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The Jewish calendar--a mix of astronomy and theology.

The lunisolar Jewish calendar is an example par excellence of "time engineering." It is based on the relation between the mean period of the lunar motion about the earth and the mean period of the earth about the sun (the tropical year). This relation is that, to six places of decimals, the number of days in 19 tropical years is equal to the number of days in 235 lunar cycles and is known as the metonic cycle. Moreover, the Jewish calendar, unlike our civil calendar, for example, is intertwined with the days of the week by virtue of the first day of Tishri (Rosh Hashanah) being allowed--for religious reasons--only to occur on Monday, Tuesday, Thursday, and Saturday. Because the metonic-cycle relation is not exact, there will eventually be a significant seasonal drift in holidays such as Passover unless the calendar construction is modified appropriately.

I. Introduction

The Jewish calendar plays a central role in the lives of Jews of various affliations and degrees of commitment. However, other than being vaguely aware of the seasons of the year in which the different holidays and festivals occur, many Jews have only a limited acquaintance with the astronomical and historical/theological roots of this calendar. This lack of knowledge is certainly not due to a dearth of authoritative calendrical references. Indeed, there are many easily available sources of information on the Jewish calendar, including monographs, book chapters, lengthy encyclopedia and journal articles, translations of traditional sources with commentaries, and a recent spate of internet websites. For a list of these sources, see Appendix A.

This paper is intended to give a reasonably concise presentation of how astronomy and theology are intertwined in the Jewish calendar. We feel that its mix and depth of discussion of essential elements of the Jewish calendar covered in a short paper is fairly unique. It is mainly intended for readers who would like an initial overview of the subject as a prelude to more in-depth studies of the Jewish calendar.

The dates of all important holidays--Passover, Rosh Hashanah, Yom Kippur, etc.--are fixed by this calendar. The first day of Rosh Hashanah, for example, occurs on the first day of the month of Tishri, and Passover begins on the 15th day of the month of Nisan. Of particular interest here is the fact that the eight days of Passover always occur in the Spring--late March/early April--and Rosh Hashanah invariably begins a half year later in the fall--September/October. In this respect the Jewish calendar mirrors our civil calendar, which is a solar calendar in which the four seasons-spring, summer, fall, winter--recur annually on the same dates.

By contrast to our civil calendar, however, the Jewish calendar is not only a solar calendar but in addition it is also a lunar calendar, in that it closely follows the cycles of the moon. Thus not only does the first night of Passover, on the 15th of Nisan, always occur in the early spring, but what is not as well known nor often as appreciated is that, on that night of the first seder, the moon is invariably full! Similarly, there is always a full moon on the first night of Succoth on the 15th of Tishri and invariably a new (crescent) moon two weeks earlier on the first night of Rosh Hashanah on the first day of Tishri.

To more fully appreciate this important aspect of the Jewish calendar, of its being synchronized with both the solar and the lunar cycles, let us briefly review some features of our civil calendar. Recall that the 21st of March (1) always marks the first day of spring (in the northern hemisphere, of course), that the 21st of June marks the first day of summer, and so on for the 21st of September and the 21st of December marking the first days of fall and winter, respectively. Thus, as designed, our civil calendar correlates fully with the four seasons. By contrast, there is no correlation whatsoever between any particular date on our civil calendar and the phases of the moon. On any particular date it is possible for the moon to be in any one of its phases, from new moon through first quarter, to full moon, to third quarter, and back to new moon.

This complete lack of correlation between our civil calendar and the lunar cycle is illustrated in Table I, which shows the number of days to the next full moon for three arbitrarily chosen dates: January 1, July 4, and September 21 for the five years 2000-2004.

Since there are very slightly more than 29.5 days in a complete lunar cycle, the entry "7" for the 1st of January 2000, for example, means that the full moon appeared 7 days afterward on the 8th of January 2000. Similarly the entry "25" for July 4, 2003 means that the moon was full about 4 (= 29.5-25) days earlier on the 30th of June, 2003 and will be full again on the 29th of July. Clearly, there is no correlation whatsoever between dates on our civil calendar and the lunar cycle. Given any particular date and any preassigned lunar phase, that phase will eventually occur on that date. Note, however, that the existence in our civil calendar of the 12 months January, February,..., December, gives evidence to the lunar origins of this calendar.

The problem facing the designer of the Jewish calendar is thus that of reconciling the capriciousness, as shown in Table I, of the lunar cycle vis-a-vis the solar cycle, and to create a calendar that is simultaneously in synchronization with both the seasons and the lunar cycle.

Before turning to the Jewish calendar itself, let us first briefly consider our civil calendar somewhat more closely. Our present civil calendar is known as the Gregorian calendar, (2) or sometimes as the "Gregorian reform of the Julian calendar." It is based on the fact that the solar (tropical) year (3) consists of 365.2422 days, (4) or slightly less than 365 1/4 days. So, for example, in order for the first day of spring to occur on the same date (March 21) each year over the course of many years, it is clear that the calendar should, in some sense, consist of exactly 365.2422 days. Since the length of the day and of the year are determined, respectively, by the period of rotation of the earth about its axis and the period of revolution of the earth about the sun--neither of which is in humankind's power to influence--and since for calendrical purposes we require a year to consist of an integral number of days, a compromise has to be made.

Some 2000 years ago, it was believed, incorrectly, that there were exactly 365 1/4 days in the year. At the behest of Julius Caesar in 45 BCE the Roman senate set up the "Julian" calendar, which was destined to be widely used for over 1600 years until late in the 16th century. This calendar collated years into groups of four, with three successive years of 365 days followed by a fourth year, known as a leap year and consisting of 366 days. In this way, on the average, each year consisted of 365 1/4 days. At some point it was decided that years of ordinal number divisible by 4, such as 1936, 2004, etc., would be leap years each of 366 days, with the remaining years with ordinal numbers not divisible by 4, such as 1938, 2001, etc., to consist of 365 days. In this way it was believed the Julian calendar of average length 365.25 days would be able to keep in step with the seasons for ever. But this was not to be!

