The Hebrew calendar (Hebrew: הלוח העברי ha'luach ha'ivri) or Jewish calendar is a lunisolar calendar used by Jews. Today, the calendar is predominantly used for religious observances and by all official institutions in the State of Israel, as well as by Jewish farmers in Israel as an agricultural framework.
The calendar 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 daily purposes. Following the conquest of Jerusalem by Pompey in 63 BCE (see also Iudaea province), Jews began additionally following the imperial civil calendar (which was decreed in 45 BCE) for civic matters such as the payment of taxes and dealings with government officials.
The principles of the Hebrew calendar are found in the Torah, which contains several calendar-related commandments, including God's commandment during the Exodus from Egypt to fix the month of Nisan as the first month of the year. The Babylonian exile in the 6th century BCE influenced the calendar, including the adoption of Babylonian names for the months.
During Temple times and through the Tannaitic period, the Hebrew calendar was observational, with the beginning of each month determined by the high court based on the testimony of witnesses who had observed a new crescent moon. Periodically, the court ordered an extra month added to keep Passover in the spring, again based on observation of natural events. Through the Amoraic period and into the Geonic period, the purely empirical calendar was displaced by calendrical rules, which finally became systematically arranged into a computed calendar. The principles and rules of the current calendar are fully described by Maimonides in the Mishneh 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 Metonic cycle of 235 lunar months, with an intercalary lunar month added according to defined rules 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. The Hebrew calendar's year is longer by about 6 minutes and 25+25/57 seconds than the present-day mean solar year, so 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.
Years in the Hebrew calendar are labeled with the era designation Anno Mundi (Latin for "in the year of the world"), abbreviated AM and A.M., (Hebrew: לבריאת העולם), and are numbered from the epoch that, by Rabbinical reckoning, is the date of the creation of Adam. 30 September 2008 through 18 September 2009 corresponded to Hebrew year 5769; the Hebrew year 5770 began at sundown on the evening of 18 September 2009 and will end on 8 September 2010.
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 determined mathematically.
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 always runs from sunset to the next sunset; the formal adjustments used to specify a standard time and time zones are not relevant to the Jewish calendar.
The twelve regular months are: 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 (29 days). In the leap years an additional month, Adar I (30 days) is added after Shevat, and the regular Adar is referred to as "Adar II".
The first month of the festival year is Nisan. 15 Nisan is the start of the festival of Pesach, corresponding to the full moon of Nisan. Pesach is a spring festival associated with the barley harvest, so the leap-month mentioned above is intercalated periodically to keep this festival in the northern hemisphere's spring season. Since the adoption of a fixed calendar, intercalations in the Hebrew calendar have been at fixed points in a 19-year cycle. Prior to this, the intercalation was determined empirically:
The year may be intercalated on three grounds: 'aviv [i.e.the ripeness of barley], fruits of trees, and the equinox. On two of these grounds it should be intercalated, but not on one of them alone.
The Bible designates Nisan, which it calls Aviv (Gezer calendar, an Israelite or Canaanite inscription ca. 900 BCE, also begins in the fall.), as the first month of the year ( ). At the same time, the season of the fall Festival of Booths (Sukkoth), is called "the end of the year" ( ). The Sabbatical year in which the land was to lie fallow, necessarily began at the time the winter barley and winter wheat would have been sown, in the fall. The
Modern practice follows the scheme described in the Mishnah: Rosh Hashanah, which means "the head of the year", and is celebrated in the month of Tishrei, is "the new year for years." This is when the numbered year changes, and most Jews today view Tishrei as the de facto beginning of the year.
Sources and history
The Torah contains several commandments related to the keeping of the calendar and the lunar cycle.
- For smaller units of time, see Measurement of hours below.
The Jewish day is of no fixed length. The Jewish day is modeled on the reference to "...there was evening and there was morning..." in the Creation story. Accordingly, it runs from sunset (start of "the evening") to the next sunset. However, some apply special rules at very high latitudes when the sun remains above or below the horizon for longer than a civil day.
There is no clock in the Jewish scheme, so that a civil clock is used. Though the civil clock incorporates local adoptions of various conventions such as time zones, standard times and daylight saving, these have no place in the Jewish scheme. The civil clock is used only as a reference point - in expressions such as: "Shabbat starts at ...". The steady progression of sunset around the world and seasonal changes results in gradual civil time changes from one day to the next based on observable astronomical phenomena (the sunset) and not on man-made laws and conventions.
