The textbook demonstrations of the well-known properties of two conics, the common self-polar triangle and the conic-envelope of lines cut in harmonic conjugates by them, usually suffer from the defects that the general case only is given and that the methods are not always adaptable to special cases. As the author feels the habit of contentment with general cases only a dangerous one and encouraging of loose thinking, he proposes to list the special cases, indicating for the sake of brevity proofs in outline only or, where very trivial, omitting them entirely. A third property—that of the conic-locus of points at which the conics subtend harmonic conjugates, as the immediate reciprocal of the second, has not been listed.

At the present time, when the reform of the root-principles of education is being considered, it may be useful to study systems, organisations, didactic and methodic experiences of other countries, and compare them with the proposals recently made by various committees and individuals, proposals that might become the basis of educational regulations of the near future. The comparison of the study and teaching of mathematics is the more justified since they will be certainly less affected by differences in national temperament and conditions than the study and teaching of any other subject. This has been one of the purposes of the meeting of school teachers from occupied countries working in British schools, in conference with British teachers, in August 1943. These intensely practical studies of comparative education are invaluable at this time.

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.

The process normally given in books for the solution of linear equation in integer unknowns is dependent on the theory of continued fraction. There is however no need for any such special theory, and the continue fraction process is not the quickest possible.

But a knowledge of a simpler process seems to be far from general, and it would seem to be of value to give an account of one such process here, and to see how it may be given a simple arrangement. In addition, as we will see this process has the good point that it may be used without difficulty to give the general solution of equations with more than two unknowns, and systems of such equations.