# CHRONOS and CALENDARS

General principles:
The location of the moments requires the definition of scales of time, but independently of the choice of a scale, it is necessary to define a system of numbering of the days (it is easy to note a change of day, it is more difficult to note a change of year; thus the day is the natural unit). The purpose of this system of numbering is to locate a given date without any ambiguity: it is the problem of the chronology.

The simplest solution, in theory, consists in choosing an origin day arbitrarily and enumerating without other artifice the later days. The former times can be possibly located while enumerating towards the past, the days that one will say being before the origin but it is simpler to introduce one day zero and negative numbers for the days located before the origin. It is by following this principle that one defined the Julian period described hereafter.

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

Case of the years:
For the historians, since the XVIIIth century, the year which precedes year 1 of the Christian era is noted " 1 before J.-C. ": it is a leap year. The leap years following one another every 4 years, they are thus years 1, 5, 9... before J.-.C; the rule of the divisibility by 4 cannot be applied any more.

The astronomers, since J. Cassini (1740), use an algebraic notation. They call year 0 the year 1 before J.-C. and count the former years negatively. As follows:
.........
2 after J.-C. = year 2
1 after J.-C. = year 1
1 before J.-C. = year 0 (leap year)
2 before J.-C. = year -1
3 before J.-C. = year -2
4 before J.-C. = year -3
5 before J.-C. = year -4 (leap year)
6 before J.-C. = year -5
.........

It is the notation of the astronomers which is used in the calculations of the ephemerides. Compared to the notation of the historians it has two advantages:
1. The rule of divisibility by 4 of the leap years is preserved, years 0, -4, -8... are leap years.
2. The calculation of the number of years which separates positive from negative years algebraically. Thus the number of years past between January 1 -45 and January 1, 1991 is equal to: 1991 - (-45) = 2036.

The Julian calendar is used by the historians for the old dates and, in particular, for those which are former to its creation (-45).
In this case, they use a fictitious calendar having the same rules of construction.

How are the calendars defined ?
The calendars are founded on considerations of astronomical nature.
The month is the approximate duration of a lunation, interval of time between two new moons.
The year is an approximation of the tropical year, interval of time which separates two consecutive passages of the Sun on the vernal equinox.
A calendar is of type lunar or solar according to whether one privileges the approximation of the month or the year.
In a lunar calendar the average duration of the month must be a good approximation of lunation (29,530589 days), whereas in a solar calendar the average duration of the year must be a good approximation of the tropical year (365,242190 days).
Thus, the Gregorian calendar which is our usual calendar is solar. It is the same for the Julian and coptic calendars.
On the other hand, the Moslem calendar is lunar and the jewish calendar is lunisolar.

Each calendar defines its own origin from which one will count, in general, years. That corresponds to one era.

ERAS

Independently of the organization of the days in months and years, each calendar needs an origin to count, in general years(it can be days in the case of the Julian period) or the groups of several years as the Hellenic olympiades which count 4 years. Such an origin defines one ERA.

The Christian era began on December 25, year 753 of the foundation of Rome according to Varron, but the beginning of the year will be on January 1, in order to be adjusted to the calendar in use at this epoch. The Gregorian and julian calendars belong to the Christian era. The Julian calendar was in use from the year 46 before J.-C. (year 708 of the era of the foundation of Rome) to year 1582 of the Christian era and the Gregorian calendar is in use since the year 1582 of the Christian era. The use of the Christian era was introduced in 532 on a proposal of the monk Denys the Small. It is him which proposed, in 525, to the pope the expression " Anno Domini " (AD). This use will be quickly adopted in the liturgical texts and documents, but it will be necessary to wait several centuries (the reign of Pépin the Brief and that of Charlemagne) so that this use becomes current in the society.

The era of Dioclétien began on August 29, 284 (Julian). It is the origin of the coptic calendar still of use in in the coptic community of Egypt.

Hégire began on Friday 16 July, 622 (Julian) which is thus the origin of the Moslem calendar.

The judaïc 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 on January 1, 1806 and was used by the republican calendar.

The era of the Julian period begins on January 1, 4713 before J.-C. at noon (Julian calendar).

The hellenistic era counts a succession of olympiades which correspond to four years. It started with the summer solstice of year 776 before J.-C. In year 1 of the Christian era, the 195th Hellenic olympiade began in summer.

Other calendars refer to more dubious eras as for their origins. It is the case of the calendars:

• Hindu, saka era beginning on March 3, 78 after J.-C., but redefined in 1957 and now in use in the official indian calendar (cf. this calendar);
• Hindu, samvat era beginning on February 23, 57 after J.-C.;
• Tamoul, nirâyana cycle of 60 years not differentiated since year 397 after J.-C.;
• Kampuchean and Laotian, small burmese era on the basis of March 21, 638 after J.-C.;
• Kampuchean and Laotian, bouddhic era beginning in April-May from year 544 before J.-C. (death of Buddha according to the singhalese tradition);
• Chinese and Vietnamese, cycles of 60 years identified with the reigns of the emperors.

