בס"ד

The Purpose of the Hebrew Calendar Rosh Hashanah Dehiyyot
(דחיות לראש השנה = Postponements of the First Day of Tishrei)

by Dr. Irv Bromberg, University of Toronto, Canada email icon

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Menu of Topics:

  1. Introduction to the Rosh Hashanah Postponement Rules (דחיות לראש השנה)
  2. When Did The Hebrew Calendar Start Using Postponement Rules?
  3. Molad Zakein (מולאד זקין)
  4. Lo ADU Rosh (לא אד"ו ראש)
  5. Third Rule
  6. Fourth Rule
  7. Frequencies of Rosh Hashanah Postponements and Weekdays
  8. The Visibility of the New Lunar Crescent on Rosh Hashanah
  9. The Net Effect of the Rosh Hashanah Postponements
  10. Summary

Introduction to the Rosh Hashanah Postponement Rules

Note that hereinafter when English weekday names are used (Sunday, Monday, etc.) it is intended that the reader understand it as referring to the Hebrew weekday (Yom Rishon, Yom Sheini, etc.) starting at mean sunset.

As shown in this chart of the Traditional Hebrew Calendar Rosh Hashanah status 96 KB, relative to the Hebrew weekday of the molad of Tishrei, in about 39% of years Rosh Hashanah is not postponed, in about 47% of years it is postponed by one day, and in the remaining approximately 14% of years Rosh Hashanah is postponed by two days.

Traditionally the moment of the molad is expressed in terms of the number of hours since sunset, which I call Talmudic Temporal Time. This method has the advantage that the Hebrew weekday is directly and easily determined by calculating the number of days elapsed since the epoch of the Hebrew Calendar + 1 (because in comparing actual lunar conjunctions to the moments of the traditional molad, where the molad of Tishrei was on Yom Sheini = Monday), then divide by 7, and add 1 day to the remainder, which is the Hebrew day number of the week.

To compare molad moments to actual lunar conjunctions for astronomical evaluations, however, it is necessary to subtract 6 hours from the traditional molad moment so that it is expressed in terms of the civil time (but then the weekday will be wrong for all molad moments between sunset and midnight).

The Rosh Hashanah postponement rules are an innovation of the fixed arithmetic Traditional Hebrew calendar. They did not apply to the original observational lunar crescent calendar. Instead, the observational calendar varied the month lengths to be 29 days if the new lunar crescent was seen at sunset at the end of the 29th day, or 30 days otherwise. On the other hand, the Talmud Bavli mentions in tractate Rosh Hashanah page 20A that the Court (Sanhedrin calendar committee) used to intimidate visible crescent witnesses to withdraw or confuse their testimony or to disqualify them if otherwise Rosh Hashanah would be sanctified on Wednesday or Friday (see rule #2, below). Likewise, the Chinese calendar, based on astronomical algorithms for the ecliptic longitudes of Sun and Moon, varies the length of months from 29 to 30 days depending on the computed moment of the actual lunar conjunctions. The Traditional Hebrew Calendar, however, has fixed month lengths except for Cheshvan and Kislev, which vary to accomodate the Rosh Hashanah postponements that set the calendar year length:

Calendar
Year Type		 Common
Year Length		 Leap Year
Length		 Cheshvan
days		 Kislev
days
Deficient 353 383 29 29
Normal 354 384 29 30
Perfect (full) 355 385 30 30

As of version 11.150(1670) revised Aug 9, 2019 (Symmetry454) my freeware calendrical calculator, Kalendis, has a built-in Hebrew calendar report List New Year Postponements that details the types and reasons of Rosh Hashanah postponements for the selected number of years (choices are 19, 3×19=57, 100, 500, or 1000 years, starting with the currently displayed Hebrew year number). The following is an example of such a report:

Click here to see the Rosh Hashanah details and postponement reasons for 500 near-present-era Hebrew calendar years

When Did The Hebrew Calendar Start Using Postponement Rules?

In several of his books and articles, Professor Sacha Stern of University College London in the United Kingdom has persistently stated his opinion, relying on tombstone inscriptions and handwritten correspondence as evidence of Jewish calendrical diversity, that the development and application of postponement rules and the finalization of the details of the traditional fixed arithmetic Hebrew calendar was a slow, gradual process that spanned several centuries. I strongly disagree, because such sources aren't reliable for the purposes of technical chronology, and because, as will be explained below, correct reckoning of these rules is mandatory for avoiding illegal Hebrew calendar year lengths (impossibly short, impossibly long). If any rule is missing or modified, the calendar will eventually break when it runs into an illegal year length.

Also as will be explained below, all of the postponement rules were discussed in the Talmud, the authorities of which pre-dated the promulgation of the traditional fixed arithmetic Hebrew calendar, and nowhere in the Torah or the Talmud is anything written about the fixed arithmetic calendar except where Amorah Shmuel of Nehardea (Shmuel the Astronomer) stated on page 20b of tractate Rosh Hashanah of the Talmud Bavli: I am able to make a calendar for the entire diaspora. No further details were provided. Shmuel lived during the century before the fixed arithmetic calendar was promulgated.

