WQ Archive 171 - 180

Question 171

Is Hebrew year 5763H a 13 month year because it begins on Shabbat 7 September 2002g?

YES!

The year 3618H (Mon 7 Sep -143g) was the first time that a 13 month Hebrew year began on that Gregorian date. Since 3865H (8 Sep 104g) any Hebrew year beginning on 7 September has always been a 13 month year.

However, it must be noted that Hebrew years are 13 months long whenever the Hebrew year divided by 19 leaves a remainder of either 0, 3, 6, 8, 11, 14, or 17. Therefore, the year 5763H, beginning on 7 Sep 2002g, is a 13 month year because 5763 divided by 19 leaves a remainder of 6.

```5763 is a 13 month year, and it does begin on Shabbat Sept 7. But is it
a 13 month year *because* of when it begins??

No, it is a 13 month year because it is year 6 of the nineteen year cycle,
and years 0, 3, 6, 8, 11, 14, and 17 are thirteen month years.
```

Correspondent Robert H. Douglass observed as follows.

```Yes, it is a 13-month year.  And according to your list of Separation
Dates, all Hebrew Years beginning on September 7 have been 13-month
years for about the last 1900 years... and such will continue into the
far distant future.
```

Thank you Larry Padwa and Robert H. Douglass for your correct observations.

The next Hebrew year will be Hebrew year 5763H beginning on Shabbat 7 September 2002g. This year will be 385 days long, and therefore, 13 months long.

The molad of Tishrei 5763H occurs on Shabbat at 12h 54m 10hl. Consequently, this Hebrew year will begin on the day of its molad.

Question 172

Can you mention at least 3 other calendar related features of the Hebrew year 5763H?

The next Hebrew year will be Hebrew year 5763H beginning on Shabbat 7 September 2002g. This year will be 385 days long, and therefore, 13 months long.

The molad of Tishrei 5763H occurs on Shabbat at 12h 54m 10hl. Consequently, this Hebrew year will begin on the day of its molad.

1. Anyone celebrating their 100th Hebrew birthday on 1 Tishrei 5763H will be exactly 36,500 days old!

2. 1 Tishrei 5763H coincides with exactly the same Gregorian date as 1 Tishrei 1H!

3. The year 5763H will be the 3rd Hebrew year in 4 consecutive years to begin on Shabbat.

4. The molad of Tevet 5763H will be preceded by a total eclipse of the sun on Wed 4 Dec 2002g. The molad itself is calculated to be on Thu at 3h 6m 13hl. (See The Moladot).

Correspondent Robert H. Douglass had a few more calendar facts up his sleeve!

```HY 5763 is the 6th Year of the 304th Lunar Cycle (19-year cycle),
therefore a Leap Year.

HY 5763 is Year 23 of the 206th Solar Cycle (28-year cycle, which will
re-commence in 2008-2009).

HY 5763 is Year 2 of the 7-year Shmitta cycle, in which the most recent
Shmitta (7th Year) began in September of 2000.

Wishing you (in advance) a Happy New Year.
```

Thank you Robert H. Douglass for these delightful facts!

Question 173

What would be the lengths possible for periods of 19 Hebrew years if 356 and 382 day years were allowed in the Hebrew calendar?

A feature wich distinguishes the Hebrew calendar from its historical competitors is its insistence that the month of Nisan never begin on Monday, Wednesday, or Friday.

That demand sparks the entry of 356 and 382 day years into the calendar.

As explained in The Keviyyot, these two particular year lengths are eliminated using the 356 and 382 day rules.

When the 356 and 382 day years are eliminated, then the lengths possible for 19 year periods are either 6938, 6939, 6940, 6941, or 6942 days.

The lengths of 6,938 days and 6,942 days depend on whether or not the 19 year period began with a 12 or a 13 month year. The lengths possible, when 356 and 382 day years are eliminated can be tabulated as follows:

19 YEAR SPANS
WITHOUT
356 and 382 Day Years
`    235 months =   6,939d 16h  595p`
M'+/-DAYSMOD 7dOCCURS
-2
`       0d`
0
`       0`
-1
`   6,938d`
1
`  11,263`
0
`   6,939d`
2
` 311,544`
1
`   6,940d`
3
` 250,123`
2
`   6,941d`
4
` 113,011`
3
`   6,942d`
5
`   3,531`
The maximum variance is 4 days

When 356 and 382 day years are allowed, then no period of 19 consecutive Hebrew years can have a length of 6,942 days. Also, the length of 6,938 days is independent of whether or not the 19 year period started with a 12 or 13 month year.

