perpetual calendar

A perpetual calendar from Sweden to compute the easter 1140-1671 according to the Julian calendar. Notice the runic writing.

A 50 year "pocket calendar" that is adjusted by turning the dial to place the name of the month under the current year. You can then deduce the day of the week, or the date.

A perpetual calendar is a calendar which is good for a span of many years, such as the Runic calendar.

General information

For the Gregorian calendar, a perpetual calendar often consists of 14 one-year calendars, plus a table to show which one-year calendar is to be used for any given year. Note that such a perpetual calendar fails to indicate the dates of moveable feasts such as Easter.

The 14 one-year calendars consists of two sets of seven calendars, seven for each common year (year that does not have a February 29) that starts on each day of the week, and seven for each leap year that starts on each day of the week, totaling fourteen.

Also certain calendar reforms may be considered to be perpetual calendars, such as The World Calendar, International Fixed Calendar and Pax Calendar. These calendars have each year and each month within the year, always beginning on the same day of the week.

The term perpetual calendar is also used in watchmaking to describe a calendar mechanism in a watch that displays the date correctly 'perpetually', taking into account the different lengths of the months as well as leap year's day.

The internal mechanism will move the dial to the next day.

Perpetual calendar formula

Following is a formula for calculating the day of the week given the date.

The formula uses the fact that each year begins one day later than the previous except for leap years. The days in a leap year are 2 days later except for January and February where it is one day later. Since the year values increase by one we can create a sequence by adding the year to the year divided by 4 dropping the fraction. This sequence increases by 1 every year except every 4 years where it increases by 2. This sequence will work for the years 1901 through 2099 only since 1900 and 2100 are not leap years.

A table is needed to get the relative day of week of the first of each month relative to the first day of a year.

Month   1  2  3   4  5  6   7  8  9  10 11 12
Rel day 0  3  3   6  1  4   6  2  5   0  3  5

Now for the formula (example for 2006-02-15).

Add the following: The 4 digit year (2006). The integer portion of the year divided by 4 (501). The relative month code (3). The day of the month (15). If it is a leap year and January or February then subtract 1 (0). Adjust the relative week day by subtracting 2 (2525-2). Divide by 7 keeping the remainder (3).

Use this number to find the day as follows:
0-Sunday 1-Monday 2-Tuesday 3-Wednesday 4-Thursday 5-Friday 6-Saturday. Thus, February 15, 2006 is a Wednesday.

Alternate version

A Perpetual Calendar formula for finding the day of the week for any given year since the inception of the Gregorian Calendar (>1751).

1. Begin with the century:

  1700s 1800s 1900s 2000s 2100s 2200s
    4     2     0     6     4     2

Following centuries continue the cyclic sequence.

2. The year: Take the last two figures of the year date and 1/4 of the number formed by them, ignoring the remainder. For example, for the year 1963, 63 divided by 4 is 15 with no remainder. So, 63 plus 15 totals 78.

3. The month: For the month number, add as follows, except in case of January and February during a leap year.

  Month         1  2  3  4  5  6  7  8  9  10  11  12
  Add           1  4  4  0  2  5  0  3  6  1   4   6
  Leap year*    0  3

The Leap Year Rule: a) If the year is divisible by 4 but not 100. b) If the year is divisible by 400.

4. Add the day number.

5. The total is divided by 7 and the remainder will be the day of the week, Sunday being the first day.

         1    2    3    4    5    6    0
        Sun  Mon  Tue  Wed  Thu  Fri  Sat

Example 1: September 11, 2001

Step 1. 2000s corresponds to 6

Step 2. 01 plus 1/4 of 01 is 1

Step 3. September corresponds to number 6

Step 4. The day of the month 11

Step 5. The sum of 24 divided by 7 leaves 3 remainder. Number 3 correponds to Tuesday.

Example 2: December 7, 1941

1. 1900s corresponds to 0

2. 41 plus 10 is 51

3. Dec corresponds to 6

4. The 7th day, add 7

5. Sum of 64 divided by 7 is 9 with 1 remainder. The first day of the week is SUNDAY.

Perpetual Gregorian calendar table

See also

• Antikythera Mechanism
• Calculating the day of the week
• Doomsday (weekday)
• Perpetual Calendar of 800 Years
• Long Now Foundation
• Year 10,000 problem

External links

• Perpetual Calendar 1753-2199 in 11 languages.
• Perpetual Calendar 1901-2001 (just click on the year)
• A Perpetual Calendar showing the day of any month corresponding to any day of the week, for the year 1775, to the year 2025
• A perpetual calendar
• U.S. Patent 248,872, "Calendar".