WQ Archive 41 - 50

Weekly Question Archive 41 - 50

by Remy Landau

Question 41

[Relative to year 5760H] When was the most recent occurrence of a molad on the 31st day of its preceding month?

The molad can arrive 30 days after the first day of full months. Such days are the first days of the subsequent new month.

Under no circumstance is it ever possible for the molad to arrive 31 days after the first day of any Hebrew month.

Calendar arithmetic shows that the molad can occur 30 days after the first day of Tishrei, Kislev, Shevat, Adar I, Nisan, Sivan, and Av.

In abundant years, only the month of Heshvan can, in some years, have its molad occur on the 31st day of its predecessor month of Tishrei.

This happened for the molad of Heshvan 5759H (1998g).
It occurred on Wednesday 21 October 1998g at 1h 39m 16hl.

The phenomenon of a molad on the 31st day of its preceding month does not occur in every Hebrew year.

Question 42

[Relative to year 5760H] When will be the next occurrence of a molad on the 31st day of its preceding month?

The molad can arrive 30 days after the first day of full months. Such days are the first days of the subsequent new month.

Under no circumstance is it ever possible for the molad to arrive 31 days after the first day of any Hebrew month.

Calendar arithmetic shows that the molad can occur 30 days after the first day of Tishrei, Kislev, Shevat, Adar I, Nisan, Sivan, and Av.

In abundant years, only the month of Heshvan can, in some years, have its molad occur on the 31st day of its predecessor month of Tishrei.

This happened for the molad of Heshvan 5759H (1998g).
It occurred on Wednesday 21 October 1998g at 1h 39m 16hl.

The phenomenon of a molad on the 31st day of its preceding month does not occur in every Hebrew year.

The next molad to occur on the 31st day of its preceding month will be the molad of Elul 5762H corresponding to Friday 9 August 2002g. The time of this molad will be 0h 10m 9hl.

Albiruni's Rosh Chodesh Tables

The following 2 tables are found in Dr. C. E. Sachau's 1879 translation of Albiruni's 11th century work The Chronology of Ancient Nations.

The tables show the week days possible for both the first day of a Hebrew month (Arabic numerals) and the first day of Rosh Chodesh (Roman numerals) when it coincides with the last day of the old month.

These tables may be found on pages 155 and 156 of the cited work.

The terms imperfect, regular, and perfect respectively refer to Hebrew years that are
353 or 383 days long, 354 or 384 days long, 355 or 385 days long.

While there has been a slight modification to the labels so as to make their spelling and content more understandable, the actual numbers, both Roman and Arabic, have not changed in the values shown.

```

----------------------------------------------------------------------------
Table Showing On What Days of the Week the Beginning of
the Months Falls Throughout the Year

Table of Common Years
======================================================================
S M
i e
g n
M              n s
a              u i
r              m s
h
T                      S          K      e              i T
a    S        N        h   T      i      s              n i
E        m    i    I   i   A    e   e      s      h              i s
l        m    v    y   s   d    v   v      l      v    Quality   t h
u    A   u    a    a   a   a    a   e      e      a    of the    i r
l    v   z    n    r   n   r    t   t      v      n    Year      i i
=====  = =====  =  ====  = =====  = =====  =====  =====  ========= ===
4 III  2 1  VI  6  5 IV  3 2   I  7 6   V  4 III  2   I  Perfect    7
2   I  7 6   V  4  3 II  1 7  VI  5 4      3      2   I  Imperfect  7
6   V  4 3  II  1  7 VI  5 4 III  2 1 VII  6   V  4 III  Perfect    2
4 III  2 1 VII  6  5 IV  3 2   I  7 6      5      4 III  Imperfect  2
6   V  4 3  II  1  7 VI  5 4 III  2 1 VII  6      5  IV  Regular    3
2   I  7 6   V  4  3 II  1 7  VI  5 4 III  2   I  7  VI  Perfect    5
1 VII  6 5  IV  3  2  I  7 6   V  4 3  II  1      7  VI  Regular    5

---------------------------------------------------------------------------

Table Showing On What Days of the Week the Beginning of
the Months Falls Throughout the Year

Table of Leap Years
============================================================================
S M
i e
g n
M              n s
a              u i
S                           r              m s
e                           h
T                  c     P   S          K      e              i T
a    S        N    u     r   h   T      i      s              n i
E        m    i    I   i  A n   A i   e   e      s      h              i s
l        m    v    y   s  d d   d m   v   v      l      v    Quality   t h
u    A   u    a    a   a  a u   a u   a   e      e      a    of the    i r
l    v   z    n    r   n  r s   r s   t   t      v      n    Year      i i
=====  = =====  =  ====  = ===== =====  = =====  =====  =====  ========= ===
6   V  4 3  II  1  7 VI  5 4 III 2   I  7 6   V  4 III  2   I  Perfect    7
4 III  2 1 VII  6  5 IV  3 2   I 7  VI  5 4      3      2   I  Imperfect  7
1 VII  6 5  IV  3  2  I  7 6   V 4 III  2 1 VII  6   V  4 III  Perfect    2
6   V  4 3  II  1  7 VI  5 4 III 2   I  7 6      5      4 III  Imperfect  2
1 VII  6 5  IV  3  2  I  7 6   V 4 III  2 1 VII  6      5  IV  Regular    3
4 III  2 1 VII  6  5 IV  3 2   I 7  VI  5 4 III  2   I  7  VI  Perfect    5
2   I  7 6   V  4  3 II  1 7   I 5  IV  3 2      1      7  VI  Imperfect  5
---------------------------------------------------------------------------
```

