בס"ד

Traditional (Fixed Arithmetic) Hebrew Calendar (הלוח העברי הקבוע)

**by Dr. Irv Bromberg, University of Toronto, Canada **

[Click here to go back to the **Hebrew Calendar Studies** home page]

Executive Summary

**Traditional and Scientific Sources****The Leap Rule and Mean Year of the Traditional Hebrew Calendar****Sources for the Hebrew Calendar Intercalation Criteria****A Simple Arithmetic Estimate of the Hebrew Calendar Drift Rate****The Reference Meridian****Methods for Computing Equinox Moments****The Northward Equinox in Relation to the Sunset at the Start of***Nisan***The Northward Equinox in Relation to the***Molad*of*Nisan***The Southward Equinox in Relation to the***Molad*of*Tishrei***Ancient History of***Tekufat Shmuel*and the*Molad*of*Nisan***Proposed Accurate Leap Rule for the Hebrew Calendar****Hebrew Calendar**Equinox Jitter

Executive Summary

Refer to the sections below for sources, references, details, explanations, and charts.

*Rabbi Shlomo Yitzhaqi*(רבי שלמה יצחקי), known as*Rashi*(רש"י), is the only traditional rabbinic authority whose interpretation ofGuard the month of

(שָמוֹר אֶת חדֶשׁ הָאָבִיב) is consistent with the traditional Hebrew calendar and also astronomically and calendrically valid.*Aviv*- According to
*Rashi*, the spring equinox moment should occur before the*omer*offering (first barley) on the 16^{th}of*Nisan*[before Noon in Jerusalem], otherwise the year should be intercalated (by renaming the 29-day previous month from*Adar*to*Adar Sheini*and inserting the 30-day leap month*Adar Rishon*before*Adar Sheini*). - To minimize its seasonal drift, the Hebrew calendar mean year should closely approximate the mean spring equinox year, but
**is actually more than 6 minutes and 25 seconds too long.** - Relative to the spring equinox the Hebrew calendar lags another day behind per 224 years (total now more than 7 days late), or
^{1}/_{19}of a*molad*interval per 3+^{1}/_{2}centuries (this special time unit is the__key__to correcting the drift). - Because each month starts within a day or two after its
*molad*moment, Hebrew calendar dates can’t be late by a week or so. - Instead, the calendar has a substantial proportion of dates
one month late

: Presently in years 8, 19, 11 (these leap years have the earliest equinoxes) of each 19-year cycle the insertion of the leap month causes Noon on the 16^{th}of*Nisan*to fall more than 30 days after the equinox (so intercalation could have been postponed to the next year).**In these years the Hebrew calendar lags one month late for 13 months from the start of***Adar Rishon*until the end of*Shevat*. *Soon*year 3 of 19, the leap year with the next earliest equinox, will permanently join years 8, 19, and 11 as one month late for 13 months from the start of*Adar Rishon*until the end of*Shevat*.**Urgent need to reform the Hebrew calendar: Today almost 17% of**__all__Hebrew calendar months are__one month late__, occurring in blocks of 13 months, and soon more than 22% will be late.- Consequences: Barley is ready for harvest near the spring equinox, and the longer it is left standing the more will be lost due to seeds dispersal, damage and consumption by insects, rodents, and birds, and in Israel there is a risk of crop devastation due to later spring hard rains (
*malkosh*, מלקוש). When the Hebrew calendar is late by a month the barley harvest may be unnecessarily delayed by 30 days, which may have an agricultural and economic impact. Furthermore,**the following festival of**.*Sukkot*will be exceptionally late in the autumn, cold, and wet - Correcting the drift: Every ≈ 3+
^{1}/_{2}centuries, omit an*octaeteris*(group of 8 years with 3 leap) from the sequence of years to shift the calendar earlier by^{1}/_{19}of a*molad*interval. There probably were 2 ancient Jewish precedents for this intervention. - The 353-year leap cycle containing 4366 lunar months, including 130 leap months, is an accurate and convenient way to
omit an*automatically**octaeteris*near the middle of each cycle. - If used with the traditional
*molad*the mean year of the 353-year cycle would be only about 5 seconds too long, but a progressively shorter*molad*interval that closely approximates the mean lunar cycle would shave off those few extra seconds. - An alternative accurate leap cycle with a superb fixed lunar cycle has 1803 years and 22300 lunar months, including 664 leap months, and
omits an*automatically**octaeteris*5 times per cycle. Its mean year is only about one second too long, its mean month is less than^{1}/_{2}second too short, and because each cycle has a whole number of weeks and lunar cycles its full repetition cycle is only 1803 years (compare with the traditional**689472-year**full repetition cycle). - Unlike reform of a solar leap day calendar, which must either skip or repeat some calendar dates when any reform is adopted, for a lunisolar calendar it’s easy to schedule use of the corrected leap rule to start during a year that matches the traditional calendar, thus avoiding any date jump or repeated dates.

The following traditional sources are cited by page or chapter and verse in the text where referred to:

*Torah*and commentaries as cited in the discussion.- The Babylonian Talmud (
*Talmud Bavli*, תלמוד בבלי). - The Jerusalem Talmud (
*Talmud Yerushalmi*, תלמוד ירושלמי). *Rambam’s*Sanctification of the New Month

(*Hilchot Kiddush HaChodesh*, הלכות קידוש החודש).*Rambam’s*Book of Service

(*Sefer Avodah*, ספר עבודה,*Hilchot Timidin OuMusafin*, הלכות תימידין אומוסאפין).

- Jean Meeus,
__Astronomical Algorithms__, 2^{nd}edition, Willmann-Bell Inc., Virginia, 1999. - Jean Meeus, More Mathematical Astronomy Morsels, Willmann-Bell Inc., Virginia, 2002.
Calendrical Calculations

(3^{rd}[2008] and 4^{th}[2018] editions, hereinafter abbreviated as**CCUE**) by Edward M. Reingold and Nachum Dershowitz, see: <http://www.cs.tau.ac.il/~nachum/calendar-book/index.shtml>.- NASA Eclipse Web Site: Polynomial Expressions for Delta T (ΔT) at <https://eclipse.gsfc.nasa.gov/SEhelp/deltatpoly2004.html>.
- Other web pages on this web site — please follow the links!
- Links to external sites, especially Wikipedia, are used for many astronomical and scientific terms — please follow the links!

- SOLEX for numerical integration, which is arguably the
gold standard

for celestial mechanics, and which is easy to do using SOLEX version 9.1 (or later: this version was the first that could automatically find and save equinox and solstice moments), a free computer program written by Professor Aldo Vitagliano of the Department of Chemical Sciences at the University of Naples Federico II, Italy. The SOLEX home page is at <http://www.solexorb.it/>. SOLEX is validated against the ephemerides of the NASA Jet Propulsion Laboratory (JPL), but SOLEX is more accurate because it can use hardware extended floating point precision (80 bits instead of the standard 64-bitdouble precision

), can integrate to higher order (if the user is willing to wait for the extra calculations), can take solar oblateness and solar mass loss into account (JPL ignores mass loss because they limit their numerical integrations to ±3000 years, sometimes less), and can take more asteroids and comets into account (although I allowed SOLEX to ignore these, for faster performance). SOLEX numerically integrated Terrestrial Time moments of equinoxes and solstices (based on ecliptic solar longitude) for ±10000 years were saved in a database for use by*Kalendis*. *Kalendis*is my freeware calendrical and astronomical calculator, available here. For the astronomical evaluations herein I used my private version that has additional functions and subprograms for executing a variety of special-purpose calculations and generating specialized reports, and that can use the numerically integrated database of equinoxes and solstices imported from SOLEX.*Kalendis*uses many algorithms and functions from Jean Meeus and CCUE, cited above. See also my

web page, here: <http://individual.utoronto.ca/kalendis/hebrew/validations.htm>.*Kalendis*Astronomy Validations- I developed and used numerous Microsoft Excel workbooks with VBA macros as cited in the text with download links.
- For exact arbitrary precision calculations and mathematical functions: <https://www.wolframalpha.com/>.
- My favorite calculator app has a very comprehensive set of functions, handles expressions, and can perform exact calculations, including fractions:
Qalculate!

<http://qalculate.github.io>. - See below for on-line (click here) Continued Fraction calculators.

The long-term alignment and synchronization of any calendar with respect to the solar year and therefore the seasons depends on the difference between the mean solar year and the calendar mean year. It is impossible to design a calendar without a leap rule, because the length of the solar year is not a whole number of days.

The traditional fixed arithmetic Hebrew calendar (לוח הקבוע) is considered to be an approximation to both the lunar month and the solar year, a lunisolar calendar. Its lunar month approximation is known as the *molad*, please see my web page at <http://individual.utoronto.ca/kalendis/hebrew/molad.htm> for an astronomical analysis of the traditional *molad*. Its calendar year contains either 12 or 13 months that are either 30 or 29 days long. A year that has 12 months is a non-leap year

(common year, also known as a simple year

in Hebrew: שנה פשוטה). A year that has 13 months is a leap year

(also known as a pregnant year

in Hebrew: שנה מעוברת). The leap month, which is the 30-day *Adar Rishon* ('אדר א), is inserted between the months of *Shevat* (שבט) and *Adar* (אדר, which is renamed to *Adar Sheini* 'אדר ב) according to a permanently fixed 19-year cycle.

The 19-year (235 lunar months) cycle that is generally known as the Metonic cycle was published by the Greek astronomer Meton of Athens in the year 432 BC, but was already known to ancient Babylonian and Chinese astronomers. In 19 non-leap years there are 12 × 19 = 228 regular months, so this cycle requires 235 – 228 = 7 leap years per cycle to make up the full complement of 235 months. Traditionally the leap years of each Hebrew 19-year cycle are years 3, 6, 8, 11, 14, 17, and 19 in that cycle. The leap status of any given year is easily calculated as follows:

It is a Leap Year only if the *remainder* of ( 7 × *HebrewYear* + 1 ) / 19 is less than 7.

The intervals between leap years can be either 3 or 2 years.

There are five 3-year intervals and two 2-year intervals per 19-year cycle.

If a leap year *remainder* is less than 2 then the next leap year will be 2 years later, otherwise 3 years later.

With 7 leap years per 19-year cycle, the average interval between leap years = ^{19}/_{7} = 2+^{5}/_{7} years = 2.714285... years.

To calculate the Hebrew calendar mean year one needs to know the average month length. All of the Hebrew calendar month lengths are permanently fixed, except for *Cheshvan* and *Kislev*, which each can have either 29 or 30 days, depending on the exact length of that calendar year. Over the long term, however, the mean Hebrew month length equals the **traditional molad interval**, which is 29 days 12 hours and 793 parts = 29 +

Why Divide Hours into 1080 Parts?at <http://individual.utoronto.ca/kalendis/hebrew/chelek.htm>.

Thus the mean length of the Hebrew calendar year equals the number of months per cycle (12 regular months per year plus 7 leap months per cycle equals 235 months in 19 years) multiplied times the average month length, all divided by the number of years per cycle:

= 235 months × ( 29+^{13753}/_{25920} days ) / 19 years

= 365+^{24311}/_{98496} days

= 365 days 5 hours 55 minutes 25+^{25}/_{57} seconds, and since each *chelek* = ^{10}/_{3} seconds...

= 365 days 5 hours 55 minutes 7+^{12}/_{19} *chalakim* (= 365 days 5 hours 997+^{12}/_{19} *chalakim*), and since each *rega* = ^{1}/_{(4×19)} or ^{1}/_{76} *chelek*...

= 365 days 5 hours 55 minutes 7 *chalakim* 48 *regaim*

= 365.246822205977907732293697... days (the 18 overscored digits repeat...)

≈ 365.246822206 days per Hebrew calendar mean year

The denominator of the exact fractional calendar mean year (365+^{24311}/_{98496} days) indicates that it takes 98496 Hebrew calendar years to accumulate a whole number of days (35975351), and because that is one day less than a whole number of weeks it therefore takes 7 times longer = 98496 × 7 = 689472 years to complete a full arithmetic repeat cycle of the traditional Hebrew calendar. Click here for more information about this profoundly long repeat cycle.

Restated in terms of traditional *molad* intervals, the mean length of the Hebrew calendar year = 235 / 19 = 12+^{7}/_{19} *molad* intervals per year.

The most often cited Hebrew calendar intercalation criterion comes from the **first commandment** in the *Torah*:

שָמוֹר אֶת חדֶשׁ הָאָבִיב

Guard the month of, and bring a paschal-lamb offering (Avivpesach) forhashem, your God, because in the month ofAvivhashem, your God took you out of Egypt, at night. (Devarim/ Deuteronomy 16:1)

In the language of the *Torah*, *aviv* always refers to the *omer* (עומר) offering of the first of the new barley crop in the Holy Temple (בית המקדש הקדוש), whereas in modern Hebrew in Israel it refers to the spring season, and the word *pesach* (פסח) always refers to the paschal-lamb offering, or *Korban Pesach* (קרבן פסח), whereas the phrase *Chag ha-Matzot* (חג המצות) or Feast of Unleavened Bread

refers to *Passover* (חג הפסח in modern Hebrew).

In the era of the ancient *observational* Hebrew calendar, all months tended to start at least one day after the lunar conjunction, because the earliest that a month could start was the day upon which qualified witnesses reported a validated sighting of the first visible lunar crescent after sunset, which astronomically can occur no earlier than about 24 hours after the conjunction. Even with the fixed arithmetic Hebrew calendar of today months tend to start about a day after the *molad*, because on average the *Rosh Hashanah* postponement rules cause a one day delay.

Near the present era, **the length of each half of the lunar cycle varies over a range of about 41 hours**, from a minimum of about 13 days and 21+^{2}/_{3} hours to a maximum of about 15 days and 14+^{2}/_{3} hours, with an average or median of about 14 days and 18+^{1}/_{3} hours. **Amazingly, the periodic astronomical variation ranges of the waxing or waning half lunar cycle lengths are about 3 times greater than the approximately 13+ ^{1}/_{2} hour periodic variation range of the duration of the full lunar cycle**, because when one half cycle gets longer the other half cycle gets shorter by a similar amount, so their sum for the full cycle cancels about

Lunar Half-Cycles: Triple the Periodic Variations!on my

Length of the Lunar Cycleweb page at <http://individual.utoronto.ca/kalendis/lunar/>.

Traditionally, the earliest moment for offering the *Korban Pesach* was at ^{1}/_{2} hour after Noon on the 14^{th} of *Nisan* (ניסן). The counting of days of the month starts from one, so this was the 13th day from the sunset at the start of *Nisan*, or 13 days and 18+^{1}/_{2} hours after the mean sunset at the start of *Nisan*, which when taken together with a 24 hour delay is close to ^{1}/_{2} of a *molad* interval (14 days 18 hours 22 minutes and 38 *regaim*), in other words quite close to the moment of the mean lunar opposition (full moon) or average duration of the waxing half of the lunar cycle. The end of the first day of *Passover*, nominally ^{1}/_{2}+5+24 = 29+^{1}/_{2} hours after the earliest *Korban Pesach*, or exactly 15 days after the mean sunset at the start of *Nisan*, when taken together with a 24 hour delay is always beyond the maximum duration of the waxing half of the lunar cycle.

*Rabbi Shlomo Yitzhaqi* (רבי שלמה יצחקי), known as *Rashi* (רש"י), wrote a comment on Guard the month of

, explaining that the Hebrew word *aviv**aviv* (אביב) refers to its ancient meaning, barley, in that a measure of fine flour prepared from newly ripened freshly harvested barley had to be available for the *omer* offering in the Holy Temple in Jerusalem on the second day of *Passover*. Nobody would reap new grains until after the *omer* had been offered. In the climate of ancient Israel, barley was planted early in the autumn and was ready for harvest near the spring equinox. *Rashi* wrote that they used to intercalate the calendar year if otherwise the required measure of newly ripened barley wouldn’t be ready in time.

