February 24, 2005



History of Our Calendar

Dr. Frank J. Collazo


Before today's Gregorian calendar was adopted, the older Julian calendar was used. It was admirably close to the actual length of the year, as it turns out, but the Julian calendar was not so perfect that it didn't slowly shift off track over the following centuries. But, hundreds of years later, monks were the only ones with any free time for scholarly pursuits -- and they were discouraged from thinking about the matter of "secular time" for any reason beyond figuring out when to observe Easter. In the Middle Ages, the study of the measure of time was first viewed as prying too deeply into God's own affairs -- and later thought of as a lowly, mechanical study, unworthy of serious contemplation.


As a result, it wasn't until 1582, by which time Caesar's calendar had drifted a full 10 days off course that Pope Gregory finally reformed the Julian calendar. Ironically, by the time the Catholic church buckled under the weight of the scientific reasoning that pointed out the error, it had lost much of its power to implement the fix. Protestant tract writers responded to Gregory's calendar by calling him the "Roman Antichrist" and claiming that its real purpose was to keep true Christians from worshiping on the correct days. The "new" calendar, as we know it today, was not adopted uniformly across Europe until well into the 18th century.



Here are a few historical questions about our calendar:

Has the year always started on 1 January?

Then what about leap years?
What is the origin of the names of the months?
How did Dionysius date Christ's birth?
Was Jesus born in the year 0?
Why do the 9th thru 12th months have names that mean 7th, 8th, 9th and 10th?
Why does February have only 28 days?



Has the year always started on 1 January?

For the man on the street, yes. When Julius Caesar introduced his calendar in 45 B.C.E., he made 1 January the start of the year, and it was always the date on which the Solar Number and the Golden Number were incremented. However, the church didn't like the wild parties that took place at the start of the new year, and in C.E. 567 the council of Tours declared that having the year start on 1 January was an ancient mistake that should be abolished.


Through the middle ages various New Year dates were used. If an ancient document refers to year X, it may mean any of 7 different periods in our present system:




1 Mar X to 28/29 Feb X+1

1 Jan X to 31 Dec X

1 Jan X-1 to 31 Dec X-1

25 Mar X-1 to 24 Mar X

25 Mar X to 24 Mar X+1

Saturday before Easter X to Friday before Easter X+1

25 Dec X-1 to 24 Dec X


Choosing the right interpretation of a year number is difficult, so much more as one country might use different systems for religious and civil needs.


The Byzantine Empire used a year starting on 1 September, but they did not count years since the birth of Christ, instead they counted years since the creation of the world which they dated to 1 September 5509 B.C.E.


Since about 1600 most countries have used 1 January as the first day of the year. Italy and England, however, did not make 1 January official until around 1750. In England (but not Scotland) three different years were used:


The historical year, which started on 1 January.

The liturgical year, which started on the first Sunday in advent.

The civil year, which from the 7th to the 12th century started on 25 December, from the 12th century until 1751 started on 25 March, and from 1752 started on 1 January.

(See the British Calendar Act of 1751.)


What About Leap Years? If the year started on, for example, 1 March (two months later than our present year) when was the leap day inserted? When it comes to determining if a year is a leap year, since AD 8 the Julian calendar has always had 48 months between two leap days. So, in a country using a year starting on 1 March, 1439 would have been a leap year, because their February 1439 would correspond to February 1440 in the January-based reckoning.


Origin of the Names of the Months: A lot of languages, including English, use month names based on Latin. Their meaning is listed below. However, some languages (Czech and Polish, for example) use quite different names.






Named after the god Janus.



Named after Februa, the purification festival.



Named after the god Mars.












Named either after the goddess Aphrodite or the Latin word aperire, to open.



Probably named after the goddess Maia.



Probably named after the goddess Juno.



Named after Julius Caesar in 44 B.C.E. Prior to that time its name was Quintilis from the word quintus, fifth, because it was the 5th month in the old Roman calendar.



Named after emperor Augustus in 8 B.C.E. Prior to that time the name was Sextilis from the word sextus, sixth, because it was the 6th month in the old Roman calendar.



From the word septem, seven, because it was the 7th month in the old Roman calendar.



From the word octo, eight, because it was the 8th month in the old Roman calendar.



From the word novem, nine, because it was the 9th month in the old Roman calendar.



From the word decem, ten, because it was the 10th month in the old Roman calendar.


How Did Dionysius Date Christ's Birth? There are quite a few theories about this. Many of the theories are presented as if they were indisputable historical facts. The following are two theories that tend to be more accepted:


According to the Gospel of Luke (3:1 & 3:23) Jesus was "about thirty years old" shortly after "the fifteenth year of the reign of Tiberius Caesar." Tiberius became emperor in C.E. 14. If you combine these numbers you reach a birth year for Jesus that is strikingly close to the beginning of our year reckoning. This may have been the basis for Dionysius' calculations.


Dionysius' original task was to calculate an Easter table. In the Julian calendar, the dates for Easter repeat every 532 years. The first year in Dionysius' Easter tables is C.E. 532. Is it a coincidence that the number 532 appears twice here? Or did Dionysius perhaps fix Jesus' birth year so that his own Easter tables would start exactly at the beginning of the second Easter cycle after Jesus' birth?


Was Jesus born in the year 0? No.

There are two reasons for this: There is no year 0, and Jesus was born before 4 B.C.E.



The concept of a year "zero" is a modern myth (but a very popular one). Roman numerals do not have a figure designating zero, and treating zero as a number on an equal footing with other numbers was not common in the 6th century when our present year reckoning was established by Dionysius Exiguus. Dionysius let the year C.E.1 start one week after what he believed to be Jesus' birthday.


Therefore, C.E.1 follows immediately after 1 B.C.E. with no intervening year zero. So a person who was born in 10 B.C.E. and died in C.E. 10, would have died at the age of 19, not 20. Furthermore, Dionysius' calculations were wrong. The Gospel of Matthew tells us that Jesus was born under the reign of king Herod the Great, who died in 4 BCE. It is likely that Jesus was actually born around 7 B.C.E. The date of his birth is unknown; it may or may not be 25 December.


Why do the 9th thru 12th months have names that mean 7th, 8th, 9th and 10th?

September thru December were the seventh thru tenth months of a calendar used by the first Romans. Ancient historian and Greek biographer Plutarch, wrote in C.E. 75, about how they became displaced to two positions higher than their names would indicate.

> Read excerpt of Plutarch's essay.

> Read more about the early Roman calendar.


Why does February have only 28 days? January and February both date from about the time of Rome's founding. They were added to a calendar that had been divided into ten month-like periods whose lengths varied from 20 to 35 or more days. A winter season was not included, so those period lengths are believed to have been intended to reflect growth stages of crops and cattle.


When introduced, January was given 29 days and put at the beginning of the calendar year. February was given 23 days and put at the end. Then, for an undetermined period shortly after Rome's founding, months were said to have begun when a new moon was first sighted. At some later time, month lengths were separated from lunations and again became fixed. At that time, February's original length was extended by five days, which gave it a total of 28.


