# Perpetual calendar

Illustration from 1881 U.S. Patent 248872, for a perpetual calendar paperweight. The upper section is rotated to reveal one of seven lists of years (splitting leap years) for which the seven calendars below apply.
A 50-year "pocket calendar" that is adjusted by turning the dial to place the name of the month under the current year. One can then deduce the day of the week or the date.
A modern automatic wristwatch with complications of: day, date, month, Retrograde hand displays: day at 12 o'clock, date on the right hand side of the dial, small second at 6 o'clock, and 24 hours second time zone on the left hand side of the dial. Moonphase display at 6 o’clock and day/night indicator at 12 o’clock.
A Quartz perpetual calendar wristwatch with complications of: 24 hours, day, date, month, with a leap-year function so you never need to adjust.

A perpetual calendar is a calendar valid for many years, usually designed to allow the calculation of the day of the week for a given date in the future.

For the Gregorian and Julian calendars, a perpetual calendar typically consists of one of two general variations:

• 14 one-year calendars, plus a table to show which one-year calendar is to be used for any given year. These one-year calendars divide evenly into two sets of seven calendars: seven for each common year (year that does not have a February 29) with each of the seven starting on a different day of the week, and seven for each leap year, again with each one starting on a different day of the week, totaling fourteen. (See Dominical letter for one common naming scheme for the 14 calendars.)
• Seven (31-day) one-month calendars (or seven each of 28–31 day month lengths, for a total of 28) and one or more tables to show which calendar is used for any given month. Some perpetual calendars' tables slide against each other, so that aligning two scales with one another reveals the specific month calendar via a pointer or window mechanism.[1]

The seven calendars may be combined into one, either with 13 columns of which only seven are revealed,[2][3] or with movable day-of-week names (as shown in the pocket perpetual calendar picture).

Note that such a perpetual calendar fails to indicate the dates of moveable feasts such as Easter, which are calculated based on a combination of events in the Tropical year and lunar cycles. These issues are dealt with in great detail in Computus.

An early example of a perpetual calendar for practical use is found in the manuscript GNM 3227a. The calendar covers the period of 1390–1495 (on which grounds the manuscript is dated to c. 1389). For each year of this period, it lists the number of weeks between Christmas day and Quinquagesima. This is the first known instance of a tabular form of perpetual calendar allowing the calculation of the moveable feasts that became popular during the 15th century.[4]

## Other uses of the term "perpetual calendar"[]

• Offices and retail establishments often display devices containing a set of elements to form all possible numbers from 1 through 31, as well as the names/abbreviations for the months and the days of the week, so as to show the current date for the convenience of people who might be signing and dating documents such as checks. Establishments that serve alcoholic beverages may use a variant that shows the current month and day, but subtracting the legal age of alcohol consumption in years, indicating the latest legal birth date for alcohol purchases. A very simple device consists of two cubes in a holder. One cube carries the numbers zero to five. The other bears the numbers 0, 1, 2, 6 (or 9 if inverted), 7 and 8. This is perpetual because only one and two may appear twice in a date and they are on both cubes.
• Certain calendar reforms have been labeled perpetual calendars because their dates are fixed on the same weekdays every year. Examples are The World Calendar, the International Fixed Calendar and the Pax Calendar. Technically, these are perennial calendars. Their purpose, in part, is to eliminate the need for perpetual calendar tables, algorithms and computation devices.
• In watchmaking, "perpetual calendar" describes a calendar mechanism that correctly displays the date on the watch 'perpetually', taking into account the different lengths of the months as well as leap days. The internal mechanism will move the dial to the next day.[5]

These meanings are beyond the scope of the remainder of this article.

## Algorithms[]

Perpetual calendars use algorithms to compute the day of the week for any given year, month, and day of month. Even though the individual operations in the formulas can be very efficiently implemented in software, they are too complicated for most people to perform all of the arithmetic mentally.[6] Perpetual calendar designers hide the complexity in tables to simplify their use.

A perpetual calendar employs a table for finding which of fourteen yearly calendars to use. A table for the Gregorian calendar expresses its 400-year grand cycle: 303 common years and 97 leap years total to 146,097 days, or exactly 20,871 weeks. This cycle breaks down into one 100-year period with 25 leap years, making 36,525 days, or one day less than 5,218 full weeks; and three 100-year periods with 24 leap years each, making 36,524 days, or two days less than 5,218 full weeks.

Within each 100-year block, the cyclic nature of the Gregorian calendar proceeds in exactly the same fashion as its Julian predecessor: A common year begins and ends on the same day of the week, so the following year will begin on the next successive day of the week. A leap year has one more day, so the year following a leap year begins on the second day of the week after the leap year began. Every four years, the starting weekday advances five days, so over a 28-year period it advances 35, returning to the same place in both the leap year progression and the starting weekday. This cycle completes three times in 84 years, leaving 16 years in the fourth, incomplete cycle of the century.