According to *Rambam* in *Sefer Avodah* (ספר עבודה), *Hilchot Timidin OuMusafin* (הלכות תימידין אומוסאפין), chapter 7, starting from paragraph 3, the barley had to grow naturally without human irrigation in one of two specially-designated fields that were used alternately, the other field lying fallow. (Today the climate in Israel is hotter and drier, which could allow barley to ripen before the spring equinox, or could cause the crop to fail without irrigation.) Three sheafs of barley were harvested, freshly cut at night from standing grain at the beginning of the 16^{th} of *Nisan* (even if *Shabbat*). In the Courtyard of the Holy Temple the barley was threshed and tossed to separate the seeds from the chaff, spread out for manual removal of any other debris, lightly roasted over a fire in a metal cylinder with holes that allowed the flames to reach the seeds, spread out to cool, and milled to make flour that was repeatedly sifted (13 sifters) to obtain only the finest flour to be used for the *omer* offering. The *omer* was offered after the *musaf* (מוּסַף) service of the second day of Passover (16^{th} of *Nisan*) but before *bein ha-arbayim* (בֵּין הָעַרְבַּיִם), the last service of the afternoon. This might imply that the spring equinox moment had to fall no later than the earliest time that the priests (כהנים) were ready to offer the *omer* on the 16^{th} of *Nisan*, which should have been before Noon in Jerusalem.

Certainly, in the agricultural society of ancient Israel it would have been advantageous to offer the *omer* as soon as possible in the spring season, because nobody would reap new grains until after the *omer* offering. Today, although the *omer* will not be offered again until the Holy Temple is rebuilt, *in theory* Jews continue the practice of postponing reaping of new grains in Israel until the 16^{th} of *Nisan*, so it was, is, and will always be advantageous to arrange the leap years such that the second day of Passover is as early as possible after the northward equinox (and never more than 30 days after the equinox). [*In practice*, it seems that many farmers do reap their grains before the 16^{th} of *Nisan*, but they package it in canvas bags imprinted with the date of harvest, then either they or the buyers store the grain until after the 16^{th} of *Nisan*. Many people believe that all grains have to pass through

Passover before they can be used, but this applies only to grains that were harvested before Passover.]

The Hebrew word אביב (*aviv*) can also be thought of as a combination of the word אב (*av* = father, head) and the Hebrew numerals for 12 = י"ב, referring to that month as the first of the 12 regular months of the calendar year. Indeed, the *Torah* instructed that the month of the exodus from Egypt shall be the first month of the year:

הַחדֶשׁ הַזֶה לָכֶם ראשׁ חֳדָשִים רִאשׁוֹן הוּא לָכֶם לְחָדְשֵי הַשָנָה

Hashemsaid to Moses and Aaron in the land of Egypt, saying,This month shall be for you the beginning of the months, it shall be for you the first of the months of the year.(Shemot/ Exodus 12:1)

The commentary of *Rashi* on the above sentence says that at that moment Moses and Aaron were shown the first visible lunar crescent and from that comes the tradition that each new month starts when the crescent is first seen.

The *Talmud Bavli* tractate *Sanhedrin* pages 12b, 13a and 13b contains a long debate about the criteria and rules for intercalation of the Hebrew calendar year, much of it concerning whether the day of an equinox or solstice is the last day of the season that ended or the first day day of the season that is starting, concluding that the day upon which an equinox or solstice moment falls is the first day of the new season. (This is like a birthday, in that the date of birth is the first day of a baby’s life regardless of what time of day the baby was born.) [Anyway, this long debate is moot because astronomically, in terms of solar insolation, the equinoxes and solstices each mark the __middle__ of their respective seasons (sunlight seasons

), not the beginning (thermal seasons

), and, depending on local conditions, the weather seasons

typically lag a few weeks after each equinox or solstice.] That debate continued, discussing what should be the intercalation rule with respect to the **autumn** equinox relative to the days from *Sukkot* (starts on the 15^{th} of *Tishrei* = תשרי) through *Hoshana Rabbah* (21^{st} of *Tishrei*), and concluding that if the summer season extends into the greater part of the month

, defined as 16 days, then the year should be intercalated.

Insertion of *Adar Rishon* after *Shevat* would have no effect on the timing of the autumn equinox in the same calendar year, because that equinox would occur well before the leap month, so this debate is puzzling unless it refers to the insertion of an extra month of *Elul* __before__ *Tishrei*, as was done by the Babylonians in year 17 of their 19-year cycle. Intercalating *Elul* is mentioned in the *Talmud Bavli*, tractate *Betzah* page 6a: from the days of Ezra onward we do not find

, which suggests that *Elul* ever intercalated*Elul* was indeed intercalated prior to the days of Ezra. A possible reason for intercalating *Elul* is suggested by the following sources:

Talmud YerushalmitractateSheqalpage 1:2:They do not intercalate the year either in the case of the Seventh Year or in the case of the year after the Seventh Year, but if they did so then the intercalation stands.

Talmud BavlitractateSanhedrinpage 12a:Our rabbis have taught on Tannaite authority: They do not intercalate the year either in the case of the Seventh Year or in the case of the year after the Seventh Year.

In the Seventh Year or *Shmitah* year no crops could be planted or harvested in Israel, the land had to lie fallow. In order to minimize hardship for farmers and the general community, intercalation of *Adar Rishon* was avoided where possible, so as not to extend the duration of the *Shmitah* year by an extra month. What could be done if it was reckoned that intercalation would be necessary in a *Shmitah* year? They could intercalate *Elul* just prior to the beginning of the *Shmitah* year, thus having the desired effect on the timing of the following spring season, without extending the *Shmitah* year itself.

The debate continued on *Sanhedrin* page 13b with the question as to whether the (southward) equinox in *Tishrei* (תשרי) must always occur prior to the 16^{th} or the 20^{th} day of *Tishrei*. Nevertheless, astronomically neither of these limits is possible, nor any day between them, not even the 15^{th} or 21^{st} of *Tishrei*, because there are two astronomic constraints that preclude any leap rule with regard to the equinox in *Tishrei*:

The first astronomic constraint is that the date of *Rosh Hashanah* (ראש השנה) is fixed relative to the *molad* of *Tishrei*. Therefore, with a day or two leeway, *Rosh Hashanah* must start 6 lunar months after the start of *Nisan* = 6 × the *molad* interval = a bit more than 177 days, which would place the 16^{th} and 20^{th} of *Tishrei* at about 192 or 196 days after the start of *Nisan*, respectively. In the present era in the northern hemiphere there is an average of about 92+^{3}/_{4} days between the northward equinox and north solstice, and an average of about 93+^{2}/_{3} days from the north solstice to southward equinox, so the total span from the average northward equinox of *Nisan* to the average southward equinox of *Tishrei* is about 186+^{2}/_{5} days. Therefore, if the average northward equinox of *Nisan* is at the start of *Nisan* then the average southward equinox of *Tishrei* can only be at *Yom Kippur* on the 10^{th} of *Tishrei*, with ±15 days of variation at both equinoxes (in parallel) due to the calendar’s 30-day leap month. Thus the astronomical southward equinox of *Tishrei* must actually range from the last week of *Elul* through to the 25^{th} day of *Tishrei*, and there is nothing that can be done to prevent a late equinox in *Tishrei* as long as the start of *Nisan* is aligned with the average northward equinox.

The second astronomical constraint is that for the past millennium the mean length of the southward equinoctial year has been shorter than the mean northward equinoctial year, and the former will continue to get shorter whereas the latter will remain essentially constant for the next 4 or more millennia. Therefore there is no way for any fixed arithmetic calendar to simultaneously maintain alignment relative to both equinoxes, and even a calendar employing accurate astronomical algorithms can’t possibly align with both equinoxes. It is also impossible for any fixed arithmetic calendar to approximate just the mean southward equinox alone, because the mean southward equinoctial year is relatively rapidly getting progressively shorter. For more information please see my web page The Lengths of the Seasons

at <http://individual.utoronto.ca/kalendis/seasons.htm>.

Near the end of this topic, on *Sanhedrin* page 13b:

שמור אביב של תקופה שיהא בחדש ניסן

The Others(Rabbi MeirBa'al HaNess), says R. Samuel son of R. Isaac, speak ofTekufat Nisan(spring equinox), for it is written (in theTorah, as quoted above)Guard the month of, that is, take heed that the spring equinox shall occur on a day inAvivNisan.

This statement has been quoted or echoed by many *Talmud* commentators and rabbinic authorities. Taken literally, this statement implies that the spring equinox can land anywhere in *Nisan*, which would place the **average** equinox moment midway through *Nisan* at the end of the first day of *Passover*, and if so then in about 50% of years *Passover* would start before the spring season! However, commentators added to the above sentence during the renewal of the Moon in

, meaning that the equinox should occur during the waxing half of the lunar cycle in *Nisan**Nisan*, an opinion that was also echoed by many rabbinic authorities. With this seemingly innocuous addition, however, this criterion is astronomically unattainable because the solar cycle runs independently of the lunar cycle, such that actual astronomical equinox moments can occur at any time during a lunar month. The best that be done is to ensure that the equinox moment is never later than the lunar opposition moment in *Nisan* (full moon = end of the waxing half of the lunar cycle), which will result in the equinox falling in *Nisan* in approximately 50% of years (the exact proportion depends on the calendar rules for starting the month), but in the remainder of years the equinox must fall during the last half of the prior month, which will be *Adar* in non-leap (common) years or *Adar Sheini* in leap years. Evaluation of the traditional fixed arithmetic Hebrew calendar is not illuminating in this regard, because the consequence of the accumulated drift of the equinox until the present era to more than 7 days prior to the start of *Nisan* (as will be demonstrated herein) is that currently the northward equinox occurs in *Nisan* in less than 25% of years.

On the other hand, the words of The Others can also be interpreted to mean that *Nisan* must be in the spring season (understanding the Hebrew word *tekufah* as referring to the season rather than the equinox). This alternative translation doesn’t present any astronomical difficulties, but it is rather vague, implying that it is acceptable for *Nisan* to occur anywhere in the spring season, or perhaps that the entire month of *Nisan* must be within the spring season, implying that *Tekufat Nisan* must occur before the beginning of *Nisan*.

The *Talmud Bavli* tractate *Rosh Hashanah* page 21a says:

Rav Huna bar Avin sent the following message to Rava: When you see that the winter season is stretching until the 16

^{th}ofNisan, intercalate a month into that year, and don’t worry about it, for it is writtenGuard the month of, guard the month of the first-ripened produce so that it will be in the month ofavivNisan.

The phrase until the 16

could be understood inclusively, meaning that the 16^{th} of *Nisan*^{th} of *Nisan* was part of the winter season, or it could be understood exclusively, meaning that the 16^{th} of *Nisan* must be within the spring season. *Rashi* explained as a commentary to Rav Huna and also citing The Others that if the equinox lands on the 15^{th} day of *Nisan* then the prior month of *Adar* could be made full (30 days), thereby avoiding making that year a leap year. This logic seems to be the basis for using a 16- instead of a 15-day limit, but such a maneuver would always cause the equinox to land in the waning half of the lunar cycle in *Nisan*.

The *Talmud Bavli* tractate *Eruvin* page 56a defined an equinox as the day on which Sun rises from the true east direction and sets to the true west direction (these are __not__ the same as the directions shown by a magnetic compass, because the magnetic poles wander relative to the global axial poles):

Our Rabbis taught: If a town is to be squared (for the purpose of measuring its

Shabbatlimits its irregular boundary is extended to form an imaginary square) then the sides of the square must be made to correspond to the four directions of the world: its northern side must face northward, and its southern side southward; and your guides are the Chariot (Ursa Major constellation) in the north and the Scorpion (Scorpius constellation) in the south.

Most of Ursa Major is between 30° and 40° degrees away from the celestial north pole, so its position is an extremely crude guide to the northerly direction! Scorpius is between 20° and 45° south of the celestial equator, and although it is the zodiac constellation that is furthest south, it rises in the south-east, passes across the southern sky, and sets in the south-west, so its position is an extremely crude guide to the southerly direction.

R. Jose said: If one does not know how to square a town so as to make it correspond to the directions of the world, one may square it in accordance with the seasons. How? The direction in which on a long day the Sun rises and sets is the north face. The direction in which on a short day the Sun rises and sets is the south face. At the spring and autumn equinoxes the Sun rises in the middle point of the east and sets in the middle point of the west.

This method was mentioned in *Eruvin* in the context of surveying a city to determine its orientation with respect to the 4 cardinal directions for determining the *eruv techumin* (עירוב תחומין), the limits outside the city bounds beyond which one shouldn’t travel on *Shabbat* or *Yom Tov*. Clearly it would not have permitted any calendar intercalation decision to be made in advance of the spring equinox, but it could have been used to monitor the accuracy of the *Tekufat Nisan* predictions.

In modern astronomical terms we could describe equinox days as starting with the sunrise azimuth at 90° east of north or ending with the sunset azimuth at 270° east of north or 90° west of north, but this is strictly true only at the equator. The *Talmud Bavli* asserts that true east is the horizon mid-point between the furthest northeast sunrise on the day of the north solstice (summer solstice for the northern hemisphere) and the furthest southeast sunrise on the day of the south solstice (winter solstice for the northern hemisphere), and similarly it asserts that true west is the horizon mid-point between the furthest northwest sunset on the day of the north solstice and the furthest southwest sunset on the day of the south solstice. This definition is astronomically valid only at the equator. At northern latitudes, such as Israel, because of atmospheric refraction near the horizon making Sun appear 1° to 2° higher than its true geometric position and also because of the approximately ^{1}/_{2}° solar diameter (we consider it to be daytime if any part of the solar disk is visible), and especially when observed at elevations above sea level (which causes Sun to be visible earlier as it rises and later as it sets), such as Jerusalem, the sunrise and sunset directions are always further north than they are at the equator. Consequently, at northern latitudes this method will always reckon true east to be modestly northeast of the correct direction and true west to be modestly northwest of the correct direction, and the converse applies at southern latitudes.

Another source of error in this method is the time difference between the moment of an equinox and the moments of sunrise and sunset. A true east sunrise is possible at the equator only if the equinox moment occurs near sunrise, and a true west sunset is possible at the equator only if the equinox moment occurs near sunset at the observer’s locale. **The actual astronomical moment of an equinox is the moment when Sun crosses the celestial equator going from south-to-north (northward equinox) or from north-to-south (southward equinox), and can occur at any time of day or night**.

Rather than requiring sunrise to be exactly from true east or sunset exactly to true west, one could define the first day of the northern hemisphere spring as the first day having a sunrise azimuth that is less than 90° east of north, or as the first day when the sunset azimuth is less than 90° west of north, at some specified location. Hebrew calendar days begin at sunset, therefore the sunset azimuth ought to be of observationally greater importance than the sunrise azimuth.

I am not aware of any evidence suggesting that any observational method was officially employed by the *Sanhedrin* (בית הדין הגדול) to determine whether to declare a leap year or not. The probable reason is that the *Sanhedrin* had to make equinox predictions and leap year declarations sufficiently far in advance so that pilgrims would know when to begin their journeys, so they relied on calculations instead. For more information about the traditional Jewish methods for predicting the moments of equinoxes and solstices, click here to see my web page

. *Rambam* and the Seasons**Hereinafter, I will assume that the reader is familiar with all 3 of the methods that Rambam documented.**

In Hebrew year 5746 (1986 AD), Israeli engineer Yaaqov Loewinger (יעקב לוינגר ,זכרונו לברכה), then a resident of Tel Aviv, published a Hebrew book entitled *Al ha-Sheminit* (על השמינית), which discussed why *Passover* is always more than a month later than the spring equinox in the 8^{th} year of each 19-year Hebrew calendar cycle (the same is also true of the 19^{th}, 11^{th} and 3^{rd} year of each 19-year cycle, as will be shown below). In that book he pointed out that an alternative version of the *Talmud Bavli* tractate *Rosh Hashanah* was discovered in the Cairo *geniza* (underground archive of holy books) which in the message of *Rav Huna* addressed to *Rava*, and explicitly in the commentary of *Rabbenu Hana'el*, says that **the criterion for intercalation was to insert a leap month if otherwise the equinox moment would land on or beyond 16 days after the molad**. Loewinger wrote that medieval Sephardic calendar experts, including

Rabbi Moshe ben Maimon (

), also commonly known by his Greek name, Moses Maimonides was the least ambiguous and most comprehensive traditional source for Hebrew calendar criteria and arithmetic. His written calendar specifications in *Rambam**Hilchot Kiddush haChodesh* (הלכות קדוש החדש) are so straightforward that one can implement them as exact computer program functions. Nevertheless, even *Rambam* didn’t tell us what were the specific criteria that guided *Hillel ben Yehudah* (הלל השני, Hillel II) when he established the fixed arithmetic Hebrew calendar. Referring to the *observational* Hebrew calendar, however, *Rambam* wrote:

When the court calculates and determines that the spring equinox will fall on the 16

^{th}ofNisanor later, the year is made a leap year.The month that would have been, so thatNisanis madeAdar SheiniPesachwill be in the spring season.This factor alone is sufficient for the court to make the year a leap year, other factors need not be considered.(Hilchot Kiddush haChodeshchapter 4, paragraph 2)

In other words, ** Rambam’s limit for the latest equinox was the end of the 15^{th} day of Nisan, exactly halfway through the month**.