Question 43

Are the Albiruni Rosh Hodesh tables as shown in the Sachau translation free of numeric typographical error?

Albiruni's Rosh Hodesh Tables

The following 2 tables are found on pages 155 and 156 in Dr. C. E. Sachau's 1879 translation of Albiruni's 11th century work The Chronology of Ancient Nations.

The tables show the week days possible for both the first day of a Hebrew month (Arabic numerals) and the first day of Rosh Chodesh (Roman numerals) when it coincides with the last day of the old month.

The terms imperfect, regular, and perfect respectively refer to Hebrew years that are
353 or 383 days long, 354 or 384 days long, 355 or 385 days long.

While there has been a slight modification to the labels so as to make their spelling and content more understandable, the actual numbers, both Roman and Arabic, have not changed in the values shown.

```

----------------------------------------------------------------------------
Table Showing On What Days of the Week the Beginning of
the Months Falls Throughout the Year

Table of Common Years
======================================================================
S M
i e
g n
M              n s
a              u i
r              m s
h
T                      S          K      e              i T
a    S        N        h   T      i      s              n i
E        m    i    I   i   A    e   e      s      h              i s
l        m    v    y   s   d    v   v      l      v    Quality   t h
u    A   u    a    a   a   a    a   e      e      a    of the    i r
l    v   z    n    r   n   r    t   t      v      n    Year      i i
=====  = =====  =  ====  = =====  = =====  =====  =====  ========= ===
4 III  2 1  VI  6  5 IV  3 2   I  7 6   V  4 III  2   I  Perfect    7
2   I  7 6   V  4  3 II  1 7  VI  5 4      3      2   I  Imperfect  7
6   V  4 3  II  1  7 VI  5 4 III  2 1 VII  6   V  4 III  Perfect    2
4 III  2 1 VII  6  5 IV  3 2   I  7 6      5      4 III  Imperfect  2
6   V  4 3  II  1  7 VI  5 4 III  2 1 VII  6      5  IV  Regular    3
2   I  7 6   V  4  3 II  1 7  VI  5 4 III  2   I  7  VI  Perfect    5
1 VII  6 5  IV  3  2  I  7 6   V  4 3  II  1      7  VI  Regular    5

---------------------------------------------------------------------------

Table Showing On What Days of the Week the Beginning of
the Months Falls Throughout the Year

Table of Leap Years
============================================================================
S M
i e
g n
M              n s
a              u i
S                           r              m s
e                           h
T                  c     P   S          K      e              i T
a    S        N    u     r   h   T      i      s              n i
E        m    i    I   i  A n   A i   e   e      s      h              i s
l        m    v    y   s  d d   d m   v   v      l      v    Quality   t h
u    A   u    a    a   a  a u   a u   a   e      e      a    of the    i r
l    v   z    n    r   n  r s   r s   t   t      v      n    Year      i i
=====  = =====  =  ====  = ===== =====  = =====  =====  =====  ========= ===
6   V  4 3  II  1  7 VI  5 4 III 2   I  7 6   V  4 III  2   I  Perfect    7
4 III  2 1 VII  6  5 IV  3 2   I 7  VI  5 4      3      2   I  Imperfect  7
1 VII  6 5  IV  3  2  I  7 6   V 4 III  2 1 VII  6   V  4 III  Perfect    2
6   V  4 3  II  1  7 VI  5 4 III 2   I  7 6      5      4 III  Imperfect  2
1 VII  6 5  IV  3  2  I  7 6   V 4 III  2 1 VII  6      5  IV  Regular    3
4 III  2 1 VII  6  5 IV  3 2   I 7  VI  5 4 III  2   I  7  VI  Perfect    5
2   I  7 6   V  4  3 II  1 7   I 5  IV  3 2      1      7  VI  Imperfect  5
---------------------------------------------------------------------------
```

The Albiruni tables are read right to left, and line by line.