Instead of the international date line convention, there are varying opinions as to where the day changes. One opinion uses the antimeridian of Jerusalem. (Jerusalem is 35°13’ east of the prime meridian, so the antimeridian is at 144°47' W, passing through eastern Alaska.) Other opinions exist as well.
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 (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 Chamishi (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, "... And there was evening and there was morning, one day". "One day" also translates to "first day" or "day one". Similarly, see , 1:13, 1:19, 1:23, 1:31 and 2.2.
Importance of lunar months
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 (), which implies that we should count the year by months and not by days."
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, , , ) it is implied that the months are thirty days long. 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 (12:2, , 23:15, 34:18, ) (first; literally "spring", which originally probably meant "the ripening of barley"); Ziv ( , 6:37) (second; literally "light"); Ethanim ( ) (seventh; literally "strong" in plural, perhaps referring to strong rains); and Bul ( ) (eighth). All of these are Canaanite names, and at least two are Phoenician (Northern Canaanite).
According to the Book of Exodus, the first commandment the Jewish people received as a nation was to determine the new moon: states, "This month [Nisan] is for you the first of months." refers to a specific month: "Observe the month of Aviv (HE: spring), and keep the passover unto the LORD thy God; for in the month of Aviv the LORD thy God brought thee forth 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, which was similar in structure to the Hebrew one.
Hebrew names and romanized transliteration may somewhat differ, as they do for חשוון / Marheshvan or כסלו / Kislev: the Hebrew words shown here are those commonly indicated e.g. in newspapers. The Syrian calendar used in the Levant countries shares many of the same names for months as the Hebrew calendar, such as Nisan, Iyyar, Tammuz, Ab, Elul, Tishri, and Adar.
|Length||Babylonian analog|| Holidays/|
|1||נִיסָן||Nīsān||Nisan||Nissan||30 days||Nisanu||Passover|| Called Abib (23:15, 34:18, ) , |
and Nisan ( ) in the Tanakh.
|2||אִיָּר / אייר||ʼIyyār||Iyyar||Iyar||29 days||Ayaru|| Pesach Sheni|
|Called Ziv in 6:37.,|
|3||סִיוָן / סיוון||Sīwān||Siwan||Sivan||30 days||Simanu||Shavuot|
|4||תַּמּוּז||Tammūz||Tammuz||Tamuz||29 days||Du'uzu||Seventeenth of Tammuz|
|5||אָב||ʼĀḇ||Av||Ab||30 days||Abu|| Tisha B'Av|
|7||תִּשׁרִי||Tišrī||Tishri||Tishrei||30 days||Tashritu|| Rosh Hashanah|
| Called Ethanim in . |
First month of civil year.
|8||מַרְחֶשְׁוָן / מרחשוון||Marḥešwān||Marẖeshwan|| Marcheshvan|
| 29 or |
|Arakhsamna||Called Bul in.|
|9||כִּסְלֵו / כסלוו||Kislēw||Kislew|| Kislev |
| 29 or |
|10||טֵבֵת||Ṭēḇēṯ||Tebeth||Tevet||29 days||Tebetu||Tenth of Tevet|
|11||שְׁבָט||Šəḇāṭ||Shevat|| Shvat |
|30 days||Shabatu||Tu B'Shevat|
|12*||אֲדָר א׳||Adar I*||30 days||*Only in leap years.|
|12 / 13*||אֲדָר / אֲדָר ב׳||ʼĂḏār||Adar / Adar II*||29 days||Adaru||Purim|
In a short (chaser) year, both Cheshvan and Kislev have 29 days. In a regular (kesidran) year, Cheshvan has 29 days and Kislev has 30 days. In a full (maleh) year, both Cheshvan and Kislev have 30 days.
The calendar rules have been designed to ensure that Rosh Hashanah does not fall on a Sunday, Wednesday or Friday. This is to ensure that Yom Kippur does not directly precede or follow Shabbat, which would create practical difficulties, and that Hoshana Rabbah is not on a Shabbat, in which case certain ceremonies would be lost for a year.
The solar year is about eleven days longer than twelve lunar months. The Bible does not directly mention the addition of "embolismic" or intercalary months. However, without the insertion of embolismic months, Jewish festivals would gradually shift outside of the seasons required by the Torah. This has been ruled as implying a requirement for the insertion of embolismic months to reconcile the lunar cycles to the seasons, which are integral to solar yearly cycles.
When the observational form of the calendar was in use, 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 (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.
The day most commonly referred to as the "New Year" is 1 Tishrei, when the formal New Year festival, Rosh Hashanah ("head of the year") is observed. (see , which uses the phrase "beginning of the year".) This is the civil new year, and the point at which the year number advances. Certain agricultural practices are also marked from this date.
