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Understanding > Fundamental concepts > Calendars I



General principles

The identification of moments requires the definition of time scales, but regardless of the choice of a scale, it is necessary to define a system for numbering days (it is easy to see a change of day, it is more difficult to see the change of year, so the day is the natural unit). This numbering system is designed to locate any event without ambiguity: it is the problem of chronology.

The simplest solution, in principle, is to choose arbitrarily one day origin and without any artifice to list successive days. Earlier periods may possibly be identified by listing to the past days but it is easier to introduce a zero day and negative numbers for days before the origin. It is this principle that defined the Julian period described below.

For reasons of tradition, but also convenience, we have been led to complicate the system by cutting in years and months. The rules adopted , which are the various calendars, differ from one civilization to another.

Years :

For historians , since the eighteenth century, the year preceding the year 1 of the Christian era is written " 1 BC " and is a leap year. Leap years succeeding every 4 years, so these are the years 1, 5, 9, ... BC. The rule for divisibility by 4 may no longer be applied.

Astronomers from J. Cassini (1740 ) use an algebraic notation. They call year 0 the year 1 BC and have noted negatively previous years as follows:
... ... ...
2 AD (2 after J.-C.) = year 2
1 AD (1 after J.-C.) = year 1
1 BC (1 before J.-C.) = year 0 (leap year)
2 BC (2 before J.-C.) = year -1
3 BC (3 before J.-C.) = year -2
4 BC (4 before J.-C.) = year -3
5 BC (5 before J.-C.) = year -4 (leap year)
6 BC (6 before J.-C.) = year -5
... ... ...

This is the notation of astronomers that is used in the calculations of ephemerides. Compared to the notation of historians it has two advantages:
1 . The rule for divisibility by 4 of leap years is preserved: years 0 , -4 , -8, ... are leap years .
2 . Calculating the number of years between a negative year a positive year is done algebraically . So, the number of years between January 1, -45 and January 1, 1991 is equal to : 1991 - ( -45) = 2036.

The Julian calendar is used by historians to ancient times and, in particular, to those prior to its creation ( -45) .
In this case, they use a dummy calendar with the same rules of construction.

How do we define calendars?

Calendars are based on considerations of astronomical nature.
The month is the approximate length of a lunation (time interval between two new moons).
The year is an approximation of the tropical year, time interval between two consecutive passages of the Sun at the spring equinox.
A calendar is of lunar or solar type depending on whether it favors the approximation of the month or of the year . In a lunar calendar the average duration of the month should be a good approximation of the lunar cycle (29.530589 days), whereas in a solar calendar the average length of the year should be a good approximation of the tropical year (365 242,190 days).
Thus, the Gregorian calendar, which is our usual calendar is solar. It is the same for Julian and Coptic calendars.
Contrarily, the Muslim calendar is lunar and the Jewish calendar is lunisolar .

Each calendar defines an origin from which we will count and number the years. This corresponds to an era.


Regardless of the organization of days, months and years, each calendar needs an origin to count the years in general (it may be days in the case of the Julian period) or groups of several years as the Olympics Greek which have 4 years. Such an origin defines an era.

The Christian era began on December 25 of the year 753 of the foundation of Rome (according to Varron) but the beginning of the year is January 1 to match with the former calendar. Julian and Gregorian calendars are currently in the Christian era. The Julian calendar was in effect from January 1 of the year 45 BC (year 709 of the era of the foundation of Rome, a leap year) to 1582, year of the Christian era and the Gregorian calendar is used since 1582, year of the Christian era. The use of the Christian era was introduced in 532 on a proposal by the monk Dionysius Exiguus. He proposed also, in 525, to the Pope the term "Anno Domini" (AD). This custom spread rapidly in the liturgical texts and documents, but it will take several centuries (the reign of Pepin and of Charlemagne) for a common use in the society.

The era of Diocletian began on August 29, 284 (Julian). This is the origin of the Coptic calendar still in use in the Coptic community in Egypt.

