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Finding the Julian Day

On the first of these pages, a perpetual calendar was shown by means of which one could obtain the last digit of the Julian Day for any date. However, obtaining the whole Julian Day Number is useful for relating specific dates, and is also a different kind of problem.

As with the perpetual calendars on the first of these pages, for simplicity one treats January and February as though they belong to the end of the preceding year.

Start from this number, giving the Julian Day number for noon GMT (or, rather, noon UTC) on March 1st on the first year of a century (strictly speaking, year 0 of the century, or the last year of the century before):

Julian:
   0  1721118     400  1867218     800  2013318    1200  2159418    1600  2305518
 100  1757643     500  1903743     900  2049843    1300  2195943    1700  2342043
 200  1794168     600  1940268    1000  2086368    1400  2232468
 300  1830693     700  1976793    1100  2122893    1500  2268993

Gregorian:
                 1600  2305508    2000  2451605    2400  2597702    2800  2743799
                 1700  2342032    2100  2488129    2500  2634226    2900  2780323
                 1800  2378556    2200  2524653    2600  2670750    3000  2816847
1500  2268983    1900  2415080    2300  2561177    2700  2707274    3100  2853371

3200  2889896    3600  3035993
3300  2926420    3700  3072517
3400  2962944    3800  3109041
3500  2999468    3900  3145565

Revised Julian:

1500  2268984
1600  2305508    2000  2451605    2400  2597702
1700  2342032    2100  2488129    2500  2634226    2900  2780323    3300  2926420
1800  2378556    2200  2524653    2600  2670750    3000  2816847    3400  2962944
1900  2415080    2300  2561177    2700  2707274    3100  2853371    3500  2999468 
                                  2800  2643798    3200  2889895    3600  3035992
                                                                    3700  3072516

As noted for the perpetual calendar for the days of the week, while there is no year 0, the entry for the year 0 is required for the years from 1 A.D. to 99 A.D.

add this number for the year of the century:

       0     1     2     3     4     5     6     7     8     9
00     0   365   730  1095  1461  1826  2191  2556  2922  3287
10  3652  4017  4383  4748  5113  5478  5844  6209  6574  6939
20  7305  7670  8035  8400  8766  9131  9496  9861 10227 10592
30 10957 11322 11688 12053 12418 12783 13149 13514 13879 14244
40 14610 14975 15340 15705 16071 16436 16801 17166 17532 17897
50 18262 18627 18993 19358 19723 20088 20454 20819 21184 21549
60 21915 22280 22645 23010 23376 23741 24106 24471 24837 25202
70 25567 25932 26298 26663 27028 27393 27759 28124 28489 28854
80 29220 29585 29950 30315 30681 31046 31411 31776 32142 32507
90 32872 33237 33603 33968 34333 34698 35064 35429 35794 36159

add this number for the day of the year:

    Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb
  1   0  31  61  92 122 153 184 214 245 275 306 337
  2   1  32  62  93 123 154 185 215 246 276 307 338
  3   2  33  63  94 124 155 186 216 247 277 308 339
  4   3  34  64  95 125 156 187 217 248 278 309 340
  5   4  35  65  96 126 157 188 218 249 279 310 341
  6   5  36  66  97 127 158 189 219 250 280 311 342
  7   6  37  67  98 128 159 190 220 251 281 312 343
  8   7  38  68  99 129 160 191 221 252 282 313 344
  9   8  39  69 100 130 161 192 222 253 283 314 345
 10   9  40  70 101 131 162 193 223 254 284 315 346
 11  10  41  71 102 132 163 194 224 255 285 316 347
 12  11  42  72 103 133 164 195 225 256 286 317 348
 13  12  43  73 104 134 165 196 226 257 287 318 349
 14  13  44  74 105 135 166 197 227 258 288 319 350
 15  14  45  75 106 136 167 198 228 259 289 320 351
 16  15  46  76 107 137 168 199 229 260 290 321 352
 17  16  47  77 108 138 169 200 230 261 291 322 353
 18  17  48  78 109 139 170 201 231 262 292 323 354
 19  18  49  79 110 140 171 202 232 263 293 324 355
 20  19  50  80 111 141 172 203 233 264 294 325 356
 21  20  51  81 112 142 173 204 234 265 295 326 357
 22  21  52  82 113 143 174 205 235 266 296 327 358
 23  22  53  83 114 144 175 206 236 267 297 328 359
 24  23  54  84 115 145 176 207 237 268 298 329 360
 25  24  55  85 116 146 177 208 238 269 299 330 361
 26  25  56  86 117 147 178 209 239 270 300 331 362
 27  26  57  87 118 148 179 210 240 271 301 332 363
 28  27  58  88 119 149 180 211 241 272 302 333 364
 29  28  59  89 120 150 181 212 242 273 303 334 365
 30  29  60  90 121 151 182 213 243 274 304 335
 31  30      91     152 183     244     305 336

