Now that the mechanism itself is ancient and broken, we have to make reconstructions and models for how it must have worked. We take what we have, and we build what we can. Here is how we know our current explanation makes sense:
In a geocentric solar system, the Sun rotates around the Earth at a rate of 1 rotation per year and the planets rotate in circles around the Sun at their mean distances and mean rates of rotation. If the solar system bodies all moved at constant rates in circular orbits, then the deferent and epicycle models of the planets previously outlined would be an exact model. In this model, (g1, -g2) is a period relation for the planet, where g1 and g2 are positive integers. This means that the mean period of the planet around the Sun, r, is defined as r = –g2/(g1 – g2) and its rotation is 1/r = 1 – g1/g2 rotations per year.
If s is a unit vector in the direction of the Sun and m is a unit vector in the direction of Mars from the Sun, then the vector between Mars and the Earth is s + pm, where p is the mean distance of Mars from the Sun. The deferent and epicycle model describes a planet’s position in the reverse order: pm + s.
It seems an obvious thing to state that a + b = b + a, but it is essential to justify this reconstruction. It is impossible for us to understand what the s were thinking when they made their device. What we know now changes the way that we think. We can’t help but know that the Earth revolves around the sun, and so we come to our conclusions through different paths. a + b = b + a. Our results are still the same.
In the gears of the Antikythera mechanism, the point s is fixed to G4, since it is the mirror image of p across the b–g line of symmetry (drawn in red), p is fixed to G3, and the gears G3, G4 have equal numbers of teeth. We want to show that s is the sum of two vectors.
The notation R(a | b) will be used to mean the relative rotation of “gear a” or “point a” relative to “gear b”. Relative to b1, G1 and G2 are gears on fixed axes. So we can calculate their rotations from the basic equation of meshing gears:
Rot (G2 | b1) = (- g1/g2)*Rot (G1 | b1)
G1 is fixed, so its rotation relative to b1 is -1, because b1 rotates at the rate 1.
Therefore, Rot (G2 | b1) = (- g1/g2)*-1 = g1/g2
Since m is the mirror of p (fixed to G2) in the bg-mirror, its rotation, relative to b1 is Rot (m | b1) = – g1/g2. The rotation of m can then be calculated as Rot (m) = Rot (m | b1) + Rot (b1) = -g1/g2 + 1 = (g2 – g1)/g2 = 1/r. This rotation is identical to the mean rotation of a planet defined earlier.
The vector joining b with m is dm, where d is the distance of the pin p from the center of G2, g. Because the bg line of symmetry (in red) and the bg’ line of symmetry
(in green) are both perpendicular to line bgg’ (in blue) and so are parallel, the points m, s and p are all the same distance from the bgg’ axis. Therefore, length ms = length gg’ = d/p.
So, if s is a unit vector in the direction of the bgg’ axis, then the point s is defined by the vector dm + (d/p)s = (d/p)(s + pm). This position is related to a planet’s position s + pm by a conversion factor, d/p, which depends on the size of the gear. The rotations of the chosen gears can therefore be made to correspond to the movements of each planet.