Our Year: Calendars are normally based on astronomical events, and the two most

Important astronomical objects are the sun and the moon. Their cycles are very important in the construction and understanding of calendars. Our concept of a year is based on the earth's motion around the sun. The time from one fixed point, such as a solstice or equinox, to the next is called a tropical year. Its length is currently 365.242190 days, but it varies. Around 1900 its length was 365.242196 days, and around 2100 it will be 365.242184 days. (This definition of the tropical year is not quite accurate, see section astronomic issues for more details.)




Illuminations of Dante's Divine Comedy
by Giovanni di Paolo (15th cent)
Dante and Beatrice reach the sun, shown as a golden wheel sending golden rays to the landscape below. The Sun, located in the middle of the orbs, with three lesser above and three below, like the heart in the middle of the body, or a wise king in the middle of his kingdom.

Our concept of a month is based on the moon's motion around the earth, although this connection has been broken in the calendar commonly used now. The time from one new moon to the next is called a synodic month, and its length is currently 29.5305889 days, but it varies. Around 1900 its length was 29.5305886 days, and around 2100 it will be 29.5305891 days.


Note that these numbers are averages. The actual length of a particular year may vary by several minutes due to the influence of the gravitational force from other planets. Similarly, the time between two new moons may vary by several hours due to a number of factors, including changes in the gravitational force from the sun, and the moon's orbital inclination.


It is unfortunate that the length of the tropical year is not a multiple of the length of the synodic month. This means that with 12 months per year, the relationship between our month and the moon cannot be maintained.


However, 19 tropical years is 234.997 synodic months, which is very close to an integer. So every 19 years the phases of the moon fall on the same dates (if it were not for the skewness introduced by leap years). Nineteen years is called a Metonic cycle (after Meton, an astronomer from Athens in the 5th century B.C.E.). So, to summarize, please note: A tropical year is 365.24219 days, and a synodic month is 29.53059 days.

Nineteen tropical years is close to an integral number of synodic months. The Christian calendar (Gregorian calendar) is based on the motion of the earth around the sun, while the months have no connection with the motion of the moon. On the other hand, the Islamic calendar is based on the motion of the moon, while the year has no connection with the motion of the earth around the sun.


Finally, the Jewish calendar combines both, in that its years are linked to the motion of the earth around the sun, and its months are linked to the motion of the moon. See also related information in another exhibit, Daylight Saving Time.


Astronomical Basis 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.   


What are different measures of the year?
What are Equinoxes and Solstices?
Did the church study astronomy?
Didn't the church condemn Galileo?
How did the observatories work?
How did Cassini prove Kepler was right?


What are Different Measures of the Year? The tropical year is defined as the mean interval between vernal equinoxes; it corresponds to the cycle of the seasons. Our calendar year is linked to the tropical year as measured between two March equinoxes, as originally established by Caesar and Sosigenes. 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 T2 + 2.64E-10 T3 [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.


Another kind of year is called the sidereal year, which is the time it takes the earth to orbit the sun. In the year 2000, the length of the Tropical Year = 365.24219 days. 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.


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. 00000021621 T - 3.64E-10 T2 [days] of the Sidereal Year = 365.2564.


NIST Exhibits



Meridan Line. S. Petronio, Bologna
In this sun calendar, a hole in the ceiling of the cathedral projects a shaft of sunlight onto this bronze strip on the pavement below which is engraved with the days of the year and signs of the zodiac.

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.



Astronomical clock
from Prague, Czech Republic


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 if the cycles are accurate.


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.

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.


What are Equinoxes and Solstices? Equinoxes and solstices are frequently used as anchor points for calendars. For people in the northern hemisphere, Winter solstice is the time in December when the sun reaches its southernmost latitude. At this time we have the shortest day. The date is near 21 December.


Summer solstice is the time in June when the sun reaches its northernmost latitude. At this time we have the longest day. The date is near 21 June.


Vernal equinox is the time in March when the sun passes the equator moving from the southern to the northern hemisphere. Day and night have approximately the same length. The date is near 20 March.


Autumnal equinox is the time in September when the sun passes the equator moving from the northern to the southern hemisphere. Day and night have approximately the same length. The date is near 22 September.


For people in the southern hemisphere these events are shifted half a year. The astronomical "tropical year" is frequently defined as the time between, say, two vernal equinoxes, but this is not actually true. Currently the time between two vernal equinoxes is slightly greater than the tropical year. The reason is that the earth's position in its orbit at the time of solstices and equinoxes shifts slightly each year (taking approximately 21,000 years to move all the way around the orbit). This, combined with the fact that the earth's orbit is not completely circular, causes the equinoxes and solstices to shift with respect to each other. The astronomer's mean tropical year is really a somewhat artificial average of the period between the time when the sun is in any given position in the sky with respect to the equinoxes and the next time the sun is in the same position.


Did the church study astronomy? Yes, they did. Although the Roman Catholic Church once waged a long and bitter war on science and astronomy (particularly condemning Galileo), in general, they were quite involved in astronomy. The church gave more financial and social support to the study of astronomy for over six centuries, from the recovery of ancient learning during the late Middle Ages into the Enlightenment, than any other, and probably, all other, institutions. The church was not necessarily seeking knowledge for knowledge's sake, a traditional aim of pure science. Rather, like many patrons, it wanted something practical in return for its investments: mainly the improvement of the calendar so church officials could more accurately establish the date of Easter.


When to celebrate the feast of Christ's resurrection had become a bureaucratic crisis in the church. Traditionally, Easter fell on the Sunday after the first full moon of spring. But by the 12th century, the usual ways to predict that date had gone awry. To set a date for Easter Sunday years in advance, and thus reinforce the church's power and unity, popes and ecclesiastical officials had for centuries relied on astronomers who pondered over old manuscripts and devised instruments that set them at the forefront of the scientific revolution.


In its scientific zeal, the church adapted cathedrals across Europe, and a tower at the Vatican itself, so their darkened vaults could serve as solar observatories. Beams of sunlight that fell past religious art and marble columns not only inspired the faithful but provided astronomers with information about the Sun, the Earth and their celestial relationship. Among other things, solar images projected on cathedral floors disclosed the passage of dark spots across the Sun's face, a blemish in the heavens, which theologians once thought to be without flaw. Over the centuries, observatories were built in cathedrals and churches throughout Europe, including those in Rome, Paris, Milan, Florence, Bologna, Palermo, Brussels and Antwerp.


Didn't the church condemn Galileo? Yes. The traditional view of the church's hostility toward science grew out of its famous feud with Galileo, condemned to house arrest in 1632 for astronomical heresy. Since antiquity, astronomers had put Earth at the center of planetary motions, a view the church had embraced. But Galileo, using the new telescope, became convinced that the planets in fact moved around the Sun, a view Nicholas Copernicus, a Polish astronomer, had championed.


                 Read Nicholas Copernicus, On the Revolutions.

