in 1582, ten days were eliminated from October

England and its American colonies did not adopt the reformed Gregorian calendar until 1752 and eleven days were eliminated from September.

   September 1752
Su Mo Tu We Th Fr Sa
       1  2 14 15 16
17 18 19 20 21 22 23
24 25 26 27 28 29 30

Alaska did not change from the Julian calendar to the New Style Gregorian calendar until 1867 because, up to that point, it was part of Russia.

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A calendar has been used over the centuries in nearly every civilization. Its purpose is to provide a method of measuring time and to allow man to record and calculate dates and events. The calendar has changed dramatically over the years, and family historians who research colonial records will soon realize that even as recently as 1750, the calendar was different. A basic knowledge of the 1752 calendar change during the colonial period of American history will help with family history research.

The Julian Calendar To better understand the 1752 calendar change, it is beneficial to review the history of major calendars that led up to it, starting with the Romans. Following the advice of his astronomer and mathematician, Julius Caesar established a calendar in 45 B.C. This calendar is known as the Julian or Old Style (O.S.) calendar. It and had three common years containing 365 days, and one year (leap year) containing 366 days (every fourth year). This twelve-month calendar, based on a solar (tropical) year, served for many years in perpetual cycle.

Under this calendar, the first day of the year was March 25th (often known as Lady Day, Annunciation Day, or Feast of the Annunciation), and the last day of the year was March 24th. March was considered the first month.

Examples of the Julian Calendar

7ber VIIber September 7th month
8ber VIIIber October 8th month
9ber IXber November 9th month
10ber Xber December 10th month

Under the Julian calendar, four of the months were written ending in "ber." The Gregorian Calendar During the Middle Ages, astronomers and mathematicians observed that the calendar year was not completely accurate with matching solar years. Errors in the Julian calendar were noted by church officials and scholars because church holidays did not occur in their appropriate seasons.

In 1582, Pope Gregory XIII (150285), who was pope from 1572 to 1585, and his astronomer and mathematician created a new, reformed calendar known as the Gregorian or New Style (N.S.) calendar. It was adopted first in Roman Catholic countries. Protestant countries adopted the calendar during the eighteenth century.

In order to make the calendar adjustment in 1582, ten days were eliminated from October. Thus 4 October 1582 was followed by 15 October 1582.

England and its American colonies did not adopt the reformed Gregorian calendar until 1752. Scotland adopted it earlier, celebrating the New Year on 1 January 1600 and subsequently on January 1st of each year. Interestingly, Alaska did not change from the Julian calendar to the New Style Gregorian calendar until 1867 because, up to that point, it was part of Russia.

In order to make the calendar adjustment, eleven days were dropped from the month of September 1752. An eleven-day adjustment in 1752 was needed because one more day had been lost since the calendar was changed in 1582. The year 1751 began on 25 March and ended on 31 December 1751. The first day of the year was now January 1st and the last day was December 31stthe calendar we use today. Thus, 2 September 1752 was followed by 14 September 1752. In this way, the Julian calendar added one day between 1582 and 1752.

Summary of the 1752 Calendar Change

31 December 1750 was followed by 1 January 1750
24 March 1750 was followed by 25 March 1751
31 December 1751 was followed by 1 January 1752
2 September 1752 was followed by 14 September 1752
31 December 1752 was followed by 1 January 1753

Note that the 1752 calendar change occurred in a series of steps. Just imagine your eighteenth-century ancestors going to bed on Wednesday, September 2nd and waking up on Thursday, September 14th. What would have been September 3rd was actually September 14th in the year 1752. They lost those eleven days from their lives. September 1752 had only nineteen days.

Other countries adopted the Gregorian calendar at different times. The standard reference source for a discussion of the 1752 calendar change is Handbook of Dates for Students of English History. It includes a list of rulers of England, Saints days and festivals used in dating, legal chronology, the Roman calendar, and other calendar details. A chart showing dates of changes from the Julian calendar to the Gregorian calendar in countries outside the British Empire is shown in Know Your Ancestors: A Guide to Genealogical Research.

