Int. J. Communications, Network and System Sciences, 2011, 4, 483-486
doi:10.4236/ijcns.2011.48059 Published Online August 2011 (
Copyright © 2011 SciRes. IJCNS
Astronomical Algorithms:
Amended Multi-millennia Calendar
Boris S. Verkhovsky
Computer Scien ce Department, New Jersey Institute of Technology,
University Heights, Newark, USA
Received May 21, 201 1; revised June 16, 2011; accepted June 20, 2011
Three new worldwide calendars are proposed and compared in this paper. None of them requires any depar-
ture from an existing tradition to divide years on lean and leap. Although all three are pretty accurate, it is
demonstrated that the Julian calendar with one additional amendment is the simplest and the most suitable
for implementation.
Keywords: Astronomic Algorithm, Julian Calendar, Gregorian Calendar, Error Accumulation, Amendments,
Multi-millennia Calendar, Synchronization
1. Introduction and Gregorian Calendar
Modern computing and communication systems require
immensely high degree of synchronization, which in its
turn requires measurement of time with great precision.
Nowadays, atomic clock can measure time with accuracy
smaller than one billionth of a second. Without such a
precision numerous navigation systems including the
GPS and other equivalents will not be able to properly
Yet, modern calendars are highly inaccurate. Human-
kind has used and is currently using many calendars:
Egyptian Calendar was introduced in the V Millennium
BCE; the first year of Jewish calendar is 3760 BCE;
Mayan chronology started from 3372 BCE. Six hundred
years later (in 2772 BCE) Egypt adopted a calendar of
365 days without adjustments. Babylonian and Chinese
astronomers learned about planetary movements in VIII
century BCE. Julian calendar with leap years was
adopted in 46 BCE, although the idea has been proposed
193 years earlier by Aristarchus from Alexandria. The
average length of a year in the Julian calendar is 365.25
days. This is significantly different from the “real” length
of the solar year. An error accumulates so fast that after
about 131 years the calendar is out of sync by one day.
By the 16th Century this affected the determination of
the date of Easter.
Gregorian calendar was introduced in 1582 initially in
France and the Netherlands, and 170 year later in Eng-
land by an Act of Parliament. French Republican calen-
dar was adopted in 1790s. Although Gregorian calendar
was adopted in 1923 in th e USSR, the Russian Orthodox
Church does not recognize it and still celebrates Christ-
mas thirteen days later. More details about various cal-
endars are provided in [1-4]. Great mathematician and
astronomer Carl F. Gauss proposed an algorithm that
calculates Easter Sundays in the Gregorian calendar [5].
The Gregorian calendar is proposed by astronomers C.
Clavius and L. Lilio. It is basically a Julian calendar with
two amendments, {historic details are provided below}.
It is reasonable to assume that when the Gregorian
calendar has been introduced in XVI century, one of
considerations was that such amendments would be easy
to implement. Indeed, even with the generally low level
of education at that time, there was no difficulty to rec-
ognize what year was divisible by 100 and what was
divisible by 400. With this pattern, a modified Julian
calendar requires not two, but either three additional
amendments {as demonstrated in (10)-(14)} or five addi-
tional amendments {as demonstrated in (17)-(19)}.
The existing Western calendar basically fluctuates
between lean an d leap years. A year is leap if it is divisi-
ble by four. However, a year ending with 00 is leap only
if it is divisible by 400. For instance, 1900 was lean year,
but 1600 and 2000 were leap years.
Let y be the y-th year of New Era. In order to find how
many days
Ny are in the y-th year, we use formula
specified by Gregorian calendar:
365 ,,,
yaf ybf yab , (1)
where ; (2)
,; 4,100zaba b
and let
1, ifmod0
,:0, ifmod0
fyz yz
. (3)
Therefore, Equations (1)-(3) specify that leap years
must be added every four years with an exception of
those years that are divisible by 100, but not divisible by
400. Assuming that 1= 365 and 2= 365.25, Formula
(1) takes into account that the astronomic year, i.e., the
period of rotation of the Earth around Sun is
= 365.242 days [6].
PFor further verification, we also calculate the number
of days at the end of the y-th year from the be-
ginning of the New Era .
365Gy y4 100400yy y 
 
