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From: Stephen Hayes
To: All
Date: 2004-03-07 03:28:42
Subject: Calendar notes

[ Part 1 of 2 ]


                        NOTES on the CALENDAR
                        ~~~~~~~~~~~~~~~~~~~~~
The intention of this document is to provide information on the calendar for
computer programmers or others who may need to do calculations involving
calendar dates, or who may need to verify that a given date validly represents
some actual historical day.

1.   TERMINOLOGY USED

The following terms, when used in this document, have the meanings given here.
Some further expansion of these definitions may be found in later sections.
The terms are presented in alphabetical order.

AD        -    Anno Domini (in the year of the Lord), prefixed to the number
               designating a year in the christian era.

AND       -    a mathematical symbol (from the programming language PASCAL)
               meaning that the conditions to the left and the right must both
               be true.

AUC       -    Ab Urbe Condita (from the foundation of the city), suffixed to
               the number designating a year in the Roman (republican and
               imperial) civil calendar.

BC        -    Before Christ, suffixed to the number designating a year in the
               pre-christian era.

Calendar  -    a system for numbering and counting the cycles of days, months
               and years.

DIV       -    a mathematical symbol (from the programming language PASCAL)
               meaning integer division of the term to the left of DIV by the
               term to the right of DIV, where integer division means division
               with any remainder (however large) ignored;  for example "a DIV
               b" means the integer quotient of `a' divided by `b', the
               remainder being ignored, or 13 DIV 7 = 1.

Gregorian -    the calendar now in use throughout the world and also known as
               the New (or New-style) Calendar.

Julian    -    the calendar in use throughout Christendom until AD 1582 and in
               England (and other places) until AD 1752 and also known as the
               Old (or Old-style) Calendar.

Leap year -    any year which is one day longer than the standard year of the
               calendar in use.

MOD       -    a mathematical symbol (from the programming language PASCAL)
               meaning the remainder which results from the division of the
               term to the left of MOD by the term to the right of MOD;  for
               example "a MOD b" means the remainder of `a' divided by `b',
               the quotient being ignored, or 13 MOD 7 = 6.

Month     -    the major subdivision of a year;  in all calendars considered
               here there are twelve months in every year with varying numbers
               of days in each month - this document uses the modern English
               names of the months (sometimes abbreviated to the first three
               letters of the name) or their modern numbers (January = 1,
               February = 2, etc) throughout.

OR        -    a mathematical symbol (from the programming language PASCAL)
               meaning that the conditions to the left and the right may
               either or both be true.

Week      -    the seven-day cycle of days starting on Sunday and ending on
               Saturday;  this is independent of the cycles of months and
               years and has Jewish and other origins - this document uses the
               modern English names of the days of the week.

<>        -    a mathematical symbol (from the programming language PASCAL)
               meaning "is not equal to".

*         -    a mathematical symbol (from the programming language PASCAL)
               meaning "multiplied by".

( )       -    parentheses and other mathematical symbols not explicitly
               defined here have their normal mathematical meanings and
               significances.

2.   BASIC ASSUMPTIONS

The following statements are assumed to be facts - the interested reader may
verify them from various reference sources.  For brevity the sources are
mostly not quoted here.

2.1  THE WEEK

The seven-day cycle is of very ancient origin (Jewish and other).

a)   The individual days are numbered and named as follows.

     1 Sunday       2 Monday       3 Tuesday      4 Wednesday    5    Thursday
     6 Friday       7 Saturday

b)   At different times and in different places the names of the days may
     vary.

c)   This cycle has not been broken or in any way disturbed within the period
     of any calendar discussed here.  Consequently any system for calculating
     the day of the week from the date must result in Friday when applied to
     3rd April AD 33 (the traditional date of the Crucifixion, Good Friday).

     [It is neither important for the purpose of this document to believe in
     the historical accuracy of any traditional dates, nor in the truth or
     otherwise of any events.  It is necessary to know what the traditions are
     and at least their mathematical significances.  Such dates are the basis
     of any calendar and are used here for the purpose of calculation.]

2.2  THE ROMAN CIVIL CALENDAR

In 46 BC Julius Caesar regularized the civil calendar of the Roman republic to
make it operate as follows.

a)   Every year was divided into twelve months which were numbered as follows
     and to which we assign the following modern names.