                 Read Galileo Galilei, Dialogue Concerning the Two Chief World Systems.


The censure of Galileo, at age 70, hurt the image of the church for centuries. Pope John Paul II finally acknowledged in 1992, 359 years later, that the church had erred in condemning the scientific giant. Although some scholars claim that Rome's handling of Galileo made Copernican astronomy a forbidden topic among faithful Catholics for two centuries, in fact, Rome's support of astronomy was considerable. The church tended to regard all the systems of the mathematical astronomy as fictions. That interpretation gave Catholic writers scope to develop mathematical and observational astronomy almost as they pleased, despite the tough wording of the condemnation of Galileo.


How did the observatories work?

Typically, the building, dark inside, needed only a small hole in the roof to allow a beam of sunlight to strike the floor below, producing a clear image of the solar disk. In effect, the church had been turned into a pinhole camera, in which light passes through a small hole into a darkened interior, forming an image on the opposite side. On each sunny day, the solar image would sweep across the church floor and, exactly at noon, cross a long metal rod that was the observatory's most important and precise part.


The noon crossings over the course of a year would reach the line's extremities -- which usually marked the summer and winter solstices, when the Sun is farthest north and south of the Equator. The circuit, among other things, could be used to measure the year's duration with great precision.


The path on the floor was known as a meridian line, like the north-south meridians of geographers. The rod, in keeping with its setting and duties, was often surrounded by rich tile inlays and zodiacal motifs. The instruments lost much of their astronomical value around the middle of the 18th century as telescopes began to exceed them in power. But the observatories still played a significant role because the solar timepieces were often used to correct errors in mechanical clocks and even to set time for railroads.

One of the observatories also impressed Charles Dickens, who in his book "Pictures from Italy" wrote that he found little to like in Bologna except "the Church of San Petronio, where the sunbeams mark the time among the kneeling people."


Today, the surviving cathedral solar instruments are lovely anachronisms that baffle most visitors, who are usually unaware of their original use or historical importance. In the book, The Sun in the Church, author Dr. Heilbron, describes his astonishment with seeing the old instruments in Bologna, Italy, at the Basilica of San Petronio. "The church itself was beautiful, somber," Dr. Heilbron recalled. "When the sun crawled across that floor, there was nothing else. That's what you had to look at. It was intense."


In the great Basilica of San Petronio, a solar observatory was erected in 1576 by Egnatio Danti, a mathematician and Dominican friar who worked for Cosimo I dei Medici, the Grand Duke of Tuscany, who also advised Pope Gregory on calendar reform. The church observatory produced data long before the telescope existed. By 1582, the Gregorian calendar had been established, creating the modern year of 365 days and an occasional leap year of 366 days. Danti was rewarded with a commission to build a solar observatory in the Vatican itself within the Torre dei Venti, or Tower of the Winds. The golden age of the cathedral observatories came later, between 1650 and 1750, and helped to disprove the astronomical dogma that the church had defended with such militancy in the case of Galileo.


How did Cassini prove Kepler was right?

Among the best known of the rebel observers was Giovanni Cassini, an Italian astronomer who gained fame for discovering moons of Saturn and the gaps in its rings that still bear his name. Around 1655, Cassini persuaded the builders of the Basilica of San Petronio that they should include a major upgrade of Danti's old meridian line, making it larger and far more accurate, its entry hole for daylight moved up to be some 90 feet high, atop a lofty vault.


"Most illustrious nobles of Bologna," Cassini boasted in a flier drawn up for the new observatory, "the kingdom of astronomy is now yours." The exaggeration turned out to have some merit as Cassini used the observatory to investigate the "orbit" of the Sun, quietly suggesting that it actually stood still while the Earth moved.


Cassini decided to use his observations to try to confirm the theories of Johannes Kepler, the German astronomer who had proposed in 1609 that the planets moved in elliptical orbits not the circles that Copernicus had envisioned.




Kepler's Model of the Universe
Another model of the heavens that we've seen before is Kepler's nested Platonic solids and another the dome. In The Dome of Heaven, Karl Lehmann, who writes, "One of the most fundamental artistic expressions of Christian thought and emotion is the vision of heaven depicted in painting or mosaic on domes..."

If true, that meant the Earth over the course of a year would pull slightly closer and farther away from the Sun. At least in theory, Cassini's observatory could test Kepler's idea, since the Sun's projected disk on the cathedral floor would shrink slightly as the distance grew and would expand as the gap lessened. Such an experiment could also address whether there was any merit to the ancient system of Ptolemy, some interpretations of which had the Earth moving around the Sun in an eccentric circular orbit.


Ptolemy's Sun at its closest approach moved closer to the Earth than Kepler's Sun did, in theory making the expected solar image larger and the correctness of the rival theories easy to distinguish.


For the experiment to succeed, Cassini could tolerate measurement errors no greater than 0.3 inches in the Sun's projected face, which ranged from 5 to 33 inches wide, depending on the time of year. No telescope of the day could achieve that precision. The experiment was run around 1655, and after much trial and error, succeeded. Cassini and his Jesuit allies confirmed Kepler's version of the Copernican theory.


Between 1655 and 1736, astronomers used the solar observatory at San Petronio to make 4,500 observations, aiding substantially the tide of scientific advance.



What years are leap years?

Leap years were introduced to keep New Year's Day on autumnal equinox. But this

Turned out to be difficult to handle, because equinox is not easy to predict. In fact, the first decree implementing the calendar (5 Oct 1793) contained two contradictory rules, as it stated that the first day of each year would be that of the autmunal equinox. Every fourth year would be a le leap year.


In practice, the first calendars were based on the equinoxial condition. To remove the confusion, a rule similar to the one used in the Gregorian Calendar (including a 4000 year rule) was proposed by the calendar's author, Charles Rommes, but his proposal ran into political problems. In short, during the time when the French Revolutionary Calendar was in use, the following years were leap years: 3, 7, and 11.


Is there a 4000-year rule?

It has been suggested (by the astronomer John Herschel (1792-1871) among others) that a better approximation to the length of the tropical year would be 365 969/4000 days = 365.24225 days. This would dictate 969 leap years every 4000 years, rather than the 970 leap years mandated by the Gregorian calendar. This could be achieved by dropping one leap year from the Gregorian calendar every 4000 years, which would make years divisible by 4000 non-leap years. This rule has, however, not been officially adopted.


Do the Greeks do it differently?

When the Orthodox church in Greece finally decided to switch to the Gregorian calendar in the 1920s, they tried to improve on the Gregorian leap year rules, replacing the "divisible by 400" rule by the following: Every year which when divided by 900 leaves a remainder of 200 or 600 is a leap year. This makes 1900, 2100, 2200, 2300, 2500, 2600, 2700, 2800 non-leap years, whereas 2000, 2400, and 2900 are leap years. This will not create a conflict with the rest of the world until the year 2800. This rule gives 218 leap years every 900 years, which gives us an average year of 365 218/900 days = 365.24222 days, which is certainly more accurate than the official Gregorian number of 365.2425 days. However, this rule is not official in Greece.