Therefore, I assert that all of the postponement rules were routinely applied to the observational Hebrew calendar from the days when the Talmud authorities lived until the promulgation of the traditional fixed arithmetic Hebrew calendar, and that the fixed calendar had all of these rules firmly and expertly established since its first release by the Sanhedrin of Hillel ben Yehudah in Hebrew year 4119 = Julian 358 AD, and probably since Shmuel developed the aforementioned fixed diaspora calendar.

Note that a Sanhedrin ruling can only be changed by a later Sanhedrin ruling, so it is implausible that the calendar rules gradually evolved to their final form. If anybody deviated from the established rules then he was outside of his authority to do so, or was mistaken.

Rule #1. Molad Zakein: If the molad of Tishrei falls at or after Noon then postpone Rosh Hashanah to the next day.

This is the second-most-commonly invoked postponement rule, but logically should be checked first because its invokation can cause Rosh Hashanah to be postponed to a disallowed weekday (which in turn will be handled by the second postponement rule). The Hebrew term molad zakein literally means old (or late) molad. For example, this is the reason for postponing from Monday to Tuesday in Hebrew year 5786.

Talmud Bavli tractate Rosh Hashanah, page 20b, says that the Court regarded the calculated molad as the moment of the lunar conjunction, and determined that if the conjunction occurred after Noon then it would be impossible to see the new lunar crescent after the coming sunset. Even though the Court continued to rely on the testimony of sighting witnesses, this calculation was used to refute false witnesses. The molad zakein postponement rule is analogous to this practice.

Note that Noon coincides with 3/4 of the day elapsed, because Hebrew calendar days start at sunset. Therefore, rather than explicitly checking the timing of the molad moment, German Mathematician Johann Carl Friedrich Gauss (Gauß) pointed out that this postponement rule could be permanently bypassed simply by adding 1/4 day to the traditional molad epoch. If the original moment had been past Noon then that extra 1/4 day would automatically advance the molad moment to the following day. Indeed, this modification would eliminate all molad zakein postponements, decreasing the number of years in which Rosh Hashanah is postponed from more than 3/5 to less than 1/2 of years, and it would also slightly simplify the arithmetic for the 3rd and 4th postponement rules. It would have the side effect, however, that all molad moments would be calculated as 6 hours later, in addition to the +6 hours due to the difference from mean sunset to civil midnight (since Hebrew dates start at sunset), increasing the uncertainty and relevance of announcing the molad moment in synogogues on Shabbat Mivarchim (the last sabbath before the start of the coming month), which might be culturally undesirable. It wouldn’t have any effect whatsoever on Hebrew calendar year length frequencies or the distribution of Rosh Hashanah starting weekdays nor the actual reckoning of dates or holidays, however, thus obviating any ritual objections to this modification.

Traditionally, this postponement was considered necessary to ensure the visibility of the New Moon on the first day of Rosh Hashanah. In reality, it doesn’t ensure that, see below for my astronomical analysis of the visibility of the lunar crescent on Rosh Hashanah.

Remy Landau (זכרונו לברכה) stated that this postponement is to ensure that the calculated time of any molad does not exceed the first day of any month in the Hebrew calendar, see <https://hebrewcalendar.tripod.com/#11>. I don’t know how to confirm this assertion, because as I explained on my molad web page, the molad moment isn’t a clock time (it’s just Hebrew calendar arithmetic that provisionally determines the starting date of Rosh Hashanah, subject to the postponements discussed herein), and also because the moment when any month starts varies from locale to locale, depending on when is the local sunset.

The Talmud Bavli, in tractate Rosh Hashanah page 20B explains that if the molad occurs after Noon then there is no way that the new lunar crescent will be visible at the coming sunset (six hours later), but it will be visible at the following sunset. This Noon cutoff time led Rabbi Zerachiah ben Isaac Ha-Levi Gerondi (the Baal HaMaor, 1125-1186 AD) to assign the meridian 90° east of Jerusalem as the date line in Jewish Law, but this was partially based on the incorrect assumption that the traditional molad moment refers to the meridian of Jerusalem. As I have shown on my web page Moon and the Molad of the Hebrew Calendar, the original historical reference meridian of longitude of the traditional molad was actually midway between the Nile River and the end of the Euphrates River, which, according to the logic of the Baal HaMaor, would shift the date line to about 94° east of Jerusalem, or almost 130° east of Greenwich, passing east of Korea, through the middle of the southern island of Japan, the Philippine Sea, the Java Sea west of New Guinea, the Timor Sea north of Australia, and just west of Darwin, Australia. Another problem with the interpretation of the Baal HaMaor is that although in his own era the world was understood to be a sphere with times of day differing between meridians of longitude, in the era of the sages of the Talmud it was thought that the world was flat, during the night Sun was outside the celestial sphere, and during rising or setting Sun moved through windows (or tunnels) between the outside and inside of the celestial sphere such that sunrise and sunset occurred at the same moment for the entire world. These concepts are discussed as such in that section of the Talmud and by its commentators. In those days there was no understanding of time zones or meridians of longitude, therefore the Noon cutoff time specified in the Talmud could not have had any relationship to any concept of a date line.

