it had previously been re-invented several times in antiquity. Working chariots that use differentials have been constructed in recent decades. Whether any such chariots existed previously is not known with certainty. Although the Antikythera mechanism is believed to have used differential gears, the first true differential gear definitely known to have been used was by Joseph Williamson in 1720. He used a differential for correcting the equation of time for a clock that displayed both mean and solar time. Geometrical properties If the south-pointing chariot were built perfectly accurately, using a differential gear, and if it travelled on an Earth that was perfectly smooth, it would have interesting properties. It would be a mechanical compass that transports a direction, given by the pointer, along the path it travels. Mathematically the device performs parallel transport along the path it travels. The chariot can be used to detect straight lines or geodesics. A path on a surface the chariot travels along is a geodesic if and only if the pointer does not rotate with respect to the base of the chariot. Because of the curvature of the Earth's surface (due to it being curved around as a globe), the chariot would generally not continue to point due south as it moves. For example, if the chariot moves along a geodesic (as approximated by any great circle) the pointer should instead stay at a fixed angle to the path. Also, if two chariots travel by different routes between the same starting and finishing points, their pointers, which were aimed in the same direction at the start, usually do not point in the same direction at the finish. Likewise, if a chariot goes around a closed loop, starting and finishing at the same point on the Earth's surface, its pointer generally does not aim in the same direction at the finish as it did at the start. The difference is the holonomy of the path, and is proportional to the enclosed area. If the journeys are short compared with the radius of the Earth, these discrepancies are small and may have no practical importance. Nevertheless, they show that this type of chariot, based on differential gears, would be an imperfect compass even if constructed exactly and used in ideal conditions. Lack of precision, and implications Real machines are never built perfectly accurately. Simple geometry shows that if the chariot's mechanism is based on a