In Memoriam:

To the late O.L. Harvey
of Silver Spring, Maryland,
who loved Julian Day Numbers.

# 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

I still get confused about this. The various editions of the Astronomical Almanac, published by the United States Naval Observatory and the United Kingdom Hydrographic Office, give the Julian Date under the section "Universal and Sidereal Times" [division "B"] and "Sun" [division "C"]. If we look up, say, 7 September 2021, we find the "Julian Day" as 245 9464.5. That the lower number in the interger part of the date contains four digits reflects the convention to state the Day Number there as "myriads," i.e. 10,000's. That our number ends in a decimal means that this is the Julian Date, not the Day Number, and it is dating the moment at Midnight of the beginning of the Civil Day of 7 September 2021. If one calculates the Day Number for that date, using the methods below, it will come out 245 9465, which would be the Noon Date (12:00h) for the Civil Day of 7 September 2021, as well as 00:00h for the Astronomical and Nautical Day. So one gets the Day Number just by adding "0.5" to the given Julian Dates in the Almanac.

As it happens, 7 September 2021 is the date of the Jewish New Year, Rōʾsh Haššānâh, for 2021 (5782 AM). But the Jewish day begins neither at Noon nor Midnight, but at the Sunset of the previous day. Thus, if one looks up the Jewish New Year in the Astronomical Almanac [division "B"], one finds two dates: (1) September 6, glossed as "Year begins at sunset" ["Chronological Cycles and Eras"], and (2) September 7, glossed as "The Jewish and Islamic dates above are tabular dates, which begin at sunset on the previous evening and end at sunset on the date tabulated" ["Religious Calendars"]. If we wanted to use the Ancient Egyptian Calendar, where the day began at Sunrise, we would need to decide whether this is the sunrise after the beginning of the Civil Day or the one 24 hours earlier. Although the Copts actually use this calendar, I am told by the The Coptic Orthodox Diocese of the Southern United States that they begin the day at sunset.

The virtue of the Astronomical Day was that the observations of a single night could be unambiguously ascribed to a single calendar day, something that might also have been nice for the night naval battles off Guadalcanal. 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.

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 [note].

The Months of the Julian and Gregorian Calendars
MonthDayMonthDayMonthDay
3. March07. July12211. November245
4. April318. August15312. December275
5. May619. September1841. January306
6. June9210. October2142. February337
For the month, a day number must be found on the accompanying 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.

CenturyCorrectionCenturyCorrection
1582-101800-12
16001900-13
1700-112000
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.
CenturyCorrectionCenturyCorrection
1582-101800-12
16001900-13
1700-112000
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
MonthDayMonthDayMonthDay
3. March07. July12211. November245
4. April318. August15312. December275
5. May619. September1841. January306
6. June9210. October2142. February337
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.

Julian Date Converter

### Julian Day Numbers, Note 1

An area where there is still a distinction between Civil, Nautical, and Astronomical practice is in the reckoning of twilight. Before sunrise, when the center of sun is 18 degrees below the horizon, that is defined as the beginning of Astronomical Twilight. When the sun is then 12 degrees below the horizon, that is the beginning of Nautical Twilight. And then at 6 degrees, it is the beginning of Civil Twilight.

Astronomers need the darkest skies possible. So it is during Astronomical Twilight that they begin to lose that. At the beginning of Nautical Twilight, the sky is too light for nightime astronomy. During Nautical Twilight, however, when the horizon becomes visible, there are still enough bright stars that their angle above the horizon can be observed and measured, and the position of the observer calculated. With Civil Twilight, the sky will be too bright either for astronomy or navigation -- except that very bright objects, like Venus or Jupiter, may still be visible. Of course, astronomy of the sun, or navigation by shooting the sun, can begin with the actual sunrise.

This process, of course, is reversed at sunset, where "dusk" replaces "dawn," and astral navigation and then astronomy can begin in reversed sequence. Maps showing areas of day and night on maps of the earth often also show boundaries with shadings for the three areas of twilight. In ordinary life, people need have little concern for any of this; and navigation now is typtically a matter of GSP signals. Casual sailors, and even the typical practice of navies, need not worry about the old forms of celestial navigation. Astronomers, however, still want dark skies.

### Julian Day Numbers, Note 2

One and only one person can give steering and engine orders at any one time....The commanding officer may take over the deck or the conn...In taking the conn from the officer of the deck, the captain should do so in such a manner that all personnel of the bridge watch will be notified of the fact.

