The Hebrew calendar From Wikipedia the free encyclopedia

 

The Hebrew calendar (Hebrew: הלוח העברי) or Jewish calendar is a lunisolar calendar used by Jews for predominantly religious purposes. It is used to reckon the Jewish New Year and dates for Jewish holidays, and also to determine appropriate public reading of Torah portions, Yahrzeits (dates to commemorate the death of a relative), and daily Psalm reading, among many ceremonial uses. Originally the Hebrew calendar was used by Jews for all quotidian purposes, but by the era of the Roman occupation (1st Century BCE), Jews were compelled to follow the imperial civil calendar for all civic matters such as the payment of taxes and dealings with government officials.

 

The Hebrew calendar's epoch (reference date), 1 Tishrei 1 anno mundi, is equivalent to Monday, October 7, 3761 BCE in the proleptic Julian calendar, the equivalent tabular date (same daylight period) and is about one year before the traditional Jewish date of Creation on 25 Elul AM 1, based upon the Seder Olam Rabbah of Rabbi Yossi ben Halafta, a second century CE sage.[1] Thus, adding 3760 or 3761[2] to any Julian/Gregorian year number after 1 CE will yield the Hebrew year. For earlier years there may be a discrepancy.

 

Two major forms of the calendar have been used. Before the destruction of the Second Temple in 70 CE, the calendar was observational, with the beginning of each month determined by the testimony of witnesses who had observed a new crescent moon. Between 70 and 1178 CE a rule-based fixed-arithmetic lunisolar calendar system was adopted to achieve the same effect.

 

The origins of the Hebrew calendar are found in the Torah, which refers to the existence of several numbered but un-named months in the Noahide (pre-Jewish) period, and which recounts several calendar-based commandments, including God's commandment during the Exodus from Egypt to fix the month of Nisan as the first month of the year.[3] The development of the calendar was likely influenced by the Babylonian exile in the 6th Century BCE, during which Babylonian names for the months were adapted; the Babylonians also employed a lunisolar calendar derived from the Sumerian calendar. Following the Jewish diaspora of Roman times (c. 1st Century CE), calculations were increasingly used to fix dates, with the principles fully described by Maimonides in 1178 CE in the Mishnah Torah.

 

Because of the roughly eleven-day difference between twelve lunar months and one solar year, the year lengths of the Hebrew calendar vary in a repeating 19-year Me-tonic cycle of 235 lunar months, with an intercalary lunar month added every two or three years, for a total of 7 times per 19 years. Seasonal references in the Hebrew calendar reflect its development in the region east of the Mediterranean Sea and the times and climate of the Northern Hemisphere. With respect to the present-day mean solar year, the Hebrew calendar's year is longer by about 6 minutes and 25+25/57 seconds, meaning that every 224 years, the Hebrew calendar will fall a full day behind the modern fixed solar year, and about every 231 years it will fall a full day behind the Gregorian calendar year. This is due to the 0.6 second discrepancy between the Calendric "Molad" (lunar conjunction interval), which is fixed by Jewish Law,[4] and the actual mean lunar conjunction interval, which itself is slowly changing over time.[5]

 

The Jewish calendar is a lunisolar calendar, or "fixed lunar year," based on twelve lunar months of twenty-nine or thirty days, with an intercalary lunar month added seven times every nineteen years (once every two to three years) to synchronize the twelve lunar cycles with the slightly longer solar year. Each Jewish lunar month starts with the new moon; although originally the new lunar crescent had to be observed and certified by witnesses, the timing of the new moon is now mathematically determined.

 

Concurrently there is a weekly cycle of seven days, mirroring the seven day period of the Book of Genesis in which the world is created. The names for the days of the week, like those in the Creation story, are simply the day number within the week, with Shabbat being the seventh day. The Jewish day runs from sunset to the next sunset, and accordingly, standard times and time zones have no place in the Jewish calendar.

 

The twelve regular months include: Nisan (30 days), Iyar (29 days), Sivan (30 days), Tammuz (29 days), Av (30 days), Elul (29 days), Tishrei (30 days), Cheshvan (29 or 30 days), Kislev (29 or 30 days), Tevet (29 days), Shevat (30 days), and Adar (30 days). In the leap years an additional month, Adar II (29 days) is added.

 

The first month of the year is Nisan, as ordained in the Bible. The 14th of Nisan is the start of the festival of Pesach, a date also prescribed in the Bible, corresponding to the full moon of Nisan. Though it is not expressly prescribed in these terms, Pesach is a spring festival, so the 14th of Nisan is the first full moon after the vernal equinox. Therefore, from the standpoint of determining the annual calendar cycle, the principal problem is that the lunar month/new moon of Nisan must occur before the spring equinox. Since at least the 12th Century, the Hebrew calendar has determined this time mathematically, but prior to this tradition held that the 1st of Nisan does not start (and an intercalary month would be added) "until the barley is ripe."

 

The day

The day is the smallest unit in the Jewish calendar. It is modeled on the Creation story and is of no fixed length. Based on the reference to "...there was evening and there was morning..."[6], the Jewish day runs from sunset (start of "the evening") to the next sunset. Accordingly, standard times and time zones have no place in the Jewish calendar. However, the steady progression of sunset around the world and seasonal changes results in gradual time changes from one day to the next based on observable astronomical phenomena (the sunset) and not on man-made laws and conventions.

 

Weeks

 

A bronze Shabbat candlestick holder made in Israel in the 1940s.The Hebrew calendar follows a seven-day weekly cycle, which runs concurrently but independently of the monthly and annual cycles. The names for the days of the week are simply the day number within the week. In Hebrew, these names may be abbreviated using the numerical value of the Hebrew letters, for example יום א׳ (Day 1, or Yom Rishon (Hebrew: יום ראשון):

 

Yom Rishon (Hebrew: יום ראשון), abbreviated יום א׳ = "first day" = Sunday

Yom Sheni (יום שני), abbr. יום ב׳ = "second day" = Monday

Yom Shlishi (יום שלישי), abbr. יום ג׳ = "third day" = Tuesday

Yom Reviʻi (יום רבעי), abbr. יום ד׳ = "fourth day" = Wednesday

Yom amishi (יום חמישי), abbr. יום ה׳ = "fifth day" = Thursday

Yom Shishi (יום ששי), abbr. יום ו׳ = "sixth day" = Friday

Yom Shabbat (יום שבת or more usually שבת - Shabbat), abbr. יום ש׳ = "Sabbath day (Rest day)" = Saturday

 

The names of the days of the week are modeled on the seven days mentioned in the Creation story. For example, Gen 1:5 "... And there was evening and there was morning, one day". "One day" also translates to "first day" or "day one". Similarly, Gen 1:8, Gen 1:13, Gen 1:19, Gen 1:23, Gen 1:31 and Gen 2.2.