What day is the leap day?

It is 24 February! Weird? Yes! The explanation is related to the Roman calendar.

From a numerical point of view, of course, 29 February is the extra day. But from the point of view of celebration of feast days, the following correspondence between days in leap years and non-leap years has traditionally been used:


Non-leap year


Leap year

22 February

22 February

23 February

23 February


24 February (extra day)

24 February

25 February

25 February

26 February

26 February

27 February

27 February

28 February

28 February

29 February


For example, the feast of St. Leander has been celebrated on 27 February in non-leap years and on 28 February in leap years. Many countries are gradually changing the leap day from the 24th to the 29th. This affects countries such as Sweden and Austria that celebrate "name days" (i.e. each day is associated with a name).



What is the Solar Cycle?

In the Julian calendar the relationship between the days of the week and the dates of the year is repeated in cycles of 28 years. In the Gregorian calendar this is still true for periods that do not cross years that are divisible by 100 but not by 400. A period of 28 years is called a Solar Cycle. The Solar Number of a year is found as: Solar Number = (year + 8) mod 28 + 1


In the Julian calendar there is a one-to-one relationship between the Solar Number and the day on which a particular date falls. (The leap year cycle of the Gregorian calendar is 400 years, which is 146,097 days, which curiously enough is a multiple of 7. So in the Gregorian calendar the equivalent of the "Solar Cycle" would be 400 years, not 7 x 400 = 2800 years as one might be tempted to believe.)


When can I reuse my 1992 calendar?

Let us first assume that you are only interested in which dates fall on which days of the week; you are not interested in the dates for Easter and other irregular holidays. Let us further confine ourselves to the years 1901-2099.


With these restrictions, the answer is as follows:

If y If year X is a leap year, you can reuse its calendar in year X+28.

                  If year X is the first year after a leap year, you can reuse its calendar in years X+6, X+17, and X+28.

                  If year X is the second year after a leap year, you can reuse its calendar in years X+11, X+17, and X+28.

                  If year X is the third year after a leap year, you can reuse its calendar in years X+11, X+22, and X+28.

Note that the expression X+28 occurs in all four items above. So you can always reuse your calendar every 28 years.


But if you also want your calendar's indication of Easter and other Christian holidays to be correct, the rules are far too complex to be put to a simple formula. Sometimes calendars can be reused after just six years. For example, the calendars for the years 1981 and 1987 are identical, even when it comes to the date for Easter. But sometimes a very long time can pass before a calendar can be reused; if you happen to have a calendar from 1940, you won't be able to reuse it until the year 5280!


What is the correct way to write dates?

The answer to this question depends on what you mean by "correct." Different countries have different customs.

In the U.S.A. a month-day-year format is common:

12/25/1998          12-25-1998

Most other countries use a day-month-year format, such as:

25.12.1998          25/12/1998          25/12-1998          25.XII.1998

International standard ISO-8601 mandates a year-month-day format, namely either

1998-12-25 or 19981225.

In all of these systems, the first two digits of the year are frequently omitted:

25.12.98          12/25/98          98-12-25


This confusion leads to misunderstandings. What is 02-03-04? To most people it is 2 March 2004; to an American it is 3 February 2004; and to a person using the international standard it would be 4 March 2002. If you want to be sure that people understand you, I recommend that you write the month with letters instead of numbers, and write the years as 4-digit numbers.


How does one count years?

In about C.E. 523, the papal chancellor, Bonifatius, asked a monk by the name of Dionysius Exiguus to devise a way to implement the rules from the Nicean council (the so-called "Alexandrine Rules") for general use.


Dionysius Exiguus (in English known as Denis the Little) was a monk from Scythia. He was a canon in the Roman curia, and his assignment was to prepare calculations of the dates of Easter. At that time it was customary to count years since the reign of emperor Diocletian, but in his calculations Dionysius chose to number the years since the birth of Christ, rather than honour the persecutor Diocletian.


Dionysius (wrongly) fixed Jesus' birth with respect to Diocletian's reign in such a manner that it falls on 25 December 753 AUC (ab urbe condita, i.e. since the founding of Rome), thus making the current era start with C.E. 1 on 1 January 754 AUC. How Dionysius established the year of Christ's birth is not known (see section 2.10.1 for a couple of theories). Jesus was born under the reign of king Herod the Great, who died in 750 AUC, which means that Jesus could have been born no later than that year. Dionysius' calculations were disputed at a very early stage.


When people started dating years before 754 AUC using the term "Before Christ," they let the year 1 B.C.E. immediately precede C.E. 1 with no intervening year zero. Note, however, that astronomers frequently use another way of numbering the years B.C.E. Instead of 1 B.C.E. they use 0, instead of 2 B.C.E. they use -1, instead of 3 B.C.E. they use -2, etc.


It is frequently claimed that it was the venerable Bede (673-735) who introduced B.C. dating. Although Bede seems to have used the term on at least one occasion, it is generally believed that B.C. dates were not used until the middle of the 17th century.


In this section I have used C.E. 1 = 754 AUC. This is the most likely equivalence between the two systems. However, some authorities state that C.E. 1 = 753 AUC or 755 AUC. This confusion is not a modern one; it appears that even the Romans were in some doubt about how to count the years since the founding of Rome.




NIST Exhibits

When did the 3rd millennium start? The first millennium started in AD 1, so the millennia are counted in this manner:


1st millennium: 1-1000
2nd millennium: 1001-2000
3rd millennium: 2001-3000


Thus, the 3rd millennium and, similarly, the 21st century started on 1 Jan 2001. This is the cause of some heated debate, especially since some dictionaries and encyclopedias say that a century starts in years that end in 00. Furthermore, the change 1999/2000 is obviously much more spectacular than the change 2000/2001.


Let us propose a few compromises: Any 100-year period is a century. Therefore the period from 23 June 2004 to 22 June 2104 is a century. So please feel free to celebrate the start of a century any day you like! Although the 20th century started in 1901, the 1900s started in 1900. Similarly, the 21st century started in 2001, but the 2000s started in 2000.


What do A.D., B.C., C.E., and B.C.E. stand for?

Years before the birth of Christ are in English traditionally identified using the abbreviation B.C. ("Before Christ"). Years after the birth of Christ are traditionally identified using the Latin abbreviation AD ("Anno Domini", that is, "In the Year of the Lord"). Some people, who want to avoid the reference to Christ that is implied in these terms, prefer the abbreviations BCE ("Before the Common Era" or "Before the Christian Era") and CE ("Common Era" or "Christian Era").


Historical Eras and Chronology: The calendars described in this exhibit, 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.E., 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.E. 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.E. 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.



Countries' Calendar Reform: 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).


When did countries change from Julian to Gregorian calendars?