19 YEAR SPANS
WITH
356 and 382 Day Years
`    235 months =   6,939d 16h  595p`
M'+/-DAYSMOD 7dOCCURS
-2
`       0d`
0
`       0`
-1
`   6,938d`
1
`  30,571`
0
`   6,939d`
2
` 288,705`
1
`   6,940d`
3
` 234,346`
2
`   6,941d`
4
` 135,850`
3
`       0d`
0
`       0`
The maximum variance is 3 days

It was noted that 1 Tishrei 5763H coincided with exactly the same Gregorian date as 1 Tishrei 1H, namely, 7 September.

On the Julian calendar, 1 Tishrei 1H coincides with 7 October. However, on the Julian calendar, 1 Tishrei 5763H coincides with the date 25 August.

Question 174

When next will Rosh Hashannah begin on the Julian calendar date 7 October?

It was noted that 1 Tishrei 5763H coincided with exactly the same Gregorian date as 1 Tishrei 1H, namely, 7 September.

On the Julian calendar, 1 Tishrei 1H coincides with 7 October. However, on the Julian calendar, 1 Tishrei 5763H coincides with the date 25 August.

Keeping the Hebrew and Julian calendar rules constant, this coincidence will next occur on Rosh Hashannah 106,550H (Thu 7 Oct 102,788j).

Measured against the Gregorian calendar, Rosh Hashannah 106,550H will occur on Thu 15 Nov 102,790g.

```Tricky question!

The Hebrew Solar Year (of Rav Ada) is 235/19 lunations of 29d 12h 793p
each, which amounts to 365.2468222 days.  This is longer than the
Gregorian year of 365.2425 days, but shorter than the Julian year of
365.25 days.  Hence over the years, the position of the Hebrew Calendar
becomes progressively later compared to the Gregorian calendar, but
progressively earlier compared to the Julian.

The last time there was a Rosh Hashannah on October 7 Julian was in
Hebrew Year 1092 (on a Saturday).  And it will not occur again for a
very long time, until the relative positions of the calendars have
shifted by one full year.  It looks like the next occurrance will be in
the Hebrew Year 106550 (on a Thursday).

This would be October 7 of the year 102788 AD Julian, equivalent to June
24 of 102789 AD on the Gregorian calendar.

Regards,   Robert H. Douglass
```

Intriguingly, correspondent Robert H. Douglass indicates a different Gregorian calendar date for Rosh Hashannah 106,550H. This difference will be left up to our readers to resolve.

Thank you Robert H. Douglass for sharing your answer and providing an interesting little diversion.

Question 175

In how many unique ways are the single year lengths arranged in the mahzor qatan (19 year cycle)?

The Rev. Sherrard Beaumont Burnaby detailed a solution to this question beginning in Chapter 6, at page 146 of his book Elements of the Jewish and Muhammadan Calendars, published by George Bell and Sons, London (1901).

He based his thinking on a paper published in 1843 by Professor Nesselman. The paper, Beitrage zur Chronologie, was published in Crelle Journal fur die Mathematik, Band 26, page 59, Berlin 1843.

The Weekly Question investigated this problem using computer programming techniques, and was able to confirm that indeed there are exactly 61 unique arrangements of the single years lengths in the mahzor qatan (19 year cycle).