 

The Jewish Shabbat has a special place in the Jewish weekly cycle. There are many special rules which relate to the Sabbath, discussed more fully in Shabbat.

 

In Hebrew, the word "Shabbat" (שַׁבָּת) can also mean "(Talmudic) week",[7] so that in ritual liturgy a phrase like "Yom Revi’i bəShabbat" means "the fourth day in the week".[8]

 

Importance of Lunar Months

Num 10:10 stresses the importance of the new moon and consequently lunar months, "... in your new moons, ye shall blow with the trumpets over your burnt-offerings,"[9]. Similarly in Num 28:11.

 

In his work Mishneh Torah of 1178, Maimonides included a chapter "Sanctification of the New Moon," in which he discusses the calendrical rules and their scriptural basis. He notes,

 

"By how much does the solar year exceed the lunar year? By approximately 11 days. Therefore, whenever this excess accumulates to about 30 days, or a little more or less, one month is added and the particular year is made to consist of 13 months, and this is the so-called embolismic (intercalated) year. For the year could not consist of twelve months plus so-and-so many days, since it is said: throughout the months of the year (Num 28:14), which implies that we should count the year by months and not by days."[10]

 

Months

 

Mosaic pavement of a zodiac in the 6th century synagogue at Beit Alpha, Israel.Biblical references to the pre-Jewish calendar include ten months identified by number rather than by name. In parts of the Torah portion Noach (Noah) (specifically, Gen 7:11, Gen 8:4-5, Gen 8:13-14) it is implied that the months are thirty days long.[11] There is no indication as to the total number of months in the annual cycle.

 

In the parts of the Tanakh (the Hebrew Bible) prior to the Babylonian exile, only four months are named: Aviv (first; literally "spring", which originally probably meant "the ripening of barley"), Ziv (second; literally "light"), Ethanim (literally "strong" in plural, perhaps referring to strong rains) I Kings 6:38: seventh month; and Bul I Kings 6:38: eighth month. All of these are Canaanite names, and at least two are Phoenician (Northern Canaanite).

 

The first commandment the Jewish people received as a nation was to determine the new moon: Exodus 12:2 states, "This month [Nissan] is for you the first of months." Deut 16:1 refers to a specific month: "Observe the month of Abib, and keep the Passover unto Yahvveh Elohim; for in the month of Abib Yahvveh Eloheinu brought you out of Egypt by night."

 

During the Babylonian exile, which started in 586 BCE, Jews adopted Babylonian names for the months, which are still in use. The Babylonian calendar also used a lunisolar calendar, derived from the Sumerian calendar.

 

Hebrew names and romanized transliteration may somewhat differ, as they do for כסלו / Kislev or חשוון / Marheshvan: the Hebrew words shown here are those commonly indicated e.g. in newspapers.

 

Measurement of the month

           

Animation of the Moon as it cycles through its phases, as seen from the Northern Hemisphere. The apparent wobbling of the Moon is known as libration.Synodic month

 

A synodic month (known in Hebrew as a "molad") is a scientifically defined lunar conjunction based on the position of the moon in the lunar orbit. It equals exactly 765433/25920 days, or 29 days, 12 hours, and 793 parts (44+1/18 minutes) (ie 29.5306 days). This interval matches the mean synodic month determined by the Babylonians before 300 BCE and as adopted by the Greek astronomer Hipparchus and the Alexandrian astronomer Ptolemy. Its remarkable accuracy is thought to have been achieved using records of lunar eclipses from the eighth to fifth centuries BCE, with a reference meridian midway between the Nile River and the end of the Euphrates River, about 4° east of Jerusalem.

 

The traditional new moon

 

A "new moon" is the day on which the first visible crescent of the moon is observed. It occurs 29 or 30 days after the preceding visible crescent and traditionally signaled the start of a Jewish lunar month.

 

Combining the observation method with the scientific lunar month length works as follows: assuming we start at a particular new month of 29 days. As the mean lunar month is 29.5306 days long, there would be a carry forward into the next month of 0.5306 days (ie 12 hours, 44+1/18 minutes). Adding that carry forward amount to the next month will make it equal 30.0612 days (30 days, 1 hour and 24+2/18 minutes). So the second month would be 30 days long, and 0.0612 days (or 1 hour 24+2/18 minutes) would be carried forward to be added to the next cycle, and so on. Then every 17 lunar months the carry forward amounts would exceed 24 hours (0.0612 x 17 = 1.0404), which would require an additional day to be added to that month. In summary, the progression becomes: year 1 | 29 - 30 - 29 -30 - 29 - 30 - 29 - 30 - 29 - 30 - 29 - 30 | year 2 | 29 - 30 - 29 - 30 - 30 - 29 - etc.

 

 

 

 

Hebrew names of the months with their Babylonian analogs

Number

Hebrew

Tiberian

Academy

Common/Other

Length

Babylonian analog

Notes

1

נִיסָן

Nīsān

Nisan

Nissan

30 days

Nisanu

called Aviv and Nisan in the Tanakh

2

אִיָּר / אייר

ʼIyyār

Iyyar

Iyar

29 days

Ayaru

called Ziv in the Tanakh

3

סִיוָן / סיוון

Sīwān

Siwan

Sivan

30 days

Simanu

 

4

תַּמּוּז

Tammūz

Tammuz

Tamuz

29 days

Du'uzu

 

5

אָב

ʼĀ

Av

Ab

30 days

Abu

 

6

אֱלוּל

ʼĔlūl

Elul

Elul

29 days

Ululu

 

7

תִּשׁרִי

Tišrī

Tishri

Tishrei

30 days

Tashritu

called Eitanim in the Tanakh.
Modern first month, Rosh Hashana is celebrated in Tishrei.

