# Perpetual Calendar

 The week of seven days is an artificial unit, though it has been used from time immemorial.  For measuring astronomical periods of time longer than the day, two systems have been in common use.  One uses a lunation (the time from one new moon to the next) as a fundamental cycle.  The other is based on the sun's year, and it is this which we shall study.

 The first difficulty is that the length of the year is not commensurable with the length of the day, for the year contains about 365.242199 days.  The history of the calendar is the history of the attempts to adjust these incommensurable units in such a way as to obtain a simple and practicable system.

 Our calendar story goes back to Romulus, founder of Rome, who introduced a year of 300 days divided into 10 months.  His successor, Numa, added two months.  This calendar was used for the following six and a half centuries until Julius Caesar introduced a more exact year of 365.25 days.  The difficulty of the extra quarter of a day was handled by making the length of the ordinary year just 365 days and making every fourth year a leap year of 366 days.  The Julian Calendar spread abroad and was generally used until 1582.

 The Julian calendar was a little too long, and by 1582 the accumulated error amounted to 10 days.  A second reform, instituted by Pope Gregory XIII in 1582, compensated for the error.  The date Friday 5 October 1582 was renamed Friday 15 October 1582, thus dropping 10 days.  In future, of centenary years only those that can be divided exactly by 400 would be leap years (that is, 1600 is a leap year but 1700 is not).  This Gregorian Calendar was adopted in 1732 by Great Britain and in 1752 by the English colonies in America.

 The following formula, which is for the Gregorian calendar only, may be used to find the day of the week corresponding to a given date:

 W   = D + M + C + Y   (mod 7)

 where W is the number of the day of the week (starting with Sunday = 1) D is the number representing the day of the month and M, C, Y are numbers based on the month, century and year, respectively

 MONTH M January 0 February 3 March 3 April 6 May 1 June 4 July 6 August 2 September 5 October 0 November 3 December 5

 FIRST TWO DIGITS OF THE YEAR (Gregorian Calendar) C 15, 19, 23, ... 1 16, 20, 24, ... 0 17, 21, 25, ... 5 18, 22, 26, ... 5

LAST TWO DIGITS OF THE YEAR Y
 00 06 17 23 28 34 45 01 07 12 18 29 35 40 46 02 13 19 24 30 41 47 03 08 14 25 31 36 42 09 15 20 26 37 43 48 04 10 21 27 32 38 49 05 11 16 22 33 39 44 50
 0 1 2 3 4 5 6
 51 56 62 73 79 84 90 57 63 68 74 85 91 96 52 58 69 75 80 86 97 53 59 64 70 81 87 92 98 54 65 71 76 82 93 99 55 60 66 77 83 88 94 61 67 72 78 89 95
 0 1 2 3 4 5 6

In the table directly above, if the year is a leap year, the Y value must be diminished by one if the month is January or February.  If the year is exactly divisible by 4, it is a leap year - with one exception.  Of centenary years, only those that can be divided exactly by 400 are leap years.

Maurits Cornelis Escher
Dutch Graphic Artist
born: 17 June 1898
Sunday
D = 17, M = 4, C = 5, Y = 3
D + M + C + Y  =  17 + 4 + 5 + 3  =  29
29 divided by 7 leaves a residue/remainder of 1
Sunday = 1

Centenary of Escher's Birth
17 June 1998
Wednesday
D = 17, M = 4, C = 1, Y = 3
D + M + C + Y  =  17 + 4 + 1 + 3  =  25
25 divided by 7 leaves a residue/remainder of 4
Sunday = 1, Monday = 2, ..., Wednesday = 4