There is an urgent need for the expansion of the R.A.F Many young men who are physically and temperamentally fit for the service are unfortunately lacking in the necessary knowledge of elementary mathematics. For some time the Regional Committees for Adult Education have been giving assistance to such men, but the time has now come when this help must be greatly increased and extended to men of lower initial attainments. Cases differ so much that only very small classes are possible, or individual tuition may be necessary Usually the only time that the men are free is in the evening.

In view of the reduction of other facilities by the war, the maintenance of the Library service seemed so desirable that the risks of loss though they could not be ignored must be taken. To reduce them, each volume issued since the outbreak has had a jacket bearing the name of the Association and an address, and a request, to which few borrowers pay any attention, for a monthly notification if the volume is being retained.

A Set of numbers aij (called elements) arranged in a rectangular array of m rows and n columns constitutes a matrix of orders m × n. For example, of the arrays

the first is a matrix of two rows and three columns, i.e. of orders 2 × 3, the second is a square matrix of orders 3 × 3.

From a square matrix of orders n × n we can form a determinant of order n. Thus from the above square matrix we can form the determinant of order 3

If L, M, N are the feet of the perpendiculars from P to the sides of a triangle ABC, the circle through L, M, N is the pedal circle of P with respect to the triangle ABC.

They wept like anything to see
Such quantities of sand.
“If this were only cleared away,”
They said, “it would be grand.”

It would be, and if those young airmen of whom we heard in the February Gazette had not spent so much lost endeavour in weaving ropes of this sand, perhaps they would not now be in such need of the help which we are so eager to give them. For although the seven-times-seven mops of our teaching committees have been sweeping for far more than half a year, there is no doubt that school Arithmetic is still cumbered with a lot of stuff that is of no use outside the class-room. In spite of the fact that teachers mostly teach so well, and pupils learn so well, we must confess that children going out into the world often cannot do the calculations that they need, that Training College lecturers find that many students have to learn Arithmetic all over again (though we camouflage it as “Method”), and that intelligent well-educated folk seem to delight in telling anyone whom they suspect of mathematical leanings that “they never could do Arithmetic”. Surely it is the Arithmetic that is wrong, not the teachers or taught.

Probably many members of the Mathematical Association are being called upon to help with the instruction of an Air Training Corps, or of attested men who have been accepted as pilots or observers and then sent home on “deferred service”. It may be of interest to give the experiences of a teacher who came into contact with this work at an early stage and so encountered some of the inherent difficulties in their most awkward form. As Klein remarked, mathematical teaching is a function of two variables, the subject and the pupil. The difficulties with the subject are not serious, for there are few ideas in air navigation which are not already known to those who teach elementary mathematics and geography. However, these ideas are clothed in unfamiliar dress, and it is necessary to learn a few technical terms, which will be given at the end of this article.

The article by Mrs. Linfoot on the “Teaching of Elementary Inequalities” published in the issue of the Mathematical Gazette for July, 1940, raises a matter of considerable interest and importance.

It is unhappily the case that the elementary treatment of inequalities remains a weak point in English mathematical teaching. While no Higher School Certificate examination paper in Algebra would be considered complete without the inclusion of questions on identities and equations, yet direct questions on inequalities and in equations occur comparatively rarely and, indeed, even when such questions are included in University Entrance Scholarship examinations, they tend to be ignored by the majority of candidates.

The rectangle ABCD is supposed to lie in a vertical plane, with AB horizontal. P is any point in the horizontal plane through CD. Prove that ∠APB < ∠CPD.

Provided that the perpendicular from P on to CD meets CD between C and D, this theorem is true and easily proved by comparing the angles into which a plane through P perpendicular to the rectangle divides the two angles considered. The sine of each of the angles into which ∠APB is divided is less than the sine of the corresponding angle in ∠CPD.

The construction, presumably used by Diocles, who flourished in the second century B.C., for the cissoid is as follows.

Let AB, CD be perpendicular diameters of a circle. Let E, F be points on the quadrants BD, BC such that the arcs BE, BF are equal. Draw EG, FH perpendicular to CD. Join CE and let P be the point of intersection of CE and FH The cissoid is the locus of P corresponding to points E, F on the quadrants BD, BC such that the arcs BE, BF are equal.

Is it always possible to determine a real number to an assigned number of decimal places? Can we always tell how many terms of a convergent sequence must be taken to determine its limit to, say, four decimal places? How much information does the convergence of a sequence give us about the decimal expansion of the limit? These are the questions we shall consider; we shall find that the answer to the first two is “no”. To the third question the answer is that convergence itself gives very little information but that “most” convergent sequences have in fact a subtler property which gives precise information about the expansion of the limit, the rate at which the successive digits of the expansion can be determined and the number of times a digit can recur. The questions we have raised are of practical importance for it is with the approximations-to-so-many-places, not with real numbers themselves, that all practical mathematics is concerned.

One of the fundamental difficulties in teaching the calculus is that we are obliged to introduce the concepts of the Theory of Functions and yet make no use of the real number theory on which the Theory of Functions is based. We must talk of limits, continuous functions, derivatives, definite integrals and of existence theorems like Rolle’s and Taylor’s, and must endeavour to hide the deficiencies of our reasoning beneath the kindly cloak of geometrical intuition. The Calculus course is usually a compromise, completely rigorous proofs of the basic theorems being deemed too difficult for the type of student for whom the course is planned, but I believe a far more rigorous account than the customary one is possible without greatly increasing the strain on the students’ powers of understanding, this can be achieved, however, only by a new approach to the subject and it is with the first steps in such an approach that we are here concerned.

Broadly speaking, the development of mathematical thought has proceeded along two main channels. On the one hand mathematicians have been studying properties of objects which in some obvious and intuitive sense form what is called a continuum, while others were concerned with objects which one conceived as being discrete and well separated from each other. An example of the former type of mathematics is the differential calculus; and the theory of the distribution of the prime numbers 2, 3, 5, 7, 11, … belongs to the mathematics of the discrete. The mysteries of the epsilontic—so dear to us all: “If epsilon is sufficiently small, then delta will be as small as we please”—have no direct counterpart in the theory of the natural numbers 1, 2, 3, …. The underlying idea of the continuum seems to be that of motion, while the discrete and separable appears to be more of a static nature.