The Annual Meeting of the British Association in 1938 will be held at Cambridge, August 17-24. A programme notice appears on the opposite cover.

The Association will mourn the untimely death of its President, Professor Louis Napoleon George Filon, less than a week before the Annual Meeting to which he had looked forward with such keen anticipation and at which he had hoped to preside and to give his Presidential Address. It is easy for those who knew him and his work to understand how strongly the objects of the Association appealed to him and how deeply he appreciated his election as its President.

Even more than the politician, the educationist must be prepared for the probable trend of events, for on him lies the responsibility of training the next generation to face those events. When civilisation is passing through a period of stability, this responsibility is not very onerous, no doubt he will conceive it to be his duty to be continually seeking for new methods of improving his teaching, but the questions, what shall he teach, and upon what principles shall his teaching be based, hardly arise, or, if they be mooted, are easily answered according to the accepted tenets of the day. In such times as these, philosophers are unlikely to propound theories which upset the whole basis upon which the current society rests. It may indeed happen that some genius, half a century before his time, attacks the foundations, metaphysical and ethical, which underlie the opinions of the day, but few are likely to heed him until changing circumstances compel the attention of the multitude, and the pedagogue, though perhaps influenced intellectually, will be unwilling to jeopardise his security by pointing out defects in a social adjustment which at the time appears to be stable, or by putting into practice theories which may well undermine that stability.

Nearly a month of the summer term remains to be got through by the somewhat jaded boys and girls after the Higher School examination is over. For many years I have been casting round for stimulating topics to discuss with my more or less advanced mathematicians, and this paper is concerned with my latest find. Its title refers to the four tangents at the points of contact of the nine-points circle of a triangle with the four circles touching the sides. I am aware that a great many properties of the points of contact themselves have been investigated, and I cannot flatter myself that I have found out anything new about the figure formed by the tangents thereat. But the literature of any branch of mathematics is so vast that it seemed best for me, instead of hunting in books, to try to find out what I could in my own way. The effort to do so resulted in a delightful mathematical adventure, which has enabled me to devise something to bring out to our pupils what analytical geometry really is about.

When it is said of the Axioms of Projective Geometry that, for the “points”, “lines” and “planes” of which the Axioms treat, we may take what elements we please (provided only that the Axioms be satisfied), it is, I believe, tacitly assumed that the elements so to be taken are classes of algebraic symbols. That in point of fact the “elements” most commonly used, at least in the introductory stages of learning geometry, are not “points”, “lines” and “planes”, but dots, dashes and blackboards, is rarely acknowledged. Only too often the so-called “figure” of a theorem is apologised for and treated as a mere concession to human weakness. Whereas in fact the “dot”, “dash” and “blackboard”, as elements satisfying the Axioms, provide, in many cases, a most excellent symbolism for the expression of geometrical theorems.

I should like to say straightaway that collecting cigarette-cards is not my hobby But recently the manufacturers of the brand of cigarette I smoke started to issue an attractive series of cards and I said to a friend, “I think I shall save these cards until I obtain the complete set of fifty.” He replied, “About how many packets do you think you will have to buy before you get the set?” And this raises an interesting problem in probability which I do not recollect having seen before.

It may not be improper here to caution learners against some inaccuracies in style and expression which are often met with in scientific works published in this country To matters of this kind much attention is paid by the French mathematicians. By this means, as may naturally be expected, accuracy is produced in thought as well as in style, and hence it is, that many of their recent works exhibit a perspicuity and simplicity, and a logical clearness of arrangement, which enhance their value in a very considerable degree. To put the student on his guard against faults of this kind, some instances of the inaccuracies above referred to are subjoined; and he will find that, by avoiding the use of them and of similar modes of expression, he will soon acquire a comparatively correct style, instead of the careless and random one that is so common in this country Some of these instances will doubtless be found in the works of authors of the present day, but they have been purposely selected from the writings of authors no longer living.

Alfred Lodge became a member of the A.I.G.T. in 1888, was Honorary Treasurer from 1891 to 1896, and first President of the Mathematical Association in 1897-1898. Until recently he was constant in his attendance at meetings of the Council; we shall not easily forget his keen interest in all the activities of the Association or the charm of his kindly manner. His first contribution to the Gazette appeared in No. 1; his last in No. 239.

A Striking thing about contemporary education is the increasing demand for mathematical training. In the eighteenth century the only departments of science which conspicuously called for it were astronomy and such branches of mechanics and optics as had arisen in connection with astronomical pursuits. Navigation was perhaps the only important profession for which any mathematical equipment was an indispensable prerequisite. Though an eighteenth century H. G. Wells might have anticipated minor avenues of future employment for professional mathematicians as teachers attached to artillery or actuarial work, he would scarcely have foreseen that chemistry, power production, genetics, psychology and social statistics would severally enlist the services of the mathematician. The educational problem which arises from the rapid mathematisation of science during the past half-century has found us unprepared, and is largely an unsolved one.

The Annual Meeting of the Mathematical Association was held at the Institute of Education on 4th and 5th January, 1938. On Tuesday, 4th January, Professor E. H. Neville took the chair for the Business Meeting at 2.30 p.m. Professor Neville spoke of the loss suffered by the Association through the death of the President, Professor L. N. G. Filon, on 29th December.

1,1. This contribution to a fundamentally important subject has a multiple origin. Its immediate occasion is the address by Professor Temple, on “The Theory of Complex Numbers”, published in the Gazette of July 1937; and its theme is another alternative to the two types of “theory” there considered (“descriptive” and “constructive”)—which may be designated descriptive-in-terms-of-discovery.

Through no fault of the administrators of the Association my October number of the Gazette has come to hand five months late, and with it the letters of Mr Tuckey and Mr. Snell in criticism of my article, “A Weighty Matter” (Gazette, May 1937). On account of pressure of other affairs, the absence of the missing number was overlooked, I trust that the gentlemen concerned have not assumed that I have succumbed to the adverse portions of their comments “hands down”, and hasten to assure them to the contrary as well as to apologise to them for apparently ignoring their replies. I am grateful for the opportunity afforded me by the publication of their open letters to reconsider not so much the substance as the wording of certain statements in my article, and at the same time to bring forward other suggestions and comments which I have received from other correspondents.

If AC, BD, EF are the diagonals of the quadrilateral and PQR is the triangle formed by the diagonals, the circles on the diagonals as diameters form a co-axal system and the circle PQR is a circle of the complementary system. If the circles on the diagonals as diameters intersect in real points S1, S2, then S1, S2 are the limiting points of the system to which the circle PQR belongs and if S1, S2 are not real points and if the straight line joining the mid-points of the diagonals cuts the circle PQR in L1, L2 then all the circles of the system to which the circle PQR belongs pass through L1, L2 and L1, L2 are the limiting points of the system to which the circles on the diagonals as diameters belong.