Double Dating Double dating was used in Great Britain, colonial British America, and British possessions to clarify dates occurring between 1 January and 24 March on years between 1582 and 1752. In the ecclesiastical or legal calendar, March 25th was recognized as the first day of the year and was not double dated.

Researchers of colonial American ancestors will often see double dating in older records. Double dates were identified with a slash mark (/) representing the Old and New Style calendars, e.g., 1690/1691. Even before 1752 in colonial America, some educated clerks knew of the calendar change in Europe and used double dating to distinguish between the calendars. This was especially true in civil records, but less so in church registers. Researchers will often see this type of double dating in New England town records, court records, church records, and wills, or on colonial gravestones or cemetery transcriptions. The system of double dating ended in 1752 in the American colonies with the adoption of the Gregorian calendar.

Double Dating Examples in Colonial Records

15 January 1690 or 15 January 1691
15 February 1745 or 15 February 1746
1 March 1749 or 1 March 1750
15 March 1700 or 15 March 1701
Quaker Dating
After 1752, Quakers adjusted to the calendar change by calling January the first month (N.S. calendar), February the second month, December the twelfth month, and so forth. However, even with the calendar change, dates will undoubtedly appear a little complicated for researchers.

Quakers almost exclusively used numbers for months. In some cases, researchers will find the number and name of the month, such as "4th month called June" or "the 10th day of the 10th month called December 1690." Any date in March was considered the first month. And Sunday was the first day of the week, Monday the second day, and so forth.

Quakers also wrote numbers in their meeting records, such as "3rd month" instead of May (an example before 1752). Saying July (Julius), after Julius Caesar, or August, after the Roman Emperor Caesar Augustus, was considered too pagan or worldly.

An example of an early Quaker date might be: 2/10/1720 (with 2 being the second month). This date should be interpreted as 10 April 1720. For examples of Quaker dating practices, see the article by Gordon L. Remington, "Quaker Preparation for the 1752 Calendar Change," National Genealogical Society Quarterly 87 (June 1999): 146-150.

Helpful Calendar Web Sites

Calendar Change

Calendar Zone

Cyndis List: Calendars and Dates

Doggett, L.E. Calendars

Gregorian Calendar

Perpetual Calendar

Roman Numeral and Date Conversion

The 10,000-Year Calendar

Time and Date.com Selected Bibliography Cheney, C.R., ed. Handbook of Dates for Students of English History. 1945. Reprint. Cambridge: Cambridge University Press, 1995.

Duncan, David Ewing. Calendar: Humanitys Epic Struggle to Determine a True and Accurate Year. New York: Bard, 1998.

Garman, Leo H. "Genealogists and the Gregorian Calendar." NEXUS 6 (April 1989): 61-62.

Haydn, Joseph. Haydns Dictionary of Dates and Universal Information Relating to all Ages and Nations. 25th ed. New York: G.P. Putnams Sons, 1911.

Herber, Mark D. Ancestral Trails: The Complete Guide to British Genealogy and Family History. 1997. Reprint. Baltimore: Genealogical Publishing Co., 1998.

Hetherington, R. "The Calendar." The Midland Ancestor 6 (December 1981): 106-108.

Jacobus, Donald Lines. Genealogy as Pastime and Profession. 2nd rev. ed. Baltimore: Genealogical Publishing Co., 1968. See Chapter 18, "Dates and the Calendar."

Pollard, A.F. "New Years Day and Leap Year in English History." English Historical Review 55 (1940): 177-93.

Prindle, Paul W. "The 1752 Calendar Change." The American Genealogist 40 (October 1964): 246-48.

Remington, Gordon L. "Quaker Preparation for the 1752 Calendar Change." National Genealogical Society Quarterly 87 (June 1999): 146-50.

Richards, Edward Graham. Mapping Time: The Calendar and Its History. New York: Oxford University Press, 1999.

Rubincam, Milton. Pitfalls in Genealogical Research. Salt Lake City: Ancestry, 1987. See Chapter 4, "The Problem of Dates" and Chapter 5, "The 1752 Calendar Change."

Smith, Kenneth Lee. Genealogical Dates: A User-Friendly Guide. Camden, Maine: Picton Press, 1994.

Smith, Mark M. "Culture, Commerce, and Calendar Reform in America." William and Mary Quarterly, 3rd series, vol. 55 (October 1998): 558-84.