Hegira began on Friday, July 16, 622 (Julian), which is the origin of the Muslim calendar.

The Jewish era began on October 7, -3760 (Julian), which is the origin of the Jewish calendar.

The era of the French began on September 22, 1792 and ended January 1, 1806 and was used by the Republican calendar.

The era of the Julian period begins on January 1, 4713 BC at noon (Julian calendar).

The Hellenistic era has a succession of Olympiads corresponding to four years. It began with the summer solstice in the year 776 BC. In the year 1 of the Christian era, the 195th Greek Olympiad began in summer.

Other calendars refer to more uncertain origins of their eras. This is the case of calendars:

  • Hindu , Saka era beginning March 3, 78 AD but codified in 1957 and used in the Indian calendar (see this calendar) ;
  • Hindu, Samvat era beginning February 23, 57 AD ;
  • Tamil, Nirayana cycle of 60 years undifferentiated since the year 397 AD ;
  • Cambodian and Laotian, Small Burmese era starting from 21 March 638 AD ;
  • Cambodian and Laotian, Buddhist era beginning in April- May in the year 544 BC (death of Buddha from the Sinhalese tradition) ;
  • Chinese and Vietnamese, cycles of 60 years identified to the reigns of the emperors.




The Julian calendar is, globally, consistent with the reformed Roman calendar introduced by Julius Caesar. In modern usage, it is used with the Christian era, the year 1 being the 47th year of this julian reform.

This calendar is solar-type. It has two kinds of years, common years of 365 days divided into 12 months of 31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30 and 31 days, and leap years of 366 days in which the second month is 29 days. Leap years are those whose number is divisible by 4: one year in 4 years is a leap year.

The average length of the Julian year (365.25 days) is a poor approximation of the tropical year. As a result, season dates are shifted about 3 days every 400 years, one month every 4000 years.

The Julian calendar was in use in most nations of Europe until the sixteenth century.

It was then replaced by the Gregorian calendar, but it is still used today to determine the orthodox religious holidays.



This is our usual calendar.

The Gregorian calendar is solar-type. It was created in 1582 by Pope Gregory XIII to correct the delay taken by the Julian calendar on the Sun, delay being of 10 days at the time of this reform.

This calendar is defined with respect to the Julian calendar in the following way : the day after Thursday, 4 October 1582 (Julian) was Friday, October 15, 1582 (Gregorian), the succession of days of the week being respected.

The Gregorian calendar differs from the Julian calendar as the division between common years (365 days) and leap years (366 days). Leap years are the same as those of the Julian calendar (calendar year whose is divisible by 4) but three years out of four centuries, those with number being a multiple of 100 without being 400, are not leap years. Thus, the years 1700, 1800, 1900 are common while 2000 is a leap year.

The average length of the year is 365 , 2,425 days . It is very close to that of the tropical year .

The Gregorian calendar was adopted in 1582 in Italy, Spain, Portugal and the Catholics Netherlands. In France the reform was implemented in December 1582, succeeding Monday, December 20 to Sunday, December 9. In Great Britain, it is only in 1752 that succeeded September 14 to September 2 and the Gregorian calendar was adopted .

Gradually adopted until the early twentieth century by all countries, this calendar is now in use worldwide.



This calendar is solar-type . The year consists of twelve months of 30 days, followed by three consecutive years with 5 epagomenal days called complementary and the 4th year having 6 epagomenal days. The average length of the year (365.25 days) is the same as in the Julian calendar.
This calendar defines the era of Diocletian, whose origin (Tout 1, Year 1) corresponds to 29 August 284 Julian. Years with 366 days correspond to those with number plus 1 is a multiple of 4. The Coptic year begins on 29 or 30 August, Julian .
This calendar is still used today in Egypt.

