Julian day

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The Julian day is the continuous count of days since the beginning of the Julian period, and is used primarily by astronomers, and in software for easily calculating elapsed days between two events (e.g. food production date and sell by date). [1]


The Julian day number (JDN) is the integer assigned to a whole solar day in the Julian day count starting from noon Universal Time, with Julian day number 0 assigned to the day starting at noon on Monday, January 1, 4713 BC, proleptic Julian calendar (November 24, 4714 BC, in the proleptic Gregorian calendar), [2] [3] [4] a date at which three multi-year cycles started (which are: Indiction, Solar, and Lunar cycles) and which preceded any dates in recorded history. [lower-alpha 1] For example, the Julian day number for the day starting at 12:00 UT (noon) on January 1, 2000, was 2 451 545. [5]

The Julian date (JD) of any instant is the Julian day number plus the fraction of a day since the preceding noon in Universal Time. Julian dates are expressed as a Julian day number with a decimal fraction added. [6] For example, the Julian Date for 00:30:00.0 UT January 1, 2013, is 2 456 293.520 833. [7] Expressed as a Julian date, this page was loaded at 2459594.5565509. [ refresh ]

The Julian period is a chronological interval of 7980 years; year 1 of the Julian Period was 4713 BC (−4712). [8] The Julian calendar year 2022 is year 6735 of the current Julian Period. The next Julian Period begins in the year AD 3268. Historians used the period to identify Julian calendar years within which an event occurred when no such year was given in the historical record, or when the year given by previous historians was incorrect. [9]


The term Julian date may also refer, outside of astronomy, to the day-of-year number (more properly, the ordinal date) in the Gregorian calendar, especially in computer programming, the military and the food industry, [10] or it may refer to dates in the Julian calendar. For example, if a given "Julian date" is "October 5, 1582", this means that date in the Julian calendar (which was October 15, 1582, in the Gregorian calendar—the date it was first established). Without an astronomical or historical context, a "Julian date" given as "36" most likely means the 36th day of a given Gregorian year, namely February 5. Other possible meanings of a "Julian date" of "36" include an astronomical Julian Day Number, or the year AD 36 in the Julian calendar, or a duration of 36 astronomical Julian years). This is why the terms "ordinal date" or "day-of-year" are preferred. In contexts where a "Julian date" means simply an ordinal date, calendars of a Gregorian year with formatting for ordinal dates are often called "Julian calendars", [10] but this could also mean that the calendars are of years in the Julian calendar system.

Historically, Julian dates were recorded relative to Greenwich Mean Time (GMT) (later, Ephemeris Time), but since 1997 the International Astronomical Union has recommended that Julian dates be specified in Terrestrial Time. [11] Seidelmann indicates that Julian dates may be used with International Atomic Time (TAI), Terrestrial Time (TT), Barycentric Coordinate Time (TCB), or Coordinated Universal Time (UTC) and that the scale should be indicated when the difference is significant. [12] The fraction of the day is found by converting the number of hours, minutes, and seconds after noon into the equivalent decimal fraction. Time intervals calculated from differences of Julian Dates specified in non-uniform time scales, such as UTC, may need to be corrected for changes in time scales (e.g. leap seconds). [6]


Because the starting point or reference epoch is so long ago, numbers in the Julian day can be quite large and cumbersome. A more recent starting point is sometimes used, for instance by dropping the leading digits, in order to fit into limited computer memory with an adequate amount of precision. In the following table, times are given in 24-hour notation.

In the table below, Epoch refers to the point in time used to set the origin (usually zero, but (1) where explicitly indicated) of the alternative convention being discussed in that row. The date given is a Gregorian calendar date if it is October 15, 1582, or later, but a Julian calendar date if it is earlier. JD stands for Julian Date. 0h is 00:00 midnight, 12h is 12:00 noon, UT unless otherwise specified. Current value is as of 01:21, Saturday, January 15, 2022 ( UTC ) and may be cached. (update)