The Baal HaMaor also claimed that the selected cutoff time ensures that somewhere in the world the actual lunar conjunction will correspond to the traditional molad moment. However, there is nothing special about that particular cutoff time that ensures any higher probability of that statement being true, in fact it would be equally true for any alternative cutoff time. Even so, of what relevance is it?

Furthermore, the reference meridian for the traditional moladot is drifting eastward at an accelerating rate, so are we to learn that the International Date Line is also drifting eastward at the same rate, to remain 90° (6 hours) ahead of it?

If the date line were to pass through populated lands then it would be possible to almost completely avoid Shabbat by taking a step to the east across the line as Shabbat starts west of the line, waiting for Shabbat to end west of the line, then taking a step back to the west across the line as Shabbat starts east of the line. Conversely one could observe two consecutive days of Shabbat by staying to the west of the date line at the start of Shabbat, then as Shabbat is ending west of the line taking a step east across the line where it will be near the start of Shabbat. Such examples make it obvious that the international date line must never pass through populated lands, and led Rabbi Avrohom Yeshaya Karelitz (the Chazon Ish) in 1941 AD to extend the halachah date line to the eastern coasts of Asia and of Australia. Nevertheless that extension still leaves the problem that a person swimming at a coastal beach, or boating in that region, will be crossing the line. Perhaps such a date line should be extended to the continental shelf?

For more information about the date line controversy see Willie Roth’s essay The International Date Line and Halachah 442KB, and The Sabbath, the International Date Line and Jewish Law by Rabbi Yehuda Shurpin at <https://www.chabad.org/library/article_cdo/print/false/aid/1736567/jewish/The-Sabbath-the-International-Date-Line-and-Jewish-Law.htm>.

There is some commentary debate as to whether the cited Talmud text refers to the actual lunar conjunction or the mean conjunction as estimated by the Hebrew Calendar molad arithmetic. In the era of Hillel ben Yehudah the molad of Tishrei was on average 4+1/5 hours late with respect to the actual mean lunar conjunctions but today it is on average 6 hours late, as shown in this chart 316KB (see also the last paragraph on this page), although today an individual molad moment can be as much as 16 hours late and in the era of the Talmud a molad moment could have been almost 14+1/2 hours late. Astronomically, in the era of the Talmud the absolute minimum elapsed time from the actual lunar conjunction (New Moon) until the moment when the visible crescent for the month of Tishrei could be visible at sunset in Jerusalem under rare ideal conditions was about 3/4 day. In other words, when the actual lunar conjunction of Tishrei occurs shortly after midnight in Jerusalem, the new lunar crescent might be seen at sunset that same day. Such an early crescent would be truly exceptional (<2% of years) and would require a highly favorable lunar altitude at sunset and Moon moving most rapidly near its orbital perigee, whereas most (85%) of the first visible crescents occur between one and 2+1/2 days after the actual lunar conjunction, 50% occur more than 1+3/4 days after conjunction, and 30% more than 2 days after conjunction. When the lunar altitude is exceptionally unfavorable and Moon is moving most slowly near its orbital apogee, which occurs in about 7.5% of years, the visible crescent of Tishrei is first seen more than 2+1/2 days after conjunction — but the crescent is essentially never first seen beyond 3 days after conjunction (unless weather conditions interfered with its visibility).

The earliest new lunar crescent sightings from the records of the present era Israeli New Moon Society (האגודה הישראלית לצפייה בירח החדש) are always more than 24 hours after the actual lunar conjunction (at sea level, without optical aids). Relative to the molad of Tishrei, which can be as much as 16 hours later than the actual conjunction in the present era, the earliest visible new lunar crescent should be at least 6 hours later than the molad moment. Therefore the 6-hour cutoff cited from the Talmud does fit the astronomy of the present era if the term molad is understood to refer to the molad arithmetic of the Traditional Hebrew Calendar. On the other hand, it would be very uncommon for the new crescent to be visible only 6 hours after the molad, and in the era of the Talmud the minimum elapsed time would have had to be at least 7+1/2 hours.

The Earth-to-Moon distance at the lunar orbital perigee is quite variable. Lesser perigee distances are associated with faster lunar orbital motion and shorter elapsed times from conjunction to visible crescent. Likewise the Earth-to-Moon distance at the lunar orbital apogee is also quite variable. Greater apogee distances are associated with slower lunar orbital motion and longer elapsed times from conjunction to visible crescent.