The difference between the actual number of days in the year of 365.2422 and the average number of days in the year according to the Julian calendar, 365.2500, is quite small; it corresponds to a difference of 0.0078 days per year or 0.78 days per century! This difference, albeit negligible compared to a human lifetime, over longer periods corresponds to an extra day every 128 (=1/.0078) years. And over the course of a thousand years or so this calendar is not stable relative to the seasons since, for example, the first day of spring would occur a day earlier every 128 years. Thus after about 1300 years the first day of spring--as determined by astronomical observations--on the Julian calendar occurred not on the 21st of March but on or about the 10th of March. And it was this slippage of the first day of spring from the 21st of March into the "winter" months that the Gregorian calendar was designed to rectify.

The Gregorian calendar, which is essentially in universal use today, was promulgated by Pope Gregory XIII in 1582 CE and gradually adopted/recognized by most countries--some only very recently; for example by the Soviet Union in 1917. It is a relatively small modification of the Julian calendar, and as introduced by Pope Gregory XIII in 1582 the Gregorian reform of the Julian calendar mandated:

(1) The elimination of 10 days from the Julian calendar by relabeling 5 October 1582 as 15 October 1582; and

(2) Continued use of this relabeled Julian calendar with the added provision that the only century years that would be leap years of 366 days would be those divisible by 400.

The latter implies that the century years 1700, 1800, 1900, which are divisible by 4 and which would have been leap years on the Julian calendar, would not be leap years of 366 days on the reformed calendar, since they are not also divisible both by 4 and 400. On the other hand, the century year 2000, which is divisible by 400, would be retained as a leap year. Thus, in every 400-year cycle of the new calendar there would be 97 leap years of 366 days and 303 regular years of 365 days.

The effect of this change, then, is that on the average the Gregorian year has 365.2425 = ((303 x 365 + 97 x 366)/400) days and not 365.2500 as did the Julian year. The difference between the Gregorian 365.2425 days and the actual year of 365.2422 days is .003 days per year, corresponding to an extra day only every 3333 years! Thus by means of this relatively minor change, the slippage of 1 day every 128 years of the Julian calendar was reduced to a slippage of 1 day every 3333 years in the Gregorian calendar! Evidentially it will not be necessary for humankind to revisit this matter for the next ten or so millennia.

It is also of interest to contrast the Gregorian solar calendar with the Islamic calendar, which is a purely lunar calendar. The latter consists of a year of 12 months of alternately 30 and 29 days just as does the Jewish calendar. A full year on this calendar thus consists of 354 (= 12 x 29.5) days. Since the period of the lunar cycle is actually 29.53059 (5) days and not 29.5 days, it is necessary from time to time to add an extra day to the year in order to keep the months in step with the phases of the moon. This is achieved by operating on a 30-year cycle, with the twelfth month of Dha al-Miljah having 30 days instead of the normal complement of 29 during the eleven years numbered 2,5, 7,10,13,16, 18, 21, 24, 26, and 29 of the cycle. In this way the average length of the month, as can be easily calculated, is 29.53056 days and compares very favorably with the actual 29.53059 days. In a given year, if the sun is at the vernal equinox on the first day of the first month of the year, Muharram, for example, then since the sun will return to this point approximately 365 1/4 days later, the following year the sun will be at the vernal equinox on the 11th day of Muharram; the year following on the 22nd day of Muharram and the one following that on the 4th day of Safar, etc. It follows that there is no simple correlation between this calendar and the seasons. The first day of spring, for example, moves progressively through the months of the Islamic year. After about 32 1/2 such years, early in the month of Muharram the sun will again be at the vernal equinox.

Returning to our earlier theme, let us note here that the introduction of the Gregorian calendar in no way affects the total lack of synchronization between our civil calendar and the lunar cycle.

II. Origins of the Jewish Calendar: The Metonic Cycle

The problem of designing a calendar which would faithfully follow both the solar and the lunar cycles seems at first glance to be a very formidable--if not impossible--task. Included among the externally imposed constraints that the Jewish calendar designer must address are the following:

1. The solar (tropical) year consists of 365.2422 days;

2. The lunar cycle consists of 29.53059 days;

3. The calendar year must consist of an integral number of days;

4. There is a calendar unit of the month, whose length on the average must be the same as that of a lunar cycle and which at the same time must consist of an integral number of days;

5. The calendar year must consist of an integral number of months;

6. There is the calendar unit of the week consisting of exactly seven days, Sunday ... Saturday. There is no requirement for a year or a month having an integral number of weeks;

7. The first day of Rosh Hashanah, the first of Tishri, cannot be on a Sunday, Wednesday, or Friday. (This singular requirement is to satisfy the religious constraints that Hashanah Rabah on the 21st of Tishri cannot fall on the Sabbath, and that Yom Kippur on the 10th of Tishri not fall on a Friday or Sunday. As an aside, it is a consequence of the first of these requirements that Yom Kippur can never fall on a Tuesday).

8. A month can only have 29 or 30 days. In particular, months of length 31 days are not permitted.

To satisfy any one or two of these eight conditions would be straightforward. For example, to satisfy #1 and #3 alone we could simply introduce the leap years mandated by the Gregorian calendar. Similarly to satisfy #2 and #4 alone, we could introduce the year as a sequence of twelve months which alternately consisted of 29 and 30 days with an occasional "leap" month to account for the "0.03059" in #2. Such a twelve-month year would have only 354 days--11 short of 365--and thus would require a 13th month about every three years. Brief reflection shows that to satisfy all eight of the above requirements simultaneously would seem to be virtually impossible! However, there is a curious relationship between the number of days in a solar year and those in a lunar cycle as given in #1 and #2, respectively, that nevertheless makes a lunar-solar (or lunisolar) calendar possible; this relationship is known as the Metonic cycle.