8

מַרְחֶשְׁוָן / מרחשוון

Marešwān

Mareshwan

Marcheshvan

29 or 30 days

Arakhsamna

often shortened to Cheshvan; called Bul in the Tanakh

9

כִּסְלֵו / כסלוו

Kislēw

Kislew

Kislev, Chisleu

30 or 29 days

Kislimu

also spelled Chislev

10

טֵבֵת

ēē

Tevet

Tebeth

29 days

Tebetu

 

11

שְׁבָט

Šəā

Shevat

Shvat, Shebat

30 days

Shabatu

 

12*

אֲדָר א׳

ʼĂār

Adar I*

 

30 days

Adaru

*Only in leap years

12 / 13*

אדר / אדר ב׳

 

Adar / Adar II*

 

29 days

 

 

Leap months

 

Detail of the Ishtar Gate of Babylon, dating from the era of the Babylonian captivity. Due to the difference in length between twelve lunar months and a solar year, a purely lunar calendar cycle would have resulted in the gradual shifting of the Hebrew calendar independently of the seasons. However, the Torah requires that certain festivals take place during certain seasons. This implies that a system of reconciling lunar months in the context of solar years and consequently seasons was in use. The Bible does not directly mention the addition of an "embolismic" or intercalary month that would prevent the drifting of the calendar year.

 

Whether or not an embolismic month was announced after the "last month" (Adar) depended on whether "the barley was ripe". It may be noted that in the Bible the name of the first month, Aviv, literally means "spring" but originally it probably meant "the ripening of barley". Thus, if Adar was over and the barley was not yet ripe, an additional month was observed. However, according to some traditions, the announcement of the month of Aviv could also be postponed depending on the condition of roads used by families to come to Jerusalem for Passover, adequate numbers of lambs to be sacrificed at the Temple, and on the ripeness of the barley that was needed for the first fruits ceremony.

 

Under the codified rules, the Jewish calendar is based on the Metonic cycle of 19 years, of which 12 are common years (12 months) and 7 leap years (13 months). The leap years are years 3, 6, 8, 11, 14, 17, and 19 of the Metonic cycle. Year 19 (there is no year 0) of the Metonic cycle is a year exactly divisible by 19 (ie when the Jewish year number, when divided by 19, has no remainder). In the same manner, the remainder of the division indicates the year in the Metonic cycle (years 1 to 18) the year is in.

 

During leap years a month, Adar II, is added before Nisan. During leap years Adar I (or Adar Aleph — "first Adar") is actually considered to be the extra month, and has 30 days. Adar II (or Adar Bet — "second Adar") is the "real" Adar, and has the usual 29 days. For this reason, during a leap year, holidays such as Purim are observed in Adar II, not Adar I.

 

New Year

 

A shofar made from a ram's horn is traditionally blown in observance of Rosh Hashanah; the beginning of the Jewish civic year. The Jewish year has four distinct starting points, according to the Mishnah, Rosh Hashanah 1:1:

 

The day most commonly referred to as the "New Year" is the first of Tishrei, when the formal New Year festival, Rosh Hashanah ("head of the year") is observed. (see Ezekiel 40:1, which uses the phrase "beginning of the year".) This is the beginning of the civil year, and the point at which the year number advances. Certain agricultural practices are also marked from this date.[12]

 

However, the first month of the year as prescribed in Exodus 12:2 is Nisan: "This month shall be to you the beginning of months". This means that the civil New Year, Rosh Hashanah, actually begins in the seventh month of the year.

 

The month of Elul is the New Year for counting animal tithes (ma'aser). Tu Bishvat ("the 15th of Shevat") marks the New Year for trees (and agricultural tithes).

 

There may be an echo here of a controversy in the Talmud about whether the world was created in Tishrei or Nisan; it was decided that the answer is Tishrei, and this is now reflected in the prayers on Rosh Hashanah.[13]

 

Epoch

 

The Jewish calendar uses a calendar era anno mundi ("in the year of the world"), abbreviated AM. Interestingly, the beginning of "year 1" is not Creation, but about one year before Creation. This caused the new moon of its first month (Tishrei) to be called molad tohu (the mean new moon of chaos or nothing).

 

Karaite interpretation

For several centuries, many Karaites, especially those outside Israel, followed the calculated Rabbinical calendar used by most Jews for the sake of convenience. However, in recent years most Karaites have chosen to again follow the observational method.

 

Karaites use the lunar month and the solar year, but the Karaite calendar differs from the Rabbinical calendar in two ways: the determination of the first day of each month (Rosh Chodesh), and the determination of the first month of the year (called aviv), which is the month in which Passover falls.

 

For Karaites, the beginning of each month, the Rosh Chodesh, can be calculated, but is confirmed by the observation in Israel of the first sightings of the new moon. This may result in an occasional variation of a maximum of one day , depending on the inability to observe the new moon. The day is usually "picked up" in the next month.

 

The addition of the leap month (Adar II) is determined by observing in Israel the ripening of barley (called abib), rather than using the calculated and fixed calendar of Rabbinic Judaism. The 4 rules of postponement are not applied, as they are not found in the Tanakh. Occasionally this results in Karaites being one month ahead of Jews using the calculated Rabbinic calendar. The "lost" month would be "picked up" in the next cycle when Karaites would observe a leap month while other Jews would not.

 

Observational principles

 

A stone (2.43x1 m) with Hebrew inscription "To the Trumpeting Place" is believed to be a part of the Second Temple.Before the destruction of the Second Temple in 70 CE and the general disruption of Jewish communal life after the Jewish-Roman Wars, and for a transitional period thereafter, the new moons/months were determined on an observational basis.

 

In Second Temple times (c. 518 BCE - 70 CE), the beginning of each lunar month was decided on the basis of two eyewitnesses testifying to having seen the new lunar crescent at sunset. Patriarch Gamaliel II (c. 100) asked the witnesses to select the appearance of the moon from a collection of drawings that depicted the crescent in a variety of orientations, only a few of which could be valid in any given month.[citation needed] According to tradition, these observations were compared against calculations made by the supreme Jewish court, the Sanhedrin. When thirty days elapsed since the last new moon, the witnesses were readily believed.

 

At first the beginning of each Jewish month was signaled to the communities of Israel and beyond by fires lit on mountaintops, but after the Samaritans and Boethusians began to light false fires, messengers were sent. The inability of the messengers to reach communities outside Israel before mid-month Mikra’Ey Kodesh (Succot, Passover) led outlying communities to celebrate scriptural festivals for two days rather than one, observing the second feast-day of the Jewish diaspora because of uncertainty of whether the previous month ended after 29 or 30 days.

 

1st-3rd Centuries CE

 

In Rome the Arch of Titus still stands, depicting the enslaved Judeans and objects from the Temple being brought to Rome. By the Roman era, some sects, such as the Essenes, used a solar calendar during the last two centuries BCE.

 

The Jewish-Roman wars of 66–73, 115–117, and 132–135 caused major disruptions in Jewish life, also disrupting the calendar. During the third and fourth centuries, Christian sources describe the use of eight, nineteen, and 84 year lunisolar cycles by Jews, all linked to the civil calendars used by various communities of Diaspora Jews, which were effectively isolated from Levant Jews and their calendar. Some assigned major Jewish festivals to fixed solar calendar dates, whereas others used epact’s to specify how many days before major civil solar dates Jewish lunar months were to begin.