A major difficulty that I have with all of the above and other traditional sources is that they seem to universally regard the spring equinox as the *beginning* of the spring season, yet astronomically it is at the *middle* of the spring season. How so? For non-tropical latitudes, the solar insolation determines the seasonal variations. Insolation is the amount of energy received as sunlight, hence we can speak of sunlight seasons

. The quarter of the year that receives the greatest solar insolation for a region is its astronomical summer, and the summer solstice is in the middle of that season. The quarter of the year that receives the least solar insolation for a region is its astronomical winter, and the winter solstice is in the middle of that season. The other two annual quarters are the spring, between winter and summer, with the spring equinox in its middle, and the autumn, between the summer and winter, with the fall equinox in its middle. (These quarter years aren’t equal in duration, due to the eccentricity of Earth’s precessing elliptical orbit.) Weather seasons, however, typically lag about 30 days or ^{1}/_{3} season after the astronomical seasons, because the surface temperature depends on the balance between solar insolation, reflectance, absorption, and passive radiation of energy from Earth into space. When there is ice and snow on the ground, and greatly increased cloud cover, radiative heat losses are increased due to reflection of solar radiation. Altogether, we can say that the spring weather season begins about (^{1}/_{2} season – ^{1}/_{3} season) = ^{1}/_{6} season or about ^{1}/_{2} month before the astronomical spring equinox, and likewise for the other seasons. If the latest allowable spring equinox was around the middle of *Nisan* then in effect that corresponds to a requirement that the *entire* month of *Nisan* had to be within the spring weather season.

Jewish communities inside and outside Israel used to depend on messengers sent at the beginning of most months by the *Sanhedrin* in Jerusalem to deliver news of calendar decisions. In Julian year 358 AD (Hebrew year 4119) the Roman Emperor Constantius II (who converted the Roman government and society to Christianity), wanted to prevent Christians from determining when to celebrate Easter by asking Jews when will be the date of *Passover*, so he outlawed New Moon announcements with the intent of quashing the Hebrew calendar. *Hillel ben Yehudah*, the second-last President of the *Sanhedrin* (his son was the last) responded by promulgating the fixed arithmetic Hebrew calendar (no doubt hoping that it would only be a temporary measure), which had probably been developed a century earlier in Babylonia by *Amora *Shmuel of Nehardea (Shmuel the Astronomer

) and which had since then been used internally by the *Sanhedrin* as a guide for calendar decisions.

Shmuel said:

I am able to make a calendar for the entire(diaspora.Talmud BavlitractateRosh Hashanahpage 20b)

Release of the fixed arithmetic calendar rules had to be carried out in a hurry, otherwise Jewish communities would not have known when to observe ritually significant days. When the Romans later realized that their attempt to quash the Hebrew calendar had failed, they raided the *Sanhedrin* headquarters and confiscated all property and records. After that raid the *Sanhedrin* ceased to exist, but __not__, as commonly taught to Jewish children in *cheider* and students in *yeshiva*, due to a lack of judges ordained with traditional Mosaic *smichah*. The Christian Romans censored any attempts to publish the truth, with dire consequences for the authors.

Constantius II was scathingly anti-semitic in his conversations, orders, speeches and writings, see <http://www.ccel.org/ccel/schaff/npnf214.vii.x.html> (there are many similar examples at that web site, written by Constantinius and by others), which lead to increasingly severe abuse of Jews throughout the Roman Empire and beyond. The Roman Catholic Church is the present era vestige of the Roman Empire, so there is hope that the stolen *Sanhedrin* property and records may yet today still exist in their archives deep within the bowels of Vatican City.

On the other hand, *Hillel ben Yehudah* may never have published any written documents outlining the rules of the fixed calendar, for the following reasons:

- release of the calendar had to be carried out in a hurry, so there wasn’t time for scribes to copy and verify the many necessary copies
- he was afraid that such documents would fall into the wrong hands (
*e.g.*, the Romans) - he only intended the fixed calendar as a temporary measure, but documenting it would have made it seem permanent

So he sent out into the diaspora his most knowledgeable colleagues and students, entrusting them with the rules of the calendar, and compelling them not to reveal those rules to the Romans. The calendar became an orally transmitted tradition, and the diaspora became greatly enriched by the dispersal of so many sages into its communities, further strengthening and ensuring the endurance of the Jewish faith.

But there was a side-effect: because so many wise colleagues and students were sent away, and only the elderly who were unable to move to the diaspora were left behind, there no longer were enough sages with *smichah* to carry on the *Sanhedrin* for more than a generation. If correct, *Hillel ben Yehudah* considered the promulgation of the fixed Hebrew calendar to be so important that he traded off the future of the *Sanhedrin* in order to ensure the future of Judaism. This decision is consistent with the *Torah* having given primary importance of the calendar by making guard the month of

the first commandment to the children of Israel.*aviv*

The oral tradition continued, and when the *Talmud* was later put down in writing the details of the fixed calendar rules were not included because the calendar was still properly considered to be only temporary, the Christians continued to inspect and censor their works, and because the sages were afraid that if the details were included in the *Talmud* then it would become a permanent calendar.

Much later, *Rambam* published the details of the calendar because he was afraid that the rules would otherwise be lost or corrupted, or that calendar disputes would arise. *Rambam*’s having done so has in a way given a degree of permanence to the Hebrew calendar that was probably not intended by *Hillel ben Yehudah* and his colleagues.

A non-traditional theory, suggesting that there was a gradual evolution to today’s fixed arithmetic Hebrew calendar, based primarily on alternative interpretations of *Talmud* sources, and relying heavily on unofficial sources such as Jewish tombstones and contractural documents with Hebrew dates (which are not reliable sources for the purposes of technical chronology), was published by J. Jean Ajdler under the title Rav Safra and the Second Festival Day: Lessons About the Evolution of the Jewish Calendar

, in *Tradition* 2004 Winter; 38(4): 3-28, and is available to subscribers from the journal’s web site at <http://www.traditiononline.org/>, and also by the same author his freely available manuscript A Short History of the Jewish Fixed Calendar: The Origin of the Molad

published in Volume 20 (Winter 2015) of *Ḥakirah* (The Flatbush Journal of Jewish Law and Thought), pages 133 to 190.

There are many other theories about the origin of the fixed calendar, but, as will be shown below, **it was only in the era of Hillel ben Yehudah that it simultaneously exactly matched the Talmud equinox criteria with respect to both the actual astronomical spring equinox as well as the traditional equinox approximation of Rav Adda bar Ahavah.**

In Jewish law (*halachah*), a ruling by any authority can only be changed or overruled by an authority of equal or higher authority. Therefore if the fixed arithmetic Hebrew calendar was a ruling of the *Sanhedrin*, as is traditionally held, then it can only be changed by a present era or future *Sanhedrin*, whereas if the calendar was a gradual evolution then it might only be necessary to obtain the consensus of respected rabbinic authorities in Israel.

As soon as *Hillel ben Yehudah* promulgated the fixed arithmetic calendar, part of the Jewish population immediately accepted it, and their descendents comprise most of the Jewish population of today, but some Jews rejected it, continuing to follow an observational calendar, although the latter have few descendents today. Similarly, if any authority approves changes to the modern Hebrew calendar, regardless of the merits of those changes, there will be those who will accept and follow, and others who will reject change and continue using the unmodified calendar, so again the new calendarists

will diverge from the old calendarists

. That is one reason why some modern rabbinic authorities are not interested in considering any modification of the Hebrew calendar. On the other hand, with a charismatic and respected rabbinic leader, modern communication and the internet may make it possible to rapidly develop a global consensus for change, thus not only avoiding a Jewish split but in fact promoting worldwide Jewish unity (on at least this issue).

Summarizing the traditional rabbinic sources relevant to Hebrew calendar intercalation, they imply *some relationship* between the spring equinox and the month of *Nisan*, but the definitive relationship is unclear. We have no original record from the era of *Hillel ben Yehudah* to indicate how he understood these sources, what other information he may have had available, what criteria he used when fixing the calendar, or which method(s) he used to determine equinox moments. In the following sections, I will use traditional calculations (as published by *Rambam*) as well as modern astronomical analyses to deduce what those criteria must have been.

Given that the Hebrew calendar leap rule has *something* to do with the **northward equinox** (spring equinox of the northern hemisphere), the calendar’s astronomical drift can be estimated by comparing the calendar mean year to the **mean northward equinoctial year** (average number of elapsed days and fraction of a day from one spring equinox to the next spring equinox of the northern hemisphere).

As shown on my web page The Lengths of the Seasons

at <http://individual.utoronto.ca/kalendis/seasons.htm>, **in the present era the average length of the northward equinoctial year is 365 days 5 hours 49 minutes 0 seconds = 365 + ^{5}/_{24} + ^{49}/_{1440} = 365 + ^{349}/_{1440} mean solar days**. Subtracting this from the Hebrew calendar mean year of 365 +

Therefore, if we assume that the solar year will remain constant in length (it won’t, but we only need an approximation at this point) then the number of years to accumulate each additional day of drift later than the equinox is the inverse of the fraction ^{2197}/_{492480}, which is ^{492480}/_{2197} or about **224+ ^{1}/_{6} solar years per day of drift**. This implies that since the era of

Another way to look at it is that if the leap rule were doing its job properly then the Hebrew calendar and the solar year would return to the same relative timing at the beginning of each 19-year cycle. In 19 Hebrew years there are 235 months, having a mean month length equal to the traditional *molad* interval. The difference between 19 mean northward equinoctial years and 19 Hebrew calendar mean years is therefore:

235 × (

moladinterval) – 19 × (mean northward equinoctial year)= 235 × (29 days 12 hours and 793 parts) – 19 × (365 days 5 hours 49 minutes 0 seconds)

= 235 × (29 +

^{12}/_{24}+^{793}/_{25920}) – 19 × (365 +^{5}/_{24}+^{49}/_{1440})=

^{2197}/_{25290}of a day = 2197 parts = 2 hours 2 minutes and 1 part

Therefore the number of 19-year cycles required to accumulate one full day of drift is the fraction ^{2197}/_{25290} taken as its inverse = ^{25290}/_{2197} = 11+^{1753}/_{2197} or just under 11+^{4}/_{5} cycles, which when multiplied by 19 years per cycle again yields ^{492480}/_{2197} or again about 224 solar years per day of drift, as above.

**Note, however, that because each Hebrew month starts within a day or two after its molad moment it is impossible for any Hebrew date to be 7 or so days late. The drift estimate of more than 7 days is in fact the average lag of the Hebrew calendar, relative to the mean northward equinox. It implies that the Hebrew calendar spends a substantial amount of time being one month late, typically after the premature insertion of a leap month (in the sense that it delayed Noon in Jerusalem on the 16^{th} of Nisan beyond 30 days after the northward equinox, hence there was no need to intercalate that year) for 13 months from Adar Rishon until the end of Shevat.**

It is logical, consistent, and valid to count the leap month,

Adar Rishon, as a month in which the Hebrew calendar is late because as soon as the leap month is inserted all ritual and calendrical events (Shabbat Zachor,Ta'anit Esther,Purim,Shushan Purim,Shabbat Parah), anniversaries, birthdays, and memorial days (yahrzeits) that were to occur inAdarare immediately postponed by 30 days toAdar Sheini.After

Shevatthe traditional calendar says that the month isAdarwhereas the rectified calendar says that the month isAdar Sheini, but all ritual and calendrical events match so those months should be counted as matching.

We can estimate the proportion of Hebrew months that are presently one month late

as the accumulated drift divided by the 30-day length of the leap month = (^{1649}/_{224}) / 30 = ^{1649}/_{6720} ≈ 24.5% of months or 57 to 58 months per 19-year cycle! These late

months are not randomly distributed or otherwise scattered about, but always occur as consecutive 13 month blocks that start from *Adar Rishon*.

The actual astronomical drift of the Hebrew calendar was a bit faster in the era of *Hillel ben Yehudah*, because at that time the mean northward equinoctial year was a few seconds shorter than it is today. After approximately Hebrew year 10500 the Hebrew calendar astronomical drift rate will accelerate, because the date of perihelion (the point in Earth’s elliptical orbit that is closest to Sun) slowly advances through the solar year and by then will have advanced past the northward equinox, causing progressive shortening of the mean northward equinoctial year. These astronomical trends are documented and explained on my web page entitled The Lengths of the Seasons

at <http://individual.utoronto.ca/kalendis/seasons.htm> and in my leap cycle web page section entitled Calendar Seasons: Stable Points in the Solar Cycle

at <http://individual.utoronto.ca/kalendis/leap/index.htm#CS>. To calculate the actual calendar drift more accurately, taking these variations into account, it is best to numerically integrate the changes over time (see below).

The actual proportion of Hebrew months that can properly be considered to be one month late

depends on the specific criteria that *Hillel ben Yehudah* intended when he fixed the calendar, which can be deduced arithmetically (see below).

Note that the so-called

length, which doesn’t refer to the mean northward equinoctial year and is significantly shorter, is almost universally yet mistakenly used by others in similar calculations, and **mean tropical year****is the wrong year length to use for Hebrew calendar purposes**, because it is too short, it is valid only in terms of Terrestrial Time (passes at the same rate as International Atomic Time, unaffected by tidal slowing of the Earth rotation rate) rather than the mean solar time that is appropriate for calendars, and because its meaning, calculation, and even its definition are rather ambiguous (for further information see <http://en.wikipedia.org/wiki/Tropical_year>). Astronomers most commonly employ any one of several published cubic or higher-order polynomials to estimate the length of the mean tropical year (in terms of atomic time), but that can’t possibly be valid in the long term, because the length of the mean tropical year must vary periodically, in parallel with periodic variations of the Earth axial tilt (obliquity), and no polynomial can approximate a periodic variation except over a time range that is shorter than a single period. Some, including myself, have proposed periodic functions (*e.g.*, based on either *sine* or *cosine*) to estimate the length of the mean tropical year (or to estimate the Earth axial tilt), but such approximations are unavoidably inexact, because the mean tropical year variation range depends on the oblateness of Earth’s not-quite-spherical shape (Earth is presently axially flattened by a factor of about ^{1}/_{298}), which varies with the poorly predictable mass of polar ice. For further information about the mean tropical year, see my web page entitled The Lengths of the Seasons

at <http://individual.utoronto.ca/kalendis/seasons.htm>.

Some prefer to see the drift expressed in terms of the Gregorian calendar, as if that were somehow the ideal

calendar cycle. The Gregorian calendar mean year is itself currently slightly too long, so relative to that the drift of the Hebrew calendar will seem a bit better. The Gregorian calendar inserts a leap day every 4 years except for 3 of 4 centurial years, so there are 97 leap years per 400-year Gregorian cycle. Therefore the Gregorian calendar mean year = 365+^{97}/_{400} = 365.2425 days = 365 days 5 hours 49 minutes and 12 seconds, which is 12 seconds longer than the mean northward equinoctial year. This discrepancy is small enough that numerical integration is necessary to properly estimate the mean Gregorian calendar drift rate. For further information, please see my Solar Calendar Leap Rules

web page at <http://individual.utoronto.ca/kalendis/leap/>, which also includes several examples of leap rules that are superior to and simpler than the Gregorian leap rule.

