The Final Verdict

1977 the Dutch Professor André Wegener Sleeswyk published the results of his analysis of the ancient texts collected by Mr. Needham. Coming from a technological background, he applied mathematical methods and upheld the validity of physics and voilá, there are solutions:

Synopsis: According to the Chinese historiography of the Sung-dynasty the courtiers Yen-Su and Wu Tê-jen constructed south-pointing chariots of which brief descriptions exist. The longer-standing problem of reconstructing these chariots is attacked anew by a method which combines analyses of the text, engineering considerations and elementary algebraic geometry. The latter is used first to derive the requirements of geometry for successfully working south-pointing chariots, and subsequently to show that the south-pointing mechanism in Yen-Su's chariot must have been a special type of intermittently working gear assembly. The historical text combined with the requirements of enginering leads to a reconstruction in which an escapement lever synchronizes two gears.

The reconstruction of the Wu Tê-jen's chariot starts from the assumption - based on the symmetry of the assembly - that two differential gears were used. The gears may  then be assembled in 15 different ways to construct south-pointing mechanism. Using algebraic geometry again, the required track-width of the chariots incorporating these 15 gear assemblies are calculated and compared to the known width of the body of the chariot. In only three instances are the two widths approximately equal; of here three, one is to be preferred on grounds of adherence to the text and good engineering. The text is too brief that more than part of the mechanism which actuated jackwork in Wu Tê-jen's chariot can be reconstructed.

Without using elementary algebraic geometry it is impossible to limit sufficiently the number of possible reconstructions, but analysis of the texts, intuition and engineering knowledge are essential too. The results show that:

The following text is not taken literally, but shortened heavily for the sake of transparency. The gear number in brackets refere to the text in Sung Shih.

Yen Su's chariot

Mr. Lanchester's reconstruction cannot be valid as from the three horizontal gearwheels specified in the Sung Shih no differential gear train can be constructed. Furthermore each differential halves the angle from input to output, whereas the transmission of the orginal maintained 12 teeth from input (subordinate gear (B)) to output (big wheel (C)).

What can be learned from the text is:

So we nead a means to transmit the rotations of (B) to (C) intermittently. Gears (E ?) alone could not faciliate this, as each gear (D) not is mesh has somehow to store the rotational information until its duty cycle. Some translatorial movement of the axles of (D) is proposed. Further investigation shows, that gears (B) just have to move the centres of gears (D), not turning them ! The most probable construction is shown right.
Now we have to build an alternator, which synchronizes the both wheels (D). The ominous 'crossbar' or 'transverse wood' comes in handy here: If set up as show left, it will have each wheel advance one tooth alternatingly as long as both want to. If one stops, the lever will stay at one side, thus unlocking the other (required if the chariot turns round one fixed point !)
If made of wood, the lever would wear quite fast. Well, we have wheels (E) left. If we employ them to roll over the teeth of (D) we meet the description in the Sung Shih very well.
Now we put all this together. The function is described as follows:

Gears (D) are mounted on swivel arms pivoting round shaft (V). Gears (B) pull these arms forward, until the hook jumps off the tooth as depicted above (Some sort of spring is needed to ths ends). 
The pulling action at the same time engages (D) to (C), thus turning it as well as long as (D) is in the locked state. The alternator (labelled balancing lever in he picture) takes care that next the other wheel gets a chance to turn (C) back by the same or another amount, effectively leaving the difference on (C) and at last in the position of shaft (V) - QED

Wu Tê-jen's chariot

Mr. Lanchester's reconstruction again fails due to the lack of two central gears. But the availability of five gears at each side of the pole allows for different solutions:

Note that the axle for the wheels (B) probably runs in front or behind the main axles. The gears (H) mesh two other gears of different pitch at the same time, which is possible if wooden wheels with long dowels for teeth are used.
Dimensions necessary for proper operation:
Gear no. teeth DP pitch dia.
addendum dia.

Just comparing the dimensions should convince everybody.

A much simplier exercise in geometry is the construction of the turning 'boys' in the corners round the hsien, using the 13 equal wheels given by the text
But one riddle still remains unsolved: What was the purpose of the 'back poles' and bamboo ropes ? As they are not used for the south-pointing action (contrary to other reconstructions !), they may be part of some fancy motion works like waving the boys' hands on turning (some early indicators ?). Anyway, one possible way to meet the ancient description is given left.

Source: Chinese Science, Volume II, January 1977