In the 5th century BCE, the Athenian astronomer Meton first pointed out that the number of days in 19 solar years is almost exactly the same as the number of days in 235 lunar cycles (or lunations). This period of time is now known as the Metonic cycle and served as the basis for the Babylonian calendar from which the Jewish calendar was originally derived. According to #1, we find 19 solar years have, to 7 decimal places, 19 x 365.2422 = 6939.602 days. Similarly according to #2, we find that 235 lunations consist of 235 x 29.53059 = 6939.689 days. Surprisingly, these agree to five places of decimals! To put the matter in an alternate form, the ratio of the number of days in 235 lunations to the number of days in 19 solar years is given by (235 x 29.53059) / (19 x 365.2422) = 6939.689 / 6939.602 = 1.000013 and is unity to five places of decimals. Disregarding the small decimal at the end of this ratio, to a high degree of accuracy, there are almost precisely 235 lunar cycles in 19 solar years. It is perhaps interesting to note that this relationship would be "exact" if there were 365.2468 days (6) in a year instead of the actual number of 365.2422.

With the Metonic cycle as background the basic structure of the Jewish calendar can be easily described qualitatively. For this purpose it is convenient to suppose, in the following, that the length of the lunar cycle is precisely 29.5 days and not the actual 29.53059 days and to fine-tune the result later by including the effects of the small decimal .03059.

Just as the Gregorian calendar groups years into 400-year cycles, so the Jewish calendar is concerned with years grouped into 19-year cycles. To satisfy the condition that there must be 235 months or lunar cycles in each 19 year solar cycle, the individual years in each such cycle are divided into one of two types: 12 ordinary years each of 12 months and the remainder of 7 leap years each consisting of 13 months. A brief calculation shows this to add up to 235 = (12 x 12 + 7 x 13) months for the full 19-year cycle. Table II lists the names of the months for both ordinary years and leap years and the number of days, 29 or 30, in each month. Note that for ordinary years the numbers of days alternate between 30 and 29 so that on the average there would be 29.5 days in each month, Thus, Tishri, Kislev, Shevat, Nisan, Sivan, and Ab each consist of 30 days and the remaining six months consist of 29 days for a total of 354 days for a regular year. For a leap year of 13 months, the line-up in Table II is the same except that the 29-day month of Adar becomes two months: Adar I and Adar II with 30 and 29 days respectively. Thus there are 384 days in a leap year. Note that the extra month has 30 days--thus upsetting the balance of the 30-29 day alternation. But this is in the right direction and brings the average number of days per month in a leap year from 29.5 to 29.53846, which is also slightly greater than the actual value in #2 of 29.53059. For the entire 19-year cycle, including both ordinary and leap years, the average number of days in a month is 29.5149, which is slightly greater than the regular year average of 29.5 days but goes in the direction of the required 29.53059 days of #2.

Which years in a given 19-year cycle are selected to be leap years are uniquely specified to be the years numbered 3, 6, 8, 11, 14, 17, and 19, i.e., seven years in total. The remaining twelve years numbered 1, 2, 4, etc. are the ordinary years. For those familiar with modular arithmetic, there is a formula which gives the leap years as follows:

(7N + 1)Mod 19 [less than or equal to] 7

where N is presumed to be one of the numbers 1,2,...,19 and is the ordinal number of the year in the cycle. So for a given cycle, years 1 and 2 are regular years while year number 3 is a leap year, etc.

To determine the ordinal number of a year on the Jewish calendar given the year of the Gregorian calendar (or vice versa), we can see that such a year is equal to the Gregorian year plus 3760 before 1 Tishri, and the Gregorian year plus 3761 after 1 Tishri. Recall that the first day of Rosh Hashanah occurs on the first day of Tishri. Accordingly, during the spring of 2000, for example, the Jewish year was 2000 + 3760 = 5760 (recall, Tishri is in the fall), while after the 30th of September--or Tishri 1--of 2000 it is 5761 (= 2000 + 3761). Similarly, before Rosh Hashanah of the Jewish year 5783 (26 September 2022), the year of the Gregorian calendar will be 5783 - 3760 = 2023, and so forth.

To determine whether any given year is a leap year or not, let us proceed using as an example the year 5761 above. Since 5761 can be written as 19 x 303 + 4, it follows that during the winter months of 2000 (November, say) we were in the fourth year of the 303rd 19-year cycle since the "beginning of time." Since the fourth year of a 19-year cycle is not a leap year, it follows that the year 5761 was a regular year. Correspondingly, the Jewish year 5689 will be a leap year since 5689 = 19 x 299 + 8 and the eighth year of a cycle is a leap year, according to the list above.

In Table III we present actual data for the 19 years of the current, 303rd, 19-year cycle of the Jewish calendar. The third column shows the civil date of the first of Tishri which, as we shall see, is almost always within a day or so of the new moon (Molad). Note that according to the fourth column, and consistent with #7 on page 6, the first day of Tishri never occurs on a Sunday, Wednesday, or Friday. The last column presents the actual number of days in the given year. We see that, as will be explained below, a regular (non-leap) year can have 353, 354, or 355 days corresponding to what is known as a defective, ordinary, or full year, respectively. Correspondingly, for leap years there can be 383, 384, or 385 days, corresponding to a defective, ordinary, or full leap year, respectively. This matter will be discussed in more detail below. If we add up the number of days in each year for all 19 years of the 303rd cycle, we find in Table III that the total number of days in this 19-year cycle is 6941, just slightly over the exact 6939.689 days required for 235 lunar cycles. If we consider other 19-year cycles, we find that on the average the number of days in each cycle is closer to 6939.689. For example the 301st and 302nd 19-year cycles have 6939 and 6940 days, respectively, so the average number of days in these three 19-year cycles is 6940! It should be emphasized that Table III represents actual data and not the data on the calendar that would result from direct application of the lengths of the months as presented in Table II.

