This information is reprinted from the Explanatory Supplement to the

Astronomical Almanac, P. Kenneth Seidelmann, editor, with permission from

University Science Books, Sausalito, CA 94965.


Another place on the WWW to look for calendar information is Calendarland.



                                 Calendars



                              by L. E. Doggett

---------------------------------------------------------------------------



1. Introduction



A calendar is a system of organizing units of time for the purpose of

reckoning time over extended periods. By convention, the day is the

smallest calendrical unit of time; the measurement of fractions of a day is

classified as timekeeping. The generality of this definition is due to the

diversity of methods that have been ised in creating calendars. Although

some calendars replicate astronomical cycles according to fixed rules,

others are based on abstract, perpetually repeating cycles of no

astronomical signifcance. Some calendars are regulated by astronomical

observations, some carefully and redundantly enumerate every unit, and some

contain ambiguities and discontinuities. Some calendars are codified in

written laws; others are transmitted by oral tradition.



The common theme of calendar making is the desire to organize units of time

to satisfy the needs and preoccupations of society. Inn addition to serving

practical purposes, the process of organization provides a sense, however

illusory, of understanding and controlling time itself. Thus calendars

serve as a link between mankind and the cosmos. It is little wonder that

calendars have held a sacred status and have served as a source of social

order and cultural identity. Calendars have provided the basis for planning

agricultural, hunting, and migration cycles, for divination and

prognostication, and for maintaining cycles of religious and civil events.

Whatever their scientific sophistication, calendars must ultimately be

judged as social contracts, not as scientific treatises.



According to a recent estimate (Fraser, 1987), there are about forty

calendars used in the world today. This chapter is limited to the

half-dozen principal calendars in current use. Furthermore, the emphasis of

the chapter is on function and calculation rather than on culture. The

fundamental bases of the calendars are given, along with brief historical

summaries. Although algorithms are given for correlating these systems,

close examination reveals that even the standard calendars are subject to

local variation. With the exception of the Julian calendar, this chapter

does not deal with extinct systems. Inclusion of the Julian calendar is

justified by its everyday use in historical studies.



Despite a vast literature on calendars, truly authoritative references,

particularly in English, are difficult to find. Aveni (1989) surveys a

broad variety of calendrical systems, stressing their cultural contexts

rather than their operational details. Parise (1982) provides useful,

though not infallible, tables for date conversion. Fotheringham (1935) and

the Encyclopedia of Religion and Ethics (1910), in its section on

"Calendars," offer basic information on historical calendars. The sections

on "Calendars" and "Chronology" in all editions of the Encyclopedia

Brittanica provide useful historical surveys. Ginzel (1906) remains an

authoritative, if dated, standard of calendrical scholarship. References on

individual calendars are given in the relevant sections.



1.1 Astronomical Bases of Calendars



The principal astronomical cycles are the day (based on the rotation of the

Earth on its axis), the year (based on the revolution of the Earth around

the Sun), and the month (based on the revolution of the Moon around the

Earth). The complexity of calendars arises because these cycles of

revolution do not comprise an integral number of days, and because

astronomical cycles are neither constant nor perfectly commensurable with

each other,



The tropical year is defined as the mean interval between vernal equinoxes;

it corresponds to the cycle of the seasons. The following expression, based

on the orbital elements of Laskar (1986), is used for calculating the

length of the tropical year:

365.2421896698 - 0.00000615359 T - 7.29E-10 T^2 + 2.64E-10 T^3 [days]

where T = (JD - 2451545.0)/36525 and JD is the Julian day number. However,

the interval from a particular vernal equinox to the next may vary from

this mean by several minutes.



The synodic month, the mean interval between conjunctions of the Moon and

Sun, corresponds to the cycle of lunar phases. The following expression for

the synodic month is based on the lunar theory of Chapront-Touze' and

Chapront (1988):

29.5305888531 + 0.00000021621 T - 3.64E-10 T^2 [days].

Again T = (JD - 2451545.0)/36525 and JD is the Julian day number. Any

particular phase cycle may vary from the mean by up to seven hours.



In the preceding formulas, T is measured in Julian centuries of Terrestrial

Dynamical Time (TDT), which is independent of the variable rotation of the

Earth. Thus, the lengths of the tropical year and synodic month are here

defined in days of 86400 seconds of International Atomic Time (TAI).



From these formulas we see that the cycles change slowly with time.

Furthermore, the formulas should not be considered to be absolute facts;

they are the best approximations possible today. Therefore, a calendar year

of an integral number of days cannot be perfectly synchronized to the

tropical year. Approximate synchronization of calendar months with the

lunar phases requires a complex sequence of months of 29 and 30 days. For

convenience it is common to speak of a lunar year of twelve synodic months,

or 354.36707 days.



Three distinct types of calendars have resulted from this situation. A

solar calendar, of which the Gregorian calendar in its civil usage is an

example, is designed to maintain synchrony with the tropical year. To do

so, days are intercalated (forming leap years) to increase the average

length of the calendar year. A lunar calendar, such as the Islamic

calendar, follows the lunar phase cycle without regard for the tropical

year. Thus the months of the Islamic calendar systematically shift with

respect to the months of the Gregorian calendar. The third type of

calendar, the lunisolar calendar, has a sequence of months based on the

lunar phase cycle; but every few years a whole month is intercalated to

bring the calendar back in phase with the tropical year. The Hebrew and

Chinese calendars are examples of this type of calendar.



1.2 Nonastronomical Bases of Calendars: the Week



[omitted]



1.3 Calendar Reform and Accuracy



In most societies a calendar reform is an extraordinary event. Adoption of

a calendar depends on the forcefulness with which it is introduced and on

the willingness of society to accept it. For example, the acceptance of the

Gregorian calendar as a worldwide standard spanned more than three

centuries.



The legal code of the United States does not specify an official national

alendar. Use of the Gregorian calendar in the United States stems from an

Act of Parliament of the United Kingdom in 1751, which specified use of the

Gregorian calendar in England and its colonies. However, its adoption in

the United Kingdom and other countries was fraught with confusion,

controversy, and even violence (BAtes, 1952; Gingerich, 1983; Hoskin,

1983). It also had a deeper cultural impact through the disruption of

traditional festivals and calendrical practices (MacNeill, 1982).



Because calendars are created to serve societal needs, the question of a

calendar's accuracy is usually misleading or misguided. A calendar that is

based on a fixed set of rules is accurate if the rules are consistently

applied. For calendars that attempt to replicate astronomical cycles, one

can ask how accurately the cycles are replicated. However, astronomical

cycles are not absolutely constant, and they are not known exactly. In the

long term, only a purely observational calendar maintains synchrony with

astronomical phenomena. However, an observational calendar exhibits

short-term uncertainty, because the natural phenomena are complex and the

observations are subject to error.



