The work presented here was prompted by curiosity concerning a structural detail of the ironwork supporting the overhead cover of the railway platform at Stockport, Cheshire. The detail – see the front cover which shows the roof support – comprises an elliptical ‘quadrant’ joined to vertical and horizontal members of the supporting structure, with a circular ring fitted interstitially so as to touch the elliptical arc and the two straight members. The question arises: what is the radius of the circular ring in terms of the parameters of the ellipse?

‘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.

This article presents interesting generalisations of three well-known results related to pedal triangles and distances, and the Simson line.

The triangle whose vertices are the feet of the perpendiculars from a point P inside a triangle ABC to each of its sides AB, BC and AC, is called a pedal triangle.

When can a triangle cover a square? More precisely, when can a square of side s fit into a triangle with sides a, b, c?

Questions about when, and whether, one shape can fit into another are fundamental to the elementary study of shapes in geometry, and yet such questions have only rarely been considered in the literature. In 1872 Cayley thought the question of determining precisely when a given triangle fits into a given circle sufficiently interesting that he posed it as a problem in the Educational Times. In 1956 Ford asked when one given rectangle fits in another given rectangle. A necessary and sufficient condition on the sides was soon supplied by Carver (see also Wetzel). The corresponding question for two triangles, asked in 1964 by Steinhaus, went unanswered for thirty years until Post supplied a set of 18 inequalities on the six sides whose disjunction is both necessary and sufficient.

When the authors met at a geometry conference at Easter 1999, they found that each had discovered and investigated an elegant result (Theorem 2.1) about an octagon inscribed in a conic, Larry Evans algebraically and John Rigby synmetically. This turned out to be a rediscovery, as the result had been discussed by E. H. Lockwood in the Gazette in 1967, in an article that complements the present article. Also, almost a century earlier in 1872, the result had been posed somewhat imprecisely by T. T. Wilkinson as a problem in the Educational Times. There is a surprising connection between a special case of the octagon theorem and the golden cross-ratio.

The defining property of a rectangle is that it is a quadrilateral whose angles are all equal (or are right angles, which amounts to the same thing).

The opposite sides of a rectangle are equal. If a and b are the lengths of adjacent sides of a rectangle, such a rectangle can be described as an abab rectangle (or an (ab)2 rectangle). The area of the rectangle is ab.

As far as I understand the latest version of the national curriculum and the new A level syllabuses, which make no serious reference to irrational numbers, students leaving school will not necessarily know that there are two sorts of rectangle: those coverable by a set of identical squares, which can therefore be drawn on squared paper; and those not, such as the golden rectangle. An (ab)2 rectangle is coverable by squares if, and only if, a/b is rational.

Conies are often presented through the following definition: The conic Γ with focus F, directrix l and eccentricity e is the set of points Y of the Euclidean plane E2 such that d(Y, F) = e d(Y, l). Of course, the main interest in this focus-directrix property is to allow a unified approach to the three conies. Surprisingly, a simple construction of Γ given F, l, e seldom follows. The aim of this note is to fill the gap by describing … just one construction for the three conies! Readers are strongly encouraged to try out this construction, either by hand, using a ruler and a set-square, or by using a computer package. As is to be expected, the tangents to Γ are also easily obtained. Moreover, the transformation underlying the whole process leads to some properties of the conies in a simple and elegant way. A few examples will be given.

Two recent problems appearing in die September 1997 and September 1999 issues of the NCTM publication Mathematics Teacher asked the following:

A. An integer is called decreasing if each digit is less than the one to its left. Determine the number of decreasing integers between 2000 and 7000.

B. A 5-digit integer is a peak number if its digits strictly increase to the middle one then strictly descend. Determine the number of 5-digit peak numbers that are greater than 70,000.

Fermat (1601-1665) is well-known for offering mathematical results without stating their proofs. In Mahoney’s fine mathematical biography, suggestions are made giving possible lines of reasoning which Fermat may have used, suggestions which are easily recognised by those familiar with number theory. This article offers some conjectured reconstructions of Fermat’s reasoning which may be more accessible to a beginner since they are linked to pattern recognition, and capitalise on the special cases with which Fermat illustrated his ideas. Generic examples played an essential part in Fermat’s exposition and may well have played a larger part in his proofs than would be respectable in a textbook nowadays.

This article grew out of a problem that I saw in an old textbook on sale in a second hand bookshop. It inspired a generalisation that leads to two sets of well known numbers (Bell and Stirling numbers), and then to a new result for one of them.