Through any point 0 in space a straight line POP1 can be drawn meeting the opposite edges AD, BC of a tetrahedron ABCD in the points P, P 1 respectively; another straight line QOQ1 can be drawn to meet DB, AC in Q, QI; and another straight line ROR1 to meet DC, AB in R, RI.

A new and general proof of theorem [1] is presented, which accommodates directed angles and covers all possible configurations of line and conic. A sign convention for the use of directed angles in this setting is developed, and examples are given showing how this fundamental theorem can be used to establish, much more directly and succinctly, the many tangent/focus properties of conies.

After the publication of the Mathematical Association’s booklet on sixth-form work in pure mathematics in 1967, P. R. Buckland published the results of a short multiple-choice test, given to graduates training to teach mathematics, in the Gazette in 1969. He discovered what he called “alarming gaps” in students’ mathematical backgrounds. Is the situation any different 15 years later? In an attempt to obtain an answer, Buckland’s test (with a few changes and a new ‘historical’ section) was given to a group of graduates and undergraduates.

In the Gazette for June 1975 (59, No. 408, 98–106) Colin Tripp showed how the familiar problem illustrated in Fig. 1 (in the isosceles triangle, for certain given values of the angles marked a, b, c, to find θ) could be tackled for most of the possible triples (a, b,c) revealed by a computer investigation.

Popular expositions of the new subject of catastrophe theory have tended to concentrate on ‘soft’ applications, but, as I hope to show, it is perfectly possible to make a serious, important, and mathematically more demanding application intelligible to sixth formers with a minimum of differential calculus. The application is in that part of engineering science which deals with the stability of structures. Catastrophe theory produces nothing really novel in this field, but, by assuming Thorn’s theorem, which will be explained below, it is possible to make the whole subject accessible to those for whom Taylor’s theorem for a number of variables, and the associated concepts, are too difficult. Even for engineers already familiar with the results, catastrophe theory offers an attractive unity within the subject and can point to analogies in related fields.

If a square pyramid is cut obliquely by a plane as shown in figure 1, a pentagonal cross-section PQSTR is generated. The “Beijing theorem” states that the only case in which this pentagon can be regular is that in which the pyramid is half an octahedron and (except for a complication to be discussed later) the coordinates of the points P, Q, R, S and T are uniquely determined.