III. The Calendar Council

To provide some perspective on the evolution of the Jewish calendar, in this section we consider some historical aspects of this calendar. As presented here, this is not a definitive historical account but rather is a more simplified or idealized discussion. For more scholarly discussions, the interested reader is referred to the work of M. Friedlander and C. Roth (see Appendix).

During the approximately 800 years beginning in the 5th century BCE, the period of the second temple, until almost 300 years after its destruction, there was no fixed Jewish calendar. Originally, the basics of the Jewish calendar, as we know it, were adapted from the Babylonians during the Babylonian captivity. As a reminder of this we note that the names of the months of the Jewish calendar in Table II have their counterparts in the names of the corresponding Babylonian months: Tashritu, Arakhasamna, Kislim, Tebetu, Shabatu, Adaru, Nisanu, Ayaru, Simanu, Diuzu, Abu, Ululu which also alternated between 30 and 29 days with a thirteenth month added periodically to keep the Babylonian calendar lunisolar. The Jewish calendar, during this 800-year period, was determined on a month-to-month basis by a group of Jewish leaders, originally the high priests but later consisting of a three member council called the Calendar Council (Sod Hadibbur). Its members were the president (Hanasi) plus two other members of the Sanhedrin, who we may presume were generally skilled in astronomy and mathematics. On occasion, the membership of the Council was increased to five and sometimes even to seven.

One of the main purposes of the Council was to have the first day of each month coincide with the new moon (Molad). It achieved this by selecting the number of days in each month, to be 29 or 30. Also by taking advantage of the Metonic cycle it continuously set the calendar by intercalating an extra month into the year as necessary to keep it synchronized with the solar cycle. In this ad hoc way, it was possible for the months of the year to remain more or less faithful to the lunar cycle while at the same time satisfying the second requirement, of consistency with the seasons. Thus the first day of Passover would always occur in the spring on the 15th of Nisan, and the first day of the harvest festival, Succoth, would invariably occur in the fall on the 15th day of Tishri. Specifically, the Calendar Council was empowered to:

(1) determine and set the first day of each month by direct observation of lunar phases and;

(2) declare a leap year by intercalating an additional leap month when necessary to maintain the balance between the lunar and solar cycles.

The institution of the Calendar Council was obviously a very rational way to keep the seasons fixed to a lunar calendar over long periods of time. Incidentally, the names of the months for a regular year as listed in Table II--but not necessarily the number of days in each month--were those used throughout this 800-year period.

While it was in session, the Calendar Council used to sit in Jerusalem and on the 30th day of any given month await the arrival of witnesses who could testify to having "seen" the new moon. The Council would interrogate these witnesses, and if it were confirmed that they had indeed seen a growing crescent moon, for example, and not a waning one and so forth, the 30th day of the given month would then become the first day of the new month. If no witnesses appeared, the next day, the 31st day of the previous month, would automatically become the first day of the new month. In this way, the months consisted of either 29 or 30 days. If on the day of the new moon it was cloudy, that month would automatically have 30 days. If three successive months had cloudy skies there would be a serious problem. The Calendar Council was given some leeway but operated under the constraint that even though an ordinary year had 354 days, they could change this only by [+ or -] 1 day, so that under no circumstances could an ordinary year have less than 353 days nor more than 355 days. This practice continues to be maintained in today's Jewish calendar, and is reflected in Table III.

In making certain that the calendar followed the solar cycle, the Council was guided by the principle that the first day of spring--the day the sun crosses the equator in a northerly direction at the vernal equinox--must precede the first day of Passover on the 15th of Nisan. If the Calendar Council determined that the sun would traverse the vernal equinox after the 16th of Nisan, it would declare a leap year by the addition of an extra month, thereby postponing Nisan by 30 days. Thus the Calendar Council had the ability and the authority to adjust the calendar at will in its responsibility to maintain the balance between a lunar and a solar year.

Curiously enough, both the king and the high priest were singled out and specifically excluded from membership on the Council, the former, presumably, because he might be tempted to vote in favor of a leap year to lessen the strain on the treasury in making annual disbursements for the civil service and the army. Similarly, we may suppose that the high priest was excluded since he might be tempted to vote against a leap year to increase the chances of Yom Kippur--when he had to immerse his body in spring water five times--falling during a warmer time of the year!

This ad hoc process of setting the calendar served the Jews for about 800 years, through the middle of the 4th century C.E. It was at this latter time, with the Diaspora escalating, that Hillel II promulgated our present system of calendar calculation, which up to then may have been known to the Calendar Council and used by them, perhaps, to check the accuracy of witnesses and to determine the precise time of the sun's passage through the vernal equinox. Hillel's algorithm, which also utilizes the Metonic cycle, is consistent with the 8 requirements on page 6 and is in the form of a complex formula that fixes the lengths of the months and the occurrence of leap years. The resultant calendar has been used for the last 1600 years and has remained substantially synchronized with both the motion of the sun and of the moon.