 
. (4)
For instance, at the end of the 100th year
at the end of t he 400th
year and
at the end of the First Millennium . Yet, as it
is demonstrated below, Gregorian calendar is correct for
a relatively small number of centuries.
10036,50025 136,524;G
40036,500 400 100G
1000365,000 25010 2G
4 1144,097;
2. Average Astronomical Year
An average astronomical year is defined as a period with
which the Earth rotates around Sun. The time from one
fixed point, such as an equinox, to the next is called a
tropical year. Its current length is 365.242190 days, but
it fluctuates. In 1900 the period was 365.242196 days,
and in 2100 it will be 365.242184 days. Thus, to get
more accurate results, we must consider that in average
P = 365.24219, [7,8]. (5)
3. Inaccuracy in Gregorian Calendar
The first five millennia: Let
:Dy yP (6)
Therefore, using P, we derive by direct computation
that for y = 5000
5000D= 1,826,211.
Yet, by Formul a (4) for G reg o ri an calendar
3655000 125050 1500021,826,.G 212 (7)
Thus, = 1. (8)
Hence, as can be seen in (8), even the more accurate
period of rotation does not eliminate
the discrepancy between the number of rotations around
Sun and the corresponding number of days computed on
the basis of Gregorian calendar.
The first ten millennia: For y = 10,000
10000D= 3,652,422. (9)
Yet, from (4)
10000G= 3,650,000 + 2500100 + 25 = 3,652,425.
Hence, a three-day discrepancy will accumulate by the
year y = 10,000 if the Gregorian calend ar is us ed.
4. Astronomic Algorithm with the Fourth
In the past, three amendments were introduced into the
World calendar:
1) The Egyptians introduced 365 days instead of 360
days (the 1st Amendment);
2) 365.25 days were introduced by Julian calendar (the
2nd Amendment);
3) Further corrections were introduced by Gregorian
calendar (the 3rd Amendment), {see (1)-(3)}.
However, even these amendments do not provide an
accurate account for a large period of time measured in
millennia. To make the counting of the days more accu-
rate, an additional (fourth) term should be introduced.
Step1.1: if 3200 divides y
then y-th year is lean; stop;
Step2.1: if 400 divides y
then y-th year is leap; stop;
Step3.1: if 100 divides y
then y-th year is lean; stop;
Step4.1: if 4 divides y
then y-th year is leap; stop;
else y-th year is lean; stop. (10)
The 4-th amendment provides a more accurate count-
ing of the days for dozens of millennia ahead. From the
fourth amendment in the astronomic algorithm (10) it
follows that the 3200-th year, 640 0-th year, 9600-th year
etc. must be the lean years. As a result,
3654100400 3200Vyy yyyy 
  
  
Finally, let
4, 100,400,3200;z (12)
and let
1, ifmod0
fyz yz
365 ,4
 
,100 ,400 ,3200fy fyfy 
Copyright © 2011 SciRes. IJCNS
However, there is a simpler solution with only one
amendment to Julian calendar.
5. Julian Calendar with One Amendment
Step1.2: if 128 divides y
then y-th year is lean; stop;
Step2.2: if 4 divides y
then y-th year is leap
else y-th year is lean; stop. (15)
Mnemonic rule: After thirty one leap years skip one.
6. Discrepancy Analysis
3654 128Byyyy 
; (16)
then is the nu mber of days at the end of y-th year
from the beginning of the New Era if the amended Julian
calendar is used.
Let’s compute for y = 2,000; 3,200; 10,000;
50,000; and 100,000; and compare them with corre-
sponding values for , and
Vy {see
Table 1}:
From Ta ble 2, it follows that Julian calendar with one
additional amendment, or the Gregorian calen dar with an
additional amendment, provides the same accuracy. Thus,
from our point of view, the former one should be imple-
mented as the simplest.
Besides, it will be easy to introduce the Amended
Julian calendar, because the first correction is required in
2048. Hence, the world community has sufficient time to
incorporate this simple and beautiful calendar. Obviously,
it is up to the internation al community to decide how the
new calendar will be called. Excerpts of this paper were
published in [9].
Table 1. Total number of days at the end of y-th year.
Errors y = 2000 y = 3200 y = 10,000 y = 50,000 y = 100,000
y 730,484.38 1,168,775.008 3652421.9 18,262,109.536,524,219
y 730,485 1,168,775 3,652,422 18,262,110 36,524,219
Gy 730,485 1,168,776 3,652,425 18,262,125 36,524,250
Vy 730,485 1,168,776 3,652,422 18,262,110 36,524,219
Table 2. Accumulated errors in days.
Errors y = 2000 y = 3200 y = 10,000 y = 50,000 y = 100,000
 