CALENDARS:

THE JULIAN CALENDAR

Definition
The Julian calendar is, in his main rules, in conformity with the Roman calendar reformed by Julius Caesar. In the modern use, one will use it with the Christian era of which year 1 was the 47th of this Julian reform.
This calendar is of solar type. It includes two kinds of year, the common year of 365 days, divided in 12 months of 31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30 and 31 days, and the leap years of 366 days in which the second month be of 29 days. The leap years are those of which the year number is divisible by 4; one year out of 4 is thus a leap year.

The average duration of the Julian year (365,25 days) is a poor approximation of that of the tropical year. It results from it that the dates of the seasons shift approximately 3 days every 400 years, i.e. one month every 4000 years.

The Julian calendar was in use in the majority of the nations of Europe until the XVIth century.

It was replaced then by the Gregorian calendar but it is still used nowadays to determine the orthodox religious festivals.

THE GREGORIAN CALENDAR

Definition
It is our usual calendar.

The Gregorian calendar is of solar type. It was created into 1582 by the pope Gregoire XIII to correct the delay which took the Julian calendar on the Sun, delay which reached 10 days at the time of this reform.
This calendar is defined in the following way compared to the Julian calendar: the next day after Thursday 4 October 1582 (Julian) was Friday 15 October 1582 (Gregorian), the succession of the days of the week being respected.
The Gregorian calendar differs from the Julian calendar only by the distribution between common years (365 days) and leap years (366 days).
The leap years are the same ones as those of the Julian calendar (years of which the year number is divisible by 4) except three last years of the century out of four, those of which the year number is a multiple of 100 without being it of 400. Thus, the years 1700, 1800, 1900 are common whereas year 2000 is a leap year.
The average duration of the year is 365, 2425 days. It is very close to that of the tropical year.

The Gregorian calendar was adopted as of 1582 in Italy, in Spain, in Portugal and in the catholic Netherlands. In France the reform was applied in December 1582, Monday 20 December succeeding Sunday 9 December. In Great Britain, it is only in 1752 that on September 14 succeeded September 2 and that the Gregorian calendar was adopted.Gradually adopted until the beginning of XXth century by all the countries, this calendar is now of use in the whole world.

COPTIC CALENDAR

Definition
This calendar is of solar type. The year is composed twelve 30-days months, followed, during three years, by 5 complementary days known as "epagomene", the 4th year by 6 "epagomene" days. The average duration of the year (365,25 days) is thus the same one as in the Julian calendar.
This calendar defines the era of Dioclétien whose the origin (1 Tout of year 1) corresponds to August 29, 284 (Julian). The yera of 366 days correspond to those of which the year number plus 1 is a multiple of 4. The coptic year begins on August 29 or 30 (Julian).
This calendar is still used nowadays in Egypt.

```Months of the coptic year

1  Tout
2  Bâbah
3  Hâtour
4  Keihak
5  Toubah
6  Amchîr
7  Barmahât
8  Barmoudah
9  Bachnas
10 Bou'nah
11 Abib
12 Masarî
```

THE MOSLEM CALENDAR

Definition
The Moslem calendar was adopted, in its current form, near 632 after J.-C. It defines the Moslem era of which the origin, 1st day of the year 1 (Hégire), corresponds to Friday 16 July, 622 (Julian) .
It is a calendar of a lunar type. The years are divided into 12 months. The lunar cycle of the Moslems is 30 years.
It includes 19 common years of 354 days and 11 abundant years of 355 days. From one year to another the beginning of the Moslem year thus shifts from 10 to 12 days compared to the seasons (approximately 10.875523... days per year).
Inside a cycle, the abundant years are the numbered years 2, 5, 7, 10, 13, 16, 18, 21, 24, 26, 29.
The current cycle began the 1 Mouharram of year 1411 of the Moslem era which corresponds to Tuesday 24 July, 1990.
The months are of duration of 30 and 29 days alternatively, the first month of the year being 30 days long and the last 29 days (common year) or 30 days (abundant year). The average duration of the month (29,530556 days) is close to that of the lunation.

```Name of the months and number of days

1  Mouharram           30 days
2  Safar               29 days
3  Rabi'-oul-Aououal   30 days
4  Rabi'-out-Tani      29 days
5  Djoumada-l-Oula     30 days
6  Djoumada-t-Tania    29 days
7  Radjab              30 days
8  Cha'ban             29 days
9  Ramadan             30 days
10 Chaououal           29 days
11 Dou-l-Qa'da         30 days
12 Dou-l-Hidjja      29 or 30 days
```

The Moslem calendar presented here is theoretical.
There are different alternatives using a different chronology. The date origin is sometimes taken on July 15, 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 the religious festivals, the beginning of the month is defined by the observation of the first crescent of the moon following a new moon. The calendars are then local and depend on the conditions of observation.