 

The Talmud notes the irregular intercalation (adding of extra months) performed in three successive years in the early second century.[citation needed]

 

The Ethiopic Christian computus (used to calculate Easter) describes in detail a Jewish calendar which must have been used by Alexandrian Jews near the end of the third century.[14] These Jews formed a relatively new community in the aftermath of the annihilation (by murder or enslavement) of all Alexandrian Jews by Emperor Trajan at the end of the 115–117 Kitos War. Their calendar used the same epacts in nineteen year cycles that were to become canonical in the Easter computus used by almost all medieval Christians, both those in the Latin West and the Hellenist East. Only those churches beyond the eastern border of the Byzantine Empire differed, changing one epact every nineteen years, causing four Easters every 532 years to differ.

 

The transitional period

 

The period between 70 and 1178 saw a gradual transition from an observation based calendar to a purely mathematically calculated one. Except for the modern year number, the modern rules reached their final form before 820 or 921, with some uncertainty regarding when. The rules were codified in 1178 by Maimonides, who described all of the modern rules, including the modern epochal year.

 

Under the patriarchate of Rabbi Judah III (300-330) the testimony of the witnesses with regard to the appearance of the new moon was received as a mere formality, the settlement of the day depending entirely on calculation. This innovation seems to have been viewed with disfavor by some members of the Sanhedrin, particularly Rabbi Jose, who wrote to both the Babylonian and the Alexandrian communities, advising them to follow the customs of their fathers and continue to celebrate two days, an advice which was followed, and is still followed, by the majority of Jews living outside of Israel.

 

Traditionally intercalations were determined at meetings of a special calendar commission of the Sanhedrin. But Constantius II, following the precedents of Hadrian, prohibited the holding of such meetings. However, the Jewish community outside the land of Israel depended on the calendar sanctioned by the Judean Sanhedrin for the proper observance of the Jewish holidays. However, danger threatened the participants in that sanction and the messengers who communicated their decisions to distant congregations. Temporarily, to relieve the foreign congregations, Huna ben Abin once advised Rava not to wait for the official intercalation:

 

When you are convinced that the winter quarter will extend beyond the sixteenth day of Nisan declare the year a leap year, and do not hesitate (R. H. 21a).

There is a popular tradition, first mentioned by Hai Gaon (d.1038) that as religious persecutions continued, Patriarch Hillel II determined to provide an authorized mathematically-based calendar for all time to come, though by so doing he severed the ties which united the Jews of the diaspora to their mother country and to the patriarchate. It is recorded that the calendar was adopted at a clandestine, and maybe final, meeting of the Sanhedrin in 358, marking the last universal decision made by that body.

 

This explanation has been questioned. It is noted that the Talmud, which did not reach its final form until c. 500, does not mention the continuous calendar or even anything as mundane as either the nineteen-year cycle or the length of any month, despite discussing the characteristics of earlier calendars, suggesting the final form of the modern calendar was fixed subsequent to the sixth century.

 

Furthermore, Jewish dates during post-Talmudic times (specifically in 506 and 776) are impossible using modern rules, and all evidence points to the development of the arithmetic rules of the modern calendar in Babylonia during the times of the Geonim (seventh to eighth centuries), under the Abbasid Caliphate. The Babylonian rules required the delay of the first day of Tishrei when the new moon occurred after noon.

 

Most of the modern rules appear to have been in place by about 820, according to a treatise by the Muslim astronomer Muammad ibn Mūsā al-wārizmī (c. 780-850 CE) a Persian polymath noted for his contributions to Islamic mathematics, Islamic astronomy, Islamic astrology and geography. Al-Khwārizmī's study of the Jewish calendar, Risāla fi istikhrāj tarīkh al-yahūd "Extraction of the Jewish Era" describes the 19-year intercalation cycle, the rules for determining on what day of the week the first day of the month Tishrī shall fall, the interval between the Jewish era (creation of Adam) and the Seleucid era, and the rules for determining the mean longitude of the sun and the moon using the Jewish calendar.

 

One notable difference between the calendar of that era and the modern form was the date of the epoch (the fixed reference point at the beginning of year 1), which at that time was one year later than the epoch of the modern calendar.

 

In 921, Aaron ben Meir, a leader of the Jewish community in Palestine otherwise unknown to history, sought to return the authority for the calendar to the Land of Israel by asserting that the first day of Tishrei should be the day of the new moon unless the new moon occurred more than 642 parts (35⅔ minutes, where a "part" is 1/1080 of an hour or 1/18 of a minute or 3⅓ seconds) after noon, when it should be delayed by one or two days. He may have been asserting that the calendar should be run according to Jerusalem time, not Babylonian. Local time on the Babylonian meridian was indeed about 642 parts (35 minutes and 40 seconds) later than (ahead of) the meridian of Jerusalem, corresponding to a longitude difference of 8° 55'.

 

An alternative explanation for the 642 parts is that Ben Meir may have believed, along with many earlier Jewish scholars, in a Creation theology placing Creation in the Spring season, and that the calendar rules had been adjusted by 642 parts to fit in with an Autumn date. If Creation occurred in the Autumn, to coincide with the observance of Rosh Hashanah, the calculated time of New Moon during the six days of creation was on Friday at 14 hours exactly (counting from the day starting at 6pm the previous evening). However, if Creation actually occurred six months earlier, in the Spring, the new moon would have occurred at 9 hours and 642 parts on Wednesday.

 

In any event he was opposed by Saadiah Gaon of the Talmudic academy of Sura. Only a few Jewish communities accepted ben Meir's opinion, and even these soon rejected it. Accounts of the controversy show that all of the rules of the modern calendar (except for the epoch) were in place before 921.

 

Codification of the rules

 

In 1000, the Muslim chronologist al-Biruni also described all of the modern rules except that he specified three different epochs used by various Jewish communities being one, two, or three years later than the modern epoch. Finally, in 1178 Maimonides described all of the modern rules, including the modern epochal year.

