Thursday, April 29, 2010

Some Reading on Calendars

Some Reading on Calendars

I was doing some reading on Calendars from an "imap" software package (SFWimap) with source code originating from http://www.Washington.EDU/imap and on an older Solaris 10 machine in this location (/opt/sfw/doc/imap/calendar.txt) . I found it so interesting that I decided to copy-paste it!

All About Calendars

                         ALL ABOUT CALENDARS

Although one can never be sure of what will happen at some future
time, there is strong historical precedent for presuming that the
present Gregorian calendar will still be in effect within the useful
lifetime of the IMAP toolkit. We have therefore chosen to adhere to
these precedents.

The purpose of a calendar is to reckon time in advance, to show
how many days have to elapse until a certain event takes place in the
future, such as the harvest or the release of a new version of Pine.
The earliest calendars, naturally, were crude and tended to be based
upon the seasons or the lunar cycle.


The calendar of the Assyrians, for example, was based upon the
phases of the moon. They knew that a lunation (the time from one full
moon to the next) was 29 1/2 days long, so their lunar year had a
duration of 354 days. This fell short of the solar year by about 11
days. (The exact time for the solar year is approximately 365 days, 5
hours, 48 minutes, and 46 seconds.) After 3 years, such a lunar
calendar would be off by a whole month, so the Assyrians added an extra
month from time to time to keep their calendar in synchronization with
the seasons.

The best approximation that was possible in antiquity was a 19-year
period, with 7 of these 19 years having 13 months (leap months). This
scheme was adopted as the basis for the lunar calendar used by the
Hebrews. The Arabs also used this calendar until Mohammed forbade
shifting from 12 months to 13 months; this causes the Muslim holy month
of Ramadan to move backwards through the seasons, completing a cycle
every 32 1/2 years.

When Rome emerged as a world power, the difficulties of making a
calendar were well known, but the Romans complicated their lives because
of their superstition that even numbers were unlucky. Hence their
months were 29 or 31 days long, with the exception of February, which
had 28 days. Every second year, the Roman calendar included an extra
month called Mercedonius of 22 or 23 days to keep up with the solar


Even this algorithm was very poor, so that in 45 BCE, Caesar,
advised by the astronomer Sosigenes, ordered a sweeping reform. By
imperial decree, the year 46 BCE was made 445 days long to bring the
calendar back in step with the seasons. The new calendar, similar to
the one we now use was called the Julian calendar (named after Julius

Months in the Julian calendar were 30 or 31 days in length and
every fourth year was made a leap year (having 366 days) by adding a day
to the end of the year. This leap year rule was not consistantly
applied until 8 CE. The year-ending month of February, never a popular
month, was presently shortened so that Julius Caesar and Emperor
Augustus could each have long months named after them.

Caesar also decreed that the year would start with the first of
January, which since 153 BCE was the day that Roman consuls took office,
and not the vernal equinox in late March. Not everyone accepted that
part of his reform, as we shall see.


Caesar's year was 11 1/2 minutes short of the calculations
recommended by Sosigenes and eventually the date of the vernal equinox
began to drift. Roger Bacon became alarmed and sent a note to Pope
Clement IV, who apparently was not impressed. Pope Sixtus IV later
became convinced that another reform was needed and called the German
astronomer, Regiomontanus, to Rome to advise him. Unfortunately,
Regiomontanus died of the plague shortly thereafter and the plans died
as well.

In 1545, the Council of Trent authorized Pope Gregory XIII to
reform the calendar once more. Most of the mathematical work was done
by Father Christopher Clavius, S.J. The immediate correction that was
adopted was that Thursday, October 4, 1582 was to be the last day of the
Julian calendar. The next day was Friday, with the date of October 15.
For long range accuracy, a formula suggested by the Vatican librarian
Aloysius Giglio was adopted. It said that every fourth year is a leap
year except for century years that are not divisible by 400. Thus 1700,
1800 and 1900 would not be leap years, but 2000 would be a leap year
since 2000 is divisible by 400. This rule eliminates 3 leap years every
4 centuries, making the calendar sufficiently correct for most ordinary
purposes. This calendar is known as the Gregorian calendar and is the
one that we now use today.

