The diversified character of time

The students in the calendar class all brought calculators and/or laptops to assist with the number-crunching, but they didn't need them on the first day, which covered the period before Hillel II's mathematical calendar.

Before things got too technical, we began with the motivations for studying the calendar:
1) As do-it-yourself Jewish communities, if we're going to use the Hebrew calendar (and even the Jew in the NJPS who just goes to a seder every year implicitly uses it) we should know how to derive it ourselves, rather than depending on funeral homes to do it for us. (This was inspired in part by Kevin Hale's mezuzah-writing class two years ago: he began by talking about how liberal Jewish communities should have people who know how to write mezuzot rather than depending on Orthodox communities as we do now.)
2) This esoteric study is actually the very foundation of Judaism. We looked at the famous passage from Heschel's The Sabbath beginning with "Judaism is a religion of time aiming at the sanctification of time," as well as other sources, including the first commandment given to the whole nation of Israel ("This month shall be to you the beginning of months") and the part in Rosh Hashanah 8b where it says that the court above does not enter into judgment on Rosh Hashanah until the court below has sanctified the new month (in other words, God observes the holidays when we say God does!).

Then we had an outline of the old-school system of witnesses observing the crescent moon and being examined by the court, who then declared the new month. Wacky stories added color. We looked at the sources in the Mishnah, Gemara, Rashi, and Rambam (Hilchot Kiddush Hachodesh) explaining the origins of two-day Rosh Chodesh, two-day Rosh Hashanah, and two-day festivals, each of which is for a different reason. Based on these explanations, people asked the question I was dreading: Why are the two days of Rosh Hashanah designated as the 1st and 2nd of Tishrei rather than the 30th of Elul and 1st of Tishrei? (This doesn't affect when Rosh Hashanah is observed, but determines whether Yom Kippur is observed 9 days or 10 days after the 1st day of Rosh Hashanah.) The lame answer I gave (which was the best answer I could find after reading all of Masechet Rosh Hashanah and the non-way-out-there parts of Hilchot Kiddush Hachodesh) was the statement in the Talmud that since the days of Ezra, Elul had always been 29 days, so this tradition had been kept alive. To which they responded, if Elul was always 29 days during the old system, then when was there ever a need for a two-day Rosh Hashanah? Which is an excellent point. So if anyone out there has a better answer, we would all appreciate it.

The second day began with the last piece of the old system: intercalating the lunar years to get them to line up with the equinoxes/seasons. Then we said goodbye to the astronomical system and had an introduction to modular arithmetic. Everyone does it whether they know it or not: 27 minutes after 4:45 is 5:12 (not 4:72). We practiced adding and multiplying with the lunar month of 29 days 12 hours 793 parts. (A part, or chelek, is 3 1/3 seconds. There are 1080 parts in an hour; this number is divisible by 2,3,4,5,6,8,9,10, and more!)

We solved the leap year problem by introducing the 19-year cycle, but showed that there is an error of 2 hours in each cycle, or about 1 day every 200 years! This means that in 20,000 years we'll be observing Pesach in the summer! In 79,000 years, we'll cycle back around all the way, and (the most useless fact available) in around 300 million years, the number of the Gregorian year will be greater than the number of the Hebrew year.

We introduced the molad (mean lunar conjunction) and the general algorithm for calculating everything there is to know about a year:

1. Leap or not? (Divide by 19 and find the remainder)
2. Find the molad of Tishrei for this year and next.
3. Find the date of Rosh Hashanah for this year and next.
4. Subtract the dates in #3, and find the number of days in the year.
5. Assign a number of days to each month based on #4.

The last two days of the class were devoted to fleshing out the details of #2 and #3. We looked at the source for molad tohu (the beginning of Tishrei in the year 1), explained in Tosafot to Rosh Hashanah 8a-b. I think this is a fabulous example of the rabbis shooting an arrow and then painting a bull's-eye around it. They knew roughly when the molad had to be, so they used the sources creatively to come up with a justification (of course Adam sanctified the new moon at the time when he was commanded not to eat from the tree!). And if the molad had to be two days earlier, they would have said that the molad happened on the 4th day (when the moon was created). Based on this, you can find the molad of any month, just by adding the number of months that have elapsed since then.

We looked at all four dechiyot (rules for postponing Rosh Hashanah later than the date of the molad). A whopping three of them come into play for this year (all of them except GaTaRaD, or as someone suggested, GaToRaDe), including the elusive BeTUTeKaPoT!!! BeTUTeKaPoT happens on average once every 190 years! The last time was in 1927, and the next won't be until 2252, so for most of us, this will be the only BeTUTeKaPoT of our lifetimes. This rule moves the upcoming Rosh Hashanah from Monday to Tuesday, in order to add an extra day to the year, so that it won't have the illegal length of 382 days. It's so rare because it only applies to the Rosh Hashanah following a leap year, and only when the molad of Tishrei falls into a specific 2.5-hour range out of the entire week.

Finally, we hit the controversy between Saadiah Gaon and Aharon ben Meir: due to political posturing between Israel and the Diaspora, some Jews observed all the holidays two days earlier than others for two years!

And then time was up.