 

In his work Mishnah Torah of 1078, Maimonides included a chapter "Sanctification of the New Moon," in which he discusses the calendrical rules and their scriptural basis. He notes,

 

"By how much does the solar year exceed the lunar year? By approximately 11 days. Therefore, whenever this excess accumulates to about 30 days, or a little more or less, one month is added and the particular year is made to consist of 13 months, and this is the so-called embolismic (intercalated) year. For the year could not consist of twelve months plus so-and-so many days, since it is said: throughout the months of the year (Num. 28:14), which implies that we should count the year by months and not by days."[15]

 

Maimonides continues, showing analytically how the scriptural procedure for determining the calendar must be flawed, something he could explain through his faith. He noted that non-Jewish savants had presented mathematically correct methods of calculating the potential visibility of the new crescent, and reasoned that since these methods exist, they must have been used by the Court and the record of their use lost.[16]

 

If one back-calculates the moments of the traditional moladot using modern astronomical calculations then the closest that their reference meridian of longitude ever got to Israel was midway between the Nile River and the end of the Euphrates River (about 4° east of Jerusalem), and that was in the era of the Second Temple.

 

From the times of the Amoraim (third to fifth centuries), calculations were increasingly used, for example by Samuel the astronomer, who stated during the first half of the third century that the year contained 365 ¼ days, and by "calculators of the calendar" circa 300. Jose, an Amora who lived during the second half of the fourth century, stated that the feast of Purim, 14 Adar, could not fall on a Sabbath nor a Monday, lest 10 Tishrei (Yom Kippur) fall on a Friday or a Sunday. This indicates a fixed number of days in all months from Adar to Elul, also implying that the extra month was already a second Adar added before the regular Adar.

 

Epoch

 

The Jewish calendar's reference point is traditionally held to be about one year before the Creation of the world.The Jewish calendar uses a calendar era anno mundi ("in the year of the world"), abbreviated AM. Interestingly, the beginning of "year 1" is not Creation, but about one year before Creation. This caused the new moon of its first month (Tishrei) to be called molad tohu (the mean new moon of chaos or nothing).

 

Its epoch (reference date), 1 Tishrei 1 AM, is equivalent to Monday, October 7, 3761 BCE in the proleptic Julian calendar, the equivalent tabular date (same daylight period). This date is about one year before the traditional Jewish date of Creation on 25 Elul AM 1, based upon the Seder Olam of Rabbi Yossi ben Halafta, a second century CE sage. (A minority opinion places Creation on 25 Adar AM 1, six months earlier, or six months after the modern epoch.) Thus, adding 3760 (from September-October through December, 3761) to any Julian/Gregorian year number after 1 CE will yield the Jewish year, ending in September-October, this roughly coincides with that Julian/Gregorian year. Owing to the slow drift of the modern Jewish calendar relative to the Gregorian calendar, this will be true for about another 20,000 years.

 

The traditional Jewish date for the destruction of the First Temple (3338 AM = 423 BCE) differs from the modern scientific date, which is usually expressed using the Gregorian calendar (586 BCE). The scientific date takes into account evidence from the ancient Babylonian calendar and its astronomical observations. In this and related cases, a difference between the traditional Jewish year and a scientific date in a Gregorian year results from a disagreement about when the event happened — and not simply a difference between the Jewish and Gregorian calendars. See the "Missing Years" in the Jewish Calendar.

 

Pattern of calendar years

The Jewish calendar is based on the Me-tonic cycle of 19 years, of which 12 are common years (12 months) and 7 leap years (13 months).

 

A Me-tonic cycle equates to 235 lunar months in each 19-year cycle. This gives an average of 6939 days, 16 hours and 595 parts for each cycle.

 

But due to the Rosh Hashanah postponement rules (see below), a cycle of 19 Jewish years can be either 6939, 6940, 6941, or 6942 days in duration. Since none of these values is evenly divisible by seven, the Jewish calendar repeats exactly only following 36,288 Me-tonic cycles, or 689,472 Jewish years. There is a near-repetition every 247 years, except for an excess of 50 minutes (905 parts).

 

There are 14 different patterns that Jewish years may take. Each of these patterns is called a "keviyah" (Hebrew for "a setting" or "an established thing"), and is distinguished by the day of the week for Rosh Hashanah of that particular year and by that particular year's length.

 

A Jewish non-leap year can only have 353, 354, or 355 days. A leap year can have 383, 384, or 385 days (always 30 days longer than the non-leap length).

 

A chaserah year (Hebrew for "deficient" or "incomplete") is 353 or 383 days long because a day is taken away from the month of Kislev. The Hebrew letter ח "het", and the letter for the weekday denotes this pattern.

A kesidrah year ("regular" or "in-order") is 354 or 384 days long. The Hebrew letter כ "kaf", and the letter for the week-day denotes this pattern.

A shlemah year ("abundant" or "complete") is 355 or 385 days long because a day is added to the month of Heshvan. The Hebrew letter ש "shin", and the letter for the week-day denotes this pattern.

 

Determining leap years

The Jewish leap years are years 3, 6, 8, 11, 14, 17, and 19 of the Me-tonic cycle. To determine whether a year is a leap year, find the remainder when dividing the Jewish year number by 19. If the remainder is 3, 6, 8, 11, 14 or 17, the year is a leap year and an extra month, Adar II, is added. If the remainder is zero, the year is also a leap year since year 19 of the Me-tonic cycle is a year exactly divisible by 19. Another way to check a specific year is to find the remainder in the following calculation: (7 x the Jewish year number + 1) / 19. If the remainder is less than 7, the year is a leap year.

 

A mnemonic word in Hebrew is GUCHADZaT "גוחאדז"ט" (the Hebrew letters gimel-vav-het aleph-dalet-zayin-tet, i.e. 3, 6, 8, 1, 4, 7, 9. See Hebrew numerals). A variant of this pattern of naming includes another letter which specifies the day of the week for the first day of Pesach (Passover) in the year.

 

Another memory aid notes that intervals of the major scale follow the same pattern as do Jewish leap years: a whole step in the scale corresponds to two common years between consecutive leap years, and a half step to one common year between two leap years.

 

Special holiday rules

Adjustments are needed to ensure certain holy days and festivals do or do not fall on certain days of the week in the coming year.

 

Rosh Hashanah postponement rules

 

Although simple math would calculate 21 patterns for calendar years, there are other limitations which mean that Rosh Hashanah may only occur on Mondays, Tuesdays, Thursdays, and Saturdays (the "four gates"), according to the following table:

 

Day of Week

Number of Days

Monday

353

355

383

385

Tuesday

354

 

 

384

Thursday

354

355

383

385

Saturday

353

355

383

385

 

 

 

 

 

 

 

 

That means that an extra day or days are added if Rosh Hashanah were to fall on the other 3 days.

 

Yom Kippur

 

Yom Kippur, on which no work can be done, can never fall on Friday (the day prior to the Sabbath), to avoid having the previous day's fast day still going on at the start of Sabbath. Thus some flexibility has been built in.