However, the first month of the year referred to in Aviv (now called Nisan): "This month shall be to you the beginning of months". This is referred to as the ecclesiastical new year, which means that the civil new year, Rosh Hashanah, actually begins in the seventh month of the ecclesiastical year.is
Josephus, in the first century CE, states that while -
Moses...appointed Nisan...as the first month for the festivals...the commencement of the year for everything relating to divine worship, but for selling and buying and other ordinary affairs he preserved the ancient order [i. e. the year beginning with Tishrei]."
In ancient Israel, the start of the ecclesiastical new year (ie. Nisan) was determined by reference to Passover. Passover begins on the 14th day of the month of Nisan, ( ) which corresponds to the full moon of Nisan. As Passover is a spring festival, the 14th of Nisan begins on the night of a full moon after the vernal equinox. To ensure that Passover did not start before spring, the tradition in ancient Israel held that the 1st of Nisan would not start until the barley is ripe, being the test for the onset of spring. If the barley was not ripe an intercalary month (Adar II) would be added.
Edwin Thiele has concluded that ancient Kingdom of Israel counted years using the ecclesiastical new year (which was the practice of Babylon, as well as other countries of the region), while the Kingdom of Judah counted years using the civil new year, a practice followed to this day.
There may be an echo here of a controversy in the Talmud about whether the world was created in Tishrei or Nisan; it was determined that the answer is Tishrei, and this is now reflected in the prayers on Rosh Hashanah.
The 1st of Nisan is the new year for kings and feasts; the 1st of Elul is the new year for the tithe of cattle... the 1st of Tishri is the new year for years, of the years of release and jubilee years, for the planting and for vegetables; and the 1st of Shevat is the new year for trees-so the school of Shammai; and the school of Hillel say: On the 15th thereof.
Since about the third century CE, the Jewish calendar has used a calendar era anno mundi ("in the year of the world"), abbreviated AM. 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).
The Jewish calendar's epoch (reference date), 1 Tishrei 1 AM, is equivalent to Monday, 7 October 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. Thus, adding 3760 before Rosh Hashanah or 3761 after to a Julian or Gregorian year number after 1 CE will yield the Hebrew year. For earlier years there may be a discrepancy (see: "Missing Years" in the Hebrew Calendar).
The Jewish year starting on Rosh Hashanah, 1 Tishrei 5769 AM is equivalent to 29 September 2008.
Before the adoption of the current year numbering system, other systems were in use. For example, during the Greek period, Seleucid era counting was used: eg. 1 Maccabees 1:54. During the Baylonian captivity, Ezekiel counted the years from the first deportation, that of Jehoiachin, (eg. ). The era year was then called "year of the captivity of Jehoiachin". (eg. )
For several centuries, many Karaite Jews, especially those outside Israel, followed the calculated Rabbinic calendar, because it was not possible during the exile to retrieve accurate new moon sightings, and aviv barley data from the land of Israel, which had to be relayed to the entire Karaite Jewish community. However, since the establishment of the State of Israel, and especially since the Six Day War, most Karaite Jews have made Aliyah, and can now again use the observational calendar.
Karaites use the lunar month and the solar year, but the Karaite calendar differs from the Rabbinic calendar in a number of ways.
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 aviv), rather than using the calculated and fixed calendar of Rabbinic Judaism. Occasionally this results in Karaites being one month ahead of other 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.
Furthermore, the seasonal drift of the Rabbinic calendar is avoided, resulting in the years affected by the drift starting one month earlier in the Karaite calendar.
Also, the four rules of postponement of the Rabbinic calendar are not applied, as they are not found in the Tanakh. This affects the dates observed for all the Jewish holidays by one day.
Change to a calculated calendar
Persian period: evidence of the papyri
Calendrical evidence for the postexilic Persian period is found in papyri from the Jewish colony at Elephantine, in Egypt. These documents show that the Jewish community of Elephantine used the Egyptian and Babylonian calendars.
Later postexilic period: evidence of the Mishnah
In the Maccabean, Herodian, and Mishnaic periods, according to the evidence of the Mishnah and Tosefta, the Hebrew calendar operated on an observational basis. The beginning of each lunar month was decided on the basis of two eye witnesses testifying to the Sanhedrin to having seen the new lunar crescent at sunset. Patriarch Gamaliel II (c. 100) would ask 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. These observations were compared against calculations. 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 began to light false fires, messengers were sent. The inability of the messengers to reach communities outside Israel before mid-month High Holy Days (Succot and 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.
