[Relative to year 5760H] When was the most recent occurrence of a

moladon 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.

[Relative to year 5760H] When will be the next occurrence of a

moladon 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**.

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

Are the Albiruni

Rosh Hodeshtables as shown in the Sachau translation free of numeric typographical error?

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

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.

From that it may be seen that The Albiruni value is 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.587 741 seconds per month fasterthan the 20th c. one The Hebrew value is.456 425 seconds per month slowerthan the 20th c. one Or, the accuracy of the Albiruni value is about1 day in 11,885 yearsand the accuracy of the Hebrew value is about1 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.

The first day of

Iyar5759H was on17 April 1999g.Saturday

The 1st day of Rosh Hashannah 5760H will be11 September 1999g.Saturday

Does the first day of

Iyaralways occur on the same day of the week as the first day of the followingRosh 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 weeksbefore 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.

Which postponement rule governs the first day of

Rosh Hashannahfollowing a Hebrew leap year whosemolad of Tishreiis 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.

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 =12,362,915days of Rosh Hodesh.

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.

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

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

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 First Paged 5 Nov 2004 Next Revised 5 Nov 2004