Before universal calendars became dominant, dates were recordedwith respect to the beginnings of reigns. Recovering a global chronology from such records is a major source of headachesfor historians, who may need the help of classic works like"L'Art de vérifier les dates des faits historiques"("On the Art of Verifying the Dates of Historical Events",first published by the Benedictines in 1750).
Here's how to express the 89th day of the year 1956 (CE)in various calendars. Note the "double dating" [sic]in the Julian calendar between January 1 and March 24,due to the fact that the New Year "Old Style" (O.S.) started on March 25.
Precise astronomical formulas have been devisedat the Bureau des Longitudes (Paris, France) which give the number of days in a synodic lunar monthor in a tropical year. (Those are valid for millenia but would fail overgeological periods.) Let's quote the lunar model ofMichelle Chapront-Touzé & Jean Chapront (1988) and the orbital elements of JacquesLaskar (1986) :
month/day = 29.5305888531 + 0.00000021621 T - 3.64E-10 T2 year/day = 365.2421896698 - 0.00000615359 T - 7.29E-10 T2+ 2.64E-10 T3
In both formulas, T is the time expressed as the number ofJulian centuries of 3652425 "atomic" days (i.e.,3155760000 s ) elapsed since 2000.0.
Those are just mean values: The actual number of days between consecutive new moons may differ from theabove by as much as 0.3 (i.e., 7 hours). The above average number of days in a tropical year may differ byseveral minutes from the actual number of days observed fromone vernal equinox to the next.
Ocean tides provide a braking mechanism which slows down the rotation ofthe Earth about its axis, thereby increasing the duration of the day. Total angular momentum is a conserved quantity. So what'slost in the spin of the Earth goes (mainly) to the orbital angular momentunof the Moon around the Earth. A lesser transfer of angular momentum affects the orbital motion of theEarth around the Sun. Those two effects contribute, respectively, to an increase in the absolutedurations of the month and the year.
By itself, tidal braking would increase the length of a day at a rate of 2.3 ms per century. However, observed historical eclipses show the actual increase to be only 1.7 ms per century.
The difference (i.e., an additional decrease of 0.6 ms per century) has been attributedto a reduction in the oblateness of the Earth since the last ice age. There may be a periodic oscillation in the shape of the Earth,in addition to its secular (irrevocable) decay.
As the Earth spins less rapidly, its buldge at the equator must be reduced. Indeed,it's been observedthat, in the main,major earthquakes tend to redistribute mass so as to reduce the equatorial buldge.
(2002-12-30) Counting days and converting days to absolute time...
From a scientific perspective, a calendar is not about measuring time,it's about counting actual solar days. No amount of averaging will ever be able to equate the two concepts over long periods oftime, because the rotation of the Earth on its ownaxis is steadily slowing down(due totidal braking): The average length of a day currently increases by about 2.3 millisecondsper century. This observation is a fairly recent discoverywhich affects the continuing accuracy of anycalendar whose structure is based on some definite value of the solar year and/or thelunar monthexpressed in actual days. (The scientificday unit of precisely 86400atomic SI secondsisnot directly relevant to calendars.)
This flaw is not present in theJulian Day numbering scheme,arguably the simplest of all calendars,because no attempt is made at counting anything but days(not years, not months, just days). However, the lengthening of the astronomical day may not be neglected when absolute timedifferences (in atomic seconds) are to be obtained from calendar dates, in this or anyother calendar.
The JDN is thus a propercalendar, a well-defined method for counting days,fully specified by the JDN assigned to some specific day in a known calendar. It's closely related to theJulian Date (JD),which is a continuous measure of time obtained by adding to the JDNthe fraction of a day elapsed since noon GMT.
This numbering scheme was invented around 1583,in the wake of theGregorian reform,by Joseph Justus Scaliger (1540-1609). Scaliger put the origin in 4713 BC because this year predatesall our recorded history and can be construed as a commonbeginning to the following three noteworthy cycles (which repeat after 7980 years).
The 28 yearcycle of the Julian calendar. The pattern of weekdays and leap years repeat after a 28 years in the JulianCalendar (it's 400 years with the modern Gregorian calendar).
The 19 yearMetonic cycle. 19tropical years (about 6939.602 days) are onlytwo hours short of235lunar months (about 6939.688 days). For any reasonably accuratesolar calendar,a given phase of the Moon will thus occur [nearly]at the same calendar date after a period of 19 years. Conversely, if we use a perfect lunar calendar and estimate the solar year tobe 235/19 lunar months,we'll drift away from the solar seasons at a rate of lessthan half a day per century (that's how theJewishcalendar is built).
The 15 yearRoman indiction cycle. This tax cycle was only abolished in 1806. It had been introduced on September 1, 312,byConstantine the Great (c.274-337),the founder of Constantinople (modernIstanbul)and the first Roman emperor to become a Christian (baptized on his deathbed). Theindiction number was used as a calendrical era(e.g., "third year of the fourth indiction").
Thelowest common multiple of these is a periodof 7980 years, which is known as Scaliger'sJulian period. Scaliger reportedly named the thing after his late father(Julius Caesar Scaliger, 1484-1558),so the etymological connection withthe Julian calendar (named after emperor Julius Caesar) is anindirect one.
Counting Seconds:
(2003-01-21) The fundamental social cycle has not always been a week of 7 days.
Second only to the natural daily rythm, a regular man-made cycle of 4 to 10 dayshas always governed human activity everywhere, throughout recorded history(and probably well before that). This period has not always been the familiarweek of 7 days, though. Here are some examples:
(2002-12-29) This solar calendar paved the road for its successors.
When Moses was alive, these pyramids were a thousand years old. Here began the history of architecture. Here, people learned to measure time by a calendar,to plot the stars by astronomy and chart the Earth by geometry. Here, they developed that most awesome of all ideas—the idea of eternity. WalterCronkite (1916-) CBS evening news (1962-1981).
The ancient Egyptian civilization lasted longer than any other. It had asolar calendar whose year consisted of 12 months of30 days (3decans of 10 days each)and 5 additional "yearly days" (epagomenes), for a total of 365 days.
Egyptian astronomers knew that a period of 365 days was about ¼ day shortof an actual tropical year, but an intercalary day wasnever added,and the calendar was allowed to drift through the seasons.
A drift of a fixed calendar date through the seasons is a flaw of asolarcalendar calledcalendar creep. The Egyptian calendar had a severe case of this,but it wasoriginally designed to match the3 seasons of the Nile (4 months each):
Akhet : "Inundation".
Proyet,Peret, orPoret : "Emergence", "Winter", or "Growing Season".
Shomu orShemu : "Harvest", "Summer", or "Low Water".
Although the Egyptian months have specific names(tabulatedbelow,in our discussion of the modernCoptic calendar),they are commonly denoted by their ranks within those fictitious calendar "seasons",whose own names are either transliterated or translated: Third month ofAkhet, first month of Harvest, etc.
1461Egyptian years are equal to 1460 years of 365¼ days(the length of what would become theJulian year). This period of 533265 days has been dubbed aSothic period,becauseSothis is the Greek name of Sirius,calledSopdet [spdt] by the Egyptians. The Egyptian civilization lived throughseveral such cycles... (It has been reported that ancient Egyptians also had another "sacred" calendarbased on a year of 365¼ days,but we found no evidence to support this claim.)
A period of 533265 days doesn't quite bring the Egyptian calendarback to the same point in the actual cycle of the seasons,because a tropical year isn't exactly equal to 365.25 days: It's more like 365.2422 days,which would imply a period of 1508Egyptian years(1507tropical years) between successive returns of the Egyptian calendar to thesame seasonal point.
(2003-01-12) "Sirius is the one consecrated to Isis, for it brings the water."--Plutarch
Aheliacal rising of a star is defined as its appearanceabove the horizon just before sunrise. In ancient times, the Egyptians observed that theheliacal risingof Sirius marked the yearly beginning of the Nile's floods.
The exact day when anheliacal rising is observed may dependon the longitude and latitude of the observer. Thealtitude is somewhat relevant too(on the equator, a star rising due east would be seenfrom a 100 m cliffabout 76.8 s earlierthan from the beach). The brightness of the star is important as well,since fainter objects disappear earlier at dawn.
(2003-01-06)
To avoid most of thecalendar creep describedabove,a reform of the Egyptian calendar was introduced at the time of Ptolemy III (Decree of Canopus, in 238 BC) which consisted in theintercalation of a 6thepagomenal day everyfourth year. However, the reform was opposed by priests,and the idea was discarded until 25 BC or so,when Roman emperor Augustus formally reformedthe calendar of Egypt to keep it forever synchronizedwith the newly introducedJulian calendar. To distinguish it from the ancient Egyptian calendar, which remained in use by someastronomers until medieval times,this reformed calendar is known as theAlexandrian calendar andit's the basis for the religiousCoptic calendar,which the Copts [the Christians from Egypt] are still using now.
Coptic years are counted from AD 284, theera of the Coptic martyrs,the year Diocletian became Roman Emperor(his reign was marked by tortures and mass executions of Christians). TheCoptic year is identified by the abbreviation "AM"(forAnno Martyrum) which is unfortunately also used forthe unrelatedJewish year (Anno Mundi). To obtain theCoptic year number,subtract from theJulian year number either 283 (before the Julian new year)or 284 (after it).
The table below shows the correspondence between theCoptic calendarand theJulian calendar. For the period between 1901 and 2099 CE, the secular (Gregorian) date isobtained by adding 13 days to the Julian day shown in the table,so that the Coptic year actually starts on September 11, onmost years. The 7 months which precede the intercalationof aJulian February 29 actually start one day later(this is what the " + " signs in the table are reminders for). Therefore, the Coptic year which starts just before aJulian leap year beginson August 30 in theJulian calendar,which corresponds to September 12 in theGregorian calendar(every fourth year, from 1903 to 2095 CE).
