It is well known that the simpler regular solids can be assembled into compounds which have, as a whole, the symmetry of a more complicated solid : two tetrahedra with a symmetry of a cube, five cubes with that of a dodecahedron, and so forth ([1] p. 134, [2] p. 134).

In 1924 The International School of Geneva was founded to cater for the families of the diplomats and officials working in the headquarters of the League of Nations. Since then, and with much greater speed since 1947, schools have been set up in many of the famous towns and cities throughout the world for the children of parents who, for one reason or another, spend some part of their working lives in foreign countries. The children who attend these schools come from very varied backgrounds, tend to move even during the later years of their schooling and have an almost undefined variety of plans for their future education or professional training.

The number π signifies the ratio of circumference to diameter of a circle, and the definition depends upon the fact that the value of this ratio is independent of the size of the circle. We can determine a similar ratio for any shape by replacing circumference by perimeter and diameter by width.

In Fig. 1 ABC is any triangle on the sides of which similar isosceles triangles are erected. For the present these are shown external to the triangle, but later on the possibility of triangles erected internally is included.

A semi-regular tessellation is one consisting of regular polygons of the same length of side, with the same ‘behaviour’ at each vertex. By this we mean that the polygons appear in the same order (though different senses are allowed) at each vertex. The three regular tessellations are for convenience included here as special cases of semi-regular ones. An example of a semi-regular tessellation is that with triangle–triangle–square–triangle–square in cyclic order, at each vertex. A shorthand notation (‘Schläfli symbol’) for this is 3.3.4.3.4.

In Gazette No. 394 (December 1971), R. H. Bromley raised an interesting question: what is the rôle most appropriate to each of the two familiar forms of the derivative, Leibniz’s dy/dx and Lagrange’s f′(x)?

In a recent paper [1], Victor Bryant shows how the number of axioms required to define a vector space can be reduced to seven (in addition to closure requirements). The main result of his article is that commutativity of addition can be deduced from the other axioms. In the present article we show how to reduce this number to six. For certain underlying fields one or more of these axioms can be deduced from the others. However, the six axioms are in general independent; we invite interested readers to show this by constructing their own counter-examples, which the editor of the Gazette will be pleased to receive.

Most elementary differential equations texts treat reduction of order and variation of parameter as separate—and seemingly unrelated—methods. The purpose of this short article is to show how these two well-known methods may be regarded as special cases of a more general algorithm, and to clarify the conditions under which they are applicable. We suggest that this algorithm be called ‘parametric reduction’.