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[sci.astro] Time (Astronomy Frequently Asked Questions) (3/9)
Section - C.07.2 Can I calculate the date of Easter?

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John Horton Conway (the Princeton mathematician who is responsible for
"the Game of Life") wrote a book with Guy and Berlekamp, _Winning
Ways_, that describes in Volume 2 a number of useful calendrical
rules, including How to Calculate the Day of the Week, Given The Date,
and Easter.  Here's a brief precis of how to calculate Easter:

  G(the Golden Number) = Year_{mod 19} + 1 (never forget to add the 1!)

  C(the Century term) = +3 for all Julian years (i.e., if using the
                         Julian Calendar)
                        
                        -4 for 15xx, 16xx        }
                        -5 for 17xx, 18xx        } Gregorian
                        -6 for 19xx, 20xx, 21xx  }

The general formula for C in a Gregorian year Hxx is

	C = -H + [H/4] + [8*(H+11)/25]  (brackets [] mean integer part)

1)  The Paschal Full Moon is given by the formula

  (Apr 19 = Mar 50) - (11*G+C)_{mod 30}

Except when the formula gives Apr 19 you should take Apr 18, and when it
gives Apr 18 and G>=12 you should take Apr 17.  Easter is then the
following Sunday, since Easter always falls on the next Sunday that is
_strictly later_ than the Paschal Full Moon.

Example: 1945 = 7 mod 19, so G = 8 and we find for the Paschal Full Moon

       Mar 50 - (88-6)_{mod 30} = Mar 50 - 22 = Mar 28.

This happens to be a Wednesday (by Horton's "Doomsday" rule for Day of
the Week, see below).  Therefore, Easter 1945 took place on Sunday,
April 1.

Conway's "Doomsday" method for finding the day of the week, given the
date, is needed for his Easter method.

To every year there is a distinguished day of the week, which Conway
calls the "Doomsday", D.  In any year, if March 0 (the last day of
February) falls on a particular DOW, then the following dates also
fall on the same DOW: 4/4, 6/6, 8/8, 10/10, 12/12.  Also 5/9, 9/5,
7/11, 11/7 (for which he has devised the mnemonic "I went to my
nine-to-five job at the Seven-Eleven.  Note to non-US readers:
"Seven-Eleven" is the name of a ubiquitous chain of convenience
stores.)  In non-leap years, Jan 3 and Feb 0 (Jan 31) also fall on
that DOW; in leap years, Jan 4 and Feb 1.  Conway calls this DOW the
"doomsday" for that year.

For example, in 1995 Doomsday is Tuesday.  Columbus Day (10/12) is two
days after 10/10, a Tuesday, so 10/12 is a Thursday.

All that remains is a rule for calculating the Doomsday for any year.
In any century, this is done by taking the last two digits of the
year, call them xx, dividing by 12 to get a quotient Q and remainder
R.  Divide R by 4 to get a second quotient Q2.  Then this century,
the Doomsday for that year is given by Wednesday + Q + R + Q2.  In
1995, for example, we have 95/12 = 7 with remainder 11; 11/4 gives
quotient 2; Wednesday + 7 + 11 + 2 = Tuesday (cf. above).

In other years on the Gregorian calendar, one uses instead of
Wednesday, the century day as follows: 16xx and 20xx: Tuesday; 17xx
and 21xx: Sunday; 18xx and 22xx: Friday; 15xx, 19xx and 23xx:
Wednesday.  The cycle repeats over a 4 century period.

If you need the DOW on the Julian calendar, the rules are the same
except that the century rule is different: for a date in the year ccxx,
use -cc for the century day of week, where Sunday = 0.  For example,
October 4, 1582 (the last day of the Julian calendar in countries that
followed Pope Gregory's institution of the Gregorian calendar) took
place as follows:

  82/12 = 6 remainder 10; 10/4 gives remainder 2; 6+10+2-15= 3,
  which is Wednesday.  10/10 was Wednesday, 10/3 was Wednesday, so
  10/4/1582 (Julian) was a Thursday.

  The following day was October 15, 1582 (Gregorian).  Again we
  can check: 6+10+2+Wed = Sunday.  10/10 was a Sunday (Gregorian)
  so 10/15/1582 (Gregorian) was a Friday.

The nice thing about these algorithms is that they can easily be done in
one's head with a little practice (OK, mod 19 for the Golden Number is a
bit hairy for me, but I can still do it!).  The DOW calculation is very
useful if you are caught without a calendar, and it makes a good party
trick.

Additional information is available at
<URL:http://quasar.as.utexas.edu/BillInfo/doomsday.html> and
<URL:http://quasar.as.utexas.edu/BillInfo/ReligiousCalendars.html>.

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Top Document: [sci.astro] Time (Astronomy Frequently Asked Questions) (3/9)
Previous Document: C.07.1 When is Easter?
Next Document: C.08 What is a "blue moon?"

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