Reply to: Antikythera Mechanism - The Book
There is a new edition of this book, in Greek: http://www.antikythera-mechanism.gr/node/763
Reply to: Does it favor a Heliocentric, or Geocentric Universe?
To reinterate what has been said above, from possibly a little different angle:
Ιον Αστρουομερ (pardon my Greek) is standing in the night sky, and wants to locate Saturn in the sky. Like all ancient Greeks, he believes that these bodies travel in circles, but to explain why they back up occasionally in the sky, he needs epicycles (his drinking buddy Hipparchus told him about them, and how they work computationally). So, he starts the indicated computation which involves a circle that Saturn is travelling along the ecliptic and an epicycle, which proportions to the size of Earth's orbit, and arrives at the location. He knows where to look.
1500 years later, his English descendant, John Astronomer, needs the same information. His friend Kepler has not yet let him into his new secret about orbits, but John is none-the-less a modern fellow, so he uses the new-fangled celestial mechanics as Tycho Brahe taught him: he determines his location on Earth, translates and rotates that into a solar-centered coordinate system, and then applies the six orbital variables of Saturn and derives the location in his sky.
Both got an equally close answer, because both of their methods were the same. The two epicycles of Ptolomy and the two orbits of Copernicus equate to the same computations, perhaps ordered differently and certainly different in viewpoint and technique, but, in essence, the same.
The upshot is that the two theories which explain who moves around whom are not relevant to the Antikythera Mechanism; it uses a mechanical equivalent which could be explained by either Ιον or John in terms of epicycles or celestial mechanics. Until Kepler expands the universe of the mind to include all conics as orbits, the two are entirely equivalent to the problem that these two have, and which the Antikythera Mechanism addresses.
Reply to: Does it favor a Heliocentric, or Geocentric Universe?
The object of the mechanism is to make specific astronomical predictions. This requires a geocentric model, since actual observations require everything to turn around the observer, think of the coordinates you use to point your telescope. In fact, there is a slight discrepancy since the observer is not at the center of the Earth, which is the actual center of the geocentric model, but this was already taken into account in antiquity and was estimated by Archimedes in his paper "The Sand Reckoner."
The geocentric/heliocentric debate is more of a conceptual issue as geocentric will eventually be required for observations. Moreover, until the 17th century, astronomy was mathematical astronomy in which one constructs the best model fitting the data without giving physical reasons why the model occurs in nature. Since the geocentric and heliocentric models are mathematically equivalent, the distinction is once again moot.
Moreover, the heliocentric theory certainly didn't have any observational support at the time (experimental verification only came in 1838) and it loses conceptual significance without physical explanations such as gravitation and Newton's Laws. Therefore, there was no scientific reason to adopt a heliocentric model in antiquity. The only exception is Archimedes who did adopt it in his paper "The Sand Reckoner" because he wanted to use the largest theoretical model of the universe, which is the case for the heliocentric model in order to avoid parallax issues. He was never persecuted for this view and there is no evidence that anyone supporting the heliocentric model was ever persecuted in Ancient Greece.
Reply to: What does it tell us about Greek Astronomy?
The Antikythera mechanism confirms that our previous understanding of Ancient Greek science was correct. Moreover, in actually implementing their knowledge by building a machine, they have proved their knowledge, to quote Donald Knuth: "Science is what we understand well enough to explain to a computer."
It also confirms the fact that, contrary to the popular misconception, Ancient Greeks scientists were very practical and did not limit themselves to theoretical investigation. To the surprise of the academic community, this misconception was strongly refuted when Archimedes' "The Method" was discovered in 1906 by Heiberg, in which Archimedes states that he systematically used physical analogies to discover his mathematical theorems.
However, after a century, this more realistic understanding of Ancient Greek science has had a difficult time entering the public consciousness. The fame of the Antikythera mechanism is therefore an important step in the general realization that Ancient Greeks had a very similar scientific process to our own.
Reply to: What are the Metonic and Callipic Cycles?
The current western (Gregorian) calendar corresponds to the return of the seasons. In other words, if you give a date of the year and a location, then you should have a good idea of the weather. For example, in Europe, July is hot and January is cold. However, our months do not represent any physical phenomenon, knowing the day of the month doesn't tell you anything about the natural surroundings.
However, there are calendars where the months keep track of the phases of the Moon, the first of the month is always a new Moon and the 15th of the month is always a full Moon. In particular, this is true for the Islamic and Jewish calendars.
A lunar month averages about 29.5 days, so in a lunar calendar months alternate between 29 and 30 days (instead of 30 and 31 in the western calendar). This means that 12 months give 354 days, so about 11 short of the calendar year. For this reason, the Islamic calendar is 11 days earlier every western year and this explains why Ramadan is shifted significantly after a few years. Therefore, given a date of the Islamic calendar, you don't know anything about the weather but you do know the phase of the Moon.
The Metonic cycle is the observation that 235 lunar months are almost exactly 19 solar years. In this way, adding a 13th month 7 times every 19 years allows a lunar calendar to synchronize with the return of the seasons. For such calendars both the date of the year and the date of the month have astronomical significance.
The Jewish calendar follows the Metonic cycle, so that the Jewish New Year on 1 Tishrei is always on the new Moon, and Pesah on 15 Nissan is always on a full Moon. This also explains why Jewish holidays can seem to shift by a month from one year to the next with respect to the western calendar. Due to the Metonic cycle, after exactly 19 years, a western calendar date and a Jewish calendar date should correspond to within very few days.