It might seem that the lunar latitude is not very important in regard to the molad of Tishrei, because Rosh Hashanah is near the southward equinox, when Sun and Moon set close to due west. If the lunar latitude is at its extreme 5° south of the ecliptic, Moon will appear to set to the left of the sunset point, or if Moon is at its extreme 5° north of the ecliptic it will appear to set to the right of the sunset point. The greater the lunar latitude is at sunset, however, either south or north of the ecliptic, the longer the arc-of-light from Sun to Moon will be, enhancing the probability that the new lunar crescent will be visible.

In the far distant future, around Hebrew year 12000 if the Hebrew calendar solar drift is left uncorrected, when the Earth orbital aphelion is close to Rosh Hashanah (it is currently in Tammuz), the lunations will tend to be shorter at that time of year. In that millennium the maximum limit will be reduced to 2+1/2 days (97.5th percentile) or absolute maximum of 2+3/4 days, but the minimum limit will increase closer to a full 24 hours (at sea level, without optical aids). Only 5% of years in that millennium will have less than 24 hours elapsed from actual conjunction to visible crescent in Jerusalem.

Back in the millennium beginning with Hebrew year 4000, when the Earth orbital aphelion was in Sivan, the lunations near Rosh Hashanah tended to be slightly longer than they are at present, so an actual lunar conjunction after midnight was essentially never associated with a visible crescent at the next sunset in Jerusalem. This leads to another possible explanation. The classical Hebrew word used to specify the cutoff time is chatzot (חֲצוֹת). The context of the cutoff time discussion is that if witnesses come to testify that they saw the new lunar crescent, but the moment of the (actual) lunar conjunction is calculated to have occurred after chatzot, then their testimony can be confidently refuted because it is still too early for the new crescent to be visible. The word chatzot, unqualified, has two possible meanings: either midnight or Noon, so it can refer either to the middle of the day or the middle of the night. To unambiguously specify the intended meaning, the qualifier haLeilah (הלילה = of the night) or haYom (היום = of the day) could have been used. Thus chatzot haLeilah (חֲצוֹת הלילה) would have unambiguously referred to midnight, and chatzot haYom (חֲצוֹת היום) would have unambiguously referred to Noon. The Talmud, in its typically terse style, only used the unqualified chatzot. All commentators and translators that I have seen have understood the word chatzot as referring to Noon, but astronomically it could only be sensible to understand it as referring to midnight, unless the intention was to err on the side of very liberally accepting doubtful testimonies.

Rule #2. Lo ADU Rosh: If the molad of Tishrei falls on Sunday, Wednesday or Friday then postpone Rosh Hashanah to the next day.

This is the most commonly invoked postponement rule, but logically should be checked after the first rule given above, because the first rule can postpone the Rosh Hashanah weekday from an allowable weekday to a disallowed weekday, in which case this second rule would cause a further one day postponement to the next allowable weekday.

For example, Hebrew years 5746, 5749, 5753, 5780 have Sunday to Monday postponements; years 5741, 5748, 5768 have Wednesday to Thursday postponements; and years 5757, 5764, 5784 have Friday to Saturday postponements.

This postponement rule is for ritual convenience: The Talmud Bavli tractate Rosh Hashanah page 20a states that excluding Wednesday and Friday is to prevent Yom Kippur from occurring on either side of Shabbat, which would be ritually inconvenient with regard to the burial of a corpse. The shofar is sounded after the end of Yom Kippur, but that ritual would not be permissible if Yom Kippur was on Friday because as it ended Shabbat would have already started. The Talmud Bavli tractate Sukkah 43a as well as Talmud Yerushalmi tractate Sukkah 4:5 says that excluding Sunday is to prevent Hoshanah Rabbah from occurring on Shabbat, which would render the lulav and willow branches muktze (not to be touched on Shabbat) so they couldn’t be carried to or around the synagogue, and the willow branches couldn’t be beaten on the floor to symbolize the final elimination of sins. Note that Succot always starts on the same weekday as Rosh Hashanah.

Remy Landau (זכרונו לברכה) <https://hebrewcalendar.tripod.com/#10> pointed out that this rule reduces the number of Qeviyyot (קווייות) or Hebrew year types from 28 to 16, and it also prevents the first day of Rosh Hashanah from falling adjacent to Shabbat (although Rosh Hashanah can and does fall on Shabbat itself).

Implementation tip: If the molad date is converted to a weekday as an integer from Sunday=0 through Saturday=6 then the molad is on a disallowed weekday if ( MoladWeekday × 3 ) MOD 7 < 3.

Rule #3. If the molad of Tishrei for a non-leap year (12 months) falls on Tuesday then if the next molad of Tishrei plus 1/4 day will fall on Sunday then postpone this Rosh Hashanah to Thursday.

There is no need to check this rule if the molad moment was already found to fall on or past Noon or on a disallowed weekday.