The 61 arrangements possible are listed below.

```
Pattern  1 first begins at        1H (  -3,760g)
Pattern  1       occurs          539 times
Pattern  1 has a length of     6,940 days

355  355  383  354  355  385  354  383  355  354
383  355  354  385  353  354  385  355  383

Pattern  2 first begins at       20H (  -3,741g)
Pattern  2       occurs        1,605 times
Pattern  2 has a length of     6,939 days

354  355  383  355  354  385  353  384  355  355
383  354  355  385  354  353  385  354  383

Pattern  3 first begins at       39H (  -3,722g)
Pattern  3       occurs          539 times
Pattern  3 has a length of     6,940 days

355  354  385  353  355  384  355  383  354  355
385  353  354  385  355  354  383  355  383

Pattern  4 first begins at       58H (  -3,703g)
Pattern  4       occurs          829 times
Pattern  4 has a length of     6,941 days

354  355  385  354  353  385  354  385  353  354
385  355  353  384  355  355  383  354  385

Pattern  5 first begins at       77H (  -3,684g)
Pattern  5       occurs          299 times
Pattern  5 has a length of     6,939 days

353  354  385  355  354  383  355  385  354  353
385  354  355  383  354  355  383  355  384

Pattern  6 first begins at       96H (  -3,665g)
Pattern  6       occurs        2,438 times
Pattern  6 has a length of     6,940 days

355  353  384  355  355  383  354  385  355  354
383  355  354  383  355  354  385  353  385

Pattern  7 first begins at      115H (  -3,646g)
Pattern  7       occurs        1,903 times
Pattern  7 has a length of     6,939 days

354  355  383  354  355  383  355  384  355  353
384  355  355  383  354  355  385  354  383

Pattern  8 first begins at      134H (  -3,627g)
Pattern  8       occurs          829 times
Pattern  8 has a length of     6,940 days

355  354  383  355  354  385  353  385  354  355
383  354  355  385  353  354  385  355  383

Pattern  9 first begins at      153H (  -3,608g)
Pattern  9       occurs          535 times
Pattern  9 has a length of     6,939 days

354  355  383  354  355  385  354  383  355  354
385  353  354  385  355  353  384  355  383

Pattern 10 first begins at      172H (  -3,589g)
Pattern 10       occurs        1,899 times
Pattern 10 has a length of     6,941 days

355  354  385  355  354  383  355  383  354  355
385  354  353  385  354  355  383  354  385

Pattern 11 first begins at      191H (  -3,570g)
Pattern 11       occurs        1,065 times
Pattern 11 has a length of     6,940 days

353  355  384  355  353  384  355  383  355  354
385  355  354  383  355  354  383  355  385

Pattern 12 first begins at      210H (  -3,551g)
Pattern 12       occurs        1,903 times
Pattern 12 has a length of     6,939 days

354  353  385  354  355  383  354  385  353  355
384  355  353  384  355  355  383  354  385

Pattern 13 first begins at      229H (  -3,532g)
Pattern 13       occurs          535 times
Pattern 13 has a length of     6,939 days

355  354  383  355  354  385  353  385  354  355
383  354  355  383  354  355  385  353  384

Pattern 14 first begins at      286H (  -3,475g)
Pattern 14       occurs        1,065 times
Pattern 14 has a length of     6,940 days

355  354  385  353  355  384  355  383  354  355
383  355  354  385  355  354  383  355  383

Pattern 15 first begins at      324H (  -3,437g)
Pattern 15       occurs          534 times
Pattern 15 has a length of     6,939 days

353  354  385  355  354  383  355  385  354  353
385  354  355  383  354  355  383  354  385

Pattern 16 first begins at      476H (  -3,285g)
Pattern 16       occurs          833 times
Pattern 16 has a length of     6,939 days

355  354  383  355  354  383  355  385  354  353
385  354  355  383  354  355  385  353  384

Pattern 17 first begins at      552H (  -3,209g)
Pattern 17       occurs          833 times
Pattern 17 has a length of     6,941 days

354  355  385  354  353  385  354  383  355  354
385  353  355  384  355  353  384  355  385

Pattern 18 first begins at      666H (  -3,095g)
Pattern 18       occurs          829 times
Pattern 18 has a length of     6,941 days

354  355  385  353  354  385  355  383  354  355
385  354  353  385  354  355  383  354  385

Pattern 19 first begins at      742H (  -3,019g)
Pattern 19       occurs          535 times
Pattern 19 has a length of     6,940 days

355  355  383  354  355  383  354  385  355  354
383  355  354  385  353  354  385  355  383

Pattern 20 first begins at      875H (  -2,886g)
Pattern 20       occurs        1,903 times
Pattern 20 has a length of     6,939 days

355  354  383  355  354  385  353  385  354  355
383  354  355  383  355  354  385  353  384

Pattern 21 first begins at      894H (  -2,867g)
