We live in days when more than ordinary care is taken to extract from jam-pots, gardens, dust-bins and ourselves the maximum of productivity; and in such times why should mathematical tables be exempt from the general squeeze? To all of us, I suppose, there come moments when we say “For this calculation I wish I had tables giving more figures”: such cases occur especially when anything is found as the difference of two much larger quantities, in which a small percentage-error may appear greatly magnified in their difference (weighing the ship’s cat by the displacements before and after she fell overboard). I shall outline here and attempt to justify the least laborious that I know of methods by which the inevitable approximation-errors can be cut down to a much smaller mean than is involved in using the crude values as printed.

Whether it is because of the utility aspect or whether it is because of some other reason, boys are invariably interested in logarithms. Whenever I take up the subject with a form, at least one boy will ask: “Please, Sir, how were logarithm tables constructed?” When a master is confronted with this question, he usually thinks of logarithmic expansions, and he is obliged to tell the boy that the explanation is beyond his understanding.

The recruitment of suitable teachers of ancillary subjects in technical schools and colleges was often difficult before the war, and even if the ideas of the Spens report did not imply a considerable increase in demand, it is time that this supply problem was faced. In the technical subjects themselves the problem is not so acute; the teacher of a branch of engineering is teaching the subject very much as he was taught it, to students whose outlook is very similar to his own. Teachers of physics or chemistry to the same students, although not in quite such a happy position, have certainly in their own studies met with frequent practical applications of their subject, and will usually be entirely sympathetic towards this aspect. With the majority of the teachers of mathematics, however, the position is very different.

Given a tetrahedron of reference ABCD, let λ, μ, ν, π be the cosines of the angles made by a line with the perpendiculars to the faces BCD, ACD, ABD, ABC respectively, these perpendiculars being drawn all inwards or all outwards; where

A, B, C, D being the areas of the faces BCD, .. If P 1 (α 1, ² 1, γ 1, δ 1) and P 2 (α 2, β 2, γ 2, δ 2) are any two points, the lines PP 1 and PP 2 can be written

to find the cosine of the angle between PP 1 and PP 2.

Several difficulties arise in the definition of the radius of curvature ρ of a plane curve. In the first place, it does not seem to have been previously noticed that the convention adopted in many textbooks for the sign of ρ leads to ridiculous consequences. Consider the very simple case of a circle of unit radius, with centre at the origin. Use A, B, C, D to denote the points (1, 0), (0, 1), ( −1, 0), (0, −1) respectively. Then the upper semicircle ABC is convex upwards and the lower semicircle CDA is concave upwards. G. A. Gibson, in his Elementary Treatise on the Calculus (p. 355), defines the sign of ρ at any point P as being that of d 2 y/dx 2 at that point, or, what is equivalent, as being positive or negative according as the curve is concave upwards or convex upwards at P. From this definition it follows that ρ = −1 on the upper semicircle, but ρ=+1 on the lower semicircle! At A and C d 2 y\dx 2 does not exist, so presumably ρ does not exist at these points! We cannot obtain a value by postulating continuity, since ρ is certainly discontinuous.

By a curious chronological coincidence, the tercentenary of Newton’s birth in 1642 is succeeded by the centenary of another outstanding event in the history of mathematics; the discovery of Quaternions in 1843 by Sir W. R. Hamilton. It seems a suitable occasion for drawing attention to the extent to which conceptions due to Irish mathematicians of that epoch are interwoven into the technique employed in recent developments in mathematical physics.

Accessible accounts of confocal coordinates are somewhat diffuse, and the following synopsis may be helpful. It will be understood that my sole concern here is with the efficient handling of these coordinates; I am not denying that confocal conicoids must be seen as a linear tangential system if they are to be appreciated, or that pure geometry throws a brilliant light on their properties.

A girl is about to leave school and is not intending to go to a university. She has always been interested in mathematics and has done quite well in that subject while at school. Are there any jobs available in which she will find her knowledge of mathematics an advantage?

Members of the Association must frequently have been asked this question, and while war-time conditions have perhaps decreased its importance, when the war is over we shall again be expected to provide an answer. The Executive Committee would be most grateful for any relevant information, and members who can make suggestions are urged to write to Mrs. E. M. Williams, 17 Belgrave Square, Nottingham. It is hoped that in this way a list of ideas can be compiled for publication in the Gazette.

Mathematics has for so long had an honoured place in the curriculum that teachers have expanded the essentials of elementary algebra, and have brought in excessive manipulation, without considering its value or whether it is needed. They have lost sight of the objects of their teaching and have allowed manipulation, of a type comparable to a difficult cross-word puzzle, to creep in. This may satisfy the puzzle instinct of a few of the more intelligent pupils but certainly causes much heart-burning to the average pupil with little distinctive mathematical ability. We should, then, from time to time review the objects of our teaching, and not only our methods, and then examine the contents of the curriculum and discard those parts which are unnecessary, and replace by mathematics which is more value.