IV. Fine Tuning the Jewish Calendar

Consider again the entries in Table III for the 303rd 19-year cycle of the Jewish calendar, 5758-5776, corresponding to the civil calendar years 1997-2015. Imagine starting a calendar with the first day of Tishri of the year 5758 on Thursday, 2 October 1997 and assigning to the succeeding days the dates in accordance with the lengths of the months in Table II. Thus each of the 235 months of the cycle would alternate between 29 and 30 days as shown in the Table. The 19 years of the cycle would then consist of 12 ordinary years, each of 354 days, plus 7 leap years each of 384 days. In this way we would not come up with an acceptable Jewish calendar as detailed in Table III! For one thing the total number of days in this 303rd 19-year cycle would have only 6936 (= 12 x 354 + 7 x 384) days and thus would be 3 or 4 days short of the 6939.689 days the moon requires to circuit the earth 235 times. For another, the first of Tishri on some of the succeeding years of the cycle would fall on the "forbidden" days of Sunday, Wednesday, or Friday, thus violating constraint #7 on page 7. The entries in Table III are correct but can obviously not be obtained by direct application of the lengths of the months as given in Table II. Clearly, something additional is required to construct the Jewish calendar.

One of the difficulties with the above procedure is that it assumed that the lunar cycle consisted of exactly 29.5 days, when we know for a fact that it consists of 29.53059 days. For an ordinary year of 12 months this difference amounts to a little more than a third of a day (12 x 0.03059 = .367) or about 8.8 hours per year, which is equivalent to about 7 days (= 19 x 8.8hrs = 6.97 days) per 19-year cycle. It is for this reason, of course, that the 19-year cycle is divided into 12 ordinary years and 7 leap years, thereby producing the 7 extra days required by the "0.03059" in the lunar cycle of 29.53059. By introducing 7 leap years, then, we have satisfied completely all of the 8 conditions on pages 6-7 except for #7 which mandates a certain unambiguous connection between the days of the week and the first of Tishri. This restriction of the day on which the first of Tishri can occur is perhaps the most unusual aspect of the Jewish calendar. It ties together various astronomical periods of time--the day, the year, and the lunar cycle--to the 7-day week, which as we know has a non-astronomical origin and is based more on ecclesiastical and/or astrological considerations.

To see the way the Jewish calendar is structured--so that the first of Tishri does not fall on a Sunday, Wednesday, or Friday--takes several steps. The first involves allowing the calendar year to have a (restricted) variable number of days. Specifically, the ordinary years of the Jewish calendar can assume lengths of 353, 354, or 355 days while the leap years can have lengths of 383, 384, or 385 days. An ordinary year of 353 days is called a defective year, one of 354 days a regular year, and one of 355 days a full year. And similarly, for defective, ordinary, and full leap years the numbers of days are 383, 384, and 385, respectively. For the 303rd 19-year cycle, the number of days in each year is given by the last column of Table III. We see that for this 19-year cycle, only the years 5761 and 5774 are ordinary defective years, while there are 6 ordinary regular years and 4 ordinary full years. Correspondingly, for the leap years, there are 2 defective years, zero regular ones and 5 full years. The total number of days in this 303rd cycle adds up to 6941, which is slightly larger than 235 lunar cycles of 6939.692 days, a matter that will be further discussed below.

Setting aside for the moment the question of determining which of the various possibilities of any given year is appropriate, let us first look at the given mechanics of how this is done. We make use of the assignment of the number of days in each month given in Table II, except by making a slight adjustment for the number of days during the months of Kislev and Marcheshvan. For an ordinary regular year of 354 days the assignment is precisely that in Table II. For an ordinary defective year of 353 days, one day is deleted from the month of Kislev, thus reducing its number of days from 30 to 29. Finally, for an ordinary full year of 355 days, we add one day to Marcheshvan, thereby increasing its number of days from 29 to 30. And precisely in the same way we modify Kislev and Marcheshvan to produce regular, defective, and full leap years. None of the other months are ever modified in any way. We already know from the paragraph below Table II which years of a given 19-year cycle are leap years, so now we need to describe how we determine which years are defective, regular, or full. In fact, depending on which day of the week of a given year the first of Tishri falls and on which day it falls in the immediately following year, there are 7 distinct types of ordinary years and 7 distinct types of leap years. These 14-year types follow from the requirement that the first day of Tishri, and thus, of Rosh Hashanah, occur on the day of the new moon (Molad) except for the following four postponements (Dehioth) (7):

1. When the Molad Tishri occurs on a Sunday, Wednesday, or Friday, the first day of Rosh Hashanah is postponed to the following day.

[As noted previously, if Rosh Hashanah were to occur on a Sunday, Hoshana Rabah on the 21st day of Tishri would fall on the Sabbath, and this would interfere with the ceremony connected with the willow branches. If it were to fall on a Sunday or a Wednesday, Yom Kippur on the tenth day of Tishri would fall next to the Sabbath on Friday or Sunday and thus cause difficulties in the preparation of meals.]

2. When the Molad Tishri occurs at noon (18h) or later, the first day of Tishri is postponed to the next day. If this day is a Sunday, Wednesday, or Friday, the first day of Rosh Hashanah is further postponed to Monday, Thursday, or the Sabbath, respectively, in accordance with Dehiah 1.

[Most scholars of the Jewish calendar have suggested that this postponement rule is introduced in order to guarantee the visibility of the New Moon on the first day of Rosh Hashanah. However, Shocken (8) has stated (without explanation) and Landau (9) has shown mathematically that it ensures that no month begins before the actual setting of the New Moon.]

3. When the Molad Tishri of an ordinary year falls on a Tuesday after 9:11:20 (i.e., 3:11:20 am) Rosh Hashanah is postponed to Wednesday, and because of Dehiah 1, is further postponed until Thursday.

[It can be shown that this two-day postponement prevents an ordinary year from having 356 days and, in combination with Dehioth 1 and 2, limits ordinary years to 353, 354, or 355 days.]