Sperry, Kip. Reading Early American Handwriting. Baltimore: Genealogical Publishing Co., 1998. See Chapter 6, "Dates and the Calendar Change."

Webb, Clifford. Dates and Calendars for the Genealogist. London: Society of Genealogists, 1989.

Wilson, George B. "Genealogy and the Calendar." Maryland Magazine of Genealogy 1 (Fall 1978): 13-20.

Kip Sperry is an associate professor of family history at Brigham Young University. He is author of Abbreviations & Acronyms: A Guide for Family Historians (Ancestry, 2000), Reading Early American Handwriting, Genealogical Research in Ohio, and other works.

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The leap-year rule of 366 days began with the Julian calendar and was later adopted by Pope Gregory XIII. At this time, one day was added to February every fourth yearleap year. The Gregorian calendar also allowed for dropping a day from every centesimal year (that is, a year ending in "00") and every year that cannot be divided by 400. If a centesimal year is divisible by 400, it is a leap year, as decreed by Pope Gregory XIII.

The century years 1700, 1800, and 1900, for example, were not leap yearseach February in those years had twenty-eight days. But the year 1600 was a leap year. Today, we add one day to February every fourth year, leap year. The year 2000 was a leap year, and February had twenty-nine days because 2000 is divisible by 400.

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Ancestry Magazine 11/1/2000 - Archive

November/December 2000 Vol. 18 No. 6

Editor's Note: This article is one of three sidebars to "Time to Take Note: The 1752 Calendar Change" by Kip Sperry. Read also "A Brief History of Time" by Jake Gehring and "The Complexities of Leap Year."

The birth date of George Washington illustrates how the Old Style calendar and New Style calendar are used. George Washington (1732-99), the first president of the United States, was actually born 11 February 1731 under the Old Style (Julian) calendar, on his father's estate in Westmoreland County, Virginia. Thus, he was born 11 February 1731/32 using double dating. That is, he was born on the 11th of the month in 1731. However, his birth date is observed on 22 February 1732 under the New Style (Gregorian) calendar.

http://astro.nmsu.edu/~lhuber/leaphist.html

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 Calendar Zone.

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 used in creating calendars. Although some calendars replicate astronomical cycles according to fixed rules, others are based on abstract, perpetually repeating cycles of no astronomical significance. 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. In 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 Britannica 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 calendar. 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 solar 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.

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 ecclesiastical 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).

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 tradition, 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 first 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 crescent. 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 influenced, 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. [Erratum: It appears that Doggett should have stated that Jum'a is Day 6, not Day 5.]

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** 30 7. Rajab** 30 2. Safar 29 8. Sha'ban 29 3. Rabi'a I 30 9. Ramadan*** 30 4. Rabi'a II 29 10. Shawwal 29 5. Jumada I 30 11. Dhu al-Q'adah** 30 6. Jumada II 29 12. 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 named 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 holidays 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 occur 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 measured 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 being 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-chen 32. 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 Longitude Approx. Greg. Date Duration S-1 Lichun Beginning of Spring 315 Feb. 4 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 Summer 45 May 6 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 Autumn 135 Aug. 8 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 Winter 225 Nov. 8 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, calculations 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 peculiarities 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 division 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]

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

Worldwide Leap Year Festival

http://www.trialsofgrizelda.com/harvest/September%201752.htm

The Time that nothing happened.....

Did you know that nothing happened between 3 and 13 September 1752?

In September 1752 the Julian calendar was replaced with the Gregorian calendar in Great Britain and its American colonies. The Julian calendar was 11 days behind the Gregorian calendar, so 14 September got to follow 2 September on the day of the change. The result was that between 3 and 13 September, absolutely nothing happened!

The calendar switch also influenced the way George Washington's birthday is celebrated. He was born on 11 February 1731, but the anniversary of his birth is on 22 February because of the 11 days eliminated from the calendar switch. At the same time, New Year's Day was changed from 25 March to 1 January, thus according to the new calendar, Washington was born in 1732.

