In spite of the great increase in the last fifteen years in the use of the Calculus in the higher forms of schools, and of its introduction into the Intermediate Examinations of many universities, it has not yet been used as much as it could be in the teaching of elementary Applied Mathematics.

It has, of course, long been realised that in the kinematics of the motion of a particle in a straight line we have some of the best examples of the ideas that are the basis of the Calculus, and the notation and machinery of the Calculus is now extensively used in dealing with this part of the subject.

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.

One of the most striking results in the theory of uniform convergence asserts that for a monotonic sequence of continuous functions, continuity of the limit function is a necessary and sufficient condition for uniform convergence. Much has been written about this theorem, but the fact that the proofs given in such works as Bromwich’s Infinite Series and De La Vallée Poussin’s Cours d’Analyse are incomplete, seems to justify the hope that this account may be of some assistance to students and teachers of convergence.

A recent conference, held at Southampton early in September, marked the end of the first stage, and of the first twelve months’ work, of this Project. Its objectives are now well-known, namely, the experimental introduction of entirely new mathematical syllabuses in certain schools, together with the setting of appropriate G.C.E. papers, and the writing of text-books and teachers’ guides. But it may be of some interest to fill in some further details of the work.

Most direct proofs of Morley’s theorem are very involved. Dick Hogg’s neat geometrical proof (Note 67.3) circumvents the problem with its slightly indirect approach, but is very pleasing, none the less. It may however not be generally known that there is a direct straightforward proof using complex numbers which is simple to those familiar with the technique involved; indeed the proof is almost inevitable, once the right start has been made.