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.