Archimedes developed a method for calculating the area of a segment of a parabola. This result was outstanding in mathematical antiquity. The idea and the method were not developed further until nearly 1800 years later. In our article his method is presented in modem notation using modem concepts. We show that the method generalises to curves of higher order. Many of Archimedes' arguments can be used without any adjustments and thus show the strength of his original approach.

The mathematician from Syracuse, Archimedes (born about 287 BC, died 212 BC), developed a method for the calculation of the area of a segment of the parabola.

Many readers are familiar with the celebrated Finsler-Hadwiger and Weitzenböck’s inequality in triangle geometry. On the other hand, Wolstenholme’s inequality is not widely known, but is equally important and in an indirect way can be used to derive these two. We will introduce it and then show its strength.

The celebrated theorem of Weierstrass, dating from 1885, states that continuous functions can be uniformly approximated by polynomials on any bounded, closed interval. But just how well can we approximate by polynomials of a certain degree? Let us introduce some notation to facilitate the discussion. For an interval I (which will usually be ), denote by C (I) the space of continuous functions on I, and write for (the notation is often used). Uniform convergence of fn to f (on I) equates to the statement that . Denote by the space of polynomials of degree not more than n. This is a linear subspace of C (I) of dimension n + 1. We write

This Association came into existence nearly one hundred years ago with the prime object of improving the teaching of geometry and although its field of interest has widened considerably the teaching of geometry remains a very important issue. Of course, the question “Whither Geometry?” is rather meaningless unless one specifies whether one is talking about geometry as taught to primary, junior or secondary school levels, to Colleges of Education and the Polytechnics, to undergraduates and postgraduates at University, or geometry as understood by professional research mathematicians. Moreover the motivation for studying geometry may well be quite different at these different levels. I shall try to deal with the question as it affects all these various levels of mathematics.