Months of the Coptic year

1  Tout
2  Babah
3  Hatour
4  Keihak
5  Toubah
6  Amchir
7  Barmahat
8  Barmoudah
9  Bachnas
10 Bou'nah
11 Abib
12 Masari



The Muslim calendar was adopted in its present form, near 632 AD. It defines the Muslim era whose origin, the first day of Year 1 (Hegira), is the Friday, July 16, 622 Julian .
This is a calendar of lunar type. Years have 12 months. The lunar cycle of Muslims is 30 years.
It consists of 19 common years of 354 days and 11 abundant years of 355 days. From one year to another the beginning of the Muslim year is thus shifted from 10 to 12 days compared to seasons (an average of 10.875523... days per year).
Within a cycle, abundant years are numbered years 2, 5, 7, 10, 13, 16, 18, 21, 24, 26, 29.
The current cycle began on 1 Muharram of the year 1411 of the Muslim era which corresponds to Tuesday, July 24, 1990.
Months are a period of 30 and 29 days alternately, the first month of the year is 30 days and the last 29 days (common year) or 30 days (abundant year). The average duration of the month (29.530556 days) is similar to that of the lunar month.

Month names and number of days

1 Muharram         30 days
2 Safar            29 days
3 Rabi'-ul-Aououal 30 days
4 Rabi'-ut-Tani   29 days
5 Djoumada-l-Oula  30 days
6 Djoumada-t-Tania 29 days
7 Rajab            30 days
8 Sha'ban          29 days
9 Ramadan          30 days
10 Chaououal       29 days 
11 Dou-l-Qa'da     30 days
12 Dou-l-Hijjah  29 or 30 days

The Muslim calendar presented here is theoretical.
There are other variants using a different chronomogy. The date of origin is sometimes fixed to 15 July 622; according to some authors the 15th year of the lunar cycle is abundant and the 16th common.
It also happens that for the determination of religious holidays, the beginning is defined by the observation of the first crescent of the Moon after the New Moon. Then, calendars are local and depend on the conditions of observation.



The Jewish calendar, in its present form, is the one of the fourth century AD. It is lunisolar type. It provides a mean duration of months (29.530594 days) very close to that of using the lunar month months lasting 29 or 30 days. It also provides an average of the year (365.2468 days) similar to that of the tropical year by alternating 12 common years of 12 months and 7 leap years of 13 months within a cycle of 19 years.
In each cycle the embolismic years are those numbered 3, 6, 8, 11, 14, 17 and 19. The current cycle began on 1 Tishrei of the year 5739 which corresponds to Monday, 2 October 1978.
Common years can last years 353, 354 or 355 days and leap years 383, 384 or 385 days. The three species of years thus defined are called, respectively , defective, regular or abundant.

The date of the origin of the Jewish calendar is 1 Tishrei of the year 1. It corresponds to 7 October -3760 Julian.

Month names and number of days

1  Tisheri      30 days
2  Hesvan       29 ou 30 days
3  Kislev       29 ou 30 days
4  Tebeth       29 days
5  Shevat       30 days
6  Adar         29 ou 30 days
7  Nissan       30 days
8  Iyar         29 days
9  Sivan        30 days
10 Tamouz       29 days
11 Ab           30 days
12 Elul         29 days

In the embolistic years, the month of 29 days, Veadar, is interposed between the months of Adar and Nissan.

Click here for the correspondence between the dates of different calendars.


The era of the French republican calendar and have been adopted by the National Convention at the meeting of October 5, 1793: "The era of the French ... [ begins ] on September 22, 1792 of the vulgar era, the day the Sun came to the true equinox of autumn ... at 9 hours 18 minutes 30 seconds in the morning at the observatory of Paris".

The definition of republican calendar adopted is as follows (Decree of 5 October 1793) :
"The beginning of each year is set at midnight, beginning on the day of the true autumnal equinox at the Paris Observatory."
"The first year of the French Republic began at midnight September 22, 1792, and ended at midnight between the 21 and the 22 of September, 1793."
"The year is divided into twelve equal months of thirty days each, followed by five days to complete the regular year and which do not belong to any month, they are called the additional days."
"Each month is divided into three equal parts each of ten days, which are called decades..."
"In memory of the revolution, which, after four years, led France to the Republican government, the leap period over 4 years is called "Franciade".
"The leap day must end this period: it is called the day of the Revolution. This day is placed after the additional 5 days..."

