I wonder why problems about map-colourings are so fascinating? I know several people who have made more or less serious attempts to prove the Four-Colour Theorem, and I suppose many more have made collections of maps in the hope of hitting upon a counter-example. I like P. G. Tait’s approach myself; he removed the problem from the plane so that it could be discussed in terms of more general figures. He showed that the Four-Colour Theorem is equivalent to the proposition that if N is a connected cubical network, without an isthmus, in the plane, then the edges of N can be coloured in three colours so that the colours of the three meeting at any vertex are all different. (A cubical network is a set of points called vertices joined in pairs by simple arcs called edges, no two of which intersect except at a common vertex, in such a way that just three edges meet at each vertex. An isthmus is an edge whose removal destroys the connection of the network.) It was at first conjectured that every cubical network having no isthmus could be “three-coloured” in this way, but this was disproved by the example of Fig. 1, for which it may readily be verified that no three-colouring exists.

The problem how to express very large numbers is discussed in ‘The Sand-Reckoner’ of Archimedes. Grains of sand being proverbially ‘innumerable’, Archimedes develops a scheme, the equivalent of a 10 n notation, in which the ‘Universe’, a sphere reaching to the Sun and calculated to have a diameter less than 1010 stadia, would contain, if filled with sand, fewer grains than ‘1000 units of the seventh order of numbers’, which is 1051. [A myriad-myriad is 108; this is taken as the base of what we should call exponents, and Archimedes contemplates 108 ‘periods’, each containing 108 ‘orders’ of numbers ; tlle final number in the scheme is 108. 1015.] The problem of expression is bound up with the invention of a suitable notation; Archimedes does not have our ab , with its potential extension to aa a . We return to this question at the end ; the subject is not exhausted.

In certain problems of X-ray crystallography it became necessary to determine the coordinates of a point P, originally at (x, y, z) in the non-rectangular, Cartesian system (X, Y, Z) after rotation of the vector about some line . This is the classical Euler transformation, and the result is well known for rectangular axes. Since, however, no reference could be found to the more general case, the results of this calculation may be of interest.

If we take any integral number, and suppose it written round the circumference of a circle, it is possible to form differences by subtracting each digit from its neighbour on the right. If the result is negative, 10 is added to bring the answer positive. The second and successive differences may be formed similarly. It is found that at some stage the lines of differences start to repeat those occurring before.

In Keynes's contribution to the ‘Newton Tercentenary Celebrations’ there is the following passage :

“Again, there is some evidence that Newton in preparing the Principia was held up almost to the last moment by lack of proof that you could treat a solid sphere as though all its mass were concentrated at the centre, and only hit on the proof a year before publication. But this was a truth which he had known for certain and had always assumed for many years.

“Certainly there can be no doubt that the peculiar geometrical form in which the exposition of the Principia is dressed up bears no resemblance at all to the mental processes by which Newton actually arrived at his conclusions.

“His experiments were always, I suspect, a means, not of discovery, but always of verifying what he knew already.”