Many everyday events give rise to a need for quantitative thinking, thus leading to the creation of numbers and number relations in living experience. Such development of number in the normal spheres of human activity is one of the fundamental modes of human thought, and early consideration of this initial number work may be confined to verbal form. At this stage, situations requiring number thought can be discussed as they arise, and two consecutive situations often have no more in common than the use of number itself. It is this realisation which enables number-learning to take place. Increasing awareness of number and number-relations brings greater complexity, and it is then that the notation of number is needed. This notation has to be learned in relation to the ideas represented, and there are two major forms :

(i) The figures 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and the notion of place position to represent other numbers.

(ii) The use of a standard measure to facilitate comparisons between certain similar quantities such as length and weight.

The following problem was recently proposed by Professor E. H. Thompson: A rigid body is freely pivoted at a point O through which pass three fixed axes, OA, OB, OC; if the body is given one finite rotation about each of these axes, in the order stated, what conditions must be satisfied by the angles between the axes if these three rotations are to be capable of moving the body into every position consistent with O remaining fixed?

It is clearly sufficient that OA and OC coincide with a perpendicular to OB, for this is the familiar configuration. for the three Euler angles which specify rotations about two fixed intersecting perpendicular axes.

The following problem was included in the paper set in the Mathematical Tripos on the morning of January 5, 1881 :

iii. Prove that, if a + b + c = 0 and x + y + z = 0, then 4(ax + by + cz)3 - 3(ax+ by + cz)( a2+ b2+ c2)(a2+ y2 + z2) - 2 (b - c) (c - a) (a - b)(y - z) (z - x) (z - y)=54abcxyz.

A solution of this problem was published by A. Cayley in June 1881 ; see the Messenger of Mathematics, XI. (1882), 23-25. Cayley's solution took the form of a long and cumbrous verification, and he admitted that he did not know the origin of the result nor did he see any simple way of proving it.

A paradox connected with the idea of a uniform field of force is recalled. The main purpose is to discuss frames of reference in the gravitational fields of systems originally studied by Milne and McCrea in developing their “newtonian cosmology” It is shown that a frame moving with any particle of the material of such a system serves as a newtonian frame for the description of mechanical phenomena throughout a region that may be taken to be arbitrarily large and that a frame of such a sort is the nearest approach possible in the system to the realization of a local newtonian frame ; any two such frames are nevertheless in mutually accelerated motion.

The purpose of this note is to suggest a system of marking chess-players that is both simple to apply and based on a theoretical background. The non-mathematical reader will not need to understand the formulae, but is recommended to read all the words.

In the British Chess Federation Year Book, 1953–4, pages 39–42, there is an article entitled “ A grading system for British players ”. It describes a system that it admits to be open to some objections. At first sight the system looks reasonable and is fairly simple to apply, though of course any system requires a fair amount of calculation if it is to be applied to hundreds of players. An objection to the system, mentioned in the article, is that it is always profitable to play an opponent whose mark exceeds one's own by more than 500 points, unless the system is modified in an inelegant manner. The publication of the article is stimulating to further thought and the system to be described here is one of the results. It is not open to an objection of the above form.