Is the leap year distribution known as

GUChADZaTan astronomical or arithmetical derivation?

The scholars **Richard A. Parker and Waldo H. Dubberstein**, in their paper **Babylonian Chronology 626 B.C.-A.D. 45**, (The University of Chicago Press 1942), indicated that the Babylonians had been using a calendar system which used a cycle of

The ancient Greek astronomer **Meton (c. 5th cent. b.c.e.)** observed that **235
lunation periods** practically equalled ** 19 solar years**. He therefore suggested a cyclical method of distributing **7 extra lunar months** into every period of **19 lunar years**. It is not known if he borrowed the idea from the ancient Babylonians or determined that independently. Also, it is not really known how Meton actually distributed the extra month lunar years within this synchronized period of time.

The fixed Hebrew calendar uses only **two** atronomical parameters. These parameters are the period of the ** molad** which was set to

What is rarely noticed or discussed in the Hebrew calendar literature is the fact that the ancient mathematicians chose to distribute the years of the **19 year cycles** so that no calendar year could *theoretically* start more than **one lunar month** from its corresponding solar year.

The leap year distribution known as ** GUChADZaT** is just one of the

Consequently, the leap year distribution known as ** GUChADZaT** is an

Which are the fundamental arithmetic rules governing the leap year distribution known as

GUChADZaT?

The two fundamental arithmetic rules governing the leap year distribution known as ** GUChADZaT** can be either

It is a well known fact that the **12-month Hebrew year** is a few days shorter than the **solar year** and that the **13-month Hebrew year** is a few days longer than the **solar year**. The exact amounts by which these years are either shorter or longer than the **solar year** have no particular relevance to the determination of the eventual ** leap year** distribution in the

Of major importance is the fact that **235 lunar months** are traditionally accepted as equalling

And of even greater importance in this derivation is the fact that, at some point in the calendar's history, it was decided to limit the start of each **lunar year** to less than **one lunar month** from its corresponding **solar year**.

These considerations define a very simple algorithmic process for distributing the leap years within the ** mahzor qatan**.

The above indicates the existence of 2 algorithms for identifying leap years.Let S = the length of a solar year Let m = the length of a lunar month By assumption, 235 * m = 19 * S Hence, 13 * m - S = (13 * m * 19 - 235 * m) / 19 = 12 * m / 19 and 12 * m - S = (12 * m * 19 - 235 * m) / 19 = -7 * m / 19

For Y years, when we proceed to add 13-month years, 13 * m * Y = S * Y + 12 * m * Y / 19 Let x = the number of months which must be sutracted from 12 * m * Y / 19 such that 0<12 * m * Y / 19 - x * m < m Then, 0<12 * Y - 19 * x < 19 which inequality represents the remainder of 12 * Y / 19. Let R(12 * Y, 19) denote the remainder of 12 * Y / 19. Now, when R(12 * Y, 19) > 6, then a lunar month must be subtracted. Since, we are proceeding in this algorithm by adding 13-month years, Y is a 13-month year whenever R(12 * Y, 19) < 7.

With a bit of additional effort, it is possible to note thatFor Y years, when we proceed to add 12-month years, 12 * m * Y = S * Y - 7 * m * Y / 19 Let x = the number of months which must be added to -7 * m * Y / 19 such that m > -7 * m * Y / 19 + x * m>0 Then, 19 > -7 * Y + x * 19>0 which inequality represents -R(7 * Y, 19). Now, when R(7 * Y, 19) > 11, then a lunar month must be added. Since, we are proceeding in this algorithm by adding 12-month years, Y is a 13-month year whenever R(7 * Y, 19) > 11.

Correspondent **Dwight Blevins** shared an interesting ** Hanukah** observation.

Of what significance is

TuesdaytoHanukah?

The First Day of The Month in the Additional Notes shows the statistical weekday distribution of the first day of each month over the full Hebrew calendar cycle of **689,472 years**.