For the usual Gregorian secular date between 1901 and 2099 CE,add 13 days to the Julian date shown(the 7 months before aJulian Feb. 29 start 1 day later).
Days
Coptic Month(Egyptian Name)
First Day (Julian)
Nile Season
1
30
Thoth, Thot, Thout, Thuthy
August 29+
Inundation Akhet
2
30
Paophi, Paapi, Paopy
September 28+
3
30
Athyr, Hathor, Hathys,Athor
October 28+
4
30
November 27+
5
30
Tybi, Tobi, Tyby
December 27+
Emergence Proyet Peret, Poret
6
30
Mesir, Mechir, Menchir,Mekhir
January 26+
7
30
Phamenoth, Paremhat,Famenoth
February 25+
8
30
Pharmouthi, Paremoude,Parmuthy
March 27
9
30
Pachons, Pakhons
April 26
Summer Harvest "Low Water" Shomu Shemu
10
30
Payni, Paoni, Paony
May 26
11
30
Epiphi, Epip, Epipy, Epep
June 25
12
30
Mesori, Mesore
July 25
13
5 or 6
Epagomena,Little Month
August 24
(2002-12-22)
The Julian calendar is still being used for religious purposes bysome Eastern Orthodox churches, such as the Russian Orthodox church.
An early form of theJulian Calendar was introduced by Julius Caesarin 46 BC, on the advice of the Egyptian astronomer Sosigenes. Officially, the first day of the Julian Calendar was the Kalends of Januarius, 709 AUC(January 1, 45 BC). At first, there was a leap year everythird year,but this was soon recognized to be a mistake: In 8 BC, the calendrical reform of Augustus gave the monthstheir modern names and lengths, and returned the calendar year backto the seasonal point intended by Julius Caesar. This was done by shunning leap years until AD 8, which would be a leap year like everyfourth year thereafter. (5 BC, 1 BC and AD 4 wereordinary years.)
Historical Leap Years (before the Regular Julian Pattern)
43 BC
40 BC
37 BC
34 BC
31 BC
28 BC
25 BC
22 BC
19 BC
16 BC
13 BC
10 BC
AD 8
AD 4n
The Julian Calendar, before and after Augustus
45 BC to8 BC
After 8 BC
Days
Month
Days
Month
I
31
Ianuarius
31
Ianuarius
II
29 or 30
Februarius
28 or29
Februarius
III
31
Martius
31
Martius
IV
30
Aprilis
30
Aprilis
V
31
Maius
31
Maius
VI
30
Iunius
30
Iunius
VII
31
Quintilis / Iulius
31
Iulius
VIII
30
Sextilis
31
Augustus
IX
31
September
30
September
X
30
October
31
October
XI
31
November
30
November
XII
30
December
31
December
New Year's Day
Julius Caesar made the year start on January 1, probably because this was the traditional beginning of the sessionin the Roman Senate (and the date when consuls used to beelected). However, it seems that thepopular use of the previous "March 1" systemsurvived at least until theAugustan Age (27 BC-AD 14). The "January 1" convention was not finally established (or restored)until the introduction of theGregorian calendar.
The most common convention in latemedieval times was thatthe beginning of a new Julian year occurred on March 25. This was thenominal date of the vernal equinox(it was theactual date of the equinox shortly before thecalendar reform of Julius Caesar). In medieval times, March 25 was thought of asthe mythical anniversary of Creation. For Christians, this is theFeast of the Annunciation,theIncarnation when Christ was conceived (the alternate nameLady Day has a pagan origin,rooted in the Celtic tradition). However, the Julian New Yearhas been celebrated at a variety of dates throughout history. The following sketchy table is only meant to show the utter lack of universal conventions:
Note : When Easter was taken as the beginning of the year,there could betwo days with thesame date, at the beginning and at the end ofsome years. The ambiguity used to be lifted by specifying "after Easter" of "before Easter".
Days of the Month:
The Roman way of numbering days was used in Latin with the Julian calendar,until the late Middle Ages. Three special days were singled out:
TheKalends: First day of the month. [Etymology of "calendar"]
TheNones: The 7th day of March, May, July, and October. The 5th day of the other months (i.e., always theninth day of the Ides).
TheIdes: The 15th day of March, May, July, and October. The 13th day of the other months.
The other days were countedbackwards andinclusively,from the next such special day. Thus, since March 13 wastwo days before the Ides of March,it was called thethird day of the Ides of March. Most of the month came after the Ides and was thus referred to the Kalendsof thenext month. In a leap year, the intercalary day was insertedafter February 23(the seventh day of the Kalends of March)so there would be a day designated asbissextilis,being the "other sixth" day of the Kalends of March... Leap years are thus still calledbissextile.
(2002-12-31)
Dionysius Exiguus was a Russian monkwho had been commissioned by pope St. John I to work on calendrical matters,including the official computation of the date of Easter. The story goes that he was confronted with theCoptic calendar in thecourse of his work with Alexandrian data. He liked the idea of a continuous count of years based on a Christian milestone,but was disturbed by the choice of the Copts, who were honoring their greatestpersecutor by counting from the year Diocletian became emperor (284 CE). Dionysius had the idea to count years from a joyous event instead, the birth of Christ. In 527, he formally declared that Jesus was born on December 25 in the year 753 AUC,equating the year 754 AUC with the year AD 1 (Anno Domini = Year of the Lord).
The guess of Dionysius may have been off by several years: Jesus was born during the census of Augustus(Luke 2:1) while Quirinius was governing Syria (Luke 2:2),under the reign of Herod the Great (Matthew 2:1). In 1583, Scaliger argued thatHerod died in 750 AUC (4 BC), so Jesus was bornat least4 years earlier than Dionysius thought. We don't know how Dionysius arrived at his result,but we may venture the guess that he simply took the Gospel of Luke literally...
(Luke 3:23)after begin baptized by John, who began preaching(Luke 3:1). As Tiberius became emperor in AD 14, the Gospel of Luke says thatJesus was baptized in AD 29 or AD 30,when he was (he may have been 34 or so).
The original task of Dionysius was to prepare a table giving the dates of Easterstarting with AD 532. In the Julian calendar, such a table has a periodicity of 532 years,so that it was tempting to place the birth of Christat the beginning of the previous cycle. Either that or Dionysius guessed the birth of Christ first,by some other argument,andthen chose to have his tables start with the second cycle.
The numbering scheme suggested by Dionysius may not have been popular untilthe time of the calendrical studies ofBede (673-735) in Britain.
The Date of Christmas
Incidentally, this calendrical focus on the nativity of Jesusturned Christmas into a major Christian festival,rivalingEaster. The birth of Christ was hardly celebrated at all by early Christians,and different communities did so on different dates... The choice of December 25 had been proposed by anti-popeSaint Hippolytus of Rome (170-236),but it was apparently not accepted until AD 336 or 364. Dionysius emphatically quoted mystical justifications for this very choice:
March 25 was considered to be the anniversary of Creation itself. It was the first day of the year in the medievalJulian Calendar and thenominal vernal equinox (it had been theactual equinox at the time when the Julian calendarwas originally designed). Considering that Christ was conceived at that date turned March 25 intotheFeast of the Annunciation which had to be followed,9 months later,by the celebration of the birth of Christ,Christmas,onDecember 25...
There may have been more practical considerations for choosing December 25. The choice would help substitute a major Christian holiday for the popularpagan celebrations around the winter solstice(RomanSaturnaliaor Brumalia). The religious competition was fierce. In 274, Emperor Aurelian had declared a civil holiday on December 25 (Sol Invicta, the Unconquered Sun) to celebrate the birth ofMithras, the Persian Sun-God whose cult predatedZoroastrianismand was then very popular among the Roman military... Finally, joyous festivalsare needed at that time of year, to fightthe natural gloom of the season. TheJews haveHanukkah, an eight-day festival beginningon the 25th day ofKislev.
Whatever the actual reasons were for choosing a December 25 celebration,the scriptures indicate that the birth of Jesus of Nazareth didnot even takeplace around that time of year,since (Luke 2:8). During cold months, shepherds broughttheir flocks into corals and did not sleep in the fields. That's about all we know directly from scriptures, besideswildspeculations.
(2002-12-22)
TheGregorian calendar is like the aboveJulian calendar,except for its pattern of leap years. Its Christian origins are all but forgotten, as it has now been adopted as asecular calendar bymost modern nations. A few countries are still officially using other traditional and/or religious calendars,but they all have to accomodate theGregorian calendar,at least in an International context...
This calendar has been dubbedGregorian because it was introduced under theauthority of pope Gregory XIII, né Ugo Boncompagni (1502-1585),Pope from 1572 to 1585. The Gregorian calendrical reform was engineered by astronomer Christopher Claviusto make the seasons correspondpermanently to what they were under theJuliancalendar in AD 325,at the time of theFirst Ecumenical Council of the Christian Church,theFirstCouncil of Nicea, when rules were adopted for the date of Easter.
TheCouncil of Trent(1545-1563) had previously urged Pope Paul III to reform the calendar,and Clavius was one of several scientistswho had been approached in the wake of that resolution. Over 20 years later, Gregory XIII finally asked Clavius to leada commission on the subject,which would be formally presided by Cardinal Guglielmo Sirleto (1514-1585),a contender for the papacy.
