With the new calendar came a new procedure for calculating the date of Easter. Each year would be assigned a number called the Epact – this was the age of the Moon on January 1 (the number could be anywhere from 1 to 29). In addition, each year was given a letter corresponding to the date of the first Sunday in January (A-G). These “Dominical Letters” (Leap Years get two) plus the Epact for that year, plus the Golden number (where you are in the Metonic cycle) are the raw material used to calculate the date of Easter. These are just the basics – in order to keep the ecclesiastical Moon and Equinox reasonably aligned to the astronomical ones, periodic adjustments have to be made which make the actual calculation much more complicated (for a good look at how things get complicated fast, check out this article on the Cycle of Epacts, which will tax your appetite for minutiae like you wouldn’t believe).
A couple of points: first, the astronomical events considered in the calculation are abstractions. The Church considers March 21 the fixed date of the Spring Equinox, but in fact, the date of the actual astronomical Equinox varies from one year to the next. Second, the astronomical full moon doesn’t always correspond to the ecclesiastical full Moon. Creating algorithms that spit out the correct date of Easter has been a diversion for mathematicians ever since Gregory XIII reformed the calendar, and even before. Karl Friedrich Gauss, who is often called the greatest mathematician of the 19th century, came up with such an algorithm in 1800, and in The Art Of Computer Programming, Donald Knuth (who famously coined the term “surreal numbers” to describe John Conway’s discovery of a set of numbers much larger than infinity) wrote that, “There are many indications that the sole important application of arithmetic in Europe in the Middle Ages was the calculation of the date of Easter.”
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. There have actually been three successive astronomical clocks there since about 1354, but the most recent was completed in 1843. Designed by Jean-Baptiste Schwilgué, it has a true mechanical computus – probably the first ever constructed. It’s not the only mechanical computus, but I haven’t been able to find anything in English on other computus devices (although a reprint of a review of a book on the Strasbourg computus mentions at least two other “similar” mechanisms).
Certainly it’s the only one of its kind in terms of operating principles; I’m actively trying to research how it works but it’s an uphill climb to put it mildly. The clock itself is a virtuoso piece of horology even without the computus – an article by Bryan Hayes, for Sciences, in 1999, mentions that there is a gear in the astronomical train of the clock that makes one rotation every 2,500 years, and that furthermore, the clock features a celestial globe that makes one rotation about an axis showing the precession of the Equinoxes only once every 25,000 years (the article was on Y2K compliance, and on how the Strasbourg clock is Y2K compliant with a vengeance).
Fortunately for the horologically curious (and maybe intellectually masochistic) you can see the computus mechanism – it’s on display in a case at the lower left hand side of the base of the clock. 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.
Once a year, on New Year’s Eve, the mechanism comes to life. Its gears turn and, on the main calendar ring next to the computus – mirabile dictu – a metal tab changes position until it comes to rest next to the correct date of Easter for that year.
Schwilgué had made a model of the computus as well, which was stolen in 1945 and hasn’t been seen since. However, clockmaker Frederic Klinghammer (1908-2006) who was employed by a company that at one time was responsible for the care of the clock, built a working model of the computus in the 1970s, and it’s that model which is the basis for what modern information there is on how the Strasbourg cathedral computus actually works.
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.” Modern fabrication techniques might make it possible to make a mechanical computus, based on Schwilgué’s design, that would fit into a large wrist or pocket watch but my guess is that even with things like LIGA and silicon fabrication, it would be pushing it (though I’d kind of love it if someone would try). A 28 year program disk seems a reasonable compromise, even if replacing it with a disk for another 28 years probably involves non-minor surgery on the Caliber 89. The program disk is an unavoidable necessity as, if you use the current rules for calculating the date of Easter, a full cycle of Easter dates only repeats itself once every 5,700,000 years.
The Strasbourg clock appears to be built to be theoretically correct until the year 10,000 AD (the year indication goes to 9,999 and Schwilgué is supposed to have helpfully suggested that in 10,000, someone might paint in a “1” to the left of the year window). However, if the computus follows a 10,000 year cycle, it will output the incorrect date for Easter in 11,999. In that year, the computus will display the date of Easter as April 4th; in fact, the correct date will be April 11nth.
As you can probably imagine, a program disk for the full cycle of Easter dates would be a wildly impractical thing as well; it would have to have 5,700,000 steps in order to encode the full cycle of Easter dates. If you assume that the 28 step disk is, say, 3 cm in diameter, this gives an approximate circumference of 9.42 cm. That means each individual step takes up about 3.364mm (94.2mm/28).
A 5,700,000 step program wheel would, therefore, be over 19 million millimeters in circumference – more exactly, about 19.176428 kilometers, which is roughly 6.1 km across. Even by pocket watch standards, that’s getting a little hefty.
There is a hidden, abstract beauty to the date of Easter – the very long period of its date cycle hides a remarkable structure. Ian Stewart explains:
“In general terms, the date of Easter slips back by about eight days each year until it hops forward again. The pattern looks irregular but actually follows the arithmetical procedure just described. In 1990 Alan Mackay, a crystallographer at the University of London, realized that this near-regular slippage ought to show up in a graph that compared the date of Easter with the number of the year. The result is approximately a regular lattice, like the arrangement of atoms in a crystal.”
“The peculiarities of the calendar, however, make the dates vary slightly as compared with the lattice. The graph more closely resembles a quasicrystal, a molecular structure built for the first time in the early 1980s. Quasicrystals are not as regular as crystals, but their arrangement of atoms is by no means random. The structure is similar to a curious class of tilings discovered by University of Oxford physicist Roger Penrose; these tilings cover the plane without repeating the same pattern periodically. The atoms of quasicrystals have the same near regularity, as do the dates of Easter. The holiday is a quasicrystal in time rather than space.”
The date of Easter encodes a strange kind of orderly disorder, and yet, even that is an expression of an abstraction that only approximates reality. Over a period of 5,700,000 years, as Bryan Hayes points out in his 1999 article on Y2k compliance and the Strasbourg clock, things like tidal drift will cause enough variation in the orbital and rotational periods of the Earth that any algorithm will require ad hoc correction anyway (assuming any humans are around by then to celebrate the holiday in the first place).
You can look at the Fake 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. There is a poignancy, whether intentional or not, in the fact that there is, at the very heart of the Caliber 89 – a monument to the dream of the music of the spheres – a mechanism that acknowledges that that beautiful dream is also an impossible one.
For a more complete discussion of date-of-Easter algorithms, including that of Gauss, and another famous method submitted anonymously to the journal Nature in 1876, see the Wikipedia article on the computus.