The papal bull of February 1582 decreed that 10 days should be dropped from October 1582 so that 15 October should follow immediately after 4 October, and from then on the reformed calendar should be used. This was observed in Italy, Poland, Portugal, and Spain. Other Catholic countries followed shortly after, but Protestant countries were reluctant to change, and the Greek orthodox countries didn't change until the start of the 1900s.

Changes in the 1500s required 10 days to be dropped. Changes in the 1600s required 10 days to be dropped. Changes in the 1700s required 11 days to be dropped. Changes in the 1800s required 12 days to be dropped. Changes in the 1900s required 13 days to be dropped. For example, when Soviet Russia undertook its calendar reform in February 1918, they moved from the Julian calendar to the Gregorian. This move resulted in a loss of 13 days, so that February 1, 1918, became February 14.


The following list contains the dates for changes in a number of countries. It is very strange that in many cases there seems to be some doubt among authorities about what the correct days are. Different sources give very different dates in some cases. The list below does not include all the different opinions about when the change took place.

(See the British Calendar Act of 1751.)





December 1912



Different regions on different dates
Brixen, Salzburg and Tyrol:
5 Oct 1583 was followed by 16 Oct 1583
Carinthia and Styria:
14 Dec 1583 was followed by 25 Dec 1583
See also Czechoslovakia and Hungary



Then part of the Netherlands



31 Mar 1916 was followed by 14 Apr 1916



Different areas changed at different times.
Newfoundland and Hudson Bay coast:
2 Sep 1752 was followed by 14 Sep 1752
Mainland Nova Scotia:
Gregorian 1605 - 13 Oct 1710
Julian 2 Oct 1710 - 2 Sep 1752
Gregorian since 14 Sep 1752
Rest of Canada:
Gregorian from first European settlement








The Gregorian calendar replaced the Chinese calendar in either 1912 or 1929 (depending on

which authorities you believe).

Czechoslovakia (i.e. Bohemia and Moravia):

6 Jan 1584 was followed by 17 Jan 1584




31 Jan 1918 was followed by 14 Feb 1918


Then part of Sweden. (Note, however, that Finland later became part of Russia, which then

still used the Julian calendar. The Gregorian calendar remained official in Finland, but some

use of the Julian calendar was made.)


9 Dec 1582 was followed by 20 Dec 1582
Alsace: 5 Feb 1682 was followed by 16 Feb 1682
Lorraine: 16 Feb 1760 was followed by 28 Feb 1760
Strasbourg: February 1682


Different states on different dates:
Catholic states on various dates in 1583-1585
Prussia: 22 Aug 1610 was followed by 2 Sep 1610
Protestant states: 18 Feb 1700 was followed by 1 Mar 1700
(Many local variations)

Great Britain and Dominions:

2 Sep 1752 was followed by 14 Sep 1752


9 Mar 1924 was followed by 23 Mar 1924
(Some sources say 1916 and 1920)


21 Oct 1587 was followed by1 Nov 1587


See Great Britain


4 Oct 1582 was followed by 15 Oct 1582


The Gregorian calendar was introduced to supplement the traditional Japanese calendar on

1 Jan 1873.


During German occupation 1915 to 1918




14 Dec 1582 was followed by 25 Dec 1582





(In the eastern parts of the country the change may not have occurred until Netherlands (including Belgium):

Zeeland, Brabrant, and the "Staten Generaal":
14 Dec 1582 was followed by 25 Dec 1582
1 Jan 1583 was followed by 12 Jan 1583
Limburg and the southern provinces (currently Belgium):
20 Dec 1582 was followed by 31 Dec 1582 or
21 Dec 1582 was followed by 1 Jan 1583
10 Feb 1583 was followed by 21 Feb 1583
Went back to Julian in the summer of 15941920)

31 Dec 1700 was followed by 12 Jan 1701
30 Jun 1700 was followed by 12 Jul 1700
Utrecht and Overijssel:
30 Nov 1700 was followed by 12 Dec 1700
31 Dec 1700 was followed by 12 Jan 1701
30 Apr 1701 was followed by 12 May 1701


Then part of Denmark.




4 Oct 1582 was followed by 15 Oct 1582


4 Oct 1582 was followed by 15 Oct 1582


31 Mar 1919 was followed by 14 Apr 1919
(The Greek Orthodox parts of the country may have changed later)


31 Jan 1918 was followed by 14 Feb 1918


Much confusion exists regarding Scotland's change. Different authorities disagree about whether Scotland changed together with the rest of Great Britain, or if they had changed



4 Oct 1582 was followed by 15 Oct 1582

Sweden (including Finland):

17 Feb 1753 was followed by 1 Mar 1753 (see note below)


Catholic cantons: 1583, 1584 or 1597



Protestant cantons:
31 Dec 1700 was followed by 12 Jan 1701
(Many local variations)


Gregorian calendar introduced 1 Jan 1927


Different areas changed at different times.
Along the Eastern seaboard: With Great Britain in 1752.
Mississippi valley: With France in 1582.
Texas, Florida, California, Nevada, Arizona, New Mexico:
With Spain in 1582
Washington, Oregon: With Britain in 1752.
Alaska: October 1867 when Alaska became part of the USA.


See Great Britain






Sweden has a curious history. Sweden decided to make a gradual change from the Julian to the Gregorian calendar. By dropping every leap year from 1700 through 1740 the eleven superfluous days would be omitted and from 1 Mar 1740 they would be in sync with the Gregorian calendar. (But in the meantime they would be in sync with nobody!)


So 1700 (which should have been a leap year in the Julian calendar) was not a leap year in Sweden. However, by mistake 1704 and 1708 became leap years. This left Sweden out of synchronization with both the Julian and the Gregorian world, so they decided to go back to the Julian calendar. In order to do this, they inserted an extra day in 1712, making that year a double leap year! So in 1712, February had 30 days in Sweden. Later, in 1753, Sweden changed to the Gregorian calendar by dropping 11 days like everyone else.


The Christian Calendar: The "Christian calendar" is the term traditionally used to designate the calendar commonly in use, although its connection with Christianity is highly debatable. This calendar is used by the United States, and most countries in the world. This section presents historical information about the Christian calendar. For more current information about how our calendar works today, see the section on Our Year.




The Christian calendar has years of 365 or 366 days. It is divided into 12 months that

have no relationship to the motion of the moon. In parallel with this system, the concept

of weeks groups the days in sets of 7. Two main versions of the Christian calendar have

existed in recent times: The Julian calendar and the Gregorian calendar. The difference

between them lies in the way they approximate the length of the tropical year and their rules

for calculating Easter.


What is the Julian calendar?
What years are leap years?
What consequences did the use of the Julian calendar have?
What day of the week was 2 August 1953?
What is the Roman calendar?
How did the Romans number days?
What is the proleptic calendar?
What is Easter?
What is the Indiction?
What is the Julian Period?
Is there a formula for calculating the Julian day number?
What is the modified Julian day number?
What is the Lilian day number?


What is the Julian calendar?