It is interesting to note that in 1582, all the Protestant princes
ignored the papal decree and so many countries continued to use the
Julian calendar until either 1698 or 1752. Britain and its American
colonies went from Wednesday, September 2, 1752 to Thursday, September
14. Prior to the changeover, the British used March 25 as the start of
the new year.

In Russia, it needed the revolution to introduce the Gregorian
calendar in 1918. Turkey didn't adopt it until 1927.


The numbering of the year is generally done according to an "era",
such as the year of a ruler's reign.

In about 525, a monk named Dionysius Exiguus suggested that the
calculated year of Jesus' birth be designated as year 1 in the Julian
calendar. This suggestion was adopted over the next 500 years and
subsequently followed in the Gregorian calendar.

For the benefit of those who seek religious significance to the
calendar millenium, note that year 1 is too late by at least 4 years.
Herod the Great, named in the Christian Bible as having all children in
Bethlehem put to death in an attempt to kill the infant Jesus, died in 4

Nothing particularly significant of an historic or religious nature
happened in Gregorian year 1; however it has become a worldwide standard
as the "common era." In modern times, the terms "CE" (common era) and
"BCE" (before common era) are preferred over the earlier (and, as we
have seen, less accurate) "AD" (anno Domini, "the year of the Lord") and
"BC" (before Christ).

The Hebrew lunar calendar begins at 3760 BCE, the year of creation
in Jewish tradition. The Muslim lunar calendar begins on July 16, 622,
when Mohammed fled from Mecca to Medina.

The Japanese, Taiwanese, and North Koreans use the Gregorian
calendar, but number the year by political era. In Japan, an era begins
when an emperor succeeds to the throne; year 1 of the Heisei era was
1989 when Emperor Akihito ascended to the throne (the first few days of
1989 was year 64 of the Shouwa era). In Taiwan, year 1 is the year of
founding of the Republic of China (1911). In North Korea, year 1 is the
year of the Juche (self-reliance) ideal, corresponding to the birth year
of founder Kim Il-Sung (1912). Thus, year 2000 is Heisei 12 (Japan),
90th year of the Republic (Taiwan), and Juche 89 (North Korea).


Despite the great accuracy of the Gregorian calendar, it still
falls behind very slightly every few years. The most serious problem
is that the earth's rotation is slowing gradually. If you are very
concerned about this problem, we suggest that you tune in short wave
radio station WWV or the Global Positioning System, which broadcasts
official time signals for use in the United States. About once every
3 years, they declare a leap second at which time you should be
careful to adjust your system clock. If you have trouble picking up
their signals, we suggest you purchase an atomic clock (not part of
the IMAP toolkit).

Another problem is that the Gregorian calendar represents a year
of 365.2425 days, whereas the actual time taken for the earth to
rotate around the Sun is 365.2422 days. Thus, the Gregorian calendar
is actually 25.92 seconds slow each year, resulting in the calendar
being one day behind every 3,333 1/3 years.

Consequently, the Gregorian calendar has been modified with a
further rule, which is that years evenly divisible by 4000 are not
leap years. Thus, the year 4000 will not be a leap year. Or, at
least we assume that's what will happen assuming that the calendar
remains unchanged for the next 2000 years.

The modified Gregorian calendar represents a year of 365.24225
days. Thus, the modified Gregorian calendar is actually 4.32 seconds
slow each year, resulting in the calendar being one day slow every
20,000 years.

There is some dispute whether the modified Gregorian calendar was
officially adopted, or if it's just a proposal. Other options (see
below) exist; fortunately no decision needs to be made for several
centuries yet.

There is code in c-client to support the modified Gregorian
calendar, although it is currently disabled. Sometime in the next
2000 years, someone will need to enable this code so that c-client is
Y4K compiliant. Then, 18,000 years from now, someone will have to
tear into c-client's code to fix the Y20K bug.


The Eastern Orthodox church in 1923 established its own rules to
correct the Julian calendar. In their calendar, century years modulo
900 must result in value of 200 or 600 to be considered a leap year.
Both the Orthodox and Gregorian calendar agree that the years 2000 and
2400 will be leap years, and the years 1900, 2100, 2200, 2300, 2500,
2600, 2700 are not. However, the year 2800 will be a leap year in the
Gregorian calendar but not in the Orthodox calendar; similarly, the
year 2900 will be a leap year in the Orthodox calendar but not in the
Gregorian calendar. Both calendars will agree that 3000 and
3100 are leap years, but will disagree again in 3200 and 3300.

