Mensuration involves the construction of measures and the methods of using them and the calculation of various quantities from the simplest possible measurements. It is in this last sense that the term is usually employed, and herein problems in heights and distances are solved, the lengths of curves and the areas of plane or other figures and the volumes and surfaces of solids are measured.

Many readers will be familiar with the sequence of Catalan numbers {Cn: n ≥ 0} and the formula

with its alternative form

These can be proved by using recurrence relations, generating functions or André's reflection principle. A good reference for all of these methods is Martin Griffiths' book [1].

However, none of these approaches strikes me as being naturally combinatorial. A formula such as (1) is often derived by making a list of all the ways of doing something, and then subdividing this list into classes of equal size, so that either one class consists entirely of ‘valid’ cases or there is exactly one ‘valid’ case in each list.

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.

At the beginning of the seventeenth century π had already been calculated to 35 places of decimals by the Archimedean method of finding in succession the perimeters of inscribed and circumscribed polygons, each having twice as many sides as its predecessor.

But at this time several mathematicians, notably Snell, were seeking shorter methods of approximation, and finally Huygens in thirteen propositions proved or disproved the assumptions they made, and in three more propositions, making use of the known position of the centre of gravity of a parabolic segment and a theorem of his own on the centre of gravity of a segment of a circle, showed how to find a closer upper limit for π from any polygon than had yet been obtained, and by a seventeenth theorem “on centre of gravity”, of which he does not give the proof or even the enunciation, he found also a closer lower limit.

During my year as President of The Mathematical Association the government began a review of the curriculum across all subjects taught in school. Given the constant tinkering with the education system by every government, this is probably a sentence that any MA president could write during their tenure. But 2013 saw government really trying to layout their vision for what education should deliver in the coming decade.

In English, an aim of the curriculum review is to develop a student's love of literature through widespread reading for enjoyment. Students will get to read Shakespeare, romantic poetry, the great novels of the nineteenth century together with seminal works of world literature. It is hoped that this will expose students to the great works of literature and allow them to acquire an appreciation of ‘our rich and varied literary heritage’ [1, p. 13].

In science, by building on fundamental ideas and concepts, the curriculum review aims to foster in students ‘a sense of excitement and curiosity about natural phenomena’ [1, p. 99]. Students will study the fundamental mechanics of the cell, learn about stem cells, photosynthesis and genomics; they will be exposed to radiation (in a theoretical sense) and the evidence for the Big Bang. Such fundamental ideas will give students the scientific knowledge to be able to understand the uses and implications for the impact of science on society.

If the bisectors of the base angles of a triangle are equal, the triangle is isosceles.

Several solutions of the problem are given in the Gazette for July 1932, and again in the issue of May 1933. It is stated in the former that the problem was mentioned in the Lady’s and Gentleman’s Diary of 1859 and of 1860. The history of the problem is interesting. It appears that it was first proposed in 1840 by Professor Lehmus of Berlin to Jacob Steiner, who had been Professor of Geometry at Berlin since 1834 and held that post until his death in 1863.

Among the various logically equivalent sets of axioms for boolean algebra ([1] discusses four of them), that referred to in the title of this article is popular because of its symmetry (self-duality) and convenience in application. The textbook presentation is essentially as follows; the qualifiers ‘left’, ‘right’ are often omitted, although the symbolic version indicates that they are intended.

A tsunami usually starts on deep ocean water as a result of a submarine earthquake. A tsunami wave is very long, even as much as tens of kilometres, but of only very small amplitude, typically less than half a metre (Bascom [1]). In mid-ocean, the passage of a tsunami is imperceptible, but on reaching a shore it can achieve great heights and can deliver massive surges of water. Before the arrival of the first surge, and between subsequent surges, the water at a shore line usually retracts for a long distance, leaving bare large areas of ocean floor that are normally under water. This paper analyses the behaviour of a tsunami, and explains how its mid-ocean character is transformed to produce such massive surges of water at a shore line.