Relative to the Gregorian calendar mean year the traditional Hebrew calendar mean year is always exactly ( 365 + ^{24311}/_{98496} ) – ( 365 + ^{97}/_{400} ) = ^{10643}/_{2462400} days too long. The Hebrew calendar drift relative to the Gregorian calendar is exactly the inverse of that fraction = ^{2462400}/_{10643} or about 231+^{1}/_{3} Gregorian years per day of drift.

The Julian calendar, which has a calendar mean year identical to that of *Tekufat Shmuel* = 365 days and 6 hours = 365+^{1}/_{4} days, is presently 11 minutes longer than the mean northward equinoctial year, so it is drifting at the relatively rapid rate of ^{1440}/_{11} = 130+^{10}/_{11} mean northward equinoctial years per day of drift. The Julian calendar mean year is exactly 4 minutes and 34+^{32}/_{57} seconds = ^{313}/_{98496} of a day longer than the Hebrew calendar mean year, so its average drift rate **toward dates that average progressively later in the Hebrew calendar year** is the inverse of that fraction = ^{98496}/_{313} = 314+^{214}/_{313} ≈ 314+^{2}/_{3} Hebrew years per day of drift.

In the context of Hebrew calendar equinox drift there is an unnamed time unit that is of special significance: ^{1}/_{19} of a *molad* interval = 1+^{272953}/_{492480} day = 1 day 13 hours 18 minutes 1 part and 72 *regaim*. This time unit is special because, as will be shown below, for each ^{1}/_{19} of a *molad* interval that the spring equinox drifts __earlier__ in the Hebrew calendar year, that amount of drift can be corrected by simply omitting an *octaeteris* (a group of 8 years that __begins__ after a leap year, __ends__ on a leap year, and __contains__ 3 leap months, for a total of 99 months) from the sequence of leap years, which will shift the equinox __later__ by ^{1}/_{19} of a *molad* interval. (Classically, the 8-year *octaeteris* contained alternating 30-day and 29-day months, plus 3 leap months having 30 days, yielding an excessively long mean year of exactly 365+^{1}/_{4} days and an unduly short mean month of 29+^{17}/_{33} days.)

It is important to know how many years it takes for a drift of ^{1}/_{19} of a *molad* interval to accumulate, calculated as follows:

Excess length of the Hebrew calendar year relative to the mean northward equinoctial year from above =

^{2197}/_{492480}days.Divide

^{1}/_{19}of amoladinterval by the excess length = (1+^{272953}/_{492480}) / (^{2197}/_{492480}) =^{765433}/_{2197}years ≈ 348.4 years ≈ 3+^{1}/_{2}centuries.

It is easy to set up arithmetic to automatically make such an adjustment if the number of years equals a multiple of 19 years minus 8 years, in other words a multiple of 19 years plus 11 years, and it will work especially well if that many years contains a whole number of mean lunar cycles. The nearest number of years that meets these criteria is (9×19)+11+(9×19) = 353 years, which contain almost exactly 4366 mean synodic lunar months (or 4719 sidereal, or 4738 draconic, or 4679+^{1}/_{10} anomalistic lunar months), making a 353-year leap cycle essentially ideal for the Proposed Accurate Leap Rule for the Hebrew Calendar

that will be presented at the end of this web page.

A portion of the excess length of the Hebrew calendar mean year is due to the excess length of the traditional *molad* interval, presently amounting to about ^{5}/_{9} of a second per month (see <http://individual.utoronto.ca/kalendis/hebrew/molad.htm> and The Length of the Lunar Cycle

at <http://individual.utoronto.ca/kalendis/lunar/>). If we were to deduct ^{5}/_{9} of a second from the traditional *molad* interval then the Hebrew calendar mean year would be:

= [ 235 months × ( 29+^{13753}/_{25920} days – ^{5}/_{9} of a second) ] / 19 years

= 365+^{145819}/_{590976} days

= 365 days 5 hours 55 minutes and 18+^{97}/_{171} seconds

= only 6+^{149}/_{171} seconds shorter than the traditional Hebrew calendar mean year.

The excess length of the Hebrew calendar mean year relative to the northward equinoctial mean year was given near the beginning of this section as 6 minutes and 25+^{25}/_{57} seconds, so the proportion of that excess that is presently due to the excess length of the traditional *molad* interval = (6+^{149}/_{171}) / (6 × 60 + 25+^{25}/_{57}) = ^{235}/_{13182}, which is less than 1.8%. Thus more than 98.2% of the excess is due to the traditional 19-year leap cycle rather than the slight excess of the traditional *molad* interval. Nevertheless, if the 353-year leap cycle is adopted but for some reason the traditional *molad* is retained then the excess length of the *molad* interval will entirely account for the remaining 5+^{25}/_{1059} seconds of excess length of the adjusted calendar mean year. For the sake of both lunar and solar astronomical accuracy, therefore, it would be best to couple the 353-year leap cycle with a progressively shorter *molad* interval that more closely approximates the astronomical mean lunar conjunction interval.

To evaluate any calendar that purports to have some relationship to astronomical events, one needs to know the reference meridian or time zone of the clock that will be used to reckon the moments of those events. I am not aware of any primary traditional source that authoritatively specifies the reference meridian for the Hebrew calendar, although many recent authors have **assumed** it to be the meridian of Jerusalem.

My astronomical analysis of the traditional *molad*, however, revealed that the original *molad* reference meridian, in the era of the Second Temple, was **at the mid-point between the Nile River and the end of the Euphrates River**, which is about **4° of longitude east of Jerusalem** = Jerusalem mean solar time **+ 16 minutes** = Israel Standard Time + 37 minutes = Universal Time + 2 hours and 37 minutes. In the present era that meridian happens to correspond to the longitude at which the borders of the modern states of Jordan, Iraq, and Saudi Arabia meet.

For proof of this assertion, please see my astronomical analysis of the traditional *molad* at <http://individual.utoronto.ca/kalendis/hebrew/molad.htm>.

The *Torah* source for selecting this meridian is from Genesis chapter 15 verse 18: On that day

This territory also corresponds to the full range of our patriarch’s travels during his lifetime, as described in the *HaShem* made a covenant with Abram, saying *To your descendants have I given this land, from the river of Egypt to the great river, the Euphrates River.**Torah*, from Ur to Egypt.

It is implausible that the Hebrew calendar might use one reference meridian for the lunar cycle yet a different meridian for the solar cycle, so wherever a reference meridian was required for the modern astronomical analyses herein I have employed the original reference meridian of the traditional *molad*.

For details about the traditional Jewish methods for predicting the moments of equinoxes and solstices, including *Tekufat Shmuel* (תקופת שמואל), *Tekufat Adda* (תקופת אדא), and the true solar longitude

method of Maimonides, click here to see my web page

.*Rambam* and the Seasons

To obtain the accurate moments of the actual astronomical northward equinox as the basis for evaluating its relationship to the Hebrew calendar, I used **numerical integration**, which is arguably the gold standard

for celestial mechanics, and which is easy to do using **SOLEX** version 9.1 (or later), a free computer program written by Professor Aldo Vitagliano of the Department of Chemical Sciences at the University of Naples Federico II, Italy. SOLEX version 9.1, released in January 2007 and available at the SOLEX web site (see the end of this section), was the first released SOLEX version that could automatically find equinox and solstice moments, logging those moments to an external text file.

The SOLEX integration was carried out in terms of Terrestrial Time (usually abbreviated TT but indicated as TDT within SOLEX), with Delta T switched off and the geographic locale set to the Equator at the Prime Meridian, starting from date January 1, 2000. SOLEX integrated forward at 1-day intervals to beyond the year 12000 AD, and then starting again from year 2000 SOLEX integrated backward at 1-day intervals to before the year 7000 BC. The numerically integrated equinox and solstice moments were stored as TT moments in a database. When retrieved for calendrical calculations, my computer algorithm converted those TT moments to mean solar time moments by subtracting an approximation to Delta T (ΔT), using the Espenak-Meeus expressions found at the NASA Eclipses web site (published January 2007) at <http://eclipse.gsfc.nasa.gov/SEcat5/deltatpoly.html> and adjusting to the appropriate time zone, except that to avoid monthly granularity in the Delta T approximation I used the following expression when calculating *y* (the fractional year number):

*y* = 2000 + ( *TTmoment* – J2000.0 ) / *MARY*

where *TTmoment* is the Terrestrial Time moment and J2000.0 is January 1, 2000 AD at Noon, Terrestrial Time, both in terms of the number of days and fraction of a day elapsed relative to a specified ordinal day numbering epoch, and *MARY* (Mean Atomic Revolution Year) = 365+^{31}/_{128} atomic days, as explained on The Lengths of the Seasons

at <http://individual.utoronto.ca/kalendis/seasons.htm>. Over the entire range from 500 BC to 2050 AD, this modification never causes more than ^{4}/_{5} second of difference compared to the unmodified arithmetic of the NASA algorithm.

The NASA Delta T polynomials assume a certain rate of tidal slowing of the Earth rotation rate, based on published papers about historical solar and lunar eclipses, but the actual Earth rotation rate does not slow down at such a perfectly steady rate, going through short-term fluctuations and long-term periodic cycles, many of which are unpredictable with our present state of knowledge. Tidal slowing was probably greater in the past when the polar ice caps were more massive with lower sea levels and axial tilt was greater than in the present era, and tidal slowing will probably diminish over the coming several millennia due to global warming (reduction of polar ice mass, rising sea levels) and due to declining axial tilt. Nevertheless, they are considered the best available prediction, and suffice for the studies shown herein.

According to the SOLEX documentation, its numerical integration takes into account:

- Starting conditions of the Jet Propulsion Laboratory DE409 ephemeris.

(I also tried the older DE406 settings but rejected those because I found that DE409 was obviously more precise.) - The masses and 3-dimensional positions and velocities of Sun, Mercury, Venus, Earth, Moon, Mars, Ceres, Vesta, Pallas, Jupiter, Saturn, Uranus, Neptune, and Pluto.
- Osculating orbital elements, precession, nutation, aberration, light time.
- First order relativistic effects (see Special Relativity, General Relativity).
- Solar oblateness, and solar mass loss (due to nuclear fusion, and the solar wind)

SOLEX Limitations:

- For the evaluated range of dates (within ±10000 years of the present era), SOLEX calculates precession and Earth axial tilt (obliquity) using formulas published by J. G. Williams in
Contributions to the Earth’s Obliquity Rate, Precession and Nutation

*Astronomical Journal*1994 Aug;**108**(2): 711-724. - SOLEX assumes a constant
*J*_{2}parameter for the Earth, so it makes no attempt to model long-term variations in Earth’s oblate shape (polar : equatorial flattening) by taking into account tectonic plate movements or the mass of the polar ice caps. - SOLEX ignores planetary satellites and rings, except for our Moon.
- By default, SOLEX also ignores comets, asteroids, and minor planets other than those mentioned above. The user can optionally include thousands of additional objects in the integration, but doing so will slow down the computation accordingly.

**For more information about SOLEX and to download the program please see its web page at <http://www.solexorb.it/>.**

Each of the following charts shows the moment of the northward equinox (spring equinox for the northern hemisphere), calculated according to one of 3 traditional methods or a modern numerical integration, as indicated in the chart title, in relation to sunset at the start of the first day of *Nisan*. The Hebrew year numbers along the *x*-axis are either from the era of *Hillel ben Yehudah* (marked by a dashed vertical line at Hebrew year 4119), with axis labels at the first year of each 19-year cycle, or from Hebrew year 4000-18000, with each ^{1}/_{2} millennium labelled.

Each equinox moment is indicated with an ×

symbol, and they are sequentially connected with light grey lines. The moment of the sunset at the start of the first day of *Nisan* (taken as 6 hours before midnight) is indicated by a horizontal green line. The average timing of the equinox moments is shown as a thick grey line, which may be horizontal or slightly sloped, depending on whether or not there is a drift between the Hebrew calendar and that method of calculating equinox moments. The earliest moment for offering the *Korban Pesach* (paschal lamb sacrifice), traditionally at ^{1}/_{2} hour after Noon on the 14^{th} day of *Nisan*, is shown as a horizontal blue line. The sunset at the end of the first day of Passover, in other words starting the second day of Passover, is indicated by a horizontal red line. If *Hillel ben Yehudah* fixed the Hebrew calendar according to the criterion of the *Talmud* and *Rambam* then equinox moments should never land below that red line.

**Each chart is linked to a higher-resolution full-page Adobe Acrobat version of itself.**

The chart above shows that many *Tekufat Shmuel* equinox moments went far beyond the end of the first day of Passover, so it is clear that *Hillel ben Yehudah* didn’t use this method in fixing the Hebrew calendar. The average, which in the era of *Hillel ben Yehudah* landed on the 7th day of *Nisan*, is slightly sloped downward, toward later dates in later years, because, as explained above, the method of *Tekufat Shmuel* drifts one day later in the Hebrew calendar year for each elapsed ≈ 314+^{2}/_{3} Hebrew years.

The following chart takes a longer-term view of *Tekufat Shmuel* spring equinox moments, for traditional Hebrew calendar years 2000 through 6000:

As expected, the spring equinox according to *Tekufat Shmuel* falls progressively later in *Nisan* as the centuries pass, with the average at the start of *Nisan* in Hebrew year 2000 and at the 13th of *Nisan* in the present era. The latest *Tekufat Shmuel* spring equinox moments fell on the 16^{th} of *Nisan* in the era of the Revelation at Mount Sinai, traditionally taken as Hebrew year 2448. I will further discuss the significance of this finding under the heading Ancient History of

, below.*Tekufat Shmuel* and the *Molad* of *Nisan*

The equinox calculation method of *Rav Adda bar Ahavah* (רב אדא בר אהבה), who was a contemporary of *Hillel ben Yehudah*, yields northward equinox moments that *almost* never exceed the first 16 days of *Nisan*:

Year 1 of each 19 year cycle always has its equinox closer to the start of *Nisan* than any other year in the cycle, although occasionally the 12th year is also very close. The average, which happens to sit at Noon on the first day of *Nisan* = ^{3}/_{4} day after the sunset at the start of *Nisan*, is perfectly drift-free, because the mean year of *Rav Adda’s* method is identical to the Hebrew calendar mean year. Nevertheless, there seem to be two violations

within the charted range of years, including the very next year after *Hillel ben Yehudah* fixed the calendar, which *Hillel ben Yehudah* couldn’t possibly have missed

.

*Tekufat Nisan* of *Rav Adda* always falls in *Adar Sheini* in leap years, always falls in *Adar* in years 1 and 9 of each 19-year cycle (because the *Tekufat Nisan* of the previous year, a leap year, always falls more than 11 days prior to the start of *Nisan*), and in all other non-leap (common) years it falls in *Nisan*.

The following longer-term chart with every 19^{th} year connected shows that similar violations periodically recur, forever, and always involve the 16^{th} year of 19 (at intervals of 19, 38, 76, or 95 years). In addition, the 16^{th} year of 19 always exceeds the earliest *Korban Pesach* moment, and the 5th year of 19 periodically exceeds that limit (at intervals of 114 or 133 years):

From the above chart it seems highly likely that *Hillel ben Yehudah* went by *Rav Adda’s* method, but his equinox limit couldn’t have been the start of the 16^{th} day of *Nisan*, otherwise we can’t explain the recurrent violations

beyond that limit. If one understands the limit to be 16 **elapsed days** from the start of *Nisan*, in other words the **end of the 16 ^{th} of Nisan**, then there are never any violations of the limit, although in the 16

I calculated that the latest that

Rav Adda’sTekufat Nisanwill land in the entire 689472-year repeat cycle of the traditional Hebrew calendar is 17h 0m 14p 36r after mean sunset = 59m 3p 40r before Noon, but that will be in the absurdly far future (Hebrew years 75066, 272058, 469050, and 567546).