JEWISH CALENDAR

Definition
The jewish calendar is in its current form since the IVth century after J.-C.
It is of lunisolar type. It ensures one average duration of the month (29,530594 days) very close to that of lunation by using a duration of the month of 29 or 30 days.
It ensures also an average duration of the year (365,2468 days) close to that of the tropical year while making alternate 12 common years of 12 months and 7 embolismic years of 13 months inside a 19-years cycle.
In each cycle the embolismic years are those numbered 3, 6, 8, 11, 14, 17 and 19. The current cycle began on 1 Tisseri of the year 5739 which corresponds to Monday October 2, 1978.
The common years may count 353, 354, or 355 days and the embolismic years, 383, 384, or 385 days. The three species of years thus defined are said, respectively, defective, regular or abundant.

The date origin of the jewish calendar is 1 Tisseri of year 1. It corresponds to October 7, -3760 (Julian).

```Name of the months and number of days

1  Tisseri            30 days
2  Hesvan             29 ou 30 days
3  Kislev             29 ou 30 days
4  Tébeth             29 days
5  Schébat            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 Elloul             29 days
```

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

THE FRENCH REPUBLICAN CALENDAR

The era of the French and the republican calendar were adopted by French National Convention at the time of the meeting of October 5, 1793:
" the era of the French... [ begins ] on September 22, 1792 of the vulgar era, day when the Sun arrived at the true equinox of autumn... at 9 hours 18 minutes 30 secondes in the morning for the observatory of Paris "

The adopted definition of the republican calendar is the following (decree of October 5, 1793):
" The beginning of each year is fixed at midnight, beginning the day when falls the true equinox of autumn for the observatory of Paris ".
" The first year of the French Republic started at midnight September 22, 1792, and ended at midnight separating the 21 from September 22, 1793 ".
" The year is divided into twelve months of thirty days each one, after which follow five days to supplement the ordinary year, and which do not belong to any month; they are called the complementary days. "
" Each month is divided into three parts of ten days each one, and which are called decades... "
" In memory of the revolution which, after four years, led France to the republican government, the bissextile period of 4 years is called " Franciade. "
" The intercalary day which must finish this period, is called day of the Revolution. This day is placed after the 5 complementary days... "

Names of the months suggested by Fabre d' Eglantine with the meeting of the National Convention of October 24, 1793.
Autumn
Vendémiaire, Brumaire, Frimaire
Winter
Nivôse, Pluviôse, Ventôse
Spring:
Germinal, Floreal, Prairial
Summer:
Messidor, Thermidor, Fructidor.

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

One will usefully consult the book "  The Republican Calendar " published by the Bureau of longitudes.

THE INDIAN CALENDAR

Definition
The national calendar in used in India at the present time is the one defined by the Calendar Reform Committee. It is applied since March 22, 1957 (1 Chaitra 1879 of the Saka era). The numbering of the year is made in the Saka era.
The months of the indian calendar are as follows:

```Months of                             Gregorian date of the
the indian calendar                    first of the month

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

(*): for the leap years
```
The rules for the leap years is the same than for the Gregorian calendar.

JULIAN PERIOD

Definition
The Julian period is a scale of time which numbers, without discontinuity, the days since Monday January 1, -4712 at 12 hours.
This Julian period was introduced by the astronomer Scaliger in 1583. It names " Julian " this classification by analogy with the Julian year.

Julian Date : it is the duration passed since January 1, -4712 at 12 hours. One expresses it in days and 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 12h, the Julian day 1 begins on January 2, -4712 at 12h, the Julian day 2451911 begins on January 1, 2001 at 12h.
The continuity of the days during the Julian period makes possible to calculate an interval of time without any risk of error even if it covers several calendars. Thus on January 1, 1001 at 0h (Julian calendar) corresponds to the Julian date 2086673.5; January 1, 2001 at 0h (Gregorian calendar) corresponds to the Julian date 2451910.5. It thus ran out between these two dates:
2451910.5 - 2086673.5 = 365237 days.

BEGINNING of the CENTURIES and the MILLENIA

The 21st century begins on January 1, 2001

The century represents one period of 100 years, the millenium one period of 1000 years. The first century begins on January 1 of year 1 and ends on December 31 of year 100. XXth century begins on January 1, 1901 and is completed on December 31, 2000. The 21st century begins on January 1 2001.

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