4. When, in an ordinary year that follows a leap year, the Molad Tishri occurs on a Monday at 15:32:43 (9:32:43 am) or later, Rosh Hashanah is postponed to Tuesday.

[This postponement prevents a leap year from having 382 days and, in combination with Dehioth 1 and 2, limits leap years to 383, 384, or 385 days.]

(Note that in stating these Dehioth and in the following we use the notation "a:b:c" to indicate a time, or time interval of a-hours b-minutes and c-seconds. A civil calendar time will be recognized by the appearance of "am" or "pm." Also note that by contrast to our civil calendar, where each day begins at midnight, i.e. 12:00:01 am, for the Jewish calendar the day begins at 6:00:01 pm of the preceding day and also runs for 24 hours. Thus, Tuesday 8:00:00 pm of our civil calendar corresponds to Wednesday 2:00:00 of the Jewish calendar. Correspondingly, Wednesday 18:00:00 on the Jewish calendar corresponds to Wednesday noon or 12:00:00 pm of our civil calendar.)

It is not necessary at this point to understand in detail these four rules for postponements, and we list them mainly for the sake of completeness. The fourteen possible years that result from the application of these postponements are presented in Table IV. This table shows the day of Rosh Hashanah of a given year, that of the immediately following year as well as the number of days in the given year. Presented are the first days of Tishri for both regular and leap years, which, it should be noted, do not necessarily correspond to the Molads of Tishri.

Let us also consider in a little more detail how the first day of Tishri of a given year is ascertained. As the four dehioth imply, it is determined in the first instance by the Molad of Tishri, i.e., the day and time of the first new moon following the month of Elul (see Table II). In turn this Molad is itself determined not by direct astronomical observation of the new moon--as in the time of the Calendar council--but rather, following the procedure laid down by Hillel II, by calculating the number of lunations or lunar cycles, since the "beginning of time." (10) The astronomical and traditional theological considerations that have been used to characterize the first several Moladot Tishri at the "beginning of time" are rather complicated. (11) According to one of the main rabbinical traditions, the world was conceived at 3:35:40 am on a Wednesday (i.e., 9:35:40 on a Wednesday), but the first new moon did not occur until the moon and the sun had been created six lunation periods later on Molad Tishri of Jewish calendar year 2 at exactly 8:00:00 am on Friday (i.e., at exactly the start of the 14th hour of Friday, Jewish time). (12) In order to make life easier for calendar calculators, Maimonides introduced an "imaginary" Molad Tishri for year 1 of the Jewish calendar that started 12 lunation periods prior to Molad Tishri for year 2, i.e., at 11:11:20 pm on Sunday (or 5:11:20 of Monday). Calendar calculators can thus conveniently start with the time of this "fictitious" Molad Tishri of year 1 and proceed to calculate the times of all subsequent moladot by simply adding the appropriate month number times 29.53059 days. In particular, we can calculate the date and time of the first Molad following the month of Elul for any given year. The actual day of 1 Tishri, and thus of the first day of Rosh Hashanah then follows by application of the above Dehioth or postponements. It has been shown mathematically by use of the latter that the only possible years are those detailed in Table IV. In particular, no ordinary years of length 352 or 356 days--nor any leap years of length 382 or 386 days--can ever occur.

Let us illustrate this procedure for determining the length of a calendar year by considering a few special cases. For this purpose it is simplest to have available the total time required for twelve and for thirteen lunation periods, as follows:

A: 12 average lunation periods = 50 weeks + 4d 8:48:40

B: 13 average lunation periods = 54 weeks + 5d 21:32:43 1/3

Consider, for example, the case that the Molad of Tishri for an ordinary year occurs on Monday at 6:00:00 (i.e., at midnight 12:00 am). Then the Molad of Tishri of the immediately following year would occur, according to A, four days later in the week on Friday at 6:00:00 + 8:48:40 = 14:48:40. It follows, according to Dehioth #1, that Rosh Hashanah would have to be postponed to Saturday and the year would be full with 355 days. Or consider the original Molad on Monday at 6:00:00 but for a leap year. This time by B the Molad for the next Tishri would occur on Saturday 6:00:00 + 21:32:43 1/3 or on a Sunday at 3:32:43 1/3. Again by Dehioth #1 Rosh Hashanah would have to be postponed to Monday and the leap year would have 385 days. Clearly one can use this procedure to work out all the possibilities and confirm that the only possible types of years are those listed in Table IV.

Because of the four Dehioth, the lengths, in days, of various Metonic-cycle periods will vary slightly from one period to another. However, extensive numerical studies of the Jewish calendar, such as the one of Remy Landau (see Appendix), show that the average length of a year remains essentially the same as the value of 365.2468 days attributed to Rav Adda. Since this value is larger than the modern value of 365.2422 days by about six minutes and thirty seven seconds, there is on average a yearly advance, by this difference, of the time of occurrence of holidays such as Passover. As a consequence, today we celebrate Passover and other holidays about four and a half days later, on average, than our ancestors did 1000 years ago, and the celebration times will advance on average by about a full month during the next 6500 years. As for all other calendars that have ever been proposed this slight shortcoming in the Jewish calendar has been known for a long time, and has not been a priority concern. The expectation of the sages is that this dilemma of seasonal drifts of holidays will be resolved by the "coming of the Messiah."

V. Concluding Remarks

Many times during their long and distinguished history, the Jewish people have adopted the knowledge and practices of other cultures in the pursuit of their unique concepts and visions. In this relatively brief overview, we have tried to describe how the Jews adapted the basic lunar-solar astronomical underpinnings and month-naming practices of the Babylonian calendar to rabbinic theology as embodied in the Torah and Talmud.

