The Junior Mathematical Association of London, of which I expect very few of you have even heard, is an association in a sense of the word rather different from that in which it is used in the title of the Association I am now addressing. It is the name adopted by some eight large schools (of the Public School type) in and around London for a joint Mathematical Society which has been running amongst the Sixth Form mathematical specialists in those schools for nearly five years. It is essentially an association of institutions rather than of persons, and this type of organisation will at once be seen to be necessary by any one who considers the effect of the changing personnel of Sixth Forms. It possesses therefore no strict membership roll, no subscription, and, incidentally, no harassed treasurer. All the same, it is alive, as I hope to be able to show you by the end of this short paper. It might also be recorded at this point that a somewhat similar association exists amongst some of the Classical Sixths.

The subject about which I have to speak to you to-day is one which was bound, in the course of time, to occupy the attention of the Mathematical Association. This Association was founded, as you know, for the purpose of improving geometrical teaching, but since its foundation, some sixty years ago, its activities have widened. It no longer confines itself to Elementary Geometry, but has issued reports on the teaching of Arithmetic, Algebra, and Mechanics. None of these, however, except possibly the last, touches on anything more advanced than School Certificate work, and even the Mechanics report does not deal with anything far in advance of this standard.

As the President has told you, the subject of this lecture was not my own choice. When first asked to speak on it my impression was that, as the phrase goes, the answer was a lemon; but, on thinking it over further, it seemed that quantum mechanics might have some bearing on school work, although that influence is in a sense only accidental, as I will explain. As far as I can see, however, statistical mechanics, the other half of the subject of the paper, has no bearing on school work, and it only occurs in the title because that title was given me!

The course of study pursued by very many boys in secondary schools who take an advanced course after matriculation is governed by the syllabus of the intermediate examination of a degree in a university other than those of Oxford and Cambridge. In London those boys who intend to specialise in mathematics generally take physics and chemistry as well. It is clear that physics is a valuable complementary study for the mathematical specialist, but it is not so clear that chemistry is. It is a pity that the intermediate qualification cannot be obtained on a syllabus in mathematics a good deal more advanced than the present one, coupled with the present syllabus in physics.

My purpose this evening is to submit to you an other sketch of the general course and the circumstances of mathematical study at (Cambridge in my student days between fifty and sixty years ago. It does not set out to be an estimate of the state mathematical knowledge at that date It is not a comparison (or would contrast be the better word ?) between the empirical natural philosophy then in vogue and the rather speculative (slightly Aristotelian) physical theories that now absorb a small world of seekers after new Truth. It is only a picture of that academic, life as it then appeared to an unfledged student, now drawn as faithfully as memory will allow.

I understand that this problem appeared many years ago in a mock examination paper in the Granta. I heard of it from Dr. A. C. D. Crommelin in 1919, when A, B, G and D (or more strictly C, C’, D and E) were about to start on the eclipse expeditions which verified Einstein’s prediction of the deflection of light. Crommelin in an after-dinner speech hinted that this problem might arise. I made use of it in my book New Pathways in Science, and rashly gave my solution-—the result being a considerable increase in my correspondence. I hope the following explanation will be sufficiently clear to avert a similar sequel to its appearance in the Mathematical Gazette.

The article on “Differentials”, in the May Gazette—based on a discussion at the last Annual Meeting of the Association, and referring back to articles of July 1931 and February 1932 —has so clarified certain issues for the writer as to confirm a long-standing conviction that the standard definition of “differentials” is quite unsound, and that the view set out below (which he had developed in his own teaching ) is much truer to all the facts of the situation.

Poincaré, dans sa conférence prononcée au Musée pédagogique de Paris en 1904, déclarait déjà qu'ily avait lieu de penser en dérivées non en différentielles. II me semble utile pour l'enseignement de se conformer résolument à ce principe et d'abandonner les explications assez compliquées qui sont classiquement données sur le symbole d. Pour la différentielle premiére, passe encore : Je puis comprendre l'égalité accroissements infiniment petits.

“ Nul esprit normal n'est fermé ni aux sciences ni aux lettres, du moins à ces éléments des sciences et des lettres qui composent les programmes de nos lycées jusqu'è la fin de la Première.” Cette constatation, que l'on trouve aux premières pages des Instructions relatives à l' application des Programmes de l'Enseignement Secondaire indique le but qu'ont recherché — et atteint — les auteurs des programmes actuellement en vigueur. Tous les élèves de notre Enseignement Secondaire (de la classe de Sixième de nos Lycées et Collèges jusqu'à la classe de Prèmiere inclusivement) reçoivent la même formation mathématique, et, plus généralement la même culture scientifique. Les programmes d'études scientifiques sont exacte-ment les mêmes pour les élèves qui étudient les Langues Anciennes oue nour ceux qui étudient les Langues Vivantes.

A popular gibe against the branch of the teaching profession to which most of you belong is that the qualifications on which an appointment is made are not always related very closely to the duties of the post which is being filled. It is athletic prowess, we are told, that really matters. If only a man has a blue, all that is expected of him in the way of learning is sufficient intelligence to keep a clear week ahead of his class, in whatever subject he is engaged to teach. I am not going to ask whether there is or ever has been any truth in this aspersion, but I can properly point out to you this afternoon that you have no right to resent it. The accidents and aptitudes that may bring a man to the Chair of your Association are many and diverse, but one consideration is certainly never raised.

The following approach to the study of geodesic ellipses and hyperbolas is instructive, and leads to a generalisation in Riemannian geometry.