Pattern 21       occurs          298 times
Pattern 21 has a length of     6,940 days

355  355  383  354  355  385  354  383  355  354
385  353  354  385  355  353  384  355  383

Pattern 22 first begins at      932H (  -2,829g)
Pattern 22       occurs          833 times
Pattern 22 has a length of     6,940 days

353  354  385  355  353  384  355  383  355  354
385  355  354  383  355  354  383  355  385

Pattern 23 first begins at    1,065H (  -2,696g)
Pattern 23       occurs          539 times
Pattern 23 has a length of     6,939 days

353  354  385  355  354  383  355  383  354  355
385  354  353  385  354  355  383  354  385

Pattern 24 first begins at    1,255H (  -2,506g)
Pattern 24       occurs          833 times
Pattern 24 has a length of     6,939 days

354  355  383  354  355  385  353  384  355  355
383  354  355  385  354  353  385  354  383

Pattern 25 first begins at    1,388H (  -2,373g)
Pattern 25       occurs        1,899 times
Pattern 25 has a length of     6,940 days

355  355  383  354  355  385  354  383  355  354
383  355  354  385  353  355  384  355  383

Pattern 26 first begins at    1,464H (  -2,297g)
Pattern 26       occurs        1,605 times
Pattern 26 has a length of     6,939 days

355  354  383  355  354  383  355  385  354  353
385  354  355  383  354  355  383  355  384

Pattern 27 first begins at    1,483H (  -2,278g)
Pattern 27       occurs          299 times
Pattern 27 has a length of     6,940 days

355  353  384  355  355  383  354  385  355  354
383  355  354  385  353  354  385  355  383

Pattern 28 first begins at    1,578H (  -2,183g)
Pattern 28       occurs          535 times
Pattern 28 has a length of     6,940 days

355  353  384  355  355  383  354  385  353  354
385  355  354  383  355  354  385  353  385

Pattern 29 first begins at    1,654H (  -2,107g)
Pattern 29       occurs          535 times
Pattern 29 has a length of     6,941 days

354  355  385  353  354  385  355  383  354  355
383  354  355  385  354  353  385  354  385

Pattern 30 first begins at    1,768H (  -1,993g)
Pattern 30       occurs          834 times
Pattern 30 has a length of     6,940 days

355  354  385  353  354  385  355  383  354  355
383  355  354  385  355  354  383  355  383

Pattern 31 first begins at    1,787H (  -1,974g)
Pattern 31       occurs        1,065 times
Pattern 31 has a length of     6,939 days

354  355  385  354  353  385  354  383  355  354
385  353  355  384  355  353  384  355  383

Pattern 32 first begins at    1,844H (  -1,917g)
Pattern 32       occurs          535 times
Pattern 32 has a length of     6,939 days

354  355  383  354  355  383  354  385  355  353
384  355  355  383  354  355  385  354  383

Pattern 33 first begins at    2,167H (  -1,594g)
Pattern 33       occurs          535 times
Pattern 33 has a length of     6,940 days

353  354  385  355  353  384  355  383  354  355
385  353  354  385  355  354  383  355  385

Pattern 34 first begins at    2,319H (  -1,442g)
Pattern 34       occurs          298 times
Pattern 34 has a length of     6,940 days

353  355  384  355  353  384  355  385  353  354
385  355  354  383  355  354  385  353  385

Pattern 35 first begins at    2,395H (  -1,366g)
Pattern 35       occurs          298 times
Pattern 35 has a length of     6,941 days

354  355  383  355  354  385  353  384  355  355
383  354  355  385  354  353  385  354  385

Pattern 36 first begins at    2,490H (  -1,271g)
Pattern 36       occurs          535 times
Pattern 36 has a length of     6,939 days

354  355  383  354  355  385  353  384  355  355
383  354  355  383  354  355  385  354  383

Pattern 37 first begins at    2,566H (  -1,195g)
Pattern 37       occurs          540 times
Pattern 37 has a length of     6,940 days

353  355  384  355  353  384  355  385  353  354
385  355  354  383  355  354  383  355  385

Pattern 38 first begins at    2,585H (  -1,176g)
Pattern 38       occurs          299 times
Pattern 38 has a length of     6,939 days

354  353  385  354  355  383  354  385  355  353
384  355  355  383  354  355  385  354  383

Pattern 39 first begins at    2,680H (  -1,081g)
Pattern 39       occurs          535 times
Pattern 39 has a length of     6,939 days

354  353  385  354  355  383  354  385  353  354
385  355  353  384  355  355  383  354  385

Pattern 40 first begins at    2,756H (  -1,005g)
Pattern 40       occurs          535 times
Pattern 40 has a length of     6,940 days

355  354  385  353  354  385  355  383  354  355
383  354  355  385  353  354  385  355  383

Pattern 41 first begins at    2,908H (    -853g)
Pattern 41       occurs          295 times
Pattern 41 has a length of     6,942 days

