Lessons on sets have become commonplace in schools today. Venn diagrams proliferate and, even in primary schools, children can be seen sorting and classifying objects by size, colour and shape and placing them in spaces marked out on the floor by chalk outlines or wooden hoops. Older children learn that such diagrams are named after the English logician, John Venn, and that through them we can represent the relations of membership and inclusion and the operations of union, intersection and complementation. A rectangle is drawn to represent the universe U: subsets of U are represented by the interiors of circles, or other closed curves within U, i.e. subspaces of the rectangle. The elements of U are represented by points within the rectangle, the elements of a subset A by points within the corresponding subspace of the rectangle and the elements of A′ by points within the rectangle but outside the region representing A.

The contribution “Mathematics of the pop charts” by E. de St. Q. Isaacson (The Mathematical Gazette, 51 (1967), 282-287) prompts me to publish a result found in 1959 during research on the same subject.

It must be remembered that one does not get a leap year at the turn of a century unless the year is divisible by 400.

Groups have for several hundred years been used in the composition of long peals in change ringing, as was mentioned on p. 399 of the Gazette. A brief outline of their use for the non-ringing mathematician may be of some interest.

The use of matrices in connection with geometric transformations is now well established in the literature devoted to the teaching of “modern mathematics” in schools. A possible way of motivating an interest in projective transformations is to be found in the problem of constructing an accurate perspective drawing of an object as seen from a selected viewing point. One of the first artists to master the technique was Albrecht Dürer, and the woodcut by him (Figure 1) shows clearly the basic idea, that such a drawing is a projective transformation from “3D” space onto a plane. These transformations can be represented by 3 = 3 matrices. This note gives an elementary derivation of such a matrix representation, and shows how it may be used to construct perspective drawings.

Engineers photographing oscilloscope traces often want to know how much film is left in a cassette after some of the original film has been exposed and cut off. The obvious way to find out is to open the cassette in a dark room and measure the length left inside. Alternatively of course the length can be found by weighing or if the amounts of film originally present and used are known. Another way, unfortunately only of academic interest, is suggested below.

The problem is to join two given straight pipes by a curved pipe made up of two circular arcs of equal radii. The case when the two given pipes are coplanar has been discussed by E. Kosko in the Gazette of May 1964, pp. 192-196. Purely geometrical constructions for this case have been provided by S. N. Collings in the Gazette of October 1965, p. 302, and by me in the Gazette of February 1966, pp. 37-39. (Unfortunately Mr. Collings’s construction does not always work. On his diagram one can put in the two straight pipes in four different ways. For two of these ways the construction gives valid solutions, but for the other two ways the joining curve has a discontinuity of sense.)

“About, about in reel and rout,
The death fires danced at night:
The water, like a witch’s oils,
Burnt green and blue and white.”
Coleridge

(1) If a small light source, S, is observed through a parallel sided glass sheet, G, such as a window, the only result is that S appears to be less brilliant. But if G is misted over by moisture, the apparent brilliancy of S is further decreased, and diffuse haloes are seen concentric with S. If S moves at a steady speed parallel with G, the haloes appear to move with it. But if S is a discharge lamp, so that it flickers steadily, a new phenomenon is noticed as it moves: these are evenly spaced parallel light and dark bands at right angles to the direction of motion of S. When S is a polychromatic source, such as a mercury-discharge lamp, both the haloes and the bands are coloured.

A weight is suspended by a rigid rod from frictionless bearings which allow it unimpeded travel of 360° in either direction. The pendulum swings in vacuo, and is kept in motion by electro-magnetic impulses at its lowest point, monitored (as regards direction of impulse) by two photo-electric cells, on either side of bottom-centre.