Many members of the Association know the Gazette only in its hampered and restricted war-time form. The time is therefore opportune for some remarks on the pre-war Gazette and on what may reasonably be expected from the post-war Gazette. The history of the Gazette, however, cannot be fully understood without some reference to the history of the Association; such reference need be only brief, since a full account is available in the admirable Presidential Address by Mr. A. W Siddons (XX, No. 237, February, 1936).

(I). The following inequality is a straight generdisation of one of the most important inequalities occurring in elementary analysis. It is consequently of some intrinsic interest, even though it has to do with a determinant.

Let n be a positive integer (≥ 2) and let a, b, . . . , h be n real numbers (unrestricted as to sign) arranged in descending order of magnitude, and no two being equal. Let x be a positive number, which will be regarded as variable.

It never ceases to amaze me how often an apparently innocent-looking problem can lead on to something altogether more deep and interesting. As an example of this, I had recently set one of my Lower Sixth classes, as part of a statistics revision lesson, the task of devising some original questions on probability when the following problem emerged:

‘A person tosses a fair coin six times. Given that 4 heads and 2 tails result, what is the probability that the proportion of tails obtained at each point before the last toss is less than one third?’

The subject of the definition of irrational numbers is so intimately connected with questions about the existence of a limit of a sequence, which has recently formed the subject of much discussion in this Gazette, that a fuller consideration of the subject, and, in particular, of how irrational numbers are to be introduced into a course introductory to ‘higher’ mathematics, may not be undesirable.

The familiar fact that we may cut any triangle into three polygonal pieces which can be rearranged to form a rectangle is one of the most elementary problems of polygonal dissection. This fact is shown in two ways in Figure 1 where both sides of the rectangle are predetermined after setting up the triangle: one side of the triangle and half of the altitude to this side, or one altitude and half of the side at right angles to it.

In the June 1974 edition of the Mathematical Gazette Dr. C. W. Puritz gave an interesting description of a school-room exercise on the calculation of π and some associated numerical error analysis. This note suggests some further relevant material which his pupils might also find interesting and instructive.

Since a link in a plane has three degrees of freedom the simplest closed plane mechanism with constrained motion must have four lower pairs, i.e. one more than the number of degrees of freedom. If the lower pairs are turning pairs, the mechanism is the well-known four-bar linkage. One or two turning pairs may be replaced by sliding pairs without affecting constrained motion, but not less than two turning pairs must remain in a closed link group.

My interest in this fascinating area of mathematics was aroused by a recent article [1] in which it was shown that

has an infinity of non-trivial solutions (i.e. u ≠ v) in positive integers, rationals and algebraic irrationals. Replacing u and v by their reciprocals, x and y, permits the equation to be written as

Morley’s theorem is well-known: if we trisect the interior angles of any triangle A0A1A2, then the common point of each pair of trisectors adjacent to a side is a vertex of the Morley triangle, M0M1M2, which is always equilateral (Figure 1). Less well-known is that the two triangles are in perspective, that is, the three lines AiMi, concur at a point M.

The pairs of trisectors further from the sides meet at W0, W1 and W2, the vertices of the anti-Morley (or perhaps the Worley) triangle. It, too, is in perspective with A0A1A2, the centre of perspective being W. Also, the Morley and anti-Morley triangles have a centre of perspective X, the points M, W and X being collinear.