When the A.I.G.T. was founded in 1871 its committee was really a teaching committee. By 1878 sub-committees had been appointed to consider different branches of Geometry; in 1884 sub-committees were appointed to consider Arithmetic and Mechanics. But it was in 1902 that the Mathematical Association first appointed a committee described as a “Teaching Committee”. What was it that led up to the appointment of this teaching committee?

Finite Galois arithmetics are well-known; finite geometries however, though more interesting to the amateur, have not really acquired professional status and do not appear to any great extent in standard works. The following example arose from a chance remark in Mathematics for T. C. Mits, by L. R. and H. G. Lieber; from this I deduce that a full theory has been worked out, but I have not seen it, and as far as I am concerned what follows is original, and I hope readers of the Gazette may find it new and stimulating.

Introduction, (a) Let M, M’ be two isogonal conjugate points with respect to a triangle (T) = ABC: MP, M’P’; MQ, M’Q’; MR, M’R’ the perpendiculars from M, M’ upon the sides BC, CA, AB of (T), respectively. The triangles (M) = PQR, (M’) = P’Q’R’ are the pedal triangles of the points M, M’ for (T), and the common circumcircle (L) of those two triangles is the pedal circle of M. M’ for (T) [1, pp. 238–242].

Some years ago, under the heading “Introductory theorems in geometrical conics” (Gazette, XVIII (1934), p. 223), an elementary geometrical treatment was offered of some general properties of conics based on the focus-directrix definition

Other definitions of the conic were not considered nor was the relation of the curve to its auxiliary circle introduced.

The literature dealing with curves of constant diameter is not very extensive, and the following notes may be of general interest.

If the distance between parallel tangents to any curve is constant, whatever their common direction, then the curve is a corve of constant diameter. Thus, if p (ψ) is the perpendicular distance from the origin to the tangent to the curve, where ψ is the angle the tangent makes with the positive x-axis, and D(ψ) is the diameter of the curve, that is, the distance between parallel tangents, then

Crawling under a very old six-cylinder car to find out why the self-starter kept jamming, I found that the large gear wheel round the outside of the fly-wheel was worn very badly in three regions spaced at 120° intervals round its outside. Presumably then the engine always stops in one of three positions, and every time the self-starter is used it crashes always into one of these three places. It is an interesting little problem to see just why the engine always stops in these places and to find out mathematically where they are.

It is nearly a hundred years since Boole published An Introduction to the Laws of Thought (1). An attempt to reduce the art of logical deduction to algebra was a bold venture, and perhaps not all of Boole’s original hopes have been fulfilled. Nevertheless, Boole algebra has become a well-established branch of mathematics which has given rise to theoretical masterpieces such as Principia Mathematica (2), and has found practical applications in electrical circuitry, switching problems, insurance law and statistics.

The fallacies in the traditional approach to the problem of evaluating

in a pre-calculus course have been well brought out in an interesting article by Mr. Turton where the limit is obtained by means of polynomial approximations to the area of a circle. On re-reading the article recently it occurred to me that, at very little extra cost in effort and depth of ideas, a rigorous evaluation of the limit was possible which was completely free from appeals to the physico-spatial properties of geometrical figures and assumptions of symmetry and continuity which Mr. Turton makes.

The solution of linear differential equations under given boundary and initial conditions has been greatly expedited in recent years by the use of the method of integral transforms. It is the purpose of this note to show that an entirely parallel method may be developed for the solution of linear difference equations and that this leads to a new view of the generating function of a sequence as a transform of the sequence, analogous to the Laplace transform of a function.