Julian Day Numbers for dates on the Gregorian and Julian Calendars

On the gangway the 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

Julian Day Numbers, or the Julian Date (JD), is the absolute count of days that have elapsed since Noon 1 January 4713 BC on the Julian Calendar, or on what may more strictly be called the Julian "Proleptic" Calendar, meaning the Julian Calendar as applied to an era prior to its actual use. That use began with Julius Caesar in 46 BC. That the Julian Day begins at Noon reflects the practice of the Astronomical or Nautical Day before 1925 -- which made the noon sighting at sea of particular importance, as commemorated by Patrick O'Brian above. The Civil Day of the same calendar date began the Midnight before the Astronomical or Nautical Day.

The device of Julian Day Number was introduced by Joseph Scalinger (1540-1609). It 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). Scalinger 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 "Scalinger 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 Scalinger 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 Scalinger 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 Scalinger 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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Islâmic Dates
with Julian Day Numbers

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

To convert an Islâmic or Annô Hegirae date to Julian Day Numbers, e.g. 12 Rajab 1418, first divide the year by 30, noting the Quotient and the Remainder of the division. With 1418, this yields a Quoteint of 47 and a Remainder of 8. 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 following table for the year number and note the corresponding number of days. Thus, year 8 corresponds to 2835 days.

Add 2835 to the previous product: 499,657 + 2,835 = 502,492.

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

Now, search this table for the month and note the corresponding day. For Rajab, the day is 177.

Add the day number (177) for the month (Rajab) and the day of the month (12) to the previous product: 502,492 + 177 + 12 = 502,681

The Julian Date or Julian Day Number is then that number plus the Islâmic Benchmark number, which is 1948,085, which is the Julian Date of 0 Muh.arram 0 AH. Thus 502,681 + 1948,085 = 2,450,766.

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 = 2,450,765.5. This corresponds to 00:00 13 November 1997 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, 2,450,766, might be used to represent 12 Rajab 1418 in its entirety.

Converting the Julian Day Number to the Annô Hegirae date repeats the above process in reverse. Given the Julian Date 2,450,713, substract the Islâmic Benchmark number, 2,450,713 - 1948,085 = 502,628.

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

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 2,450,713.

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

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 2,450,713 is thus 18 Jumaadaa l'uulaa 1418.

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