and the total is the Julian Day number for noon GMT on the day for which one has consulted the table.

For years B.C., the table for the year of the century needs to be changed slightly:

       9     8     7     6     5     4     3     2     1     0
90     0   365   731  1096  1461  1826  2192  2557  2922  3287
80  3653  4018  4383  4748  5114  5479  5844  6209  6575  6940
70  7305  7670  8036  8401  8766  9131  9497  9862 10227 10592
60 10958 11323 11688 12053 12419 12784 13149 13514 13880 14245
50 14610 14975 15341 15706 16071 16436 16802 17167 17532 17897
40 18263 18628 18993 19358 19724 20089 20454 20819 21185 21550
30 21915 22280 22646 23011 23376 23741 24107 24472 24837 25202
20 25568 25933 26298 26663 27029 27394 27759 28124 28490 28855
10 29220 29585 29951 30316 30681 31046 31412 31777 32142 32507
00 32873 33238 33603 33968 34334 34699 35064 35429 35795 36160

and the table for centuries becomes:

2999  626098    1999  991348     999 1356598
2899  662623    1899 1027873     899 1393123
2799  699148    1799 1064398     799 1429648
2699  735673    1699 1100923     699 1466173
2599  772198    1599 1137448     599 1502698
2499  808723    1499 1173973     499 1539223
2399  845248    1399 1210498     399 1575748
2299  881773    1299 1247023     299 1612273
2199  918298    1199 1283548     199 1648798
2099  954823    1099 1320073      99 1685323

These three tables make it simple to calculate a Julian Day Number by adding only three numbers together.

For dates outside the range of the table, one can use the fact that a Julian century is exactly 36,525 days long, or the fact that four Gregorian centuries are exactly 146,097 days long. In the former case, one also needs to remember that going 100 years back from 100 A.D. brings you to 1 B.C., and going another 100 years back reaches 101 B.C., and so on.

Essentially, these tables are constructed by adding the number of days in the major cycles of the Gregorian calendar.

A year is normally 365 days long. But four years, including one leap year, is normally 1,461 days long. A hundred years, in the Gregorian calendar, normally includes one span of four years without a leap year, so it is 36,524 days long. But four hundred years in the Gregorian calendar includes one century where every span of four years has its leap year, so they are 146,097 days long.

Thus, one might construct smaller tables, each involving only one of these rules, starting from Julian Day 2305507, one day before March 1st, 1600 AD.

Thus, the first table, adding 146,097 each time, would be:

1600  2305507
2000  2451604
2400  2597701
2800  2743798

Then, for intermediate centuries, we use the preceding rule to construct this table:

  0       0
100   36524
200   73048
300  109572

and combining these two tables for convenience produces the first compound table above.

Similarly, for years within the century, we can start with a table that shows the number of days for every fourth year:

 0     0   20  7305   40 14610   60 21915   80 29220
 4  1461   24  8766   44 16071   64 23376   84 30681
 8  2922   28 10227   48 17532   68 24837   88 32142
12  4383   32 11688   52 18993   72 26298   92 33603
16  5844   36 13149   56 20454   76 27759   96 35064

combined with a short table allowing us to handle individual years:

0    0
1  365
2  730
3 1095

Combining these two tables to save another addition gives us the second combined table above.

And the final table can be replaced by just one of its rows,

 Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb
   0  31  61  92 122 153 184 214 245 275 306 337

since one can always add the day of the month to the month number. And since the first day of the month is numbered "1", not zero, that's why the starting century numbers were one smaller in the first of these tables than in the combined tables.

