Julian Day Numbers for dates on the Gregorian and Julian Calendars

On the gangway the [sailing] master lowered his sextant, walked aft to Mr Hervey and said, 'Twelve o'clock, sir: fifty-eight minutes [of latitude] north.'

The first lieutenant turned to [Captain] Jack [Aubrey], took off his hat and said, 'Twelve o'clock, sir, if you please, and fifty-eight minutes north.'

Jack turned to the officer of the watch and said, 'Mr Nicholls, make it twelve.'

The officer of the watch called out to the mate of the watch, 'Make it twelve.'

The mate of the watch said to the quartermaster, 'Strike eight bells'; the quartermaster roared at the Marine sentry, 'Turn the glass and strike the bell!'

Patrick O'Brian, H.M.S. Surprise, W.W. Norton & Company, 1973, p.118

The virtue of the Astronomical Day was that the observations of a single night could be unambiguously ascribed to a single calendar day. Otherwise, the astronomer must use a double date, e.g. the night of 14-15 January, to avoid ambiguity -- exactly the practice that is now required. The Nautical Day might take advantage of the astronomical practice, as navigators shoot the stars to fix a ship's location, but the convention had more to do with the ritual of the Noon Sighting, which, at least in the days of sail, was the moment that the ship's clocks whould be reset to Local Apparent Time and a new calendar date recorded in the ship's log. The Noon Sighting called for some considerable skill with the sextant, since the sun appears to "hang" at its maximum elevation for a few moments, and the navigator must exercise his experience and judgment to determine when the Meridian was reached. As commemorated by Patrick O'Brian above, the Sailing Master was the one to perform this determination, which is then communicated through no less than six levels of command before the hour glass is turned [note].

The device of Julian Day Numbers was introduced by the polymath Joseph Justus Scaliger (1540-1609). He named the "Julian Period," not after the Julian Calendar or even directly after Julius Caesar, but in memory of his father, who happened to be named Julius Caesar Scaliger (1484-1558). The relation of father and son sounds like that between James Mill (1773-1836) and John Stuart Mill (1806-1873), who were among the principal exponents of Utilitarianism. Where John Stuart Mill was being taught Greek at three, Scaliger's father required him as a child to give a short speech in Latin every day. The elder Scaliger, however, for some reason forbade the study of Greek, which the son took up on the father's death, determining that "those who do not know Greek know nothing at all." As the younger Mill seems to have been plagued by his father's memory the rest of his life, Scaliger was also troubled, suffering from strange dreams and insomia and sometimes forgetting to eat. He thought that he had once encountered the Devil. But Scaliger was also one of Europe's first Arabists, having studied with Guillaume Postel (1510-1581), himself a very eccentric scholar, ruled insane by the Inquisition, who obtained the first Chair of Arabic at the Collège de France in 1539. Scaliger was invited to teach Arabic at Leiden in 1592. He hated lecturing but was instrumental in establishing a Chair of Arabic at the University in 1599. One of Scaliger's students, Thomas van Erpe, or Erpenius (1584-1625), produced the first modern grammar of Arabic, the Grammatica Arabica (1613).

Julian Day Numbers effectively ended the use of the Egyptian calendar and the Era of Nabonassar for astronomical purposes, as had been introduced by Claudius Ptolemy (c.100-c.170 AD). Scaliger picked 4713 BC because it was the first year on a number of different calendar cycles and was earlier than any possible historical dates that he knew of. "Julian Day Numbers" may refer to integer numbers corresponding to whole days, while the "Julian Date" may mean an integer plus decimal that brings the Julian count down to precise parts of a day.

To convert dates from the Julian or Gregorian calendars to Julian Day Numbers, first the year of the Julian Period must be determined. An AD year is simply added to 4713. Thus, 1997 yields 6710. Years BC must be expressed as negatives of AD years. 747 BC corresponds to -746 AD (since 1 BC = 0 AD) = 3967. But the year of the Julian Period is awkward for purposes of calculation. If 4713 BC is set to Year 0 instead of Year 1, this is more convenient. The "Scaliger Year" is thus one less than the year of the Julian Period, and may be obtained by adding 4712 instead of 4713 to the year of the AD era.

For the year 1997, the Scaliger Year is 6709. Also for purposes of calculation, the calendar year is taken to begin on 1 March instead of January 1. January and February 1997 are thus reckoned to be in 1996 (6708). The Scaliger Year is then divided by 4. 6709/4 = 1677 with a remainder of 1. 1677 is the number of four years cycles in the Julian Calendar and 1 is the year (0-3) in the current cycle.

1677 is then multipled by the number of days in four Julian years, 1461, and 1 is multiplied by the number of days in a common Julian year, 365. 1677 x 1461 + 1 x 365 = 2,450,462

For the month,

The Months of the Julian and Gregorian Calendars
Month	Day	Month	Day	Month	Day
3. March	0	7. July	122	11. November	245
4. April	31	8. August	153	12. December	275
5. May	61	9. September	184	1. January	306
6. June	92	10. October	214	2. February	337

a day number must be found on the following table. If we are in the month of September, the corresponding day number is 184

With the day of the month, let's say 21, the number for the month (184) is added to the previous sum: 2,450,462 + 184 + 21 = 2,450,667

Two things must be done to 2,450,667 before we are finished. First, since we are using 1 March as the beginning of the year, the number of days elapsed from 1 January 4713 to 0 March 4713 must added. That is 59.

