To the mathematician, at least in the field of elementary mathematics, it is immediately apparent what is true and what is false, and for the teacher of mathematics it is the truth, the correct answer, the right reasoning, that are all-important.

In this article, I propose to examine the other side of the picture, to consider the mistakes made by our pupils and to see how they can be turned to advantage in our work as teachers.

Let ABC be a plane triangle with D, E, F the midpoints of the sides BC, CA, AB respectively. Let L, L’ be two real intersecting straight lines in the plane. Then it can easily be seen, as in § 1 below, that if parallelograms are described on the three sides as diagonals, each with its sides parallel to L, L’, then the other diagonals of these parallelograms are concurrent at a finite point of the plane, which may be denoted by P(L, L’). An attempt to determine those points of the plane which are such points of concurrence has led to the following results.

The Sturmian theory is concerned with the number of zeros that a solution of a linear differential equation of the second order may have in a given interval of the independent variable. As usually presented (e.g. Ince (1)) it appears as a very advanced theory, involving elaborate manipulations and reasoning. The purpose of this paper is to simplify the theory by an approach that makes the results intuitive and somewhat reduces the assumptions made.

Tonight I shall deal with some examples of mathematical reasoning which have in common only one feature: that of being surprising, or, even, of once having been surprising. We may picture them as monsters, waiting to leap on our loose ideas about mathematics and to tear them to shreds.

I shall not concern ourselves with elementary errors. Every one of you, no doubt, can “prove” that all triangles are isosceles, and that 2 = 1 by simple algebra.

Some of the properties proved in this essay can hardly have escaped notice, but judging from the lack of reference, they seem to have been forgotten; others are given for the sake of completeness, and a few are probably new. It is hoped that in any case all are of sufficient novelty to be of interest.

The locus of the point from which the tangents to two conics form a harmonic pencil is in general a conic which passes through the eight points of contact of their four common tangents; it is called the harmonic locus of the conics, and it plays an important role in projective geometry.

The mathematics teacher has from time to time to make a decision as to how much he should say to a class about a topic which is too difficult to discuss with full mathematical rigour. This situation may arise with students who are too young or too immature in their mathematical development to appreciate a rigorous discussion, or with older students who are interested in mathematics purely as a tool which they require in the study of other subjects such as engineering and have not time to spare even if they would be interested in a rigorous development of the particular topic.

We say that a bilinear transformation w=f(z) leaves a given locus C in the complex plane “invariant” if C coincides, as a point set, with its image f(C) under the transformation.

The stream function, ψ, is a function specially suited for dealing with two-dimensional flow while the velocity potential, ϕ, is a function which may be used with either two- or three-dimensional flow.

In two-dimensional flow, the chief utility of ψ and ϕ occurs when they satisfy Laplace's equation ∇2ψ = 0 or ∇2ϕ = 0, for then, standard solutions of these equations are known which can be fitted to meet the needs of the particular problem under consideration. The connexion between ψ and ϕ is not clearly set out in aerodynamics books; especially is this the case when it is desired to pass from the study of incompressible flow to the study of compressible flow. In this paper, an attempt is made to make clear the connexion between ψ and ϕ.