1.4 Historical Eras and Chronology



The calendars treated in this chapter, except for the Chinese calendar,

have counts of years from initial epochs. In the case of the Chinese

calendar and some calendars not included here, years are counted in cycles,

with no particular cycle specified as the first cycle. Some cultures eschew

year counts altogether but name each year after an event that characterized

the year. However, a count of years from an initial epoch is the most

successful way of maintaining a consistent chronology. Whether this epoch

is associated with an historical or legendary event, it must be tied to a

sequence of recorded historical events.



This is illustrated by the adoption of the birth of Christ as the initial

epoch of the Christian calendar. This epoch was established by the

sixth-century scholar Dionysius Exiguus, who was compiling a table of dates

of Easter. An existing table covered the nineteen-year period denoted

228-247, where years were counted from the beginning of the reign of the

Roman emperor Diocletian. Dionysius continued the table for a nineteen-year

period, which he designated Anni Domini Nostri Jesu Christi 532-550. Thus,

Dionysius' Anno Domini 532 is equivalent to Anno Diocletian 248. In this

way a correspondence was established between the new Christian Era and an

existing system associated with historical records. What Dionysius did not

do is establish an accurate date for the birth of Christ. Although scholars

generally believe that Christ was born some years before A.D. 1, the

historical evidence is too sketchy to allow a definitive dating.



Given an initial epoch, one must consider how to record preceding dates.

Bede, the eighth-century English historian, began the practice of counting

years backward from A.D. 1 (see Colgrave and Mynors, 1969). In this system,

the year A.D. 1 is preceded by the year 1 B.C., without an intervening year

0. Because of the numerical discontinuity, this "historical" system is

cumbersome for comparing ancient and modern dates. Today, astronomers use

+1 to designate A.D. 1. Then +1 is naturally preceded by year 0, which is

preceded by year -1. Since the use of negative numbers developed slowly in

Europe, this "astronomical" system of dating was delayed until the

eighteenth century, when it was introduced by the astronomer Jacques

Cassini (Cassini, 1740).



Even as use of Dionysius' Christian Era became common in ecclesiastical

writings of the Middle Ages, traditional dating from regnal years continued

in civil use. In the sixteenth century, Joseph Justus Scaliger tried to

resolve the patchwork of historical eras by placing everything on a single

system (Scaliger, 1583). Instead of introducing negative year counts, he

sought an initial epoch in advance of any historical record. His

numerological approach utilized three calendrical cycles: the 28-year solar

cycle, the nineteen-year cycle of Golden Numbers, and the fifteen-year

indiction cycle. The solar cycle is the period after which weekdays and

calendar dates repeat in the Julian calendar. The cycle of Golden Numbers

is the period after which moon phases repeat (approximately) on the same

calendar dates. The indiction cycle was a Roman tax cycle. Scaliger could

therefore characterize a year by the combination of numbers (S,G,I), where

S runs from 1 through 28, G from 1 through 19, and I from 1 through 15.

Scaliger noted that a given combination would recur after 7980 (= 28*19*15)

years. He called this a Julian Period, because it was based on the Julian

calendar year. For his initial epoch Scaliger chose the year in which S, G,

and I were all equal to 1. He knew that the year 1 B.C. was characterized

by the number 9 of the colar cycle, by the Golden Number 1, and by the

number 3 of the indiction cycle, i.e., (9,1,3). He found that the

combination (1,1,1) occurred in 4713 B.C. or, as astronomers now say,

-4712. This serves as year 1 of Scaliger's Julian Period. It was later

adopted as the initial epoch for the Julian day numbers.

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2. The Gregorian Calendar



The Gregorian calendar today serves as an international standard for civil

use. In addition, it regulates the ceremonial cycle of the Roman Catholic

and Protestant churches. In fact, its original purpose was ecclesiastical.

Although a variety of other calendars are in use today, they are restricted

to particular religions or cultures.



2.1 Rules for Civil Use



Years are counted from the initial epoch defined by Dionysius Exiguus, and

are divided into two classes: common years and leap years. A common year is

365 days in length; a leap year is 366 days, with an intercalary day,

designated February 29, preceding March 1. Leap years are determined

according to the following rule:

 Every year that is exactly divisible by 4 is a leap year, except for years

                     that are exactly divisible by 100;

 these centurial years are leap years only if they are exactly divisible by

                                    400.



As a result the year 2000 is a leap year, whereas 1900 and 2100 are not

leap years. These rules can be applied to times prior to the Gregorian

reform to create a proleptic Gregorian calendar. In this case, year 0 (1

B.C.) is considered to be exactly divisible by 4, 100, and 400; hence it is

a leap year.



The Gregorian calendar is thus based on a cycle of 400 years, which

comprises 146097 days. Since 146097 is evenly divisible by 7, the Gregorian

civil calendar exactly repeats after 400 years. Dividing 146097 by 400

yields an average length of 365.2425 days per calendar year, which is a

close approximation to the length of the tropical year. Comparison with

Equation 1.1-1 reveals that the Gregorian calendar accumulates an error of

one day in about 2500 years. Although various adjustments to the leap-year

system have been proposed, none has been instituted.



Within each year, dates are specified according to the count of days from

the beginning of the month. The order of months and number of days per

month were adopted from the Julian calendar.



                                Table 2.1.1

                     Months of the Gregorian Calendar



                     1. January   31 7. July       31

                     2. February  28*8. August     31

                     3. March     31 9. September  30

                     4. April     30 10. October   31

                     5. May       31 11. November  30

                     6. June      30 12. December  31



                  * In a leap year, February has 29 days.



2.2 Ecclesiastical Rules



The ecclesiastical calendars of Christian churches are based on cycles of

movable and immovable feasts. Christmas is the principal immovable feast,

with its date set at December 25. Easter is the principal movable feast,

and dates of most other movable feasts are determined with respect to

Easter. However, the movable feasts of the Advent and Epiphany seasons are

Sundays reckoned from Christmas and the Feast of the Epiphany,

respectively.



In the Gregorian calendar, the date of Easter is defined to occur on the

Sunday following the ecclesiastical Full Moon that falls on or next after

March 21. This should not be confused with the popular notion that Easter

is the first Sunday after the first Full Moon following the vernal equinox.

In the first place, the vernal equinox does not necessarily occur on March

21. In addition, the ecclesiastical Full Moon is not the astronomical Full

Moon -- it is based on tables that do not take into account the full

complexity of lunar motion. As a result, the date of an ecclesiastical Full

Moon may differ from that of the true Full Moon. However, the Gregorian

system of leap years and lunar tables does prevent progressive departure of

the tabulated data from the astronomical phenomena.



