Catalan's constant, named after E. C. Catalan (1814-1894) and usually denoted by G, is defined by

It is, of course, a close relative of

The numerical value is G ≈ 0.9159656. It is not known whether G is irrational: this remains a stubbornly unsolved problem. The best hope for a solution might appear to be the method of Beukers [1] to prove the irrationality of ζ (2) directly from the series, but it is not clear how to adapt this method to G.

Have you ever looked at a scenic photograph and wondered where it was taken from? The problem of calculating the location of the camera turns out to have a rather elegant solution. In the first place we want to make as few assumptions as possible about the optical systems used to produce the photograph. In particular, we will not, in general, know the focal length of the camera lens (or that of the enlarger). Neither will we know the precise direction in which the camera was pointing: the photograph we are looking at may be an enlargement of just a portion of the original photograph, so the centre of the photograph may not correspond to the centre of the optical system of the camera. However, we have to assume that there is no distortion, as is often evident on photographs of interiors or buildings taken with a wide angle lens. Fortunately, scenic photography usually involves bright light conditions with the camera stopped down to a small aperture so that it is a good approximation to an ideal “pin-hole” camera.

The substance of this note arose out of an illustration in the Eighteenth Yearbook of the (American) National Council of Teachers of Mathematics. On p. 299 of this work, which is full of many good things, there appears a diagram of a number of compound polyhedra. One of them is an icosahedron surmounted by an octahedron on every face, producing a solid which is simple and singly-connected (not counting the faces in common with the icosahedron itself). Since the dihedral angle of the icosahedron is 138° 11′ and that of the octahedron is 109° 28′, the total angle at each edge of the icosahedron is 357° 7′, and the solid thus has narrow fissures between each pair of octahedra. A slight deformation would close these fissures and produce a solid which consists entirely of equilateral triangular faces, which is neither regular nor Archimedean, but can be thought of as an Archimedean icosidodecahedron with pentagonal pyramids indented in its pentagonal faces.

I have never seen any explanation of why it is possible for a fast bowler to make a new cricket ball swerve, and why it becomes more difficult or impossible to do this when the shine has worn off the ball. Of course a sliced golf ball swerves because of the spin it has, and this swerve is a well-known illustration of the Magnus effect. Spin cannot be the explanation in the case of fast swerve bowling, because if it were the effect would increase as the ball became older and rougher. In any case a fast bowler imparts little or no spin to the ball.

We start from two simple identities:

For any p > 0 and 0 ≥ b ≥ a, now let

Can we formulate statements about Gp(a, b) that generalise (1) and (2)? We cannot hope for equalities, but perhaps we can establish inequalities which somehow reproduce (1) when p = 1 and (2) when p = 2. For (1), this might mean an inequality of the form Apap ≤ Gp (a, b) ≤ Bpap for certain constants Ap and Bp, and for (2) a similar statement with ap replaced by ap + bp. However, these are not the only possibilities, as we shall see.

The purpose of this paper is to present some results regarding the limit is of the sequence

In the problem proposed by A. G. Ioachimescu in 1895 [1], it is asked to be shown that the sequence , defined by , for each , is convergent and its limit lies between -2 and -1.

There have been given many generalisations and other results regarding Ioachimescu's problem in the literature (see, for example, [2, problem 3.1, p. 431], [3,4], [5, Theorem 1, parts (a) and (b)], [6, problem P2, parts (i) and (ii)], [7, 8]).

To anyone remotely interested in the large number of megalithic stone ‘circles’ that are to be found in Britain, any discussion of their geometry, metrology or astronomy brings one name to mind—Professor Alexander Thom. The professor has spent a lifetime scouring these islands, and has produced surveys for many sites, from which he has evolved a most sophisticated picture of the society of the late Neolithic Age. His best known claim is the discovery of a universal unit of length, the megalithic yard (2·72 ft), which he says was used with extreme accuracy (±0·003 ft) during the whole period of circle construction, approximately 3000 to 1500 BC.

Huygens’ approximation to the length of a circular arc is , where l0 is the chord of the arc and l1 is the chord of half the arc. This result is most easily obtained from the series for sin and sin by neglecting powers of beyond the third, L being the true length of the arc and R the radius of the circle. For arcs less than 45° Huygens’ rule is very accurate, the error for an arc of 30° being less than 1 in 300,000.

Cyclically repeating patterns are a common feature of mathematical films. The study of the symmetries which arise in such configurations has a theoretical interest to mathematicians watching the film, and it also has a practical interest to the film maker, because the symmetries present in a diagram can often be used to reduce the labour of drawing. It does not seem that these symmetries have been commented on before, and in this article we attempt a complete enumeration of the cases which can arise.

These past two years have seen the bicentenaries of Michael Faraday, Charles Babbage and John Frederick Herschel. A fourth contemporary, who deserves to rank with these, is George Green, the bicentenary of whose birth will be marked by the dedication of a plaque in Westminster Abbey. His memorial will be in proximity to those commemorating Newton, Kelvin, Faraday and Clerk Maxwell. Those to the Herschels (William and John) and Stokes are close by. Green’s memorial designates him “Mathematician and Physicist”. Most mathematicians will know of Green’s theorem and Green’s functions; physicists find his papers seminal to the study of, for example, solid state physics and elasticity and, since the mid-twentieth century, Green’s functions have become an indispensable technique for those working in nuclear 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.

Group theory has been used in the study of various properties of musical scales. The cyclic group C12 provides a mathematical model for the tempered intervals (see for example Budden [1] Chapter 23, where the diminished seventh chords are shown to correspond to an appropriate subgroup of C12 and its cosets).

In a comparatively short paper, no complete survey of the range of agricultural problems whose solution involves statistical methods can be given. I have therefore chosen to confine myself to one particular theme, the development of the modern field experiment, even though this has meant that I must present a rather unbalanced picture on account of the omission of other important topics such as the use of sampling methods. During six years in the Statistical Department of the Rothamsted Experimental Station, I spent much time in the planning and analysis of a large annual programme of field experiments, and I have illustrated my paper with examples drawn from the work of this Station, at which many of the new ideas in field-experimentation were first tried. I have not attempted to discuss mathematical details at all fully, for the pure mathematics of statistics is a large subject in itself; my present concern is to show some of the practical applications of the theory.