Name Epoch CalculationCurrent valueNotes
Julian date12:00 January 1, 4713 BC2459594.55625
Reduced JD12:00 November 16, 1858JD − 240000059594.55625 [13] [14]
Modified JD0:00 November 17, 1858JD − 2400000.559594.05625Introduced by SAO in 1957
Truncated JD0:00 May 24, 1968floor (JD − 2440000.5)19594Introduced by NASA in 1979
Dublin JD12:00 December 31, 1899JD − 241502044574.55625Introduced by the IAU in 1955
CNES JD 0:00 January 1, 1950JD − 2433282.526312.05625Introduced by the CNES [15]
CCSDS JD 0:00 January 1, 1958JD − 2436204.523390.05625Introduced by the CCSDS [15]
Lilian date day 1 = October 15, 1582 [lower-alpha 2] floor (JD − 2299159.5)160435Count of days of the Gregorian calendar
Rata Die day 1 = January 1, 1 [lower-alpha 2] proleptic Gregorian calendar floor (JD − 1721424.5)738170Count of days of the Common Era
Mars Sol Date 12:00 December 29, 1873(JD − 2405522)/1.0274952625.80703Count of Martian days
Unix time 0:00 January 1, 1970(JD − 2440587.5) × 864001642209686Count of seconds, [16] excluding leap seconds
.NET DateTime0:00 January 1, 1 proleptic Gregorian calendar (JD − 1721425.5) × 8640000000006.3777806485998E+17Count of 100-nanosecond ticks, excluding ticks attributable to leap seconds [17]

The Heliocentric Julian Day (HJD) is the same as the Julian day, but adjusted to the frame of reference of the Sun, and thus can differ from the Julian day by as much as 8.3 minutes (498 seconds), that being the time it takes light to reach Earth from the Sun. [lower-alpha 3]


Julian Period

The Julian day number is based on the Julian Period proposed by Joseph Scaliger, a classical scholar, in 1583 (one year after the Gregorian calendar reform) as it is the product of three calendar cycles used with the Julian calendar:

28 (solar cycle) × 19 (lunar cycle) × 15 (indiction cycle) = 7980 years

Its epoch occurs when all three cycles (if they are continued backward far enough) were in their first year together. Years of the Julian Period are counted from this year, 4713 BC, as year 1, which was chosen to be before any historical record. [28]

Scaliger corrected chronology by assigning each year a tricyclic "character", three numbers indicating that year's position in the 28-year solar cycle, the 19-year lunar cycle, and the 15-year indiction cycle. One or more of these numbers often appeared in the historical record alongside other pertinent facts without any mention of the Julian calendar year. The character of every year in the historical record was unique – it could only belong to one year in the 7980-year Julian Period. Scaliger determined that 1 BC or year 0 was Julian Period (JP) 4713. He knew that 1 BC or 0 had the character 9 of the solar cycle, 1 of the lunar cycle, and 3 of the indiction cycle. By inspecting a 532-year Paschal cycle with 19 solar cycles (each year numbered 1–28) and 28 lunar cycles (each year numbered 1–19), he determined that the first two numbers, 9 and 1, occurred at its year 457. He then calculated via remainder division that he needed to add eight 532-year Paschal cycles totaling 4256 years before the cycle containing 1 BC or 0 in order for its year 457 to be indiction 3. The sum 4256 + 457 was thus JP 4713. [29]

A formula for determining the year of the Julian Period given its character involving three four-digit numbers was published by Jacques de Billy in 1665 in the Philosophical Transactions of the Royal Society (its first year). [30] John F. W. Herschel gave the same formula using slightly different wording in his 1849 Outlines of Astronomy. [31]

Multiply the Solar Cycle by 4845, and the Lunar, by 4200, and that of the Indiction, by 6916. Then divide the Sum of the products by 7980, which is the Julian Period: The Remainder of the Division, without regard to the Quotient, shall be the year enquired after.