By Dy 0.62 0.008 0.1 0.5 0
 
Gy Dy 0. 62 0.992 3.1 15.5 31.0
 
Vy Dy 0.62 0.992 0.1 0.5 0
7. Alternative Algorithm for Calendar
Step1.3: If 100,000 divides y
then y-th year is lean; stop;
Step2.3: if 5000 divides y + a
then y-th year is lean; stop;
Step3.3: if 2000 divides y
then y-th year is lean; stop;
Step4.3: if 400 divides y
then y-th year is leap; stop;
Step5.3: if 100 divides y
then y-th year is lean; stop;
Step6.3: if 4 divides y
then y-th year is leap
else y-th year is lean; stop. (17)
In this case
20005000 100000AyGyyy ay 
where a is an even positive integer such that mod 40a
or a = 100.
,2000Sy Ny fy
,5000 ,100000fya fy. (19)
8. Historic Details
The Julian reform: Originally the Romans numbered
years ab urbe condita , that is, “from the founding of the
city” (of Rome). After his conquest of Egypt in 48 B. C.
Julius Caesar realized that the a.u.c. calendar was totally
inappropriate to the needs of the new empire.
Peculiarities in modern calendar: Originally the Julian
calendar was simple: all odd months had odd number of
days (31 days each), and all even months, except February,
had even number of days (30 days each). February had 29
days in lean years and 30 days in leap years. The seventh
month was named in honor of Julius Caesar. Later the first
Roman Emperor Caesar Augustus renamed the eighth
month as Augustus and the number of days in August,
previously 30, now became 31 (the same as the number of
days in July), “so that Augustus Caesar would not be re-
garded as inferior to Julius Caesar”.
Consequently in the remaining months (September-
December) the numbers of days were swapped from 31
to 30 and vice verse. The extra day needed for August
was taken from the end of February.
Remark: Maybe it will never become known why
Caesar Augustus did not decide to rename the fifth or
ninth month, which had thirty one days. Amazingly, this
historic nonsense perpetuates more than two millennia.
Copyright © 2011 SciRes. IJCNS
Copyright © 2011 SciRes. IJCNS
Table 3. Number of days in months.
Jan Feb Mar Apr May Jun
31 30 31/30 30 31 30
Jul Aug Sep Oct Nov Dec
31 30 31 30 31 30
The Gregorian reform: Pope Paul III recruited several
astronomers to come up with a solution; one of them was
Christopher Clavius (1537-1612). Various calendar reforms
were proposed. When Pope Gregory XIII has been elected,
he decided in favor of Clavius’ reform [10,11].
9. Alternate System of Counting Days in
The self-explanator y Table 3 provides an alternate sys-
tem of counting days in every month of each leap year;
where March has thirty days in lean years .
10. Acknowledgements
I express my appreciation to J. W. Jones and J. Scher for
comments, to W. Ambers for assistance in numerous
computer experiments, to anonymous reviewers and the
typesetter for their suggestions that improved this paper.
11. References
[1] L. E. Doggett, “Calendars,” In: P. K. Seidelmann, Ed.,
Explanatory Supplement to the Astronomical Almanac,
US Naval Observatory, University Science Books Com-
pany, Mill Valley, 1992.
[2] D. D. McCarthy and P. K. Seidelmann, “TIME from
Earth Rotation to Atomic Physics,” Chapter 10 & 11,
Wiley-VCH, Weinheim, 2009.
[3] J. H. Meeus, Astronomical Algorithms,” 2nd Edition,
Willmann-Bell, Inc., Richmond, D ecember 1999.
[4] B. Crun, “The Timetables of History: A Horizontal Link-
age of People and Events,” Si mo n and Schuster, New Yor k,
[5] C. F. Gauss, “Berechnung des Osterfestes,” Monatliche
Correspondenz zur Beförderung der Erd- und Himmels-
Kunde, Vol. 2, August 1800, pp. 121-130.
[6] K. Diem and C. Lentner, “Scientific Tables: Documenta
Geigy,” 8th Edition, Ciba-Geigy, Tarrytown, 1986, pp.
[7] B. A. Archinal, “Terrestrial Coordinates and the Rotation
of the Earth,” Chapter 4, In: P. K. Seidelmann, Ed., Ex-
planatory Supplement to the Astronomical Almanac, US
Naval Observatory, University Science Books Company,
Mill Valley, 1992, p. 253.
[8] C. Y. Hohenkerk, B. D. Yallop, C. A. Smith and A. T.
Sinclair, “Celestial Reference Systems,” Chapter 4, In: P.
K. Seidelmann, Ed., Explanatory Supplement to the As-
tronomical Almanac, US Naval Observatory, University
Science Books Company, Mill Valley, 1992, p. 116.
[9] B. Verkhovsky, “Amended Julian Worldwide Calendar,”
Annals of European Academy of Sciences, ISSN 1379-
1982, European Academy of Sciences, Brussels, 2003, pp.
[10] C. Clavius, “Romani Calendarii a Gregorio XIII Restituti
Explicatio,” Opera Mathematica, Vol. 5, 1603, p. 13.
[11] Gregory XIII, “Inter Gravissimas,” W. Spenser and R. T.
Crowley, Translation, International Organization for Stan-
dardization, Geneva, 2002 (Original work published in