בס"ד

The Seasonal Drift of the
Traditional (Fixed Arithmetic) Hebrew Calendar (הלוח העברי הקבוע)

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

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Menu of Topics:

  1. Traditional and Scientific Sources
  2. The Leap Rule and Mean Year of the Traditional Hebrew Calendar
  3. Sources for the Hebrew Calendar Intercalation Criteria
  4. A Simple Arithmetic Estimate of the Hebrew Calendar Drift Rate
  5. The Reference Meridian
  6. Methods for Computing Equinox Moments
  7. The Northward Equinox in Relation to the Sunset at the Start of Nisan
  8. The Northward Equinox in Relation to the Molad of Nisan
  9. The Southward Equinox in Relation to the Molad of Tishrei
  10. Ancient History of Tekufat Shmuel and the Molad of Nisan
  11. Proposed Accurate Leap Rule for the Hebrew Calendar
  12. Hebrew Calendar Equinox Jitter
  13. Conclusions and Executive Summary

Traditional and Scientific Sources

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

  1. Torah and commentaries as cited in the discussion.
  2. The Babylonian Talmud (Talmud Bavli, תלמוד בבלי).
  3. The Jerusalem Talmud (Talmud Yerushalmi, תלמוד ירושלמי).
  4. Rambam’s Sanctification of the New Month (Hilchot Kiddush HaChodesh, הלכות קידוש החודש).
  5. Rambam’s Book of Service (Sefer Avodah, ספר עבודה, Hilchot Timidin OuMusafin, הלכות תימידין אומוסאפין).
The following astronomical and scientific sources were used:
  1. Jean Meeus, Astronomical Algorithms, 2nd edition, Willmann-Bell Inc., Virginia, 1999.
  2. Jean Meeus, More Mathematical Astronomy Morsels, Willmann-Bell Inc., Virginia, 2002.
  3. Calendrical Calculations (3rd [2008] and 4th [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>.
  4. NASA Eclipse Web Site: Polynomial Expressions for Delta T (ΔT) at <https://eclipse.gsfc.nasa.gov/SEhelp/deltatpoly2004.html>.
  5. Other web pages on this web site — please follow the links!
  6. Links to external sites, especially Wikipedia, are used for many astronomical and scientific terms — please follow the links!
The following computer programs were used:
  1. 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-bit double 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.
  2. 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 Kalendis Astronomy Validations web page, here: <http://individual.utoronto.ca/kalendis/hebrew/validations.htm>.
  3. I developed and used numerous Microsoft Excel workbooks with VBA macros as cited in the text with download links.
  4. For exact arbitrary precision calculations and mathematical functions: <https://www.wolframalpha.com/>.
  5. 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>.
  6. See below for on-line (click here) Continued Fraction calculators.

The Leap Rule and Mean Year of the Traditional Hebrew Calendar

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 + 12/24 + 793/25920 = 29+13753/25920 days = 29.530594135802469... days (in the decimal representation of the exact fraction the 9 overscored digits repeat forever). For further information about the numeric length of the molad interval, see 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.

Sources for the Hebrew Calendar Intercalation Criteria

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

שָמוֹר אֶת חדֶשׁ הָאָבִיב

Guard the month of Aviv, and bring a paschal-lamb offering (pesach) for hashem, your God, because in the month of Aviv hashem, 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 2/3 of the variations. For more information and charts depicting the lunar half-cycle variations, please see Lunar Half-Cycles: Triple the Periodic Variations! on my Length of the Lunar Cycle web 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 14th 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 aviv, explaining that the Hebrew word 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 16th 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 (16th 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 16th 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 16th 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 16th 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 16th 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

Hashem said 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 15th of Tishrei = תשרי) through Hoshana Rabbah (21st 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 Elul ever intercalated, which suggests that Elul was indeed intercalated prior to the days of Ezra. A possible reason for intercalating Elul is suggested by the following sources:

Talmud Yerushalmi tractate Sheqal page 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 Bavli tractate Sanhedrin page 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 16th or the 20th day of Tishrei. Nevertheless, astronomically neither of these limits is possible, nor any day between them, not even the 15th or 21st 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 16th and 20th 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 10th 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 25th 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 Meir Ba'al HaNess), says R. Samuel son of R. Isaac, speak of Tekufat Nisan (spring equinox), for it is written (in the Torah, as quoted above) Guard the month of Aviv, that is, take heed that the spring equinox shall occur on a day in Nisan.

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 Nisan, meaning that the equinox should occur during the waxing half of the lunar cycle in 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 16th of Nisan, intercalate a month into that year, and don’t worry about it, for it is written Guard the month of aviv, guard the month of the first-ripened produce so that it will be in the month of Nisan.

