# Boulder Community Network Last Update: January 1, 1999.

## Julian and Gregorian Day Numbers

### Introduction

Astronomers and chronologists use a system of numbering days called Julian days or Julian day numbers. The temporal sequence of days is mapped onto the sequence of integers, -2, -1, 0, 1, 2, 3, etc. This makes it easy to determine the number of days between two days (just subtract one Julian day number from the other).

For example a solar eclipse is said to have been seen at Ninevah on Julian day 1,442,454 and a lunar eclipse is said to have been observed at Babylon on Julian day number 1,566,839 (these numbers correspond to the Julian Calendar dates -763-03-23 and -423-10-09 respectively). Thus the lunar eclipse occurred 124,384 days after the solar eclipse.

Generally speaking, an integer date is any system of assigning a one-to-one correspondence between the days of our experience (more strictly, solar days) and the integers. Such systems differ only in the day chosen to correspond to day 0 or day 1. For example, in some applications NASA uses the Truncated Julian Date, which is the number of days since midnight on 1968-05-24 (at which time the Apollo missions to the Moon were underway).

Other starting dates popular with computer programmers are, or have been, 1601-01-01 (Gregorian), 1900-01-01, 1901-01-01 and 1980-01-01. The choice is usually a consequence of the temporal precision required (days to microseconds), the application period (a decade, a century, a millennium, etc.) and the number of bytes available for storing the date.

The system of Julian days should not be confused with the simpler system of the same name which associates a date with the number of days elapsed since January 1st of the same year (according to which 2000-12-31 is day 366 ů of the year 2000).

### History

The Julian Day number system was invented by Joseph Justus Scaliger (born 1540-08-05 J in Agen, France, died 1609-01-21 J in Leiden, Holland), who during his life immersed himself in Greek, Latin, Persian and Jewish literature. His invention (in 1583) of the Julian day number system is considered by some the foundation of the science of chronology.

Although the term Julian Calendar derives from the name of Julius Caesar, the term Julian day number probably does not. Most say that this system was named, not after Julius Caesar, but after its inventor's father, Julius Caesar Scaliger (1484-1558). Perhaps it was simply named after the Julian Calendar.

The younger Scaliger combined three traditionally recognized temporal cycles of 28, 19 and 15 years to obtain a great cycle, the Scaliger cycle, or Julian period, of 7980 years (7980 is the least common multiple of 28, 19 and 15). According to the Encyclopedia Britannica:

1. The length of 7,980 years was chosen as the product of 28 times 19 times 15; these, respectively,
2. are the numbers of years in the so-called solar cycle of the Julian calendar in which dates recur on the same days of the week;
3. the lunar or Metonic cycle, after which the phases of the Moon recur on a particular day in the solar year, or year of the seasons;
4. and the cycle of indiction, originally a schedule of periodic taxes or government requisitions in ancient Rome.

The first Scaliger cycle began with Year 1 on -4712-01-01 (Julian) and will end after 7980 years on 3267-12-31 (Julian), which is 3268-01-22 (Gregorian). 3268-01-01 J is the first day of Year 1 of the next Scaliger cycle.

### Astronomical System

Little mention seems to be made as to whether Joseph Scaliger regarded -4712-01-01 J as day 0 or as day 1 in the first Julian period. Astronomers adopted this system and adapted it to their own purposes, and they took noon GMT -4712-01-01 as their zero point.

For astronomers a day begins at noon and runs until the next noon (so that the nighttime falls conveniently within one "day"). Thus they defined the Julian day number of a day as the number of days (or part of a day) elapsed since noon GMT (or more exactly, UTC) on January 1st, 4713 B.C., in the Proleptic Julian Calendar. Thus the Julian day number of noon GMT on -4712-01-01 (Julian), or more casually, the Julian day number of -4712-01-01 itself, is 0.

(Note that 4713 B.C. is the year -4712 according to the astronomical year numbering.) The Julian day number of 1996-03-31 is 2,450,174 ů meaning that on 1996-03-31 2,450,174 days had elapsed since -4712-01-01 (or more exactly, that at noon on 1996-03-31 2,450,174 days had elapsed since noon on -4712-01-01).