The Jewish calendar continues to interest many people, especially in the present computer age when many calculations that could previously be done only with great difficulty can now be easily implemented. For extensive technical details and numerical studies of computer-based algorithms for calendrical calculations and conversions, see, e.g., Dershowitz and Reingold, and Landau, cited in the Appendix. (13)

Appendix

The most important primary source on the technical details of the Jewish calendar is

Maimonides (Moshe ben Maimon), Mishneh Torah: Sefer Zemanim-Hilhot Kiddush HaHodesh, 1177. English translation by S. Gandz (with commentary by J. Obermann and O. Neugebauer) as Code of Maimonides, Book Three, Treatise Eight, Sanctification of the New Moon. Yale Judaica Series, Vol. XI, (New Haven:Yale University Press, 1956). Addenda/corrigenda by E.J. Weisenberg are added at the end of Code of Maimonides, Book Three, The Book of Seasons, translated by S. Gandz and H. Klein. Yale Judaica Series, Volume XIV (New Haven: Yale University Press, 1961). Another English translation of, and commentary on, this source is Maimonides, Mishneh Torah, Hilchot Shekalim, The Laws of Shekalim and Hilchot HaChodesh, The Laws of the Sanctification of the New Moon, translated by E. Touger (New York: Moznaim Publishing Corporation, 1993).

Useful secondary sources also consulted by the authors in English include:

Burnaby, S. B. Elements of the Jewish and Muhammadan Calendars. London: George Bell & Sons, 1901.

Bushwick, N., Understanding the Jewish Calendar. New York: Moznaim Publishing Company, 1989.

Encyclopedia Britannica (see, in particular the classic Calendar article by W. S. B. Woolhouse in eighth through the 11th editions).

Feinstein, D. The Jewish Calendar, Its Structure and Laws, Brooklyn: Mesorah Publications, Ltd., 2004.

Feldman, W. M. Rabbinical Mathematics and Astronomy, 3rd corrected edition. New York: Sepher-Hermon, 1978.

Friedlander, M. "Calendar." The Jewish Encyclopedia. New York: Funk and Wagnalls, 1906 (this classic encyclopedia is now available on the internet: http://www.jewishencyclopedia.com/index/jsp).

Gabai, H. Judaism, Mathematics, and the Hebrew Calendar. Northvale, NJ: Jason Aronson Inc., 2002.

Hastings, J., ed. Encyclopedia of Religion and Ethics. New York: Charles Scribner's Sons, 1911.

Roth, C., ed. Encyclopedia Judaica. New York: MacMillan, 1971;

Spier, A. The Comprehensive Jewish Calendar, 3rd ed. New York: Feldheim Publishers, 1986.

Schamroth, J., A Glimpse of Light. A Discussion of the Hebrew Calendar and Judaic Astronomy (Based on Maimonides' Kiddush Ha'chodesh). Southfield, MI: Targum Press, 1998.

Shoken, W.A. The Calculated Confusion of Calendars. New York: Vantage Press, 1976, p. 38.

Zinberg, G. Jewish Calendar Mystery Dispelled. New York: Vantage Press, 1963.

Additional references may be found in the excellent book:

Richards, E. G. Mapping Time. The Calendar and its History. Oxford: Oxford University Press, 1998.

which discusses calendars from around the world, and the extremely detailed book on the mathematics of calendars in terms of modern numerical algorithms:

Dershowitz, N. and E. M. Reingold. Calendrical Calculations. Cambridge: Cambridge University Press, 1997, chapter 7.

Finally we mention the magnificent website, Hebrew Calendar Science and Myths, maintained by Remy Landau, which contains voluminous numerical detail:

http://www.geocities.com/Athens/1584

Solomon Gartenhaus and Arnold Tubis

Purdue University

(1) Although we often think of the first day of the four seasons as occurring on the 21st of March, June, September, and December, respectively, since the number of days in the year is not an integer the actual dates can deviate slightly from these. It should be emphasized that the first days of the seasons are determined not by humans but via astronomical measurements by the instants when the sun, in its annual journey around the celestial sphere, crosses the four fixed points in the sky known as: the vernal equinox, the summer solstice, the autumnal equinox, and the winter solstice.

(2) See, for example, G. Moyer, "The Gregorian Calendar," Scientific American, Vol. 246, No. 5 (May 1982): 144-152.

(3) The "tropical" year is defined as the time interval between successive passages of the sun through the vernal equinox. It has the value of 365.2422 days. The "day," in turn, is defined as the minimum time interval it takes the sun to appear at the highest point in the sky on two successive days. This is to be contrasted with the sideral day which is about 4 minutes shorter and is the time it takes the earth to rotate completely about its axis.

(4) This is the modern value of the tropical year. In the ancient world, observers were limited to an accuracy of about six hours in their estimates of the true solstice and equinoctial times. However, by measuring solstice times separated by many years, surprisingly good estimates were obtained. For example, Hipparchus (c. 140 BC) obtained an estimate of 365.2467 days for the tropical year, based on an observation of the sun at summer solstice in 280 BC (by Aristarchus) and his own observation in 135 BC. This value was generally accepted up to the middle ages, and differs from the modern value by only six minutes and 26 seconds!

(5) This is the modern value and (perhaps surprisingly) also the value determined by the Babylonians during the last three centuries BCE. They achieved this by counting the number of days between two widely spaced lunar eclipses and dividing by the total number of full (or new) moons that occur between the eclipses. This value for the average lunation cycle was the one used by Maimonides in his detailed presentation of Jewish calendar calculations.

(6) The value of 365.2468 days, which makes the Metonic-cycle relationship exact, is associated with Rav Adda in the Jewish rabbinic literature, and was adopted by Maimonides. As will be discussed later in this paper, the discrepancy between this value and the modern one leads over time to significant seasonal drifts in the occurrence of some of the Jewish holidays.

