Wednesday, August 10, 2005

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.


  1. You asked "if Elul was always 29 days during the old system, then when was there ever a need for a two-day Rosh Hashanah?" I think what happened in the old system was that the day after the 29th of Elul was observed as Rosh Hashana initially, even if there weren't any witnesses yet, in case witnesses would come. If, by the end of the day, no witnesses had come, then the following day was also observed as Rosh Hashanah, and would be considered 1 Tishrei. So in that sense, Elul always had 29 days. The 30th of Elul was never designated as such until after it was over.

    Regarding the reason for the timing of molad tohu--I'm not sure what the gemara says, but it seems obvious to me what the real reason is. Molad tohu was chosen to be when it was, so that Molad Tishrei of year 2 (which was the first molad after the creation of the world, on 25 Elul of year 1) would come out exactly at 8:00 a.m. on a Friday. Why that time? Because it is the time that you would calculate for the mean new moon for that month based on Ptolemy's formula, rounded off (or rounded down maybe, I forget) to an exact hour. Ptolemy's formula was based, in turn, on lunar eclipse observations going back to the 700s BCE, a period of more than 800 years. They could estimate the time at which the middle of eclipse was observed to within about 10 minutes, based on the positions of the stars, and if you work it out, that allows them to measure the length of the synodic month to within about 0.1 seconds. The present synodic month is about 0.5 seconds shorter than Ptolemy's figure, but that's because the earth's rotation has slowed down slightly since the period in which the eclipse data was taken, not because of any errors in Ptolemy's data or calculations.

    Incidentally, you can use Ptolemy's data to disprove the theory, expounded in Seder Olam, of the missing 135 years in the Hebrew calendar. Ptolemy gives his years in terms of the reigning Babylonian and Persian kings, so we can show that Bayit Rishon was destroyed in (or close to) 586 BCE, and not 135 years later.

  2. Thanks for your comments! I didn't think anyone would post to this thread. :)

    Re Elul: How is what you have described different from any other Rosh Chodesh (in that the day after the 29th is observed as Rosh Chodesh, and then either becomes the 1st or not)? The statement in Rosh Hashanah 19b/32a is "Mimot Ezra va'eilach lo matzinu Elul m'ubar" - since the days of Ezra, we have never seen a pregnant/intercalated/30-day Elul. This seems to say that it so happened that the witnesses always came on the day after 29 Elul, which became 1 Tishrei. How is this empirical report consistent with Rosh Hashanah occasionally being 2 days? And if this statement about Elul is a red herring (because it didn't apply after some other time), then why don't we refer to the two days of Rosh Hashanah as 30 Elul and 1 Tishrei (like any other two-day Rosh Chodesh)?

    Re molad tohu: Tosafot cites the chronology in Sanhedrin 38b, which says that Adam was commanded not to eat from the tree in (the beginning of) the 9th hour of the day, i.e. 8 hours after sunrise, i.e. 2 PM. And clearly this is the time that he sanctified the new month. Since it takes 6 hours from the molad to the time that the moon is visible (see Rosh Hashanah 20b), the molad must have been at 8 AM on Friday! Bull's eye!

    And thanks for the info about the lunar eclipse - i didn't realize that that was how they did it. I figured that they could get the length of the synodic month much more precisely than the time of the molad, since it's easy enough just to find how many days are in 1000 months and divide by 1000, and presto, you have it to the nearest thousandth of a day, but it's harder to pinpoint the moment of the lunar conjunction (but eclipses make it much easier!). Thus we have the length of the month to the nearest second, whereas the time of the molad of Tishrei of year 2 just gets rounded to the nearest hour. (Or maybe it's Nisan of year 1 that gets rounded, according to one of at least three interpretations of the Saadiah/ben Meir dispute.)

  3. I don't have any sources for this, but maybe, starting at the time of Ezra, they always declared the day after 29 Elul as 1 Tishrei, whether or not witnesses came. I suppose they would have the power to do that, if they chose. If witnesses didn't come during that day, then they would observe the second day (the "real" Rosh Chodesh) as a second day of Rosh Hashanah, but would call it 2 Tishrei. This way, people wouldn't feel, on the first day of Rosh Hashanah, that maybe it wasn't really Rosh Hashanah, but only 30 Elul, and that they didn't need to have full kavanah when davening and doing teshuvah. This also explains why, the calendar was fixed, the two days of Rosh Hashanah were called 1 Tishrei and 2 Tishrei, rather than 30 Elul and 1 Tishrei.

  4. Thanks! That makes a lot of sense.

  5. Look here in the comments section for more.