Lecture 11: The Calendar
Our calendars are based on the motions of the Sun and Moon.
Types of Calendars:
- Lunar Calendars
- Solar Calendars
The Julian & Gregorian Calendars
The phases of the moon provide a convenient way to keep track
- Phases are easily visible and distinctive.
- 12 lunar months is 354 days, 11 days short of a year of
The oldest recognizable ancient calendars are lunar calendars.
- Common among nomadic peoples and those without written languages.
The Metonic Cycle
A particular difficulty of pure lunar calendars is that the
354d Lunar Year is about 11d short of the
365d Solar Year.
- This causes the seasons to drift among the months in lunar calendars.
The Babylonians discovered what was later called the Metonic Cycle,
a whole-number coincidence between lunar months and solar years:
The Babylonians built a complex, but very precise hybrid luni-solar
calendar based on the Metonic Cycle.
- 235 lunar months is almost exactly 19 solar years.
Lunar Calendars in use Today
For example, in 2007, the Muslim holy month of Ramadan began on
September 13 and runs through October 12. In 2006, Ramadan was
September 24 through October 21. Each successive year, Ramadan starts
about 11 or 12 days earlier in the Gregorian Calendar.
- Purely lunar calendar, 354d in the calendar year.
- Months occur in different seasons (e.g., Ramadan).
An example of how aspects of the Lunar calendar come into play is the
holiday of Rosh Hashanah, the Jewish New Year, which begins at sunset on
the day of the New Moon that marks the start of the month of Tishrei in
the Hebrew Calendar. This year (2007) Rosh Hashanah began at sundown on
September 12. Next year it will begin at sundown on 2008 September 29.
At the earliest it occurs on Sept 5, the latest is October 5.
- Luni-solar calendar.
- Add an extra 13th month (Adar I) every 2 or 3 years to
keep the calendar aligned with the seasons.
- Repeats on the 19-year Metonic Cycle (7 leap years per cycle).
There are also traditional Chinese, Vietnamese, and Japanese, Hindu, and
Tibetan luni-solar calendars, though these are now primarily used only
to set the dates of traditional holidays (for example: Chinese New Year)
since all of these nations have adopted the Gregorian calendar.
Solar Calendars mark time by the Seasons
- Set the calendar by the Equinoxes and Solstices.
Arrival of seasons often has practical and/or cultural importance:
- Knowing when to plant or harvest.
- Annual flood of the Nile Valley.
- Religious festivals associated with specific seasons (e.g., Easter,
May Day, and Christmas)
Egyptian Solar Calendar
The Egyptians developed the first recorded a solar calendar in about
- Divided year into 12 months of 30d each
- Added an extra 5 days to make up 365d.
- Year began in July when the star Sirius rose exactly
in line with the rising Sun ("Heliacal Rising").
- By 300 BC, they measured the year as 365.25d,
only 11m 14s longer than the true
The Roman Calendar
The Romans had a complex luni-solar calendar tied to important holidays
(fasti) associated with specific seasons:
- 10 lunar months, with extra months and days added to keep
it aligned with the seasons.
- The priests in charge of it were continually tweaking it
for religious and political reasons.
- They usually got it wrong...
The Julian Calendar (46 BC)
Julius Caesar asked the Alexandrine astronomer Sosigenes to reform the
He started with the solar year of 365.25d:
- Year divided in 12 months of 30 and 31d,
(except February which had 29), adding up to 365 days.
- Every 4 years, add a day to February to make 366 days.
- "Leap Year" makes up the difference between
365.25 and 365 days per year.
Caesar started the new calendar in 46 BC:
- Added 80 extra days to 46 BC so that 45 BC would start
correctly aligned with the seasons.
- 46 BC had 445 days in it!
Caesar called 46 BC the ultimus annus confusionis
("The final year of confusion")
Roman wits, however, called it the
annus confusionis ("Year of Confusion").
Missed it by that much...
Sosigenes and the others knew, however, that the year was not exactly
365.25 days long.
- True solar year is about 365.242199 days:
- Calendar gets ahead about 1d every 128 years.
- Causes a slow slip of the seasons against the dates of
the Julian Calendar.
By the Middle Ages, this slip became about 10 days.
A Moveable Feast
In 325 A.D., the Council of Nicaea established a formula to compute the
date of Easter.
- Most important holy day in Christianity.
- Important to celebrate it on the correct day.
The Council adopted a fixed March 21 equinox:
- Easter is the first Sunday after the first full moon of
the Vernal (Spring) Equinox in the North.
- But, it must not coincide with Passover.
Gregorian Calendar Reform
By the 1570's, the Julian Calendar was out of alignment with the seasons
by 10 days.
- Easter was being computed incorrectly, and so celebrated
on the wrong day.
- Important holy days were being celebrated on the wrong days.
Pope Gregory XIII appointed a commission to develop an improved
A New Leap Year Formula.
An elegant formula was invented the Italian physician
Aloysius Lilius (sometimes known by his Italian name: Luigi Lillio).
- Keep the Julian formula for leap years in every year divisble
- Exception is that century year is not a leap year unless
it is divisible by 400.
- In this system, 2000 is a leap year
- but 1700, 1800, & 1900 are not leap years.
This has the effect of removes 3 days every 400 years
- Eliminates all but ~3 hours of error per 400 years.
The Lost Ten Days
Pope Gregory XIII instituted the new calendar in 1582. This required
taking 10 days out of October 1582 to realign the calendar with the
seasons. The day after October 4, 1582 was October 15.
The Gregorian Calendar was adopted by Catholic countries within 2
- Some discontent about the "lost days", especially
over payment of wages & rents ("Give us back our 11 days!"), but
apparently no riots or other unreast.
- Adopted all over continental Europe by mid 1700s
A (nearly) Universal Calendar
Other countries worldwide eventually adopted the Gregorian Calendar:
- England and her American Colonies waited until 1752.
- Japan adopted it in 1873 (Meiji period)
- Soviet Russia adopted it in 1918.
- Communist China adopted it in 1949.
Eastern Orthodox Church still uses a Julian Calendar, and voted as
recently as 1971 to reject switching to the Gregorian Calendar. Some
groups within the Eastern Church have adopted some or all of the
Gregorian solar year, but still compute Easter using the Julian
Calendar. The situation is very complex.
Still off by a little bit...
The Gregorian Calendar formula is equivalent to a year of
- This is ~0.0003 days longer than the length of the true
solar year (about 365.2422 days).
- Gets ahead of the true solar year by 1 day every 3327
The Gregorian Calendar will be ahead of the true solar year by 1 day in
The story of the development of the calendar can only be covered in its
essential outlines within the scope of one lecture, which is unfortunate
as it is a rich and fascinating topic in itself. For a lively and
engaging history of the calendar, an excellent recent book is
Calendar, by David Ewing Duncan (1998, Avon Books).
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Updated: 2007 September 29
Copyright © Richard W. Pogge, All Rights