355  354  385  353  355  384  355  383  354  355
385  353  354  385  355  354  383  355  385

Pattern 42 first begins at    2,984H (    -777g)
Pattern 42       occurs            5 times
Pattern 42 has a length of     6,939 days

354  355  383  354  355  385  353  384  355  353
384  355  355  383  354  355  385  354  383

Pattern 43 first begins at    3,079H (    -682g)
Pattern 43       occurs          534 times
Pattern 43 has a length of     6,939 days

354  353  385  354  355  383  354  385  355  353
384  355  355  383  354  355  383  354  385

Pattern 44 first begins at    3,174H (    -587g)
Pattern 44       occurs            5 times
Pattern 44 has a length of     6,939 days

354  353  385  354  353  385  354  385  353  354
385  355  353  384  355  355  383  354  385

Pattern 45 first begins at    3,269H (    -492g)
Pattern 45       occurs          540 times
Pattern 45 has a length of     6,939 days

354  355  385  354  353  385  354  383  355  354
385  353  354  385  355  353  384  355  383

Pattern 46 first begins at    9,634H (   5,873g)
Pattern 46       occurs            5 times
Pattern 46 has a length of     6,940 days

353  354  385  355  353  384  355  383  355  354
385  353  354  385  355  354  383  355  385

Pattern 47 first begins at   23,124H (  19,363g)
Pattern 47       occurs            5 times
Pattern 47 has a length of     6,939 days

355  354  383  355  354  385  353  385  354  353
385  354  355  383  354  355  385  353  384

Pattern 48 first begins at   35,873H (  32,112g)
Pattern 48       occurs            4 times
Pattern 48 has a length of     6,941 days

354  355  385  353  354  385  355  383  354  355
385  354  353  385  354  353  385  354  385

Pattern 49 first begins at   42,352H (  38,591g)
Pattern 49       occurs            5 times
Pattern 49 has a length of     6,941 days

355  354  385  353  354  385  355  383  354  355
385  354  353  385  354  355  383  354  385

Pattern 50 first begins at   55,652H (  51,891g)
Pattern 50       occurs            5 times
Pattern 50 has a length of     6,941 days

354  355  385  353  354  385  353  384  355  355
383  354  355  385  354  353  385  354  385

Pattern 51 first begins at   75,051H (  71,290g)
Pattern 51       occurs            4 times
Pattern 51 has a length of     6,940 days

355  354  385  353  354  385  355  383  354  355
383  355  354  385  353  354  385  355  383

Pattern 52 first begins at   75,792H (  72,031g)
Pattern 52       occurs            5 times
Pattern 52 has a length of     6,940 days

355  354  385  353  354  385  353  385  354  355
383  354  355  385  353  354  385  355  383

Pattern 53 first begins at   82,081H (  78,320g)
Pattern 53       occurs            5 times
Pattern 53 has a length of     6,940 days

355  353  384  355  353  384  355  385  353  354
385  355  354  383  355  354  385  353  385

Pattern 54 first begins at   88,541H (  84,780g)
Pattern 54       occurs            5 times
Pattern 54 has a length of     6,939 days

353  354  385  355  354  383  355  385  354  353
385  354  353  385  354  355  383  354  385

Pattern 55 first begins at  108,491H ( 104,730g)
Pattern 55       occurs            4 times
Pattern 55 has a length of     6,940 days

355  353  384  355  355  383  354  385  355  354
383  355  354  385  353  354  385  353  385

Pattern 56 first begins at  114,780H ( 111,019g)
Pattern 56       occurs            4 times
Pattern 56 has a length of     6,941 days

354  355  385  354  353  385  354  385  353  354
385  355  353  384  355  353  384  355  385

Pattern 57 first begins at  161,159H ( 157,398g)
Pattern 57       occurs            5 times
Pattern 57 has a length of     6,940 days

355  355  383  354  355  385  354  383  355  354
385  353  354  385  353  355  384  355  383

Pattern 58 first begins at  167,448H ( 163,687g)
Pattern 58       occurs            5 times
Pattern 58 has a length of     6,939 days

354  353  385  354  355  383  354  385  355  353
384  355  353  384  355  355  383  354  385

Pattern 59 first begins at  213,827H ( 210,066g)
Pattern 59       occurs            5 times
Pattern 59 has a length of     6,941 days

354  355  385  354  353  385  354  385  353  354
385  353  355  384  355  353  384  355  385

Pattern 60 first begins at  246,545H ( 242,784g)
Pattern 60       occurs            4 times
Pattern 60 has a length of     6,940 days

353  354  385  353  355  384  355  383  354  355
385  353  354  385  355  354  383  355  385

Pattern 61 first begins at  305,483H ( 301,722g)
Pattern 61       occurs            4 times
Pattern 61 has a length of     6,939 days

355  354  383  355  354  385  353  385  354  355
383  354  355  385  353  354  385  353  384

