The Science Museum, London, has recently acquired four fragments of a portable sundial with associated calendrical gearing. All the fragments are made of low zinc brass of substantially the same composition. The sundial is of a type known in other examples, some the products of recent archaeological excavations and all dated to the Late Antique or Early Byzantine period. Dating by the place names included in the latitude table, by the style of the heads of the planetary gods used to (...) identify the days of the week, and by the style of the script, all indicate that the instrument dates from the late fifth century of the Christian era or the first half of the sixth.The gearing, of which two arbors survive, carrying four toothed wheels and one ratchet, seems to have resembled that of a calendrical device described by al-Bīrūnī in about AD 1000. Like al-Bīrūnī's device and the geared calendar associated with a Persian astrolabe of AD 1221/2 now in the Museum of the History of Science, Oxford, the Byzantine instrument displayed the shape of the Moon and its age in days. Probably the calendar further resembled its Islamic counterparts in also displaying the positions of Sun and Moon in the Zodiac.The emergence of Byzantine mathematical gearing indicates that the Hellenistic tradition attested by the Antikythera machine continued to be active in the Byzantine period, and suggests that it may have influenced the Islamic tradition. Once again an artefact has shown up the inadequacy of the evidence derived from literary sources. (shrink)