The first Roman Calendar (introduced in 535BC) had 10 months, with 304 days in a year that began in March. January and February were added only later. In 46BC, Julius Caesar created "The Year of Confusion" by adding 80 days to the year making it 445 days long to bring the calendar back in step with the seasons. The solar year - with the value of 365 days and 6 hours - was made the basis of the calendar. To take care of the 6 hours, every 4th year was made a 366-day year. It was then that Caesar decreed that the year begins with the 1st of January.

In 325AD Constantine the Great, the first Christian Roman emperor, introduced Sunday as a holy day in a new 7-day week. He also introduced movable (Easter) and immovable feasts (Christmas).

In 1545 the Council of Trent authorised Pope Paul III to reform the calendar once more. Advised by astronomer Father Christopher Clavius and physician Aloysius Lilius, Pope Gregory XIII ordered that Thursday, 4 October 1582 was to be the last day of the Julian calendar. The next day was Friday, 15 October. For long-term accuracy, every 4th year was made a leap year unless it is a century year like 1700 or 1800. Century years can be leap years only when they are divisible by 400 (e.g. 1600). This rule eliminates three leap years in four centuries, making the calendar sufficiently correct for all ordinary purposes.

Protestant rulers ignored the new calendar that the Pope ordered. It was not until 1698 that Germany and the Netherlands changed to the Gregorian calendar. As mentioned, Britain made the change only in 1752. Russia adopted the new calendar in 1918, China in 1949.

In spite of the leap year, the Gregorian year is about 26 seconds longer than the earth's orbital period. Thus the beginning of the third millennium should have been celebrated at 9:01pm on 31 December 1999. But considering that the Gregorian calendar starts with Year 1, and not Year 0, adding 2000 years means that the third millennium started at 21h00:34s on 31 December 2000. However, because Dionysis Exeguus - the 6th Century monk whose task it was to pivot the calendar around the birth of Jesus Christ - miscalculated the founding of Rome by about 4 years (and left out the year 0), the TRUE THIRD MILLENNIUM actually started on 31 December 1995

Clock background copywrite LisaKay Allen

http://www.astro-tom.com/time/calendars.htm

There are two basic sources for calendars presently in use: the monthly motion of the Moon (Lunar calendars) and the yearly motion of the Sun (Solar Calendars). Examples of Lunar calendars still in use are the traditional Jewish and Chinese calendars. The difficulty with Lunar calendars is that the seasons are correlated with the Sun, not the Moon. Thus, Lunar calendars require elaborate adjustments or translations to relate to the seasons. That calendars correlate with seasons is now primarily a matter of convenience, but in more ancient cultures keeping track of the seasons was serious business: it could be a matter of survival to know things like the proper time to plant to ensure a bountiful harvest.

The Roman Lunar Calendar Our present calendar (called the Gregorian Calendar) is a basically solar calendar that grew from what was originally a Lunar calendar used by the Romans. The original calendar contained 10 months of length 29 or 30 days. This was later modified to a 12 month calendar, but 12 months of average length 29.5 days gives only 354 days in the year, whereas the orbital period of the Earth is 365.242199 days. Thus, at the end of each year this calendar was 11 days out of step with the seasons and at the end of 3 years it was almost a month out of step. This was initially corrected in an arbitrary way by adding 13th months, but this was used for various political purposes and soon threw the calendar into severe confusion.

The Julian Calendar In 46 B.C., Julius Caesar reformed the calendar by ordering the year to be 365 days in length and to contain 12 months. This forced some days to be added to some of the months to bring the total from 354 up to 365 days, so the months now were out of phase with the cycles of the Moon: although the Julian Calendar retained monthly divisions, it was no longer a Lunar calendar. The Julian Calendar improved things tremendously, but there was still about a quarter day difference between the true length of the year and the 365 days assumed for the Julian Calendar. Thus, February was given an additional day every 4 years (leap years) and the average length of the Julian year with leap years added was 365.25 days.

The Gregorian Calendar However, the Julian year still differs from the true year of 365.242199 days by 11 minutes and 14 seconds each year, and over a period of 128 years even the Julian Calendar was in error by one day with respect to the seasons. By 1582 this error had accumulated to 10 days and Pope Gregory XIII ordered another reform: 10 days were dropped from the year 1582, so that October 4, 1582, was followed by October 15, 1582. In addition, to guard against further accumulation of error, in the new Gregorian Calendar it was decreed that century years not divisible by 400 were not to be considered leap years. Thus, 1600 was a leap year but 1700 was not. This made the average length of the year sufficiently close to the actual year that it would take 3322 years for the error to accumulate to 1 day.