A JavaScript program to convert to and from Julian Day Numbers is available on this page.

A brief, MAD, note

October 1, 1952 is Julian Day 214 + 18993 + 2415080 = 2,434,287 from the tables above; that is, it began on J. D. 2,434,286.5 and ended on J. D. 2,434,287.5. Some sources will simplify to an integer by calling it Julian Day 2,434,286, instead of 2,434,287 as I have chosen to do in my tables above; that is, for example, the usual convention in discussions of an epoch for the Mayan calendar.

Subtracting 2,434,286 from the Julian Day for a given date (or 2,434,285, if this other convention is used) will give the number of the day, or Clarke (named after MAD artist Bob Clarke, not Arthur C. Clarke, even if it is the period of a geosynchronous orbit) in the calendar given in the Potrzebie system of units.

Thus, I write these words on February 9, 2008, which is Julian Day 2451605 + 2556 + 345 = 2,454,506. Subtracting 2,434,286, one obtains 20,220, which would make today the 10th Clarke of the second Mingo (the Mingo of Calculated) of Cowznofska Madl 203 (not 202, as the Potrzebie calendar also begins with Cowznofski 1, not Cowznofski 0).

In this connection, one might also note that the 9th Clarke of the sixth Mingo of Cowznovska Madl 12 would be Julian Day 2,434,286 plus 1159, or December 3, 1955, while the 6th Clarke of the ninth Mingo of Cowznovska Madl 12 would be Julian Day 2,434,286 plus 1186, or December 30, 1955. Thus, as the latter date would fall during the Christmas holidays, during which the Physics Lab of Miluwakee Lutheran High School would be closed and inaccessible, we can take it that the historic date of 6-9-12, on which the measurement of MAD Magazine issue #26 was made, on which the Potrzebie system of measurments is based, is December 3, 1955, which, however, fell on a Saturday.

MAD Magazine #26 was the November, 1955 issue, and so it would likely have been available on newsstands sometime in October, 1955.

To the Julian Day through the Vague Year

Before the Julian Day system was invented, when Copernicus wanted a uniform sequence of days for calculating the motions of the planets, he used the ancient Egyptian calendar, which gave 365 days to each year without any leap years; 12 months of 30 days, and 5 extra epagomenal days. The ancient Mayan calendar included a component, the Haab, which worked similarly, but with 18 months of 20 days, and an extra period of 5 days called Uayeb.

First, one can go from the Gregorian calendar to the Julian calendar by noting that the difference in days between them changes by a day three times in every 400 years:

   0  100  200  300
  -2   -1    0    1

 400  500  600  700
   1    2    3    4

 800  900 1000 1100
   4    5    6    7

1200 1300 1400 1500
   7    8    9   10

1600 1700 1800 1900
  10   11   12   13

2000 2100 2200 2300
  13   14   15   16

The differences shown come into effect on March 1st of the Gregorian calendar, it being the more complicated one that includes the skipped leap year day that causes the difference to change. (Since the Julian calendar has more days, this difference is how many days sooner a given date occurs on the Gregorian calendar, or how many days later a given day is by the Gregorian calendar: thus, at the present time, with a difference of 13 days, Christmas on the Julian calendar falls on January 7th by the Gregorian calendar.)

The difference between the Julian calendar and the Egyptian calendar changes by one day every four years, and thus can be calculated by simple arithmetic.

July 20 on the Julian Calendar, the 141st day after March 1 of the year, or the 200th day of the year in a year that is not a leap year, was the Egyptian New Year's Day in 139 A.D. according to both Censorinus and Wikipedia. To avoid dealing with an exception for things happening in January and February of leap years, then, I will use a Julian calendar starting on March 1st, so that I can say the Egyptian calendar starts 141 days later than the Julian for the years 136, 137, 138, and 139, and then only 140 days later than the Julian for the years 140, 141, 142, and 143, and so on.

So this difference is 175 days for the years 1 BC, 1 AD, 2, and 3, and from this difference, one subtracts the year divided by four; the year 2000 divided by four yielding 500 days to subtract. Currently, then, this being 2007, the Julian calendar starts 13 days later than the Gregorian, and the Egyptian calendar starts 326 days earlier than March 1 of the Julian calendar, or 267 days earlier than the Julian calendar using January 1 again as the starting day.