The Julian calendar was introduced by Julius Caesar in 45 B.C.E. Author David Duncan

says the Julian calendar was born of Caesar's tryst with Cleopatra. Before the Julian

calendar was introduced, priests in the Roman Empire exploited the calendar for political

ends, inserting days and even months into the calendar to keep the politicians they favored

in office. Tired of the chaos that this undependable system eventually gave rise to, Julius

Caesar finally set out to put the long-abused calendar back on track.


It was in common use until the 1500s, when countries started changing to the Gregorian

calendar (section the modern year). However, some countries (for example, Greece and

Russia) used it into the 1900s, and the Orthodox church in Russia still uses it, as do some

other Orthodox churches. In the Julian calendar, the tropical year is approximated as 365

1/4 days = 365.25 days. This gives an error of 1 day in approximately 128 years. The

approximation 365 1/4 is achieved by having 1 leap year every 4 years.


What years are leap years?

The Julian calendar has 1 leap year every 4 years. Every year divisible by 4 is a leap year.

However, the 4-year rule was not followed in the first years after the introduction of the

Julian calendar in 45 B.C.E. Due to a counting error, every 3rd year was a leap year in the

first years of this calendar's existence. The leap years were:

45 B.C.E., 42 B.C.E., 39 B.C.E., 36 B.C.E., 33 B.C.E., 30 B.C.E., 27 B.C.E., 24 B.C.E.,

21 B.C.E., 18 B.C.E., 15 B.C.E., 12 B.C.E., 9 B.C.E., C.E. 8, C.E. 12, and every 4th year

from then on.









Authorities disagree about whether 45 B.C.E. was a leap year or not. There were no leap years between 9 B.C.E. and C.E. 8 (or, according to some authorities, between 12 B.C.E. and C.E. 4). This period without leap years was decreed by emperor Augustus in order to make up for the surplus of leap years introduced previously, and it earned him a place in the calendar as the 8th month was named after him.

It is a curious fact that although the method of reckoning years after the (official) birth year of Christ was not introduced until the 6th century, by some stroke of luck the Julian leap years coincide with years of our Lord that are divisible by 4.


What consequences did the use of the Julian calendar have?

The Julian calendar introduces an error of 1 day every 128 years. So every 128 years the tropical year shifts one day backwards with respect to the calendar. Furthermore, the method for calculating the dates for Easter was inaccurate and needed to be refined. In order to remedy this, two steps were necessary: 1) The Julian calendar had to be replaced by something more adequate. 2) The extra days that the Julian calendar had inserted had to be dropped.







NIST Exhibits


The solution to problem 1 was the Gregorian calendar described in the section about the modern year. The solution to problem 2 depended on the fact that it was felt that 21 March was the proper day for vernal equinox (because 21 March was the date for vernal equinox during the Council of Nicaea in C.E. 325). The Gregorian calendar was therefore calibrated to make that day vernal equinox. By


1582 vernal equinox had moved (1582-325)/128 days = approximately 10 days backwards. So 10 days had to be dropped.


What is the Roman calendar?

Before Julius Caesar introduced the Julian calendar in 45 B.C.E., the Roman calendar was a mess, and much of our so-called "knowledge" about it seems to be little more than guesswork. Originally, the year started on 1 March and consisted of only 304 days or 10 months (Martius, Aprilis, Maius, Junius, Quintilis, Sextilis, September, October, November, and December). These 304 days were followed by an unnamed and unnumbered winter period. The Roman king Numa Pompilius (c. 715-673 B.C.E., although his historicity is disputed) allegedly introduced February and January (in that order) between December and March, increasing the length of the year to 354 or 355 days. In 450 B.C.E., February was moved to its current position between January and March.


In order to make up for the lack of days in a year, an extra month, Intercalaris or Mercedonius, (allegedly with 22 or 23 days though some authorities dispute this) was introduced in some years. In an 8 year period the length of the years were:


1: 12 months or 355 days
2: 13 months or 377 days
3: 12 months or 355 days
4: 13 months or 378 days
5: 12 months or 355 days
6: 13 months or 377 days
7: 12 months or 355 days
8: 13 months or 378 days


A total of 2930 days corresponding to a year of 366 1/4 days. This year was discovered to be too long, and therefore 7 days were later dropped from the 8th year, yielding 365.375 days per year.


This is all theory. In practice it was the duty of the priesthood to keep track of the calendars, but they failed miserably, partly due to ignorance, partly because they were bribed to make certain years long and other years short. Furthermore, leap years were considered unlucky and were therefore avoided in time of crisis, such as the Second Punic War.


In order to clean up this mess, Julius Caesar made his famous calendar reform in 45 B.C.E. We can make an educated guess about the length of the months in the years 47 and 46 B.C.E.:


47 B.C.E.

46 B.C.E.








































Duo December










The length of the months from 45 B.C.E. onward were the same as the ones we know today.


Occasionally one reads the following story: "Julius Caesar made all odd numbered months 31 days long, and all even numbered months 30 days long (with February having 29 days in non-leap years). In 44 B.C.E. Quintilis was renamed 'Julius' (July) in honor of Julius Caesar, and in 8 B.C.E. Sextilis became 'Augustus' in honor of emperor Augustus. When Augustus had a month named after him, he wanted his month to be a full 31 days long, so he removed a day from February and shifted the length of the other months so that August would have 31 days." This story, however, has no basis in actual fact. It is a fabrication possibly dating back to the 14th century.


How did the Romans number days?

The Romans did not number the days sequentially from 1. Instead they had three fixed points in each month:

"Calendar" (or "Calendar") which was the first day of the month.

"Ides" which was the 13th day of January, February, April, June, August, September, November, and December, or the 15th day of March, May, July, or October.

"Nonie" which was the 9th day before Ides (counting Ides itself as the 1st day).


The days between Calendar and Nonie were called "the 5th day before Nonie," "the 4th day before Nonie," "the 3rd day before Nonie," and "the day before Nonie." (There was no "2nd day before Nonie." This was because of the inclusive way of counting used by the Romans: To them, Nonie itself was the first day, and thus "the 2nd day before" and "the day before" would mean the same thing.)


Similarly, the days between Nonie and Ides were called "the TX day before Ides," and the days after Ides were called "the TX day before Calendar (of the next month)."


Julius Caesar decreed that in leap years the "6th day before Calendar of March" should be doubled. So in contrast to our present system, in which we introduce an extra date (29 February), the Romans had the same date twice in leap years. The doubling of the 6th day before Calendar of March is the origin of the word bissextile. If we create a list of equivalence between the Roman days and our current days of February in a leap year, we get the following:

7th day before Calendar of March



6th day before Calendar of March



6th day before Calendar of March



5th day before Calendar of March



4th day before Calendar of March



3rd day before Calendar of March



The day before Calendar of March



Calendar of March




You can see that the extra 6th day (going backwards) falls on what is today 24 February. For this reason 24 February is still today considered the "extra day" in leap years. However, at certain times in history, the second 6th day (25 Feb) has been considered the leap day.