As explained by Remy Landau (זכרונו לברכה), see <https://hebrewcalendar.tripod.com/#12>, this rule eliminates all of the 356-day years that would otherwise be caused by the disallowed weekday rule.

For the fixed arithmetic traditional Hebrew calendar, if the molad of Tishrei of a non-leap year falls on Tuesday at or after 9h 204p then the next molad of Tishrei will be 12 molad intervals later, which will place it at or after Noon on Shabbat (Saturday), which will be postponed to Sunday, but that weekday is not allowed, or if the molad falls at or after 15h 204p then the next molad of Tishrei will land directly on Sunday — either way Rosh Hashanah will be postponed to Monday. Either of these postponements would cause this year length to be 356 days, but the longest possible length of a non-leap year is 355 days (where Cheshvan and Kislev both are full 30-day months). Therefore Rosh Hashanah this year is postponed from Tuesday to Wednesday, and because Wednesday is not allowed (see rule #2, above) it is postponed again to Thursday, making this year an acceptable 354 days.

Rather than explicitly checking the molad time against the fixed cutoff time of 9h 204p, which is only a calculation shortcut, and which is incompatible with any variable interval molad such as the progressive molad or the actual lunar conjunction, a generic alternative is simply to postpone Rosh Hashanah if the next molad of Tishrei will be at or after Noon on Shabbat or land directly on Sunday. The Gauss shortcut (adding 1/4 day) can also be used on that molad moment, triggering the 2-day postponement for this year if the next molad of Tishrei plus 1/4 day will land on Sunday. Alternatively, postpone this Rosh Hashanah by two days if the following expression is true:

floor( NextTishreiMolad + 1/4 day ) – floor( ThisTishreiMolad + 1/4 day ) = 355

This rule is invoked in only about 3.31% of years (for example, Hebrew years 5620, 5640, 5647, 5667, 5718, 5745, 5789, 5796, 5816, 5867, 5887, 5894 — the boldfaced year numbers were cases where the next molad of Tishrei landed directly on Sunday), and such years are always non-leap years having 354 days. Although the intervals between these uncommon postponements average to about once per 30 years, only 8 intervals are actually possible within the Traditional Hebrew Calendar:

Interval (years): 7 20 27 31 44 51 64 71
% Occurrence: 18.7% 35.2% 10.1% 2.6% 1.9% 24.1% 1.1% 6.3%

Rule #4. If the molad of Tishrei after a leap year (13 months) falls on Monday then if the previous molad of Tishrei fell at or after Noon on Tuesday then postpone this Rosh Hashanah to Tuesday.

There is no need to check this rule if the molad moment was already found to fall on or past Noon or on a disallowed weekday.

As explained by Remy Landau (זכרונו לברכה), see <https://hebrewcalendar.tripod.com/#13>, this rule eliminates all of the 382-day years that would otherwise be caused by the disallowed weekday postponement rule.

For the fixed arithmetic traditional Hebrew calendar, if the molad of Tishrei falls on Monday at or after 15h 589p and the prior year was a leap year, then the previous molad of Tishrei would have been 13 molad intervals earlier, which would have placed it at or after Noon on Tuesday, which would have been postponed to Wednesday, but that weekday is not allowed, so it would have been again postponed to Thursday. This would have caused that prior year length to be only 382 days, but the shortest possible length of a leap year is 383 days (where Cheshvan and Kislev both are deficient 29-day months). Therefore Rosh Hashanah this year is postponed from Monday to Tuesday, making the prior year an acceptable 383 days.

Rather than explicitly checking the molad time against the fixed cutoff of 15h 589p, which is only a calculation shortcut, and which is incompatible with any variable interval molad such as the progressive molad or the actual lunar conjunction, a generic alternative is simply to postpone Rosh Hashanah if the prior molad of Tishrei was at or after Noon on Tuesday. Again, the Gauss shortcut (adding 1/4 day) can be used on that molad moment, triggering a postponement for this year if the prior molad of Tishrei plus 1/4 day landed on Wednesday. Alternatively, postpone this Rosh Hashanah by one day if the following expression is true:

floor( ThisTishreiMolad + 1/4 day ) – floor( PriorTishreiMolad + 1/4 day ) = 383

This rule is invoked in only about 0.54% of years (for example, Hebrew years 5096, 5194, 5441, 5519, 5688, 5766, 6013, 6111), and again such years are always non-leap years having 354 days. Although the intervals between these rare postponements average to about once per 186 years, only 5 intervals are actually possible within the Traditional Hebrew Calendar:

Interval (years):7898169247345
% Occurrence:22.1%15.5%12.3%41.2%8.9%

Frequencies of Rosh Hashanah Postponements and Weekdays

The overall frequencies of applicability of the Rosh Hashanah postponement rules are shown below, and are similar to those given by Remy Landau (זכרונו לברכה) at <https://hebrewcalendar.tripod.com/#24.3>. Note that it is least common for Rosh Hashanah to start on Tuesday, and most common for it to start on Thursday.