The ecclesisatical Full Moon is defined as the fourteenth day of a tabular

lunation, where day 1 corresponds to the ecclesiastical New Moon. The

tables are based on the Metonic cycle, in which 235 mean synodic months

occur in 6939.688 days. Since nineteen Gregorian years is 6939.6075 days,

the dates of Moon phases in a given year will recur on nearly the same

dates nineteen years laters. To prevent the 0.08 day difference between the

cycles from accumulating, the tables incorporate adjustments to synchronize

the system over longer periods of time. Additional complications arise

because the tabular lunations are of 29 or 30 integral days. The entire

system comprises a period of 5700000 years of 2081882250 days, which is

equated to 70499183 lunations. After this period, the dates of Easter

repeat themselves.



The following algorithm for computing the date of Easter is based on the

algorithm of Oudin (1940). It is valid for any Gregorian year, Y. All

variables are integers and the remainders of all divisions are dropped. The

final date is given by M, the month, and D, the day of the month.

                                C = Y/100,

                            N = Y - 19*(Y/19),

                             K = (C - 17)/25,

                   I = C - C/4 - (C - K)/3 + 19*N + 15,

                            I = I - 30*(I/30),

          I = I - (I/28)*(1 - (I/28)*(29/(I + 1))*((21 - N)/11)),

                      J = Y + Y/4 + I + 2 - C + C/4,

                             J = J - 7*(J/7),

                                L = I - J,

                           M = 3 + (L + 40)/44,

                          D = L + 28 - 31*(M/4).



2.3 History of the Gregorian Calendar



The Gregorian calendar resulted from a perceived need to reform the method

of calculating dates of Easter. Under the Julian calendar the dating of

Easter had become standardized, using March 21 as the date of the equinox

and the Metonic cycle as the basis for calculating lunar phases. By the

thirteenth century it was realized that the true equinox had regressed from

March 21 (its supposed date at the time of the Council of Nicea, +325) to a

date earlier in the month. As a result, Easter was drifting away from its

springtime position and was losing its relation with the Jewish Passover.

Over the next four centuries, scholars debated the "correct" time for

celebrating Easter and the means of regulating this time calendrically. The

Church made intermittent attempts to solve the Easter question, without

reaching a consensus.



By the sixteenth century the equinox had shifted by ten days, and

astronomical New Moons were occurring four days before ecclesiastical New

Moons. At the behest of the Council of Trent, Pope Pius V introduced a new

Breviary in 1568 and Missal in 1570, both of which included adjustments to

the lunar tables and the leap-year system. Pope Gregory XIII, who succeeded

Pope Pius in 1572, soon convened a commission to consider reform of the

calendar, since he considered his predecessor's measures inadequate.



The recommendations of Pope Gregory's calendar commission were instituted

by the papal bull "Inter Gravissimus," signed on 1582 February 24. Ten days

were deleted from the calendar, so that 1582 October 4 was followed by 1582

October 15, thereby causing the vernal equinox of 1583 and subsequent years

to occur about March 21. And a new table of New Moons and Full Moons was

introduced for determining the date of Easter.



Subject to the logistical problems of communication and governance in the

sixteenth century, the new calendar was promulgated through the

Roman-Catholic world. Protestant states initially rejected the calendar,

but gradually accepted it over the coming centuries. The Eastern Orthodox

churches rejected the new calendar and continued to use the Julian calendar

with traditional lunar tables for calculating Easter. Because the purpose

of the Gregorian calendar was to regulate the cycle of Christian holidays,

its acceptance in the non-Christian world was initially not at issue. But

as international communications developed, the civil rules of the Gregorian

calendar were gradually adopted around the world.



Anyone seriously interested in the Gregorian calendar should study the

collection of papers resulting from a conference sponsored by the Vatican

to commemorate the four-hundredth anniversary of the Gregorian Reform

(Coyne et al., 1983).

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3. The Hebrew Calendar



As it exists today, the Hebrew calendar is a lunisolar calendar that is

based on calculation rather than observation. This calendar is the official

calendar of Israel and is the liturgical calendar of the Jewish faith.



In principle the beginning of each month is determined by a tabular New

Moon (molad) that is based on an adopted mean value of the lunation cycle.

To ensure that religious festivals occur in appropriate seasons, months are

intercalated according to the Metonic cycle, in which 235 lunations occur

in nineteen years.



By ttradition, days of the week are designated by number, with only the

seventh day, Sabbath, having a specific name. Days are reckoned from sunset

to sunset, so that day 1 begins at sunset on Saturday and ends at sunset on

Sunday. The Sabbath begins at sunset on Friday and ends at sunset on

Saturday.



3.1 Rules



Years are counted from the Era of Creation, or Era Mundi, which corresponds

to -3760 October 7 on the Julian proleptic calendar. Each year consists of

twelve or thirteen months, with months consisting of 29 or 30 days. An

intercalary month is introduced in years 3, 6, 8, 11, 14, 17, and 19 in a

nineteen-year cycle of 235 lunations. The initial year of the calendar,

A.M. (Anno Mundi) 1, is year 1 of the nineteen-year cycle.



The calendar for a given year is established by determining the day of the

week of Tishri 1 (first day of Rosh Hashanah or New Year's Day) and the

number of days in the year. Years are classified according to the number of

days in the year (see Table 3.1.1).



                   --------------------------------------



                               Table 3.1.1

                   Classification of Years in the Hebrew

                                 Calendar



                                Deficient Regular Complete

                 Ordinary year  353       354     355

                 Leap year      383       384     385



                   --------------------------------------



                                Table 3.1.2

                       Months of the Hebrew Calendar



                      1. Tishri  30    7. Nisan   30

                      2. Heshvan 29*   8. Iyar    29

                      3. Kislev  30**  9. Sivan   30

                      4. Tevet   29    10. Tammuz 29

                      5. Shevat  30    11. Av     30

                      6. Adar    29*** 12. Elul   29

                * In a complete year, Heshvan has 30 days.

                ** In a deficient year, Kislev has 29 days.

  *** In a leap year Adar I has 30 days; it is followed by Adar II with 29

                                   days.



                   --------------------------------------



                                Table 3.1.3

                    Terminology of the Hebrew Calendar



 Deficient (haser) month: a month comprising 29 days.

 Full (male) month: a month comprising 30 days.

 Ordinary year: a year comprising 12 months, with a total of 353, 354, or

 355 days.

 Leap year: a year comprising 13 months, with a total of 383, 384, or 385

 days.

 Complete year (shelemah): a year in which the months of Heshvan and

 Kislev both contain 30 days.

 Deficient year (haser): a year in which the months of Heshvan and Kislev

 both contain 29 days.

 Regular year (kesidrah): a year in which Heshvan has 29 days and Kislev

 has 30 days.

 Halakim(singular, helek): "parts" of an hour; there are 1080 halakim per

 hour.

 Molad(plural, moladot): "birth" of the Moon, taken to mean the time of

 conjunction for modern calendric purposes.