Why did Caesar choose to double the 6th day before Calendar of March? It appears that the leap month Intercalaris/Mercedonius of the pre-reform calendar was not placed after February, but inside it, namely between the 7th and 6th day before Kalendae of March. It was therefore natural to have the leap day in the same position.


What is the proleptic calendar?

The Julian calendar was introduced in 45 BC, but when historians date events prior to that year, they normally extend the Julian calendar backward in time. This extended calendar is known as the "Julian Proleptic Calendar".

Similarly, it is possible to extend the Gregorian calendar backward in time before 1582. However, this "Gregorian Proleptic Calendar" is rarely used. If someone refers to, for example, 15 March 429 BC, they are probably using the Julian proleptic calendar.

In the Julian proleptic calendar, year X BC is a leap year, if X-1 is divisible by 4. This is the natural extension of the Julian leap year rules.


What is Easter?

In the Christian world, Easter (and the days immediately preceding it) is the celebration of the death and resurrection of Jesus in (approximately) C.E. 30. See additional information on Easter.


What is the Indiction?

The Indiction was used in the middle ages to specify the position of a year in a 15-year taxation cycle. It was introduced by Emperor Constantine the Great on 1 September 312 and ceased to be used in 1806.


The Indiction may be calculated thus:

Indiction = (year + 2) mod 15 + 1

The Indiction has no astronomical significance.

The Indiction did not always follow the calendar year. Three different Indictions may be identified:

The Pontifical or Roman Indiction, which started on New Year's Day (being either 25 December, 1 January, or 25 March).
The Greek or Constantinopolitan Indiction, which started on 1 September.
The Imperial Indiction or Indiction of Constantine, which started on 24 September.


What is the Julian Period?

The Julian period (and the Julian day number) must not be confused with the Julian calendar. The French scholar Joseph Justus Scaliger (1540-1609) was interested in assigning a positive number to every year without having to worry about B.C.E. / C.E. He invented what is today known as the Julian Period. The Julian Period probably takes its name from the Julian calendar, although it has been claimed that it is named after Scaliger's father, the Italian scholar Julius Caesar Scaliger (1484-1558).


Scaliger's Julian period starts on 1 January 4713 B.C.E. (Julian calendar) and lasts for 7980 years. C.E. 2000 is thus year 6713 in the Julian period. After 7980 years the number starts from 1 again.


Why 4713 B.C.E. and why 7980 years? Well, in 4713 B.C.E. the Indiction (see above), the Golden Number (see section on Easter) and the Solar Number (see above) were all 1. The next times this happens is 15 x 19 x 28 = 7980 years later, in C.E. 3268.


Astronomers have used the Julian period to assign a unique number to every day since 1 January 4713 B.C.E. This is the so-called Julian Day (JD). JD 0 designates the 24 hours from noon UTC on 1 January 4713 B.C.E. to noon UTC on 2 January 4713 B.C.E.


This means that at noon UTC on 1 January C.E. 2000, JD 2,451,545 will start.

This can be calculated thus: From 4713 B.C.E. to C.E. 2000 there are 6712 years. In the Julian calendar, years have 365.25 days, so 6712 years correspond to 6712 x 365.25=2,451,558 days. Subtract from this the 13 days that the Gregorian calendar is ahead of the Julian calendar, and you get 2,451,545. Often fractions of Julian day numbers are used, so that 1 January C.E. 2000 at 15:00 UTC is referred to as JD 2,451,545.125.


Note that some people use the term "Julian day number" to refer to any numbering of days. NASA, for example, uses the term to denote the number of days since 1 January of the current year.


Is there a formula for calculating the Julian day number?

Try this one (the divisions are integer divisions, in which remainders are discarded):

a = (14-month)/12
y = year+4800-a
m = month + 12*a - 3

For a date in the Gregorian calendar:
JD = day + (153*m+2)/5 + y*365 + y/4 - y/100 + y/400 - 32045

For a date in the Julian calendar:
JD = day + (153*m+2)/5 + y*365 + y/4 - 32083

JD is the Julian day number that starts at noon UTC on the specified date.


The algorithm works fine for AD dates. If you want to use it for BC dates, you must first convert the BC year to a negative year (e.g., 10 BC = -9). The algorithm works correctly for all dates after 4800 BC, i.e. at least for all positive Julian day numbers.

To convert the other way (i.e., to convert a Julian day number, JD, to a day, month, and year) these formulas can be used (again, the divisions are integer divisions):


For the Gregorian calendar:
a = JD + 32044
b = (4*a+3)/146097
c = a - (b*146097)/4

For the Julian calendar:
b = 0
c = JD + 32082

Then, for both calendars:
d = (4*c+3)/1461
e = c - (1461*d)/4
m = (5*e+2)/153

day = e - (153*m+2)/5 + 1
month = m + 3 - 12*(m/10)
year = b*100 + d - 4800 + m/10



What is the modified Julian day number?

Sometimes a modified Julian day number (MJD) is used which is 2,400,000.5 less than the Julian day number. This brings the numbers into a more manageable numeric range and makes the day numbers change at midnight UTC rather than noon. MJD 0 thus started on 17 Nov 1858 (Gregorian) at 00:00:00 UTC.


What is the Lilian day number?

The Lilian day number is similar to the Julian day number, except that Lilian day number 1 started at midnight on the first day of the Gregorian calendar, that is, 15 October 1582. The Lilian day number is named after Aloysius Lilius.












The Jewish calendar

The current definition of the Jewish calendar is generally said to have been set down by

the Sanhedrin president Hillel II in approximately C.E. 359. The original details of his calendar are, however, uncertain. The Jewish calendar is used for religious purposes by Jews all over the world, and it is the official calendar of Israel. The Jewish calendar is a combined solar/lunar calendar, in that it strives to have its years coincide with the tropical year and its months coincide with the synodic months. This is a complicated goal, and the rules for the Jewish calendar are correspondingly fascinating.


Lunisolar calendars use months to approximate the tropical year. Examples are the Jewish and Chinese calendars. Since 12 months are about 11 days shorter than the tropical year, a leap month (also called intercalary month) is inserted about every third year to keep the calendar in tune with the seasons. The big question is how to do this. A simple method is to just base it on nature. In ancient Israel, the religious leaders would determine the date for Passover each spring by seeing if the roads were dry enough for the pilgrims and if the lambs were ready for slaughter. If not, they would add one more month. An aboriginal tribe in Taiwan would go out to sea with lanterns near the new moon at the beginning of spring. If the migrating flying fish appeared, there would be fish for New Year's reunion dinner. If not, they would try their luck next month.


What does a Jewish year look like?

An ordinary (non-leap) year has 353, 354, or 355 days. A leap year has 383, 384, or 385 days. The three lengths of the years are termed, "deficient," "regular," and "complete," respectively. An ordinary year has 12 months, a leap year has 13 months. Every month starts (approximately) on the day of a new moon. The months and their lengths are:





Length in a
 deficient year  

Length in a
regular year

Length in a
 complete year 





















Adar I




Adar II





























353 or 383

354 or 384

355 or 385



The month Adar I is only present in leap years. In non-leap years Adar II is simply called "Adar."