     1    January
     2    February
     3    March
     4    April
     5    May
     6    June
     7    July
     8    August
     9    September
     10   October
     11   November
     12   December

b)   The years were numbered AUC (from the date of the foundation of the city
     of Rome).

c)   Every fourth year was to have one extra day in February.  However, the
     Roman civil authorities responsible for the calendar mismanaged it after
     Caesar's death, keeping a leap year every third year, so by 8 BC three
     days had been gained.

2.3  THE JULIAN CALENDAR

In 8 BC the Emperor Augustus adjusted the Roman civil calendar by dropping the
days which had been gained - omitting leap years until the year we now know as
AD 12.  Thus AD 12 was the first leap year of the christian era.  (At the same
time Augustus gave the months their present numbers of days.)  This calendar
is known today as Julian after Julius Caesar who first devised it as mentioned
in the foregoing section.  From 8 BC the calendar operated as stated in
(a)-(e) and has the continuing history given in (f)....

a)   Every year was divided into twelve months which were numbered and named
     as follows and each month had the following numbers of days:

     1    January   31 days
     2    February  28 days except in leap years when it had 29 days
     3    March     31 days
     4    April     30 days
     5    May       31 days
     6    June      30 days
     7    July      31 days
     8    August    31 days
     9    September 30 days
     10   October   31 days
     11   November  30 days
     12   December  31 days

b)   Every normal year had 365 days;  every leap year had 366 days.

c)   Every year began on 1st January and ended on the following 31st December.

d)   A leap year occured in AD 12 and every fourth year thereafter (AD 4 and
     AD 8 were not leap years because of Augustus' adjustment).

e)   The years were still numbered AUC (from the date of the foundation of the
     city of Rome).

f)   With the christianization of the Roman empire this calendar continued to
     be used for all purposes throughout both eastern and western parts of the
     empire.  It was the Emperor Constantine who first officially introduced
     the seven-day cycle (week) into the Roman system.  It was unchanged then
     (and since) from the Jewish seven-day cycle.

g)   When the AD system of numbering the years was introduced calculations
     made at the time fixed that year as AD 532.  AD 1 was reckoned to begin
     on 1st January when the infant Christ would have been seven days old.
     The year 754 AUC was reckoned as AD 1.  The year in which Christ was
     conceived and born became known as 1 BC.  For the purposes of calculation
     it may be convenient to regard that year as "AD 0".

h)   At some times and in some places within the currency of this christian
     form of the Julian calendar (particularly in England and its colonies at
     the time but not universally) the year was regarded as beginning on Lady
     Day so that 25th March was regarded as the first day of the year and the
     same year was regarded as ending on 24th March next following.  This
     still meant that the conception and birth of Christ were in the year 1
     BC.

i)   A leap year still occured every fourth year - when the AD year-number was
     exactly divisible by four.

     Mathematically the year AD `yr' is a leap year if and only if:

     (`yr' MOD 4) = 0

     and, assuming the year to begin on 1st January, the February in that year
     had 29 days.  In those places where the year was reckoned to begin on
     25th March the February which had 29 days was the February previous to
     the 25th March on which the year was reckoned to begin.  This meant that
     all places had a 29th February together although their reckoning of the
     AD year-number differed.  For those places which started the year on 25th
     March one could calculate a year to be a leap year if and only if:

     ((`yr'+1) MOD 4) = 0

     for the February in the year AD `yr' to have 29 days.  The years AD 4 and
     AD 8 (starting 1st January) were special cases and were not leap years
     because of Augustus' adjustments.  The calendar followed its own rules
     without irregularity from 1st March AD 8 (the day following the
     effectiveness of Augustus final adjustment) throughout its remaining
     period of use.

j)   This calendar was eventually found (by astronomical observation) to be
     inaccurate.  By the sixteenth century it had gained ten days because too
     many years were being kept as leap years.

k)   As described in the next section (2.4) the Gregorian calendar was devised
     (with fewer leap years) at the instigation of the pope of the time and
     the last effective day of the Julian calendar in Rome was 4th October AD
     1582 which was a Thursday.  (The ten days were dropped without altering
     the weekly cycle of days and the next day was Friday 15th October AD 1582
     on the Gregorian calendar.)  At the same time the first day of the year
     was fixed as 1st January to avoid the former confusion.

l)   Some places (including England and its colonies at the time) continued
     to use the Julian calendar officially - not recognizing the authority of
     the pope.  Some of these places (again including England and colonies)
     were those which started the year on 25th March - they continued to do
     so.