Building on the work of Luigi Lilio,this commission recommended dropping 10 calendar days immediately,and reducing the number of future leap years(to avoid a new drift of the calendar with respect to the seasons). Thus, a Papal Bull (Inter Gravissimas)decreed that, October 4, 1582 would be followed by October 15. Furthermore, futureleap years would be multiples of 4 (as inthe Julian calendar)except for years evenly divisible by 100 but not by 400(so that 1600 and 2000were indeed leap years). For each Gregorian period of 400 years,This reduces the number of leap years to 97 (down from 100 in the Julian scheme). Thus:
400 Gregorian years = 146097 days ( = 20871 weeks
Various countries adopted the "new" calendar only much later(see tablebelow). In particular, the earliest valid Gregorian date in England(and itsAmerican Colonies) is September 14, 1752,which followed September 2, 1752 (the difference between the two calendars hadgrown from 10 to 11 days by then, since 1700 wasn't a leap yearin the Gregorian calendar).
Some Official Transitions to the Gregorian Calendar
On 4 October 2016, Google celebrated, one day ahead, the passing of 434 Gregrorian years,using the "Doodle" below, in a few selected countries:
(2007-05-26) How to go back and forth between a Gregorian date and a straight count.
We present a way to go from a count of days to a Gregorian date and vice-versa,using only simple arithmetic formulas. This makes it easy to determine how many days there are between two distant dates.
Making Mincemeat of Monthly Irregularities :
The key trick to deal with the not-so-regular pattern of varying numbers ofday per month is to pretend that the year starts on March 1 (asit didwhen the months got their current Latin names). This makes it possible to work out the following simple formula,which I designed back in July 1978.
Counting Days and/or Months from the previousMarch 1st
Mar.
Apr.
May
June
July
Aug.
Sep.
Oct.
Nov.
Dec.
Jan.
Feb.
m
0
1
2
3
4
5
6
7
8
9
10
11
Nm
0
31
61
92
122
153
184
214
245
275
306
337
Day N corresponds to month number m = floor ( (N+0.5) / 30.6 ). Conversely, the first day of month m is Nm = ceiling ( 30.6 m - 0.5 ).
For the record, here's a full analysis which determines exactly how far one can wanderaway from the values 30.6 and 0.5 which appear in the above formulas (In 1978, I wanted to obtain safe binary fixed-point values.)
If N is the number of days (from 0 to 365) elapsed since the previous March 1,we want the number of months elapsed (0 to 11)to be floor ((N+y)/x).
The result will be correct for March if an N from 0 to 30 gives a result of 0,which means that (N+y)/x is between 0 (included) and 1 (excluded) when N is between0 and 30. This is true if and only if y is 0 or more and 30+y is less than x. With similar constraints for the other months, we have a total of 24 inequalitiesto satisfy. However, only 4 (or 5) of those are "critical",as they define the inside ofa small quadrilateral in the (x,y) plane where all24 inequalities are satisfied. (Thefifth "critical" inequality is y < 6x-183. It corresponds to the last day of August. Its constraining line grazes the following convexsolution quadrilateral at corner B.)
Upper boundary (excluded): y < x - 30 (last day of March) AB y < 11x - 336 (last day of January) BC
Lower boundary (included): y ≥ 4x - 122 (first day of July) CD y ≥ 9x - 275 (first day of December) DA
A = (30.625, 0.625) B = (30.6, 0.6) C = ( 30 4/7 ,2/7) D = (30.6, 0.4)
Corners A, B and C are excluded. The point D is included, so the value y = 0.4 is barely acceptable with x = 30.6. The best decimal value is, of course, the middle of BD,namely: x = 30.6 and y = 0.5. For low-level binary routines (my original concern, in 1978) we may retain y = 0.5 and use any valueof x between 673/22 = 30.59090909... and 551/18 = 30.611111... This is the interval represented by the in the above diagram. Inbinary numeration:
11110.10010111010001011101000... At least.11110.10011100011100011100011... At most.11110.10011 (hex 3d3) is thus the coarsest usable value.
(In other words, we may use 30+19/32 = 30.59375 instead of 30.6.) With y = 0.5, 1/x should be between 18/551 and 22/673. In binary, this is:
0.000010000101110011101100... At least.0.000010000101111001010101... At most. 10000101111 is the coarsest usable binary value.
If we multiply that 11-bit integer (42F in hexadecimal) byone plus twice the number of days, we obtain the month number bydiscarding the lower 16 bits of the product. So, the following piece of68000 assembly languageturns a number of days (0 to 365) from the lower 16-bit word ofD0 (a 32-bit register) into the corresponding 0-11 month number (the other half of D0 becomes junk).
E340 N2MONTH ASL.W #1,D0 Multiply by 25240 ADDQ.W #1,D0 Add 1 (i.e., add 0.5)C0FC 042F MULU.W #$42F,D0 Multiply by 1/30.64840 SWAP D0 Get integer result RTS
Conversely, if the lower half of D0 contains a month number (0 to 11) we mayobtain the day number (0 to 337) of the first day of that month, using the code:
C0FC 03D3 MONTH2N MULU.W #$3D3,D0 Multiply by 30.60640 0010 ADDI.W #$10,D0 Add 0.5 (= 1.0 - 0.5)EA48 LSR.W #5,D0 Get integer result RTS
If I may say so, I'm proud of my younger self for pioneering this,almost 30 years ago (time flies). I just had a nice time retracing my own footsteps,as my 1978 notes are lost (I did remember the quadrilateral's shape and the 30.6 value).
Complete Conversion Algorithms :
With the issue of months out of the way, other Gregorian calendrical computationsare straightforward if we consistently put exceptions at the endof their respective periods,just like we put February at the end of each year in the above... A leap year (366 days) is at the end of an olympiad of 1461 days. A short olympiad of 1460 days (no leap year) is at the end of a normal century(36524 days). A long century (36525 days = Julian century) is at the endof each Gregorian period of 400 years (exactly 146097 days).
With those conventions, everything falls into place if we start counting daysfrom what would have been the Gregorian date March 1 of year 0, if the Gregorian schemehad been in place back then (in theproleptic Julian calendaractually used for that period of history,"year 0" is called "1 BC" or "1 BCE"). Because of our original trick (which made it so easy to count months within a year) we merely have to increment the year for the monthsof January and February so they belong to the same year as the following month ofMarch, as is the modern usage. That's all there is to it!
TheModified Julian Day Number is 0 for November 17, 1858 which came 678881 days after the above arithmetically convenient origin. Therefore, we'll use that offset in an actual implementation which turnsour counting of days into straight conversions to and from MJDN dating.
The screenshot at right shows how this can be implemented on anhandheld calculator,like the TI-92, TI-89 or Voyage 200 from Texas Instruments. The gdate function takes aninteger (although it also allows fractional numbers) interpretedas MJDN and returns the corresponding Gregorian date in theformat used by the calculator itself to read its own real-time clock [ when getDate ( ) is called ] namely a list of the form { yearmonthday }.
MJDN to Gregorian conversion,with early proleptic Julian dates
As the Gregorian calendar is never used for dates beforeOctober 15, 1582 (a negative MJDN of100840) we must modify the aboveto use the proleptic Juliancalendar for all earlier dates (by skipping Gregorian century rulesand using an offset matching the Julian calendar). The proper code shown at left can accomodate any switch date: Simply replace100840 by the MJDNof theearliest Gregorian dateacceptable to you, if it's not October 15, 1582.
For Gregorian to Julian conversions, it is useful to have a version of theabove which never switches to the Gregorian calendar. We call it jdate (for "Julian date") and the simplecode for it is given by the screenshot at right.
The two functions,dubbed jday and day, do theopposite of the above, namely they take aJulian or Gregorian date (respectively) and return the correspondingday number (MJDN).
We do allow months outside of the 1-12 range formonths of the previous or following year(s). Likewise, the number of days can be outside the 1-31 range and may befractional.(Fractional years or monthsare not allowed.) Year 0 is 1 BC,Year -1 is 2 BC,Year -2 is 3 BC, etc. All of this is compatible withastronomical standards.
The function day calls jday when it finds an MJDN that's below the earliestacceptable Gregorian date (again, you may change the -100840value to the switching MJDN of your choice). Therefore, day correctly interprets {1582,10,14} as a deprecated Julian date, 9 daysafter {1582,10,15}. Nice.
The above four functions present great computational flexibility. For example:
date ( jday ( {yyyy,mm,dd} )) obtains a Gregorian date from a Julian one. jdate ( day ( {yyyy,mm,dd} )) obtains a Julian date from a Gregorian one. date ( day ( {yyyy,mm,dd} )) puts a"generalized" date in standard form. day({y2,m2,d2}) - day({y1,m1,d1}) is the difference in days between two dates. jday({Y,1,1}) - day({Y,1,1}) is the Julian lag at the beginning of year Y. day(getDate()) - day({1956,3,29}) is my current age, in days. date(10000 + day({1956,3,29})) is when I was 10,000 days old (Aug. 15, 1983). date(20000 + day({1956,3,29})) is when I'll be 20,000 days old (Dec. 31, 2010). date(-2400000.5) is {-4712,1,1.5}. That's Julian date 0.0 (defined by the IAU). date(0) is {1858,11,17} namely MJD = 0.0 (Nov. 17, 1858 at 0:00 GMT).