Note that in a regular year the numbers 30 and 29 alternate. A complete year is created by adding a day to Heshvan, whereas a deficient year is created by removing a day from Kislev. The alteration of 30 and 29 ensures that when the year starts with a new moon, so does each month.


What years are leap years?

A year is a leap year if the number year mod 19 is one of the following: 0, 3, 6, 8, 11, 14, or 17. The value for year in this formula is the 'Anno Mundi' described below.


What years are deficient, regular, and complete?

That is the wrong question to ask. The correct question to ask is: When does a Jewish year begin? Once you have answered that question (see below) the length of the year is the number of days between 1 Tishri in one year and 1 Tishri in the following year.




When is New Year's Day?

That depends. Jews have 4 different days to choose from:

1 Tishri:

Rosh HaShanah. This day is a celebration of the creation of the

world and marks the start of a new calendar year. This will be

the day we shall base our calculations on in the following sections.

15 Shevat:

Tu B'shevat. The new year for trees, when fruit tithes should

be brought.



1 Nisan:



New Year for Kings. Nisan is considered the first

month, although it occurs 6 or 7 months after the start

of the calendar year.

1 Elul:

New Year for Animal Tithes (Taxes).


Only the first two dates are celebrated today.


When does a Jewish day begin?

A Jewish-calendar day does not begin at midnight, but at either sunset or when three medium-sized stars should be visible, depending on the religious circumstance.

Sunset marks the start of the 12 night hours, whereas sunrise marks the start of the 12-day hours. This means that night hours may be longer or shorter than day hours, depending on the season.


When does a Jewish year begin?

The first day of the calendar year, Rosh HaShanah, on 1 Tishri is determined as follows:


The new year starts on the day of the new moon that occurs about 354 days (or 384 days if the previous year was a leap year) after 1 Tishri of the previous year.

                  If the new moon occurs after noon on that day, delay the new year by one day. (Because in that case the new crescent moon will not be visible until the next day.)

                  If this would cause the new year to start on a Sunday, Wednesday, or Friday, delay it by one day. (because we want to avoid that Yom Kippur (10 Tishri) falls on a Friday or Sunday, and that Hoshanah Rabba (21 Tishri) falls on a Sabbath (Saturday)).

                  If two consecutive years start 356 days apart (an illegal year length), delay the start of the first year by two days.

                  If two consecutive years start 382 days apart (an illegal year length), delay the start of the second year by one day.

Note: Rule 4 can only come into play if the first year was supposed to start on a Tuesday. Therefore a two-day delay is used rather that a one-day delay, as the year must not start on a Wednesday as stated in rule 3.


When is the new moon?

A calculated new moon is used. In order to understand the calculations, one must know that an hour is subdivided into 1080 'parts'. The calculations are as follows: The new moon that started the year AM 1, occurred 5 hours and 204 parts after sunset (i.e. just before midnight on Julian date 6 October 3761 B.C.E.). The new moon of any particular year is calculated by extrapolating from this time, using a synodic month of 29 days 12 hours and 793 parts.



Note that 18:00 Jerusalem time (15:39 UTC) is used instead of sunset in all these cases.


How does one count years?

Years are counted since the creation of the world, which is assumed to have taken place in 3761 B.C.E. In that year, AM 1 started (AM = Anno Mundi = year of the world). In the year C.E. 1998 we have witnessed the start of Jewish year AM 5759.


The Islamic Calendar: The Islamic calendar (or Hijri calendar) is a purely lunar calendar. It contains 12 months that are based on the motion of the moon, and because 12 synodic months is only 12 x 29.53=354.36 days, the Islamic calendar is consistently shorter than a tropical year, and therefore it shifts with respect to the Christian calendar.


The calendar is based on the Qur'an (Sura IX, 36-37) and its proper observance is a sacred duty for Muslims. The Islamic calendar is the official calendar in countries around the Gulf, especially Saudi Arabia. But other Muslim countries use the Gregorian calendar for civil purposes and only turn to the Islamic calendar for religious purposes. So you can't print an Islamic calendar in advance? How does one count years?


Due to different transliterations of the Arabic alphabet, other spellings of the months are possible. Each month starts when the lunar crescent is first seen (by a human observer's eye) after a new moon. Although new moons may be calculated quite precisely, the actual visibility of the crescent is much more difficult to predict. It depends on factors such as weather, the optical properties of the atmosphere, and the location of the observer. It is therefore very difficult to give accurate information in advance about when a new month will start. Furthermore, some Muslims depend on a local sighting of the moon, whereas others depend on a sighting by authorities somewhere in the Muslim world. Both are valid Islamic practices, but they may lead to different starting days for the months.


So you can't print an Islamic calendar in advance?

Not a reliable one. However, calendars are printed for planning purposes, but such calendars are based on estimates of the visibility of the lunar crescent, and the actual month may start a day earlier or later than predicted in the printed calendar.


Different methods for estimating the calendars are used.


Some sources mention a crude system in which all odd numbered months have 30 days and all even numbered months have 29 days with an extra day added to the last month in 'leap years' (a concept otherwise unknown in the calendar). Leap years could then be years in which the number year mod 30 is one of the following: 2, 5, 7, 10, 13, 16, 18, 21, 24, 26, or 29. (This is the algorithm used in the calendar program of the Gnu Emacs editor.)


Such a calendar would give an average month length of 29.53056 days, which is quite close to the synodic month of 29.53059 days, so on the average it would be quite accurate, but in any given month it is still just a rough estimate. Better algorithms for estimating the visibility of the new moon have been devised.


How does one count years?

Years are counted since the Hijra, that is, Mohammed's emigration to Medina in AD 622. On 16 July (Julian calendar) of that year, AH 1 started (AH = Anno Hegirae = year of the Hijra). In the year AD 2003 we have witnessed the start of Islamic year AH 1424. Note that although only 2003-622=1381 years have passed in the Christian calendar, 1423 years have passed in the Islamic calendar, because its year is consistently shorter (by about 11 days) than the tropical year used by the Christian calendar.


When will the Islamic calendar overtake the Gregorian calendar?

As the year in the Islamic calendar is about 11 days shorter than the year in the Christian calendar, the Islamic years are slowly gaining in on the Christian years. But it will be many years before the two coincide. The 1st day of the 5th month of C.E. 20874 in the Gregorian calendar will also be (approximately) the 1st day of the 5th month of AH 20874 of the Islamic calendar.


Saudi Arabia doesn't rely on a visual sighting of the crescent moon to fix the start of a new month. Instead they base their calendar on a calculated astronomical moon.

Since 1999 (1420 AH) the rule has been as follows: On the 29th day of an Islamic month, the times when the sun and the moonset are compared. If the sunsets before the moon, the next day will be the first of a new month; but if the moon sets before the sun, the next day will be the last (30th) of the current month. The times for the setting of the sun and the moon are calculated for the coordinates of Mecca.