I was interested to see in a recent issue of the Gazette a note from Hoare and Wright [1] about the distribution of leading significant digits, l.s.d., and “Benford’s Law” that this distribution is logarithmic. This distribution has fascinated many people since Benford’s original observation [2] in 1938 of the dirty pages at the beginning of a book of log tables.

Ask anyone who has studied mathematics to a moderate level how many trigonometric functions there are and one is likely to be presented with a range of answers depending on what the person being asked is most likely to remember. Perhaps the ‘calculator button’ three of sine, cosine, and tangent will come to mind as these are the three trigonometric functions found on any standard scientific calculator. At a stretch, perhaps the names for their respective reciprocals, cosecant, secant and cotangent, will be recalled. Beyond the modern standard six, looking at calculus or trigonometric texts published prior to 1900 one soon discovers others going by strange names such as versine, haversine, or coversine (see, for example, [1, pp. 53, 63]). There are at least six others with as many as perhaps ten to twelve having received a name at one time or another. Today all these additional trigonometric functions considered important enough to grace the pages of texts in centuries past have fallen by the wayside, to be largely forgotten in favour of the modern standard six. Of course the pedant amongst us would say there is only one trigonometric function, the sine function, which currently stands as the preferred fundamental trigonometric entity, with all others being simple variations of this function, and they would not be incorrect in asserting this. But having the current standard six seems about the right balance between the minimalistic on the one hand and convenience on the other hand.

In Note 1568 (V for Victory) very brief reference was made to Generalised Chess, that is, an expansion of the ordinary game as played on the 8 × 8 board. In that note attention was called to the existence of the 5-leaper, whose moves are from (0, 0) to (5, 0); (0, 5); (3, 4); (4, 3). Here it is proposed to give some account of chess in three and four dimensions. In anticipation it must be pointed out that fractions do not exist in chess geometry—a special kind of lattice geometry; but that lengths such as √(a2 + b2) and √(a2 + b2 + c2), which are not integral, but where a, b, c are, do exist, since they can be constructed from integral lattices.

It was at the Fifth Southeast Asian Conference on Mathematical Education (June 1990, in Brunei Darussalam) that Brian Bolt posed the question: Can the numbers 1 to 12 be arranged within the hexagram below to give equal sums in the six indicated directions? This is the Magic Hexagram Problem.

The process of cutting squares from rectangles described by D. G. Ball in the June 1974 Gazette, and its connection with continued fractions given by Professor Goodstein in Mathematical note 3359 (December 1974), has an interesting application to electrical circuits.

In this note we will discuss five topics from triangle geometry and occasionally encounter something new. Areal coordinates (presented as Appendix A in [1]) will be used throughout, with the exception of the final section where it is advantageous to employ Cartesians.

The topics are:

(1) examples of collinearities where the distance ratios for important points vary with triangle shape;

(2) a pencil of lines which includes the Fermat, Napoleon and Vecten axes;

(3) two more circles related to the Lester circle;

(4) the family of rectangular hyperbolae which circumscribe the triangle of reference ABC and whose principal member is the Kiepert hyperbola [2];

(5) Cartesian coordinates leading to a solution of the ‘Euler disc problem’ [3].

A topic of current interest is that of finding defining sets (preferably smallest defining sets) for various combinatorial structures such as latin squares and block designs, so it seemed interesting (though of no practical use) to try to solve the same problem for magic squares of small size. (The germ of this idea was suggested in the Gazette Note [1] of 1988 under the sub-heading ‘Squares from clues’.)