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The Hebrew calendar works on a 19-year cycle of leap years and a 247-year cycle of varying types of years. This is discussed more fully in Shvili d'Rakia, printed in all Mishnayos containing Tiferes Yisrael at the beginning of Moed Aleph, and you can locate a chart of this 247-year cycle in the Tur, between OC Simanim 428 and 429.

In theory, just the 19-year cycle keeps the Hebrew and English years in balance, such that Pesach and Sukkos don't drift too early (the Hebrew calendar is usually 10-12 days shorter than the English calendar, with the exception of leap years). Comparing any given year in the 247-year cycle with its corresponding year in the next or previous 247-year cycle will show that the moladim of those years differ from those of their corresponding years by less than an hour.

In light of all of this, one would expect that if Hebrew date X and English date Y were the same in any given year, 19 years later, or at least 247 years later, they would coincide once more. And indeed, most Jews would tell you that this is indeed the case (at least in my experience).

However, the test fails for Hebrew leap years. In any year except for a leap year, the above statement is accurate. But in leap years, every 19 years, the Hebrew date will toggle back and forth between one of two consecutive English dates. Sometimes it will stick to one of those two days for two years at a time, but it seems to alternate for the most part.

Can anyone explain this phenomenon? You can confirm it for yourself by going to http://www.hebcal.com/converter and plugging in any Hebrew date from last year (5776 was a leap year), then keep adding 19 to the Hebrew year and seeing what the corresponding Gregorian year becomes.

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    Did you check the net molad shift for a 19 year cycle? – Double AA Jul 16 '17 at 17:50
  • As DoubleAA mentioned, I think the Molad shift is a main factor. I suspect that another factor, here, has to do with an accumulation of time difference by the fact that the Hebrew date begins at night time while the Gregorian date always starts at 12:00 A.M. Regarding the Molad and dates - it's a given that Rosh Hodesh never began exactly at the time of the Molad. It was never that way even prior to our current "fixed" calendar. – DanF Jul 16 '17 at 18:04
  • @DanF That's fair - that RC is six hours after the molad is what causes the molad zakein dechiya. According to the numbers presented in Shvili d'Rakia, the "she'eiris kol hamachzor" is 2 days, 16 hours, and 595 chalakim, while the shift over an entire 247-year cycle is -905 chalakim. – DonielF Jul 16 '17 at 18:12
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    I'm unfamiliar with Shvili d'Rakia. However, it seems that you may have found the answer to your question. The difference of 6 hours of RC affecting Molad Zakein is just one part of the problem. Don't forget that there are other Rosh Hashanna postponement causes esp. Lo Adu Rosh which is more common than Molad Zaken. But, I think, eventually, the difference does somewhat "balance out" over the 19 year cycle. Regardless, every Rosh Hodesh will be after the molad. – DanF Jul 16 '17 at 18:47
  • @DanF Shvili d'Rakia is the Tiferes Yisrael's treatise on how the calendar works. It's an amazing piece; you really should take a look at it at some point. I'm sure that the fact that Rosh Chodesh cannot be declared until six hours after the molad is relevant; I'm just unsure of the math behind it. (Hence the mathematics tag on the question.) – DonielF Jul 16 '17 at 20:20
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I think I may have discovered the answer. After mulling over the problem for a bit, I realized that not all cycles are created equally. The 19-year cycle just determines which years are leap years; it's the 247-year cycle which determines the lengths of Cheshvan and Kislev.

To test out my theory, I built an Excel spreadsheet, then went, column by column, down the Tur chart cited in my OP, recording just the number of days in each year.

The Tur codifies years by three criteria:

  1. The starting date of the year: ב for Monday, ג for Tuesday, ה for Thursday, and ז for Shabbos
  2. The lengths of Cheshvan and Kislev: ח if they're both 29, כ if one is 29 and the other 30, and ש if they're both 30
  3. Whether Adar II exists: פ for non-leap years and מ for leap years

As I don't care about the day of the week, I only looked at the latter two criteria, and listed each year by its length (353, 354, 355, 383, 384, or 385).

