Trigonometry, like most abstract sciences, has both an operational side and a quantitative side. The customary discussion is mainly quantitative, based as it is on the concept of the radian and the definition of sine and cosine as ratios of the sides of a right-angled triangle. The operational side has been touched upon by V. Naylor in his “Note on the Nature of Trigonometry” (1). Starting with the hypothesis that a function λ(x) exists which has the property

Because of the many papers that teachers must grade, it sometimes develops that a problem solved incorrectly is checked quickly by the teacher as wrong, and returned to the pupil without the teacher’s having had a sufficient opportunity to know what the pupil was thinking when he did the problem. Some errors may be classified as pure carelessness, but in other cases there may be some basic fundamentals that the pupil has not grasped, and these are important to clarify

In Note 2449 (M.G. 1954, p. 213), Dr. F. G. Maunsell gave an account of two interesting figures, the flexagon (F), and a modified “hexahexaflexagram” (H). I have been prompted to investigate this subject further, and find that there is really an infinite family of such figures, whose properties become more vivid when they are studied as a group rather than in isolation.

Most of us here present today have been involved (facing one way or the other in the class-room) in one of the most far-reaching and important revolutions in teaching theory and practice that any national system of education has ever known. The first step was Rawdon Levett’s famous letter to Nature in 1870, though it was thirty years before any solid progress could be claimed. Another twenty years of steady endeavour, in which some well-known members of this Association took a leading part, was necessary before the first of our major reports (that on Geometry) was published. Yet another twenty years, in which time four more reports were issued, bring us to the greatest change of all: the Jeffery Report and the acceptance by the schools and the Examining Authorities of the Alternative Syllabus. So far as can be judged after ten years’ experience this revolution, far-reaching in its effect on the mathematical work in schools, has been entirely bloodless—for Euclid, if deposed, is neither exiled nor ostracised.

Digital Computers can be thought to have 3 parts, the memory, which stores numbers and instructions, the arithmetic unit, which does the arithmetic, and the control, which selects, interprets, and acts upon the instructions. Numbers are fed into the machine by some means such as paper tape, which is prepared by a device like a typewriter. The instructions are put on the tape in a coded form, and may look like something like this:

The scope of the additive prime number theory is evident from its name. In this part of the theory of numbers we are concerned with the representation of integers as sums of primes; and here the central problem consists in the proof (or possibly refutation) of a celebrated conjecture made by Goldbach in 1742 to the effect that every even integer ≥4 can be represented as the sum of 2 primes. The truth of this conjecture is still unsettled; but though progress in this field has been slow, attempts to deal with the problem have led to a whole series of striking results.

A certain type of transformation of a set of numbers may be represented as the multiplication of a vector by a square matrix. Repetition of the operation is equivalent to multiplying the original vector by a power of the matrix. Properties of powers of matrices are thus of considerable importance, and many general properties have been established for some time.

The values of the Radii of Gyration have been worked out for some solid polyhedra, polyhedra with faces only, (shell or box-like), polyhedra with edges only (skeleton), polyhedra with vertices only. The results are given in the tables.