 Dehiyyah(plural, dehiyyot): "postponement"; a rule delaying 1 Tishri

 until after the molad.



                   --------------------------------------



The months of Heshvan and Kislev vary in length to satisfy requirements for

the length of the year (see Table 3.1.1). In leap years, the 29-day month

Adar is designated Adar II, and is preceded by the 30-day intercalary month

Adar I.



For calendrical calculations, the day begins at 6 P.M., which is designated

0 hours. Hours are divided into 1080 halakim; thus one helek is 3 1/3

seconds. (Terminology is explained in Table 3.1.3.) Calendrical

calculations are referred to the meridian of Jerusalem -- 2 hours 21

minutes east of Greenwich.



Rules for constructing the Hebrew calendar are given in the sections that

follow. Cohen (1981), Resnikoff (1943), and Spier (1952) provide reliable

guides to the rules of calculation.



3.1.1 Determining Tishri 1



The calendar year begins with the dirst day of Rosh Hashanah (Tishri 1).

This is determined by the day of the Tishri molad and the four rules of

postponements (dehiyyot). The dehiyyot can postpone Tishri 1 until one or

two days following the molad. Tabular new moons (maladot) are reckoned from

the Tishri molad of the year A.M. 1, which occurred on day 2 at 5 hours,

204 halakim (i.e., 11:11:20 P.M. on SUnday, -3760 October 6, Julian

proleptic calendar). The adopted value of the mean lunation is 29 days, 12

hours, 793 halakim (29.530594 days). To avoid rounding and truncation

errors, calculation should be done in halakim rather than decimals of a

day, since the adopted lunation constant is expressed exactly in halakim.



                               Table 3.1.1.1

                     Lunation Constants for Determining

                                 Tishri 1



                   Lunations   Weeks-Days-Hours-Halakim

                   1         =        4-1-12-0793

                   12        =       50-4-08-0876

                   13        =       54-5-21-0589

                   235       =       991-2-16-0595



Lunation constants required in calculations are shown in Table 3.1.1.1. By

subtracting off the weeks, these constants give the shift in weekdays that

occurs after each cycle.



The dehiyyot are as follows:

(a) If the Tishri molad falls on day 1, 4, or 6, then Tishri 1 is postponed

one day.

(b) If the Tishri molad occurs at or after 18 hours (i.e., noon), then

Tishri 1 is postponed one day. If this causes Tishri 1 to fall on day 1, 4,

or 6, then Tishri 1 is postponed an additional day to satisfy dehiyyah (a).



(c) If the Tishri molad of an ordinary year (i.e., of twelve months) falls

on day 3 at or after 9 hours, 204 halakim, then Tishri 1 is postponed two

days to day 5, thereby satisfying dehiyyah (a).

(d) If the first molad following a leap year falls on day 2 at or after 15

hours, 589 halakim, then Tishri 1 is postponed one day to day 3.



3.1.2 Reasons for the Dehiyyot



Dehiyyah (a) prevents Hoshana Rabba (Tishri 21) from occurring on the

Sabbath and prevents Yom Kippur (Tishri 10) from occurring on the day

before or after the Sabbath.



Dehiyyah (b) is an artifact of the ancient practice of beginning each month

with the sighting of the lunar crescebt. It is assumed that if the molad

(i.e., the mean conjunction) occurs after noon, the lunar crescent cannot

be sighted until after 6 P.M., which will then be on the following day.



Dehiyyah (c) prevents an ordinary year from exceeding 355 days. If the

Tishri molad of an ordinary year occurs on Tuesday at or after 3:11:20

A.M., the next Tishri molad will occur at or after noon on Saturday.

According to dehiyyah (b), Tishri 1 of the next year must be postponed to

Sunday, which by dehiyyah (a) occasions a further postponement to Monday.

This results in an ordinary year of 356 days. Postponing Tishri 1 from

Tuesday to Thursday produces a year of 354 days.



Dehiyyah (d) prevents a leap year from falling short of 383 days. If the

Tishri molad following a leap year is on Monday, at or after 9:32:43 1/3

A.M., the previous Tishri molad (thirteen months earlier) occurred on

Tuesday at or after noon. Therefore, by dehiyyot (b) and (a), Tishri 1

beginning the leap year was postponed to Thursday. To prevent a leap year

of 382 days, dehiyyah (d) postpones by one day the beginning of the

ordinary year.



A thorough discussion of both the functional and religious aspects of the

dehiyyot is provided by Cohen (1981).



3.1.3 Determining the Length of the Year



An ordinary year consists of 50 weeks plus 3, 4, or 5 days. The number of

excess days identifies the year as being deficient, regular, or complete,

respectively. A leap year consists of 54 weeks plus 5, 6, or 7 days, which

again are designated deficient, regular, or complete, respectively. The

length of a year can therefore be determined by comparing the weekday of

Tishri 1 with that of the next Tishri 1.



First consider an ordinary year. The weekday shift after twelve lunations

is 04-08-876. For example if a Tishri molad of an ordinary year occurs on

day 2 at 0 hours 0 halakim (6 P.M. on Monday), the next Tishri molad will

occur on day 6 at 8 hours 876 halakim. The first Tishri molad does not

require application of the dehiyyot, so Tishri 1 occurs on day 2. Because

of dehiyyah (a), the following Tishri 1 is delayed by one day to day 7,

five weekdays after the previous Tishri 1. Since this characterizes a

complete year, the months of Heshvan and Kislev both contain 30 days.



The weekday shift after thirteen lunations is 05-21-589. If the Tishri

molad of a leap year occurred on day 4 at 20 hours 500 halakim, the next

Tishri molad will occur on day 3 at 18 hours 9 halakim. Becuase of dehiyyot

(b), Tishri 1 of the leap year is postponed two days to day 6. Because of

dehiyyot (c), Tishri 1 of the following year is postponed two days to day

5. This six-day difference characterizes a regular year, so that Heshvan

has 29 days and Kislev has 30 days.



3.2 History of the Hebrew Calendar



The codified Hebrew calendar as we know it today is generally considered to

date from A.M. 4119 (+359), though the exact date is uncertain. At that

time the patriarch Hillel II, breaking with tradition, disseminated rules

for calculating the calendar. Prior to that time the calendar was regarded

as a secret science of the religious authorities. The exact details of

Hillel's calendar have not come down to us, but it is generally considered

to include rules for intercalation over nineteen-year cycles. Up to the

tenth century A.D., however, there was disagreement about the proper years

for intercalation and the initial epoch for reckoning years.



Information on calendrical practices prior to Hillel is fragmentary and

often contradictory. The earliest evidence indicates a calendar based on

observations of Moon phases. Since the Bible mentions seasonal festivals,

there must have been intercalation. There was likely an evolution of

conflicting calendrical practices.



