Doomsday Calendar, Part 2

This internet thing, though. It seems to be all rabbit holes, all the way down. In my previous post, I wrote about John Conway’s “Doomsday Algorithm” for figuring out what day of the way “any” day falls on. It seems that there are quite a few people who have spent a lot more time thinking about calendars in general, and mental calendars in particular, than I have. Most of what’s in this post is based on this excellent piece by Chamberlain Fong.

The first helpful thing Fong does is to give some history. Conway’s method has two parts: the identification of particular days of the year as “Doomsdays,” which all fall on the same day of the week. Once you know what day of the week any give year’s Doomsdays fall on, it is easy to figure out the day of the week of an arbitrary date. The other part is to figure out what day of the week the given year’s Doomsdays fall on.

The first part of the algorithm seems to be entirely due to Conway. The second part, interestingly, was apparently worked out by Lewis Carroll—one of the few people in the orbit of mathematicians who may have been more playful that Conway. Martin Gardner, the legendary author of Scientific American’s Mathematical Games column, was a serious student of Carroll, and when he came across Carroll’s work on perpetual calendars, challenged Conway to come up with something simpler.

The second helpful thing is some clarification of how the leap year issue arises:

Earth spins around its axis approximately 365.24219 times annually. This rate is known as the mean tropical year. The true number of days in the mean tropical year actually varies slightly over time.

He couples this with some helpful history:

Our modern calendar system of 12 months started with the Julian calendar circa 45 B.C. The Julian calendar approximated the mean tropical year as 365.25 days and proposed 366-day leap years for every one divisible by four. This approximation made the Julian calendar fairly accurate in tracking the annual seasons for several centuries. But eventually the approximation errors accumulated. By 1582 the Julian calendar was out of sync with the seasons by several days. Pope Gregory XIII mandated a reform to the calendar; his Gregorian calendar approximated the mean tropical year as 365.2425 days.

The Gregorian rule says that years ending in 00 are not leap years, unless they are divisible by 400. So every four centuries, the Julian calendar will advance by 3 days compared to the Gregorian.

Of course, not everyone adopted the Gregorian calendar immediately, and Church schisms seem to have a lot to do with it. Henry VIII separated the Church of England from Catholicism in 1534, about 50 years before the Papal calendar reform. Fong goes on:

The Gregorian calendar was not widely adopted across the world until much later. In fact, the British Empire and its colonies did not use the updated calendar system until 1752. By then the Julian calendar was out of sync with the seasons by 11 days. In order to synchronize with the Gregorian calendar, it was mandated that Wednesday, September 2, 1752, be followed by Thursday, September 14, 1752.

Russia did not adopt the “New Style” dates until their revolution in 1918; I have been told that their “October Revolution” (10/25/1917, Old Style) is celebrated annually on November 7! So the bottom line is: Conway’s Doomsday algorithm won’t help you with any date before 1582; in the anglophone world it won’t help you until 1752; and in Russia it won’t help you until 1918.

The final valuable thing that Fong does is to point out an even simpler Doomsday algorithm that he and Michael Walters developed to calculate the Doomsday for any given year, which I will discuss in Part 3.

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