Evaluation of the Mishnaic evidence
It has been noted that the procedures described in the Mishnah and Tosefta are all plausible procedures for regulating an empirical lunar calendar. Fire-signals, for example, or smoke-signals, are known from the pre-exilic Lachish ostraca. Furthermore, the Mishnah contains laws that reflect the uncertainties of an empirical calendar. Mishnah Sanhedrin, for example, holds that when one witness holds that an event took place on a certain day of the month, and another that the same event took place on the following day, their testimony can be held to agree, since the length of the preceding month was uncertain. Another Mishnah takes it for granted that it cannot be known in advance whether a year's lease is for twelve or thirteen months. Hence it is a reasonable conclusion that the Mishnaic calendar was actually used in the Mishnaic period.
The accuracy of the Mishnah's claim that the Mishnaic calendar was also used in the late 2nd temple period is less certain. One scholar has noted that there are no laws from Second Temple period sources that indicate any doubts about the length of a month or of a year. This led him to propose that the priests must have had some form of computed calendar or calendrical rules that allowed them to know in advance whether a month would have 30 or 29 days, and whether a year would have 12 or 13 months.
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.
Most of the present rules of the calendar were in place by 823, according to a treatise by the Muslim astronomer al-Khwarizmi (c. 780–850 CE). Al-Khwarizmi's study of the Jewish calendar, Risāla fi istikhrāj taʾrīkh 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.
In 921, Aaron ben Meir proposed changes to the calendar. Though the proposals were rejected, it indicates that all of the rules of the modern calendar (except for the epoch) were in place before that date. In 1000, the Muslim chronologist al-Biruni described all of the modern rules of the Hebrew calendar, except that he specified three different epochs used by various Jewish communities being one, two, or three years later than the modern epoch.
Between 70 CE and 1178 CE, the observation based calendar was gradually replaced by a mathematically calculated one. Except for the epoch year number, the calendar rules reached their current form by the beginning of the 9th century, as described by al-Khwarizmi in 823.
There is a tradition, first mentioned by Hai Gaon (d.1038 CE), that Hillel b. R. Yehuda "in the year 670 of the Seleucid era" (i.e. 358–359 CE) was responsible for the new calculated calendar with a fixed intercalation cycle. Later writers, such as Nachmanides, explained Hai Gaon's words to mean that the entire computed calendar was due to Hillel b. Yehuda. Maimonides, in the 12th century, stated that the Mishnaic calendar was used "until the days of Abaye and Rava", who flourished ca. 320–350 CE, and that the change came when "the land of Israel was destroyed, and no permanent court was left." Taken together, these two traditions suggest that Hillel b. Yehuda (whom they identify with the mid-4th century Jewish patriarch Ioulos, attested in a letter of the Emperor Julian, and the Jewish patriarch Ellel, mentioned by Epiphanius) instituted the computed Hebrew Calendar because of persecution. H. Graetz attempted to link the introduction of the computed calendar to a sharp repression following a failed Jewish insurrection that occurred during the rule of Constantius and Gallus. A later writer, S. Lieberman, argued instead that the introduction of the fixed calendar was due to measures taken by Roman authorities to prevent the Jewish patriarch from sending calendrical messengers.
Both the tradition that Hillel b. Yehuda instituted the complete computed calendar, and the theory that the computed calendar was introduced due to repression or persecution, have been questioned. Furthermore, two Jewish dates during post-Talmudic times (specifically in 506 and 776) are impossible under the rules of the modern calendar, indicating that its arithmetic rules were developed in Babylonia during the times of the Geonim (seventh to eighth centuries). The Babylonian rules required the delay of the first day of Tishrei when the new moon occurred after noon.
The Talmuds do, however, indicate at least the beginnings of a transition from a purely empirical to a computed calendar. According to a statement attributed to Yose, an Amora who lived during the second half of the third century, 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 that, by the time of the redaction of the Jerusalem Talmud (ca. 400 CE), there were 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. In another passage, a sage is reported to have counseled "those who make the computations" not to set the first day of Tishrei or the Day of the Willow on the sabbath. This indicates that there was a group which "made computations" and was in a position to control, to some extent, the weekday on which Rosh Hashanah would fall.