Jacques de Billy

Carl Friedrich Gauss introduced the modulo operation in 1801, restating de Billy's formula as:

Julian Period year = (6916a + 4200b + 4845c) MOD 15×19×28

where a is the year of the indiction cycle, b of the lunar cycle, and c of the solar cycle. [32] [33]

John Collins described the details of how these three numbers were calculated in 1666, using many trials. [34] A summary of Collin's description is in a footnote. [35] Reese, Everett and Craun reduced the dividends in the Try column from 285, 420, 532 to 5, 2, 7 and changed remainder to modulo, but apparently still required many trials. [36]

The specific cycles used by Scaliger to form his tricyclic Julian Period were, first, the indiction cycle with a first year of 313. [lower-alpha 4] [37] Then he chose the dominant 19-year Alexandrian lunar cycle with a first year of 285, the Era of Martyrs and the Diocletian Era epoch, [38] or a first year of 532 according to Dionysius Exiguus. [39] Finally, Scaliger chose the post-Bedan solar cycle with a first year of 776, when its first quadrennium of concurrents, 1 2 3 4, began in sequence. [lower-alpha 5] [40] [41] [42] Although not their intended use, the equations of de Billy or Gauss can be used to determined the first year of any 15-, 19-, and 28-year tricyclic period given any first years of their cycles. For those of the Julian Period, the result is AD 3268, because both remainder and modulo usually return the lowest positive result. Thus 7980 years must be subtracted from it to yield the first year of the present Julian Period, −4712 or 4713 BC, when all three of its sub-cycles are in their first years.

Scaliger got the idea of using a tricyclic period from "the Greeks of Constantinople" as Herschel stated in his quotation below in Julian day numbers. [43] Specifically, the monk and priest Georgios wrote in 638/39 that the Byzantine year 6149 AM (640/41) had indiction 14, lunar cycle 12, and solar cycle 17, which places the first year of the Byzantine Era in 5509/08 BC, the Byzantine Creation. [44] Dionysius Exiguus called the Byzantine lunar cycle his "lunar cycle" in argumentum 6, in contrast with the Alexandrian lunar cycle which he called his "nineteen-year cycle" in argumentum 5. [45]

Although many references say that the Julian in "Julian Period" refers to Scaliger's father, Julius Scaliger, at the beginning of Book V of his Opus de Emendatione Temporum ("Work on the Emendation of Time") he states, "Iulianam vocauimus: quia ad annum Iulianum accomodata", [46] [47] which Reese, Everett and Craun translate as "We have termed it Julian because it fits the Julian year." [36] Thus Julian refers to the Julian calendar.

Julian day numbers

Julian days were first used by Ludwig Ideler for the first days of the Nabonassar and Christian eras in his 1825 Handbuch der mathematischen und technischen Chronologie. [48] [49] John F. W. Herschel then developed them for astronomical use in his 1849 Outlines of Astronomy, after acknowledging that Ideler was his guide. [50]

The period thus arising of 7980 Julian years, is called the Julian period, and it has been found so useful, that the most competent authorities have not hesitated to declare that, through its employment, light and order were first introduced into chronology. [51] We owe its invention or revival to Joseph Scaliger, who is said to have received it from the Greeks of Constantinople. The first year of the current Julian period, or that of which the number in each of the three subordinate cycles is 1, was the year 4713 BC, and the noon of January 1 of that year, for the meridian of Alexandria, is the chronological epoch, to which all historical eras are most readily and intelligibly referred, by computing the number of integer days intervening between that epoch and the noon (for Alexandria) of the day, which is reckoned to be the first of the particular era in question. The meridian of Alexandria is chosen as that to which Ptolemy refers the commencement of the era of Nabonassar, the basis of all his calculations. [52]

At least one mathematical astronomer adopted Herschel's "days of the Julian period" immediately. Benjamin Peirce of Harvard University used over 2,800 Julian days in his Tables of the Moon, begun in 1849 but not published until 1853, to calculate the lunar ephemerides in the new American Ephemeris and Nautical Almanac from 1855 to 1888. The days are specified for "Washington mean noon", with Greenwich defined as 18h 51m 48s west of Washington (282°57′W, or Washington 77°3′W of Greenwich). A table with 197 Julian days ("Date in Mean Solar Days", one per century mostly) was included for the years –4713 to 2000 with no year 0, thus "–" means BC, including decimal fractions for hours, minutes and seconds. [53] The same table appears in Tables of Mercury by Joseph Winlock, without any other Julian days. [54]

The national ephemerides started to include a multi-year table of Julian days, under various names, for either every year or every leap year beginning with the French Connaissance des Temps in 1870 for 2,620 years, increasing in 1899 to 3,000 years. [55] The British Nautical Almanac began in 1879 with 2,000 years. [56] The Berliner Astronomisches Jahrbuch began in 1899 with 2,000 years. [57] The American Ephemeris was the last to add a multi-year table, in 1925 with 2,000 years. [58] However, it was the first to include any mention of Julian days with one for the year of issue beginning in 1855, as well as later scattered sections with many days in the year of issue. It was also the first to use the name "Julian day number" in 1918. The Nautical Almanac began in 1866 to include a Julian day for every day in the year of issue. The Connaissance des Temps began in 1871 to include a Julian day for every day in the year of issue.