Scaliger preceded the astronomers in introducing the notion of decimal times, designating midnight as .00, 6 a.m. as .25, midday as .50 and 6 p.m. as .75, thus allowing easier calculation involving dates and times. Astronomers, as noted above, preferred to use .00 to mean midday and .50 to mean midnight.

### Modified Julian Day Number

This was not to the liking of all scholars using the Julian day number system, in particular, historians. For chronologists who start "days" at midnight, the zero point for the Julian day number system is 00:00 at the start of -4712-01-01 J, and this is day 0. This means that 2000-01-01 G is 2,451,545 JD.

Since most days within about 150 years of the present have Julian day numbers beginning with "24", Julian day numbers within this 300-odd-year period can be abbreviated. In 1975 the convention of the modified Julian day number was adopted:

Given a Julian day number JD, the modified Julian day number MJD is defined as MJD = JD - 2,400,000.5. This has two purposes:

1. Days begin at midnight rather than noon.
2. For dates in the period from 1859 to about 2130 only five digits need to be used to specify the date rather than seven.

MJD 0 thus corresponds to JD 2,400,000.5, which is twelve hours after noon on JD 2,400,000 = 1858-11-16. Thus MJD 0 designates the midnight of November 16th/17th, 1858, so day 0 in the system of modified Julian day numbers is the day 1858-11-17.

### Lilian Day Number

This concept is similar to that of the Julian day number. It is named after Aloysius Lilius (an advisor to Pope Gregory XIII) who, together with his brother, is said to have invented the Gregorian Calendar. The Lilian day number is defined as "the number of days since 14 October 1582".

This was the time of the introduction of the Gregorian Calendar, when it was decreed by Pope Gregory XIII that the day following 4 October 1582 (which is 5 October 1582, in the Julian Calendar) would thenceforth be known as 15 October 1582.

Strictly speaking, the definition should be "the number of days since 14 October 1582 in the Proleptic Gregorian Calendar". Thus 15 October 1582 (Gregorian) is Lilian day 1 (the first day of the Gregorian Calendar), 16 October 1582 is Lilian day 2, and so on. The relation between Julian day numbers and Lilian day numbers is: LD = JD - 2,299,160

### Computation

Mathematicians and programmers have naturally interested themselves in mathematical and computational algorithms to convert between Julian day numbers and Gregorian dates.

In 1968 in a letter to the editor of Communications of the ACM (CACM, volume 11, number 10, October 1968, p.657) Henry F. Fliegel and Thomas C. Van Flandern presented such an algorithm.

The Julian day (jd) is computed from Gregorian day, month and year (d, m, y) as follows:

``` jd = ( 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
```

Division is to be understood as in integer arithmetic, with the remainders discarded.

Converting from the Julian day to the Gregorian day is performed thus:

```    l = jd + 68569
n = ( 4 * l ) / 146097
l = l - ( 146097 * n + 3 ) / 4
i = ( 4000 * ( l + 1 ) ) / 1461001 (that's 1,461,001)
l = l - ( 1461 * i ) / 4 + 31
j = ( 80 * l ) / 2447
d = l - ( 2447 * j ) / 80
l = j / 11
m = j + 2 - ( 12 * l )
y = 100 * ( n - 49 ) + i + l       (that's a lower-case L)
```

Days are integer values in the range 1-31, months are integers in the range 1-12, and years are positive or negative integers. This algorithm is valid only for dates from -4900-03-01 G onward when converting from a Julian day number to a date, or from -4800-03-01 G onward (when converting from a date to a Julian day number). It should be noted that these algorithms are valid only in the Gregorian Calendar and the Proleptic Gregorian Calendar (after the dates given above). They do not handle dates in the Julian Calendar.

This algorithm is implemented in two C functions in DATLIB, a library of C functions for converting and manipulating calendar dates.

A completely different calendar date / Julian day number computational algorithm (developed by the present writer) is implemented as two other C functions in the same function library. This algorithm handles dates in the Julian Calendar as well as dates in the Gregorian Calendar. In addition it is valid for any calendar date with a year within the range of about -5,800,000 to about +5,800,000.

In each of these algorithms Julian day number 0 corresponds to -4713-11-24 G (= -4712-01-01 J).