A method of calculating real roots by a simple division process when an integral approximation is known is described. It is very powerful and economical in labour as well as being of extreme simplicity It surpasses Horner’s method in these respects and in many applications it is preferable to the Newton-Raphson method and others. A root is expressed directly in any scale of notation or, alternatively, as a continued fraction, and the processes involved would appear to be suitable for use in an automatic computer.

The theorem to be discussed in this note depends only on the plane incidence axioms (with Desargues), the Pappus axiom, and the property that a point and its harmonic conjugate are in general distinct. When algebra is applied to the geometry it may be taken over any commutative field of characteristic other than 2.

In the autumn of 1693 the Writing Master of Christ’s Hospital, one John Smith, asked Samuel Pepys the diarist, who was a Governor of Christ’s Hospital and eventually became its Vice-President, for the answer to the following question:

It was conjectured by Fermat that if n > 2, the equation has no solution in positive integers. If n is composite, say n = lm, the equation implies Am + Bm = Cm, where A = al, etc. So the insolubility of the equation with exponent m ensures the insolubility of the equation with exponent n.

This problem has been treated in many Geometries (Casey, Forder, Steiner) and in the Mathematical Gazette, and by Frederick Soddy.

Perhaps I can add a little of interest.