```

The answer to Weekly Question 175 suggested that, measured against the Gregorian calendar, Rosh Hashannah 106,550H would occur on Thu 15 Nov 102,790g.

Correspondent Robert H. Douglass, however, indicated that Rosh Hashannah 106,550H would be June 24 of 102789 AD on the Gregorian calendar.

Robert H. Douglass resolved the difference as follows:

```I stand corrected.  From Oct 7 julian to the next June 24
gregorian (my proposed date) would mean a julian-to-gregorian offset of
260 days.

From Oct 7 to Nov 15 (of the second following year... your date)
would mean an offset of 769 days.

The year in question, 102788 CE is in the century that began 102700.
In the century beginning 200 CE the julian and gregorian calendars were
in agreement.  So the time elapsed between these dates is 102500 years.

This represents 256 intervals of 400 years (during each of which the
offset increases by 3 days) plus 100 years (with an additional offset of 1
day).

256 x 3+ 1 = 769 days of calendar offset, confirming your date.
```
Thank you Robert H. Douglass for providing this intriguing insight into Julian to Gregorian calendar conversion.

Question 176

By what fraction of the lunar month does the 12 month Hebrew year deviate from the traditional mean solar year?

A little bit of very elementary algebra is needed to answer this question.

```
Let  S = the traditonal length of the solar year
Let  m = the traditonal length of an average lunar month
Let dc = the difference between a 12 month lunar year and the traditional solar year
= 12 * m - S
Let dl = the difference between a 13 month lunar year and the traditional solar year
= 13 * m - S

Then   dl - dc = 13 * m - S - (12 * m - S)
= m

Since, dl - dc = m
dl = m + dc

In 19 years, by definition, all of the differences must accumulate to zero.

Hence, 12 * dc +  7 * dl = 0 (since there are 12 common years and 7 leap years)

12 * dc + 7 * (m + dc) = 0
19 * dc = -7 * m
dc = -7 * m / 19
```
Therefore, the 12 month Hebrew year deviates from the traditional mean solar year by a negative 7/19ths of one lunar month.

It is positively interesting to note that the length of the solar year was not required in order to solve this particular problem.

Fred Espenak's Lunar Tables show that the year 2099g will experience a total solar eclipse on Monday 14 September at 16:50 UTC.

Rosh Hashannah 5680H will begin on Tuesday 15 September 2099g.

The molad of Tishrei 5680H will occur on Tuesday at 6h 1m 4hl.

Interestingly, a number of Jewish communities will have begun their holiday observances at the time of that particular solar eclipse.

However, Fred Espenak's Lunar Tables indicate that 5860H will experience another total solar eclipse. The second eclipse will take place on Shabbat 4 September 2099g at 8:48 UTC.

The molad of Elul 5860H will take place on Shabbat 4 September at 14h 49m 16hl.

The predicted molad time comes close to the predicted eclipse time in this month.

Consequently, the year 5860H is very interesting in terms of the Hebrew calendar because it will experience 2 total solar eclipses, both of which will occur reasonably close to the times predicted for the corresponding moladot in that year.

Referring to Weekly Question 176, correspondent Avrohom Veisz wanted to know why 12 * dc + 7 * dl = 0.