2.4  GREGORIAN CALENDAR

This calendar replaced the Julian calendar, operates as stated in (a)-(d) and
has the history given in (e)....

a)   Every year is divided into twelve months which are numbered and named as
     follows and each month has the following numbers of days:

     1    January   31 days
     2    February  28 days except in leap years when it has 29 days
     3    March     31 days
     4    April     30 days
     5    May       31 days
     6    June      30 days
     7    July      31 days
     8    August    31 days
     9    September 30 days
     10   October   31 days
     11   November  30 days
     12   December  31 days

     "Thirty days hath September, April, June and November;
     All the rest have thirty-one excepting February alone;
     Which hath twenty-eight days clear and twenty-nine in each leap year."

b)   Every normal year has 365 days;  every leap year has 366 days.

c)   Every year begins on 1st January and ends on the following 31st December.

d)   A year is a leap year if and only if:

     i)   the year-number is exactly divisible by 400;  or

     ii)  the year-number is not exactly divisible by 100 and is exactly
          divisible by 4.

     Mathematically the year AD `yr' is a leap year if and only if:

     ((`yr' MOD 400) = 0) OR (((`yr' MOD 100) <> 0)) AND ((`yr' MOD 4) = 0))

e)   If the Gregorian calendar is projected back to 1st January AD 1 this is
     the same day as Julian 1st January AD 1 but the Gregorian leap year rule
     has no special cases.  This means that the earliest day which has
     different dates in the two systems is Gregorian 29th February AD 4 which
     is Julian 1st March AD 4.  In each calendar it is the first day after
     28th February AD 4.

f)   This calendar was devised as a result of astronomical observation and
     calculation on behalf of Pope Gregory XIII whose name is associated with
     it.

g)   The first year of this calendar (AD 1) is reckoned to be the year
     following the birth of Christ - ie. the year identical with Julian AD 1,
     beginning on 1st January when Christ was seven days old.  This meant that
     the AD year-numbers were not changed by the Gregorian reform of the
     calendar if the year is regarded as beginning on 1st January.

h)   It came into force in the Papal States (Rome and other parts of central
     Italy) and in some other parts of Europe (and their colonies at the time)
     on Friday Gregorian 15th October AD 1582.  (The previous day had been
     Thursday Julian 4th October AD 1582.)  Dates in the range 5th October AD
     1582 through 14th October AD 1582 are therefore not valid in places which
     adopted the new calendar immediately.  The weekly cycle of days was not
     altered.

i)   Many parts of Europe (including England) did not adopt it immediately.

j)   Most countries of the world have now adopted it at least for civil
     purposes.

k)   England and its colonies at the time brought this calendar into use for
     all purposes with effect from 14th September AD 1752 which was a
     Thursday.

2.5  THE CHANGE IN ENGLAND

In England and its colonies of the time the change to the Gregorian system was
effected as follows.

a)   Until AD 1751 (which began on 25th March) the Julian calendar was the
     official calendar although some events had already been (and continued to
     be) recorded with two year-numbers.  For example, "3rd Feb.
1750/1" would
     record a day in the year which was still officially AD 1750 in England
     but was recognized as AD 1751 elsewhere.

b)   AD 1751 officially ended in England on 31st December and AD 1752
     officially began on the following day, 1st January.

c)   By this time the difference between the English (Julian) and the more
     accurate (Gregorian) calendar was eleven days, so this was the number of
     days which had to be dropped.

     "Give us back our eleven days."

d)   In England the next day after Wednesday Julian 2nd September AD 1752 was
     officially made Thursday Gregorian 14th September AD 1752.  Dates in the
     range 3rd September AD 1752 through 13th September AD 1752 are therefore
     not valid in England or any place which was its colony in AD 1752.

e)   The English financial year had previously begun on the 25th March with
     each new AD year.  Because the financial community of the time refused to
     acceptSEEN-BY: 5/0 140/1 7105/1 7106/20 22 46

SEEN-BY: 633/267 270
@PATH: 7106/22 140/1 106/2000 633/267


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