The French mathematician and astronomer Pierre-Simon Laplace first expressed the time of day as a decimal fraction added to calendar dates in his book, Traité de Mécanique Céleste, in 1823. [59] Other astronomers added fractions of the day to the Julian day number to create Julian Dates, which are typically used by astronomers to date astronomical observations, thus eliminating the complications resulting from using standard calendar periods like eras, years, or months. They were first introduced into variable star work in 1860 by the English astronomer Norman Pogson, which he stated was at the suggestion of John Herschel. [60] They were popularized for variable stars by Edward Charles Pickering, of the Harvard College Observatory, in 1890. [61]

Julian days begin at noon because when Herschel recommended them, the astronomical day began at noon. The astronomical day had begun at noon ever since Ptolemy chose to begin the days for his astronomical observations at noon. He chose noon because the transit of the Sun across the observer's meridian occurs at the same apparent time every day of the year, unlike sunrise or sunset, which vary by several hours. Midnight was not even considered because it could not be accurately determined using water clocks. Nevertheless, he double-dated most nighttime observations with both Egyptian days beginning at sunrise and Babylonian days beginning at sunset. [62] Medieval Muslim astronomers used days beginning at sunset, so astronomical days beginning at noon did produce a single date for an entire night. Later medieval European astronomers used Roman days beginning at midnight so astronomical days beginning at noon also allow observations during an entire night to use a single date. When all astronomers decided to start their astronomical days at midnight to conform to the beginning of the civil day, on January 1, 1925, it was decided to keep Julian days continuous with previous practice, beginning at noon.

During this period, usage of Julian day numbers as a neutral intermediary when converting a date in one calendar into a date in another calendar also occurred. An isolated use was by Ebenezer Burgess in his 1860 translation of the Surya Siddhanta wherein he stated that the beginning of the Kali Yuga era occurred at midnight at the meridian of Ujjain at the end of the 588,465th day and the beginning of the 588,466th day (civil reckoning) of the Julian Period, or between February 17 and 18JP 1612 or 3102 BC. [63] [64] Robert Schram was notable beginning with his 1882 Hilfstafeln für Chronologie. [65] Here he used about 5,370 "days of the Julian Period". He greatly expanded his usage of Julian days in his 1908 Kalendariographische und Chronologische Tafeln containing over 530,000 Julian days, one for the zeroth day of every month over thousands of years in many calendars. He included over 25,000 negative Julian days, given in a positive form by adding 10,000,000 to each. He called them "day of the Julian Period", "Julian day", or simply "day" in his discussion, but no name was used in the tables. [66] Continuing this tradition, in his book "Mapping Time: The Calendar and Its History" british physics educator and programmer Edward Graham Richards uses Julian day numbers to convert dates from one calendar into another using algorithms rather than tables. [67]

Julian day number calculation

The Julian day number can be calculated using the following formulas (integer division rounding towards zero is used exclusively, that is, positive values are rounded down and negative values are rounded up): [lower-alpha 6]

The months January to December are numbered 1 to 12. For the year, astronomical year numbering is used, thus 1 BC is 0, 2 BC is −1, and 4713 BC is −4712. JDN is the Julian Day Number. Use the previous day of the month if trying to find the JDN of an instant before midday UT.

Converting Gregorian calendar date to Julian Day Number

The algorithm is valid for all (possibly proleptic) Gregorian calendar dates after November 23, 4713. Divisions are integer divisions towards zero, fractional parts are ignored. [68]

JDN = (1461 × (Y + 4800 + (M − 14)/12))/4 +(367 × (M − 2 − 12 × ((M − 14)/12)))/12 − (3 × ((Y + 4900 + (M - 14)/12)/100))/4 + D − 32075

Converting Julian calendar date to Julian Day Number

The algorithm [69] is valid for all (possibly proleptic) Julian calendar years ≥ −4712, that is, for all JDN ≥ 0. Divisions are integer divisions, fractional parts are ignored.

