (for example processes of selection), and then seek to determine how economically or how effectively these processes can be carried out. Questions of this nature are quantitative and the solution, if it is known, is generally expressed by a number. It is for this reason that I speak of 'arithmetical' geometry. The problems of the type I shall discuss are usually simple to state and require very little in the way of mathematical equipment for their understanding. However, their solution often presents formidable difficulties and is in many cases as yet far from complete. Let us now consider in some detail a few of the characteristic problems in this field.

The evaluation of moments of inertia of simple solids is frequently regarded merely as a convenient source of problems in the integral calculus. From the point of view of the student of dynamics it is, however, perhaps even more instructive to make these calculations without any actual integration, for in this way he may obtain a deeper insight into the physical significance of the underlying concept. Whether this be so or not it seemed to the writer an amusing task (performed during a moment of inertia caused by a bacteriological infection) to see how many of the well known formulae could be derived by such methods.