This led to the following spreadsheet:

Spreadsheet of years and their respective lengths

Yes, it was more intuitive to me as well to do it left-to-right. But I was copying off of the Tur's right-to-left chart, and I didn't feel like switching the columns around after.

The top row labels which of the 13 cycles (247/19=13) that column is. The next 19 rows are the lengths of those respective years, with the orange rows being leap years (3, 6, 8, 11, 14, 17, 19).

The final row is what this spreadsheet leads to: the total sum of days in a 19-year cycle.

The problem that leads to the phenomenon noted in the OP is that although most cycles are 6940 days long, there are a few 6939-day cycles (and cycle 6 confuses things with 6941 days).

Well, you may ask, if years aren't the same length, then why should it be limited to Jewish leap years? Well, there's two calendars we're dealing with here, so let's figure out how long the corresponding Gregorian years are.

According to the Tur's chart, 5549/1788-89 was the start of the current 247-year cycle, so I decided to use that one. (Besides, if an urban legend is circulating now, it probably surfaced from recent data.) I didn't feel like doing the math myself, so I took the Gregorian dates for RH of year 1 and 29 Elul of year 19 and outsourced the calculations.

Rosh HaShanah 5549 was October 2, 1788. 29 Elul 5567, the last day of the first 19-year cycle, was, uh, October 2, 1807. That's... 6938 days. One less than the corresponding Hebrew cycle. So the 19th birthday of anyone born in that first cycle would have been one off.

Going through these calculations (read: making my computer go through these calculations) for these cycles yielded the following differences between the Gregorian cycle and the Hebrew cycle:

  1. -1
  2. 0
  3. 0
  4. -1

At this point I realized that I just needed to compare the Rosh HaShanah dates, and I didn't need to go through the second website and then compare it to my spreadsheet. But that led to another realization: this just compares the first year of each cycle with each other. It doesn't address the rest of the cycle.

But it did lead to an answer to the original question. It has nothing to do with being born specifically in a leap year that results in the 19 year cycle being off by a day. It's that the Gregorian leap year cycle doesn't follow the Hebrew 19 year cycle, which means that sometimes the 19-year cycles line up, while sometimes they don't.

It doesn't matter whether you're born in a leap year. It matters whether those 19 Gregorian years are the same length as the 19 Hebrew years.

I will have to look into the math at another time if Jewish leap years just happen to be more susceptible to being out of sync with the Gregorian calendar, and that's what led to the rumor I mentioned in the OP. However, this does explain the underlying phenomenon.

  • Also consider that much of the world switched from the Julian calendar to the Gregorian calendar within the past 247 years. – Daniel Jul 17 '17 at 3:16
  • @Daniel Sweden was the last country that previously kept the Julian calendar and switched to the Gregorian one, and [that was 1753.] (en.wikipedia.org/wiki/Adoption_of_the_Gregorian_calendar). 247 years ago - 18 years before this 247-year cycle began, just to be clear - was only 1770. Close, but no cigar. You are correct in that most of the world switched significantly later than when the Catholics adopted it in 1582. – DonielF Jul 17 '17 at 3:26
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    Russia adopted the Gregorian calendar in 1918 and various other countries also held out until as late as the earliest 20th century. – Daniel Jul 17 '17 at 3:31
  • @Daniel Okay, if you're including countries that used their own calendars previously, Turkey/then-Ottoman Empire also held out until 1917. But I'm not sure I would say "much of the world" - a few countries here and there. – DonielF Jul 17 '17 at 3:34
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    Not sure what you mean by "used their own calendars previously." Russia was using the Julian calendar before it switched. There was also Greece in 1923, Estonia in 1913, and plenty of other countries in the last 247 years. Perhaps not most of the world but certainly much of the world. Russia alone would probably be sufficient for that description. – Daniel Jul 17 '17 at 3:47

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