Start of Month Distribution by Week Day | ||||||||
---|---|---|---|---|---|---|---|---|

Sun | Mon | Tue | Wed | Thu | Fri | Sat | Totals | |

Tishrei | 0 | 193280 | 79369 | 0 | 219831 | 0 | 196992 | 689472 |

Heshvan | 0 | 196992 | 0 | 193280 | 79369 | 0 | 219831 | 689472 |

Kislev | 151093 | 68738 | 69853 | 127139 | 79369 | 193280 | 0 | 689472 |

Tevet | 193280 | 26677 | 124416 | 138591 | 0 | 206508 | 0 | 689472 |

Shevat | 0 | 193280 | 26677 | 124416 | 138591 | 0 | 206508 | 689472 |

Adar | 0 | 206508 | 0 | 193280 | 26677 | 124416 | 138591 | 689472 |

v'Adar | 0 | 85899 | 0 | 72576 | 0 | 68864 | 26677 | 254016 |

Nisan | 79369 | 0 | 219831 | 0 | 196992 | 0 | 193280 | 689472 |

Iyar | 0 | 193280 | 79369 | 0 | 219831 | 0 | 196992 | 689472 |

Sivan | 196992 | 0 | 193280 | 79369 | 0 | 219831 | 0 | 689472 |

Tammuz | 219831 | 0 | 196992 | 0 | 193280 | 79369 | 0 | 689472 |

Av | 0 | 219831 | 0 | 196992 | 0 | 193280 | 79369 | 689472 |

Elul | 193280 | 79369 | 0 | 219831 | 0 | 196992 | 0 | 689472 |

Totals | 1033845 | 1463854 | 989787 | 1345474 | 1153940 | 1282540 | 1258240 | 8527680 |

Leap Adar | 0 | 72576 | 0 | 68864 | 26677 | 0 | 85899 | 254016 |

Amazingly, the month of ** Kislev** can begin on every day of the week other than

Since ** Hanukah** begins on the

Thank you correspondent **Dwight Blevins** for having shared this very intriguing
** Hanukah** fact!

How many leap years would have been in the

mahzor qatanif 240 lunar months had equalled 19 solar years?

Therefore, ifLet C = the number of 12-month years in amahzor qatan, i.e., a 19-year cycle. Let L = the number of 13-month years in the samemahzor qatan. Then C + L = 19; Hence, C = 19 - L. Now, 12 * C + 13 * L = 240 (by hypothesis) Hence, 12 * (19 - L) + 13 * L = 240 and, L = 240 - 12 * 19 = 12

Correspondent **Glenn Leider** sent the following response and comments:

Were there NO leap years in a 19-year cycle, there would be 228 lunar months, as is the case with the Islamic calendar. With 240 lunar months, there would be 240-228 or 12 leap years in the 19-year cycle in question. (Since there in reality 235 lunar months, there are instead 235-228 or 7 leap years in a 19-year cycle.) Just as note: I am one who endorses W.M. Feldman's suggestion for combining 17 cycles of 19 years (each containing 7 leap years) with one cycle of 11 years (containing 4 leap years) for a total of 17x19 + 11 or 334 years. There would be 17x7 + 4 or 123 leap years in this cycle. 334 lunar (Islamic) years total 4,008 lunar months, so when figuring in the leap years 334 solar years would total 4,008+123 or 4,131 lunar months. I propose the following lengths, the first two in the ratio of 334 to 4,131: 1 lunar month = 2,551,442.70" = 29d 12h 792.810p (44' 2.70"); 1 solar year = 31,556,915.55" = 365d 5h 874.665p (48' 35.55"); 1 cycle = 10,540,009,793.7" = 121990d 20h 538.11p (29' 53.7"). My lunar month is only about 1/10" shorter than the astronomical lunar month. My solar year is 10" shorter, but the year is getting shorter and will eventually (in a couple of thousand years or so) be the length I have assigned. What do you think?

Thank you **Glenn Leider** for sharing these magnificent observations!

Correspondent **Avi Veisz** noted that the answer to ** Weekly Question 203** had not included

The nextKislev 25, it appears, never (as in not ever) falls on Tuesday, which I never before stopped to observe. This fact, I assume would have to imply that Heshvan is NEVER elognated to 30 days in a 384 day leap year. Otherwise, the Festival of Lights ALWAYS falls on either the same day of the week as Tishrei 1 or the day before. Therefore, when Tishrei 1 falls on Tuesday, Kislev 25 ALWAYS falls on Monday and can never fall on Tuesday, making that (ie, Tue) the only day of the week which does not host Kislev 25.

