1. There is a curiously simple and compact formula

which may be used to derive from a, a first approximation to the nth root a number R, a second approximation b, very much closer to the true value Applications of this formula when n has the value 3 are the purpose of the note.

To a reader of the Gazette the formula is possibly not new. It is not to found, so far as I can ascertain, in any treatise or textbook, and has remained unknown; but in 1929 the Gazette (Vol. XIV., p. 308) printed in a footnote statement made by Reuben Burrow in the preface to his Theory of Gunnell (1779) that a rule for extracting cube root (equivalent to (ii) below) was “for more exact and expeditious” than the common method. Three years late this “gleaning” inspired Mr. G. W. Ward to send two contributions to Vol XVII. (pp. 52 and 127). In the first he showed that the order of the error b was the cube of that in a; and in the second, by an improvement upto Newton’s method of approximation, he obtained the explicit value of b give by (i) for any value of n, anticipating the present paper by ten years. I have thank Mr. T. A. A. Broadbent for drawing my attention to these facts and dates.

David Pagni drew attention to a result which is ascribed by Dickson [2, p. 286] to Liouville (1857), that the sum of the cubes of the number of divisors of each of the divisors of an integer, is equal to the square of their sum. For example, the divisors of 6 are 1, 2, 3, and 6, which have 1, 2, 2, and 4 divisors respectively, and

Pagni observed, as have others, including Mason et al. [4, p. 179], that the reason Liouville’s result works is because the numbers generated are element-by-element products of sequences of the form {1, 2, ..., t}, which are well known to have the sum of their cubes equal to the square of their sum.

For many years candidates (and teachers) involved in A-level work have quoted Newton’s experimental law of restitution as one of the principles enabling collision problems to be solved. It had always seemed to me that it is fortunate that such a simple law holds and also that it is a great pity that there is no simple relationship between the initial and final kinetic energies. Recently, however, I realised that, viewed in terms of invariance, the law is not at all surprising and also that there is a very simple way of dealing with the loss of kinetic energy.

We have already explained pp. (25-31) that there are two types of uniform three-dimensional space, viz., finite or elliptic space, in which the geodesies or straight lines are closed and of constant length, and infinite or hyperbolic space, in which the geodesies are unclosed and of infinite length. Euclidean or parabolic space may be regarded as a special third type, or as the dividing case between the two general types; but it is best to regard it as a variety of hyperbolic space, owing to the reality of the infinite. In elliptic space all geodesies through a point either return without intersecting again, or have at most one other fixed point of intersection. In hyperbolic space it does not appear necessary that a geodesic should consist of only a single branch, though this is of course the simplest case; if the geodesies have several branches, all geodesies through a point will have a fixed point on each branch, and two geodesies which intersect will have as many points of intersection as they have branches. When we consider two-dimensional space we find that, even in the elliptic type, the geodesies may have any number of intersections, and may be unclosed and infinite. The proof to the contrary on p. 31 does not apply when the geometry is confined to a surface.

The exponential law of increase has been described by Professor D. J. de Solla Price as “the fundamental law of any analysis of science”, and considered to hold over long periods of time [1]. The same writer suggested that low-grade work doubled in about ten years, but that the period taken was about twice as long for very high quality activity. It is useful to remember that a doubling every fifteen years corresponds to a multiplication by about a million in three centuries, or an annual increase of nearly five per cent.