Watch Officer's Guide, A Handbook for all Deck Watch Officers, Revised by K.C. Jacobsen, Commander, U.S. Navy, 11th Edition [Naval Institute Press, Annapolis, Maryland, 1981, pp. 68-69, color added]; the "conn" is to "conduct" the ship by way of the steering and engine orders; the "deck" involves all other commands on the Watch, e.g. "Open fire."

The Sailing Master was a Warrant Officer, not a Commissioned Officer, often an older man. He was legally responsible for the nagivation of the ship, not the Captain, who might be young and relatively inexperienced (Horatio Nelson was a Captain at 21). The men called "Master's Mates" were those in training under the Sailing Master. Since the captain of a merchant ship was often called "Master" rather than "Captain," we can see that the Naval practice might occasion moments of confusion among those unfamiliar with the conventions. Unlike Naval Sailing Masters, civilian Masters were often woefully unfamiliar with the full demands of navigation, and there is no telling how many ships were lost because of that.

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 (as referenced in the epigraph), 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. Everyone can hear the order being passed down, and any misreport can be immediately corrected.

What perhaps might also be mentioned is that the "First Lieutenant" on Jack Aubrey's warships is the "First Officer" because he is the Senior Lieutenant. Modern warships have introduced the "Executive Officer" (the "XO," as opposed to the "CO," the Commanding Officer), who is generally of Command Rank (Lieutenant Commander or Commander), close to the Captain. As with many things, Star Trek does not seem to know what an Executive Officer is for; but the XO is there to handle administrative details, leaving military issues to the Captain. Thus, where Lieutenants are Watch Officers, the Executive Officer is not. But if there is a shortage of toilet paper, that is reported to the XO.

Military Rank

### Julian Day Numbers, Note 3

American usage, where the comma separates thousands and the period is used to indicate decimals, is observed.

# Islâmic Dateswith Julian Day Numbers

The Months of the Moslem Calendar
MonthsDaysMonthsDays
1.
ʾal-Muḥarram
307.
Rajab
30
2. Ṣafar298.
Shaʿbân
29
3.
Rabîʿu l-ʾawwal
309.
Ramaḍân
30
4.
Rabîʿu θ-θânî
2910.
Shawwâl
29
5.
Jumâdâ l-ʾûlâ
3011.
Ðû l-Qaʿdah
30
6.
Jumâdâ l-ʾâḫirah
2912.
Ðû l-Ḥijjah
29/30
The calendar of Islâm is the traditional calendar that was used in Arabia as reformed by the Prophet Muḥammad. 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, Muḥammad 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 , Ramaḍân, when Muslims are supposed to Fast the daylight hours, and , Ðû l-Ḥijjah, when the Pilgrimage, the , Ḥajj, may be undertaken to Mecca, are not fixed in particular seasons. While one might prefer that Ramaḍân 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 Ramaḍân would correspondingly always be in summer in South Africa or Australia. With Ramaḍân 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).
YearDayYearDayYearDay
00010*3543207087
0135411389821*7441
02*708124252227796
03106313*4606238150
04141714496124*8504
05*1771155315258859
06212616*566926*9213
07*2480175315279568
08283518*6378289922
09318919673329*10276
3010631
* = leap years

The Hijjrah, , or Annô Hegirae (AH) Era, is benchmarked to the day in 622 AD that Muḥammad 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 Ðû l-Qaʿdah 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 Ðû l-Qaʿdah, the day is 295. Add the day number (295) for the month (Ðû l-Qaʿdah) and the day of the month (7) to the previous product: 507,453 + 295 + 7 = 507,755.

The Months of the Moslem Calendar
MonthDayMonthDay
1. al-Muḥarram07. Rajab177
2. Ṣafar308. Shaʿbân207
3. Rabîʿu l-ʾawwal599. Ramaḍân236
4. Rabîʿu θ-θânî8910. Shawwâl266
5. Jumâdâ l-ʾûlâ11811. Ðû l-Qaʿdah295
6. Jumâdâ l-ʾâxirah14812. Ðû l-Ḥijjah325
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 Muḥarram 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 Ðû l-Qaʿdah 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 Ðû l-Ḥijjah, 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.

YearDayYearDayYearDay
00010*3543207087
0135411389821*7441
02*708124252227796
03106313*4606238150
04141714496124*8504
05*1771155315258859
06212616*566926*9213
07*2480175315279568
08283518*6378289922
09318919673329*10276
3010631
* = 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 Jumâdâ lʾûlâ.

The month thus will be Jumâdâ lʾûlâ, 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 Jumâdâ lʾûlâ 1418 AH.

The Jewish and Moslem Calendars with the Era of Nabonassar