On average, about how many years does it take for the Hebrew calendar to drift by one full lunar month?

Correspondent **Glenn Leider** sent the following response and comments:

As stated in The Rosh Hashannah Drift: "The Hebrew calendar moves more slowly in time than does the Gregorian calendar. As a result, the earliest possible start day for Rosh Hashannah is moving later and later into the Gregorian year. The average Hebrew year length is365.246822... daysThe average Gregorian year length is365.2425 daysHence, the Hebrew calendar is drifting through the Gregorian calendar at an average rate of about1 day in every 231.374 years." How long will it take for this drift to reach a full lunar month? Using your figures, it should take29.5306 x 231.374 or 6832.61 years, ...

Once again, thank you **Glenn Leider** for sharing these magnificent observations!

In the Gauss

Pesachformula, which year will first producezerofor the March offset?

The Gauss *Pesach* Formula was published without commentary in **1802g**. Given a Hebrew year, the formula automatically calculates the corresponding Julian date for ** Pesach** in that year.

In the computer language QBASIC, the March offset may be calculated as follows:-

Because the average Hebrew year is shorter than the average Julian year, the date ofda = (12 * dyear + 17) MOD 19 db = dyear MOD 4 dm = 32 + 4343 / 98496 + da + da * (272953 / 492480) + db / 4 dm = dm - dyear * (313 / 98496) Marchoffset = INT(dm)

The **March offset** first becomes **zero** for the Hebrew year **9877H (6117j)**.

Since **March 0** can represent either **February 28** or **February 29**, the Gauss formula needs additional work so as to provide a correct Julian date for ** Pesach** in that year.

For the Gauss

Pesachformula, what would be the simplest way to avoid azeroMarch offset as early as year 9877H (6117j)?

The Gauss *Pesach* Formula was published without commentary in **1802g**. Given a Hebrew year, the formula automatically calculates the corresponding Julian date for ** Pesach** in that year.

In the computer language QBASIC, the March offset may be calculated as follows:-

Because the average Hebrew year is shorter than the average Julian year, the date ofda = (12 * dyear + 17) MOD 19 db = dyear MOD 4 dm = 32 + 4343 / 98496 + da + da * (272953 / 492480) + db / 4 dm = dm - dyear * (313 / 98496) Marchoffset = INT(dm)

The **March offset** first becomes **zero** for the Hebrew year **9877H (6117j)**.

The simplest way to extend the life of the Gauss formula appears to be to add **304 days to dm**. This appears to yield consistently Julian dates for the **24th day of Shevat** up the year

Subtracting **304 days** from this particular date provides us the ** Pesach** date which in the Julian calendar is

Since this year marks **Maimonides' 800th yahrzeit**, the following question is derived from his

In what year will the

molad of Nisanbe onSundayat 17h 107p, as shown in chapter 6:7 of Mamonides'?Hilkhot Qiddush HaHodesh

Often cited as the *repetition of the law*, **Maimonides' 14 volume** work called **Mishneh Torah** was compiled over a

In order to explain the actual calculation of the ** moladot**,

Theoretically, this particular ** molad of Nisan** will first occur as

The following originally appeared as ** Weekly Question 72**. The Persian born Muslim scholar

Was Al-Biruni correct in stating that 2 deficient years cannot follow each other because

"the perfect months among the months of the cycle (Enneadecateris) preponderate over the imperfect ones"?

The Persian born Muslim scholar **Al-Biruni** documented the fixed Hebrew calendar **175 years** before **Maimonides**.

The medieval scholar **Al-Biruni** claimed that 2 deficient years could
not follow each other because there are more 30 day months than 29 day months
in the Hebrew calendar's 19 year cycle.

As stated on page 153 lines 15-18, in Dr. E.C. Sachau's 1879g translation of
Al-Biruni's year 1000g work *The Chronology of Ancient Nations*

The reason why two imperfect years cannot follow each other is this, that the perfect months among the months of the cycle (Enneadecateris) preponderate over the imperfect ones. For the small cycle comprises 6,940 days,i.e.125 perfect months and only 110 imperfect ones.