A further modification to the Gregorian Calendar has been suggested: years evenly divisible by 4000 are not leap years. This would reduce the error between the Gregorian Calendar Year and the true year to 1 day in 20,000 years. However, this last proposed change has not been officially adopted; there is plenty of time to consider it, since it would not have an effect until the year 4000.

Adoption of the Gregorian Calendar An interesting historical sidelight on the Gregorian Calendar is that not all countries adopted it immediately. In particular, it was adopted uniformly in Catholic countries, but Protestant countries often still used the Julian Calendar. Thus, the date could change by 10 days simply by crossing certain country borders! England and its American colonies did not adopt the Gregorian Calendar until 1752, when 11 days were removed from the calendar, and Russia resisted this change until after the 1917 Revolution. One consequence of the British adoption of the Gregorian Calendar in 1752 is that George Washington was born on February 11, 1731, according to the calendar in use on his birthday, but we now celebrate his date of birth as February 22, 1731 (actually, even that is no longer true with the advent of Presidents Day).

This Calendar Program allows you to get a calendar for an arbitrary year in the United States and England (if you submit it with no entry it will return the calendar for the present year, by default). Look at the calendar for the year 1752 and note the missing days in September associated with the transition to the Gregorian calendar in England and its colonies.

http://mindprod.com/jgloss/leapyear.html

Java Glossary : leap year Last updated 2004-05-06 by Roedy Green 1996-2004 Canadian Mind Products

Java definitions: 0-9 A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

You are here : home : Java Glossary : L words : leap year.

leap year The earth takes 365.242190 days to orbit the sun, though this varies from 365.242196 in 1900 to 365.242184 in 2100. For the following calculations of error, we presume 1900 as the standard year. This is not an integral number of days, so we fiddle with the calendar adding leap years to keep the calendar in sync with the sun. There is a great deal of confusion about how you calculate the leap years. Julius Caesar created the modern calendar in 46 BC. Prior to that it was a mess well beyond your wildest dreams. The Julian calendar is still used by the Russian Orthodox church and by programmers who are ignorant of the official Gregorian calendar. It has leap years every four years without exception. It corrects to 365.25. It gets ahead 1 day every 128 years.

public static final boolean isLeapViaJulian ( int yyyy) { if ( yyyy < 0) return (yyyy +1) % 4 == 0; return yyyy % 4 == 0; } Joseph Justus Scaliger (1540-1609) invented an Astronomical Julian calendar, (named after his father Julius Caesar Scaliger). This Julian calendar uses the offset in days since noon, Jan 1st 4713 BC. In that scheme, 2000 January 1 noon is the start of day number 2,451,545. This calendar follows the original Julian scheme of always adding leap years every four years. The next major correction was the Gregorian calendar. By 1582, this excess of leap years had built up noticeably. At the suggestion of astronomers Luigi Lilio and Chistopher Clavius, Pope Gregory XIII dropped 10 days from the calendar. Thursday 1582 October 4 Julian was followed immediately by Friday 1582 October 15 Gregorian. He decreed that every 100 years, a leap year should be dropped except that every 400 years the leap year should be restored. Only Italy, Poland, Portugual and Spain went along with the new calendar immediately. One by one other countries adopted it in different years. Britain and its territories (including the USA and Canada) adopted it in 1752. By then, 11 days had to be dropped. 1752 September 2 was followed immediately by 1752 September 14. The Gregorian calendar is the most widely used scheme. This is the scheme endorsed by the US Naval observatory. It corrects the year to 365.2425. It gets ahead 1 day every 3289 years.