I calculated that the latest *Tekufat Nisan* of *Rav Adda* in the first 10000 years of the Hebrew calendar was 16h 51m 6p 36r after mean sunset in Hebrew year 16 = 1h 8m 11p 40r __before__ Noon. **The fact that the Tekufat Nisan moment never exceeds Noon on the 16^{th} of Nisan was probably significant with respect to the original calendar intercalation criteria, in particular because the omer sacrifice was offered in the Holy Temple after the musaf offering of the day, and according to Rashi, it had to be offered after the spring equinox.**

Another way to interpret the pattern in the chart above is that it is consistent with *Rashi*’s interpretation of Guard the month of

together with the *Aviv**Talmud Bavli* tractate *Sanhedrin* pages 12b-13a debate and conclusion that the entire day upon which an equinox or solstice moment falls is considered the first day of the new season. That is, even when the *Tekufat Nisan* of *Rav Adda* falls on the 16^{th} of *Nisan*, that entire day is considered to be the first day of spring, and the *omer* would be offered in its proper time. Furthermore, by this arrangement the proportion of equinoxes that fall within the month of *Nisan* (about 55+^{3}/_{4}% of years) is greater than the proportion that fall prior to the start of *Nisan* (about 44+^{1}/_{4}% of years).

The absence of any long-term drift between the method of *Rav Adda* and the Hebrew calendar, and the fact that the sages considered *Rav Adda’s* method to be the most accurate simple estimate of equinox and solstice moments, probably explains why for centuries many sages believed that the Hebrew calendar arithmetic was perfectly free of any drift. Note, however, that although they parallel each other perfectly, ** Tekufat Adda is not actually used in Hebrew calendar arithmetic — there is no need, because its mean year is identical to the mean year of the Hebrew calendar — instead the calendar follows a fixed 19-year leap cycle**.

*Rambam*’s even more accurate method, based on his true solar longitude

algorithm, reveals that the calendar drift is rather substantial, as shown next:

In the era of *Hillel ben Yehudah*, year 1 of each 19 year cycle had its equinox closer to the start of *Nisan* than any other year in the cycle. The thick grey line in the chart above is sloped distinctly upwards, indicating the long-term drift of the equinox to earlier dates on the Hebrew calendar. Although the same two violations stick out

below the red line, due to the drift of the calendar there were only a few later violations, as shown next:

(The PDF version of the above chart has every 19^{th} year connected, which reveals a finely detailed repeating sawtooth pattern that is due to the traditional *Rosh Hashanah* postonement rules.)

So according to *Rambam*’s method, from the era of *Hillel ben Yehudah* to the present era the Hebrew calendar has drifted 7 to 8 days, and it will continue to drift at about the same rate. The drifting pattern points back rather definitively to the era of *Hillel ben Yehudah*, so those who argue that the traditional Hebrew calendar was a subsequent gradual evolution would have a hard time explaining why somebody

made it so.

For the first 10000 years of the Hebrew calendar, *Rambam*’s method is in good agreement with a modern astronomical numerical integration (SOLEX), which is shown next:

In the era of *Hillel ben Yehudah*, year 1 of each 19 year cycle had its equinox closer to the start of *Nisan* than any other year in the cycle. The further into the past that one looks, the later the equinox landed in *Nisan* in year 1 of each cycle. The closer to the present and further into the future that one looks, the earlier the equinox lands in the month before *Nisan* in year 1 of each cycle. This evidence is strongly suggestive that at least the fixed leap cycle of the traditional Hebrew calendar was started in the era of *Hillel ben Yehudah*.

The average equinox here was slightly later in *Nisan* than it was on the Maimonides chart, the slope is nearly the same, but the two violations below the red line were slightly worse. The long-term view of the modern numerical integration looks very similar to the Maimonides chart until Hebrew year 10500, but after that the modern calculation reveals an accelerating drift that was missed by the algorithms of *Rambam*:

(Again, the PDF version of the above chart has every 19^{th} year connected, revealing the finely detailed repeating zig-zag pattern that is due to the traditional *Rosh Hashanah* postonement rules.)

The **black** least squares statistical linear regression line has, for the Hebrew years 4000 to 8000, a slope of about ^{1}/_{222} of a day per year, the inverse of which is about **222 solar years per day of drift**, which is in excellent agreement with our approximate value obtained arithmetically above. At that rate we can estimate that relative to the astronomical mean equinox the calendar has drifted (5768 – 4119) / 222 = about 7+^{1}/_{3} days compared to the alignment that it had in the era of *Hillel ben Yehudah*. The actual accumulated astronomical drift is slightly greater, because in the era of *Hillel ben Yehudah* and for several centuries thereafter the mean northward equinoctial year was several seconds shorter than it is today. We don’t need a more accurate drift evaluation at this point, because first we need to understand exactly what was the intercalation criterion of *Hillel ben Yehudah* (hold on, we're almost there...).

The thick grey line beyond Hebrew year 10500 is actually a straight-line extension of the thick black linear regression line from Hebrew year 4000 to 10500. The thick, curved magenta line is a second-order (quadratic) polynomial regression of the points from Hebrew year 10500 to 18000. The future accelerating drift will commence when perihelion advances past the northward equinox, and will get progressively worse as perihelion advances relentlessly towards the north solstice, causing the northward equinoctial year to get progressively shorter, with the tidal slowing of the Earth rotation rate contributing to this effect. These astronomical trends are documented and explained on my web page entitled The Lengths of the Seasons

at <http://individual.utoronto.ca/kalendis/seasons.htm> and in my leap cycle web page section entitled Calendar Seasons: Stable Points in the Solar Cycle

at <http://individual.utoronto.ca/kalendis/leap/index.htm#CS>.

In this section, the above evaluations are repeated, this time evaluating the relationship of the northward equinox to the moment of the __ molad of Nisan__, in keeping with the alternative version of the

Each of the following charts shows the moment of the northward equinox (spring equinox for the northern hemisphere), calculated according to one of 3 traditional methods or a modern numerical integration, as indicated in the chart title, in relation to the moment of the *molad* of *Nisan*. The Hebrew year numbers along the *x*-axis are either from the era of *Hillel ben Yehudah* (marked by a dashed vertical line at Hebrew year 4119), with axis labels at the first year of each 19-year cycle, or from Hebrew year 4000-18000, with each ^{1}/_{2} millennium labelled.

Each equinox moment is indicated with an ×

symbol, and they are sequentially connected with light grey lines. The moment of the *molad* of *Nisan* is indicated by a horizontal green line. The average timing of the equinox moments is shown as a thick grey line, which may be horizontal or slightly sloped, depending on whether or not there is a drift between the Hebrew calendar and that method of calculating equinox moments. The alternative *Talmud* limit of 16 days after the *molad* is indicated by a horizontal red line. If *Hillel ben Yehudah* fixed the Hebrew calendar according to this alternative *Talmud* criterion then equinox moments should never land below that red line.

These charts ignore the astronomical drift of the traditional *molad* of *Nisan* relative to the actual mean lunar conjunction, which is small in comparison with the solar drift of the Hebrew calendar. This is appropriate anyway, because here the *molad* of *Nisan* is being used as a surrogate for the Hebrew calendar, and is valid because the *molad* that is considered to be the *molad* of *Nisan* is determined by the traditional 19-year leap cycle.

**Each chart is linked to a higher-resolution full-page Adobe Acrobat version of itself.**

Again, many *Tekufat Shmuel* equinox moments went far beyond the *molad* + 16 days limit, so it is clear that *Hillel ben Yehudah* didn’t use this method in fixing the Hebrew calendar. The average, which in the era of *Hillel ben Yehudah* landed slightly beyond *molad* + 7 days, **always passes exactly through the equinox moment of year 12 of each 19-year cycle**, and is slightly sloped downward, further away from the *molad* in later years, because, as explained above, the method of *Tekufat Shmuel* drifts one day later in the Hebrew calendar year for each elapsed ≈ 314+^{2}/_{3} Hebrew years.

The following chart takes a longer-term view of *Tekufat Shmuel* spring equinox moments relative to the *molad* of *Nisan*, for traditional Hebrew calendar years 2000 through 6000:

Again as expected, the spring equinox according to *Tekufat Shmuel* falls progressively later relative to the *molad* of *Nisan* as the years pass, with the average (every year 12 of the 19-year cycle) near the *molad* of *Nisan* in Hebrew year 2000 and at about 12+^{1}/_{2} days after the *molad* of *Nisan* in the present era. The latest *Tekufat Shmuel* spring equinox moments fell just before the 16^{th} day after the *molad* of *Nisan* in the era of the Revelation at Mount Sinai, traditionally taken as Hebrew year 2448, which I will discuss further under the heading Ancient History of

, below.*Tekufat Shmuel* and the *Molad* of *Nisan*

**By contrast, the Tekufat Nisan calculation of Rav Adda bar Ahavah yields northward equinox moments that never exceed the molad + 16 days limit!**

Year 1 of each 19 year cycle always has its equinox closer to the *molad* of *Nisan* than any other year in the cycle.

The average always passes exactly through year 12 of each 19-year cycle and is always exactly 1 day 3 hours 42 minutes 7 parts and 72 *regaim* after the *molad* of *Nisan*, with zero long-term drift (because the mean year of *Rav Adda’s* calculation is identical to the Hebrew calendar mean year).

The timing of *Rav Adda’s* *Tekufat Nisan* moments repeats exactly in each 19-year cycle, forever, according to the following detailed pattern:

Now we can see that it is impossible for the *Tekufat Nisan* in any other year of the 19-year cycle to get closer to the *molad* of *Nisan* than it does in the first year.

Notice that the equinox wobble range

(latest minus earliest equinox) is the minimum possible = the traditional *molad* interval minus ^{1}/_{19} of a *molad* interval = ^{18}/_{19} of a *molad* interval. The chart shows that 9 of 19 = 47+^{7}/_{19}% of years have the *Tekufat Nisan* falling prior to the *molad*, and 10 of 19 = 52+^{12}/_{19}% of years have the *Tekufah* falling in *Nisan*.

*Tekufat Nisan* in leap years always advances ^{12}/_{19} of a *molad* interval *earlier* (about 18+^{2}/_{3} days) than it was in the previous year, whereas in non-leap years it always lags ^{7}/_{19} of a *molad* interval (about 10+^{7}/_{8} days) *later* than it was in the previous year.

*Tekufat Nisan* of the first year of each 19-year cycle is always the closest to the *molad*, landing exactly 9 hours 35 minutes and 12 parts before the *molad* moment. The eighth year of each 19-year cycle always has the earliest *Tekufat Nisan*, and the sixteenth year of each 19-year cycle always has the latest *Tekufat Nisan*. The *Tekufat Nisan* moments follow a uniformly spaced diagonal pattern. Listing all of the years of the 19-year cycle from earliest to latest *Tekufat Nisan* moment, the sequence is year 8=earliest, 19, 11, 3, 14, 6, 17, 9, 1, 12=average, 4, 15, 7, 18, 10, 2, 13, 5, and 16=latest, and their vertical spacing is always exactly ^{1}/_{19} of the traditional *molad* interval. The latest *Tekufat Nisan* moments never reach the *molad* + 16 days limit, in fact they are just shy of 15+^{1}/_{7} days after the *molad*, which is about ^{3}/_{8} of a day or 9 hours in excess of ^{1}/_{2} of a *molad* interval, but if one more step of ^{1}/_{19} of a *molad* interval were taken then it would be beyond the 16-day limit. The difference in *Tekufat Nisan* timing from year 1 to year 16 of each 19-year cycle is always exactly ^{10}/_{19} of a *molad* interval, which is the fraction of 19 that is closest to ^{1}/_{2} of a *molad* interval. Nevertheless, it is year 12 that is always exactly 9 steps away from the earliest and latest *Tekufat Nisan* moments, exactly at the average mid-point.

Although it seems most likely that *Hillel ben Yehudah* fixed the Hebrew calendar using the intercalation rule that *Tekufat Nisan* according to the method of *Rav Adda* shall never exceed 16 days after the *molad* of *Nisan*, this is a problematic criterion, because the timing of *Tekufat Nisan* in the 16^{th} year of every 19-year cycle always exceeds ^{1}/_{2} of a *molad* interval (14 days 18 hours 22 minutes and 38 *regaim*), implicitly landing beyond the waxing half of the mean lunar cycle. Astronomically, year 5 of 19 never reaches the waning half of the lunar cycle, but year 16 of 19 is beyond the waxing half in about 2 of 3 years, and is always beyond the mean full moon moment.

According to Yaaqov Loewinger, as written in his on-line book cited above, page 116, footnote 72A, a renowned Jewish astronomy book entitled *Yesod Olam* (Basis of the Universe) and written by Yitzhak Yisraeli ben Yoseph gives a *bereitah* in section 4, beginning of chapter 2, the original source for which is unknown today, listing 3 variants of the Hebrew calendar leap rule:

Attributed To | Leap Pattern | K |
mm |
Astronomically Appropriate Start |
---|---|---|---|---|

Rabbi Elazar | 3 2 3 3 3 2 3 | 3 | 232 | middle of Second Temple era |

Chachamim |
3 3 2 3 3 2 3 | 2 | 233 | last century of Second Temple era |

Rabban Gamliel | 3 3 2 3 3 3 2 | 1 | 234 | era of Hillel ben Yehudah |

I have added the columns containing the * K* and

It is a Leap Year only if the remainder of ( 7 ×

HebrewYear+) / 19 is less than 7.K

and in the Hebrew calendar elapsed months expression used in the traditional *molad* calculation (click here to see the traditional expression):

IF

hMonth<TishreiTHENTheYear=TheYear+ 1 ELSETheYear=hYearwhere the Hebrew months are numbered from

Nisan=1 toTishrei=7 toAdar=12,etc.

ElapsedMonths=hMonth–Tishrei+quotient( 235 ×TheYear–, 19 )mm

I also added the

column to indicate the era in which that leap rule variant would have started to yield optimal astronomical equinox alignment on the fixed arithmetic Hebrew calendar, had it been in use at the time. In each case it would have been optimal to continue using that leap rule variant for about 3+**Astronomically Appropriate Start**^{1}/_{2} centuries, as explained at the end of the section A Simple Arithmetic Estimate of the Hebrew Calendar Drift Rate

, above. An astronomical evaluation of this assertion is presented at the end of this web page.

When evaluated according to the arithmetic of *Rav Adda*, the variant attributed to Rabbi Elazar has earliest/average/latest *Tekufat Nisan* moments that are ^{2}/_{19} of a *molad* interval *earlier* than those of the traditional Hebrew calendar, as shown below:

As above, it was impossible for the *Tekufat Nisan* in any other year of the 19-year cycle to get closer to the *molad* of *Nisan* than it did in the first year.

With Rabbi Elazar’s variant the leap years would have been exactly the same as the traditional pattern, except that **years 5 and 16 would have been leap years instead of years 6 and 17 respectively**, causing year 5 to have the earliest rather than the second latest equinox moment, and causing year 13 to take over

as the year having the latest equinox moment. The average equinox would be in year 10 of 19, at almost 2 days before the *molad*.

The leap year interval sequence of the variant attributed to *Chachamim* (the Sages

) is directly obtained from Rabbi Elazar’s variant simply by omitting the first 8 years of a 19-year cycle, once only, as shown below:

Several Rabbi Elazar cycles: [ 3 2 3 3 3 2 3 ] [ 3 2 3 3 3 2 3 ] [ 3 2 3 3 3 2 3 ] ...

Omit the first 8 years of a cycle: [

3 3 2 3 ] [ 3 2 3 3 3 2 3 ] [ 3 2 3 3 3 2 3 ] ...~~3 2 3~~Sequence after omission is that of

Chachamim: [ 3 3 2 3 3 2 3 ] [ 3 3 2 3 3 2 3 ] [ 3 3 2 3 3 2 3 ] ...

When evaluated according to the arithmetic of *Rav Adda*, the variant attributed to *Chachamim* has earliest/average/latest *Tekufat Nisan* moments that are ^{1}/_{19} of a *molad* interval *earlier* than those of the traditional Hebrew calendar, as shown below:

Once again, it was impossible for the equinox in any other year of the 19-year cycle to get closer to the *molad* of *Nisan* than it did in the first year.

