1. It is necessary to challenge Mr. F H. V Gulasekharam’s Note 1448 (May, 1940) in which he claims to give simpler methods than usual for dealing with the circle

S ≡ (Σlx)(Σx) − Σa2yz = 0.

Objection must be taken at the very beginning where he says “Suppose the circle cuts the sides of the triangle of reference ABC at D1D2” The arguments of §§ 1 and 2 are sound when these points are real, but it is asking too much of a young student to expect him to understand what is meant by the ratios D1C/BD1 and D2C/BD2 when D1 and D2 are imaginary Moreover the writer considers that the arguments of §§ 3 and 4 are not simple but very difficult.

Introduction to the principles of elasticity is often received by the average student in a rather piecemeal fashion, the ideas of strain, shear, etc., being indicated from consideration of a number of very special cases. There is a danger that the student may not see the forest for the trees for a considerable time, and a more general approach to the analysis of strain, if not too difficult, would have obvious advantages. The general presentation as in Love's Elasticity is probably too lengthy for a first approach to the subject, and the following method is suggested as being perhaps suitable for the purpose. Any difficulty that a junior student might find with the mathematics used is compensated by the extreme conciseness of the method. Moreover, the mathematical methods used are all important ones, and the treatment might well serve also as a first introduction to some of these methods. The incidental derivation of the rotation formula would have the further advantage that the student would be shown at an early stage the significance of the curl of a vector.

It sometimes happens that teachers want missing figure problems and coded sums, for instance, at the end of term. These given here may be of use in saving teachers from having to make them up themselves. To save space, the missing figure problems are given as a whole, with the data printed in clarendon type. In setting these problems to a class the exact position of every missing digit must, of course, be indicated : this applies to remainders as well as to the main sum.

Any plane KLPQN cuts the edges of a tetrahedron ABCD in six points forming a complete quadrilateral. If UVW is the triangle formed by the diagonals of this quadrilateral, then

1. (i) the triangular f aces BCD, ACD, . of the tetrahedron are each of them in perspective with the triangle UVW from points A1, B1, C1, D1 respectively

2. (ii) the tetrahedra ABCD, A1B1C1D1 are in perspective f rom U, V, W and from a fourth point Ω.

Here again we are confronted with a variety of methods. For purposes of work in the early stages at least two of these are worthy of consideration. They are the factorial or Ten-ten and the whole-sale methods. This latter is sometimes called the “ Ballard Box Method “ although it was known and used (but not extensively) long before Dr. Ballard's work on this subject.

From four given lengths as sides it is possible in general to construct three distinct cyclic quadrilaterals, whose circumcircles are equal. Any pair of these quadrilaterals has one diagonal-length in common.

The algebra taught at present in schools and colleges and discussed in recent reports is essentially that of Euler and Chrystal as regards both content and method. Must this tradition continue? It is hoped that this short note will suggest the contrary and indicate the possibility of incorporating the ideas and methods of modern algebra in elementary teaching. To this end attention will be drawn to some of the simpler concepts of the subject.

Mr. Gibbins’ paper on the “Feuerbach Quadrilateral” (Math. Gazette, No. 249) deals with a delightful topic. The device employed is certainly ingenious. May I, however, be permitted to discuss the topic from an entirely different point of view, but with the same object in mind, namely of stimulating our pupils’ interest in Analytical Geometry I shall also bring out some interesting properties of the diagonal triangles and the diagonal-point triangles of certain quadrilaterals associated with a triangle, employing general rather than special methods to arrive at the results.

We will consider first the second order partial differential equation

where α, β, γ, δ, ε, η are all functions of x or t only

Suppose the six coefficients are functions of x. Then since

we have

for none of the six coefficients contains t.

It is generally accepted among teachers that there are certain forms of arrangement in arithmetic with which pupils are more comfortable and more sure. Addition, for example, is usually considered to be more rapid and more accurate when the addenda are placed beneath each other as in vertical tots than when they are placed side by side as in across tots.

When considering the advantages and disadvantages of the different methods used in long multiplication of money I decided to commence with an investigation into the various forms of setting out short multiplication of money. The surprising results obtained led me to extend the work to other items of the arithmetic syllabus. Each item of the whole investigation is recorded in the chronological order with which it was dealt.

In many School Certificate courses of Algebra inequalities are scarcely mentioned, and yet in Higher School Certificate work it is often assumed that the pupil can, without any further teaching, carry out easy manipulations of inequalities. It is true that the gap is a small one, but even the most intelligent pupils may fail to bridge it satisfactorily for themselves, and difficulties will then arise. One example of such a difficulty is noted in the Report of the Mathematical Association on the Teaching of Mechanics in the section on friction. The Report reads “There is a tendency among some pupils to proceed at once to the limiting case when equilibrium is about to be broken, to write down the equations for this case, manipulate them, and finally change the resulting equality into an inequality, using > or <, according to the answer required.”

An article by Lawrence Crawford in the Gazette for February 1934 (see also Gazette, February 1935) pointed out that the textbooks rarely make any attempt to relate coordinates of foci to the equations of corresponding directrices, etc. The article went on to show how this could be done.

In what follows an entirely different method is adopted. The method leads to actual formulae for all the principal parts of the conic.

“Please Sir, what’s the use of learning this?”

I wonder how many times we have all—that is, if we are at all “approachable”—been asked this or similar questions? To-night I hope to show why the question is ever asked and, in my opinion, how to stop it arising in the minds of the pupils.

But first let me make clear this one fact. Although this is being presented to you in the month of February it was commissioned by your committee in December last, and though in the interim the Spens Report has been published it has not been used in any way in the preparation of this paper.