A continued fraction is an expression of the form

The expression can continue for ever, in which case it is called an infinite continued fraction, or it can stop after some term, when we call it a finite continued fraction. For irrational numbers, a continued fraction expansion often reveals beautiful number patterns which remain obscured in their decimal expansion. The interested reader is referred to [1] for a collection of many interesting continued fractions for famous mathematical constants.

‘After Jordan came Lebesgue, and we enter on the subject of another Book’, declared Bourbaki. J.C. Burkill remarked in, ‘It cannot be doubted that (Lebesgue's thesis) is one of the finest which any mathematician has ever written.’ Loève in picturesquely sets the scene for us thus, ‘… the Archimedes of the extension (i.e. modern theory of measure) period was Henri Lebesgue. He took the decisive step in his thesis … . In fact contemporary (measure theory) still dances to Lebesgue's tunes.’ Arguably, before 1902, mathematicians had yet to develop a theory of integration; Lebesgue's great thesis of that year changed this state of affairs irrevocably. In it, difficulties which had begun to plague the Riemannn integral were swept away as Lebesgue boldly extended the concept of the integral by what later came to be regarded as a completion process as profound as that leading from the rational numbers to the reals. Lebesgue created no School but his influence on twentieth century mathematics was profound. In this article celebrating the centenary of Lebesgue's thesis, we look first at Lebesgue's life and then in more detail at his seminal work on integration.

A rational number is said to be expressed in Egyptian form (or Egyptian presentation) if it is written as a sum of fractions where each fraction has the numerator 1 and all of the denominators are distinct. The individual terms of die sum are often referred to as unit fractions. The problem of finding such expressions, and many related questions, has a long history, apparently beginning with Ahmes in the Rhind Papyrus. It is for this reason that we talk of ‘Egyptian’ form.

In 1942, R. C. Lyness noted that some recurrence relations generate cycles, irrespective of the initial values. For example, the order 2 recurrence relation

generates a cycle of period 5 for almost all values of u1 and u2 [1].

The globally periodic nature of sequences generated by this recurrence relation can be seen by setting u1 = x and u2 = y. The sequence is then

Lyness gave other examples of such recurrence relations but had been unable to find one with period 7 and challenged readers of the Gazette to find such a recurrence relation or prove it to be impossible.

No answer to this challenge was forthcoming. However, since Lyness's time, interest in these cycles has been maintained due to links with cross-ratios and elliptic curves. In recent years, Jonny Griffiths has done much to popularise these cycles [2].

Although high school textbooks from early in the 20th century show that spherical trigonometry was still widely taught then, today very few mathematicians have any familiarity with the subject. The first thing to understand is that all six parts of a spherical triangle are really angles — see Figure 1.

This shows a spherical triangle ABC on a sphere centred at O. The typical side is a = BC is a great circle arc from to that lies in the plane OBC; its length is the angle subtended at O. Similarly, the typical angle between the two sides AB and AC is the angle between the planes OAB and OAC.

Mathematicians are fascinated by relatios and have worked our extensive theories about them. The treatments tend to be abstract and sometimes the basic ideas are lost in the abstraction. Here we investigate some common ground between mathematical relations and down-to-earth genealogy, the study of family relations.

Family relations have been studied by mathematicians, perhaps none more playfully than Thomas P. Kirkman (the nineteenth century expert in combinatorial mathematics) in his little puzzle rhyme he sent to the Educational Times [1, p.117]:

Geometry is gradually coming back into the school syllabus, but so far only 2-dimensional geometry. I would like to make a case for including some 3-dimensional geometry as well, because the latter is vital for describing the world throughout science, engineering and architecture. Higher-dimensional geometry also comprises a major part of modern research within mathematics itself. Also 3-dimensional geometry fosters both our intuitive understanding and our geometric imagination. It teaches us to see things in the round. It also trains us to see all sides of an argument simultaneously, as opposed to algebra and computing which emphasise thinking sequentially.

As a rule graphical methods are more tedious than an analytical treatment; but they are welcomed by the weaker student on account of the illumination which they shed upon relations that would otherwise remain somewhat obscure.

Two notes which have recently appeared in the Mathematical Gazette dealing with the graphical discrimination of the cubic tempt me to publish methods of discussing the roots of the cubic and of the quartic which have been in use here for some time and have stood the test of experience.