The Babylonian exile, in the first half of the sixth century B.C., greatly

influenced the Hebrew calendar. This is visible today in the names of the

months. The Babylonian influence may also have led to the practice of

intercalating leap months.



During the period of the Sanhedrin, a committee of the Sanhedrin met to

evaluate reports of sightings of the lunar crescent. If sightings were not

possible, the new month was begun 30 days after the beginning of the

previous month. Decisions on intercalation were influences, if not

determined entirely, by the state of vegetation and animal life. Although

eight-year, nineteen-year, and longer- period intercalation cycles may have

been instituted at various times prior to Hillel II, there is little

evidence that they were employed consistently over long time spans.

---------------------------------------------------------------------------



4. The Islamic Calendar



The Islamic calendar is a purely lunar calendar in which months correspond

to the lunar phase cycle. As a result, the cycle of twelve lunar months

regresses through the seasons over a period of about 33 years. For

religious purposes, Muslims begin the months with the first visibility of

the lunar crescent after conjunction. For civil purposes a tabulated

calendar that approximates the lunar phase cycle is often used.



The seven-day week is observed with each day beginning at sunset. Weekdays

are specified by number, with day 1 beginning at sunset on Saturday and

ending at sunset on Sunday. Day 5, which is called Jum'a, is the day for

congregational prayers. Unlike the Sabbath days of the Christians and Jews,

however, Jum'a is not a day of rest. Jum'a begins at sunset on Thursday and

ends at sunset on Friday.



4.1 Rules



Years of twelve lunar months are reckoned from the Era of the Hijra,

commemorating the migration of the Prophet and his followers from Mecca to

Medina. This epoch, 1 A.H. (Anno Higerae) Muharram 1, is generally taken by

astronomers (Neugebauer, 1975) to be Thursday, +622 July 15 (Julian

calendar). This is called the astronomical Hijra epoch. Chronological

tables (e.g., Mayr and Spuler, 1961; Freeman-Grenville, 1963) generally use

Friday, July 16, which is designated the civil epoch. In both cases the

Islamic day begins at sunset of the previous day.



For religious purposes, each month begins in principle with the first

sighting of the lunar crescent after the New Moon. This is particularly

important for establishing the beginning and end of Ramadan. Because of

uncertainties due to weather, however, a new month may be declared thirty

days after the beginning of the preceding month. Although various

predictive procedures have been used for determining first visibility, they

have always had an equivocal status. In practice, there is disagreement

among countries, religious leaders, and scientists about whether to rely on

observations, which are subject to error, or to use calculations, which may

be based on poor models.



Chronologists employ a thirty-year cyclic calendar in studying Islamic

history. In this tabular calendar, there are eleven leap years in the

thirty-year cycle. Odd-numbered months have thirty days and even-numbered

months have twenty-nine days, with a thirtieth day added to the twelfth

month, Dhu al-Hijjah (see Table 4.1.1). Years 2, 5, 7, 10, 13, 16, 18, 21,

24, 26, and 29 of the cycle are designated leap years. This type of

calendar is also used as a civil calendar in some Muslim countries, though

other years are sometimes used as leap years. The mean length of the month

of the thirty-year tabular calendar is about 2.9 seconds less than the

synodic period of the Moon.



                                Table 4.1.1

                    Months of Tabular Islamic Calendar



                  1. Muharram** 307. Rajab**          30

                  2. Safar      298. Sha'ban          29

                  3. Rabi'a I   309. Ramadan***       30

                  4. Rabi'a II  2910. Shawwal         29

                  5. Jumada I   3011. Dhu al-Q'adah** 30

                  6. Jumada II  2912. Dhu al-Hijjah** 29*

               * In a leap year, Dhu al-Hijjah has 30 days.

                              ** Holy months.

                           *** Month of fasting.



4.1.1 Visibility of the Crescent Moon



[omitted]



4.2 History of the Islamic Calendar



The form of the Islamic calendar, as a lunar calendar without

intercalation, was laid down by the Prophet in the Qur'an (Sura IX, verse

36-37) and in his sermon at the Farewell Pilgrimage. This was a departure

from the lunisolar calendar commonly used in the Arab world, in which

months were based on first sightings of the lunar crescent, but an

intercalary month was added as deemed necessary.



Caliph 'Umar I is credited with establishing the Hijra Era in A.H. 17. It

is not known how the initial date was determined. However, calculations

show that the astronomical New Moon (i.e., conjunction) occurred on +622

July 14 at 0444 UT (assuming delta-T = 1.0 hour), so that sighting of the

crescent most likely occurred on the evening of July 16.

---------------------------------------------------------------------------



5. The Indian Calendar



As a result of a calendar reform in A.D. 1957, the National Calendar of

India is a formalized lunisolar calendar in which leap years coincide with

those of the Gregorian calendar (Calendar Reform Committee, 1957). However,

the initial epoch is the Saka Era, a traditional epoch of Indian

chronology. Months are names after the traditional Indian months and are

offset from the beginning of Gregorian months (see Table 5.1.1).



In addition to establishing a civil calendar, the Calendar Reform Committee

set guidelines for religious calendars, which require calculations of the

motions of the Sun and Moon. Tabulations of the religious holidays are

prepared by the India Meteorological Department and published annually in

The Indian Astronomical Ephemeris.



Despite the attempt to establish a unified calendar for all of India, many

local variations exist. The Gregorian calendar continues in use for

administrative purposes, and holidats are still determined according to

regional, religious, and ethnic traditions (Chatterjee, 1987).



5.1 Rules for Civil Use



Years are counted from the Saka Era; 1 Saka is considered to begin with the

vernal equinox of A.D. 79. The reformed Indian calendar began with Saka Era

1879, Caitra 1, which corresponds to A.D. 1957 March 22. Normal years have

365 days; leap years have 366. In a leap year, an intercalary day is added

to the end of Caitra. To determine leap years, first add 78 to the Saka

year. If this sum is evenly divisible by 4, the year is a leap year, unless

the sum is a multiple of 100. In the latter case, the year is not a leap

year unless the sum is also a multiple of 400. Table 5.1.1 gives the

sequence of months and their correlation with the months of the Gregorian

calendar.



                                Table 5.1.1

                    Months of the Indian Civil Calendar



                    Days Correlation of Indian/Gregorian

      1. Caitra     30*  Caitra 1                        March 22*

      2. Vaisakha   31   Vaisakha 1                      April 21

      3. Jyaistha   31   Jyaistha 1                      May 22

      4. Asadha     31   Asadha 1                        June 22

      5. Sravana    31   Sravana 1                       July 23

      6. Bhadra     31   Bhadra 1                        August 23

      7. Asvina     30   Asvina 1                        September 23

      8. Kartika    30   Kartika 1                       October 23

      9. Agrahayana 30   Agrahayana 1                    November 22

      10. Pausa     30   Pausa 1                         December 22

      11. Magha     30   Magha 1                         January 21

      12. Phalguna  30   Phalguna 1                      February 20

* In a leap year, Caitra has 31 days and Caitra 1 coincides with March 21.