The day of theweek(0=Sunday, 1=Monday, 2=Tuesday, 3=Wednesday, 4=Thursday,5=Friday, 6=Saturday) is given by the equivalent expressions:
mod ( 3 + day ({yyyy, mm, dd}) , 7 ) mod ( 3 + jday ({yyyy, mm, dd}) , 7 ) mod ( 3 +hday ({yyyy, mm, dd}) , 7 )
There's almost no legitimate need for projecting the Gregorian scheme into the distantpast (before 1582) as theproleptic Julian calendar isuniversally used for that purpose by astronomers and historians alike. The one useful purpose for a "pure" Gregorian scheme (as first presentedin ourintroductory gdate screenshot) would be to find out the correct seasonal date for a yearly celebration ofsome event that happened well before the Gregorian calendar ever existed...
This would be similar to what George Washington did when he adjusted his own birthdayto a Gregorian date (February 22, 1732) although it had first been recorded as February 11, in the Julian calendarused at the time in Great-Britain and inthe "American Colonies". In what would become the U.S.,the switch occurred on September 14, 1752, when Washington was a young adult.
Of course, at the time of Washington's birth, the Gregorian calendar was alreadylegitimate somewhere else, This is whythe abovedate andjday functions are sufficient tocheck Washington's computation.
(2007-06-01) The average synodic month is 29.530588853 days.
For calendrical purposes, we may consider only the average motion of the Moon based on the above period.
In traditional lunar calendars, a month starts with the actual observation of the thin crescent of a new moon, which typically takes placesa day or two after the astronomical new Moon (when the Moon is invisible).
Let's define the latter as halfway between two full moons and take the middle of a recent total lunar eclipse asan "accurate" full moon. Using the lunar eclipse ofMarch 3, 2007 (which started at 22:43 UTC and ended at 23:58) we obtainthe following formula for the age of the Moon, in days:
mod ( 10.8927 + n , 29.530588853 ) moon (n)
That's how a TI-92 functionmay be defined whose argument is thenumber n (theModified Julian Date) prominentlyfeatured in theprevious article. This is to be used jointly with the calendrical functions presented there.
The Islamic Month
Thearithmetical version of the Islamiccalendar isbased on a cycle of 10631 days, divided into 360 months. This yields an average month of :
10631 / 360 = 29.530555555... days
That's about 2.8769 s short of the astronomical average. It would take about 2428 (tropical) years to build up a discrepancy of a whole day.
The Jewish Month
The arithmeticalJewish calendar(the Hillel calendar) isbased on the following estimate of the time between consecutive new moons:
765433 / 25920 = 29.530594135802469135802469... days
This is about 0.4564 s short, compared to the astronomical average. It would take about 15305 years to build up a discrepancy of a whole day.
Incidentally, the (long term)average of the Hillel year is obtained bymultiplying the above by 235/19 (theMetonicapproximation to the number ofsynodic lunar months in a tropical year). This boils down to 35975351 / 98496 or:
365.246822205977907732293697205977907732293697... days
That's longer than the tropical year (at epoch 1900.0) by399.4639 s. Thus, the Hillel calendar drifts with respect to the solar seasons ata rate of about one day in 216 years (more precisely,3 days in 649 years, 7 days in 1514 years, 31 days in 6705 yearsor 69 days in 14924 years). The Jewish Spring festival of Passover is movingtoward the Summer at that rate. On the average, Passover now occurs more than one weeklater (with respect to the solar seasons) than it did in the timesof Hillel II. The Jews are thus facing a problem similar tothe 10-day offset which was bugging Christian authorities before theGregorian reform of 1582.
The synchronization of the Jewish calendar with the seasonsis not nearly as critical as its synchronization with the lunarcycle (a new moon occurs near the beginning of every month). The large effect of intercalary months on Jewish festivals drowns thetiny drift of those festivals, at a rate of less than half aday per century...
(2017-11-04, Beaver Moon) There are 12 or 13 full moons in each calendar year.
There's at least one full moon in each calendar month (except, on rare occasions, inFebruary). Usually, there's just one. The first full moons in calendar months bear the following names:
American Names of the Full Moons (inherited from theAlgonquian tribes)
Month
First Full Moon
January
Wolf Moon, Old Moon, Ice Moon
February
Snow Moon, Sorm Moon, Hunger Moon
March
Worm Moon, Death Moon, Crust Moon, Sap Moon
April
Pink Moon, Sprouting Grass Moon, Egg Moon, Fish Moon
May
Flower Moon, Hare Moon, Corn Planting Moon, Milk Moon
June
Strawberry Moon, Rose Moon, Hot Moon
July
Buck Moon, Thunder Moon, Hay Moon
August
Sturgeon Moon, Green Corn Moon, Grain Moon, Red Moon
September
Harvest Moon, Corn Moon, Barley Moon
October
Hunter's Moon, Travel Moon, Dying Grass Moon
November
Beaver Moon, Frost Moon
December
Cold Moon, Long Nights Moon, Oak Moon
Henry Porter Trefethen (1887-1957) was the editor of 26 issues (for the years 1932 to 1957) of the popular Maine Farmer's Almanac which was published from1819 to 1972 (154 issues). He didn't tryto innovate in the 1932 issue but thereafter his Almanac clearly reflected hisown growing interest in Pagan lore and folklore (he was of Celtic ancestry). In the 1933 issue, Trefethen introduced the names Harvest Moon and Hunter's Moon for the full moons which occurred that year on Sept. 4 and Oct. 3, respectively (he explained the names only in the 1934 issue). The 1933 almanac also contained an essay on the Native American names for all the full moons occuring in various seasons (the author being only identified by the initials C.G.F.).
Traditional rural societies recorded the full moons separately in each of the four seasons.
Trefethen's Names of the Full Moons
Winter
Spring
Summer
Autumn
First
Yule Moon
Egg Moon
Hay Moon
Harvest Moon
Middle
Wolf Moon
Milk Moon
Grain Moon
Hunter's Moon
Last
Lenten Moon
Honey Moon
Fruit Moon
Ice Moon
After the passing of Trefethen (1957) the editors of The Maine Farmer's Almanac didn't apply his system reliably. Note that this defunct almanac (1819-1972) bears no relation with the extant Farmer's Almanac, whichis slightly older (1818) and movedits offices toLewiston, Maine, in 1955.
February :
Because the interval between two full moons (the lunar month of 29.530588853 days) is greater than the duration of the month of February (28 or 29 days) it's possible for February to contain no full moons. This happens in February 2018, which is surrounded by two months with two full moons (there's a new-style blue moon on January 31 and on March 31,those two are separated by only 59 days, which is the minimum possible). The same situation last happened in 1999. It will happen again in 2037.
(2017-11-06) There are two definitions for what a calendrical blue moon is.
Blue Moon (New Style): Second full moon in a calendar month. Blue Moon (Old Style): Third full moon in a season which has four.
In our Gregorian calendar, those two definitions are utterly incompatible.
The locution blue moon was first used in printby Pierce Egan (1772-1849) in his 1821 book Life in London, which begat astage playand awarm Christmas drink,both called Tom and Jerry, also in 1821. The drink inspired the titles of two series of cartoons: Van Beuren's Tom and Jerry (1931-1933) and MGM's hugely popular Tom and Jerry cat-and-mouse cartoons by Hanna andBarbera (1940-1958, with ongoing spinoffs).
In 1821, Pierce Egan had to give an explanatory note for this early use.
The precise usage of blue moon to denote the second full moon in a monthis far more recent. It has been traced to an honest mistake made in 1946...
The board game Trivial Pursuit (Genus II expansion pack #730077, 1986) was instrumental in the popularization of this new definition. According to the publisher's own records, their source was a children's book which first appeared the previous year (1985):
Have You Ever Wondered? [Bottom of page 229]
What is a "blue moon"? When there are two full moons in a month,the second one is called a blue moon. This is a rare occurrence.
The authors may well have heard about that in January 1980 on the NPR (National Public Radio) program StarDate produced by Deborah Byrd (1951-) as Byrd herself pointed out in the December 1990 issue of Astronomy. Her own source was a mistake made by the amateur astronomer James Hugh Pruett (1886-1955) in an article entitled Once in a Blue Moon published in theMarch 1946 issue of Sky & Telescope.
Pruett had misinterpreted the prior definition of a blue moonand became inadvertently responsible for the introduction of the simpler new style version, by hastily issuing the following comment in print:
Incidentally, that summary is incorrect, since February sometimes doesn't possess a full moon, in which case there are two full moons in January andtwo in March (that's still 13 full-moons for the whole year). This happened in 1999, this is happening in 2018 and this will happen again in 2037.
The above words of Pruett lay dormant for more than thirty years in a yellowing copy of themagazine at thePéridier Library (in the Astronomy Department of UT Austin). That's where Deborah Byrd found them in the late 1970s. She made this new style definition her ownand would naturally share it with her radio audience a couple of years later, thus ushering it into folklore, before she--or anyone else--botheredto chronicle its actual genesis.
The "source" quoted by Pruett in his 1946 essaywas just an earlier (1943) Sky and Telescope column by Laurence LaFleur which referred tothe following comment given for the month of August 1937 in The Maine Farmer's Almanac (the very publication which had introduced the old style of blue moons to the general public):
Now, the first sentence of that passage is as misleading as can be. By itself, that sentence could only be a prelude to a new style definition! It's utterly irrelevant to the --then current--old style definition; especially for the year 1937, which only had 12 full moons! The essay had indeed been prompted by the full moon on 21 August 1937 which was an old style blue moon (since the next full moon (on20 September 1937) would be the fourth full moon of the Summer of 1937).
Most of the above chain of events was unraveled by Phillip Hiscock who has published hisown account several times in the media, starting with a newspaper column he was prompted to writeon the occasion of the "Blue Moon" (new style) on 31 December 1990.