Vendémiaire, Brumaire, Frimaire
Nivôse, Pluviôse, Ventôse
Germinal, Floreal, Prairial
Messidor, Thermidor, Fructidor.

The days of the decade are the following names: Primidi , duodi , tridi , quartidi , quintidi , sextidi , septidi , octodi , nonidi and decadi.
The 5 additional days are called days sanculottides.
Leap years are called "sextiles" years.
Unfortunately, the definition of this calendar is a contradiction: the year begins on the day of the autumnal equinox, and a sextile year is intercalated every four years.
This error will never be corrected and will be one of the arguments used to return to the Gregorian calendar on January 1, 1806 (11 nivôse, year 14).
This inconsistency does not pose a problem on the period of use of the Republican calendar, the first contradiction would have appeared in the year 19.

Click here to correspondence between the dates of the Republican calendar and the Gregorian dates.

It useful to consult the book " The republican calendar " published by the Bureau des Longitudes.

Credit : Bureau des longitudes/IMCCE
Above and right, the month "Pluviôse" of the republican calendar. Each day correspond to a plant, an animal or a usual tool.


The national calendar currently in use in India is that defined by the Calendar Reform Committee. It is officially applied since 22 March 1957 (1 Chaitra 1879 Saka Era). The numbering of years is in the Saka Era.
Month of the Indian calendar have 30 or 31 days. They are given below:

   Month of the                       Gregorian date
  Indian calendar                 of the first of the month

Chaitra    (30 ou 31(*) days)          22 or 21(*) March
Vaisakha   (31 days)                   21 April
Jyaistha   (31 days)                   22 May
Asadha     (31 days)                   22 June
Sravana    (31 days)                   23 July
Bhadra     (31 days)                   23 August
Asvina     (30 days)                   23 September
Kartika    (30 days)                   23 October
Agrahayana (30 days)                   22 November
Pausa      (30 days)                   22 December
Magha      (30 days)                   21 January
Phalguna   (30 days)                   20 February

(*) : for leap years

The leap year rule is the same as the Gregorian calendar.


A perpetual calendar allows to determine the day of the week corresponding to a given date. Click here to access interactive software and click here to access the manual calculation method from a table.



The Julian period is a time scale numbering without interruption days since Monday, January 1 -4712 at noon.
This julian period was introduced by astronomer Scaliger in 1583. He named "julian" this period by analogy to Julian year and also in honor of his father Julius Caesar Scaliger.

Julian date : this is the period since 1 January -4712 at 12 hours. It is expressed as days and a decimal fraction of day.
Julian day : it is the integer part of the Julian date.
The Julian day 0 begins on January 1, -4712 at 12 hours , the Julian day 1 starts January 2 -4712 at 12 hours, the Julian day 2451911 begins on January 1, 2001 at 12pm.
The continuity of days in the Julian period allows to calculate an interval of time without error even if it covers multiple calendars. So, January 1, 1001 at 0h (Julian calendar) is the Julian date 2086673.5,: Jan. 1, 2001 at 0h (Gregorian calendar) is the Julian date 2451910.5. The elapsed time between these two dates is:
2451910.5-2086673.5 = 365.237 days.
Click here to calculate a correspondance between the Julian day and the Gregorian date.


The twenty-first century begins on January 1, 2001.
A century represents a period of 100 years, the millennium a period of 1000 years. The first century began on January 1, Year 1 and ends on December 31 of the year 100. The twentieth century began January 1, 1901 and ends December 31, 2000. The twenty-first century begins January 1, 2001.

The first millennium ends on December 31 of the year 1000. The second millennium begins January 1, 1001, the third millennium begins January 1, 2001.

Credit : IMCCE