5.2 Principles of the Religious Calendar



Religious holidays are determined by a lunisolar calendar that is based on

calculations of the actual postions of the Sun and Moon. Most holidays

occur on specified lunar dates (tithis), as is explained later; a few occur

on specified solar dates. The calendrical methods presented here are those

recommended by the Calendar Reform Committee (1957). They serve as the

basis for the calendar published in The Indian Astronomical Ephemeris.

However, many local calendar makers continue to use traditional

astronomical concepts and formulas, some of which date back 1500 years.



The Calendar Reform Committee attempted to reconcile traditional

calendrical practices with modern astronomical concepts. According to their

proposals, precession is accounted for and calculations of solar and lunar

position are based on accurate modern methods. All astronomical

calculations are performed with respect to a Central Station at longitude

82o30' East, latitude 23o11' North. For religious purposes solar days are

reckoned from sunrise to sunrise.



A solar month is defined as the interval required for the Sun's apparent

longitude to increase by 30o, corresponding to the passage of the Sun

through a zodiacal sign (rasi). The initial month of the year, Vaisakha,

begins when the true longitude of the Sun is 23o 15' (see Table 5.2.1).

Because the Earth's orbit is elliptical, the lengths of the months vary

from 29.2 to 31.2 days. The short months all occurr in the second half of

the year around the time of the Earth's perihelion passage.



                                Table 5.2.1

               Solar Months of the Indian Religious Calendar



                   Sun's Longitude Approx. Duration Approx. Greg. Date

                   deg min         d

     1. Vaisakha   23 15           30.9             Apr. 13

     2. Jyestha    53 15           31.3             May 14

     3. Asadha     83 15           31.5             June 14

     4. Sravana    113 15          31.4             July 16

     5. Bhadrapada 143 15          31.0             Aug. 16

     6. Asvina     173 15          30.5             Sept. 16

     7. Kartika    203 15          30.0             Oct. 17

     8. Margasirsa 233 15          29.6             Nov. 16

     9. Pausa      263 15          29.4             Dec. 15

     10. Magha     293 15          29.5             Jan. 14

     11. Phalgura  323 15          29.9             Feb. 12

     12. Caitra    353 15          30.3             Mar. 14



Lunar months are measures from one New Moon to the next (although some

groups reckon from the Full Moon). Each lunar month is given the name of

the solar month in which the lunar month begins. Because most lunations are

shorter than a solar month, there is occasionally a solar month in which

two New Moons occur. In this case, both lunar months bear the same name,

but the first month is described with the prefix adhika, or intercalary.

Such a year has thirteen lunar months. Adhika months occur every two or

three years following patterns described by the Metonic cycle or more

complex lunar phase cycles.



More rarely, a year will occur in which a short solar month will pass

without having a New Moon. In that case, the name of the solar month does

not occur in the calendar for that year. Such a decayed (ksaya) month can

occur only in the months near the Earth's perihelion passage. In

compensation, a month in the first half of the year will have had two New

Moons, so the year will still have twelve lunar months. Ksaya months are

separated by as few as nineteen years and as many as 141 years.



Lunations are divided into 30 tithis, or lunar days. Each tithi is defined

by the time required for the longitude of the Moon to increase by 12o over

the longitude of the Sun. Thus the length of a tithi may vary from about 20

hours to nearly 27 hours. During the waxing phases, tithis are counted from

1 to 15 with the designation Sukla. Tithis for the waning phases are

designated Krsna and are again counted from 1 to 15. Each day is assigned

the number of the tithi in effect at sunrise. Occasionally a short tithi

will begin after sunrise and be completed before the next sunrise.

Similarly a long tithi may span two sunrises. In the former case, a number

is omitted from the day count. In the latter, a day number is carried over

to a second day.



5.3 History of the Indian Calendar



The history of calendars in India is a remarkably complex subject owing to

the continuity of Indian civilization and to the diversity of cultural

influences. In the mid-1950s, when the Calendar Reform Committee made its

survey, there were about 30 calendars in use for setting religious

festivals for Hindus, Buddhists, and Jainists. Some of these were also used

for civil dating. These calendars were based on common principles, though

they had local characteristics determined by long-established customs and

the astronomical practices of local calendar makers. In addition, Muslims

in India used the Islamic calendar, and the Indian government used the

Gregorian calendar for administrative purposes.



Early allusions to a lunisolar calendar with intercalated months are found

in the hymns from the Rig Veda, dating from the second millennium B.C.

Literature from 1300 B.C. to A.D. 300, provides information of a more

specific nature. A five-year lunisolar calendar coordinated solar years

with synodic and sidereal lunar months.



Indian astronomy underwent a general reform in the first few centuries

A.D., as advances in Babylonian and Greek astronomy became known. New

astronomical constants and models for the motion of the Moon and Sun were

adapted to traditional calendric practices. This was conveyed in

astronomical treatises of this period known as Siddhantas, many of which

have not survived. The Surya Siddhanta, which originated in the fourth

century but was updated over the following centuries, influenced Indian

calendrics up to and even after the calendar reform of A.D. 1957.



Pingree (1978) provides a survey of the development of mathematical

astronomy in India. Although he does not deal explicitly with calendrics,

this material is necessary for a full understanding of the history of

India's calendars.

---------------------------------------------------------------------------



6. The Chinese Calendar



The Chinese calendar is a lunisolar calendar based on calculations of the

positions of the Sun and Moon. Months of 29 or 30 days begin on days of

astronomical New Moons, with an intercalary month begin added every two or

three years. Since the calendar is based on the true positions of the Sun

and Moon, the accuracy of the calendar depends on the accuracy of the

astronomical theories and calculations.



Although the Gregorian calendar is used in the Peoples' Republic of China

for administrative purposes, the traditional Chinese calendar is used for

setting traditional festivals and for timing agricultural activities in the

countryside. The Chinese calendar is also used by Chinese communities

around the world.



