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.

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.

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.

The general solution in positive integers of the equation

(1) 2a(x2−y2) + l =z2

where a is a prime number (not 2).

In recent years there has been a great development in the design and use of machines for carrying out calculations of various kinds, and this is a technical advance of some importance, not only because of the saving of time and labour in calculations which would be carried out in any case, but still more because it makes it possible to carry out extensive calculations which would be too laborious and time-consuming to undertake without such mechanical or electrical aids.

If the bisectors of the base angles of a triangle are equal, the triangle is isosceles.

Several solutions of the problem are given in the Gazette for July 1932, and again in the issue of May 1933. It is stated in the former that the problem was mentioned in the Lady’s and Gentleman’s Diary of 1859 and of 1860. The history of the problem is interesting. It appears that it was first proposed in 1840 by Professor Lehmus of Berlin to Jacob Steiner, who had been Professor of Geometry at Berlin since 1834 and held that post until his death in 1863.

It was announced in the May issue of the Gazette, that the General Teaching Committee of the Association has appointed a subcommittee to enquire further into the teaching of mathematics to technical students. Many teachers (especially those elementary and secondary teachers who also do part-time evening work) lack the time to explore the vast field of the application of mathematics to other subjects, the knowledge of which would, as they realise, do so much to increase the efficiency and interest of their work.

Questions on differentiating from first principles many of the elementary functions, such as sin x, xn, log x, ax, are still often set in examination papers on elementary Differential Calculus, in spite of the difficulties of evaluating rigorously the necessary limits.