There is code in c-client to support the Orthodox calendar. It
can be enabled by adding -DUSEORTHODOXCALENDAR=1 to the c-client
CFLAGS, e.g.

The Orthodox calendar represents a year of 365.24222222... days.
Thus, the Orthodox calendar is actually 1.91 seconds slow each year,
resulting in the calendar being one day slow every 45,000 years. The
Eastern Orthodox church has not yet made any statements on how the
Y45K bug will be fixed.


The effect of leap seconds also needs to be considered when
looking at the Y20K and Y45K problems. Leap seconds put the clock
back in line with the Earth's rotation, whereas leap years put the
calendar back in line with the Earth's revolution. Since leap seconds
slow down the clock (and hence the calendar), they actually bring the
day of reckoning for the Gregorian and Orthodox calendars sooner.

Another factor is that the next ice age (technically, the end of
the current interglacial period; we are in the middle of an ice age
now!) is due around Y25K. It is not known what perturbations this will
cause on the Earth's rotation and revolution, nor what calendar
adjustments will be necessary at that time.


The names of days of the week from a combination of Roman and
Germanic names for celestial bodies:
. Sunday Latin "dies solis" => "Sun's day"
. Monday Latin "dies lunae" => "Moon's day"
. Tuesday Germanic "Tiw's day" => "Mars' day"
. Wednesday Germanic "Woden's day" => "Mercury's day"
. Thursday Germanic "Thor's day" => "Jupiter's day"
. Friday Germanic "Frigg's day" => "Venus' day"
. Saturday Latin "dies Saturni" => "Saturn's day"


The names of the months are from the Roman calendar:
. January Janus, protector of doorways
. February Februalia, a time for sacrifice to atone for sins
. March Mars, god of war
. April Latin "aperire" => "to open" buds
. May Maia, goddess of plant growth
. June Latin "juvenis" => "youth"
. July Julius Caesar
. August Augustus Caesar
. September Latin "septem" => "seven"
. October Latin "octo" => "eight"
. November Latin "novem" => "nine"
. December Latin "decem" => "ten"

As you'll notice, the last four months are numbered 7 to 10, which
is an artifact of the time when the new year started in March.


There's another reason why the historical starting of the new year
is significant. Starting with March, the length of months follows a
mathematical series:
31 30 31 30 31 31 30 31 30 31 31 28

This means that you can calculate the day of week for any
arbitrary day/month/year of the Gregorian calendar with the following
formula (note all divisions are integral):
_ _
| 7 + 31*(m - 1) y y y |
dow = | d + -------------- + y + - - --- + --- | MOD 7
|_ 12 4 100 400_|
d := day of month (1..31)
m := month in old style (March = 1..February = 12)
y := year in old style
dow := day of week (Tuesday = 0..Monday = 6)

To convert from new style month/year to old style:
if (m > 2) m -= 2; /* Mar-Dec: subtract 2 from month */
else m += 10,y--; /* Jan-Feb: months 11 & 12 of previous year */

Here's another fun formula. To find the number of days between two
days, calculate a pair of calendar days with the formula (again, all
divisions are integral), using new style month/year this time:
m + -
8 y y y
d + 30 * (m - 1) + ----- + y * 365 + - - --- + --- - ld
2 4 100 400

d := day of month (1..31)
m := month in new style (January = 1..December = 12)
y := year in new style
ld := leap day correction factor:
0 for January and February in non-leap years
1 for January and February in leap years
2 for all other months in all years

In C code, the leap day correction factor is calculated as:
(m < 3) ? !(y % 4) && ((y % 100) || !(y % 400)) : 2

It's up to you to figure out how to adapt these formulas for the
Y4K bugfix and the Orthodox calendar. If you're really clever, try to
use these formulae to implement the C library ctime(), gmtime(), and
mktime() functions. Most C library implementations use a table of the
number of days in a month. You don't need it.


The original version is from an old Digital Equipment Corporation SPR
answer for VMS. Modifications for c-client, and additional information
added by Mark Crispin.

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