When Rosh Hashanah does start on Thursday there are two days of Yom Tov (High Holy Days) followed by Shabbat for Rosh Hashanah, plus, outside Israel, the same for the first two and last two days of Sukkot.

Traditional Rosh Hashanah Postponements and Weekdays
(number of years per 1000 years, disallowed weekdays are Sunday, Wednesday, Friday)

Year Range No Shift Delayed 1 Delayed 2 Monday Tuesday Thursday Saturday
4001-5000 394 469 137 277 116 318 289
5001-6000 388 469 143 282 114 319 285
6001-7000 390 468 142 280 117 316 287
7001-8000 390 470 140 280 114 323 283
8001-9000 387 471 142 280 115 318 287
9001-10000 390 469 141 278 116 318 288

The Visibility of the New Lunar Crescent on Rosh Hashanah

Rabbi Moshe ben Maimon (רבי משה בן מימון, Rambam, רמב״ם), also commonly known by his Latin name, (Moses) Maimonides, wrote a book entitled Hilchot Kiddush haChodesh (הלכות קידוש החודש, title translated as Sanctification of the New Month or alternatively as Sanctification of the New Moon) around the Julian year 1178 or Hebrew year 4938, which is the eighth treatise in Zmanim (ספר הזמנים, The Book of Seasons), the third book of his Mishneh Torah collection (משנה תורה, code of Jewish Law). In this book, as item 7 and item 8 of chapter 7, Rambam wrote:

עִברִית English
וּמִפְּנֵי מָה אֵין קוֹבְעִין בְּחֶשְׁבּוֹן זֶה בִּימֵי אד''ו. לְפִי שֶׁהַחֶשְׁבּוֹן הַזֶּה הוּא לְקִבּוּץ הַיָּרֵחַ וְהַשֶּׁמֶשׁ בְּהִלּוּכָהּ הָאֶמְצָעִי לֹא בַּמָּקוֹם הָאֲמִתִּי כְּמוֹ שֶׁהוֹדַעְנוּ. לְפִיכָךְ עָשׂוּ יוֹם קְבִיעָה וְיוֹם דְּחִיָּה כְּדֵי לִפְגֹּעַ בְּיוֹם קִבּוּץ הָאֲמִתִּי. כֵּיצַד. בִּשְׁלִישִׁי קוֹבְעִין בִּרְבִיעִי דּוֹחִין. בַּחֲמִישִׁי קוֹבְעִין בְּשִׁשִּׁי דּוֹחִין. בְּשַׁבָּת קוֹבְעִין אֶחָד בְּשַׁבָּת דּוֹחִין. בְּשֵׁנִי קוֹבְעִין 7. Why is [Rosh Hashanah] not confirmed [on the day of the conjunction] when it falls on Sunday, Wednesday, or Friday? Because these calculations determine the conjunction of the sun and the moon only according to their mean [rate of] progress, and do not [necessarily] reflect their true positions, as explained. Therefore, they instituted that [Rosh Hashanah] would be confirmed on one day but postponed on the following day, so that they would ascertain the day when the true conjunction takes place. What is implied? [When according to our calculations, the conjunction occurs on] Tuesday, we confirm [that day as Rosh Hashanah]. On Wednesday, we postpone. On Thursday, we confirm. On Friday, we postpone. On Shabbat, we confirm. On Sunday, we postpone. On Monday, we confirm.
וְעִקַּר שְׁאָר הָאַרְבַּע דְּחִיּוֹת אֵלּוּ הוּא זֶה הָעִקָּר שֶׁאָמַרְנוּ שֶׁהַחֶשְׁבּוֹן הַזֶּה בְּמַהֲלָךְ אֶמְצָעִי. וּרְאָיָה לַדָּבָר שֶׁהַמּוֹלָד יִהְיֶה בְּלֵיל שְׁלִישִׁי וְיִדָּחֶה לַחֲמִישִׁי פְּעָמִים רַבּוֹת לֹא יֵרָאֶה יָרֵחַ בְּלֵיל חֲמִישִׁי וְלֹא בְּלֵיל שִׁשִּׁי מִכְּלַל שֶׁלֹּא נִתְקַבְּצוּ הַשֶּׁמֶשׁ וְהַיָּרֵחַ קִבּוּץ אֲמִתִּי אֶלָּא בַּחֲמִישִׁי 8. This same principle, that the calculations are based on the mean rate of progress, is also the motivating factor for the four reasons for the postponement [of Rosh Hashanah]. As proof of this, there are times when [according to the calculations] the conjunction takes place on Tuesday, and [Rosh Hashanah] is postponed until Thursday, and yet the moon will not be seen Thursday night, nor even Friday night. This indicates that the true conjunction of the sun and the moon did not take place until Thursday.