A major complicating factor in constructing a perpetual calendar algorithm is the peculiar and variable length of February, which was at one time the last month of the year, leaving the first 11 months March through January with a five-month repeating pattern: 31, 30, 31, 30, 31, ..., so that the offset from March of the starting day of the week for any month could be easily determined. Zeller's congruence, a well-known algorithm for finding the day of week for any date, explicitly defines January and February as the "13th" and "14th" months of the previous year in order to take advantage of this regularity, but the month-dependent calculation is still very complicated for mental arithmetic:

${\displaystyle \left\lfloor {\frac {(m+1)26}{10}}\right\rfloor \mod 7,}$

Instead, a table-based perpetual calendar provides a simple look-up mechanism to find offset for the day of week for the first day of each month. To simplify the table, in a leap year January and February must either be treated as a separate year or have extra entries in the month table:

 Month Add For leap years Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 0 3 3 6 1 4 6 2 5 0 3 5 6 2

## Perpetual Julian and Gregorian calendar table[]

For Julian dates before 1300 and after 1999 the year in the table which differs by an exact multiple of 700 years should be used. For Gregorian dates after 2299, the year in the table which differs by an exact multiple of 400 years should be used. The values "r0" through "r6" indicate the remainder when the Hundreds value is divided by 7 and 4 respectively, indicating how the series extend in either direction. Both Julian and Gregorian values are shown 1500–1999 for convenience.

For each component of the date (the hundreds, remaining digits and month), the corresponding numbers in the far right hand column on the same line are added to each other and the day of the month. This total is then divided by 7 and the remainder from this division located in the far right hand column. The day of the week is beside it. Bold figures (e.g., 04) denote leap year. If a year ends in 00 and its hundreds are in bold it is a leap year. Thus 19 indicates that 1900 is not a Gregorian leap year, (but 19 in the Julian column indicates that it is a Julian leap year, as are all Julian x00 years). 20 indicates that 2000 is a leap year. Use Jan and Feb only in leap years.

100s of Years Remaining Year Digits Month D
o
W
#
Julian
(r ÷ 7)
Gregorian
(r ÷ 4)
r5 19 16 20 r0 00 06   17 23 28 34   45 51 56 62   73 79 84 90 Jan    Oct Sa 0
r4 18 15 19 r3 01 07 12 18 29 35 40 46 57 63 68 74 85 91 96   May Su 1
r3 17
N/A
02   13 19 24 30   41 47 52 58   69 75 80 86   97 Feb  Aug M 2
r2 16 18 22 r2 03 08 14   25 31 36 42   53 59 64 70   81 87 92 98 Feb Mar Nov Tu 3
r1 15
N/A
09 15 20 26   37 43 48 54   65 71 76 82   93 99   Jun W 4
r0 14 17 21 r1 04 10   21 27 32 38   49 55 60 66   77 83 88 94   Sep Dec Th 5
r6 13
N/A
05 11 16 22 33 39 44 50 61 67 72 78 89 95 Jan Apr Jul F 6

Example (Gregorian calendar): On what day does Feb 3, 4567 (Gregorian) fall?
1) The remainder of 45 / 4 is 1, so use the r1 entry: 5.
2) The remaining digits 67 give 6.
3) Feb (not Feb for leap years) gives 3.
4) Finally, add the day of the month: 3.
5) Adding 5 + 6 + 3 + 3 = 17. Dividing by 7 leaves a remainder of 3, so the day of the week is Tuesday.

### Revised Julian calendar[]

Note that the date (and hence the day of the week) in the Revised Julian calendar and Gregorian calendar is the same from 14 October 1923 (the date of the change from the Julian calendar to the Revised Julian calendar which advanced 13 days to align with the Gregorian calendar) until 28 February AD 2800 inclusive,[7]

The Julian table above may be used to compute the day of the week for the Revised Julian calendar if the procedure is modified to account for dropped leap years.

For simplicity with large years, subtract 6300 (the least-common multiple of the 900-year period of the leap years and the 7-day week) or a multiple thereof before starting so as to reach a year that is less than 6301.[citation needed]

To look up the weekday of any date for any year using the table, subtract 100 from the year, divide the difference by 100, multiply the resulting quotient (omitting fractions) by seven and divide the product by nine. Note the quotient (omitting fractions). Enter the table with the Julian year, and just before the final division add 50 and subtract the quotient noted above.

Example (Revised Julian calendar): What is the day of the week of 27 January 8315?

8315-6300=2015, 2015-100=1915, 1915/100=19 remainder 15, 19x7=133, 133/9=14 remainder 7. 2015 is 700 years ahead of 1315, so 1315 is used. From table: for hundreds (13): 6. For remaining digits (15): 4. For month (January): 0. For date (27): 27. 6+4+0+27+50-14=73. 73/7=10 remainder 3. Day of week = Tuesday.

### Sunday Letter[]

To find the Sunday Letter, calculate the day of the week for either 1 January or 1 October. If it is Sunday, the Sunday Letter is A, if Saturday B, and similarly backwards through the week and forwards through the alphabet to Monday, which is G.

Leap years have two Sunday Letters, so for January and February calculate the day of the week for 1 January and for March to December calculate the day of the week for 1 October.

Leap years are all years which divide exactly by four with the following exceptions:

In the Gregorian calendar – all years which divide exactly by 100 (other than those which divide exactly by 400).

In the Revised Julian calendar – all years which divide exactly by 100 (other than those which give remainder 200 or 600 when divided by 900).

### Check the result[]

A result control is shown by the calendar period from 1582 October 15 possible, but only for Gregorian calendar dates.

A genuinely perpetual calendar, which allows its user to look up the day of the week for any Gregorian date.