There is no real connection between the number of months in a **19 year** cycle
and the inability of **2 deficient months** to follow each other.

**353 day years** can begin only on ** Mondays or Saturdays**. Two such years together would cause the

**383 day years** can begin on ** Mondays, Thursdays, and Saturdays**. If such years are followed by a

The need exists to examine the possibility of **383+353** or **353+383** day years beginning on ** Monday**.

The span of time of the ** moladot** for

Hence, the earliest possible ** molad** for the beginning of the

On the other hand, a **383 day year** beginning on ** Monday** followed by a

would necessarily end on

at least

Therefore **two imperfect years** cannot follow each other.

This is also explained on **pages 11-12** of **Remy Landau**'s recently published article, **Al-Biruni's Hebrew Calendar Enigmas** in the journal

The following originally appeared as ** Weekly Question 76**.

Which year, or years, of the

mahzor qatan(19 year cycle) cannot begin a 19 year period of 6,942 days?

Measured from the first day of Tishrei, 19 year periods can be either
**6938, 6939, 6940, 6941, or 6942** days long.

The longest of the 19 year periods, **6942 days**, cannot occur if the
** first year** of the

Correspondent **Larry Padwa** not only sent the correct answer, but he
also backed it up with a mathematical proof, which will be shown next week.
Larry Padwa's proof also touches on the next ** Weekly Question**.

Thank you **Larry Padwa** for sharing with us your great insights.

The following originally appeared as ** Weekly Question 77**.

Measured from the 1st day of

Tishrei, on which day, or days, of the week can the longest period of 19 years begin?

The longest possible periods of 19 years are launched from the closing
**1h 21m 12p** of **Shabbat**'s molad of Tishrei.

Correspondent **Winfried Gerum** provided the correct answer.

The answer to Q77 is, that the longest 19-year periods (6942 days), as measured from Tishrei 1 to Tishrei 1, always commence on a shabbat.

Correspondent **Larry Padwa** not only provided the correct answer, but
also proved that answer!

1) Since 6942 is congruent to 5 (mod 7), then if year x+19 begins 6942 days later than year x, then the day of the week that begins year x+19 must be 5 days later (or 2 days earlier) than that of year x.Consider the four cases:

a) Year x begins on Monday. Then year x+19 must begin on Saturday. b) Year x begins on Tuesday. Then year x+19 must begin on Sunday. This is impossible. c) Year x begins on Thursday. Then year x+19 must begin on Tuesday. d) Year x begins on Saturday. Then year x+19 must begin on Thursday.

At this point, we are left with cases a, c, and d.

2) The molad of year x+19 is 2d 16h 595p later than molad of year x. (This is always the case).

Case a: If year x begins on Monday, then molad year x is no later than 2d 17h 1079p. (else Dehiyyah Molad Zaken would postpone the beginning of year x to Tuesday). Therefore molad of year x+19 would be no later than 5d 10h 595p which would mean RH of year x+19 would be Thursday. But case a) requires RH of x+19 to be Saturday, so case a) is impossible.

Case c: By reasoning exactly similar to case a), if year x begins on Thursday, then year x+19 would begin on Monday (not Tuesday as required by case c). Thus case c is impossible.

Case d: If year x begins on Saturday, then molad year x is no later than 0d 17h 1079p (else Dehiyyah Molad Zaken would postpone the beginning of year x to Monday).

Now consider a year x whose molad is on Saturday after 16h 689p and before 18h. This would leave RH for year x on Saturday, and the molad of year x+19 will be on Tuesday after 9h 204p. If years x and x+19 are leap years, then RH for year x+19 will be on Tuesday, and the requirement for case d) fails.

However, if years x and x+19 are common years, then Dehiyyah GaTaRad kicks in and RH for year x+19 will be on Thursday, satisfying the requirement of case d.

Thus the only time that a 19 year interval has a number of days which is congruent to 5 (mod 7) is when the starting year is a common year whose molad is between 0d 16h 689p, and 0d 18h--a period of about an hour and twenty-two minutes!

