The earlier sections of this account are concerned with some standard properties of pure recurring decimals. These include the cyclic property, illustrated for example by the decimals representing the six proper fractions with denominator 7, each of which has a period using a cyclic permutation of the digits 142857 It was a question concerning this property, raised by a student, which led to the writer's interest in the properties considered in the later sections. References are given in Dickson's History of the Theory of Numbers, Vol. I, Ch. 6.

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.

The most important properties of any distribution are its moments. The first four moments, for example, can be used describe any data set fairly well. Moments can also be used for estimation.

There have been numerous attempts to model the governing dynamics between the two ostensibly competing concepts of brilliance and mechanical steadiness. One interesting study is given by the English mathematician G. H. Hardy in his model [1] describing two characteristically different golfers playing a match against each other. The model challenges the apparently accepted doctrine of the ‘Brilliant player’ having the advantage over the ‘Steady player’ in a long series of golf matches by holes. Hardy defines ‘brilliance’ as the capacity to produce ingenious results as well as the capacity to make mistakes, compared to ‘steadiness’ being completely mechanical producing the same average result all the time. The two players in his model are equal in performance on average, only the brilliant player has a higher standard deviation whereas the consistent player has a standard deviation of 0.

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].