Belewe Mone
In Old and Middle English, the adjective belewe meant either blueor treacherous (to belewe meantto betray). The earliest extant reference to a belewe mone is found in a famous 1528 pamphlet by William Barlow, Bishop of Chichester,entitled The Treatyse of the Buryall of the Masse but more commonly known by its first words, Rede Me and Be Nott Wrothe :
The three full moons in a regular season were called first, middle and last. Occasionally, the third full moon in a season wasn't the last one andmay thus have been perceived as misleading or belewe. That's one possible etymologicalexplanation for what may have been the origin of the term.
A dubious urban legend is that this extra full moon was printed in blue in some almanacs, possibly including The Maine Farmer's Almanac (1819-1972) presumably to draw attention to the fact that a given third full moon wasn't the last of the season. I haven't seen any examples of that.
In borderline cases, it's extremely complicated to predict astronomically the season a given full moon will belong to (it even depends on the location on the surface of the Earth). Thus, calendar makers had to rely on arithmetical approximations sanctioned by some central authority. In particular, Christian monks had to determine the date of the first full moon after the vernal equinox. This happens to be precisely what the Paschal full moon is meant to approximate (in the computation of the date ofEaster, described in the next section).
Those almanac makers probably synchronized the other equinox and the twosolstices on the ecclesiastical spring equiquinox. Before the calendrical reform of1582, this was rather silly. This goes a long way toward explaining the aforementioned complaint of Bishop Barlow aboutthose who were defining the belewe mone by some obscure arbitrary process...
At the time, the calendar was off by about ten days with respect to the seasons. As farmers were still synchronizing their work with the lunar cycle, mistakes in the naming of the full moons may have resulted in lesser crop yields. The calendrical reform of 1582 wasn't just about church festivals; it was beneficial to agriculture as well.
(2003-02-20) The resurrection of Christ is celebrated onEaster Sunday,reckoned as the Sunday following the Paschal Full Moon.
According to Christian tradition, Jesus Christ was crucified on a Fridaywhich fell just before the festival of Passover (15 Nissan) which is always near a full moon. The 14th of Nissan actually fell on a Friday on the following Julian dates:
7th of April in AD 30.
3rd of April in AD 33. (The correct date of the Crucifixion.)
In1733,Sir Isaac Newton had argued for the next year (AD 34)but this would only be possible with a Jewish calendricalrule of postponementthat was not yet enforced at that time! On the other hand, the baptism of Christ is clearly statedto have occurred duringthe 15th year of Tiberius Caesar (Luke 3:1). which corresponds to AD 29 in the Julian calendar. As the ministry of Christ covered 3 full years from that point on, the day of the Crucifixion would be firmly established to beApril 3rd of AD 33. Apartiallunar eclipse was visible at moonrise from Jerusalem on that veryday, so that the Moon appeared like blood. Everything fits.
At theFirst Ecumenical Council of the Christian Church(held in Nicea, in 325 AD), it was decided to celebrate Easteron the Sunday following the so-calledPaschal full moon:
ThePaschal full moon is an arithmeticalapproximationto the first full moon after the vernal equinox. John H. Conway (1937-2020) expressed it as follows in terms of the so-calledGolden number (G) andCentury term (C):
Paschal full moon (PFM) = (April 19, or March 50) (C+11G) mod 30
... except in two cases where the PFM is one day earlier than this, namely:
When (C+11G) is 0 modulo 30, then PFM = April 18 (not April 19).
When (C+11G) is 1 modulo 30,and G ≥ 12, PFM = April 17 (not 18).
Some famous algorithms, like the so-called Gauss formula, are wrong because theyfail to incorporate those two exceptional cases(e.g., in 1981 the PFM was Saturday April 18, and Easter Sunday was April 19). TheGolden number (G)is the same for both Julian and Gregorian computations,but theCentury termis constant (C = +3) inJulian computations:
G = 1 + (Y mod 19) in year Y (Julian or Gregorian).
C = -H+ H/4+ 8(H+11)/25 with H = Y/100(Gregorian year Y)
As the Sundayfollowing the PFM, Easter is one week after the PFM when thePFM happens to fall on a Sunday...
You should work entirely within theJulian calendar(C = +3) to find when Easter is celebrated by Orthodox churches. If it doesn't take place on the same Sunday, such a celebration currentlyoccurs 1, 4 or 5 weeks after the Gregorian date of Easter...
This will not always be so in the distant future, as the calendarsdrift apart and the Julian Pascal Full Moon is no longer a good approximation of an actual full moon. The above pattern is first broken in 2437,when Gregorian Easter occurs on March 22, whereas the Julianversion would be scheduled 6 weeks later, on May 3 (that's April 17, in the Julian calendar). TheGregorian reformwas precisely engineered to avoid this slow creep of Easter toward summertime.
Ecclesiastical Calendar :
For Christians, Fixed Holidays occur at fixed dates in the Gregorian calendar (or in the Julian calendar for Orthodox churches) whereas Moveable Holidays depend on the date of Easter (as computed above).
Lent is the period of 40 days between Ash Wednesday and Easter.
The Advent is the period from Advent Sunday to Christmas (Dec. 25).
"Ordinary times" are counted fromTrinityand end withAdvent Sunday.
TheCourts of England and Wales divide their yearinto 4 terms whose names are borrowed fromthe above ecclesiastical calendar: Hilary (celebrated onJanuary 14), Easter, Trinity and Michaelmas. So do British universities with slightly different term names,as summarized below. In 2004,Newcastledecided to drop traditional names in favor of "culturally neutral" ones (Autumn, Spring, Summer) like most American universities (Fall,Spring and Summer quarters). The traditional British academic year starts with the Michaelmas term.
Lent (Cambridge), Epiphany or Hilary Term (Oxford): January to March.
Easter Term (Trinity Term in Oxford only): From April to June.
Trinity Term (judicial system only): From June to September.
(2002-12-28) The Islamic calendar is called Hijri (orHijrah calendar).
The origin of the Muslim calendar is "1 Muharram 1 AH" (i.e., Friday, July 16, 622 CE)and predates by a few weeks the"flight from Mecca"(Hijra, Latin:Hegira) which,according to Muslim tradition, took place in September 622 CE.
The numbering of years from the date of the Hegira was introduced in AD 639 (17 AH) by the second Caliph, 'Umar ibn Al-KHaTTab (592-644). The monthly Islamic calendar itself had already been in use since AD 631 (10 AH) as the Quran prescribes a lunar calendar without embolismic months (9:36-37).
Since anIslamic year (12 lunar months)falls shorts of atropical year by almost 11 days,the Islamic calendar isn't related to the seasons. Muslim festivals simply drift backwards and returnroughly to the same seasonal point after a period of 33Islamic years(which is about a week longer than 32tropical years).
Traditionally, the beginning of a new Islamic monthis defined locally from the time when the thin crescentof the young moon actually becomesvisible again at dusk,a day or soafter the new moon. If the moon can't be observed for any reason,the new month is said to begin 30 days after the last one did.
Tabular Islamic Calendars :
Printed Islamic calendars are based onstandard arithmetic predictions of moon sightings. We present the most common ofeight extant variants.
A regular cycle of 30 years is used, which includes 19 years of 354 days and 11 years of355 days (modulo 30, thelong years are:2, 5, 7, 10, 13, 16, 18, 21, 24, 26, and 29). The average Islamic month is thus 29.53055555... days,which is about 2.9 s shorter than the actualmean synodic lunar monthof 29.530588853 days(it would take about 2428 tropical years to build up a discrepancy of a whole day).Thestandard Islamic year is tabulated below:
The average Islamic year (12 months) is 10631 / 30 = 354.36666... days. If day 0 (zero) is the first day of the above cycle of 30 Islamic years,then the number of the year to which day N belongs equals floor ((30N+k)/10631) provided k is between 26 (included) and 27 (excluded).
Using k = 26, we obtain a formula (valid for an indefinite number of 30-year Islamic cycles) giving the Islamic year Y corresponding to day N:
Y = floor ( [ 30 N + k ] / 10631 )
Conversely, the number NY corresponding to the first day of year Y is:
NY = ceiling ( [ 10631 Y - k ] / 30 )
Subtracting this quantity from the original day number (N) we obtaina number N' from 0 to 354within the Islamic year. From this number N', the Islamic month is not difficult to obtain. Conversely, we may also get the number of the first day of the month. The number within the month is obtained by subtracting that from N'.
All this can be embodied into two computer routines which convert a day numberto an Islamic date (hdate) and vice-versa (hday). For compatibility with thesimilar routinesfor the Gregorian and Julian calendars, we count days from the MJDN origin,with an offset of 451915 days. The following implementations are for the TI-92,TI-89 and Voyage 200handheld calculators.
hday({y2,m2,d2}) - hday({y1,m1,d1}) isthe number of days between two Hijri dates. hdate ( hday ( {yyyy,mm,dd} )) puts a"generalized" Hijri date in standard form. date ( hday ( {yyyy,mm,dd} )) obtains a Gregorian date from a Hijri date. hdate (day ( {yyyy,mm,dd} )) obtains a Hijri date from a Gregorian one.
CompetingVariants of the Tabular Islamic Calendar :
The aforementioned date of Friday July 16, 622 CE is by far the most common starting point of the Hegira calendar,but it may also be reckoned from Thursday July 15, 622 CE. This so-called "Thursday" calendar would be obtained with an offsetof 451916 days (instead of 451915) in both of the above routines.
There are no fewer than 30 regular intercalatory patterns which would be basedon the same 30-year period (of 10631 days) as above. Apparently, only four of those have ever been advocated(as tabulated below).