 

Leap years

 

In leap years, a 30 day month called Adar I is inserted immediately after the month of Shevat, and the regular 29 day month of Adar is called Adar II. This is done to ensure that the months of the Jewish calendar always fall in roughly the same seasons of the solar year, and in particular that Nisan is always in spring. Whether either Chesvan or Kislev both have 29 days, or both have 30 days, or one has 29 days and the other 30 days depends upon the number of days needed in each year. Thus a leap year of 13 months has an average length of 383½ days, so for this reason alone sometimes a leap year needs 383 and sometimes 384 days.

 

The 265 days from the first day of the 29 day month of Adar (i.e. the twelfth month, but the thirteenth month, Adar II, in leap years) and ending with the 29th day of Heshvan forms a fixed length period that has all of the festivals specified in the Bible, such as Pesach (Nisan 15), Shavuot (Sivan 6), Rosh Hashanah (Tishrei 1), Yom Kippur (Tishrei 10), Sukkot (Tishrei 15), and Shemini Atzeret (Tishrei 22).

 

Pesach

 

The festival period from Pesach up to and including Shemini Atzeret is exactly 185 days long. The time from the traditional day of the vernal equinox up to and including the traditional day of the autumnal equinox is also exactly 185 days long. This has caused some unfounded speculation that Pesach should be March 21, and Shemini Atzeret should be September 21, which are the traditional days for the equinoxes. Just as the Hebrew day starts at sunset, the Hebrew year starts in the Autumn (Rosh Hashanah), although the mismatch of solar and lunar years will eventually move it to another season if the modern fixed calendar isn't moved back to its original form of being judged by the Sanhedrin (which requires the Beit Hamikdash)

 

Measurement of hours

 

Every hour is divided into 1080 halakim or parts. A part is 3⅓ seconds or 1/18 minute. The ultimate ancestor of the helek was a small Babylonian time period called a barleycorn, itself equal to 1/72 of a Babylonian time degree (1° of celestial rotation). Actually, the barleycorn or she was the name applied to the smallest units of all Babylonian measurements, whether of length, area, volume, weight, angle, or time. But by the twelfth century that source had been forgotten, causing Maimonides to speculate that there were 1080 parts in an hour because that number was evenly divisible by all numbers from 1 to 10 except 7. But the same statement can be made regarding 360. The weekdays start with Sunday (day 1) and proceed to Saturday (day 7). Since some calculations use division, a remainder of 0 signifies Saturday.

 

While calculations of days, months and years are based on fixed hours equal to 1/24 of a day, the beginning of each halachic day is based on the local time of sunset. The end of the Shabbat and other Jewish holidays is based on nightfall (Tzeis Hacochavim) which occurs some amount of time, typically 42 to 72 minutes, after sunset. According to Maimonides, nightfall occurs when three medium-sized stars become visible after sunset. By the seventeenth century this had become three second-magnitude stars. The modern definition is when the center of the sun is 7° below the geometric (airless) horizon, somewhat later than civil twilight at 6°. The beginning of the daytime portion of each day is determined both by dawn and sunrise. Most halachic times are based on some combination of these four times and vary from day to day throughout the year and also vary significantly depending on location. The daytime hours are often divided into Shaos Zemaniyos or Halachic hours by taking the time between sunrise and sunset or between dawn and nightfall and dividing it into 12 equal hours. The earliest and latest times for Jewish services, the latest time to eat Chametz on the day before Passover and many other rules are based on Shaos Zemaniyos. For convenience, the day using Shaos Zemaniyos is often discussed as if sunset were at 6:00pm; sunrise at 6:00am and each hour were equal to a fixed hour. However, for example, halachic noon may be after 1:00pm in some areas during daylight saving time.

 

Irregularities and "Missing Years"

Main article: Missing Years

The traditional dates of events in Jewish history are often used interchangeably with the modern secular dates according to the Gregorian calendar. For example, year 3338 AM on the Hebrew calendar is typically equated to 586 BCE. Implicit in this practice is the view that if all the differences in structure between the Hebrew and Gregorian calendars are taken into consideration, the two dates can be derived from each other. This is not the case. If the traditional dates of events before the Second Temple era are assumed to be using the standard Hebrew calendar, they refer to different objective years than those of the secular dates. The discrepancy is some 165 years.

 

The conflict does not necessarily imply that either the traditional dates or the secular dates must be objectively wrong. It is possible that the traditional dates did not use a consistent calendar matching the year count of the standard Hebrew calendar. For instance, it could be that one or more substantial calendar shifts have occurred, or the years counted might in certain periods have differed from astronomical years. Taking into account the possibility of a changing structure of the Hebrew calendar, theoretically, both the traditional dates and those of secular scholars could be correct. Even so, the account of history in the traditional sourcebook Seder Olam Rabba, and in particular its description of the period of Persian domination, seems to be irrevocably at odds with modern scientific understanding.

 

Furthermore, the Modern Hebrew calendar cannot be used to calculate Biblical dates because new moon dates may be in error by ±2 days, and months may be in error by ±2 months. The latter accounts for the irregular intercalation (adding of extra months) that was performed in three successive years in the early second century, according to the Talmud.[citation needed]

 

Nevertheless, Harold Camping has studied the timeline frameworks derived from the Hebrew, lunar and, Gregorian calendars and has arrived at dates for the creation of the world, Noah’s flood, Israel's exodus from Egypt, and the Messiahs second coming[17].

 

Mean year length

 

The Hebrew calendar mean year is 365.2468 days long, or 365 days, 5 hours 55 minutes, and 25+25/57 seconds (the molad/monthly interval × 235 months per 19-year cycle ÷ 19 years per cycle). As the present-era mean northward equinoctial year is about 365 days 5 hours 49 minutes and zero seconds long, the Hebrew calendar mean year is about 6 minutes and 25+25/57 seconds longer than the modern fixed solar year. This amounts to approximately one day in every 224 years.

 

In addition, since the mean Gregorian calendar year is 365.2425 days (exactly 365 days 5 hours 49 minutes and 12 seconds) and the mean Hebrew calendar year is 365.2468 days, the Hebrew calendar falls behind the Gregorian calendar by about a day every 231 years.