Question 178

Referring to Weekly Question 176, why is 12 * dc + 7 * dl = 0?

Question 177

Relative to the Hebrew calendar, what makes the year 5860H particularly interesting?

Weekly Question 176 derived a number of conclusions using the following elementary algebra statements.

```
Let  S = the traditonal length of the solar year
Let  m = the traditonal length of an average lunar month
Let dc = the difference between a 12 month lunar year and the traditional solar year
= 12 * m - S
Let dl = the difference between a 13 month lunar year and the traditional solar year
= 13 * m - S

Then   dl - dc = 13 * m - S - (12 * m - S)
= m

Since, dl - dc = m
dl = m + dc

In 19 years,

12 * dc =  12 * (12 * m - S)
7 * dl =   7 * (13 * m - S)
12 * dc + 7 * dl =  12 * (12 * m - S) + 7 * (13 * m - S)
= 235 * m - 19 * S
= 0                (since 235 * m = 19 * S by definition)

```
Therefore, 12 * dc + 7 * dl = 0.

The next weekly question is a timely one, even though it first appeared as Weekly Question 27.

Question 27

How often does the reading of Parshah Vayyeshev coincide with Shabbat Hanukah?

Since there are 14 ways of laying out the Hebrew years (14 keviyyot), there exist only 14 ways of dividing the annual Torah reading cycle. As a result, the 14 different divisions can be easily tabulated in very compact form.

One such tabulation may be found at the back of certain editions of the Humash (Pentateuch) as translated by Alexander Harkavy, and published by the Hebrew Publishing Co. in New York (1928).

Shabbat Hanukah is any Shabbat which occurs anywhere from Kislev 25 through Tevet 2 or 3 (if the year is deficient, ie, 353 or 383 days).

From the Torah reading tables, it can be easily found that Parshah Vayyeshev is read on Shabbat Hanukah only when the preceding Rosh Hashannah began on Shabbat!

Since exactly 2/7 of all of the Hebrew years begin on Shabbat, Parshah Vayyeshev is read on Shabbat Hanukah in two out of every seven years, or on 28.57% of all of the Hanukah's.

```If I am incorrect about it being in years in which RH begins on Shabbat,

However if I am correct about that, but it is the frequency with which
you take issue, then please realize that the 29% is a rounding (to the
nearest full per cent) of the fraction 196992/689472.
```

Weekly Question 178 deals with an amazing Hebrew calendar observation made by correspondent Dwight Blevins.

Question 178

What makes year 5766H interesting in terms of the Hebrew calendar?

The molad of Tishrei 5766H will occur on Monday 3 October 2005g at 16h 48m 12hl.

According to Fred Espenak's Lunar Tables, an annular solar eclipse will also take place on that day at 10:27 UTC.

Thus Rosh Hashannah 5766H will be preceded by a solar eclipse.

The molad of Nisan 5766H will take place on Wednesday 29 March 2006g at 21h 13m 0hl.

According to Fred Espenak's Lunar Tables, a TOTAL solar eclipse will also take place on that day at 10:16 UTC.

For those who enjoy converting times from one longitutude to another, both of these celestial events are within 3 hours of the predicted times of the moladot, assuming that the times of the moladot represented time at the Jerusalem coordinates.

This approximate separation between the molad of Shevat 5766H and the corresponding new moon on 29 January 2006g will also hold.

The molad of Adar 5766H will occur on Tuesday 28 February 2006g at the traditonally calculated time of 8h 28m 17hl. The lunar tables show that the new moon on that date will occur at 00:32 UTC.

What makes this an awesome result is that if we assume that the times of the moladot represent times at the Jerusalem coordinates, then the time of the molad of Adar 5766H is actually within 4 minutes of the modern scientifically calculated time of the new moon for that month!

Thank you correspondent Dwight Blevins for pointing out these rather intriguing lunar phase times.

The following question first appeared as Weekly Question 28 and bears relevance to Weekly Question 178

Question 28

What is the present difference between the time of the molad and the corresponding mean lunar conjunction?

According to information referenced from the US Naval Observatory web page,

```
the period of the molad = 29.5305 941 358 ... (approx)
the astronomical  mean  = 29.5305 888 531 ... (approx)
```

The difference between the period of the molad and the above given astronomical value of the mean lunar conjunction is 0.0000 052 827 days per mean lunar period.

