Editing South-pointing chariot (section)

==Chariots with differential gears==

===Background and explanation===
[[File:Transmission diagram.JPG|thumb|An illustration of a differential between the drive shaft (at bottom right) and driving wheels of an automobile]]

The English engineer [[George Lanchester]] proposed that some south-pointing chariots employed differential gears.<ref>[[Joseph Needham]] (1986). ''Science and Civilization in China: Volume 4, Part 2'', page 296. Taipei: Caves Books, Ltd.</ref> A differential is an assembly of gears, nowadays used in almost all automobiles except some electric and hybrid-electric versions, which has three shafts linking it to the external world. They are conveniently labelled A, B, and C. The gears cause the rotation speed of Shaft A to be proportional to the ''sum'' of the rotation speeds of Shafts B and C. There are no other limitations on the rotation speeds of the shafts.

In an automobile, Shaft A is connected to the engine (through the transmission), and Shafts B and C are connected to two road wheels, one on each side of the vehicle. When the vehicle turns, the wheel going around the outside of the turning curve has to roll further and rotate faster than the wheel on the inside. The differential permits this to happen while both wheels are being driven by the engine. If the sum of the speeds of the wheels is constant, the speed of the engine does not change.

In a south-pointing chariot, according to the hypothesis, Shaft B was connected to one road wheel and Shaft C was connected ''through a direction-reversing gear'' to the other road wheel. This made Shaft A rotate at a speed that was proportional to the ''difference'' between the rotation speeds of the two wheels. The pointing doll was connected (possibly through intermediate gears) to Shaft A. When the chariot moved in a straight line, the two wheels turned at equal speeds, and the doll did not rotate. When the chariot turned, the wheels rotated at different speeds (for the same reason as in an automobile), so the differential caused the doll to rotate, compensating for the turning of the chariot.

The hypothesis that there were south-pointing chariots with differential gears originated in the 20th century. People who were familiar with modern (e.g. automotive) uses of differentials interpreted some of the ancient Chinese descriptions in ways that agreed with their own ideas. Essentially, they re-invented the south-pointing chariot, as 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#Investigations and reconstructions|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.<ref name="Needham vol 4 part 2 p298">Needham, Volume 4, Part 2, 298.</ref> He used a differential for correcting the [[equation of time]] for a [[clock]] that displayed both [[Local mean time|mean]] and [[solar time]].<ref name="Needham vol 4 part 2 p298" />

===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 differential gear and if, for example, the width of the track of the chariot (the separation between its wheels) is three metres, and if the wheels are intended to be identical but actually differ in diameter by one part in a thousand, then if the chariot travels one kilometre in a straight line, the "south-pointing" figure will rotate nearly twenty degrees. If it initially points exactly to the south, at the end of the one-kilometre trip it will point almost to the [[Cardinal direction|south-southeast]] or [[Boxing the compass|south-southwest]], depending on which wheel is the larger. If the chariot travels nine kilometres, the figure will end up pointing almost due north. Obviously, this would make it useless as a south-pointing compass. To be a useful navigational tool, the figure would have to rotate no more than a couple of degrees over a journey of a hundred kilometres, but this would require the chariot's wheels to be equal in diameter to within one part in a million. Even if the process of manufacturing the wheels were capable of this precision (which would not be possible with ancient Chinese methods), it is doubtful that the equality of the wheels could be maintained for long as they are subjected to the wear and tear of travelling across open country. Irregularity of the ground would add further errors to the device's functioning.

Considerable scepticism is therefore warranted as to whether this type of south-pointing chariot, using a differential gear for the whole time, was used in practice to navigate over long distances. Conceivably, the south-pointing doll was fixed to the body of the chariot while it was travelling in straight lines, and coupled to the differential only when the chariot was turning. The charioteer could have operated a control to do this just before and after making each turn, or maybe shouted commands to someone inside the chariot who connected and disconnected the doll and the differential. This could have been done without stopping the chariot. If turns were brief and rare, this would have greatly reduced the pointing errors, since they would have accumulated only during the short periods when the doll and differential were connected. However, it raises the problem of how the chariot could have been kept travelling in straight lines with sufficient accuracy without using the pointing doll.

If the real purposes of the chariot and the accounts of it were amusement and impressing visiting foreigners, rather than actual long-distance navigation, then its inaccuracy might not have been important.  Considering that a large mechanical wagon or chariot would be obligated to travel on roads, the destination in question would typically not be in an unknown direction.   The fact that the sources cited above mention that the chariot was placed at the front of processions, its high level of mechanical complexity and fragility, and that it was 'reinvented' several times contribute to the conclusion that it was not used for navigation, as a truly practical and useful navigational tool would not be forgotten or left unused.