JDN = 367 × Y − (7 × (Y + 5001 + (M − 9)/7))/4 + (275 × M)/9 + D + 1729777

Finding Julian date given Julian day number and time of day

For the full Julian Date of a moment after 12:00 UT one can use the following. Divisions are real numbers.

So, for example, January 1, 2000, at 18:00:00 UT corresponds to JD = 2451545.25

For a point in time in a given Julian day after midnight UT and before 12:00 UT, add 1 or use the JDN of the next afternoon.

Finding day of week given Julian day number

The US day of the week W1 (for an afternoon or evening UT) can be determined from the Julian Day Number J with the expression:

W1 = mod(J + 1, 7) [70]
Day of the weekSunMonTueWedThuFriSat

If the moment in time is after midnight UT (and before 12:00 UT), then one is already in the next day of the week.

The ISO day of the week W0 can be determined from the Julian Day Number J with the expression:

W0 = mod (J, 7) + 1
Day of the weekMonTueWedThuFriSatSun

Julian or Gregorian calendar from Julian day number

This is an algorithm by Edward Graham Richards to convert a Julian Day Number, J, to a date in the Gregorian calendar (proleptic, when applicable). Richards states the algorithm is valid for Julian day numbers greater than or equal to 0. [71] [72] All variables are integer values, and the notation "a div b" indicates integer division, and "mod(a,b)" denotes the modulus operator.

Algorithm parameters for Gregorian calendar

For Julian calendar:

  1. f = J + j

For Gregorian calendar:

  1. f = J + j + (((4 × J + B) div 146097) × 3) div 4 + C

For Julian or Gregorian, continue:

  1. e = r × f + v
  2. g = mod(e, p) div r
  3. h = u × g + w
  4. D = (mod(h, s)) div u + 1
  5. M = mod(h div s + m, n) + 1
  6. Y = (e div p) - y + (n + m - M) div n

D, M, and Y are the numbers of the day, month, and year respectively for the afternoon at the beginning of the given Julian day.

Julian Period from indiction, Metonic and solar cycles

Let Y be the year BC or AD and i, m and s respectively its positions in the indiction, Metonic and solar cycles. Divide 6916i + 4200m + 4845s by 7980 and call the remainder r.

If r>4713, Y = (r − 4713) and is a year AD.
If r<4714, Y = (4714 − r) and is a year BC.


i = 8, m = 2, s = 8. What is the year?

(6916 × 8) = 55328; (4200 × 2) = 8400: (4845 × 8) = 38760. 55328 + 8400 + 38760 = 102488.
102488/7980 = 12 remainder 6728.
Y = (6728 − 4713) = AD 2015. [73]

Julian date calculation

As stated above, the Julian date (JD) of any instant is the Julian day number for the preceding noon in Universal Time plus the fraction of the day since that instant. Ordinarily calculating the fractional portion of the JD is straightforward; the number of seconds that have elapsed in the day divided by the number of seconds in a day, 86,400. But if the UTC timescale is being used, a day containing a positive leap second contains 86,401 seconds (or in the unlikely event of a negative leap second, 86,399 seconds). One authoritative source, the Standards of Fundamental Astronomy (SOFA), deals with this issue by treating days containing a leap second as having a different length (86,401 or 86,399 seconds, as required). SOFA refers to the result of such a calculation as "quasi-JD". [74]