(7) See e.g., Spier, The Comprehensive Jewish Calendar and, as the primary source, Maimonides, Sefer Zemanim-Hilhot Kiddush HaHodesh. Note that according to the method of construction of the Jewish calendar, the term Molad Tishri here refers not to the astronomical new moon time but rather that determined from the time of the Molad Tishri of year two (or the fictitious Molad Tishri of year one), the number of the lunar conjunction, and the average lunation period (see note 10 below). This calculated time of the Molad Tishri may differ from the true astronomical new moon time by as much as several days.

(8) Shocken, The Calculated Confusion of Calendars, p. 38.

(9) R. Landau "Romancing the old Moon," Mehqere Hag, Vol. 10 (1999): 6-19. See also Landau, website in Appendix.

(10) On the basis of biblical chronology as calculated in the Talmud (see, e.g., Spier, The Comprehensive Jewish Calendar) and the Chronology articles in the Jewish Encyclopedia and Roth, Encyclopedia Judaica. Jewish calendar year 1 corresponds to 3760 BCE. See also E. Frank, Talmudic and Rabbinical Chronology, The Systems of Counting Years in Jewish Literature (New York: Feldheim Publishers, 1956).

(11) For a clear and critical analysis of various traditional rabbinic arguments in this regard, see J. Landa, Torah and Science (Hoboken: KTAV Publishing House, 1991), pp. 297-327.

(12) At first glance, these times for the Molad Tishri of year 2 and the "creation of the world" may appear extremely puzzling to the reader. However, as pointed out by Landa (see note 11), they may be simply obtained via an interesting mix of rabbinic theology and basic astronomy. Mairnonides and many other rabbinic authorities apparently assumed that the first new moon and the completion of the creation of Adam occurred together on the sixth day (Friday) of creation. Furthermore, rabbinic accounts of hour-by-hour occurrences on the day of Adams creation have his creation completed at the start of the 14th hour. These (purely theological) assumptions thus account for the time of Molad Tishri of Jewish calendar year 2 being determined exactly at the start of hour 14 of Friday. Once this time is established, the time for the "beginning of time" and the imaginary Molad Tishri of Jewish calendar year 1 are determined by subtraction of 6 and 12 lunation periods, respectively. It is interesting to note that the time of Molad Tishri of year 2, a key ingredient in the construction of the Jewish calendar as it exists even to the present time, is based on purely theological speculation, in contrast to the other aspects of the calendar that are grounded in astronomical facts.

(13) We wish to thank our many colleagues and friends at Purdue University, Congregation Sons of Abraham in Lafayette, Indiana, and Temple Solel in Cardiff by the Sea, California for interesting discussions on various aspects of this paper. We also thank Irwin Rubenstein and an anonymous reviewer for constructive criticism of an earlier version of this paper.
Table I The number of days to full moon for selected dates for the years
2000-2004

              2000  2001  2002  2003  2004

January 1      7    25    12     3    22
July 4        28    16     5    25    15
September 21   9    27    20     6    24

Table II The months and the number of days in each for ordinary years
and for the leap years

    Ordinary Year           Leap Year
Month        # of days  Month        # of days

Tishri        30        Tishri        30
Marcheshvan   29        Marcheshvan   29
Kislev        30        Kislev        30
Tebeth        29        Tebeth        29
Shevat        30        Shevat        30
Adar          29        Adar I        30
--            --        Adar II       29
Nisan         30        Nisan         30
Iyar          29        Iyar          29
Sivan         30        Sivan         30
Tammuz        29        Tammuz        29
Ab            30        Ab            30
Elul          29        Elul          29
Total        354        Total        384

Table III The current 303rd 19-Year cycle

Ordinal No.         Civil Date of    First day of  No. of days
of year      Year   first of Tishri  Rosh Hashana  in the year

 1           5758   Oct. 2,1997      Th             354
 2           5759   Sept. 21, 1998   Mon            355
 3           5760*  Sept. 11, 1999   Sat            385
 4           5761   Sept. 30, 2000   Sat            353
 5           5762   Sept. 18, 2001   Tues           354
 6           5763*  Sept. 7, 2002    Sat            385
 7           5764   Sept. 27, 2003   Sat            355
 8           5765*  Sept. 16, 2004   Th             383
 9           5766   Oct. 4, 2005     Tues           354
10           5767   Sept. 23, 2006   Sat            355
11           5768*  Sept. 13, 2007   Th             383
12           5769   Sept. 30, 2008   Tues           354
13           5770   Sept. 19, 2009   Sat            355
14           5771*  Sept. 9, 2010    Th             385
15           5772   Sept. 29, 2011   Th             354
16           5773   Sept. 17, 2012   Mon            353
17           5774*  Sept. 5, 2013    Th             385
18           5775   Sept. 25, 2014   Th             354
19           5776*  Sept. 14, 2015   Mon            385
                                     TOTAL:        6941

*Leap Year

Table IV The fourteen possible years of the Jewish calendar

          ORDINARY YEARS                        LEAP YEARS
Day of       Day of                    Day of 1   Day of         # of
1 Tishri of  1 Tishri of    # of days  Tishri of  1 Tishri of    Days in
a given yr.  following yr.  in year    given yr.  following yr.  year

Monday       Thursday       353        Monday     Saturday       383
Monday       Saturday       355        Monday     Monday         385
Tuesday      Saturday       354        Tuesday    Monday         384
Thursday     Monday         354        Thursday   Tuesday        383
Thursday     Tuesday        355        Thursday   Thursday       385
Saturday     Tuesday        353        Saturday   Thursday       383
Saturday     Thursday       355        Saturday   Saturday       385
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