The standard tape–recorder consists outwardly of two spools on which the magnetic tape is wound, from the left to the right, between least and greatest radii of a and b. On my machine, a is about 1.05 in. and b is about 2.75 in. Apart from what is on the tape, the only indication at any intermediate stage as to how much playing time is still left is provided by a revolution counter geared to the left hand spool, the one which starts full and finishes empty. The numbers registered are about half the actual number of revolutions made on the left: this arrangement allows some 1400 revolutions to be registered by only three digits. It can be assumed that the tape is fed out at a constant length per unit of time: so the time still to run is proportional to the length still to run. If the total number of revolutions to run is N, how much of the tape ha3 been run off when the reading is n?

It is possible to project the four-dimensional hypercube γ4 into three dimensions in a symmetrical manner so that the edges lie mainly along the edges of a rhombi-dodecahedron but with two of the vertices coincident at the centre of the figure.

It was encouraging to receive some letters following the article, in the 1969 December Gazette (Vol. LIII, No. 386), entitled “Playing with Probability”. Most of the letters made reference to the use of “a steam hammer to crack a walnut”. The purpose of the article was to review the teaching of probability at three levels in a secondary school, and since stochastic matrices are to be found in the further mathematics of the School Mathematics Project it seemed appropriate to give an example which used them. The fact that the particular example chosen could be answered in one line does not detract from the interest of the method used in the article—although, of course, as mathematicians we are always looking for elegance.

As A. M. Cohen’s paper has reminded me, Euler’s transformation for summation of the slowly convergent alternating series

is commonly derived by the following method, without enquiring into its validity which is distinctly dubious.

It has been customary when Euclid, considered as a text-book, is attacked for his verbosity or his obscurity or his pedantry, to defend him on the ground that his logical excellence is transcendent, and affords an invaluable training to the youthful powers of reasoning. This claim, however, vanishes on a close inspection. His definitions do not alwaysdefine, his axioms are not always indemonstrable, his demonstrations require many axioms of which he is quite unconscious. A valid proof retains its demonstrative force when no figure is drawn, but very many of Euclid’s earlier proofs fail before this test.

No one can possibly welcome more cordially than I do the widening of the school curriculum in mathematics that has taken place during the last twenty years. Even I can remember the days when “Conics” and “Mechanics” were the privilege of a select few, and only a prodigy was initiated into the mysteries of the Differential (much less the Integral) Calculus. All this is changed now, and it is fairly safe (at least that is my experience) to assume that a boy who has won a scholarship has learnt something about the Integral Calculus at school.