The first day of Rosh Hodesh is always 29 days following the first day of a Hebrew month. Since 29 is 1 more than an even multiple of 7, the weekday of the subsequent Rosh Hodesh always falls one weekday later than the first day of the current month.

As an example, if the first day of the month falls on Tuesday, then the first day of the subsequent Rosh Hodesh will be Wednesday.

When a given month is 29 days long, then both the first day of the next month and the subsequent Rosh Hodesh coincide. That is why the Albiruni tables only show a single Arabic numeral as the week day following a 29 day month.

When a month is 30 days long, then the first day of the subsequent month is one day later in the week than the first day of the subsequent Rosh Hodesh. For example, if the first day of a 30 day month is Friday, then the subsequent Rosh Hodesh is on a Saturday, and the first day of the next month is on Sunday.

The Albiruni tables show this distinction using Roman numerals for the first day of Rosh Hodesh followed immediately by an Arabic numeral that represents a value one higher than the Roman numeral.

When the numeric values of the week days for Rosh Hodesh and the first day of the months are listed side by side, as is done in the Albiruni tables, the numbers read continuously from some starting point on to 7, and recycle starting at 1.

This pattern becomes evident in the Albiruni tables when reading all the numbers on a line by line, right to left basis.

That's why the two typographical errors in these tables can be so easily spotted.

The first error is in the first table. On the first line, under the heading of Tammuz, the Roman numeral value VI must be VII.

The second error is in the second table. On the last line, under the heading of Adar Secundus, the Roman numeral value I must be VI.

Page 143 ln 28 of the Sachau translation shows Albiruni's favoured value of the period of a mean lunar conjunction, namely

```   29d 12h + 1/60*(44 + 1/60*( 2 + 1/60*(17 + 1/60*(21 + 1/60*(12)))))h
```

Question 44

How much more accurate than the traditional 29d 12h 793p was Albiruni's stated value for the period of the molad?

Page 143 ln 28 of the Sachau translation shows Albiruni's favoured value of the period of a mean lunar conjunction, namely

```   29d 12h + 1/60*(44 + 1/60*( 2 + 1/60*(17 + 1/60*(21 + 1/60*(12)))))h
```

Converting the Hebrew and the Albiruni values of the period of the molad into decimal form makes possible their comparison with an accepted 20th century value.

```
The Hebrew   value is 29.5305 941 358 ... days
The Albiruni value is 29.5305 820 505 ... days
The 20th c.  value is 29.5305 888 531 ... days

From that it may be seen that

The Albiruni value is .587 741 seconds per month faster than the 20th c. one
The Hebrew   value is .456 425 seconds per month slower than the 20th c. one

Or, the accuracy of the Albiruni value is about 1 day in 11,885 years
and the accuracy of the Hebrew   value is about 1 day in 15,305 years.
```

In other words, the Albiruni value for the period of the molad is far less accurate than the Hebrew value when measured against 20th century evaluations of the mean lunar conjunction.

Question 45

The first day of Iyar 5759H was on Saturday 17 April 1999g.
The 1st day of Rosh Hashannah 5760H will be Saturday 11 September 1999g.

Does the first day of Iyar always occur on the same day of the week as the first day of the following Rosh Hashannah?

Correspondent Larry Padwa sent the following very correct answer.

Unless I'm missing something the answer to your question is YES.

The first of Iyar is exactly 21 weeks before the first of Tishrei so they are always on the same day.

Good work Larry Padwa!

There are 5 months between the first day of Iyar and the first day of Rosh Hashannah. These are the months of Iyar, Sivan, Tammuz, Av, and Elul. Each of these months is 29, 30, 29, 30, and 29 days long respectively. Hence, there are a total of 147 days between the first day of Iyar and the first day of the following Rosh Hashannah. That number of days represents exactly 21 weeks.

Question 46

Which postponement rule governs the first day of Rosh Hashannah following a Hebrew leap year whose molad of Tishrei is 4d 20h 500p?

Correspondent Larry Padwa sent the following very correct answer.

The second postponement--Moled Zaken or the third postponement (GaTaRad) could apply. Both have the same effect of postponing the RH date from Tuesday to Thursday. I reached that answer as follows:

The molad Tishrei of a year following a leap year is an integral number of weeks (which doesn't affect the calculation) + 5d 21h 589p after the leap year's molad Tishrei. Thus for the new year, molad Tishrei would be 4d 20h 500p + 5d 21h 589p which reduces to 3d 18h 9p (unless I've made an arithmetic error).