                                Table 6.1.1

                      Chinese Sexagenary Cycle of Days

                                 and Years



                     Celestial Stems Earthly Branches

                     1. jia          1. zi (rat)

                     2. yi           2. chou (ox)

                     3. bing         3. yin (tiger)

                     4. ding         4. mao (hare)

                     5. wu           5. chen (dragon)

                     6. ji           6. si (snake)

                     7. geng         7. wu (horse)

                     8. xin          8. wei (sheep)

                     9. ren          9. shen (monkey)

                     10. gui         10. you (fowl)

                                     11. xu (dog)

                                     12. hai (pig)



                                Year Names



          1. jia-zi     16. ji-mao   31. jia-wu    46. ji-you

          2. yi-chou    17. geng-chen32. yi-wei    47. geng-xu

          3. bing-yin   18. xin-si   33. bing-shen 48. xin-hai

          4. ding-mao   19. ren-wu   34. ding-you  49. ren-zi

          5. wu-chen    20. gui-wei  35. wu-xu     50. gui-chou

          6. ji-si      21. jia-shen 36. ji-hai    51. jia-yin

          7. geng-wu    22. yi-you   37. geng-zi   52. yi-mao

          8. xin-wei    23. bing-xu  38. xin-chou  53. bing-chen

          9. ren-shen   24. ding-hai 39. ren-yin   54. ding-si

          10. gui-you   25. wu-zi    40. gui-mao   55. wu-wu

          11. jia-xu    26. ji-chou  41. jia-chen  56. ji-wei

          12. yi-hai    27. geng-yin 42. yi-si     57. geng-shen

          13. bing-zi   28. xin-mao  43. bing-wu   58. xin-you

          14. ding-chou 29. ren-chen 44. ding-wei  59. ren-xu

          15. wu-yin    30. gui-si   45. wu-shen   60. gui-hai



6.1 Rules



There is no specific initial epoch for counting years. In historical

records, dates were specified by counts of days and years in sexagenary

cycles and by counts of years from a succession of eras established by

reigning monarchs.



The sixty-year cycle consists of a set of year names that are created by

pairing a name from a list of ten Celestial Stems with a name from a list

of twelve Terrestrial Branches, following the order specified in Table

6.1.1. The Celestial Stems are specified by Chinese characters that have no

English translation; the Terrestrial Branches are named after twelve

animals. After six repetitions of the set of stems and five repetitions of

the branches, a complete cycle of pairs is completed and a new cycle

begins. The initial year (jia-zi) of the current cycle began on 1984

February 2.



Days are measured from midnight to midnight. The first day of a calendar

month is the day on which the astronomical New Moon (i.e., conjunction) is

calculated to occur. Since the average interval between successive New

Moons is approximately 29.53 days, months are 29 or 30 days long. Months

are specified by number from 1 to 12. When an intercalary month is added,

it bears the number of the previous month, but is designated as

intercalary. An ordinary year of twelve months is 353, 354, or 355 days in

length; a leap year of thirteen months is 383, 384, or 385 days long.



The conditions for adding an intercalary month are determined by the

occurrence of the New Moon with respect to divisions of the tropical year.

The tropical year is divided into 24 solar terms, in 15o segments of solar

longitude. These divisions are paired into twelve Sectional Terms (Jieqi)

and twelve Principal Terms (Zhongqi), as shown in Table 6.1.2. These terms

are numbered and assigned names that are seasonal or meteorological in

nature. For convenience here, the Sectional and Principal Terms are denoted

by S and P, respectively, followed by the number. Because of the

ellipticity of the Earth's orbit, the interval between solar terms varies

with the seasons.



Reference works give a variety of rules for establishing New Year's Day and

for intercalation in the lunisolar calendar. Since the calendar was

originally based on the assumption that the Sun's motion was uniform

through the seasons, the published rules are frequently inadequate to

handle special cases.



The following rules (Liu and Stephenson, in press) are currently used as

the basis for calendars prepared by the Purple Mountain Observatory (1984):



(1) The first day of the month is the day on which the New Moon occurs.

(2) An ordinary year has twelve lunar months; an intercalary year has

thirteen lunar months.

(3) The Winter Solstice (term P-11) always falls in month 11.

(4) In an intercalary year, a month in which there is no Principal Term is

the intercalary month. It is assigned the number of the preceding month,

with the further designation of intercalary. If two months of an

intercalary year contain no Principal Term, only the first such month after

the Winter Solstice is considered intercalary.

(5) Calculations are based on the meridian 120o East.



The number of the month usually corresponds to the number of the Principal

Term occurring during the month. In rare instances, however, there are

months that have two Principal Terms, with the result that a nonintercalary

month will have no Principal Term. As a result the numbers of the months

will temporarily fail to correspond to the numbers of the Principal Terms.

These cases can be resolved by strictly applying rules 2 and 3.



                                Table 6.1.2

                            Chinese Solar Terms



 Term*            Name               Sun's LongitudeApprox. Greg. Duration

                                                    Date



 S-1  Lichun      Beginning of       315            Feb. 4

                  Spring

 P-1  Yushui      Rain Water         330            Feb. 19       29.8

 S-2  Jingzhe     Waking of Insects  345            Mar. 6

 P-2  Chunfen     Spring Equinox     0              Mar. 21       30.2

 S-3  Qingming    Pure Brightness    15             Apr. 5

 P-3  Guyu        Grain Rain         30             Apr. 20       30.7



 S-4  Lixia       Beginning of       45             May 6

                  Summer

 P-4  Xiaoman     Grain Full         60             May 21        31.2

 S-5  Mangzhong   Grain in Ear       75             June 6

 P-5  Xiazhi      Summer Solstice    90             June 22       31.4

 S-6  Xiaoshu     Slight Heat        105            July 7

 P-6  Dashu       Great Heat         120            July 23       31.4



 S-7  Liqiu       Beginning of       135            Aug. 8

                  Autumn

 P-7  Chushu      Limit of Heat      150            Aug. 23       31.1

 S-8  Bailu       White Dew          165            Sept. 8

 P-8  Qiufen      Autumnal Equinox   180            Sept. 23      30.7

 S-9  Hanlu       Cold Dew           195            Oct. 8

 P-9  Shuangjiang Descent of Frost   210            Oct. 24       30.1



 S-10 Lidong      Beginning of       225            Nov. 8

                  Winter

 P-10 Xiaoxue     Slight Snow        240            Nov. 22       29.7

 S-11 Daxue       Great Snow         255            Dec. 7

 P-11 Dongzhi     Winter Solstice    270            Dec. 22       29.5

 S-12 Xiaohan     Slight Cold        285            Jan. 6

 P-12 Dahan       Great Cold         300            Jan. 20       29.5

    * Terms are classified as Sectional (Jieqi) or Principal (Zhongqi),

                    followed by the number of the term.



In general, the first step in calculating the Chinese calendar is to check

for the existence of an intercalary year. This can be done by determining

the dates of Winter Solstice and month 11 before and after the period of

interest, and then by counting the intervening New Moons.



Published calendrical tables are often in disagreement about the Chinese

calendar. Some of the tables are based on mean, or at least simplified,

motions of the Sun and Moon. Some are calculated for other meridians than

120o East. Some incorporate a rule that the eleventh, twelfth, and first

months are never followed by an intercalary month. This is sometimes not

stated as a rule, but as a consequence of the rapid change in the Sun's

longitude when the Earth is near perihelion. However, this statement is

incorrect when the motions of the Sun and Moon are accurately calculated.



