The Sixth General Meeting of the Association for the Improvement of Geometrical Teaching appointed a Committee for Geometrical Conies, and one for “Higher Plane Geometry, including such subjects as Transversals, Projection, Anharmonic Eatio, etc.”

Dr. Hirst in his Presidential Address has said of the latter subjects, “Until these notions become more familiar ones, I, for my part, believe that Geometrical Conies will always remain in its present unsatisfactory condition. . . . It will be, of course, a question for this Association to decide whether, pending the introduction of the more thorough treatment based on the notions to which I have alluded, some improvement may not be introduced into the subject of Geometrical Conies, as at present understood, with a view of enabling examiners at all events to examine with greater facility and purpose. At present the definitions are so multiform and the sequence of propositions so varied in different text-books, that it is found to be an exceedingly difficult task to examine satisfactorily in the subject at all.”

The cover picture shows a 22° halo seen in Preston on Sunday 28 October 2001 at 10 am. In contrast with a rainbow, the red band is on the inside rather than the outside, the colour bands are much thinner, and the arc is centred round the sun instead of being opposite to the sun. Also, it is not raining. Haloes occur on still sunny days when there is plenty of cirrus cloud. It is possible to see a full circle depending on how high the sun is in the sky. This never happens with a rainbow except when seen from an aeroplane. Sometimes the colours are particularly vivid at the 3 o’clock and 9 o’clock points. These vivid areas are called parhelia or mock suns. Sometimes the parhelia are all that can be seen.

What is the most symmetric quadrilateral you can think of? Most people would say ‘a square’, but having given this (or any other) answer, can you create your own definition of a ‘quadrilateral’, and of ‘most symmetric’, so that you can justify your answer as a rigorous piece of mathematics? This is a challenging investigation because the answer may not be the obvious one.

Often the main difficulty in solving a problem lies in finding appropriate definitions; here we must first consider the questions ‘what is a quadrilateral?’, and ‘what is a symmetry of a quadrilateral?’.