Note that item 8 contradicts item 7. Item 7 vaguely suggests that the postponements bring the date of Rosh Hashanah closer to the actual lunar conjunction. Item 8, however, hints that the postponements increase the likelihood that the lunar crescent will be visible at sunset at the beginning of Rosh Hashanah. These attributes are mutually exclusive. Evidently Rambam discounted the ritual reasons for the postponements given in the Talmud.

The actual lunar conjunctions precede the visible new lunar crescent by 1 to 3 days. If the postponements were to bring Rosh Hashanah closer to the actual lunar conjunction then they would have to decrease the probability of observing the new crescent on the holiday. On the other hand, if they were to increase the likelihood of observing the new lunar crescent on Rosh Hashanah then that could only be accomplished by delays relative to the actual lunar conjunctions. Astronomical analysis of the visible crescent frequencies on Rosh Hashanah will reveal which of these alternatives is correct, as follows:

I used the computer algorithms for the lunar cycle, solar longitude, and visible crescent criteria from Calendrical Calculations by Nachum Dershowitz and Edward M. Reingold, third edition published in 2008 by Cambridge University Press, hereinafter referred to by the mnemonic CC3, plus some algorithms from Astronomical Algorithms by Jean Meeus, second edition, published in 1998 by Willmann-Bell, Richmond, Virginia, USA.

Lunar crescent visibility varies with weather conditions, clouds, atmospheric dust and clarity (especially in the westerly direction), temperature, humidity, nearby and westerly light pollution, and local elevation with unobstructed view of the horizon. Astronomically it depends on the apparent lunar size and brightness, elongation from Sun, and altitude above the horizon at sunset. Human factors include observer maturity, truthfulness, sanity, visual acuity and stereoscopic perception, iris pigmentation and pupil diameter, experience and preparation, and the use of optical aids such as telescopes or binoculars.

It also helps a lot to know exactly when and where to look, and then to actually look at the correct position in the sky! When the new lunar crescent is very dim, it may be visible only to more light-sensitive peripheral vision, rather than the sharpest central color vision. In such cases, it might be seen only when one looks slightly askance of its position, then away, then back again.

The probability of false sighting for a typical observer has been estimated at 15%, hence observational calendars that depend on a few positive sightings from a large number of observers will almost invariably start early by mistake. False sightings can, however, be refuted if the Moon was actually below the horizon or ahead of the actual lunar conjunction at the moment when it is claimed the new lunar crescent was seen. If the moment of the actual lunar conjunction is known to good accuracy (better than ±1 minute is easy to calculate), then testimony claiming a sighting less than 18 hours later is highly doubtful.

For the prediction of the visibility of the new lunar crescent I use the following mathematical criteria as recommended by Dershowitz & Reingold in CC3, as implemented in their visible-crescent function:

Using the above criteria, the absolute minimum elapsed time between actual lunar conjunction and first visibility of the new lunar crescent at sunset in Jerusalem is about 3/4 day. Published crescent visibility reports, such the records of the Israeli New Moon Society (האגודה הישראלית לצפייה בירח החדש) rarely document reliable sightings any earlier than 24 hours after conjunction, even with optical aids. Therefore the above criteria are probably optimistic and applicable to ideal observing conditions.

The above criteria ignore the apparent lunar diameter (Moon can appear up to 25% larger at closest perigee than at furthest apogee), and the CC3 algorithms ignore atmospheric refraction and lunar parallax when calculating the lunar altitude. When Moon is near the horizon, atmospheric refraction makes the apparent lunar position as seen from the surface of Earth (topocentric position) about 1/2 degree higher than its position as calculated for the center of Earth (geocentric position), and lunar parallax makes Moon appear about 1 degree lower, so the net effect of atmospheric refraction and lunar parallax will make Moon appear about 1/2 degree lower than I calculate. This partially accounts for optimistic crescent predictions.

Using the above methods and criteria I computed when the lunar crescent ought to be visible from Jerusalem at sunset at the beginning of the first and second days of Rosh Hashanah (click here or on the graphs below to view a high-quality PDF 91KB):

For the present era, the crescent ought to be visible from Jerusalem at sunset on the first day of Rosh Hashanah in only about 1/6 of years, but this proportion will more-or-less steadily increase in coming years, reaching 50% of years a few centuries after the year 10000. Of the visible crescents, most occur in years where Rosh Hashanah is delayed by two days, and the new lunar crescent will never be visible when Rosh Hashanah is not postponed — at least not until beyond the year 9000.

Naturally, the crescent is much more likely to be visible at sunset on the second day of Rosh Hashanah: in almost 2/3 of all years at present (increasing to almost 90% of years by year 10000), including almost all two-day postponement years, almost 2/3 of the one-day postponement years (increasing to almost all such years by year 10000), and more than 1/4 of the non-postponed years (increasing to almost 3/4 of such years by year 10000).

Even though the postponements of Rosh Hashanah do make it more likely that the new lunar crescent will be visible, it would be hard to defend any claim that that was an original consideration in the development of the postponement rules, because in the era of Hillel ben Yehudah the crescent ought to have been visible on the first day of Rosh Hashanah in only about 16% of years (although they should have seen the crescent in more than 60% of years on the second day of the holiday). Furthermore, they would not have known that the crescent would more often be visible in distant future years.