Hence, the difference between the molad period and the astronomical period is about
0.456 425 seconds per lunar month, equivalent to 0.1369 parts per lunar month.

From that it can be seen that the Hebrew month differs from the current astronomical value by about 1 day in every ((86,400/0.456425)/235)*19 = 15,304.883 years.

Up to Rosh Hashannah 5763H (2002g) 71,266 Hebrew months had elapsed. Consequently, the difference between the elapsed moladot and the astronomical periods had widened to 9.035440014 hours, equivalent to 9h 2m 2.28p.

Because of the constantly widening gap between the time of the molad and the time of the astronomical mean conjunction, it is impossible to suggest the geographical location over which the time of any molad does take place.

The Gregorian calendar, which today is the universally used secular calendar, was introduced on Fri 15 Oct 1582g.

Question 179

Up to now, how many 12-month Hebrew years have elapsed since the inception of the Gregrorian calendar?

The Gregorian calendar, which today is the universally used secular calendar, was introduced on Fri 15 Oct 1582g, corresponding to 19 Tishrei 5343H.

Since the current Hebrew year is 5763H, 420 Hebrew years have elapsed since the introduction of the Gregorian calendar.

420 Hebrew years consist of 22 complete 19 year cycles, plus two additional years. That is, 420 = 22 * 19 + 2.

Since there are 12 12 month years in every 19 year cycle, the total number of 12 month years = 22 * 12 + 2 = 266.

A total of 5,194 months elapsed in the 420 Hebrew year period between 5343H and 5763H.

Question 180

After 5343H (1582/3g), when next will a 420 Hebrew year period have 5,194 months?

Properties of Hebrew Year Periods - Part 2 shows that except for exact multiples of 19 years, all Hebrew year periods have 2 possible length in months, differing from each other by exactly one molad period.

Periods of 420 Hebrew years therefore are either 5,194 months or 5,195 months long.

Further analysis reveals that the shorter periods of 420 years always begin with years 1, 4, 9, 12, and 15 of the GUChADZaT mahzor qatan (the 19 year cycle in which years 1, 3, 6, 8, 11, 14, 17, and 19 are 13 month years).

Since the year 5343H (1582/3g) is the 4th year of the GUChADZaT mahzor qatan, the earliest year that can begin another 5,194 month period is 5348H (1587/8g), because that year is the 9th year of the GUChADZaT mahzor qatan.

Correspondent Winfried Gerum sent the following responses in answer to the question.

```Periods of 420 years can have either 5194 months or 5195 months.

The shorter length occurs, if both the first and the last year of the
cycle are not a leap year. The longer span occurs if either the first
or the last year of the 420 years is a leap year. However, it is
not possible that both the first and the last year of a 420 year cycle
is a leap year.

The frequency of these length over a full calendar cycle is

Length of 5194 months occurs 181440 times (  5 out of 19 )
Length of 5195 months occurs 508032 times ( 14 out of 19 )

The next periods of 420 years with 5194 months are:

Years 5343 through 5762 have 5194 months
Years 5348 through 5767 have 5194 months
Years 5351 through 5770 have 5194 months
Years 5354 through 5773 have 5194 months
Years 5359 through 5778 have 5194 months
Years 5362 through 5781 have 5194 months
Years 5367 through 5786 have 5194 months
Years 5370 through 5789 have 5194 months
Years 5373 through 5792 have 5194 months
Years 5378 through 5797 have 5194 months
Years 5381 through 5800 have 5194 months
Years 5386 through 5805 have 5194 months
Years 5389 through 5808 have 5194 months
Years 5392 through 5811 have 5194 months
Years 5397 through 5816 have 5194 months
Years 5400 through 5819 have 5194 months
Years 5405 through 5824 have 5194 months
Years 5408 through 5827 have 5194 months
Years 5411 through 5830 have 5194 months

Winfried Gerum

```

Thank you very much Winfried Gerum for sharing your excellent response.

Winfried Gerum's answer contained a rather interesting statistical fact. Periods of 5,194 months occur 5 out 19 times, and periods of 5,195 months occur 14 out of 19 times in the full Hebrew calendar cycle of 689,472 years.

``` First  Begun 21 Jun 1998