I first came across these dials at the Old Ashmolean Museum when looking at astrolabes. Miss Kathleen Higgins, then writing a thesis on dials, asked if I could explain the operation of a dial described by Regiomontanus (1436-1476) of which several examples are shown in British and European museums. That was in 1950; since then I have repeatedly tried to evolve an explanation of some way in which dials of this kind may have been invented. Only recently have I been satisfied with my results.

1. About ten years ago Neugebauer and Sachs published their Mathematical Cuneiform Texts in which the contents of Plimpton 322 were analysed for the first time. The numerical values contained in this text are given in Table I. The theory put forward by Neugebauer and Sachs was in my view not consistent with known facts and in the Proceedings of the Academy at Amsterdam I put forward a different interpretation. Since then I have made several short comments in connection with this text, but up to the present I have not given an explicit interpretation of the errors occurring on Plimpton 322.

There are many questions in mathematics to which an answer can be found by a purely mechanical procedure. For instance we can determine the valid statements of the class of the statements

n is prime

by the sieve of Eratosthenes; the numbers 2, 3, 4 and so on up to any assigned n are written in a row, and we begin by striking out every second number, starting with 2. Of those which are left, the first is 3 and 3 is prime. Starting with 3, we then strike out every third number, and the first of those which remain is 5, which is prime. Starting with 5 we strike out every fifth number, determining the prime 7, and so on.

A helicoid is the surface generated by a curve which is simultane rotated about a fixed axis and translated in the direction of the axis w velocity proportional to the angular velocity of rotation.[1] An Archime Screw [2] is strictly a finite helicoid in which the generating curve is c (and normally circular) ; but screws, helical in shape, for which the gene curve is a simple finite arc can be—and are—used in practice as Archime Screws, and are considerably easier to manufacture than tubular ones.