Outside of Rabbinic circles, the evidence shows a diversity of Jewish practice. The Sardica paschal table shows that the Jewish community of some eastern city, possibly Antioch, used a calendrical scheme that kept Nisan 14 within the limits of the Julian month of March. Some of the dates in the document are clearly corrupt, but they can be emended to make the sixteen years in the table consistent with a regular intercalation scheme. Peter, the bishop of Alexandria (early 4th century CE), mentions that the Jews of his city "hold their Passover according to the course of the moon in the month of Phamenoth, or according to the intercalary month every third year in the month of Pharmuthi", suggesting a fairly consistent intercalation scheme that kept Nisan 14 approximately between the Phamenoth 10 (March 6 in the 4th century CE) and Pharmuthi 10 (April 5). Jewish funerary inscriptions from Zoar, south of the Dead Sea, dated from the 3rd to the 5th century CE, indicate that when years were intercalated, the intercalary month was at least sometimes a repeated month of Adar. But the inscriptions reveal no clear pattern of regular intercalations, nor do they indicate any consistent rule for determining the start of the lunar month.
In 1178, Maimonides included all the rules for the calculated calendar and their scriptural basis, including the modern epochal year in his work, Mishneh Torah. The rules detailed in Maimonides' code are those used throughout the Jewish world today.
Measurement of month
- Synodic month
A synodic month is the period between two lunar conjunctions, such as between two new moons. Since the actual length of a synodic month varies by several hours from month to month, the calendar is based on a long-term average length called the mean synodic month. The virtual lunar conjunctions at the start of each mean synodic month are called molads. The mean synodic month used in the Hebrew calendar is exactly days, or 29 days, 12 hours, and 793 parts (44+1/18 minutes) (ie 29.5306 days). This interval exactly matches the mean synodic month determined by the Babylonians before 250 BCE and as adopted by the Greek astronomer Hipparchus and the Alexandrian astronomer Ptolemy. Its remarkable accuracy (less than one second from the true value) is thought to have been achieved using records of lunar eclipses from the eighth to fifth centuries BCE.
- 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. Assume one begins 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.
Pattern of calendar years
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). A Metonic 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 Metonic 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" / "קביעה" (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. Both Kislev and Cheshvan have 29 days. 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. Kislev has 30 days and Cheshvan has 29 days. 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. Both Kislev and Cheshvan have 30 days. 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 Metonic 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 I, is added, preceding Adar II (sometimes called "the real Adar"). If the remainder is zero, the year is also a leap year since year 19 of the Metonic 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, with do corresponding to year 19 (or 0): 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. This connection with the major scale is more remarkable in the context of 19 equal temperament.
Special holiday rules
Adjustments are made to ensure certain holy days and festivals do or do not fall on certain days of the week.
- Yom Kippur
Adjustments are made to ensure that Yom Kippur, on which no work can be done, does not fall on Friday (the day prior to the Sabbath) to avoid having Yom Kippur's restrictions still going on at the start of Sabbath, or on Sunday (the day after Shabbat) to avoid having the Shabbat restrictions still going on at the start of Yom Kippur.
The Rosh Hashanah postponement rules are the mechanism used to make the adjustments. As Yom Kippur falls on Tishrei 10, and Rosh Hashanah falls on the 1st, the adjustment is made so that Rosh Hashanah does not fall on a Wednesday or Friday.
- Rosh Hashanah postponement rules
|Day of week||Number of days|
To ensure that Yom Kippur does not directly precede or follow Shabbat, and that Hoshana Rabbah is not on a Shabbat, in which case certain ceremonies would be lost for a year, the first day of Rosh Hashanah may only occur on Mondays, Tuesdays, Thursdays, and Saturdays (the "four gates"). Adjustments are made to ensure that Rosh Hashanah does not fall on the other three days. To achieve that result the year may be made into a short (chaser) year (both Kislev and Cheshvan have 29 days) or full (maleh) year (both Kislev and Cheshvan have 30 days). (see table)
- Short or full years
A leap year (ie. one which has 13 months) has an average length of 383½ days, so that in discrete numbers a leap year may have either 383 or 384 days.
Also, 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.
The period from 29 Adar (or Adar II, in leap years) to 29 Heshvan contains all of the festivals specified in the Bible - Pesach (15 Nisan), Shavuot (6 Sivan), Rosh Hashanah (1 Tishrei), Yom Kippur (10 Tishrei), Sukkot (15 Tishrei), and Shemini Atzeret (22 Tishrei). This period is fixed, during which no adjustments are made.