With use of the Sages’ variant the leap years would have been exactly the same as the traditional pattern, except that **year 16 would have been a leap year instead of year 17**, causing year 16 to have the earliest rather than the latest equinox moment, and causing year 5 to take over

as the year having the latest equinox moment. **The Sages’ variant seems technically and astronomically superior, because the average equinox would be as close as possible to the molad of Nisan at year 1 of each cycle, the latest equinox would never exceed the waxing half of the lunar cycle, and the equinox moments would be distributed as perfectly symmetrically as possible before and after the molad of Nisan.**

The leap year interval sequence of the variant attributed to Rabban Gamliel, which is identical to that of the traditional fixed Hebrew calendar, is directly obtained from the sequence of *Chachamim* simply by omitting the first 8 years of a 19-year cycle, once only, as shown below:

Several

Chachamimcycles: [ 3 3 2 3 3 2 3 ] [ 3 3 2 3 3 2 3 ] [ 3 3 2 3 3 2 3 ] ...Omit the first 8 years of a cycle: [

3 3 2 3 ] [ 3 3 2 3 3 2 3 ] [ 3 3 2 3 3 2 3 ] ...~~3 3 2~~Sequence after omission is that of Rabban Gamliel: [ 3 3 2 3 3 3 2 ] [ 3 3 2 3 3 3 2 ] [ 3 3 2 3 3 3 2 ] ...

Rather than regarding these as simultaneous conflicting opinions, they can be viewed as **a series of progressive adjustments to the Hebrew calendar leap rule, periodically compensating for accumulated calendar drift**. A similar effect could be automatically obtained by employing a leap cycle that inherently omits an an 8-year group (an *octaeteris)* from a 19-year cycle once per 353 years, as was alluded to above at the end of the section A Simple Arithmetic Estimate of the Hebrew Calendar Drift Rate

, and as will be discussed at the end of this web page under the heading Proposed Accurate Leap Rule for the Hebrew Calendar

.

**After each omission of an octaeteris the earliest/average/latest Tekufat Nisan moments shift ^{1}/_{19} of a molad interval later in relation to the Hebrew calendar, which is exactly the difference between the earliest/average/latest Tekufat Nisan moments that correspond to the leap year patterns of Rabbi Elazar, Chachamim, and the traditional Hebrew calendar, respectively in series, and is the appropriate shift, carried out at intervals of about 3+^{1}/_{2} centuries, that is necessary to prevent long-term drift of the Hebrew calendar with respect to the astronomical spring equinox.**

*Rambam*’s own spring equinox method, based on searching for the moment when his true solar longitude

algorithm yields zero degrees in each year, which for the first 10 millennia of the Hebrew calendar agrees to within a fraction of a day with modern astronomical calculations, is shown next:

In the era of *Hillel ben Yehudah*, year 1 of each 19 year cycle had its equinox closer to the *molad* of *Nisan* than any other year in the cycle.

The thick grey line in the chart above is sloped distinctly upwards, indicating the average long-term drift of the equinox to earlier dates on the Hebrew calendar, again passing exactly through year 12 of each 19-year cycle, with year 8 the earliest and year 16 the latest. **There are no violations beyond the molad + 16 days limit** in the era of

Interestingly, instead of the chaotic sawtooth pattern that we saw in the PDF version of the earlier Maimonides chart, which was caused by the Hebrew calendar jitter

of the *Rosh Hashanah* postponement rules, here we see a neat and tidy jitter-free set of 19 parallel lines, each corresponding to a specific year of each 19-year cycle. Year 12 of 19 is plotted as a thicker black line to clearly indicate the average trend. Selecting the year 12 of 19 that was closest to 4119 (4116), and the one closest to the present era (5769), allows very precise calculation of the average drift between those years, as shown in the bottom right rectangle in the chart, amounting to 7 days 14 hours.

The matter is sealed by the following evaluation of the astronomical northward equinox relative to the *molad* of *Nisan*:

In the era of *Hillel ben Yehudah*, year 1 of each 19 year cycle had its equinox closer to the *molad* of *Nisan* than any other year in the cycle. The further into the past that one looks, the later the equinox landed relative to the *molad* of *Nisan* in year 1 of each cycle. The closer to the present and further into the future that one looks, the earlier the equinox lands in the lunar cycle before the *molad* of *Nisan* in year 1 of each cycle. This evidence is strongly suggestive that at least the fixed leap cycle of the traditional Hebrew calendar was started in the era of *Hillel ben Yehudah*.

Although the astronomical equinox moments landed on average a little bit later relative to the *moladot*, there were never any violations. Once again, the average always passes exactly through year 12 of each 19-year cycle, with the earliest equinoxes occurring in year 8 and the latest equinoxes in year 16.

Whereas the latest that *Tekufat Nisan* of *Rav Adda* can land is about 15+^{1}/_{7} days after the *molad* of *Nisan* (always in the 16^{th} year of each 19-year cycle), which never exceeds Noon on the 16^{th} of *Nisan*, if an astronomical equinox were allowed to fall up to 16 days (24-hour periods) after the *molad* of *Nisan* then it could land up to ^{6}/_{7} day later, which would exceed the end of the 16^{th} of *Nisan* by more than 14+^{1}/_{2} hours. Therefore it seems to me that the relevant *Tekufat Nisan* latest cutoff time for the traditional Hebrew calendar was near Noon on the 16^{th} of *Nisan*, rather than 16 days after the *molad* of *Nisan*, to ensure that the *omer* sacrifice was offered after the spring equinox.

The thick blue line beyond Hebrew year 10500 is a straight-line extrapolation of the thick black year 12-average line for Hebrew years 4000 through 10500, making obvious the curvature of the thick magenta line in future years, which passes through year 12 of each 19-year cycle. **The astronomical drift of the Hebrew calendar from year 4116 to 5769 can be calculated to very good accuracy, shown in the rectangle at the bottom right of the chart as 7 days and 10 hours. In the black plotted region the drift rate averages about one day of drift per 222.5 years**, which agrees very well with the simple arithmetic estimate, which was one day of drift per 224 years.

Also shown is a dashed red horizontal line indicating the early limit

, calculated as the year 4116 equinox moment minus ^{1}/_{2} of a traditional *molad* interval. **Any present era equinox moments plotted above that early limit indicate years in which the 13 months from Adar Rishon until the end of Shevat are **

one month late, in the sense that Noon on the 16^{th} of Nisan is more than 30 days after the northward equinox, so there is no need to intercalalate that year.

**For the present era, this shows that years 8, 19, 11 and 3 of each 19-year cycle are always one month late from the start of Adar Rishon, year 3 of 19 having only recently emerged above the limit!** So although the average drift of the calendar amounts to

only7 days and 10 hours,

on time! This implies that presently the Hebrew calendar lags one month late in

Leap years in each 19-year cycle | Non-leap (common) years in each 19-year cycle |
---|---|

08 19 11 03 14 06 17 |
09 01 12 04 15 07 18 10 02 13 05 16 |

year 08 = Earliest Equinox |
year 16 = Latest Equinox |

years 08, 19, 11, 03 = In these years the spring equinox falls in the first half of Adar Sheini, so the prematureinsertion of the leap month pushes Noon on the 16 ^{th} of Nisan to more than one month after the equinox. The Hebrew calendar remains one month latefor 13 months from the start of Adar Rishon until the end of Shevat. |
year 12 = Average Equinox |

Presently the Hebrew calendar lags one month latein ^{(4×13)}/_{235} = more than 22% of all months. |
In non-leap (common) years the Hebrew calendar is presently always on timefrom Nisan onwards |

As shown above, evaluating the astronomical equinox timing relative to the *molad* of *Nisan* indicates that year 3 of each 19-year cycle is *already* one month late. When the equinox timing is evaluated relative to Noon on the 16^{th} of *Nisan*, however, that isn’t quite happening yet, but it will *soon* (depending on the coefficients adopted for the calendar leap rule and elapsed months expressions). As soon as the first instance of year 3 of 19 being one month late occurs, it will join years 8, 19, and 11 as a permanently late year (for the 13 months from the start of *Adar Rishon* until the end of *Shevat*).

Recall from the sources discussed above that the sages debated the timing of the autumn equinox at considerable length before the discussion was abruptly terminated by the bottom line instruction to go by the spring equinox. If the seasons were truly equal in length, as was the traditional belief for both *Tekufat Shmuel* and *Tekufat Adda*, then intercalation on the basis of the autumn equinox ought to have been equally valid, and would have had the advantage of plenty of advance warning for pilgrims to arrange their voyages to Jerusalem.

Accordingly, the following chart shows the repeating 19-year cyclic relationship between *Rav Adda’s* *Tekufat Tishrei* and the *molad* of *Tishrei*:

The relationship is very similar to that documented above for *Tekufat Adda* and the *molad* of *Nisan*, except that the year numbers in the cycle have been incremented, as they refer to the year starting from *Rosh Hashanah*. The *relative* timing between years holds for any method of reckoning the southward equinox. The nearer the equinox lands to the top of the chart shown above, the colder and wetter the weather will be in *Tishrei*, and the nearer the equinox lands to the bottom of that chart, the warmer and dryer the weather will be in *Tishrei*. Therefore, the three years in which the southward equinox always occurs earliest (actually in *Elul*) and hence have the coldest and wettest weather at *Sukkot* are years 9, 1, and 12 of every 19-year cycle, especially so for years in which *Rosh Hashanah* is postponed by two days (such as Hebrew year 5823, a 9th year that will start on Thursday, October 5, 2062). Likewise, the three years in which the southward equinox always occurs latest in *Tishrei* and hence have the warmest and driest weather at *Sukkot* are years 17, 6, and 14 of every 19-year cycle, especially so for years in which *Rosh Hashanah* is not postponed (such as Hebrew year 5774, a 17^{th} year that started on Thursday, September 5, 2013, or Hebrew year 5869, a 17^{th} year that will start on Thursday, September 6, 2108).

The latest *Tekufat Tishrei* according to *Rav Adda*’s method is always less than 20 days and 14 hours after the *molad*, but in the era of *Hillel ben Yehudah* that method was almost 5 days too early relative to the actual astronomical southward equinox, a discrepancy which surely was observationally obvious and might explain the abrupt *Talmud* switch to the northward equinox. (In the present era *Tekufat Tishrei* of *Rav Adda* is about 3 days too late relative to the actual astronomical southward equinox.)

The following chart shows the long-term relationship between *Tekufat Tishrei* of *Rav Adda* and the Hebrew calendar date in or before the month of *Tishrei*:

The above chart shows that *Tekufat Tishrei* of *Rav Adda* *almost* never exceeds the end of *Hoshanah Rabbah*, which is the 21^{st} of *Tishrei*, except at irregular intervals of about once per century when it reaches at most a few hours into *Shemini Atzeret* on the 22nd of *Tishrei*. One might propose that *Hillel ben Yehudah* didn’t notice excursions that occur so rarely, but in fact one of the largest such excursions, slightly more than 6 hours and 49 minutes, occurred in Hebrew year 4121 (at most 2 years after he fixed the calendar and within the same 19-year cycle), so it seems extremely unlikely that that might have been overlooked. In addition, the *Talmud Bavli* tractate *Sanhedrin* page 13b ruling is that the entire day upon which an equinox moment lands belongs to the new season, so *Shemini Atzeret* would still be in the autumn even if *Tekufat Tishrei* occurred during its evening or early night, and certainly the daytime would be in the autumn. This arrangement is also consistent with the *Talmud Bavli* tractate *Rosh Hashanah* page 21a ruling that only the spring equinox is important.

In the case of the actual astronomical southward equinox, charted below, there won’t be an accelerating future drift like the one that will occur relative to the northward equinox. Instead there already is an acceleration **in the present era** (examine the curvature of the lines), because the mean southward equinoctial year is currently rapidly getting shorter, as shown on my web page entitled The Lengths of the Seasons

at <http://individual.utoronto.ca/kalendis/seasons.htm>, whereas in a few millennia it will enter a multi-millennium era of reasonable stability, similar to the one presently enjoyed by the mean northward equinoctial year:

**In the era of Hillel ben Yehudah the latest southward equinox moments actually landed 26 days after the molad of Tishrei, and taking one molad interval before that as the **

early limitthe chart indicates that in years 9, 1, 12, and 4 of each 19-year cycle Tishrei is presently

one month latewith respect to the southward equinox. In those years Sukkot is exceptionally late, cold, and wet, and even in Israel the rainy season starts before the prayer for rain on Shemini Atzeret.

If *Tekufat Shmuel* is so obviously inaccurate then why does it have a place in our tradition? *Rambam* wrote that it was because its calculations are simpler than the more accurate method, *Tekufat Adda*, but I don’t think that was the real reason, because the calculations are nearly identical, only differing in the assumed epoch (one week difference) and the assumed mean year.

Compared to the Hebrew calendar mean year the mean year of *Tekufat Shmuel* is ^{313}/_{98496} of a day = exactly 4 minutes 34+^{32}/_{57} seconds longer, so as time passes the latest *Tekufat Shmuel* spring equinox moments fall progressively later relative to the *molad* of *Nisan*. As shown above, in the era of the Revelation at Mount Sinai (exodus from Egypt) *Tekufat Shmuel* fell within the limit of *molad*+16 days, and I believe that to be the real reason why *Tekufat Shmuel* retains some importance in our tradition. The following chart shows a close-up of the difference between *Tekufat Shmuel* and the *molad* of *Nisan* in the era of the Revelation at Mount Sinai (click here or on the chart to open a higher-resolution PDF version, 20 KB):

Year 16 of every 19-year cycle always has the latest equinox moments. The traditional year of the Revelation at Mount Sinai, Hebrew year 2448, was the 16^{th} year of 19, and its *Tekufat Nisan* according to Shmuel was 15 days 22 hours 2 minutes and almost 57 seconds after the *molad* of *Nisan*! Traditionally, that was the year of the biblical exodus from Egypt. The __last__ time that the 16^{th} year of the 19-year cycle fell within the *molad*+16 days limit was 19 years later, when it fell 15 days 23 hours 29 minutes and a bit more than 53 seconds after the *molad* of *Nisan*.

By the way, some hold that the Revelation at Mount Sinai was two years later, in Hebrew year 2450, but the special relationship between

Tekufat Shmueland year 2448 suggests that 2448 is the correct year.

The ancient Egyptians based their calendar year on the heliacal rising of Sirius, starting a new year on the first morning when Sirius was visible rising ahead of Sun, some number of days after the north solstice. The Nile used to flood their land soon afterward. The Sirius heliacal rising mean year length is to good accuracy 365+^{1}/_{4} mean solar days, which equals the *Tekufat Shmuel* mean year! Apparently it wasn’t realized that the mean northward equinoctial year (from spring equinox to spring equinox) is significantly shorter than the mean Sirius heliacal rising year.

One should not falsely conclude from the above chart that *Tekufat Shmuel* was accurate

in the era of the exodus, because in fact at that time, compared to the mean astronomical northward equinox, it was running more than a week too early. Nevertheless the relationship between the *molad* of *Nisan* and *Tekufat Shmuel* in that era can’t be coincidental. Nor can it be a coincidence that the year of the exodus was the 16^{th} year of the 19-year cycle, for the timing of *Tekufat Shmuel* in that year likely very purposefully indicates exactly what is the latest traditional *Tekufat Nisan* limit of the Hebrew calendar.