Century	Correction	Century	Correction
1582	-10	1800	-12
1600		1900	-13
1700	-11	2000

Second, if we are using the Gregorian Calendar, a correction must be added to reduce the date on the Julian calendar to that on the Gregorian. For the period of the use of the Gregorian calendar, the corrections are listed in the following table. The corrections are listed as negative numbers so that ALL numbers may be added to produce the Julian Day Number.

Thus, the Julian Day Number for 21 September 1997 on the Gregorian Calendar is 59 + 2,450,667 + -13 = 2,450,713. The Julian Date of the corresponding Civil Day, beginning the previous midnight, may be obtained by subtracting 0.5 from the Julian Day Number: 2,450,667 - 0.5 = 2,450,712.5

Julian Day Numbers or Julian Dates are commonly stated in "myriads," i.e. 10,000s, instead of thousands. Thus, JD 2,450,713 may be seen expressed as JD 2450 713 or as JD 245 0713.

Converting a Julian Day Number to Julian or Gregorian dates proceeds in reverse from the procedure above. For example, JD 2,450,766. First 59 is subtracted from this = 2,450,707.

Century	Correction	Century	Correction
1582	-10	1800	-12
1600		1900	-13
1700	-11	2000

Then the entire number is to be divided by 1461. This yields a Quotient of 1677 and a Remainder of 610. The Remainder is then to be divided by 365, with the quotient and remainder noted, in this case 1 and 245. The Quotient (1677) is then multiplied by 4 and added to the quotient (1) = 6709. This is the Scaliger Year, unless the month turns out to be January or February. The AD year may be obtained by subtracting 4712 = 1997. If we wish a date on the Gregorian Calendar, the Gregorian Correction should be Subtracted from the remainder of the above procedure (245). The Gregorian correction for 1997 is -13: 245 - -13 = 245 + 13= 258.

The Months of the Julian and Gregorian Calendars
Month	Day	Month	Day	Month	Day
3. March	0	7. July	122	11. November	245
4. April	31	8. August	153	12. December	275
5. May	61	9. September	184	1. January	306
6. June	92	10. October	214	2. February	337

The table of months is then searched to find a day number smaller than the remainder of the procedure above (258). This proves to be 245, for November.

Substracting 245 from 258 gives us the day of the month = 13. JD 2,450,766 is thus Noon 13 November 1997 on the Gregorian Calendar. If we had found January or February in the month table, we would reckon the date as in the following year (i.e. 1998)

Dates on the Julian Calendar are obtained simply by ignoring the factor of the Gregorian Correction.

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This full elaborate ritual of passing down commands can still be found on modern warships, particularly when the Captain is on the bridge but has assumed control of neither the Deck nor the Conn, i.e. he only gives orders to the Officer of the Watch. In the example from Patrick O'Brian, it is noteworthy that the First Lieutenant, who has formally received the report of the Sailing Master and communicated it to the Captain, is not the Officer of the Watch. The Captain then issues his order to the latter.

The full ritual associated with the chain of command is rarely to never seen in movies, where it would waste time and possibily begin to look ridiculous, especially to people familiar with the representation of more casual goings on, as in Star Trek. It's purpose, of course, is to prevent confusion.

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Islâmic Dates
with Julian Day Numbers

The Months of the Moslem Calendar
Months	Days	Months	Days
1. al-Muh.arram	30	7. Rajab	30
2. S.afar	29	8. Sha'baan	29
3. Rabii'u l'awwal	30	9. Ramad.aan	30
4. Rabii'u ttaanii	29	10. Shawwaal	29
5. Jumaadaa l'uulaa	30	11. Duu l-Qa'dah	30
6. Jumaadaa l'aaxirah	29	12. Duu l-H.ijjah	29/30

The calendar of Islâm is the traditional calendar that was used in Arabia as reformed by the Prophet Muh.ammad. Since, like many ancient luni-solar calendars, which depending on the occasional intercalation of extra months, the received calendar was badly out of step with the seasons, Muh.ammad simply cut that connection altogether by abolishing the intercalation of months.

This step might seem like a strange "reform," but it ended up suiting the ritual requirements of the calendar quite nicely. With a common year of only 354 days, the calendar runs fast against the solar year; and dates move entirely through the cycle of the seasons every 32 or 33 years. This means that the months of , Ramad.ân, when Muslims are supposed to Fast the daylight hours, and , Duu l-H.ijjah, when the Pilgrimage may be undertaken to Mecca, are not fixed in particular seasons. While one might prefer that Ramad.ân always occurred in winter, when the days are short, it must be remembered that the seasons are reversed in the Southern Hemisphere, which means that Ramad.ân would correspondingly always be in summer in South Africa or Australia. With Ramad.ân moving forward about ten days every year, the times of greater hardship are shared by all (although summer in the Arctic or Antarctic would be particularly demanding -- something that might occasion the allowed postponement of the Fast).

