‘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.

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.

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.

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.

Anyone who has taken first-semester calculus has spent a considerable amount of time studying polynomial functions and their graphs. Calculus I students typically learn how to locate certain special points with respect to the graphs of polynomials, such as their x- and y-intercepts, their local maxima and minima, and their points of inflection (if any). They then learn that these points, together with information provided by the first and second derivatives of the functions, can be used to sketch the graphs of polynomials.