- Days of week of holidays
|10 Tevet||Tu Bishvat|
|Sun||Tue||Wed||Tue||Thu||Sat||Wed or Thu||Wed, Thu, or Fri||Tue, Wed, or Thu|
|Tue||Thu||Fri||Thu||Sat||Mon||Fri or Sat||Fri or Sun||Thu or Sat|
|Thu||Sat||Sun||Sun*||Mon||Wed||Sun or Mon||Sun or Tue||Sat or Mon|
|*Postponed from Shabbat|
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 (Tzeth haKochabim) 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 Sha`oth Zemaniyoth or "Halachic hours" by taking the time between sunrise and sunset or between dawn and nightfall and dividing it into 12 equal hours. The nighttime hours are similarly divided into 12 equal portions, albeit a different amount of time than the "hours" of the daytime. 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 Sha`oth Zemaniyoth. For convenience, the modern day using Sha`oth Zemaniyoth is often discussed as if sunset were at 6:00pm, sunrise at 6:00am and each hour were equal to a fixed hour. For example, halachic noon may be after 1:00pm in some areas during daylight saving time. Within the Mishnah, however, the numbering of the hours starts with the "first" hour after the start of the day. 
The Hebrew calendar's mean year is 365.2468 days long (exactly 365 days 5 hours 55 minutes and 25+25/57 seconds - ie. the molad/monthly interval × 235 months per 19-year cycle ÷ 19 years per cycle). As the present-era mean northward equinoctal year is about 365.2424 days long, the Hebrew calendar mean year is slightly longer than this tropical year. This results in a "drift" of the Hebrew calendar of about a day every 224 years.
Also, the mean Gregorian calendar year is 365.2425 days long (365 days 5 hours 49 minutes and 12 seconds), resulting in a drift of the Hebrew calendar in relation to the Gregorian calendar of about a day every 231 years.
The impact of the drift is reflected in the drift of the date of Passover from the vernal full moon:
|Year||Astronomical vernal full moon||Passover*|
|2001||8 April||8 April|
|2002||28 March||28 March|
|2003||16 April||17 April|
|2004||5 April||6 April|
|2005||25 March||24 April|
|2006||13 April||13 April|
|2007||2 April||3 April|
|2008||21 March||20 April|
|2009||9 April||9 April|
|2010||30 March||30 March|
|2011||18 April||19 April|
|2012||6 April||7 April|
|2013||27 March||26 March|
|2014||15 April||15 April|
|2015||4 April||4 April|
|2016||23 March||23 April|
|2017||11 April||11 April|
|2018||31 March||31 March|
|2019||21 March||20 April|
|2020||8 April||9 April|
|*Passover commences at sunset preceding the date indicated.|
The value 29d12h793p for the molad interval is identical to the value in the Babylonian System B (about 300 BCE), and in Ptolemy's Almagest (2nd century CE), and is approximately equal to 29.530594 days. This is as close to the correct value of 29.530589 days as it is possible for a value to come that is rounded off to whole parts (1/18 minute). So the molad interval is about 0.6 seconds too long. Put another way, if the molad is taken as the time of mean conjunction at some reference meridian, then this reference meridian is drifting slowly eastward. If this drift of the reference meridian is traced back to the mid-4th century CE, the traditional date of the introduction of the fixed calendar, then it is found to correspond to a longitude 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, a variation 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.
Furthermore, the discrepancy between the molad interval and the mean synodic month is accumulating at an accelerating rate, since the mean synodic month is progressively shortening due to gravitational tidal effects. 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 the tides slow Earth's rotation rate even more, the mean synodic month is becoming gradually shorter in terms of mean solar time.
Implications for Jewish ritual
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 (a modern Sanhedrin) 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.
Irregularities and "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, 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). 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. 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.
Usage in contemporary Israel
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 one of its official calendars (along with the Gregorian 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 5 Iyar, Jerusalem Reunification Day on 28 Iyar, and the Holocaust Commemoration Day on 27 Nisan (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, although the Hebrew calendar is still widely acknowledged, appearing in public venues such as banks (where it is legal for use on checks and other documents, though only rarely do people make use of this option) and on the mastheads of newspapers.
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 "Silvester Night" — "ליל סילבסטר") by holding all-night parties on the night between 31 December and 1 January. 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 12 Heshvan. 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 4 November. 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 — organised 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.
- ↑ Exodus 12:2
- ↑ The Babylonians also employed a lunisolar calendar derived from the Sumerian calendar.
- ↑ Josephus, Antiquities 3.248-251, Loeb Classical Library, 1930, pp. 437-438.
- ↑ Tosefta Sanhedrin 2.2, Herbert Danby, Trans., Tractate Sanhedrin Mishnah and Tosefta, Society for Promoting Christian Knowledge, London and New York, 1919, p. 31. Also quoted in Sacha Stern, Calendar and Community: A History of the Jewish Calendar Second Century BCE-Tenth Century CE, Oxford University Press, 2001, p. 70.