Since the Haab section of the ancient Mayan calendar (which was called the xiuhpohualli by the Aztecs) is a cycle that is also exactly 365 days long every year, it has a fixed relation with the Egyptian vague year.

Thus, the traditional Egyptian calendar (ignoring the fact that leap years were added to it, bringing the Egyptian calendar into harmony with the Julian calendar, by the emperor Augustus, leading to its first leap year day in the year 22 BC) from April 2006 to December 2009, and the Haab section of the ancient Mayan calendar for that period, would proceed as shown in the table below:


April 9, 2006/7      * Thoth 1               9 Muan         11 Xocotlhuetzin               Mareri 1         April 8, 2008/9
April 20, 2006/7        Thoth 12          0 Pax              2 Ochpaniztli                  Mareri 12       April 19, 2008/9
May 9, 2006/7          Paophi 1             19 Pax        1 Teoleco                        Margach 1        May 8, 2008/9
May 10, 2006/7          Paophi 2          0 Kayab            2 Teoleco                      Margach 2       May 9, 2008/9
May 30, 2006/7          Paophi 22         0 Cumhu            2 Tepeihuitl                   Margach 22      May 29, 2008/9
June 8, 2006/7         Hathor 1              9 Cumhu        11 Tepeihuitl                  Hrotich 1        June 7, 2008/9
June 19, 2006/7         Hathor 12         0 Uayeb            2 Quecholli                    Hrotich 12      June 18, 2008/9
June 24, 2006/7         Hathor 17       * 0 Pop              7 Quecholli                    Hrotich 17      June 23, 2008/9
July 8, 2006/7         Choiak 1             14 Pop        1 Panquetzalitli                 Aweleach 1       July 7, 2008/9
July 13, 2006/7         Choiak 6            19 Pop           6 Panquetzalitli            * Nawasardi 1      July 12, 2008/9
July 14, 2006/7         Choiak 7          0 Uo               7 Panquetzalitli               Nawasardi 2     July 13, 2008/9
August 3, 2006/7        Choiak 27         0 Zip              7 Atemotzli                    Nawasardi 22    August 2, 2008/9
August 7, 2006/7       Tybi 1                4 Zip          11 Atemotzli                    Nawasardi 26    August 6, 2008/9
August 23, 2006/7       Tybi 17           0 Zotz             7 Tititl                       Hori 12         August 22, 2008/9
September 6, 2006/7    Mechir 1             14 Zotz       1 Izcalli                         Hori 26         September 5, 2008/9
September 12, 2006/7    Mechir 7          0 Tzec             7 Izcalli                      Sahmi 2         September 11, 2008/9
September 26, 2006/7    Mechir 21           14 Tzec       1 Nemontemi                       Sahmi 16        September 25, 2008/9
October 1, 2006/7       Mechir 26           19 Tzec     * 1 Atlcahualo                      Sahmi 21        September 30, 2008/9
October 2, 2006/7       Mechir 27         0 Xul              2 Atlcahualo                   Sahmi 22        October 1, 2008/9
October 6, 2006/7      Phamenoth 1           4 Xul           6 Atlcahualo                   Sahmi 26        October 5, 2008/9
October 21, 2006/7      Phamenoth 16        19 Xul        1  Tlacaxipehualiztli             Tre 11          October 20, 2008/9
October 22, 2006/7      Phamenoth 17      0 Yaxkin           2 Tlacaxipehualiztli           Tre 12          October 21, 2008/9
November 5, 2006/7     Pharmouthi 1         14 Yaxkin       16 Tlacaxipehualiztli           Tre 26          November 4, 2008/9
November 11, 2006/7     Pharmouthi 7      0 Mol              2 Tozoztontli                  K'aloch 2       November 10, 2008/9
December 1, 2006/7      Pharmouthi 27     0 Chen             2 Hueytozoztli                 K'aloch 22      November 30, 2008/9
December 5, 2006/7     Pachon 1              4 Chen          6 Hueytozoztli                 K'aloch 26      December 4, 2008/9
December 21, 2006/7     Pachon 17         0 Yax              2 Toxcatl                      Arach 12        December 20, 2008/9 
January 4, 2007/8      Payni 1              14 Yax          16 Toxcatl                      Arach 26        January 3, 2009
January 10, 2007/8      Payni 7           0 Sac              2 Etzalcualiztli               Mehekani 2      January 9, 2009
January 30, 2007/8      Payni 27          0 Ceh              2 Tecuilhuitontli              Mehekani 22     January 29, 2009
February 3, 2007/8     Epiphi 1              4 Ceh           6 Tecuilhuitontli              Mehekani 26     February 2, 2009
February 19, 2007/8     Epiphi 17         0 Mac              2 Hueytecuilhuitl              Areg 12         February 20, 2009
March 5, 2006/7        Mesore 1             14 Mac          16 Hueytecuilhuitl              Areg 26         March 4, 2008/9
March 11, 2006/7        Mesore 7          0 Kankin           2 Tlaxochimaco                 Ahekani 4       March 10, 2008/9
March 31, 2006/7        Mesore 27         0 Muan             2 Xocotlhuetzin                Ahekani 24      March 30, 2008/9
April 4, 2006/7        Epagomenes 1          4 Muan          6 Xocotlhuetzin                Ahekani 26      April 3, 2008/9
April 9, 2006/7      * Thoth 1               9 Muan         11 Xocotlhuetzin               Mareri 1         April 8, 2008/9
April 20, 2006/7        Thoth 12          0 Pax              2 Ochpaniztlin                 Mareri 12       April 19, 2008/9
May 9, 2006/7          Paophi 1             19 Pax        1 Teoleco                        Margach 1        May 8, 2008/9
May 10, 2006/7          Paophi 2          0 Kayab            2 Teoleco                      Margach 2       May 9, 2008/9
May 30, 2006/7          Paophi 22         0 Cumhu            2 Tepeihuitl                   Margach 22      May 29, 2008/9
June 8, 2006/7         Hathor 1              9 Cumhu        11 Tepeihuitl                  Hrotich 1        June 7, 2008/9
June 19, 2006/7         Hathor 12         0 Uayeb            2 Quecholli                    Hrotich 12      June 18, 2008/9