In comparing actual lunar conjunctions to the moments of the traditional molad, where the molad is expressed in the traditional manner as the number of hours since sunset, which I call Talmudic Temporal Time, it is necessary to subtract 6 hours from the molad moment so that it is in terms of the civil time used for the actual lunar conjunctions.

The reason for the distant future increases in crescent visibility is that the molad interval is now 2/3 second longer than the actual mean duration of lunations (mean synodic month). The mean synodic month is continuing to get shorter at a steady rate of about 1/3 mean solar second per thousand years, due to tidal slowing of the Earth rotation rate, which was not recognized until the beginning of the 20th century and then was accurately measured only after the advent of Atomic Time (1955) and Laser Lunar Ranging (1969). This progressive shortening of the mean synodic month is causing an accelerating lateward drift of the molad, such that future moladot will occur at progressively later moments with respect to the actual mean lunar conjunctions. In the era of Hillel ben Yehudah (Hebrew year 4119 = Julian 358 AD), the molad of Tishrei was about 4+1/5 hours later than the actual mean lunar conjunctions. In the present era that molad is about 6 hours late. By year 8000 it will be 11+3/4 hours late, and by year 10000 it will be about 21+1/3 hours late. The probability of the crescent being visible on Rosh Hashanah (or on any Rosh Chodesh, especially for months that follow a postponed Rosh Hashanah) progressively increases as the molad gets later and later with respect to the actual lunar conjunctions. Please see my analysis of the molad drift, what is causing it, how it varies between months and over the years, and what can be done to adjust the molad arithmetic, presented on my Hebrew Calendar studies web page.

The Net Effect of the Rosh Hashanah Postponements

During the development of the Rectified Hebrew Calendar, I experimented with allowing any month to have 29 or 30 days, whichever yielded the best alignment relative to the molad. I then compared the date agreement against the Traditional Hebrew Calendar, and found that the best agreement was obtained by adding 24 hours to the molad moments used for the Rectified Hebrew Calendar, and this offset was optimal for both the traditional molad and the progressive molad. This observation suggests that the net effect of the Rosh Hashanah postponements, on average, is the equivalent of delaying month starts by 24 hours relative to the molad moment. This is an interesting observation, because in the present era, as mentioned above, the earliest time that the new lunar crescent can be seen is 24 hours after the actual lunar conjunction. I therefore conclude that the net effect of the Rosh Hashanah postponements is to prevent the old lunar crescent from still being visible when a month starts, and to enhance the likelihood that the new lunar crescent will be visible when months start (but not before it starts).

Obviously there are exceptions to this average net effect, because the nominally alternating 29- or 30- day fixed month lengths of the Hebrew Calendar can get out of phase with respect to the actual lunar conjunctions, especially when there are 3 deficient or full months in a row (when Cheshvan and Kislev are either both deficient 29-day or both full 30-day months), and there is the superimposed ±14 hours of periodic variability of the actual lunar conjunctions relative to the secular mean lunar conjunction moments. (The word secular, derived from the Latin saeculum, in this context refers to a variation that spans centuries, that is the progressively shorter mean synodic month).

When people first study the traditional Hebrew calendar, they are always surprised to learn that the traditional molad moments that are announced during the Sabbath morning synagogue services prior to the start of each new month (except Tishrei) actually have nothing to do with the Hebrew calendar itself, for it is only the molad of Tishrei that has any relevance to the calendar. Most people mistakenly belief that months start on the day of the molad, or on the day after the molad, but that would have required allowing any month to have 29 or 30 days. The establishment of fixed lengths for most Hebrew calendar months was a major side-effect of adopting the Rosh Hashanah postponement rules, because allowing 2-day postponements of Rosh Hashanah created a constraint precluding variable month lengths, for otherwise such a postponement could have extended the month of Elul to 30 or 31 days with truncation of the following Tishrei to 29 or only 28 days!

Summary

The molad zakein rule is analogous to the ancient practice of refuting the testimony of witnesses if the calculated molad moment was at or after Noon, the disallowed weekday postponement rule serves ritual convenience, and the 3rd and 4th rules prevent impossible year lengths that would otherwise be caused by the effect of the molad zakein rule on Rosh Hashanah of the prior or coming year, respectively. The net effect of the Rosh Hashanah postponement rules is to cause months to start on average one day after the molad moment. The establishment of fixed lengths for most Hebrew calendar months was a major and necessary side-effect of formally adopting the Rosh Hashanah postponement rules.

Updated 9 Tammuz 5782 (Traditional) = 9 Tammuz 5782 (Rectified) = Jul 5, 2022 (Symmetry454) = Jul 5, 2022 (Symmetry010) = Jul 8, 2022 (Gregorian)