Looking for further astronomical correlations, it is also intriguing that a **total solar eclipse** crossed the populated area of northern Egypt shortly after midday on Saturday, May 14, 1338 BC (Julian date, no year zero) = 29 *Iyar* 2423 (25 years before the traditional date of the Revelation at Mount Sinai), as shown in this map from the NASA Eclipses web site <http://eclipse.gsfc.nasa.gov/SEatlas/SEatlas-2/SEatlas-1339.GIF>, with further details shown in this schematic diagram <http://eclipse.gsfc.nasa.gov/5MCSEmap/-1399--1300/-1337-05-14.gif>. Note that NASA web pages show the year number as 1337 BC or -1337 because their calculations include a year zero. That eclipse was the only total solar eclipse that passed through Egypt during that era. That event occurred about 41+^{2}/_{3} days after the northward equinox, so by astronomical criteria that was a Hebrew leap year so it was *Rosh Chodesh Iyar* not *Sivan*. Moon was near perigee (the point in its orbit when it is closest to Earth) so its disk diameter was larger than Sun, and Earth was near aphelion (the point in its orbit when it is furthest from Sun, hence the apparent solar disk diameter was near its minimum, for example see this photo at NASA: <http://antwrp.gsfc.nasa.gov/apod/ap070709.html>), making it a very dark and near-maximal-length eclipse. The duration of the total eclipse phase was just under 7 minutes, however, not quite a plague of utter darkness lasting 3 days

— could it be that legend grossly exaggerated that event?

For the following reasons, it would be preferable to correct the Hebrew calendar seasonal drift as soon as possible:

- The seasonal drift has accumulated to a highly significant degree: currently in every 8
^{th}, 19^{th}, and 11^{th}year of the traditional 19-year cycle, and soon also the 3^{rd}year of every cycle, Passover is more than a month after the spring equinox,*Purim*is celebrated on the day before the day thatshould have been

Passover, and the traditional Hebrew calendar continues to lagone month late

for 13 months from the start of*Adar Rishon*until the end of*Shevat*. - To allow ripe new grain crops to be reaped one month earlier in those years in which otherwise the 16
^{th}of*Nisan*would fall more than a month after the spring equinox (applies to the years just mentioned). Note that this continues to apply in the present era even though the*omer*sacrifice is not offered in the absence of the Holy Temple, as observant Jews also refrain from consuming food items containing new grain crops until the 16^{th}of*Nisan*. When the Hebrew calendar is late by a month the reaping of new grains is unnecessarily postponed by 30 days, which has an agricultural and economic impact on the availability of new grains. Barley is ready for harvest near the spring equinox, and the longer it is left standing the more will be lost due to seeds dispersal, damage and consumption by insects, rodents, and birds, and in Israel there is a risk of crop devastation due to later spring hard rains (*malkosh*, מלקוש). - When Passover is more than a month after the spring equinox then also the following
*Sukkot*(Feast of Booths) is a month later and, in the northern hemisphere, is exceptionally late in the autumn, cold, and wet (this applies to years 9, 1, 12, and soon also year 4 of the 19-year cycle, the above mentioned year numbers having been incremented at*Rosh Hashanah*), which makes dwelling in the traditional*sukkah*uncomfortable or impractical, notwithstanding the self-deprecating belief by many that time spent in the*sukkah*isnot

. In such years, the rainy autumn season in Israel begins well in advance of the*supposed*to be comfortable*geshem*prayer for rain on*Shemini Atzeret*. - On the principle that a
*mitzvah*should be performed as soon as possible (this applies to all calendar-related commandments and High Holy Days). - In the past the amount of equinox drift and the solution to the problem was unknown or uncertain, so it could not be confidently corrected, but today, due to an excellent understanding of the celestial mechanics of Sun, Moon, Earth, and the other planets of our solar system, we do accurately know how much drift has accumulated, and the solution for correcting it and preventing further drift is simple and straightforward.
- The newly re-established
*Sanhedrin*does have consideration of Hebrew calendar correction on its web page of agenda items for future sessions (click here). Many expect the*observational*Hebrew calendar to be reinstated for ritual purposes, but these are generally the same people who refuse to accept the authority of or even to acknowledge the existence of the new*Sanhedrin*! Even if that does happen, its observational component must only be concerned with the monthly sighting of the first visible new lunar crescent at sunset, whereas the decision to declare a leap year was always non-observational. The accurate leap rule proposed here would serve well as the basis for the*Sanhedrin*to declare leap years for many millennia to come. Furthermore, the Government of Israel would need a predictable, calculable Hebrew calendar for non-ritual purposes, for which a fixed arithmetic calendar along the lines proposed here would be suitable. Nevertheless, it remains quite possible that the new*Sanhedrin*will decide for a fixed arithmetic calendar, in which case the ritual and non-ritual Hebrew calendars will be the same.

Traditionally the leap years of each 19-year cycle are years 3, 6, 8, 11, 14, 17, and 19:

Leap Month is required if the *remainder* of ( 7 × *hYear* + 1 ) / 19 is less than 7.

The intervals between leap years can be either 3 or 2 years.

There are five 3-year intervals and two 2-year intervals per 19-year cycle.

If a leap year *remainder* is less than 2 then the next leap year will be 2 years later, otherwise 3 years later.

With 7 leap years per 19-year cycle, the average interval between leap years = ^{19}/_{7} ≈ 2.7142857 years.

An accurate Hebrew calendar needs a whole number of lunar months that fit into a whole number of solar years. The required ratio of months to years is obtained by dividing the length of the actual astronomical average solar year by the length of the actual astronomical average lunar month:

Mean Northward Equinoctial Year / Mean Synodic Month

≈ ( 365 days 5 hours 49 minutes 0 seconds ) / ( 29 days 12 hours 44 minutes 2+^{11}/_{15} seconds )

= ( 365 + ^{5}/_{24} + ^{49}/_{1440} days ) / ( 29 + ^{12}/_{24} + ^{44}/_{1440} + ( ^{41}/_{15} ) / 86400 days )

= ( ^{525949}/_{1440} ) / ( ^{38271641}/_{1296000} )

= ^{473354100}/_{38271641} = 473,354,100 months in 38,271,641 years!

This result, although representing the target ratio with perfect

accuracy, would be absurdly inconvenient as a calendar leap cycle, so we need to find a numerically simpler fraction that approximates this ratio using a reasonably short leap cycle. Such an approximation is easily obtained as the 6^{th} convergent of the continued fraction for the above ratio:

^{473354100}/_{38271641} = ≈ = ** ^{4366}**/

The **4366 lunar months in 353 solar years is an excellent approximation**, with an error

of less than one part in 10 million.

By way of comparison, note that the traditional Metonic cycle with 235 months in 19 years is the 5^{th} convergent of the same ratio, but the large jump in *a _{n}* value (

Dr. William Moses Feldman (1880-1939, originator of the term biomathematics

) proposed an improved Hebrew calendar leap cycle of 334 years having 123 leap years and a total of 4131 months per cycle, which he also found by the continued fraction method (page 208, __Rabbinical Mathematics and Astronomy__, Hermon Press, New York, 1931). He calculated a shorter optimal cycle length because he used what he called the Tropical Year length of 365 days 5 hours 48 minutes 46 seconds = 365+^{10463}/_{43200} = 365.242199074... days (about 14 seconds too short relative to the present-era mean northward equinoctial year) and he used a Lunar Year length of 354 days 8 hours 48 minutes 36 seconds = 354+^{881}/_{2400} = 354.367083... days, which corresponds to 29+^{15281}/_{28800} = 29 days 12 hours 44 minutes 3 seconds ≈ 29.53059027... days per lunar cycle (about ^{1}/_{3} second shorter than the traditional *molad* interval, yet almost ^{1}/_{3} second too long relative to the present era mean synodic month). Rather than the Mean Tropical Year (which applies only to atomic time), it is the mean northward equinoctial year (in terms of the mean solar time that is appropriate for calendrical purposes) that is the appropriate year length to keep *Nisan* aligned with the northward equinox, so for Hebrew calendar purposes the 334-year cycle is not as accurate as the 353-year cycle.

For further information about continued fractions, see <http://mathworld.wolfram.com/ContinuedFraction.html>. An easier-to-understand and more comprehensive explanation is at <http://en.wikipedia.org/wiki/Continued_fraction>.The following are continued fraction calculators that you can freely use on-line:

This calculator shows intermediate values used to compute the continued fraction, and its introductory web page at <https://r-knott.surrey.ac.uk/Fibonacci/cfINTRO.html> offers a full explanation about continued fractions:

<https://r-knott.surrey.ac.uk/Fibonacci/cfCALCbn.html>This calculator displays interesting information when the user hovers the mouse pointer over each convergent value:

<http://wims.unice.fr/wims/wims.cgi?module=tool/number/contfrac.en>

In 353 non-leap years there are 12 × 353 = 4236 regular months, so this cycle requires 4366 – 4236 = 130 leap months per cycle to make up the full set of 4366 months. The cycle mean year, using the present era mean synodic month rather than the traditional *molad* interval, is presently equal to ( 29 days 12 hours 44 minutes 2+^{11}/_{15} seconds ) days × 4366 months / 353 years ≈ 365 days 5 hours 48 minutes 57+^{3}/_{5} seconds per year, or only about 2+^{2}/_{5} seconds too short per year. If the __traditional__ *molad* interval were used instead of the actual mean synodic month then the mean year would be about 5 seconds too long at present. For comparison, recall that the Gregorian cycle mean year, with 97 leap days per 400 years, is currently about 12 seconds too long.

The number 353 is a prime number. The whole numbers that divide into 4366 are 2, 37, 59, 74, 118, and 2183, of which 2, 37 and 59 are prime numbers. The whole numbers that divide into 130 are 2, 5, 10, 13, 26, and 65, of which 2, 5 and 13 are prime numbers.

The leap status of the year is calculated according to a 353-year leap cycle having 130 leap years that are at intervals as uniformly spread as possible. The leap rule is:

Leap Month is required if the *remainder* of ( 130 × *hYear* + 269 ) / 353 is less than 130.

The intervals between leap years can be either 3 or 2 years.

There are 93 three-year intervals and 37 two-year intervals per 353-year cycle.

If a leap year *remainder* is less than 37 then the next leap year will be 2 years later, otherwise 3 years later.

With 130 smoothly spread leap years per 353-year cycle, the average interval = ^{353}/_{130} ≈ 2.7153846 years.

The whole number constant 269 controls the fine-tuning of the northward equinox alignment. Each unit change in this constant adjusts the long-term equinox alignment by a step equal to ^{1}/_{353} of a *molad* interval. **The value 269 yields optimal northward equinox alignment and a symmetrical leap cycle**, which will be explained below.

Similarly, appropriate coefficient changes are necessary to the traditional elapsed months expression when calculating the *molad* moment:

IF

hMonth<TishreiTHENTheYear=TheYear+ 1 ELSETheYear=hYearwhere the Hebrew months are numbered from

Nisan=1 toTishrei=7 toAdar=12,etc.

ElapsedMonths=hMonth–Tishrei+quotient( 4366 ×TheYear– 4097, 353 )

where the first two lines are the same as the traditional expression, 4366 is the number of months per 353-year cycle, and negative 4097 is a constant given by 269 (as above) minus the number of leap years per cycle (130) minus the number of months per non-leap year (12) times the number of years per cycle (353), that is: 269 – 130 – (12 × 353) = – 4097. [I mainly detail the derivation of this coefficient in case somebody wants to experiment with using a value other than 269 in the leap year expression above.]

Note that this leap rule and its corresponding elapsed months expression are not more complicated

than the traditional leap rule and elapsed months expression of the 19-year cycle. The same arithmetic operations are required, with mere substitution of accurate numeric constants. Either way, most people would need the assistance of a calculating device.

Complexity

, whether perceived or real, is not a valid reason to avoid a necessary calendar reform. Traditional oriental lunisolar calendars (for example as employed in China, Korea, Viet Nam, and Japan) use extremely complicated and accurate astronomical algorithms, yet those calendars have proven to be practical, workable, and universal for billions of people over a span of over a thousand years! The fact is that the man on the street

doesn’t fuss with calendrical calculations, he just follows the officially published calendar. The Jewish people would not notice anything different when using this corrected calendar, other than eliminating cases where Passover is exceptionally warm (one month too late in the spring season) and the following *Sukkot* is exceptionally cold and wet (one month too late in the autumn season).

**The effect of this leap cycle change is to slightly alter the distribution of leap years, such that when the traditional cycle inserts a leap month that causes Noon on the 16 ^{th} of Nisan to land more than 30 days after the northward equinox this corrected leap cycle instead in effect postpones the insertion of that leap month to the following year, where it properly belongs.**

**Unlike reform of a solar leap day calendar, which must either skip or repeat some calendar dates when any reform is adopted, with a lunisolar calendar it is easy to schedule use of the corrected leap rule to start during a year that agrees perfectly with the traditional calendar, thus entirely avoiding any objectionable date jump or repeated dates.**

In each cycle of the traditional Hebrew calendar the leap years are the 3^{rd}, 6^{th}, 8^{th}, 11^{th}, 14^{th}, 17^{th}, and 19^{th} years. The intervals between these leap years, counting from the beginning of the cycle are 3 3 2 3 3 3 2 years, repeating for each cycle. That is the most uniform spread of 7 leap years that is possible in a 19-year cycle (shifting the pattern left or right doesn’t change the spread, nor the calendar mean year, provided that the shifted pattern repeats for each cycle). Those intervals can be arithmetically grouped into a subcycle of 3+3+2 = 8 years and a subcycle of 3+3+3+2 = 11 years. Thus the traditional 19-year leap cycle has an 8-year subcycle (historically known as an *octaeteris)* alternating with an 11-year subcycle, repeating for each cycle.

The mean year of the 8-year subcycle (*octaeteris*) equals the number of months in the subcycle multiplied by the traditional *molad* interval, divided by 8 years:

= ( 8 × 12 + 3 ) ×

MoladInterval/ 8= 99 ×

MoladInterval/ 8 = 365 +^{10163}/_{23040}days = 365 days 10 hours 35 minutes 11+^{1}/_{4}seconds

Thus the mean year of the 8-year subcycle (*octaeteris*) is **much longer** than the mean northward equinoctial year length.

The mean year of the 11-year subcycle equals the number of months in the subcycle multiplied by the traditional *molad* interval, divided by 11 years:

= ( 11 × 12 + 4 ) ×

MoladInterval/ 11= 136 ×

MoladInterval/ 11 = 365 +^{3761}/_{35640}days = 365 days 2 hours 31 minutes 57+^{19}/_{33}seconds

Thus the mean year of the 11-year subcycle is **much shorter** than the mean northward equinoctial year length of about 365 days plus 5 hours 49 minutes and 0 seconds.

The mean year of the full 19-year cycle, however, is intermediate between these two extremes:

= ( 19 × 12 + 7 ) ×

MoladInterval/ 19= 235 ×

MoladInterval/ 19 = 365 +^{24311}/_{98496}days = 365 days 5 hours 55 minutes 25+^{25}/_{57}seconds

Nevertheless, the mean year of the full 19-year traditional Hebrew calendar leap cycle is more than 6 minutes and 25 seconds too long, relative to the mean northward equinoctial year.

The leap year interval pattern of the 353-year leap cycle is almost identical, the only difference is that ** once per 353 years an 11-year subcycle occurs without an accompanying 8-year subcycle**, in other words

[ ( 11 × 12 + 4 ) + 18 ( 19 × 12 + 7 ) ] ×

MoladInterval/ 353= 4366 ×

MoladInterval/ 353 = 365 +^{1109039}/_{4574880}days = 365 days 5 hours 49 minutes 5+^{25}/_{1059}seconds

using the traditional *molad* interval, yielding a mean year that is only about 5 seconds too long per year.

Although the cycle appears symmetrical at the level of its 19- and 11-year subcycles, detailed inspection of the leap year intervals reveals that it is not a *perfectly* symmetrical distribution, because perfect symmetry would only be possible if the leap month were inserted __after__ *Nisan* (that is __after__ the northward equinox), as explained on my web page entitled Solar Calendar Leap Rules

at <http://individual.utoronto.ca/kalendis/leap/> under the topic heading Symmetrical Leap Cycles

, where the astronomical advantages of perfect symmetry are also explained.