- ↑ , , , , , and .
- ↑ "In higher latitudes, where during the summer the sun does not sink below the horizon, and during the winter does not rise above it, the days are counted in summer from midday., i.e., from one upper crossing of the meridian by the sun to the next crossing; in the winter, from midnight to midnight, i.e., from one lower crossing of the meridian by the sun to the next," Entry "Calendar" in The Jewish Encyclopedia Volume 3, Funk and Wagnalls, New York, 1916.
- ↑ 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 (shavuʻa), according to the same "מילון מורפיקס".
- ↑ For example, when referring to the daily psalm recited in the morning prayer (Shacharit).
- ↑ Numbers 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
- ↑ says "... on the seventeenth day of the second month—on that day all the springs of the great deep burst forth..." and 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.
- ↑ See Maaser Rishon, Maaser Sheni, Maaser Ani.
- ↑ Josephus, Antiquities 1.81, Loeb Classical Library, 1930.
- ↑ The barley had to be "eared out" (ripe) in order to have a wave-sheaf offering of the first fruits according to the Law. Jones, Stephen (1996). Secrets of Time.
- ↑ Edwin Thiele, The Mysterious Numbers of the Hebrew Kings, (1st ed.; New York: Macmillan, 1951; 2d ed.; Grand Rapids: Eerdmans, 1965; 3rd ed.; Grand Rapids: Zondervan/Kregel, 1983). ISBN 082543825X, 9780825438257
- ↑ 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.
- ↑ M. Rosh Hashanah 1, in Herbert Danby, trans., The Mishnah, Oxford University Press, 1933, p. 188.
- ↑ A minority opinion places Creation on 25 Adar AM 1, six months earlier, or six months after the modern epoch.
- ↑ The Karaite Korner: The New Moon in the Hebrew Bible
- ↑ The Karaite Korner: Aviv (Barley)
- ↑ Sacha Stern, "The Babylonian Calendar at Elephantine", Zeitschrift für Papyrologie und Epigraphik 130 , 159–171(2000).
- ↑ Lester L. Grabbe, A History of the Jews and Judaism in the Second Temple Period, Volume 1: Yehud: A History of the Persian Province of Judah, T&T Clark, London, 2004, p. 186.
- ↑ M. Rosh Hashanah 1.7
- ↑ M. Rosh Hashanah 2.6-8
- ↑ b. Rosh Hashanah 20b: "This is what Abba the father of R. Simlai meant: 'We calculate the new moon's birth. If it is born before midday, then certainly it will have been seen shortly before sunset. If it was not born before midday, certainly it will not have been seen shortly before sunset.' What is the practical value of this remark? R. Ashi said: Confuting the witnesses." I. Epstein, Ed., The Babylonian Talmud Seder Mo'ed, Soncino Press, London, 1938, p. 85.
- ↑ M. Rosh Hashanah 2.2
- ↑ b. Betzah 4b
- ↑ Sacha Stern, Calendar and Community, Oxford University Press, 2001, pp. 162ff.
- ↑ James B. Pritchard, ed., The Ancient Near East: An Anthology of Texts and Pictures, Vol. 1, Princeton University Press, p. 213.
- ↑ M. Sanhedrin 5.3: "If one testifies, 'on the second of the month, and the other, 'on the third of the month:' their evidence is valid, for one may have been aware of the intercalation of the month and the other may not have been aware of it. But if one says, 'on the third', and the other 'on the fifth', their evidence is invalid."
- ↑ M. Baba Metzia 8.8.
- ↑ Solomon Gandz, "The origin of the Two New Moon Days", Jewish Quarterly Review (New Series), v. 40, 1949-50. Reprinted in Shlomo Sternberg, ed., Studies in Hebrew Astronomy and Mathematics by Solomon Gandz, KTAV, New York, 1970, pp. 72-73.
- ↑ 33.0 33.1 E.S. Kennedy, "Al-Khwarizmi on the Jewish calendar", Scripta Mathematica 27 (1964) 55–59.
- ↑ 34.0 34.1 "al-Khwarizmi", Dictionary of Scientific Biography, VII: 362, 365.
- ↑ See The Remaining Signs of Past Centuries.
- ↑ Sacha Stern, Calendar and Community.
- ↑ Julian, Letter 25, in John Duncombe, Select Works of the Emperor Julian and some Pieces of the Sophist Libanius, Vol. 2, Cadell, London, 1784, pp. 57-62.