See also


  1. Both of these dates are years of the Anno Domini or Common Era (which has no year 0 between 1 BC and AD 1). Astronomical calculations generally include a year 0, so these dates should be adjusted accordingly (i.e. the year 4713 BC becomes astronomical year number 4712, etc.). In this article, dates before 15 October 1582 are in the (possibly proleptic) Julian calendar and dates on or after 15 October 1582 are in the Gregorian calendar, unless otherwise labelled.
  2. 1 2 This is an epoch starting with day 1 instead of 0. Conventions vary as to whether this is based on UT or local time.
  3. To illustrate the ambiguity that could arise from conflating Heliocentric time and Terrestrial time, consider the two separate astronomical measurements of an astronomical object from the Earth: Assume that three objects—the Earth, the Sun, and the astronomical object targeted, that is whose distance is to be measured—happen to be in a straight line for both measures. However, for the first measurement, the Earth is between the Sun and the targeted object, and for the second, the Earth is on the opposite side of the Sun from that object. Then, the two measurements would differ by about 1000 light-seconds: For the first measurement, the Earth is roughly 500 light seconds closer to the target than the Sun, and roughly 500 light seconds further from the target astronomical object than the Sun for the second measure. An error of about 1000 light-seconds is over 1% of a light-day, which can be a significant error when measuring temporal phenomena for short period astronomical objects over long time intervals. To clarify this issue, the ordinary Julian day is sometimes referred to as the Geocentric Julian Day (GJD) in order to distinguish it from HJD.
  4. All years in this paragraph are those of the Anno Domini Era at the time of Easter
  5. The concurrent of any Julian year is the weekday of its March 24, numbered from Sunday=1.
  6. Doggett in Seidenmann 1992, p. 603, indicates the algorithms are inspired by Fliegel & Van Flanderen 1968. That paper gives algorithms in Fortran. The Fortran computer language performs integer division by truncating, which is functionally equivalent to rounding toward zero.

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Every year that is exactly divisible by four is a leap year, except for years that are exactly divisible by 100, but these centurial years are leap years if they are exactly divisible by 400. For example, the years 1700, 1800, and 1900 are not leap years, but the years 1600 and 2000 are.

The proleptic Gregorian calendar is produced by extending the Gregorian calendar backward to the dates preceding its official introduction in 1582. In nations that adopted the Gregorian calendar after its official and first introduction, dates occurring in the interim period of 15 October 1582 to the date on which the pertinent nation adopted the Gregorian calendar and abandoned the Julian calendar are sometimes 'Gregorianized' also. For example, the birthday of President George Washington was originally dated 11 February 1731 because Great Britain, of which he was born a subject, used the Julian calendar and dated the beginning of English years as 25 March. After Great Britain switched to the Gregorian calendar, Washington's birthday was dated 22 February 1732 proleptically, according to the Gregorian calendar applied backward. This remains the modern dating of his birthday.

A tropical year is the time that the Sun takes to return to the same position in the sky of a celestial body of the solar system such as the Earth, completing a full cycle of seasons; for example, the time from vernal equinox to vernal equinox, or from summer solstice to summer solstice. It is the time of a year as a unit of tropical solar calendars and calendar eras which use tropical solar calendars. The solar year is one type of astronomical year and particular orbital period..

Maya astronomy

Maya astronomy is the study of the Moon, planets, Milky Way, Sun, and astronomical phenomena by the Precolumbian Maya Civilization of Mesoamerica. The Classic Maya in particular developed some of the most accurate pre-telescope astronomy in the world, aided by their fully developed writing system and their positional numeral system, both of which are fully indigenous to Mesoamerica. The Classic Maya understood many astronomical phenomena: for example, their estimate of the length of the synodic month was more accurate than Ptolemy's, and their calculation of the length of the tropical solar year was more accurate than that of the Spanish when the latter first arrived. Many temples from the Maya architecture have features orientated to celestial events.