Finally, of the possible lengths of 19 year intervals (6939-6942 days), only 6942 is congruent to 5 (mod 7). Thus when case d) is satisfied, the number of days in the interval is in fact 6942.

QED

Nice work **Larry and Winfried**!

Correspondents **Winfried Gerum** and **Larry Padwa** both noticed
and shared additional facts governing the 19 year periods as related to
the ** shortest** periods of

The following originally appeared as ** Weekly Question 78**.

Measured from the first day of Tishrei, which year, or years, of the

mahzor katan(19 year cycle) can begin a 19 year period of 6,938 days?

The shortest of the **19 year** periods, **6938 days**, only can occur if the
first year of the **19 year** period is a ** LEAP** year.

Correspondents **Larry Padwa, Winfried Gerum, and Dwight Blevins**
provided correct answers to this question.

Correspondent **Larry Padwa** not only sent the correct answer, but he
also backed it up with a mathematical proof, which will be shown next week.
Larry Padwa's proof also touches on the next ** Weekly Question**.

Thank you **Larry Padwa** for sharing with us your great insights.

The following originally appeared as ** Weekly Question 79**.

Measured from the 1st day of

Tishrei, on which day, or days, of the week can the shortest period of 19 years begin?

Measured from the first day of Tishrei, 19 year periods can be either
**6938, 6939, 6940, 6941, or 6942** days long.

The shortest of the 19 year periods, **6938 days**, only can occur if the
first year of the 19 year period is a ** LEAP** year
and begins on a

Correspondents **Larry Padwa, Winfried Gerum, and Dwight Blevins**
provided correct answers to this question.

Correspondent **Larry Padwa** not only sent the correct answer, but he
also backed it up with a mathematical proof, which is as follows:

Since 6938 is congruent to 1 (mod 7), then if year x+19 begins 6938 days later than year x, then the day of the week that begins year x+19 must be one day later than that of year x. Since the only days on which a year can begin are (Mon, Tue, Thu, Sat), the only possibility of two consecutive days are that year x begins on a Monday and year x+19 begins on a Tuesday. Furthermore, since 6938 is the only number in the set {6938,6939,6940,6941,6942} of allowable days for 19 year intervals that is congruent to 1 (mod 7), it follows that x beginning on Monday and x+19 beginning on Tuesday is necessary and sufficient for the 19 year interval to contain 6938 days.The molad of year x+19 is always 2d 16h 595p later than the molad of year x.

For RH of year x to be on Monday, its molad can be as early as 0d 18h 0p, and as late as 2d 15h 588p (if x-1 is a leap year), or as late as 2d 17h 1079p (if x-1 is not a leap year).

This means that if RH for year x is on Monday, then the earliest that the molad of x+19 could be is 3d 10h 595p. Now, if x (and x+19) are common years, then Dehiyyah GaTaRad would cause RH of x+19 to be on Thursday, thus failing our requirement of a Tuesday RH. However if x (and x+19) are leap years, then GaTaRad would not apply, and RH for x+19 would be on Tuesday if the Molad of x+19 is no later than 3d 17h 1079p. This would occur if the Molad of x is no later than 1d 1h 484p (which would of course still leave RH of year x on Monday).

To summarize: If the molad for a leap year is between 0d 18h 0p and 1d 1h 484p, then the 19 years beginning with year x will contain 6938 days.

Correspondent **Winfried Gerum** made the following observations:

The answer to Q77 is, that the longest 19-year periods (6942 days) always commence on a shabbat.Looking at a full calender cycle, one finds, that if one counts just cycles starting at year 1H one gets cycles of length 6939 commencing on Shabbat or Tuesday or Thursday length 6940 commencing on Shabbat or Monday length 6941 commencing on Thursday or Tuesday length 6942 commencing always on a Shabbat If one considers 19-year periods starting with any year there are also periods of length 6938 days commencing always on a Monday (see 5790..5808)

Correspondent **Dwight Blevins** sent in the following answer:

In the upcoming discussions of the 6938 period question on the web--the only examples I've found occur in the 19th year of the cycle, and re-occur only at intervals of 247 years. Examples are 1464 - 1483 ce, 1711 - 1730, 1958 - 1977, 2205 - 2224, etc. The set-up limits are Monday for the first year of the period, and Tueday for the declaration of Tishrie 1, day one of the next period. Thus a 6938/7 = 991.14285 week, or a 0.14285 x 7 = 1 day rotation or advance from the first day of the period.Thank you correspondents

What would have happened to the Hebrew calendar if at least 2 consecutive weekdays had been specified as postponement (

dehiyot) days forRosh HaShannah?