public static final boolean isLeapViaPopeGregory ( int yyyy) { if ( yyyy < 0) return (yyyy +1) % 4 == 0; if ( yyyy < 1582 ) return yyyy % 4 = = 0; if ( yyyy % 4 != 0 ) return false; if ( yyyy % 100 != 0 ) return true; if ( yyyy % 400 != 0 ) return false; return true; } Astronomer John Herschel (1792-1871) suggested dropping a leap year every 4000 years. This scheme never received official support. It corrects to 365.24225. It gets ahead 1 day every 18,519 years. public static final boolean isLeapViaHerschel ( int yyyy) { if ( yyyy < 0) return (yyyy +1) % 4 == 0; if ( yyyy < 1582 ) return yyyy % 4 = = 0; if ( yyyy % 4 != 0 ) return false; if ( yyyy % 100 != 0 ) return true; if ( yyyy % 400 != 0 ) return false; if ( yyyy % 4000 != 0 ) return true; return false; } The Greek Orthodox church drops the 400 rule and in its place uses a rule that any year that when divided by 900 gives a remainder of either 200 or 600 is a leap year. This is the official system in Russia. It corrects to 365.24222. It gets ahead 1 day every 41,667 years. public static final boolean isLeapViaGreek ( int yyyy) { if ( yyyy < 0) return (yyyy +1) % 4 == 0; if ( yyyy < 1582 ) return yyyy % 4 == 0; if ( yyyy % 4 != 0 ) return false; if ( yyyy % 100 != 0 ) return true; int remdr = yyyy % 900; return remdr == 200 || remdr == 600; } The SPAWAR group in the US Navy propose the following algorithm where a leap year is dropped every 3200 years. This is most accurate of the schemes, and also has the desirable property of undercorrecting, leaving room for leap second correction. It corrects to 365.2421875. It gets behind 1 day every 117,647 years. Leap seconds are added on average every 3 out of 4 years to correct for the ever lengthening day. It is best to have a scheme with slightly too few leap days than too many, since the leap seconds can compensate also. Leap seconds add up to roughly an extra day every 115,000 years. When you consider the effects of leap seconds, this scheme is bang on, within the limits of the varying length of an astronomical year. public static final boolean isLeapViaSPAWAR ( int yyyy) { if ( yyyy < 0) return (yyyy +1) % 4 == 0; if ( yyyy < 1582 ) return yyyy % 4 == 0; if ( yyyy % 4 != 0 ) return false; if ( yyyy % 100 != 0 ) return true; if ( yyyy % 400 != 0 ) return false; if ( yyyy % 3200 != 0 ) return true; return false; } I will leave as an exercise for the reader the staircase leap year algorithm that is the most accurate possible. If you get it, I will post it here with an attribution to you. Hints: Consider how you use pixels to draw a diagonal line that is almost horizontal. What if you actually went outside and looked at the heavens at midnight each February 28, or every fourth February 28, and made the decision at that point whether to have a February 29. Integrate the function that gives the length of the year in days given the year that considers both leap seconds and the gradual shortening of the year. Perhaps some future earth citizens will drag the orbit of the earth slightly so no further adjustments will be necessary. More likely, sentient life will convert to a less terracentric calendar, perhaps time quanta since the big bang. year Julian Leap? Gregorian Leap? Herschel Leap? Greek Leap? SPAWAR Leap? 1 no no no no no 4 yes yes yes yes yes 1580 yes yes yes yes yes 1582 no no no no no 1584 yes yes yes yes yes 1600 yes yes yes no yes 1700 yes no no no no 1800 yes no no no no 1900 yes no no no no 1996 yes yes yes yes yes 1997 no no no no no 1999 no no no no no 2000 yes yes yes yes yes 2100 yes no no no no 2200 yes no no no no 2300 yes no no no no 2400 yes yes yes yes yes 2800 yes yes yes no yes 2900 yes no no yes no 3200 yes yes yes no no 3300 yes no no yes no 3600 yes yes yes no yes 3800 yes no no yes no 4000 yes yes no no yes 4200 yes no no yes no 4400 yes yes yes no yes 4700 yes no no yes no 4800 yes yes yes no yes 5100 yes no no yes no 5200 yes yes yes no yes 6400 yes yes yes no no 6500 yes no no yes no 6800 yes yes yes no yes 6900 yes no no yes no 7200 yes yes yes no yes 7400 yes no no yes no 7600 yes yes yes no yes 7800 yes no no yes no

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