About the year 1700, in the course of heretofore unrecorded travels, Gulliver met islanders who used a rather odd arithmetical vocabulary They counted.

Astronautics is the scientific study of the many problems which arise when we contemplate the possibility of releasing man from the Earth’s gravitational field, setting him free to explore the Solar System and later the regions beyond. Not so long ago, to admit having such ambitions was regarded as a symptom of megalomania. Mathematical “proofs” of the impossibility of escape from the Earth’s field were legion and ranged from the simple statement “a rocket will not develop thrust in a vacuum” to a (perfectly correct) numerical calculation showing that the most powerful fuel possesses insufficient energy to release even its own mass from the Earth’s attraction. It was made very clear to the early enthusiasts that few advances in this field could be made until the climate of scientific opinion changed and astronautics became “respectable”. It is, of course, quite normal for a line of scientific enquiry to evolve from the stage at which its ideas are the subject of ridicule to the stage at which they are universally acclaimed. The idea of the periodic classification of the elements caused our great-grandfathers to laugh hugely into their beards. Marconi was a notorious striver after the unattainable. The Wright brothers achieved the “impossible” and, in defiance of contemporary opinion, the aeroplane proceeded to change our way of life irrevocably. Astronautics has been singularly fortunate. After spending comparatively few years in the wilderness, the advent of the V.2 and other rocket weapons brought about that change in the attitude of scientists towards the subject which had been seen as a necessary preliminary to the serious consideration of the problems involved. Greatly decimated, the doubters now depend upon the slogan—“it can be done, but it is not worth doing”. However, it is not the purpose of this article to discuss the value, in human terms, of conquering space but only to explain briefly why and how it can be done if man so desires. In any case, it is unlikely that the real significance for mankind of such a tremendous advance can be appreciated before the event. Only the passing of the years can reveal to us the value of our actions.

Recent articles on equations characterising the trigonometric functions ([1], [2]) prompt a consideration of still another equation, which might be classified under the heading of “Students’ Mathematical Mythology” In [3], Mr. Heafford multiplies the “obvious” sin (x + y) = sin x + sin y by the equally “clear” sin (x − y) = sin x − sin y to obtain the (correct!) result, sin (x+y) sin(x − y) = sin2x − sin2y.

I will illustrate the kind of problem I am going to discuss by showing how it applies to the case of addition. Suppose we add a series of numbers, each correct to the nearest unit ; this may be tenths, hundredths or any other unit. The maximum error in the sum of n such numbers is ½n units. The argument, almost universally applied, is that as the maximum error is ½n units, the answer is unreliable to that extent. This is bad logic. Let us take an analogy from electricity. When a current is switched on the maximum current is reached only after a time which is infinity. Actually, after a very short time, the difference of the current from the maximum is negligible. Likewise to argue that the sum of n items is unreliable to the extent of ½n units is not merely bad logic, it is completely wide of the truth.

Magic squares might reasonably be supposed to be amongst the most useless of mathematical diversions. Yet, like other apparently fruitless amusements such as Eulerian squares and Kirkman’s school-girls problem, they have found applications in the field of the design of experiments. In an investigation of the effects of two variables upon a phenomenon, where the observations are obtained one after the other, magic squares make it possible to take account of linear trend in the results. For example (the writer is only competent to offer illustrations in the field of psychology : others in bis own field will doubtless occur to the reader) Miller and Selfridge investigated how much subjects could remember of series of words which varied both in their length and the degree to which they resembled everyday English. Again Fitts and Annett, Golby and Kay investigated how fast a subject could alternately tap with a metal stylus two separate metal plates, and how his speed was affected by the width of the plates and the distance between them. In both cases the subjects performed the task a number of times under different combinations of levels of the variables (length and resemblance to English, and width and distance respectively), so that any analysis of the data is liable to be vitiated by the fact that each subject will probably improve with practice throughout the experiment, unless special account is taken of this trend.

The standard procedure for turning right at a cross-road is first to move as far over to the right as possible, while remaining on one’s side of the road, which leaves other traffic free to go straight on or to turn left, provided the road is wide enough.

On connaît le problème classique ou le problème simple de Dirichlet. On donne une surface fermée S qui partage l’espace en deux domaines, l’un intérieur, l’autre extérieur. On suppose que la surface satisfait à certaines conditions de régularité, par exemple, être douée d’un plan tangent. On assigne une fonction continue U(M) d’un point M sur la surface S, et l’on demande de déterminer la fonction U(P), harmonique et bornée dans l’un des deux domaines précédents, qui tend vers U(M) quand le point P tend vers le point M de la frontière. Le problème est intérieur ou extérieur selon le domaine considéré.

In the March 1996 issue of the Gazette, [p. 36] the following conjecture by Sir Bryan Thwaites appeared: ‘Take any set of n rational numbers. Form another set by taking the positive differences of successive members of the first set, the last such difference being formed from the last and first members of the original set. Iterate. Then in due course the set so formed will consist entirely of zeroes if and only if n is a power of two.’

A graph consists of a set of vertices some pairs of which are joined by a single edge. A path is a sequence of distinct vertices (x1, x2, …, xm) such that consecutive vertices xi and xi+1 are joined by an edge (xi, xi+1), for i = 1, 2, .., m — 1. One of the unsolved problems of graph theory is to obtain criteria for determining whether any given graph has a Hamilton path, i.e., a path which passes through each vertex exactly once. (For pertinent references and further definitions, see [1].)

1. The reviewer of Berg’s Operational Calculus as applied to Engineering and Physics (2nd ed. 1936) in the Gazette (Oct. 1937, p. 309) remarks that “Heaviside’s methods deserve more attention in this country than they appear to receive”, and a reference is made to articles contributed by Bromwich, Jeffreys and myself to Vol. XIV in 1928.

A theorem of Van Aubel, which first appeared in concerns the figure obtained by erecting squares on the sides of a quadrilateral, and considering various line segments such as the joins of the centres of opposite squares. In parts of the theorem are extended to the case of (i) four similar rhombi, and (ii) four similar rectangles erected on the sides of a quadrilateral. Further, these two results are described as in some sense ‘dual’ to each other. Here we show that they are in fact both special cases of a further extension of the original theorem (see Proposition 6, and the remarks that follow it), and we also find several extra squares in the original Van Aubel figure (Proposition 11).