6.2 History of the Chinese Calendar



In China the calendar was a sacred document, spopnsored and promulgated by

the reigning monarch. For more than two millennia, a Bureau of Astronomy

made astronomical observations, calculated astronomical events such as

eclipses, prepared astrological predictions, and maintained the calendar

(Needham, 1959). After all, a successful calendar not only served practical

needs, but also confirmed the consonance between Heaven and the imperial

court.



Analysis of surviving astronomical records inscribed on oracle bones

reveals a Chinese lunisolar calendar, with intercalation of lunar months,

dating back to the Shang dynasty of the fourteenth century B.C. Various

intercalation schemes were developed for the early calendars, including the

nineteen-year and 76-year lunar phase cycles that came to be known in the

West as the Metonic cycle and Callipic cycle.



From the earliest records, the beginning of the year occurred at a New Moon

near the winter solstice. The choice of month for beginning the civil year

varied with time and place, however. In the late second century B.C., a

calendar reform established the practice, which continues today, of

requiring the winter solstice to occur in month 11. This reform also

introduced the intercalation system in which dates of New Moons are

compared with the 24 solar terms. However, calcularions were based on the

mean motions resulting from the cyclic relationships. Inequalities in the

Moon's motions were incorporated as early as the seventh century A.D.

(Sivin, 1969), but the Sun's mean longitude was used for calculating the

solar terms until 1644 (Liu and Stephenson, in press).



Years were counted from a succession of eras established by reigning

emperors. Although the accession of an emperor would mark a new era, an

emperor might also declare a new era at various times within his reign. The

introduction of a new era was an attempt to reestablish a broken connection

between Heaven and Earth, as personified by the emperor. The break might be

revealed by the death of an emperor, the occurrence of a natural disaster,

or the failure of astronomers to predict a celestial event such as an

eclipse. In the latter case, a new era might mark the introduction of new

astronomical or calendrical models.



Sexagenary cycles were used to count years, months, days, and fractions of

a day using the set of Celestial Stems and Terrestrial Branches described

in Section 6.1. Use of the sixty-day cycle is seen in the earliest

astronomical records. By contrast the sixty-year cycle was introduced in

the first century A.D. or possibly a century earlier (Tung, 1960; Needham,

1959). Although the day count has fallen into disuse in everyday life, it

is still tabulated in calendars. The initial year (jia-zi) of the current

year cycle began on 1984 February 2, which is the third day (bing-yin) of

the day cycle.



Western (pre-Copernican) astronomical theories were introduced to China by

Jesuit missionaries in the seventeenth century. Gradually, more modern

Western concepts became known. Following the revolution of 1911, the

traditional practice of counting years from the accession of an emperor was

abolished.

---------------------------------------------------------------------------



7. Julian Day Numbers and Julian Date



[omitted]

---------------------------------------------------------------------------



8. The Julian Calendar



The Julian calendar, introduced by Juliius Caesar in -45, was a solar

calendar with months of fixed lengths. Every fourth year an intercalary day

was added to maintain synchrony between the calendar year and the tropical

year. It served as a standard for European civilization until the Gregorian

Reform of +1582.



Today the principles of the Julian calendar continue to be used by

chronologists. The Julian proleptic calendar is formed by applying the

rules of the Julian calendar to times before Caesar's reform. This provides

a simple chronological system for correlating other calendars and serves as

the basis for the Julian day numbers.



8.1 Rules



Years are classified as normal years of 365 days and leap years of 366

days. Leap years occur in years that are evenly divisible by 4. For this

purpose, year 0 (or 1 B.C.) is considered evenly divisible by 4. The year

is divided into twelve formalized months that were eventually adopted for

the Gregorian calendar.



8.2 History of the Julian Calendar



The year -45 has been called the "year of confusion," because in that year

Julius Caesar inserted 90 days to bring the months of the Roman calendar

back to their traditional place with respect to the seasons. This was

Caesar's first step in replacing a calendar that had gone badly awry.

Although the pre-Julian calendar was lunisolar in inspiration, its months

no longer followed the lunar phases and its year had lost step with the

cycle of seasons (see Michels, 1967; Bickerman, 1974). Following the advice

of Sosigenes, an Alexandrine astronomer, Caesar created a solar calendar

with twelve months of fixed lengths and a provision for an intercalary day

to be added every fourth year. As a result, the average length of the

Julian calendar year was 365.25 days. This is consistent with the length of

the tropical year as it was known at the time.



Following Caesar's death, the Roman calendrical authorities misapplied the

leap-year rule, with the result that every third, rather than every fourth,

year was intercalary. Although detailed evidence is lacking, it is

generally believed that Emperor Augustus corrected the situation by

omitting intercalation from the Julian years -8 through +4. After this the

Julian calendar finally began to function as planned.



Through the Middle Ages the use of the Julian calendar evolved and acquired

local peculliarities that continue to snare the unwary historian. There

were variations in the initial epoch for counting years, the date for

beginning the year, and the method of specifying the day of the month. Not

only did these vary with time and place, but also with purpose. Different

conventions were sometimes used for dating ecclesiastical records, fiscal

transactions, and personal correspondence.



Caesar designated January 1 as the beginning of the year. However, other

conventions flourished at different times and places. The most popular

alternatives were March 1, March 25, and December 25. This continues to

cause problems for historians, since, for example, +998 February 28 as

recorded in a city that began its year on March 1, would be the same day as

+999 February 28 of a city that began the year on January 1.



Days within the month were originally counted from designated division

points within the month: Kalends, Nones, and Ides. The Kalends is the first

day of the month. The Ides is the thirteenth of the month, except in March,

May, July, and October, when it is the fifteenth day. The Nones is always

eight days before the Ides (see Table 8.2.1). Dates falling between these

division points are designated by counting inclusively backward from the

upcoming dividion point. Intercalation was performed by repeating the day

VI Kalends March, i.e., inserting a day between VI Kalends March (February

24) and VII Kalends March (February 23).



By the eleventh century, consecutive counting of days from the beginning of

the month came into use. Local variations continued, however, including

counts of days from dates that commemorated local saints. The inauguration

and spread of the Gregorian calendar resulted in the adoption of a uniform

standard for recording dates.



Cappelli (1930), Grotefend and Grotefend (1941), and Cheney (1945) offer

guidance through the maze of medieval dating.

---------------------------------------------------------------------------



9. Calendar Conversion Algorithms



[omitted]

---------------------------------------------------------------------------



10. References



[to be added later]

---------------------------------------------------------------------------

This information is reprinted from the Explanatory Supplement to the

Astronomical Almanac, P. Kenneth Seidelmann, editor, with permission from

University Science Books, Sausalito, CA 94965.



Page author: Lyle Huber <lhuber@nmsu.edu>



Worldwide Leap Year Festival