The phrase until the 16th of Nisan could be understood inclusively, meaning that the 16th of Nisan was part of the winter season, or it could be understood exclusively, meaning that the 16th 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 15th 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 Shabbat limits 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 8th year of each 19-year Hebrew calendar cycle (the same is also true of the 19th, 11th and 3rd 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 Savasorda, Ibn Ezra, and R. Yizhaq ha-Yisraeli all explicitly mention this as the only criterion for intercalation. Note that this criterion is much simpler to calculate than 16 days after the start of Nisan. The Al ha-Sheminit (על השמינית) book is accessible in scanned PDF format at <http://www.daat.ac.il/daat/vl/tohen.asp?id=162>. This book actually is a broadly comprehensive review and analysis of the traditional sources relevant to the Hebrew calendar, and an excellent source for citations and references of the original works.

Rabbi Moshe ben Maimon (Rambam), 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 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 16th of Nisan or later, the year is made a leap year. The month that would have been Nisan is made Adar Sheini, so that Pesach will 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 haChodesh chapter 4, paragraph 2)

In other words, Rambam’s limit for the latest equinox was the end of the 15th day of Nisan, exactly halfway through the month. Rambam then went on to detail a variety of agricultural and environmental factors that may or may not be considered by the court. For example, occasionally, in a year with an exceptionally cold or wet winter, the leap month was inserted even when it was not yet due, to allow extra time for the winter to end, the barley crop to ripen, roads to dry out their winter wetness so that pilgrims could travel to Jerusalem, etc.

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 Bavli tractate Rosh Hashanah page 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:

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 aviv the first commandment to the children of Israel.

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.

A Simple Arithmetic Estimate of the Hebrew Calendar Drift Rate

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 + 24311/98496 days as calculated above, the excess length of the Hebrew calendar year is 2197/492480 days, or a bit more than 6/1345 days, or slightly less than 1/224 of a day. Multiplying the difference by 1440 minutes per day gives an excess of 6+145/342 minutes, or multiplying the difference by 86400 seconds per day gives an excess of 6 minutes and 25+25/57 seconds.

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 Hillel ben Yehudah to the present era, a span of 5768 – 4119 = 1649 years, the average Hebrew calendar solar drift has accumulated to about 1649/224 ≈ 7+2/5 days.

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 × (molad interval) – 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 16th 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 in Adar are immediately postponed by 30 days to Adar Sheini.

After Shevat the traditional calendar says that the month is Adar whereas the rectified calendar says that the month is Adar 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 mean tropical year 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 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 a molad interval 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.

The Reference Meridian

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 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. This territory also corresponds to the full range of our patriarch’s travels during his lifetime, as described in the 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.

Methods for Computing Equinox Moments

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:

SOLEX Limitations:

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

The Northward Equinox in Relation to the Sunset at the Start of Nisan

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 14th 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 PDF 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 16th 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 Tekufat Shmuel and the Molad of Nisan, below.

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 19th year connected shows that similar violations periodically recur, forever, and always involve the 16th year of 19 (at intervals of 19, 38, 76, or 95 years). In addition, the 16th 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 16th 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 16th of Nisan, then there are never any violations of the limit, although in the 16th year of each 19-year cycle the Rav Adda’s Tekufat Nisan always lands in the waning half of the lunar cycle, as will be discussed further below.

I calculated that the latest that Rav Adda’s Tekufat Nisan will 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 16th 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 Aviv together with the 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 16th 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 19th 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 19th 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>.

The Northward Equinox in Relation to the Molad of Nisan

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 Talmud Bavli that was discovered in the Cairo genizah, which specified the intercalation limit as 16 days after the molad.

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 PDF version of itself.

PDF

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 16th 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 Tekufat Shmuel and the Molad of Nisan, below.

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 16th 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 mm coefficients, used in Hebrew calendar arithmetic as follows:

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

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

IF hMonth < Tishrei THEN TheYear = TheYear + 1 ELSE TheYear = hYear

where the Hebrew months are numbered from Nisan=1 to Tishrei=7 to Adar=12, etc.

ElapsedMonths = hMonthTishrei + quotient( 235 × TheYearmm , 19 )

I also added the Astronomically Appropriate Start 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+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 2 3 3 3 2 3 ] [ 3 2 3 3 3 2 3 ] [ 3 2 3 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 Chachamim cycles: [ 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 2 3 ] [ 3 3 2 3 3 2 3 ] [ 3 3 2 3 3 2 3 ] ...

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 Hillel ben Yehudah, nor later, as shown next:

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 16th year of each 19-year cycle), which never exceeds Noon on the 16th 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 16th 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 16th 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 16th 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 only 7 days and 10 hours, presently 4 out of every 19 years are one month late, and the remaining 15 years are on time! This implies that presently the Hebrew calendar lags one month late in (4×13)/235 = more than 22% of all months! This figure is in good agreement with the simple arithmetic estimate of 24.5%, obtained above.