The four extant intercalatory patterns agree that years 2, 5, 13, 21 and 24 (modulo 30) are "long" years of 355 days,but disagree on some of the remaining 6 long years in the 30-year cycle of 10631 days. This amounts to different values of k for the calendrical formulas introducedabove (with k = 26) :
The 8 Extant Variants of the Tabular Islamic Calendar
Change the constant 26 (which appears 3 times) into 25,26, 29 or 1.
Use451915 for a Friday calendar, or 451916 for a Thursday calendar.
The above numbering of the 8 extant variants of the Tabular Islamic Calendar followsthe classification given byRobert Harry van Gent,who calls "civil" (c) the tabular calendarbased on the usual starting point of July 16, 622 CE and "astronomical" (a) the "Thursday" calendar based on a July 15 starting point.
- Ahmad ibn 'AbdallahHabash al-Hasibal-Marwazi (d. ca. AD 870). - Abu Arrayhan Muhammad ibn Ahmadal-Biruni (AD 973-1048). - Elias of Nisibis or Elias bar Senaya, PatriarchElias Iof Tirhan (1028-1049).
Many of the above authors do point out that such arithmetic approximationsare not a substitute for the actualobservational Islamic calendarsanctioned by religious authorities. Reingold and Dershowitz (Calendrica) also provide an "observational" [sic] Islamic calendar, based on more precise astronomicalcomputations to better predict the religious beginning of each Islamic month.
(2002-12-29)
The Jewish calendar is calledlunisolar, because it useslunar months(of either 29 or 30 days, following the phases of the Moon)while keeping the year roughly synchronized with thesolar seasons through theregularintercalation of a 13th (embolismic) month: Inleap years,this extra month (Adar I, or Adar aleph)occurs just before the monthwhenPurim is celebrated, the regular month of Adar(called Adar II, or Adar bet, in leap years). This compensates for the fact that12lunar months are nearly 11 days short of atropical year.
The Hebrew calendar is also known as theHillel calendar,because the first version of its modern rules was established(in 358-359 CE) under the authority of Hillel II,president (nasi) of the Great Sanhedrin (the highest Jewish courtof law, which existed until the rabbinic patriarchate wasabolished c. 425 CE). Before this arithmetical calendar was established, the Sanhedrin wasissuing a monthly ruling to determine the beginning of the month, based (at least in part) oneyewitness accounts of actual sightings of the thin crescent of the new moon.
A Jewish day begins in the evening. For calendrical computations, Rambam time is used which begins (0h) precisely a quarter of a day afterhigh noon (solar time) in Jerusalem. The calendar computed for Jerusalem is simply applied to other parts of the Worldaccording to local time.
The day is divided into 24 fixed hoursof 1080 "parts" (halakim) each. An interval of 10 seconds is thus 3 halakim(also spelledhalaqim,chalakim orchalaqim,the singular form ishelek,heleq,chalak orchalaq). Thishelek of 3seconds is subdivided into76 rega'im. Therega is thus 5/114 of a second, or about 43.386 ms (that unit of time isnot used in calendrical computations).
The Jewish tradition (Mesorah) gives the average duration of a lunar month to the nearest helek (there are 25920halakim in a day) :
29 days, 12 hours and 793 halakim
That value, of 765433halakim per month,is said to date back to the time of Moses in the Sinai, but it's also givenprosaically in Ptolemy'sAlmagest with an attribution toHipparchusof Rhodes (190-120 BC) using the Babylonian sexagesimal fractional notation(which originated in the Seleucid era, after 312 BC) whereby1' is 1/60 of a day, 1'' is 1/60 ofthat(1/3600 of a day) etc.
That duration remains a whole number ofhalakim although the notationis capable of a precision 500 times greater. We may infer that Ptolemy and/or Hipparchus were quoting the valuerecorded to the nearest helek by ancient astronomers.
That traditional value is less than half a second (456.4 ms) above themodernmean synodic lunar month value ofabout 29.530588853 days, which equals 29 days, 12 hours, 792.863 halakim. In Babylonian terms, this would be:
29 days 31' 50'' 7''' 11'''' ½
Years are counted since the mythical creation of the world, in 3761 BCE. Jewish year numbers are best suffixed with "AM"(Anno Mundi; year of the world).
In eachMetonic cycle of 19 years,there are 12simple years of 12 months,which may contain 353, 354 or 355 days. The remaining 7leap years have 13 months and contain 383, 384 or 385 days. Modulo 19, theleap years are 0, 3, 6, 8, 11, 14, or 17. In other words, Y is a leap year if and only if
mod ( 12 Y - 2 , 19 ) > 11
Either type of year comes in three different lengths,calleddefective (H forHaser, 353 or 383 days),regular ornormal (K forKesidra, 354 or 384 days),andperfect orcomplete (S forShalem, 355 or 385 days).
The months are traditionally numbered as shown in the table below(Esther 2:16, 3:7, 3:12),but the year number changes onRosh HaShanah("Jewish New Year"), the first day ofTishri. Formerly, the older "sacred year" started with the first day of Nissan (not Tishri),whereas the above convention applied only to thecivil year. Apparently, the former tradition faded away in the 3rd century (CE).
The names of the months are derived from the ancient Babylonian calendar, dating back to the days of the70-yearcaptivity in Babylon (c. 600 BC).
The ancient names shown in italics are obsolete.
Number
Month Name(s)
H
K
S
Season
1
Nissan, Nisan,Abib
30
March-April
2
Iyar,Ziv
29
April-May
3
Sivan
30
May-June
4
Tammuz
29
June-July
5
Av, Ab
30
July-August
6
Elul
29
August-Sept.
7
Tishri, Tishrei,Ethanim[New Year]
30
Sept.-Oct.
8
Cheshvan, Heshvan, Marheshvan,Bul
29
29
30
Oct.-Nov.
9
Kislev
29
30
30
Nov.-Dec.
10
Tevet, Tebet, Tebeth
29
Dec.-January
11
Shevat, Sebat, Shebat
30
January-Feb.
12
Adar I (leap years only)
30
Feb.-March
12 or 13
Adar (Adar II in leap years)
29
March-April
Shabbat is a time of weekly rest which lasts about25 hours, from Friday evening to Saturday evening. The beginning and end of Shabbat is a function of local solar time. Shabbat begins on Friday evening, 18 minutes before sunset (Sheqiya) itself defined as the timewhen the center of the Sun is 50" (i.e., an angle of 5°/6) below the horizon. Shabbat ends Saturday evening a few minutesafter nightfall (tzeit hacokhavim, the birth of stars) at a time often said to be when 3 stars should become visible (in clear wheather). This is computed as the time when the center of the Sun is 8.5 ° below the horizon. The same rules are used to define the beginning and the end ofany Jewish festival (Yom Tov).
The first day of any Jewish month is a minor festival (Rosh Hodesh) except, of course, for the beginningof Tishri (Rosh HaShanah) which marks the beginning of the Jewish year. Rosh HaShanah is a strongly observed 2-day celebration. The 3 "pilgrimage festivals" (Sukkot, Passover andShavuot) were occasions for masspilgrimages to theTemple in Jerusalem before its destruction (in 70 CE). Passover is second only to Yom Kippur in traditional observance.
The above table is for celebration of Jewish festivals in Israel. The tradition is that most Jewish holidays are extended by one additionalday for the JewishDiaspora outside Israel. Exceptions includeRosh HaShanah (2 days for everybody) and Yom Kippur (one day for everybody).
The reason for this general rule dates back to the times when the Jewish calendarwas not yet arithmetical (before 359 CE). Far from Jerusalem, the calendrical decision of the Sanhedrin for the current monthmight not be known early enough to allow the celebrations to take place on the"correct" day, so they were held for two days instead. ForRosh Hashanah, which occurs at the very beginning of the month,the Sanhedrin's decision could not reachanybody in time(as messengers wouldn't be dispatched during holidays)so,everybody celebrates for two days. On the other hand, the observance of Yom Kippur fortwo days would be too much of a hardship... So, there's only one "Day of Atonement" for everybody!
Jewish Calendrical Formulas : (2007-06-24)
First, we treat Jewish embolismic months with the method we usedforIslamic embolismic days. Namely, we observe that thepattern of Jewish leap years is such that months can benumbered continously (starting with number 0 for Tishriof Jewish Year 0) so that the year Y to which month X belongs is simply:
Y = floor ( (19 X + k) / 235 ) where 5 ≤ k < 6
Conversely, the number X of the first month (Tishri) of year Y is the least X which verifies the above equation, namely:
X = ceiling ( (235 Y - k) / 19 ) [let's use k=5]
The precise key to the Jewish calendar is Molad Tishri,namely the exact time M of the new moon which occurs on Rosh HaShanah (or slightly before, as explained below) expressed in days and fraction of a dayto the nearesthelek (in Rambam time at Jerusalem). A new moon (molad) is defined as the timewhen the apparent longitudes of the Moon and the Sun coincide.
The Hillel calendar is based on an arithmetical approximation to M, whereby consecutive new moons are exactly 765433/25920 days apart ( 29d 12h 793 ).
The periodic pattern must be anchored at some reference Molad. In practice,Tracey R. Rich recommendsone of the following [equivalent] starting points:
Y = 5558 : X = 68744. (1797-09-21) + 12487 / 25920 (= 11h 607)
Y = 5661 : X = 70018. (1900-09-24) + 11889 / 25920 (= 11h 9)
Y = 5759 : X = 71230. (1998-09-21) + 13965 / 25920 (= 12h 1005)
So calibrated, the molad corresponding to month number X is found to be:
M =
765433 X + 8255
25920
For X=13, this does give a value M = 384 + 5604/25920 (namely, 5h 204) for Molad Tishri of Year 1, known as Molad Tohu (the molad of creation). For X=25 (Molad Tishri of Year 2) we obtain a whole number of hours (14h). This "coincidence" would normally happen only once in 1080 years,butwe're toldthat the calendar was actually cooked to make it so... Since Tishri of Year 2 used to be counted in the first year AM (when the sacred year started with Nissan) it was thought that the "first Molad Tishri"ought to be a roundnumber.
