In this paper we will discuss the lemniscate curve and show that its arc length can be bisected and trisected using classical ruler and compasses construction. The method dates back to 1718 when Count Giulio Fagnano (1682-1766) first published these constructions [1]. Fagnano was self-educated in mathematics and treated the subject as a hobby. Euler was impressed by his work on this topic and recommended his admission to the Berlin Academy of Science. Euler then generalised Fagnano's work on integrals. In addition, Fagnano was employed to assist in reinforcing the dome of Saint Peter's which was in danger of collapse. Pope Benedict IV rewarded him for his work by publishing his mathematical papers. Fagnano achieved considerable international fame as a mathematician, and rightly so given the outstanding contributions which he made on a number of different topics.

Since due recognition is now being more and more widely given to the importance of Vector Analysis for three-dimensional work in mechanics, geometry and mathematical physics, the time is opportune to consider the place which that subject merits in a University degree course. A student who undertakes anything like research work in applied subjects is considerably handicapped if he has had no training in Vector Analysis; but the need for it is felt much earlier than the stage at which research is generally begun. Every University course contains subjects which are less useful than Vector Analysis; and on the ground of utility alone there is no question that the latter deserves a place in the curriculum. It will, however, be found quite unnecessary to displace anything else, because the time occupied in teaching the essential parts of vector algebra and calculus will be saved, even during a three years’ course, by the application of this method to the three-dimensional parts of mechanics and mathematical physics.

Figure 1 shows a triangle ABC with the midpoints A′, B′ and C′ of its sides. The line segments AA′, BB′ and CC′ are called the medians, and the point G of their intersection the centroid. The line segments AG, BG and CG will be called, for lack of a better name, the semi-medians. It is interesting that the medians of any triangle can serve as the side lengths of some triangle. This property of the medians is referred to as the median triangle theorem in [1, §473, page 282], and is discussed, together with generalisations to tetrahedra and higher dimensional simplices, in [2].

Suppose A is an n × n matrix with real elements, and that λ = λ1 is a dominant double real eigenvalue. We will assume for simplicity that none of the order eigenvalues are repeated. Then there are two possibilities

• (a) matrix is non-defective, and we can find two linearly independent eigenvectors corresponding to the double root λ1,

• (b) matrix is defective, and we have only one linearly independent eigenvectors corresponding to the double root λ1.

The sequence 0,1, a, aa, a(aa), . . . has certain unexpected features which may easily be demonstrated with the aid of a programmable pocket calculator. The investigation of the convergence of this sequence provides an exercise which is both interesting and informative.

There are various ways of introducing a discussion of the completeness of the system of real numbers. In this article we take the unusual approach suggested by the provocative question of the title.

In any elementary treatment of the convergence of sequences and series, a discussion of series of positive and negative terms will lead to the theorem that an absolutely convergent series is convergent.

The exponential law of increase has been described by Professor D. J. de Solla Price as “the fundamental law of any analysis of science”, and considered to hold over long periods of time [1]. The same writer suggested that low-grade work doubled in about ten years, but that the period taken was about twice as long for very high quality activity. It is useful to remember that a doubling every fifteen years corresponds to a multiplication by about a million in three centuries, or an annual increase of nearly five per cent.

By the elementary plane constructions we mean the bisection of angles and segments of lines and the drawing of perpendiculars. The constructions given below are extremely simple, and it is very possible that some or all of them have been given before; but they seem to be of sufficient interest to be reproduced, for the reason that, if they are not new, they are not generally known.

Most tennis players have two types of serve in their repertory. One is a safe serve, which is relatively accurate, but also relatively easy for the opponent to return effectively. The other is a risky serve, which is more likely to be a fault, but also more difficult to return if it is good.

One of the most striking results in the theory of uniform convergence asserts that for a monotonic sequence of continuous functions, continuity of the limit function is a necessary and sufficient condition for uniform convergence. Much has been written about this theorem, but the fact that the proofs given in such works as Bromwich’s Infinite Series and De La Vallée Poussin’s Cours d’Analyse are incomplete, seems to justify the hope that this account may be of some assistance to students and teachers of convergence.