 

Molad intervals

 

The source of the discrepancy is the difference between the molad interval and the actual lunar conjunction cycle. The molad interval is currently about 0.6 seconds too long, and the discrepancy is accumulating at an accelerating rate, since the mean lunation interval is progressively shortening due to gravitational tidal effects. The accumulated "error" since the era of Hillel II is such that the molad moments are now almost 1 hour and 40 minutes late relative to the mean lunar conjunctions at the original reference meridian, which was midway between the Nile River and the end of the Euphrates River. The modern molad moments match the mean solar times of the lunar conjunction moments near the meridian of Kandahar, Afghanistan, more than 30° east of Jerusalem.

 

In the present era actual lunar conjunction intervals can be as short as 29 days 6 hours and 30 minutes to as long as 29 days and 20 hours, astonishing variations range of about 13 hours and 30 minutes. Furthermore, due to the eccentricity of Earth's orbit, series of shorter lunations alternate with series of longer lunations, consequently the actual lunar conjunction moments can range from 12 hours earlier than to 16 hours later than the molad moment, in terms of Jerusalem mean solar time (make the conjunction moments 16 minutes earlier if referred to the original molad reference meridian midway between the Nile River and the end of the Euphrates River, about 4° east of Jerusalem). Today, in terms of the mean solar time at the meridian of Qandahar, Afghanistan the actual lunar conjunctions vary ±14 hours relative to the traditional moladot.

 

Measured on a strictly uniform time scale, such as that provided by an atomic clock, the mean synodic month is becoming gradually longer, but since due to the tides the Earth rotation rate slowing even more the mean synodic month is becoming gradually shorter in terms of mean solar time. The value 29-12-793 was almost exactly correct at the time of Hillel II and is now about 0.6 seconds per month too long. However, it is still the most correct value possible as long as only whole parts (1/18 minute) are used.

 

Implications for Jewish ritual

 

This figure, in a detail of a medieval Hebrew calendar, reminded Jews of the palm branch (Lulav), the myrtle twigs, the willow branches, and the citron (Etrog) to be held in the hand and to be brought to the synagogue during the holiday of sukkot, near the end of the autumn holiday season. Although the molad of Tishrei is the only molad moment that is not ritually announced, it is actually the only one that is relevant to the Hebrew calendar, for it determines the provisional date of Rosh HaShanah, subject to the Rosh Hashanah postponement rules. The other monthly molad moments are announced for mystical reasons. With the moladot on average almost 100 minutes late, this means that the molad of Tishrei lands one day later than it ought to in (100 minutes) ÷ (1440 minutes per day) = 5 of 72 years or nearly 7% of years!

 

Therefore the seemingly small drift of the moladot is already significant enough to affect the date of Rosh Hashanah, which then cascades to many other dates in the calendar year and sometimes, due to the Rosh Hashanah postponement rules, also interacts with the dates of the prior or next year. The molad drift could be corrected by using a progressively shorter molad interval that corresponds to the actual mean lunar conjunction interval at the original molad reference meridian. Furthermore, the molad interval determines the calendar mean year, so using a progressively shorter molad interval would help correct the excessive length of the Hebrew calendar mean year, as well as helping it to "hold onto" the northward equinox for the maximum duration.

 

If the intention of the calendar is that Passover should fall near the first full moon after the northward equinox, or that the northward equinox should land within one lunation before 16 days after the molad of Nisan, then this is still the case in about 80% of years, but in about 20% of years Passover is a month late by these criteria (as it was in Hebrew year 5765, an 8th year of the 19-year cycle = Gregorian 2005 AD). Presently this occurs after the "premature" insertion of a leap month in years 8, 19, and 11 of each 19-year cycle, which causes the northward equinox to land at exceptionally early moments in such years. This problem will get worse over time, and so beginning in Hebrew year 5817 the 3rd year of each 19-year cycle will also be a month late. Furthermore, the drift will accelerate in the future as perihelion approaches and then passes the northward equinox, and if the calendar is not amended then Passover will start to land on or after the summer solstice around Hebrew year 16652, or about 10885 years from the present. (The exact year when this will begin to occur depends on uncertainties in the future tidal slowing of the Earth rotation rate, and on the accuracy of predictions of precession and Earth axial tilt.)

 

The seriousness of the spring equinox drift is widely discounted on the grounds that Passover will remain in the spring season for many millennia, and the text of the Torah is generally not interpreted as having specified tight calendrical limits. On the other hand, the mean southward equinoctial year length is considerably shorter, so the Hebrew calendar has been drifting faster with respect to the autumn equinox, and at least part of the harvest festival of Sukkot is already more than a month after the equinox in years 9, 1, 12 and 4 of each 19-year cycle (these are the same year numbers as were mentioned for the spring season in the previous paragraph, except that they get incremented at Rosh HaShanah). This progressively increases the probability that Sukkot will be cold and wet, making it uncomfortable or impractical to dwell in the traditional succah during Sukkot. The first winter seasonal prayer for rain is not recited until Shemini Atzeret, after the end of Sukkot, yet it is becoming increasingly likely that the rainy season in Israel will start before the end of Sukkot.

 

"Rectifying" the Hebrew calendar

Given the importance, in Jewish ritual, of establishing the accurate timing of monthly and annual times, some futurist writers and researchers have considered whether a "corrected" system of establishing the Hebrew date is required, due to the small but accelerating changes in the actual lunar cycle interval. Further religious questions include how such a system might be implemented and administered throughout the diverse aspects of the world Jewish community.

 

It is traditionally held that the fixed arithmetic Hebrew calendar was established on the authority of Hillel ben Yehudah, President of the Sanhedrin in Hebrew year 4119, and therefore only an equal authority (the modern Sanhedrin) or a higher authority (the Messiah) can either amend it or reinstate the observational Hebrew calendar.

 

A 353-year leap cycle of 4366 months, including 130 leap months, along with use of a progressively shorter molad interval, could keep an amended fixed arithmetic Hebrew calendar from drifting for more than 7 millennia.[18]

 

Early Zionist pioneers were impressed by the fact that the calendar preserved by Jews over many centuries in far flung diasporas, as a matter of religious ritual, was geared to the climate of their original country: the Jewish New Year marks the moment of transition from the Dry Season to the Rainy one, and major Jewish Holidays such as Sukkot, Passover, or Shavuot correspond to major points of the country's agricultural year such as planting and harvest.

 

Accordingly, in the early 20th Century the Hebrew Calendar was re-interpreted as an agricultural rather than religious calendar. The Kibbutz movement was especially inventive in creating new rituals fitting this interpretation.

 

With the creation of the State of Israel the Hebrew Calendar was made its official calendar. New holidays and commemorations not derived from previous Jewish tradition invariably were to be defined according to their Hebrew dates - notably the Israeli Independence Day on Iyar 5, Jerusalem Reunification Day on 28 Iyar, and the Holocaust Commemoration Day on Nisan 27 (close to the Hebrew date of the start of the Warsaw Ghetto Uprising).