Once per 353-year cycle the pattern of leap year intervals slips ahead of the traditional 19-year cycle by 8 years. The maximum number of consecutive years for which the leap year intervals of the 19- and 353-year cycles can match perfectly equals the last 9 years of an 11-year subcycle, plus the next 19 consecutive 19-year subcycles, for a total of 9+(19×19) = 370 years of perfect agreement. Such 370-year periods of perfect leap status agreement recur every 6707 years, and after every 6707 years the 19-year cycle will have permanently fallen another month late relative to the 353-year cycle. It so happens that if the 353-year leap rule is properly fine-tuned for northward equinox alignment then the first 370-year period of perfect leap status agreement was several centuries ago, and the next will be in the distant future, so the existence of such perfect agreement periods is a curiosity that is of no concern for the present era or the near future.

The omission of one *octaeteris* per 353-year cycle is not by design, and it doesn’t increase the calendar’s equinox jitter

. Actually it is the equinox drift that increases that jitter beyond the 30 days that are due to the leap month, accumulating to ^{1}/_{19} of a *molad* interval over 353 years. **Each omission of an octaeteris cancels ^{1}/_{19} of a molad interval of equinox drift.**

In contrast, truncation of a 19-year cycle to only 8 years, by omitting 11 years including 4 leap months, would have the *opposite* effect, *increasing* the equinox drift by causing it to land ^{1}/_{19} of a *molad* interval *earlier*. Such a maneuver would only make sense if the target mean year were *longer* than the mean year of the 19-year cycle, as would be the case for a lunisidereal calendar intended to approximate the sidereal mean year, for example using a 160-year cycle with 59 leap months per cycle: (4 × 19) + 8 + (4 × 19) = 160 years.

The pattern of leap year intervals is just an observation made after-the-fact, and is simply a natural consequence of distributing the leap years of the 353-year cycle as uniformly as possible, generating a repeating pattern that is predictable. Nevertheless, its recognition does lead to the prediction of other longer and shorter leap cycles:

This Microsoft Excel

spreadsheet 30KB shows the grouping of leap year interval subcycles for the a variety of leap month cycles, including the 19- and 353-year leap cycles as well as some cycles that have intermediate or shorter mean years. (This simple spreadsheet is compatible with Excel for Windows or macOS, as well as LibreOffice CALC.) It shows that a series of cycles can be derived by omitting an **Leap Month Cycles***octaeteris* progressively more often, generating shorter and shorter calendar mean years. Such shorter cycles are not particularly useful today, but in the very distant future it will be necessary to switch to the shorter Feldman cycle (which has one fewer 19-year subcycle) and later the Future

cycle (another 19-year subcycle dropped), in order to compensate for the progressively shorter mean northward equinoctial year length. For comparison, a present era north solstice cycle is also included.

So: Does it work? Definitely. The following chart shows the drift of the astronomical northward equinox date in or prior to *Nisan* in the traditional Hebrew calendar from the era of the Second Temple to the present era, and shows what will happen when we switch to the 353-year leap cycle:

The choice of *Nisan* 5769 as switchover date was not arbitrary. Beginning in that month and continuing for 7 years, the traditional and rectified Hebrew calendars agreed on all dates, so therefore at anytime during this period **we could have switched the leap rule without any jump or repetition of Hebrew calendar dates** (any date jump or repetition would be socially and ritually objectionable). Well, we missed that opportunity to switch, but don’t worry, there will be plenty of future opportunities.

The above chart shows that for years prior to *Hillel ben Yehudah*, according to traditional Hebrew calendar arithmetic, the astronomical northward equinox often landed too late in *Nisan*. If, however, we assume that when *Hillel ben Yehudah* fixed the calendar he made a switch to the leap rule of Rabban Gamliel, and for 3+^{1}/_{2} centuries prior to that the Sages’ leap rule was used, and for 3+^{1}/_{2} centuries prior to that Rabbi Elazar’s leap rule was used, then we obtain the following chart, which shows that for 7 centuries prior to *Hillel ben Yehudah* the astronomical northward equinox would not have strayed too late, taking Noon on the 16^{th} of *Nisan* as the cutoff limit (prior to the *omer* offering):

Having every 19^{th} year connected, this chart shows two places prior to the present era where the year with earliest equinox swung down to become the year with the latest equinox: when the Hebrew calendar was fixed by *Hillel ben Yehudah* in Hebrew year 4119, and also 3+^{1}/_{2} centuries earlier. It furthermore confirms that the leap rule of Rabbi Elazar was astronomically appropriate for the middle of the Second Temple era, and the Sages’ leap rule was astronomically appropriate from the last century of the Second Temple until *Hillel ben Yehudah* established the traditional fixed arithmetic Hebrew calendar.

(I made no attempt to insert an additional *octaeteris* for back-calculating dates prior to the Second Temple, so those cases around Hebrew year 3200 where the chart shows that the equinox landed on or after the 17^{th} of *Nisan* should be ignored.)

This astronomical evaluation further supports my suggestion above that the 3 variant leap rules were not simultaneous opinions but rather were sequential modifications of the leap rule that were adopted to adjust the latest *Tekufat Nisan* alignment, by omitting an *octaeteris* from the leap year sequence once per 3+^{1}/_{2} centuries. This suggests that there was actually an ancient precedent for the strategy employed by the proposed accurate leap rule for the modern era, which automatically omits an *octaeteris* once in a middle of each 353-year cycle.

For more information about its arithmetic and calendrical calculations, click here to see the web page of my proposed ** rectified Hebrew calendar**, based on the 353-year leap cycle with 130 leap years per cycle. If the

hold onto the equinox until perihelion reaches the northward equinox in the 11

If, in addition to adopting the 353-year leap cycle, the *molad* interval is also progressively shortened to match the actual duration of mean lunar cycles, which are getting progressively shorter, as explained on my *molad* web page, using *progressive molad* arithmetic as detailed on my *rectified* Hebrew calendar web page, then the *progressive molad* will better track the astronomical mean lunar conjunctions, and because the *molad* interval plays a role in determining the Hebrew calendar mean year it will also be progressively shortened in parallel with the progressively shorter *molad* interval. According to astronomical numerical integration evaluation, use of the *progressive molad* together with the 353-year leap cycle will therefore allow the *rectified* Hebrew calendar to retain its alignment relative to the northward equinox **for an extra 3 millennia!**

To see what one month late

looks like in practice, **click here to view a table of matching months** of the traditional Hebrew calendar (based on the 19-year cycle with traditional *molad*) to the rectified Hebrew calendar (based on the 353-year cycle with progressive *molad*).

Don’t confuse the number 353 = years per leap cycle with the number of days in *shanah chaserah* (שנה חסרה), the deficient year, in which *Cheshvan* and *Kislev* each have 29 days.

Although the Continued Fraction method is very useful for approximating a target value with progressively more accurate fractions, it can miss nearby useful or interesting fractions if they are slightly too far away from the target value. By contrast the *Mediant Fraction Method*, coupled with the drawing of *Ford Circles*, finds all possible fractions within a wider target range, up to a specified maximum denominator. I won’t discuss this method further here, but if interested please see my web page section here: **Mediant Fractions, Farey Pairs, and Ford Circles**.

Another method, practical only with the aid of a computer program, is **to try every possible cycle length up to a specified maximum number of years**, looking for potential leap cycles that meet certain criteria. Usually I ignore cycles longer than 1000 years, but one day I ran my Excel **Fixed-Leap-Cycle-Finder** workbook, which uses this brute force method

, and asked it for *leap week* cycles up to 2000 years. **Among the best longer cycles that it found for the northward equinox was an 1803-year cycle**, with a mean year of 365+^{437}/_{1803} days = 365d 5h 49m 1+^{59}/_{601}s with either 320 leap weeks or 437 leap days. **Strikingly, because the program also looks for lunar relationships, the spreadsheet showed that the 1803-year cycle contains exactly 22300 mean synodic months!** The well-known *saros* eclipse cycle has 223 mean synodic months, so the 1803-year solar cycle is also a ^{22300}/_{223} = 100-*saros* lunar cycle, making it an excellent choice for a fixed arithmetic lunisolar calendar having 1803×12 = 21636 regular months and leaving 22300 – 21636 = 664 leap months.

The 1803-year cycle was missed by the Continued Fraction method above because its mean year is slightly too long and its mean month is slightly too short, compared to the specified targets. The mediant fraction method finds it if the specified maximum number of years per cycle is high enough (≈2000 years), but gives no indication of its relationship to the superb 100-

saroslunar cycle.

The program calculated that there are 658532 days per 1803-year cycle (this is exactly 94076 weeks because I had told the program to search only for *leap week* cycles), and the mean month is ^{(658532 days per cycle)}/_{(22300 months per cycle)} days per month = ^{164633}/_{5575} days = 29+^{2958}/_{5575} days = 29d 12h 44m 2+^{82}/_{223}s, which is almost ^{1}/_{2} second shorter than the present era mean synodic month (in terms of the mean solar days that are required for calendrical purposes). Since the denominator 22300 reduced by a factor of 4 to 5575, the cycle contains exactly 4 repeats of a 25-*saros* lunar cycle. Nevertheless, since quarter- and half-days are involved in *molad* calculations and 5575 isn’t divisible by 4 or 2, as a potential lunar cycle for a Hebrew calendar calculations would be simplified by working with the non-reduced mean month = 29+^{11832}/_{22300} days, which implies 22300 parts per day, each of duration ^{86400}/_{22300} = 3+^{195}/_{223} seconds, which is comparable to the *chelek* of the traditional *molad* = 3+^{1}/_{3} seconds.

The Traditional *Molad* mean month = 29+^{13753}/_{25920} days, so the 25-*saros* mean month is ^{13753}/_{25920} – ^{2958}/_{5575} = ^{323}/_{28900800} days shorter, or, multiplying by 24×60×60=86400 seconds per day = ^{646}/_{669} of a second shorter or almost 966 milliseconds (almost one second) shorter.

If the 1803-year cycle were used with the traditional *molad* (not recommended) then its mean year would be about 13 seconds too long.

As decimal values the mean year is approximately 365.2423738214 days (the exact decimal value has 300 repeating digits) and the mean month is approximately 29.53058295964 days (the exact decimal value has 222 repeating digits after the first two decimal points).

The smoothly distributed perfectly symmetrical subcycle pattern of the 1803-year lunisolar cycle is: (9×19)+11+(19×19)+11+(18×19)+11+(18×19)+11+(19×19)+11+(9×19) = 1803 years. It *automatically* omits an *octaeteris* at the 5 places where there is a stand-alone 11-year subcycle, at intervals of 18 or 19 repeats of 19 years, or an average of ^{1803}/_{5} = 360+^{3}/_{5} years (slightly longer than 353 years because of the 100 *saros* cycle’s slightly shorter mean month). This subcycle information is preliminary because the pattern for maintaining the optimal equinox alignment, to be finalized, isn’t perfectly symmetrical.

One can calculate the number of times that an *octaeteris* is omitted from a lunisolar calendar cycle by multiplying 235 times the number of years per cycle then subtracting 19 times the number of months per cycle. Applying this formula to the 100-*saros* cycle: 235 × 1803 years – 19 × 22300 months = 5, as above. For the Rectified Hebrew calendar we have 235 × 353 years – 19 × 4366 months = 1, as above. Lunisolar cycles that are suitable for approximating the shorter mean North Solstitial year more frequently omit an *octaeteris*. For example, the 1556-year cycle with 19245 lunar months also has 235 × 1556 years – 19 × 19245 months = 5 omissions, but they are at average intervals of only 311+^{1}/_{5} years. [Thanks to Karl Palmen of the UK for providing this formula.]

**The 1803-year cycle is unique because it can be used simultaneously (in parallel) for a leap day, leap week, and leap month calendar, all having exactly the same mean year, and all potentially sharing the 100- saros lunar cycle.**

**The whole number of years, weeks, and lunar months per cycle ensures that its full repetition cycle is only 1803 years** (compare with 689472 years for the Traditional Hebrew Calendar).

The 1803-year cycle performs surprisingly well as an alternative Future

Hebrew calendar leap cycle / *molad* cycle: **click here to see its astronomical evaluation chart, which shows that it should remain drift-free until Hebrew year 10000.** With its simple fixed *molad* cycle and elimination of *molad zakein* postponements, its calendar arithmetic is appreciably *simpler* than my proposed Rectified Hebrew Calendar, but as of the dates shown below development and further evaluation is on-going.

For another look at what one month late

looks like in practice, **click here to view a table of matching months** of the traditional Hebrew calendar (based on the 19-year cycle with traditional *molad*) compared to one full cycle of the Future

Hebrew calendar (based on the 1803-year cycle with fixed 100-*saros* lunar cycle). Observe how towards the end of the list more and more traditional years will be one month late.

Equinox Jitter

In addition to the long-term drift of the equinox, there are a few minutes of year-to-year jitter

of the equinox moment, due to the gravitational influence of Moon, Jupiter, Sun, Venus, and to a minor extent, Mars.

An unavoidable major cause of equinox jitter is the insertion of the 30-day leap month, *Adar Rishon*, every 3 or 2 years. In the traditional Hebrew calendar this jitter is 30×^{18}/_{19} = 28+^{8}/_{19} days. In the Rectified Hebrew calendar, or any other calendar based on the 353-year lunisolar cycle discussed above, this jitter is 30×^{352}/_{353} = 29+^{323}/_{353} days. In a Hebrew calendar based on the 1803-year lunisolar cycle discussed above this jitter is 30×^{1802}/_{1803} = 29+^{1773}/_{1803} = 29+^{591}/_{601} days.

With the 353-year and 1803-year leap cycles one might expect an additional equinox jitter component that has a repetition period of about 3+^{1}/_{2} centuries, during which time the equinox drifts up to ^{1}/_{19} of a *molad* interval earlier, then that amount of drift is cancelled by automatic omission of an *octaeteris* from the sequence of years, resulting in an equinox jitter component spanning ^{1}/_{19} of a *molad* interval. Click here to see a chart depicting an astronomical evaluation of a Hebrew calendar based on the 1803-year lunisolar cycle (with 100-*saros* *molad* cycle): the total jitter range is less than 31+^{3}/_{4} days, the *x-axis* major tick interval is 360 years, and one gets the impression that there is a sawtooth pattern at the top and bottom edges that repeats at intervals that correlate with the major tick marks. Nevertheless, this jitter component is already included in the greater leap month jitter ranges that were calculated for these cycles in the previous paragraph. The extra nearly 2 days of jitter is due to other jitter components.

There is always a component of Hebrew calendar equinox jitter that is due to the traditional *Rosh Hashanah* postponements, which can postpone the start of the civil year 1 or 2 days, and this jitter component may be increased or decreased by the usual adjustments to the lengths of the months of *Cheshvan* and *Kislev* (each can have 29 or 30 days, as appropriate for the length of the year). As extreme examples, the postponement may be zero days in a deficient

year with 29 days in *Cheshvan* and *Kislev*, or the postponement could be 2 days in an abundant

year with 30 days in *Cheshvan* and *Kislev*. A 2-day postponement could be offset in a deficient

year by 29 days in *Cheshvan* and *Kislev*. A 1-day postponement could be offset in a normal

year by 29 days in *Cheshvan* with 30 days in *Kislev*.

The *total jitter range* can only be determined by astronomical evaluation (in comparision with astronomical algorithms or numerical integration), as in the example of the 1803-year cycle above, and because the components are at least partially correlated the total is always less than the simple sum of the above jitter components.

Except for the unavoidable major jitter due to the leap month, the small jitter components could be minimized by using some reasonably accurate reckoning of the equinox moment relative to Noon on the 16th of *Nisan* (in Jerusalem or at the meridian of the mid-point between the Nile and the Euphrates rivers) to decide whether to intercalate the year (instead of using any fixed leap cycle), preferably while retaining the traditional postponement rules and without making the calendrical arithmetic much more complicated. (Contrary to the mistaken belief of many people, the arithmetic of the traditional Hebrew calendar makes no attempt to calculate any equinox moments.)

For more information about calendar jitter please see my web page entitled Fixed Arithmetic Calendar Cycle Jitter

at http://individual.utoronto.ca/intervalmath/jitter/.

Updated
5 *Cheshvan* 5783 (Traditional) =
5 *Cheshvan* 5783 (Rectified) =
Oct 28, 2022 (Symmetry454) =
Oct 28, 2022 (Symmetry010) =
Oct 30, 2022 (Gregorian)