The Caliber 89 date-of-Easter mechanism knows the right date for Easter thanks to a notched program wheel. Basically, the program wheel advances one step per year and each step has a different depth. Depending on the depth, the hand showing the Easter date will jump to the correct date for that year. The mechanism is reasonably straightforward:
Calculating the date of Easter didn't used to be quite so complicated. The rule according to the Julian calendar was fairly straightforward. A full cycle of full moon dates was thought to follow a 19 year cycle (the so-called Metonic cycle, which you might remember from our coverage of the Vacheron ultra-complication 57260) consisting of 235 lunar months. A fully cycle of the Julian calendar was 76 years (after four Metonic cycles -- 19 x 4 = 76 -- a full leap year cycle was completed also). Easter dates repeated, in the Julian calendar, every 536 years [...] Then Pope Gregory XIII came along.
Incidentally, when they say "which you might remember from our coverage of the Vacheron ultra-complication 57260", WOW, they were not even joking! This is amazeballs!
A method for calculating the Easter date is called a computus; is it possible to make a true mechanical computus, rather than relying on a program disk? The answer is, "sort of." The first true mechanical computus appears to have been made not long after Gauss came up with his algorithm, and it currently resides in a place more horological enthusiasts should know about: the great astronomical clock in the cathedral at Strasbourg, in Alsace, France.
You'll notice that among the otherwise gnomic assembly of gears is a display for the Epact of the current year, as well as the current Domenical Letter. The "Nombre D'Or" or Golden Number is the number corresponding to the current year's position in the Metonic cycle (one through 19, as shown) which is also necessary for the calculation. [...]
At this point you can understand why the trio who designed the date-of-Easter complication for Patek might have looked at each other and said, "Okay, guys, look ... let's just go with a program wheel." [...]
You can look at the Patek Caliber 89 and see its date-of-Easter complication as a compromise, but it isn't -- not really. Yes, it's true that the whole structure of astronomical mechanical complications -- whether in the Strasbourg cathedral clock, or in watches like Caliber 89 -- is a manifestation of a world view. That worldview -- of an orderly clockwork universe, with tidy nests of ratios that can be encoded in gear trains -- never really existed; the real universe is chaotic and probabilistic. But it is a beautiful vision, albeit it says more ultimately about how we would like the universe to be than how it actually is.