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 premature insertion of the leap month pushes Noon on the 16th of Nisan to more than one month after the equinox. The Hebrew calendar remains one month late for 13 months from the start of Adar Rishon until the end of Shevat. year 12 = Average Equinox
Presently the Hebrew calendar lags one month late in (4×13)/235 = more than 22% of all months. In non-leap (common) years the Hebrew calendar is presently always on time from 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 16th 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).

Clearly there is an urgent need to reform the traditional Hebrew calendar.


The Southward Equinox in Relation to the Molad of Tishrei

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 17th year that started on Thursday, September 5, 2013, or Hebrew year 5869, a 17th 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 21st 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 limit the chart indicates that in years 9, 1, 12, and 4 of each 19-year cycle Tishrei is presently one month late with 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.

Ancient History of Tekufat Shmuel and the Molad of Nisan

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 16th 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 16th 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 Shmuel and 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 16th 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?

Proposed Accurate Leap Rule for the Hebrew Calendar

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

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 6th convergent of the continued fraction for the above ratio:

473354100/38271641 = = 4366/353

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 5th convergent of the same ratio, but the large jump in an value (a6 = 18) indicates that the 6th convergent is a much more accurate approximation. The 7th convergent has a7 = 8, yielding the approximation 35163/2843, but a 2843-year leap cycle seems inconveniently long for the modest further improvement in accuracy, and, because future astronomical changes in the equinoctial year and synodic month are inevitable, it would be futile to force the leap cycle approximation beyond the 6th convergent (unless there are other good reasons for doing so, as is the case for the novel 1803-year cycle discussed below).

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 < Tishrei THEN TheYear = TheYear + 1 ELSE TheYear = hYear

where the Hebrew months are numbered from Nisan=1 to Tishrei=7 to Adar=12, etc.

ElapsedMonths = hMonthTishrei + 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 16th 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 3rd, 6th, 8th, 11th, 14th, 17th, and 19th 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 one octaeteris is omitted every 353 years. The full pattern in ideal arrangement for optimal spring equinox alignment has 9 repeats of the traditional 19-year subcycles, then one 11-year subcycle (octaeteris omitted), then 9 more repeats of the traditional 19-year subcycles: (9×19)+11+(9×19) = 353. Thus the subcycle truncated from 19 to 11 years is optimally situated at the middle of each 353-year cycle, making each cycle symmetrical with respect to the subcycles that it contains. Since the 11-year subcycle has a shorter mean year than the omitted octaeteris, the mean year length of the full 353-year cycle is accordingly shorter:

[ ( 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 Leap Month Cycles spreadsheet Excel30KB 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 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 16th of Nisan as the cutoff limit (prior to the omer offering):

Having every 19th 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 17th 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 molad interval is not adjusted then the accurate leap rule will, according to astronomical numerical integration evaluation, allow the Hebrew calendar to hold on to the equinox until perihelion reaches the northward equinox in the 11th millennium of the Hebrew calendar.

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+59/601s 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-saros lunar 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+82/223s, 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 13753/259202958/5575 = 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.


Hebrew Calendar 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/.


Conclusions and Executive Summary

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

  1. Rabbi Shlomo Yitzhaqi (רבי שלמה יצחקי), known as Rashi (רש"י), is the only traditional rabbinic authority whose interpretation of Guard the month of Aviv (שָמוֹר אֶת חדֶשׁ הָאָבִיב) is consistent with the traditional Hebrew calendar and also astronomically and calendrically valid.
  2. According to Rashi, the spring equinox moment should occur before the omer offering (first barley) on the 16th 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).
  3. To minimize its seasonal drift, the Hebrew calendar mean year should closely approximate the mean spring equinox year, but is more than 6 minutes and 25 seconds too long.
  4. 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).
  5. 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.
  6. 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 16th 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.
  7. 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.
  8. 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% of all months will be one month late.
  9. Consequences: When the Hebrew calendar is late by a month the reaping of new grains is unnecessarily delayed by 30 days, which has an agricultural and economic impact. 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). Furthermore, the following festival of Sukkot will be exceptionally late in the autumn, cold, and wet.
  10. 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.
  11. The 353-year leap cycle containing 4366 lunar months, including 130 leap months, is an accurate and convenient way to automatically omit an octaeteris near the middle of each cycle.
  12. If used with the traditional molad the mean year of the 353-year cycle would be only about 5 seconds too long, but with a progressively shorter molad interval that closely approximates the mean lunar cycle those few extra seconds would be shaved off.
  13. An alternative accurate leap cycle with a superb fixed lunar cycle has 1803 years and 22300 lunar months, including 664 leap months, and automatically omits an 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 having whole numbers of weeks and lunar cycles, its full repetition cycle is only 1803 years (compare with the traditional 689472-year full repetition cycle).
  14. 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 matches the traditional calendar, avoiding any date jump or repeated dates.

Updated 17 Sivan 5782 (Traditional) = 17 Sivan 5782 (Rectified) = Jun 11, 2022 (Symmetry454) = Jun 13, 2022 (Symmetry010) = Jun 16, 2022 (Gregorian)