Properties of Hebrew Year Periods - Part 1
shows that the length **L** of any period of Hebrew years is given by

4.5 L = D" - D' = M + INT(f+m) + p" - p'

where M = the integer portion of the molad period for the given period of Hebrew years m = the fractional portion of the molad period for the given period of Hebrew years f = any fractional part of a day such that 0 <= f <= d' p' = the number of days that Tishrei 1 is postponed at the start of the year p" = the number of days that Tishrei 1 is postponed at the end of the year D' = the day of 1 Tishrei H' D" = the day of 1 Tishrei H"

Assuming any **2 consecutive days** as forbidden days for **1 Tishrei**, then, whenever the day of the

When **p' = 0, INT(f+m) = 1, and p" = 2**, then the single year length becomes
**L = 354 + 1 + 2 = 357 days**.

Thus, the **357-day single Hebrew year** would be created were **2 consecutive weekdays** allowed for the postponement of **1 Tishrei**.

Correspondent **Dennis Kluk** made an observation which helped to suggest the next ** Weekly Question**.

When again, in the next 1000 years, will the 30th day of

Adarcoincide with the 29th day of February in the Gregorian calendar?

The answer to this question is actually found in First Day Hebrew-Gregorian Coincidences.

The **30th day of Adar** is really only possible in

Since **5260H (1500g)**, these coincidences are seen to have occured as follows

Due to the The5280H (1520g) Shevat-Jan v'Adar-Mar 5318H (1558g) v'Adar-Mar 5356H (1596g) Shevat-Jan v'Adar-Mar 5508H (1748g) Shevat-Jan v'Adar-Mar 5546H (1786g) v'Adar-Mar 5584H (1824g) Shevat-Jan v'Adar-Mar

Correspondent **Dennis Kluk** asked the question and correspondent **Glenn Leider** sent a correct answer.

**Dennis Kluk** made the following remarks

We had a civil and a Jewish leap day recently but they weren't the same day. However, I noticed that on Sunday 29 February 1824 it was Adar 30 5584. When will this coincedence happen again?

Note: I inserted a Roman numeral one (I) after Adar since it's only Adar I of a Jewish leap year has 30 days. The last time the above condition happened was on the 30th of Adar I 5584H or the 29th of February 1824g. To determine the next possible occurrence, the following rules apply: 1. The earliest Gregorian date occurs in the 17th year of a 19-year Jewish cycle. In the 294th such cycle the 17th year was 5548H. 2. Gregorian leap years usually occur every 4 years, such as in 1824g. So for the next possible occurrence, weneed to proceed 19 x 4 or 76 years, bringing us to 5660H or 1900g. Unfortunately, 1900g is a century year which isn't divisible by 400 so is NOT a Gregorian leap year! So let's go another 76 years to 5736H or 1976g. Alas, that year the 30th of Adar I fell on March 2. 76 years later, the 30th of Adar I 5812H will fall on March 1, 2052g. The 30th of Adar I 5888H will fall on March 2, 2128g. And the 30th of Adar I 5964H will fall on March 4, 2204g. It only gets worse after that! Ten more 76-year cycles later the 30th of Adar I 6724H will fall on March 6, 2964g. That means that the 30th of Adar I NEVER falls on February 29 during the next 1000 years. Sorry!

The only comment that should be made with regards to **Glenn Leider**'s response is that Hebrew and Gregorian calendar dates do not necessarily match up in **19 year cycles**.

Both correspondents **Dennis Kluk** and **Glenn Leider** are to be congratulated for making these excellent observations!

*Thank you!*

First Begun 21 Jun 1998 First Paged 2 Feb 2005 Next Revised 2 Feb 2005