- ↑ Epiphanius, Adversus Haereses 30.4.1, in Frank Williams, trans., The Panarion of Epiphanius of Salamis Book I (Sects 1-46), Leiden, E. J.Brill, 1987, p. 122.
- ↑ H. Graetz, Popular History of the Jews, (A. B. Rhine, trans.,) Hebrew Publishing Company, New York, 1919, Vol. II, pp. 410-411. Quoted in Sacha Stern, Calendar and Community, p. 216.
- ↑ S Lieberman, "Palestine in the 3rd and 4th Centuries", Jewish Quarterly Review, New Series 36, pp. 329-370(1946). Quoted in Sacha Stern, Calendar and Community, pp. 216-217.
- ↑ Sacha Stern, Calendar and Community: A History of the Jewish Calendar Second Century BCE-Tenth Century CE, Oxford University Press, 2001. In particular section 5.1.1, discussion of the "Persecution theory."
- ↑ Samuel Poznanski, "Ben Meir and the Origin of the Jewish Calendar", Jewish Quarterly Review, Original Series, Vol. 10, pp. 152-161(1898).
- ↑ "While it is not unreasonable to attribute to Hillel II the fixing of the regular order of intercalations, his full share in the present fixed calendar is doubtful." Entry "Calendar", Encyclopedia Judaica, Keter, Jerusalem, 1971.
- ↑ Samuel Poznanski, "Calendar (Jewish)", Encyclopaedia of Religion and Ethics, vol. 3.
- ↑ Yerushalmi Megillah 70b.
- ↑ Yerushalmi Sukkah 54b.
- ↑ Eduard Schwartz, Christliche und jüdische Ostertafeln, (Abhandlungen der königlichen Gesellschaft der Wissenschaften zu Göttingen. Philologisch-Historische Klasse. Neue Folge, Band viii, Berlin, 1905.
- ↑ Peter of Alexandria, quoted in the Chronicon Paschale. Corpus Scriptorum Historiae Byzantinae, Chronicon Paschale Vol. 1, Weber, Bonn, 1832, p. 7
- ↑ Sacha Stern, Calendar and Community, pp. 87-97, 146-153.
- ↑ Neugebauer, Astronomical cuneiform texts, Vol 1, pp 271-273
- ↑ G. J. Toomer, Hipparcus' Empirical Basis for his Lunar Mean Motions, Centaurus, Vol 24, 1980, pp. 97-109
- ↑ 52.0 52.1 Otto Neugebauer, "The astronomy of Maimonides and its sources", Hebrew Union College Annual 23 (1949) 322–363.
- ↑ See, for example, Berachot chapter 1, Mishnah 2.
- ↑ Towards a common date of Easter World Council of Churches, 1997.
- ↑ Bromberg, Irv. ""The Rectified Hebrew Calendar."". http://individual.utoronto.ca/kalendis/hebrew/rect.htm. Retrieved 2007-10-31.
- al-Biruni. The Chronology of Ancient Nations, Chapter VII. tr. C. Edward Sachau. London, 1879.
- Ari Belenkiy. "A Unique Feature of the Jewish Calendar — Dehiyot". Culture and Cosmos 6 (2002) 3-22.
- 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 * Sherrard Beaumont Burnaby. Elements of the Jewish and Muhammadan Calendars. George Bell and Sons, London, 1901.
- Nathan Bushwick. Understanding the Jewish Calendar. Moznaim, 1989. ISBN 0940118173
- 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.
- 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.
- Samuel Poznanski. "Calendar (Jewish)". Encylopædia of Religion and Ethics, 1911.
- Edward M. Reingold and Nachum Dershowitz. Calendrical Calculations: The Millennium Edition. Cambridge University Press; 2 edition (2001). ISBN 0-521-77752-6
- L.A. Resnikoff. "Jewish calendar calculations", Scripta Mathematica 9 (1943) 191-195, 274-277.
- Eduard Schwartz, Christliche und jüdische Ostertafeln, (Abhandlungen der königlichen Gesellschaft der Wissenschaften zu Göttingen. Philologisch-Historische Klasse. Neue Folge, Band viii, Berlin, 1905.
- Arthur Spier. The Comprehensive Hebrew Calendar. Feldheim, 1986.
- Sacha Stern, Calendar and Community: A History of the Jewish Calendar Second Century BCE-Tenth Century CE, Oxford University Press, 2001.
- 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.
- F.H. Woods. "Calendar (Hebrew)", Encylopædia of Religion and Ethics, 1911.
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