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 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.

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.

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.

In the previous note, I illustrated how to construct an approximation to the factorial function x! by starting from some approximation f0 (x) on 0 ⩽ x ⩽ 1, and then extending it by the functional relation

so that

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.

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.

George Boole gave (in his books, listed under [1] and [2] in the references at the end of this article) a method for solving any equation in his algebra. This part of his work is now largely ignored in the many modern books on boolean algebra which are making their appearance (I can think of several reasons why). Nevertheless, the method can be adapted to conform with our standards of rigour.

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

Suppose that a1, …, an and c1, …, cn are two sets of n positive numbers. If the arithmetic mean of the ci is at least equal to that of the ai, when can we say non-trivially that the geometric mean of the ci is at least equal to that of the ai and that equality will require equality (in some order) of the ai and ci?

A well-known answer of course occurs when each c is equal to the arithmetic mean of the ai. When the two sets each consist of three numbers I can give an answer including this important case. To obtain this result it is sufficient to confine the ci to the interval from the least to the greatest of the ai.