Our previous calendrical formula turns the above expressioninto a formula giving the Molad Tishri of Jewish year Y (Anno Mundi) namely:
M =
765433ceiling ( (235 Y - 5) / 19 ) + 8255
25920
In our straight count of days, floor (M) is the day number for Rosh HaShanah, except when otherwise specified by one of 4 so-called "rules of postponement" (dehhioth,dehioth,dehiyyotordechiyot is the plural ofdehhiah ordechiyah).
Dechiyah #1 :Molad Zakein ("Old Molad") Molad Tishri must occur before noon (18h,Rambam time) or else Rosh HaShanah is postponed to the next day.
The traditional origin for that rule was that the thin crescent ofthe young moon had to be observable at sunset onRosh Hashanah.
Dechiyah #2 :Lo A"DU Rosh ("No Beginning on Alef-Dalet-Vav") Rosh HaShanah is postponed tothe following day if it would otherwise fall on Sunday, Wednesday or Friday.
The word A"DU is amnemonic forthe hebrew letters alef, dalet and vav, whose numerical values (1, 4 and 6)correspond to Sunday, Wednesday and Friday. Rosh HaShanah is only allowed to fall on a Monday, a Tuesday, a Thursdayor a Saturday; one of the "4 gates" through which the new year must be entered.
Thatdechiyah is designed to avoid certain days of the weekfor some festivals. For example, the seventh day of the Feast of Tabernacles (Hoshana Rabbah) shouldn't fall on a Saturday since the ceremonyof "beating the willow twigs" involves work not permitted onShabbat. That rules out Sunday forRosh HaShanah, 20 days before. Similarly, the Day of Atonement (Yom Kippur) would fall immediately before or after Shabbat ifRosh HaShanah was allowedto occur on a Wednesday or a Friday...
Dechiyah #3 :Gatarad ("Tuesday, 9h, 204") Gatarad is a Hebrew mnemonic (Gimel-Teit-Reish-Dalet) forwhat this rule states, namely that Rosh HaShanah is to be postponedwhen Molad Tishri of a 12-month year falls on a Tuesday (Gimel = 3 = Tuesday) on or after 9 hours (Teit = 9) and 204halakim (Reish = 200, Dalet = 4).
Dechiyah #4 :Betutkafot ("Monday, 15h, 589") Again,Betutkafot is a Hebrew mnemonic for that rule, which statesthat Rosh HaShanah is to be postponedif Molad Tishri of a year following a 13-month year occurs on Monday (Beit = 2) on or after 15 hour (Teit-Vav = 9+6 = 15) and 589halakim(Tav-Qof-Fe-Teit = 400+100+80+9 = 589).
(2002-12-22)
Zoroastrianism is a monotheistbelief system basedon righteousness (good thoughts, good words, good deeds). When it was first preached in Persia byZarathustra (c.628-c.551 BC),it was opposed to the prevalent cult of Mithras(which demanded sacrifices andadvocated the consumption of narcotics and/or intoxicating beverages,then known asHaoma). Some scholars have considered Zoroastrianism to be a precursor of Christianity. Although Jews claim him as one of their own,it is generally believed thatZarathustra (orZoroaster)was Indo-Iranian (Aryan). He was most probably born in Mazar-I-Sharif (which is now in northern Afghanistan)and was "the son of Pourushaspa, of the Spitaman family". Zoroaster is said to have given his very first teaching just after being born,in the form of an unusual laughter,telling believers thathuman life is worth living...
Zoroastrianism is still practiced by about 18 000 people in Iran,chiefly in Shiraz. It is thriving in India (chiefly around Bombay)and Pakistan (chiefly in Karachi) amongParsis orParsees,literally "Persians" whose ancestors fled Persia in the wake of the Arab conquest,and subsequent Islamization ( 7th century AD). The total number of Zoroastrians is currently estimated to be around 140 000.
The Zoroastrian calendar is based onmonthsof 30 days and has the same basic structure asthe ancientEgyptian calendar(and/or the modernCoptic calendar),including 5 extra days after the 12th month, thegatha days.
In the year 1006 CE, the first day of the Zoroastrian year (Noruz)occupied once again its original position at thevernal equinox. (Incidentally, this would imply that the Zoroastrian calendar originatedin 500 BC or so.) It was then decided to intercalate a whole month every 120 years,to make the long-term average of the Zoroastrian year equal to 365¼ days,and avoidcalendar creep with the exact same accuracy as theJulian calendar (in the long run, at least). This unusual intercalation scheme may have been chosen for religious reasons,which made it difficult to have anything but 5gatha days at the endofevery year.
However, this rule was remembered onlyonce,about 120 years later, and only by the Parseesof India, whose calendar (now calledShahanshahi orShenshai)has been 30 days late ever since,relative to the original calendar (Qadimi orKadmi)still kept by Iranian Zoroastrianists. (Curiously, the discrepancy is said to have gone unnoticed until 1720.) Both theShenshai andKadmi calendars are thus effectively variants of theEgyptian calendar, featuring a constant year of 365 days, without any intercalations.
TheFasli calendar, on the other hand,is a modern Zoroastrian calendar, designed in 1906,in strict alignment with theGregorian calendar. The Fasli yearalways starts on March 21(the nominal Gregorian vernal equinox)and it consists of 12 months of 30 daysand a 13th "month" ofeither 5 or 6 days.
Zoroastrianism was made the official religion of Persia by Shapur I,who reigned from 241 to 272,as the second king of the Sassanian Dynasty (AD 224-641). Regnal years were then used with the Zoroastrian calendar. ThePersian empirewas conquered by the Arabsafter the battle of Nehavand in 641 CE, about 10 years after the coronation ofthe last of the Sassanids,Yazdegird III [also known (?) asYazdegerd,Yazdazard,orYazdegar Sheheryar].
The era of this last Zoroastrian king is abbreviated YZ and has been continuedup to the present time: Year 1 YZ was 631 CE.
(2002-12-22)
This calendar would be totally obsolete,if it was not for the fact that astrologers still use it. In the last column of the table below is the Gregorian correspondencemost often used by "modern" astrologers...
Zodiacal Sign
Persian Month
First Day
Aries
Farvardin
March 21
Taurus
Ordibehesht
April 20
Gemini
Khordad
May 21
Cancer
Tir
June 22
Leo
Mordad
July 23
Virgo
Shahrivar
August 23
Libra
Mihr
September 23
Scorpio
Aban
October 23+
Sagittarius
Azar
November 22
Capricorn
Day
December 22
Aquarius
Bahman
January 20
Pisces
Esphand
February 19
About 2000 years ago, when this calendar was presumably devised,the eponymousconstellation indicated the correct position of theSun for the month corresponding to a givenzodiacal sign. Because of the so-calledprecession of equinoxes, this is nolonger true at the present epoch.
In this context, it's important to maintain a clear distinction between 3 relatedconcepts that are often confused:signs,constellations andhouses: Zodiacalsigns are simply names given to months within the regular calendar year(synchronized with thetropical year) as tabulated above. On the other hand, it's clear that 12 constellations were once defined which,unlike modern constellations, dividedevenly theecliptic(the apparent path of the Sun against the background of "fixed stars"). Suchtraditional constellations are best referred to as "houses". We are not aware of any precisehistorical definition of the exact boundariesbetweenhouses (if you know better,let us know). The 88 constellations of theentire celestial spheredo have precise modern definitions,but these are virtually irrelevant with respect tohouses: There are 13 (!) modern zodiacal constellationswith uneven shares of the ecliptic. The 13th zodiacal constellation is Ophiuchus, the Serpent Bearer, which spans the ecliptic between Scorpio and Sagittarius.
) traditionally used in various diagrams. The precession of the Earth's axis of rotation makes thisvernal pointgo a full circle around the Zodiac in about 26000 years. Please, donot believe themany sources which tell you that this period isprecisely 25920 years. This would be the caseonly if theaverage yearly precession wasexactly 50" (1°/ 72),because25920 = 360 72. Thelatestdata available to us at this writing(MHB2000 nutation model) give an average yearly precession of 50.28792(2)",corresponding to a precession period of25771.597(11)years [about 25772.126(11)Gregorian years].
Some ancient Babylonian astronomers must have known about this,but Hipparchus of Rhodes (190-120 BC)is credited for the first precise description of the phenomenon,which Copernicus would correctly attribute, in 1543, to thechanging directionof the Earth's axis of rotation. The actual dynamicalreason for this precessionwas given by Isaac Newton in 1687: The Earth "bulges at the Equator", and this oblateness implies that a distant body,like the Moon or the Sun, exerts a nonzero gravitationaltorque on the Earth,(except, ideally, in the rare symmetrical case when the Earth axis isprecisely perpendicular to the direction of the body in question;for the Sun, this would be the configuration at either equinox). This torque is always "trying" to reduce the tiltof the axis with respect to the direction of the body. However, the Earth reacts like any rotating body would: It changes its rotational axis toward the direction of the applied torque(the torque is avector perpendicular to both the axis of rotation and thedirection to the influencing body). This causes a precession of the axis,instead of the naively expected reduction in tilt.
