To the observant and intelligent boy or girl who has ever sailed on the Broads or steered a motor-boat in a busy harbour, the ordinary text-book treatment of Relative Velocity must be unsatisfying. The velocity of a ship (or other moving body) P relative to a ship Q is defined as the velocity of P as it “appears” to an observer on Q. But the exact meaning of “appears” is seldom explained. The fact that in all practical applications, observations are being made from one or other of the moving platforms, not from a stationary airship above them, is generally obscured. Thus the relative velocity of P with respect to Q is presented as something mathematical, obtained by the trick of reversing Q's velocity and adding it vectorially to that of P. Now, anyone who has steered a boat and observed another approaching on a converging course, knows that he is not directly concerned with the “actual” velocities (meaning velocities relative to the earth) of either boat. On the other hand, the velocity relative to him of the other boat is a very real thing. But to obtain it he does not have to go through the mental process of imagining that he is going astern when he knows that he is going ahead. All he does is to look over the side and observe. If the other boat's relative bearing does not change, or if its rate of change is small, he knows that it is time to do something with the tiller.

FIRST of all I should like to say that I am not responsible for the subject of this paper. Some member of the Association proposed that a discussion should be held on this subject and the Committee that arranges the programme for this meeting asked me to read a paper.

This Association was founded a little over sixty years ago, mainly by schoolmasters, and its main object was to advance mathematical knowledge by improving the teaching of mathematics in schools, so that I am only too glad to welcome any discussion on actual school work, and I hope that what I have to say may provoke a discussion to which many of you will contribute.

After promising to read the paper I sat down to consider what ground it should cover. It seems to me that the subject covers nearly the whole ground of geometry teaching up to the School Certificate stage, so the subject is wide.

It will not be uninteresting to look at the past for a few minutes. I started teaching in the last century, and there was much teaching at that time that had made little advance since the date of the foundation of the A.I.G.T. But for a moment I will go back still earlier and tell you two stories of the middle of the last century

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.

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.

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.

In this article an attempt has been made to indicate the ordinary methods of integration of the total differential equation

Then the condition for integrability of (1) , i.e, the condition that (1) may possess a single integral equivalent of the form ϕ(x, y, z)= c, is

Considering the pertinent remarks on “Approximations” in The Teaching of Arithmetic in Schools, the steady change of treatment of the subject in text-books and the recent article by Mr. Inman in the Gazette, it seems that considerable independent investigation of this matter has been going on. Also few people will disagree with Mr. Inman when he says that the teaching of the subject leaves room for improvement. That here is matter worth spending time on in the classroom is clear—but the same may be said of many more topics than we can ever have time to deal with. It is in the belief that the subject of the accuracy of figures, or approximations, is one we really ought not to omit that I venture further remarks.

In the following note, Stirling’s formula for a factorial is first found by a very simple process. The expression so obtained has a meaning for non-integral as well as integral values of the variable. The function it defines is easily seen (§§ 3 ff.) to possess the fundamental properties of the Gamma function, and provides an easy and natural avenue of approach to its study.