The following extension of Van der Mond’s Theorem enables us to express as an infinite series one of the solutions of the equation

xr+ axs+ b=0.

It is true that the results can all be obtained from Lagrange’s Theorem, but the process is interesting in that it enables us to obtain the series by an algebraical method.

In this paper, we consider a sequence lr(x) (r = 1, 2, ..) of rational functions of a positive rational variable x, which approach log x as r→∞. Some properties of these functions are analogous to properties of the natural logarithm, and go over into the latter, when the limit r→∞ is taken. Except for pointing out such limiting results, we shall avoid transcendental methods.

When we write a number in the ordinary way, such as 673, the symbols such as 6, 7, 3 in this example stand for what we call the digits of the number, and a piece of mechanical or electrical equipment which operates with and records the discrete digits of each number is often called a digital calculating machine. In the past twelve years there has been a remarkable development of such machines with two important features. First, they can carry out lengthy and intricate calculations quite automatically once they have been supplied with a specification, in suitable form, of the calculation to be carried out, and secondly, they are very versatile, so that the same machine can be used for many quite different kinds of calculation, for example for evaluating values of a function from its power-series expansion, for solving large systems of linear simultaneous algebraic equations, for finding the characteristic values of matrices, for the step-by-step integration of ordinary differential equations. To express these two features, such machines are sometimes called general-purpose, automatic, digital calculating machines, and my purpose is to give an account of some aspects of such machines, both of the machines themselves and of the way in which they are used. I can only touch on some aspects, since there is not nearly enough time in a single lecture to deal with the subject fully.

Dr. John Wallis, who lived through the greater part of the seventeenth century (1616-1703), was a typical child of that energetic age. Son of a Kentish parson, he began his academic career at Cambridge, but ended as Savilian professor at Oxford. He was a scholar who apparently read mathematics in Arabic as easily as he wrote on them in Latin; a founder of the Royal Society; a grammarian; and (needless to say) a vigorous theologian. Into the middle of his famous Algebra he “spatch-cocked” (with apologies for its irrelevance) one of the earliest modern inquiries in experimental psychology : in the course of a sleepless night he had amused himself by extracting mentally the square root of 3 to 20 places and, a month later, showed in the presence of a distinguished foreigner that he could remember the result correctly. Nor did these accomplishments exhaust his versatility For as a supporter of the Parliamentarians he confounded the Royalists by his astonishing facility in unravelling their cipher messages. Those who are offended by his adherence to the cause of the Parliament may be reconciled by the knowledge that he was one of the remonstrants against the execution of the King—a defection which, happily, did not blind the authorities to his great merits when the Savilian professorship fell vacant in the following year.

To attach, in relation to a curve f(x, y) = 0, a geometrical significance to the value of f(x, y) at an arbitrary point of the plane, it is necessary to impose a condition which has the effect of standardising the function f(x, y) by means of some divisor independent of the variables. This article, another outcome of Mr. Durell’s query on the semi-cubical parabola, is concerned with normalisation of the equation of an algebraic curve, and with Laguerre’s interpretation of a standard form of f(x, y). The axes are rectangular throughout.

The following paper arose out of an attempt to improve upon the familiar “match puzzles” one sees in the popular press and which are generally too puerile to be interesting. Following some desultory experiments, it developed that the methods of construction postulated are capable of determining all points obtainable with ruler and compass, but no others. The sequence of results below is arranged to establish this conclusion.

Trigonometry, like most abstract sciences, has both an operational side and a quantitative side. The customary discussion is mainly quantitative, based as it is on the concept of the radian and the definition of sine and cosine as ratios of the sides of a right-angled triangle. The operational side has been touched upon by V. Naylor in his “Note on the Nature of Trigonometry” (1). Starting with the hypothesis that a function λ(x) exists which has the property

If, like David Copperfield, you are on the road to Dover, for the first time and with no one to guide you, and you come to a point where the road forks, then of all your mathematical knowledge, only the method of reductio ad absurdum can help you. For if you choose the wrong turning, and find yourself in Folkestone, you may be quite sure you ought to have taken the other road.

Given a positive continuous increasing function f(x), 0 ≤ x < 1, approximations by a convex function g(x) from below and from above are considered. The approximation is measured by the ratio of the integrals

The activities described in this paper are highly addictive. They can be injurious to work, mental health, personal relationships, serious research and the future of the rain forest. The authors can accept no responsibility for the subsequent behaviour of people who read this paper.

Geometric dissection theory is connected with ‘cutting up’ plane and solid objects and reassembling the pieces to form other objects. It is a very ancient and venerable mathematical activity going back to Euclid (circa 300 BC) and perhaps beyond. In Euclid’s Elements we find a square dissected into four pieces to illustrate the algebraic identity (a + b)2 = a2 + b2 + 2ab.

It is a trite but true remark to say that there are two fundamental random processes, one being the Poisson process and the other being Brownian motion. They are each fundamental in two ways; first, they do seem to describe much of the natural world very well, and second, the student is well-advised to study and master these two before going on to look at harder and more recondite processes. This note gives a brief glimpse of some aspects of the Poisson process, which is technically by far the more tractable of the two.

While reading a book by David Wells on curious and interesting geometry, I came across the following remarkable theorem named after Holditch.

In Figure 1 the point R divides a straight stick ST into lengths p and q, where p, q 0. We restrict the end points of the stick, S and T, to lie on a plane, simple, closed, convex contour, C1, and ST slides around C1. Assuming C1 is such that ST can pass around C1 once, the locus of R is another plane closed contour, C, inside C1.

In this article, we are going to be looking at certain finite sequences composed of zeros and ones.