Year	Day	Year	Day	Year	Day
00	0	10*	3543	20	7087
01	354	11	3898	21*	7441
02*	708	12	4252	22	7796
03	1063	13*	4606	23	8150
04	1417	14	4961	24*	8504
05*	1771	15	5315	25	8859
06	2126	16*	5669	26*	9213
07*	2480	17	5315	27	9568
08	2835	18*	6378	28	9922
09	3189	19	6733	29*	10276
				30	10631
* = leap years

The H.ijjrah or Annô Hegirae (AH) Era, is benchmarked to the day in 622 AD that Muh.ammad fled Mecca to take up the secular rule of Medina. The full sophisticated mechanism of the calendar, with its cycle of intercalation of 11 days every 30 years, is the product of a later generation and thus of the flowering of Islâmic philosophy and science in Baghdâd under the Abbasid Caliphs.

To convert an Islâmic or Annô Hegirae date to Julian Day Numbers, e.g. 7 Duu l-Qa'dah 1432 AH, first divide the year by 30, noting the Quotient and the Remainder of the division. With 1432, this yields a Quoteint of 47 and a Remainder of 22. Multiply the Quotient by 10631, the number of days in the Islâmic 30-year calendar cycle. This yields 499,657. With the Remainder, which is the year within the cycle, search the table at left for the year number and note the corresponding number of days. Thus, year 22 corresponds to 7796 days.

Add 7796 to the previous product: 499,657 + 7796 = 507,453. Now, search the following table for the month and note the corresponding day. For Duu l-Qa'dah, the day is 295. Add the day number (295) for the month (Duu l-Qa'dah) and the day of the month (7) to the previous product: 507,453 + 295 + 7 = 507,755.

The Months of the Moslem Calendar
Month	Day	Month	Day
1. al-Muh.arram	0	7. Rajab	177
2. S.afar	30	8. Sha'baan	207
3. Rabii'u l'awwal	59	9. Ramad.aan	236
4. Rabii'u ttaanii	89	10. Shawwaal	266
5. Jumaadaa l-'uulaa	118	11. Duu l-Qa'dah	295
6. Jumaadaa l-'aaxirah	148	12. Duu l-H.ijjah	325

The Julian Date or Julian Day Number is then that number plus the Islâmic Benchmark number, which is 1948,085 -- the Julian Date of 0 Muh.arram 0 AH (cf. discussion of zero in calendars). Thus 502,681 + 1948,085 = JD 2455,840. This corresponds to 5 October 2011 on the Gregorian calendar.

Since the Julian Day begins at Noon (the pre-1925 convention of the Astronomical or Nautical Day), the Day Number for the corresponding Civil Day may be obtained by substracting 0.5 = 2455,839.5. This corresponds to 00:00h, midnight, 5 October 2011 on the Gregorian calendar. The Islâmic Calendar Day itself begins at the previous Sunset whose civil time will depend on the time of year, the latitude, and the time zone. This is not easily represented with fractional Day Numbers (averaging N.25), so the integer day, JD 2455,840, is best used in "tabular" fashion, to represent 7 Duu l-Qa'dah 1432 AH in its entirety.

As noted, the calendar intercalates a leap day 11 times in 30 years. This is added to the very end of the year, to the month of Duu l-H.ijjah, and thus has no effect on the position of any other day within the year. This avoids the complications that attend intercalations in the Gregorian and Jewish years.

Year	Day	Year	Day	Year	Day
00	0	10*	3543	20	7087
01	354	11	3898	21*	7441
02*	708	12	4252	22	7796
03	1063	13*	4606	23	8150
04	1417	14	4961	24*	8504
05*	1771	15	5315	25	8859
06	2126	16*	5669	26*	9213
07*	2480	17	5315	27	9568
08	2835	18*	6378	28	9922
09	3189	19	6733	29*	10276
				30	10631
* = leap years

Converting the Julian Day Number to the Annô Hegirae date repeats the above process in reverse. Given the Julian Date 2450,713, or 21 September 1997, substract the Islâmic Benchmark number, 2450,713 - 1948,085 = 502,628.

This number must be divided by 10631 with Quotient and Remainder noted. The Quotient is 47 and the Remainder 2971. The year table is then examined for a day number smaller than the Remainder, 2971.

In this case, day number 2835, with year 8, is smaller than 2971. The Quotient of the previous division, 47, is then multiplied by 30 and added to the year number (8), 47 x 30 = 1410, 1410 + 8 = 1418. This is the Year of the Annô Hegirae era corresponding to our Julian Day 2450,713.

Then day number 2835 is subtracted from the Remainder of the previous division (2971), yielding 136. The Month table is then examined for a number smaller than 136. This turns out to be 118, the number corresponding to the month Jumaadaa l'uulaa.

The month thus will be Jumaadaa l'uulaa, and the day of the month will just be the month's day number (118) substracted from the previous difference (136). This yields 18. The Annô Hegirae era date corresponding to our Julian Day 2450,713 is 18 Jumaadaa l'uulaa 1418 AH.

The Jewish and Moslem Calendars with the Era of Nabonassar

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