Henry Baker: Does anyone here know what the fundamental _period_ of the antikythera device is?
I.e., by all reports, this device is a "planetarium in a box". So how many years of simulated time does it take for this device to return to its initial configuration ? If we start the antikythera device in 70 B.C., which appears to be approximately the date of its demise, how far into the future can this device "see" ? --I do not know. However, if my speculation the device was intended by its designer (Archimedes?) as an experimental navigation aid, is correct, then this does not terribly matter. The way you then would want to use such a device, is you would reset it to the truth, whenever you had access to the truth, i.e. whenever your ship came to a place (1) whose location was known, and (2) where they were making accurate astronomical observations -- such as Greenwich. The device then would only need to keep good accuracy during the duration of your sea voyage (<2 months?) which is a precision and math level the Greeks might have been capable of. The Greeks were unaware of Newton & Kepler laws of motion hence tremendous accuracy was not possible, but they were aware of a correction to the "constant speed circular" model which made the speed and radius nonconstant, and they evidently had enough observational accuracy to be confident of the need for that, and the device is known to incorporate a "pin in slot" mechanism which seems intended to cause that effect. If this device had been further developed (which apparently it was not) in concert with astronomical records being kept, then it seems to me it would have been possible, working semi-empirically, for them to get one capable of enough accuracy for useful navigation, at least in principle. Whether that level was ever actually achieved, I do not know.
Let's assume that the device was used on a daily, weekly or monthly basis. Presumably, the position of the wheels would be indicative of the last time it was used.
This is essentially equivalent to being able to determine the exact time of an earthquake from the position of the hands of clocks which stopped as a result of the earthquake. --since some parts of the device are missing, corroded, damaged, broken off, I doubt that is possible. Also, related question: using today's computers and data, are we capable of extrapolating 2000 years with enough accuracy to tell this? Here's some data I got from some random web page http://www.princeton.edu/~willman/planetary_systems/Sol/ giving alleged orbital periods in "years." Mercury := 0.2408467; Venus := 0.61519726; Earth := 1.0000174; Mars := 1.8808158; Jupiter := 11.862615; Saturn := 29.447498; #Greeks knew the above planets only Uranus := 84.016846; Neptune := 164.79132; Pluto := 248.0208; eccentricities are known to only 3-5 figures, relative masses to only 4-5. Using continued fractions, you may enjoy spotting near-rational ratios Venus/Earth=0.61518656 vs 8/13=0.61538461 Mars/Earth=1.88078307 vs 1057/562=1.88078292 ... Neptune/Uranus=1.9614081 vs 51/26=1.9615385 Pluto/Neptune=1.50506 vs 3/2=1.5 and 149/99=1.505050505... etc. How many of these are "flukes" vs how many are "real"? One way to tell would be, I suppose, to examine the statistics of partial quotients & compare with Gauss-Kuzmin distribution expected for generic reals. Due to tidal effects presumably the solar system has ratios "closer to rationals" than generic reals. Such rational approximations would have been one thing limiting the accuracy of a Greek device; and other effects like inter-planet interactions, noncircularity, etc. too.