Traditionally, the time when thevernal point enters a newhouse marksthe dawning of a new "age"(like theAge of Aquarius)which lasts for about 2148 years. A poor definition of the traditional Zodiacalhousestranslates into a fuzzy beginning for each suchage(a misalignment of 1° corresponds to an error of about 72 years).
(2009-08-17) The Persian year (Anno Persico or Anno Persarum).
Like the lunarIslamic calendar, the current Iraniancalendar counts its years from the flight from Mecca (July 622). However, it is strictly a solar calendar (the Iranian year begins at the Spring equinox) whose offset with the Gregorian year remains constant at 622 (it's only 621 between the Gregorian New Year, January 1,and the Persian New Year,Nowruz).
In March 1925, the Persian parliament enacted calendrical rules which revivedthe names of the ancient Persian names of the months without giving themtheir traditionalzodiacal duration. Instead, the first 6 months (Farvardin to Shahrivar) have 31 days, the following 5 months (Mehr to Bahman) have 30 days and the last month (Esphand) has either 29 days or 30 days.
The SH calendar year begins at midnight between the two solar noons (on the Tehran standard meridian at 51.5°E) which bracket the vernal equinox.
(2018-09-11) In September of year N, Enkutatash begins Ethiopian year N-7.
In the current Gregorian calaendar, Enkutatash occurs on September 11, except when N+1 is a leap year (in which case, it occurs on September 12). So, September 11, 2001 was the first day of Ethiopian Year 1994.
This calendar is still the main calendar for civil life in Ethiopia and is also observed by Erhiopiansestablished abroad. It's used to set the lithurgical calendar of several related religious communities:
Eritrean Orthodox Tewahedo Church.
Ethiopian Orthodox Tewahedo Church.
Eastern Catholic Churches.
Coptic Orthodox Church of Alexandria.
The Enkutatash celebration in September marks the beginning of a new Ethiopian yearwhose number is 7 years behind the current Gregorian year. For most of the year (between the Gregorian New Year and Enkutatash) the two counts are 8 years apart.
The Ethiopian months are identical to the Copt months.
(2002-12-29)
The Mayan civil year, thehaab consisted of 18 "months" (uinals), of20 days each, and 5 extra days (which were believed to be unlucky ones),for the same total of 365 days as theEgyptian year. The Maya knew that the tropical year was closerto 365¼ days, but they chose to keep a constant number of daysineach year, and shunned intercalary days(just like the ancient Egyptians).
The Mayan sacred year, thetzolkin, was a cycle of 260 days(the combination of aregular cycle of 13 numbers and of a regular cycle of 20 different signs).
When both calendars are used concurrently, a day is uniquely identified withinany period of 18980 days known as a MayanCalendar Round (18980 is the lowest common multiple of 365 and 260; it's equal to 52haabsor 73tzolkins).
Thesynodic period of Venusis about583.9214 days. The Maya estimated it to be 584 days,which happens to be 8/5 of theirhaab of 365 days. Therefore, twice the aboveCalendar Round is a multiple of theMayan value of the Venus period. This period of 37960 days is the MayanVenus Round,which is equal to 104 haabs, or 146 tzolkins,or [roughly] 65 synodic periods of Venus.
The Long Count
In addition to the above, the Maya used a so-calledLong Countto keep track of their historical events. This was simply the number of days elapsed since the Mayanmythical creation of the World, using the following 5 units:
Abaktun is 144000 days (20 katuns).
Akatun is 72000 days (20 tuns).
Atun is 360 days (18 uinals).
Auinal is 20 days (20 kins).
Akin is one day.
Each Mayan vigesimal "digit" could represent a number from 0 to 19,and aLong Count was expressed as a string of 5 such digits,usually transliterated as 5 numbers separated by dots( baktuns.katuns.tuns.uinals.kins ).
It has been argued that the Maya considered a "Great Cycle" to be13baktuns, or 1872000 days(exactly 7200 tzolkins, or over 5125tropical years). 13baktuns after its mythical beginning,the Mayan World comes to an end of sorts: The Mayan tradition would simply reset the long count to 0.0.0.0.0when it reaches 13.0.0.0.0, on December 21, 2012 CE.
In other words, it seems that the Maya would only give the leading digit of a Long Countmodulo 13... We prefer to ignore that line of thought and advocate the use of leading elementsbeyond 13 for the Long Count as needed, in thenear future.
The "5 digit" Long Count system goes beyond 13baktuns witout any difficulty, at leastuntil 19.19.19.17.19 (Thursday, October 12, 4772). Thisgives scholars a couple of millenia to decide what's to be done at that pointwith the calendrical legacy from the Maya. The next day (Friday the 13th ;-) would require some innovation, likea sixth "digit" as a coefficient of a counting unit larger than the baktun. The Maya themselves devised no less than three such units: Thepiktun,kinchiltun andalautun,worth respectively 20, 400 and 8000baktuns. An extended 8-digit Long Count based on those 3 additional units would spanmore than 60 million future years... By that time, the Sun will still be just as bright as today,but the human species will (most probably) be longgone.
Mayan Calendrical Formulas :
The regularity of the Mayan Long Count makes calendrical formulas trivial... On a TI-92, TI-89 orVoyage 200handheldcalculator, the function mayaday which takes a Long Count(as a list of 5 numbers) and returns the correspondingMJDNcan be given the following one-line definition:
Conversely, the screenshot at right shows how todefine the function mayadate which takes an MJDN and returnsthe corresponding Long Count, as a list of 5 numbers.
Use either function with thefunctionsday or date to turn a Long Count into a Gregorian date, or vice-versa.
(2003-01-01)
TheChinese calendar is anastronomical calendar,which explicitly depends onactual observations and/or delicatepredictions of astronomical events.
It's currently used by about one fourth of the World's population(at least for traditional festivals). Its modern form dates back to 1645 and is due toFather Schall(Johann Adam Schall von Bell, 1591-1666),a catholic missionary who was summoned to Peking in 1630after the death of Father Terrentius (John Schreck)to take over the task of reforming the traditional Chinese calendar.
(2003-01-12)
The latest periods in the traditional Japanese calendar system arecalled Edo, Meiji, Taisho, Showa and Heisei. Starting with Meiji (1868-1912 CE), the period changeswhen the Emperor passes away, and years are numbered from the beginning of the period. In the Edo period (1603-1868 CE),the Japanese calendar was based on its Chinese counterpart,with significant discrepancies due to the different longitudes usedfor critical observations. Years were then named using the Chinese 12-year cycle (Rat, Ox, Tiger, Hare, Dragon, Snake, Horse, Sheep, Monkey, Bird, Dog, Pig). This tradition remainspopular today,although Japan adopted the Gregorian calendar in1873.
There was also a so-calledKoki calendar based on a continuous count ofyears from the founding of the Japanese dynasty ofEmperor Jimmu Tenno, in 660 BC. The last two digits of this count were once used by the Japanese militaryfor new or revised equipment. This iswhy the "Zero" was so named,since this famous WW II fighter plane ( Mitsubishi A6M )appeared in 1940, Koki year 2600.
(2003-01-03)
TheNational Calendar of India was last reformed in 1957: Itsleap years coincide with those of theGregorian calendar,but years begin at the vernal equinox and are counted from theSaka Era(the spring equinox of 79 CE).
(2003-01-10)
There areintercalation patterns of leap years which could make theGregorian calendar even more accurate in the verylong term, while being consistent with the Gregorian rules for dates of thepast (back to 1582 CE) and the near future. However, proposals for suchmillenarian rules must be carefullyevaluated in the framework presented here.
The Gregorianyear is currently the best calendar approximation there isto thetropical year (which governs our seasons). In a Gregorian cycle of 400 years, there are 97leap yearsof 366 days and 303 regular years of 365 days, which makes themean Gregorian year equal to 365.2425 days.
Asolar calendar should be engineered to makethe long-term ratio of the number of days to the number of elapsed calendar years(365.2425 for the Gregorian calendar) as close aspossible to the observed number of days in a tropical year,which is slightly less than 365.2422. At first, it would seem easy to reform the Gregorian calendar (by droppinga leap year once in a great while) in order to make the mean calendar yearcloser to this target number.
However, all such efforts may be misguided, since the above target is amoving one(mainly because tidal braking keeps making our days longer). To put it bluntly, amillenarian rule for leap yearscould be all but obsolete before coming into play,as long as it remains based only on thecurrent number of days in a tropical year... Let's see what the actual numbers are:
The definition of theephemeris seconds makestheinstantaneous value of the tropical year"at epoch 1900.0" exactly equal to 31556925.9747ephemeris seconds. Since the definition of the modernSI second was precisely engineeredto makeit virtually indistinguishable from anephemeris second,we may as well take the above as the exact durationof the 1900.0 tropical year, in SI seconds.
There areexactly86400ephemeris seconds in anephemeris day (by definition of the latter),but thisephemeris day is an abstract unit of time,which is irrelevant to the calendar structure. What we need is a precise estimate of the 1900.0 duration of amean solar day,because actualsolar days is what calendars are meant to count. In fact, forhistorical reasons,themean solar day was precisely equal to 86400 seconds around 1820 or 1826,and has been increasing at a rate of roughly 2 ms per century ever since. In this context, a "second" (s) is anSI second,a unit now defined in atomic terms,which is virtually indistinguishable from theephemeris second(it'snot thesolar second,which is defined as 1/86400 of themean solar day,whose variable duration we are evaluating). All told, the mean solar day of 1900.0 would have been about 86400.0016 s.
(2003-02-20) 
after the battle of Nehavand in 641 CE, about 10 years after the coronation ofthe last of the Sassanids,Yazdegird III [also known (?) asYazdegerd,Yazdazard,orYazdegar Sheheryar].
