I. Let BOY and COZ be the bisectors of the angles B and C of the triangle ABC

Draw OH, CH parallel to AB, OB respectively; OK, BK parallel to AC, OC respectively

The method explained by Dr. C. V Boys has great merit. I for one fully sympathise with his preference for vulgarity in fractions. What here follows is meant to be to some extent supplementary to Dr. Boys’ article.

If we denote by P1/Q1, P2/Q2, … the successive convergents to √M in Dr. Boys’ series which are arithmetic means, and P1= p, Q1=q, the law connecting them is

Mathematical students are usually familiar with the parametric coordinates (at2, 2at) of a point on a parabola, and with the eccentric-angle coordinates of a point on an ellipse, and with similar coordinates for a hyperbola. Such coordinates are probably welcomed by the student just because they reduce by one the number of letters and the number of equations with which he has to deal.

If at2 is substituted for x1, and 2at for y1, but everything else is done in just the same way as before, parametric methods are not really being used. Parametric equations are something more than convenient coordinates.

The game of Banker’s Clock provides an interesting question in mathematical probability In this game the banker turns up in sequence the first twelve cards of a well-shuffled ordinary pack of 52 cards. He backs himself to turn up at least one card of which the face value corresponds to its position in the sequence, an Ace ranking as one, a Jack as eleven and a Queen as twelve. The interest in the question from the mathematical point of view is in finding the probability that the event will happen.

The ease with which certain series involving finite difference operators can be summed by the method of separation of symbols has long been recognised. It does not appear to be universally known, however, that many forms of algebraic series which at first sight do not seem to lend themselves to this treatment can be summed very simply by the application of operators.

Statics is founded upon two outstanding principles the Parallelogram of Forces and the Principle of Moments. The Parallelogram Law is universally acknowledged as fundamental and is given its rightful place in both analysis and graphics; in the former it furnishes two equations connecting the forces keeping a body at rest while the Principle of Moments furnishes a third equation. In graphical statics the Funicular Polygon is used as the equivalent of the third equation derived from the Principle of Moments.

“ … And so the impulse of the blow is equal to the change of momentum of the body.”

This phrase, coming at the conclusion of a more or less standard lesson in Elementary Mechanics, must surely have struck many teachers with a feeling of shame for having participated in a piece of not entirely straightforward work, and their pupils with an idea that they have witnessed some downright “wangling”. The usual impulse-momentum treatment cannot but be unsatisfactory to all concerned in the lesson, for it follows on the introduction of a new quantity of measurement of which, as so many of the text-books plaintively remark, “no special name has been given to the unit”.

This paper is devoted principally to the exposition of a new calculating process which may be useful for what are called “natural” dynamical problems. A few examples, mostly trivial, are worked out to illustrate the process. But it is possible that the method may be used with advantage for several important problems, and a paper concerning its use for problems of gravitational astronomy will be shortly published elsewhere.

The following extension of Van der Mond’s Theorem enables us to express as an infinite series one of the solutions of the equation

xr+ axs+ b=0.

It is true that the results can all be obtained from Lagrange’s Theorem, but the process is interesting in that it enables us to obtain the series by an algebraical method.

The transformation y=f'(x) on the roots of

is intersting to study on account of the fact that it leads to a transformed equation whose constant term is, the discriminant of f(x)=0. It is easy to see that this property holds, for, writing f(x) in the factored form

As this is the last paper of our meeting, you may be expecting me to provide a bright and witty epilogue to conclude the proceedings. Further, since Lewis Carroll is recognised as a humorist, who did not hesitate to include pantomime humour in his mathematical essays, you may be expecting a kind of harlequinade. But, unfortunately, I cannot fulfil these expectations the title is Lewis Carroll—mathematician, and as he took himself very seriously as a mathematician it would be a poor tribute if I treated this aspect of his work in a light-hearted manner.

During the past twelve months many tributes have been paid to Lewis Carroll. His literary genius, his imaginative powers, his love of children, his interest in logic, ”his penchant for puns, puzzles, parodies and palaver“, all these have been the subject of newspaper and magazine articles, as well as of the more ambitious essays and books in praise of one who always avoided publicity, and was almost ashamed of being recognised as the author of Alice in Wonderland. But little has been heard of the Rev C. L. Dodgson, mathematical lecturer of Christ Church, Oxford, the author of mathematical pamphlets and books, which he was not ashamed to publish without a pseudonym.

The problem under discussion is the relation between the behaviour of a function ɸ (t) near a particular point t = x and the behaviour of the partial sums sn of its Fourier series at that point. We start from Hardy and Littlewood’s theorem † that if ɸ(t) tends to a limit in the Ceaáro sense as t → x, then the Fourier series at that point is summable in the Cesàro sense, and conversely. The exact order of the Cesàro means in each case is unspecified.

Psychology has been described as the science which tells us what everyone knows, in language that no one can understand. It has a fatal attraction for those whose paucity of ideas is concealed by an exuberance of rocablilary The leading authorities not only ignore each others’ conclusions, but even refuse to use words in the same sense. “When I use a word”, said Humpty Dumpty, “it means just what I choose it to mean-neither more nor less.” The discussion of such a point as the nature of intelligence, sufficiently difficult in itself, becomes hopeless when everyone attaches a different meaning to that term. It is the object of this article to show how, by the application of mathematical methods, Professor C. Spearman has found a way of escape from this jungle of verbal entanglements. As a preliminary we shall require some account of the idea of correlation.

1. An article by my friend, Mr. R. J A. Barnard, of this university, in the October number of the Gazette, impels me to join in the discussion of which it is a part. For, while the article in itself is an interesting application, to “constant coefficient” equations, of methods of wider applicability, the thesis of “avoiding” what is plainly the right method for such equations is, I feel sure, a mistaken one: it recalls the fin de siècle game of “dodging the calculus” (from which we seem now to be swinging to the opposite extreme—of teaching infinitesimal calculus without sufficient regard to the requisite maturity of mental development of the pupil).

As a rule graphical methods are more tedious than an analytical treatment; but they are welcomed by the weaker student on account of the illumination which they shed upon relations that would otherwise remain somewhat obscure.

Two notes which have recently appeared in the Mathematical Gazette dealing with the graphical discrimination of the cubic tempt me to publish methods of discussing the roots of the cubic and of the quartic which have been in use here for some time and have stood the test of experience.

Periodically until the end of time there will appear published works purporting to prove the parallel postulate, and it has fallen to the writer's lot to read what is perhaps the most recent of these. While the exact mathematical value of such attempts can always be gauged a priori, it is useful to put on record any new method followed, if only to warn future generations of parallel-postulate-provers against trying it. But such attempts are not without all value , some readers may find it a congenial occupation to spend a little time in discovering the inherent fallacy , alternatively, the task may be given to a student as a profitable and novel exercise in geometry

For these reasons, then, the following résumé of this most recent attempt is given here. And in order that the full interest of discovering the fallacy may not be denied to the reader or the luckless student to whom he delegates the task, it has been thought better to give only the proof as the author gives it, and not to point out where the fallacy lies. The writer has, however, allowed himself to make one or two comments on certain noteworthy points.

1. Whence these “products”? Why one a scalar (actually a real number—see below) and the other a vector? These are interesting questions, if once we shake ourselves free from inherited knowledge of the facts and become curious as to the why and the wherefore.