 

Nevertheless, since the 1950s the Hebrew calendar steadily declined in importance in Israeli daily life, in favor of the worldwide Gregorian calendar. At present, Israelis - except for the minority of religiously observant - conduct their private and public life according to the Gregorian calendar.

 

The Jewish New Year (Rosh Hashanah) is a two-day public holiday in Israel. However, since the 1980s an increasing number of secularist Israelis had taken up the habit of celebrating the Gregorian New Year (usually known as "Sylvester Night" - "ליל סילבסטר") by holding all-night parties on the night between December 31 and January 1. Prominent Rabbis have on several occasions sharply denounced this practice, but with no noticeable effect on the secularist celebrants.

 

The disparity between the two calendars is especially noticeable with regard to commemoration of the assassinated Prime Minister Yitzchak Rabin. The official Day of Commemoration, instituted by a special Knesset law, is marked according to the Hebrew Calendar - on Heshvan 12. However, left-leaning Israelis, who revere Rabin as a martyr for the cause of peace and who are predominantly secularist, prefer to hold their own mass memorial rallies on November 4. In some years, the two competing Rabin Memorial Days are separated by as much as two weeks.

 

The wall Calendars commonly used in Israel are hybrids - organized according to Gregorian rather than Jewish months, but beginning in September, where the Jewish New Year usually falls, and providing the Jewish date in small characters.

 

This article incorporates text from the 1901–1906 Jewish Encyclopedia, a publication now in the public domain.

 

1 A minority opinion places Creation on 25 Adar AM 1, six months earlier, or six months after the modern epoch.

 

2 Between September-October and December, ie, after Rosh Hashana, add 3761

 

3 Exodus 12:2

 

4 The time interval between two consecutive calendric moladot is fixed by halakha at a constant 29 days, 12 hours, 44 minutes and one heleq (=1 part = 3.33 seconds)

 

5 This interval grows longer by approximately 0.1 SI seconds in 500 years. The great astronomer, Rabbi Raphael Ha-Lewi of Hanover calls this molad the "correct molad" (Luhot Ha-Ibbur, part 1, 1756, title page).

 

6 Gen 1:5, Gen 1:8, Gen 1:13, Gen 1:19, Gen 1:23, Gen 1:31 and Gen 2.2.

 

7 For example, according to Morfix מילון מורפיקס, Morfix Dictionary, which is based upon Prof. Yaakov Choeka's Rav Milim dictionary. But the word meaning a non-Talmudic week is שָׁבוּע (shavua), according to the same "מילון מורפיקס".

 

8 For example, when referring to the daily psalm recited in the morning prayer (Shacharit).

 

9 Numbers 10:10.

 

10 Sanctification of the New Moon. Translated from the Hebrew by Solomon Gandz; supplemented, introduced, and edited by Julian Obermann; with an astronomical commentary by Otto Neugebauer. Yale Judaica Series, Volume 11, New Haven: Yale University Press, 1956

 

11 Gen 7:11 says "... on the seventeenth day of the second month—on that day all the springs of the great deep burst forth..." and Gen 8:3-4 say "...At the end of the hundred and fifty days the water had gone down, (4) and on the seventeenth day of the seventh month the ark came to rest on the mountains of Ararat..." There is an interval of 5 months and 150 days, making each month 30 days long.

 

12 See Maaser Rishon, Maaser Sheni, Maaser Ani.

 

13 The Code of Maimonides (Mishneh Torah), Book Three, Treatise Eight: Sanctification of the New Moon. Translated by Solomon Gandz. Yale Judaica Series Volume XI, Yale University Press, New Haven, Conn., 1956.

 

14 See also:History of the Jews in Egypt

 

15 Sanctification of the New Moon. Translated from the Hebrew by Solomon Gandz; supplemented, introduced, and edited by Julian Obermann; with an astronomical commentary by Otto Neugebauer. Yale Judaica Series, Volume 11, New Haven: Yale University Press, 1956

 

16 op.cit.

 

17 Camping, Harold (2005). Time Has an End: A Biblical History of the World 11,013 B.C.- 2011 A.D. Vantage Press. 

 

18 Bromberg, Irv. "The Rectified Hebrew Calendar.". Retrieved on 2007-10-31.

 

The Code of Maimonides (Mishneh Torah), Book Three, Treatise Eight: Sanctification of the New Moon. Translated by Solomon Gandz. Yale Judaica Series Volume XI, Yale University Press, New Haven, Conn., 1956.

 

Ernest Wiesenberg. "Appendix: Addenda and Corrigenda to Treatise VIII". The Code of Maimonides (Mishneh Torah), Book Three: The Book of Seasons. Yale Judaica Series Volume XIV, Yale University Press, New Haven, Conn., 1961. pp.557-602.

 

Samuel Poznanski. "Calendar (Jewish)". Encylopædia of Religion and Ethics, 1911.

 

al-Biruni. The Chronology of Ancient Nations, Chapter VII. tr. C. Edward Sachau. London, 1879.

 

F.H. Woods. "Calendar (Hebrew)", Encylopædia of Religion and Ethics, 1911.

 

Sherrard Beaumont Burnaby. Elements of the Jewish and Muhammadan Calendars. George Bell and Sons, London, 1901.

 

W.H. Feldman. Rabbinical Mathematics and Astronomy,3rd edition, Sepher-Hermon Press, 1978.

 

Otto Neugebauer. Ethiopic astronomy and computus. Österreichische Akademie der Wissenschaften, philosophisch-historische klasse, sitzungsberichte 347. Vienna, 1979.

 

Ari Belenkiy. "A Unique Feature of the Jewish Calendar — Dehiyot". Culture and Cosmos 6 (2002) 3-22.

 

Arthur Spier. The Comprehensive Hebrew Calendar. Feldheim, 1986.

 

Nathan Bushwick. Understanding the Jewish Calendar. Moznaim, 1989. ISBN 0940118173

 

L.A. Resnikoff. "Jewish calendar calculations", Scripta Mathematica 9 (1943) 191-195, 274-277.

 

Edward M. Reingold and Nachum Dershowitz. Calendrical Calculations: The Millennium Edition. Cambridge University Press; 2 edition (2001). ISBN 0-521-77752-6

 

Bonnie Blackburn and Leofranc Holford-Strevens. The Oxford Companion to the Year: An Exploration of Calendar Customs and Time-reckoning. Oxford University Press; USA, 2000. pp 723-730.