and one could add a year, the dates up to February 9, 2007 shown above being valid for 2007/2008 as well as 2006/2007. Since 2008 is a leap year, Epiphi 27 and 0 Ceh would fall on February 29, 2008, and following dates would also arrive a day earlier by our calendar, as the table above illustrates.

It should be noted, though, that some sources claim that Haab component of the Mayan calendar was kept in harmony with the seasons; every fourth year, there were six, rather than five, epagomenal days, so that the day 0 Pop corresponded, at least most of the time, to July 16th in the Julian calendar. Others claim that the Mayan calendar was a more accurate representation of the tropical year than even the Gregorian calendar. This is not the case for the Haab calendar as generally used, 365 days without any intercalation, but it is possible that the Maya did keep track of the advance of the tropical year relative to the Haab using an approximation closer than that on which the Gregorian calendar was built. But while the adance of the seasons was noted, it was not used to modify the calendar itself.

In the absence of intercalation, a year of 365 days is not relatively prime to the 20-day week of the Tzolkin component of the Mayan calendar. As a result, the year can only begin on the days Kan, Muluc, Ix, or Cauac.

The claim of a cycle like the Julian for the Mayan calendar is accompanied by the statement that years beginning on Cauac are leap years. Adding a day to the Haab calendar alone would, of course, result in succeeding years beginning with Chicchan, Oc, Men, and Ahau in order. This means that, for this to be the rule, the intercalary day must stand outside the Tzolkin cycle as well as being intercalary to the Haab cycle, thus making it invisible: that is, one cannot tell that intercalation has taken place by its effect on the day of the week on which the year starts.

There is reason to believe that at least some Aztecs did adopt the intercalation rule of the Julian calendar in this fashion for a time after the Conquest, but no reason to think this was done prior to the Conquest by the Maya.