  1. "Julian date" n.d.
  2. Dershowitz & Reingold 2008, 15.
  3. Seidelman 2013, 15.
  4. "Astronomical Almanac Online" 2016, Glossary, s.v. Julian date. Various timescales may be used with Julian date, such as Terrestrial Time (TT) or Universal Time (UT); in precise work the timescale should be specified.
  5. McCarthy & Guinot 2013, 91–2
  6. 1 2 "Resolution B1" 1997.
  7. US Naval Observatory 2005
  8. Astronomical Almanac for the year 2017 p. B4, which states 2017 is year 6730 of the Julian Period.
  9. Grafton 1975
  10. 1 2 USDA c. 1963.
  11. Resolution B1 on the use of Julian Dates of the XXIIIrd International Astronomical Union General Assembly, Kyoto, Japan, 1997
  12. Seidelmann 2013, p. 15.
  13. Hopkins 2013, p. 257.
  14. Pallé, Esteban 2014.
  15. 1 2 Theveny 2001.
  16. Astronomical almanac for the year 2001, 2000, p. K2
  17. "System.DateTime.Ticks documentation". Microsoft. n.d. Retrieved January 14, 2022. The value of this property represents the number of 100-nanosecond intervals that have elapsed since 12:00:00 midnight, January 1, 0001 in the Gregorian calendar,
  18. "38 Why Is Wednesday November 17, 1858 The Base Time For VAX/VMS?". Digital Equipment Corporation-Customer Support Center. Colorado Springs. June 6, 2007. Archived from the original on June 6, 2007.
  19. Winkler n. d.
  20. Chi 1979.
  21. SPD Toolkit Time Notes 2014.
  22. Ransom c. 1988
  23. Ohms 1986
  24. IBM 2004.
  25. "datetime — Basic date and time types — date Objects" (Dec 5, 2021). The Python Standard Library.
  26. Dershowitz & Reingold 2008, 10, 351, 353, Appendix B.
  27. Date n.d.
  28. Richards 2013, pp. 591–592.
  29. Grafton 1975, p. 184
  30. de Billy 1665
  31. Herschel 1849
  32. Gauss 1966
  33. Gauss 1801
  34. Collins 1666
  35. Calculation of 4845, 4200, 6916
    by Collins
    Try 2+ until
    7980/28 = 19×15 = 285285×Try/28 =
    remainder 1
    285×17 = 19×15×17 = 4845
    7980/19 = 28×15 = 420420×Try/19 =
    remainder 1
    420×10 = 28×15×10 = 4200
    7980/15 = 28×19 = 532532×Try/15 =
    remainder 1
    532×13 = 28×19×13 = 6916
  36. 1 2 Reese, Everett and Craun 1981
  37. Depuydt 1987
  38. Neugebauer 2016, pp. 72–77, 109–114
  39. Dionysius Exiguus 2003/525
  40. De argumentis lunæ libellus, col. 705
  41. Blackburn and Holford-Strevens, p. 821
  42. Mosshammer 2008, pp. 80–85
  43. Herschel 1849, p. 634
  44. Diekamp 44, 45, 50
  45. Dionysius Exiguus 2003/525
  46. Scaliger 1629, p. 361
  47. Scaliger used these words in his 1629 edition on p. 361 and in his 1598 edition on p. 339. In 1583 he used "Iulianam vocauimus: quia ad annum Iulianum duntaxat accomodata est" on p. 198.
  48. Ideler 1825, pp. 102–106
  49. The Nabonassar day was elapsed with a typo – it was correctly printed later as 1448638. The Christian day (1721425) was current, not elapsed.
  50. Herschel, 1849, p. 632 note
  51. Ideler 1825, p. 77
  52. Herschel 1849, p. 634
  53. Peirce 1853
  54. Winlock 1864
  55. Connaissance des Temps 1870, pp. 419–424; 1899, pp. 718–722
  56. Nautical Almanac and Astronomical Ephemeris 1879, p. 494
  57. Berliner Astronomisches Jahrbuch 1899, pp. 390–391
  58. American Ephemeris 1925, pp. 746–749
  59. Laplace 1823
  60. Pogson 1860
  61. Furness 1915.
  62. Ptolemy c. 150, p. 12
  63. Burgess 1860
  64. Burgess was furnished these Julian days by US Nautical Alamanac Office.
  65. Schram 1882
  66. Schram 1908
  67. Richards 1998, pp. 287–342
  68. L. E. Doggett, Ch. 12, "Calendars", p. 604, in Seidelmann 1992. "These algorithms are valid for all Gregorian calendar dates corresponding to JD >= 0, i.e, dates after 4713 November 23."
  69. L. E. Doggett, Ch. 12, "Calendars", p. 606, in Seidelmann 1992
  70. Richards 2013, pp. 592, 618.
  71. Richards 2013, 617–9
  72. Richards 1998, 316
  73. Heath 1760, p. 160.
  74. "SOFA Time Scale and Calendar Tools" 2016, p. 20