Now, 3d 18h 9p is after 18h, so by moled zaken, RH is postponed to Thurs. Also, 3d 18h 9p is a Tuesday in a common year later than 9h 204p, so by GaTaRad RH is postponed to Thursday. If you were looking for a unique answer, I would go with moled zaken, but either one applies.

Good work Larry Padwa!

The example surfaced in Section 3.1.2 of the article CALENDARS written by the late L.R. Doggett. The article can be found by linking to the US Naval Observatory web site.

Larry Padwa was also puzzled as to exactly which of two possible dehiyyot applied in this situation. Either one could trigger the postponement. However, recognizing that the Molad Zakein rule places a maximum time limit of 17h 1089p to the molad's arrival time for any one of the legitimate days for the start of Rosh Hashannah, consistency would dictate that rule as the one applied. Larry got that right too!

The festival of Rosh Hodesh always begins 29 days after the first day of a Hebrew month. When the Hebrew month has 30 days, the observance is extended to the next day which is the first day of the subsequent month.

Question 47

In the full Hebrew calendar cycle of 689472 years, how many days are observed as Rosh Hodesh?

The festival of Rosh Hodesh always begins 29 days after the first day of a Hebrew month. When the Hebrew month has 30 days, the observance is extended to the next day which is the first day of the subsequent month.

Consequently, using the tables found in The Keviyyot in the Additional Notes the following calculation can be made:-

Hebrew years of 353, 354, 355, 383, 384, and 385 days each have respectively
16, 17, 18, 18, 19, and 20 days consecrated to the festival of Rosh Hodesh.

The total number of days for the full 689472 year cycle is then given as

```
16*69222 + 17*167497 + 18*198737 + 18*106677 + 19*36288 + 20*11051
= 12,362,915 days of Rosh Hodesh.
```

Question 48

Do the regular leap years of 384 days occur at least once in all of the Hebrew calendar's 19 year cycles?

The 384 day long Hebrew years occur 36,288 times in the full Hebrew calendar cycle of 689,472 years. The full Hebrew calendar cycle also contains exactly 36,288 of its 19 year cycles. It is therefore surprising to find that some of the 19 years cycles do not have a year which is 384 days long.

Winfried Gerum sent in this very correct answer.

```According to my reckoning none of the years 5614 through 5656
is of length 384. That is 43 consecutive years!

On average each 19-year cycle contains exectly one year of 384 days.
```

Good work Winfried Gerum!

Mr. Gerum discovered a period of time larger than one 19 year cycle in which could be found not a single 384 day year.

Question 49

What is the most recent 19 year cycle in which no 384 day year can be found?

The 384 day long Hebrew years occur 36,288 times in the full Hebrew calendar cycle of 689,472 years. The full Hebrew calendar cycle also contains exactly 36,288 of its 19 year cycles. It is therefore surprising to find that some of the 19 year cycles do not have a year which is 384 days long.

Larry Padwa sent the following observation...

```The following years had (or will have) 384 days:

5711
5738
5755
5782.

Thus the most recent cycle of 19 or more years without a 384 day year
was the 26 year interval from 5712 through 5737.

We are currently in the midst of another such 26 year interval which
includes the years from 5756 through 5781.
```

Good work Larry Padwa!

Larry Padwa concluded quite correctly that we are presently in the midst of a 26 year period of time in which no 384 day Hebrew year can be found.

Therefore, the most recent 19 year cycle (mahzor katan) not to include a 384 day year is the present one which began with Rosh Hashannah 5758H on Monday 21 September 1998g.

Question 50

Besides the 384 day years, which other year length(s) might not be found in all of the Hebrew calendar's 36,288 19-year cycles?

The 384 day long Hebrew years occur 36,288 times in the full Hebrew calendar cycle of 689,472 years. The full Hebrew calendar cycle also contains exactly 36,288 of its 19 year cycles. It is therefore surprising to find that some of the 19 year cycles do not have a year which is 384 days long.

All of the other Hebrew year lengths have frequencies that far exceed 36,288 occurrences in the full Hebrew calendar. These frequencies shown below may also be found in the Additional Notes under the topic of The Keviyyot .

YEAR LENGTH IN DAYS
DAY 353 354 355 383 384 385 TOTALS
Mon 39369 0 81335 40000 0 32576 193280
Tue 0 43081 0 0 36288 0 79369
Thu 0 124416 22839 26677 0 45899 219831
Sat 29853 0 94563 40000 0 32576 196992
TOTALS 69222 167497 198737 106677 36288 111051 689472

From the above table, it is entirely unexpected that 383 day years actually do go missing from some of the 19 year cycles.

Besides the regular leap years of 384 days, the deficient leap years of 383 days are the only other kind of year to skip some of the 19 year cycles.

``` First  Begun 21 Jun 1998