The year of the seasons was monitored, rather than simply relying on the Haab to supply the dates for planting and harvesting; this was done by noting the day on which the Pleiades first rose before sunrise. Thus, unless the precession of the equinoxes was noted, one would expect any approximation to the offset between the seasons and the Haab to be based on the sidereal year instead of the tropical year.

In addition to the Haab and Tzolkin calendars, which were used by the Aztecs (and the Toltecs, who may have been an Aztec myth) after the Maya, and by the Zapotec and Olmec before them, the Maya used a system of dating unique to themselves, called the Long Count which resembles our system of Julian Day numbers closely.

In this system, the date 0.0.0.0.0 may have stood for September 6, 3114 B.C. by the Julian calendar (Julian day 584284 as I note it here, since on that day the actual Julian Day number goes from 584283.5 to 584284.5, others will note that date as 584283). The numbers are recorded in a modified base-20 system, the second-from last digit being in base 18 so that the last two digits give the date in a span of 360 days.

Despite the fact that the Haab calendar also used by the Maya had 18 months of 20 days, plus 5 intercalary days, the Long Count did not note the intercalary days as ending in numbers from 18.0 to 18.4, but did instead count the days in multiples of 360 rather than counting the years.

And What's So Special About 4713 B.C.?

For the purpose of easily converting to Julian Day Numbers, what one could simply do is note that just as a vague year is a regular and unvarying succession of 365-day periods, the Julian calendar is a regular and unvarying succession of 1,461-day periods, since there is a leap year in every four, because the difference between the Gregorian calendar and the Julian calendar changes only infrequently; conversion to that calendar being therefore much simpler than conversion to a vague year.

In the Julian calendar, the sequence of four years making up 1,461 days is completely unbroken, and thus the sequence of calendars for 28 successive years always repeats.

If we look up the year 4714 B.C. in the table on this earlier page, since, for simplicity, it is based on a year that starts on the first of March, we will find that January 1st, 4713 B.C. falls on a Sunday, and, further, that it is a leap year.

Note that since 1 B.C. was the year before 1 A.D., it, too, was a leap year, so the fact that this year is odd-numbered should not be surprising.

The Julian period, on which the Julian Day Number was later based by William Herschel, was based on the recurrence of three calendrical cycles; one was the 28-day cycle of repetitions of the Julian calendar, called the solar cycle, so we see here that its starting point was taken to be a leap year that started on a Sunday. Incidentally, the year immediately following a leap year that would start on a Sunday, as might be more likely to be thought of as the natural starting point for this cycle, would be the sixth year of the cycle actually used by Joseph Justus Scaliger.

The next one of the three cycles used was the Metonic cycle, the 19-year cycle which approximately harmonizes the lunar months with the solar calendar.

A 19-year cycle which begins in 4713 B.C. and ends in 4695 B.C. would coincide with one that starts in 1 B.C. and ends in 18 A.D., and it would also, then, coincide with one that starts in 1919 and ends in 1937, or one that began in 1995 and ends in 2013. And, indeed, for the 19 years beginning in 1995 and ending in 2013, their Golden Numbers start with 1 and end with 19, in order.

These two cycles, a 28-year cycle and a 19-year cycle, create a period which repeats every 532 years.

In Canada, the national census is held every 10 years, on years ending with the digit 1; so a complete census was performed in 1961, 1971, and 1981 and so on; the census day is currently in the middle of May, although that appears to have varied historically. The census of Great Britain is also performed on such years, but near the end of April.

In the United States, censuses are also held every 10 years, but one year earlier, on years ending in the digit 0; the census day is near the start of April.

At one time in the Roman Empire, the amount of taxes to be paid by each of the various provinces to the Empire was set once every fifteen years. This period was called the Indiction,

The first year of an Indiction would be a year whose number is equal to 13 modulo 15, so the years 1996 through 2010 represent one indiction (or, historically, the period from September 23, 1995 through September 22, 2010), the years from 3 B.C. to 12 A.D. would be another indiction, and, since 4710 is a multiple of 30, the years 4713 B.C. through 4708 B.C would also be one indiction.

This multiplies the length of the cycle by 15, yielding a cycle of 7.980 years. Thus, a single cycle extends from 4713 B.C. to 3267 A.D., and is long enough to cover all of recorded history.


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