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A Perpetual Calendar: Some Lessons in History and Mathematics
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by Mike Contino
CMC & CalifSU East Bay
Download Perpetual Calendar
"We
cannot tell a lie," said Augustine and Mary Ball Washington. "Our
oldest child, George, was born on February 11, 1731." So why do the
history books give a date that is apparently 376 days later, February
22, 1732? It is a fascinating story, replete with science, mathematics,
politics, religion and history.
Most people think that we have a
leap year every four years. This was true in the old style or Julian
calendar, developed by the Greek astronomer and mathematician
Sosigenes, where the average length of a year was 365.25 days or
31,557,600 seconds. But the astronomical phenomenon known as the
tropical year has a length, at present, of 365 days, 5 hours, 48
minutes, and 46 seconds. This error of 11 minutes, 14 seconds per year
may not seem like much, but from the time this calendar was promulgated
by Julius Caesar (hence, the Julian calendar) in the year -45, also
known as 46 BC or BCE (Before the Common Era), until the middle of the
Sixteenth Century CE (Common Era or AD), the error had accumulated into
more than 12 days! This is about 3/4 of a day per century.
Such
an error is what caused Julius Caesar to fix the Roman 365-day calendar
in the first place. The vernal equinox was supposed to occur on March
25, New Year's Day. In his time it was months off, so the senate
legislated a bissextile or 366-day year once every four years. They
also ordered a onetime fix to bring the equinox back to March 25 by
creating a single year of some 445 days; moved the start of the new
year to the Kalends of January (January 1); renamed Quintilis, the old
fifth month, as Julius; left the names of the old seventh through tenth
months (Septem-, Octo-, Novem-, and Decem-), even though they were now
months 9 through 12; and distributed the 365 days by alternating months
of 31 and 30 days, except that February had only 29.
When
Julius' adopted son Octavian succeeded him as Augustus Caesar, he also
got a month named after him (the former 30-day month of Sextilis), and
a day was taken from February to give the new month 31 days.
The
Julian calendar persisted for more than 1500 years, although New Year's
Day varied from place to place. The Gregorian calendar, the one we use
now, was promulgated in 1582 when the vernal equinox occurred on March
11. Because the old Julian calendar was 3/4 of a day too long per
century, the new Gregorian calendar skipped the leap year in three out
of four century years and created what could be called "leap
centuries." By the new calendar, 1600 was a leap year, but 1700, 1800,
and 1900 were not. Pope Gregory established this new calendar so that
the date of Easter and other religious holy days could be more properly
set. But it was not enough to simply fix the length of the year; he
wanted the vernal equinox back where it belonged. Where it belonged was
not, however, the March 25 of Rome, but rather March 21 as it had been
in 325 CE. So Gregory XIII made ten dates disappear! He decreed that
the day after Thursday, October 4, 1582, Julian, was Friday, October
15, Gregorian.
In the Julian calendar there was a leap year
every four years and thus a year was 365.25 days long. But we use the
Gregorian calendar which drops a leap year every 100 years ( — .01) and
adds it back in every 400 years ( + .0025). Years now average to
365.2425 days.
But what does this have to do with George
Washington's birthday? Unfortunately, perhaps, Pope Gregory was a
little too late–by about a half century–to bring about a smooth
transition. Sixty-five years earlier, in 1517 CE, Martin Luther had
tacked his 95 Theses up in Wittenberg. Then in 1534, Henry VIII went
from Defender of the (Roman Catholic) Faith to head of the Church of
England. There was now a major religious and cultural split in Europe.
So while some European countries dropped those ten days in October,
others did not. And, of course, there was no compelling reason for most
of Asia and Africa to follow suit.
To say the least, there was
great confusion. In 1700, Spanish California was on the Gregorian
calendar; the colonies were on the Julian, and there were undoubtedly
also many Native American calendars. The discrepancy between English
and French Canada was even more dramatic. Instead of an International
Date Line to figure out whether a day was lost or gained, almost two
weeks would disappear or reappear in traveling a very short distance!
After
170 years of this confusion, a law was passed in British Parliament
under George II. It adopted the Gregorian reforms, moved the start of
the year back from March 25 to January 1, and set September 2, 1752,
Julian, as the date of change. The next day was September 14, Gregorian.
So
when George Washington was born in Westmoreland County, Virginia, it
was February 11, 1731, about six weeks before the end of the Julian
year. But at that exact moment it was February 22 on the Gregorian
Calendar, and the year had already turned to 1732 some 53 days earlier.
The
Gregorian year is very accurate. But, continuing the pattern, we could
propose to drop a leap century once every four millennia. For example,
2000 and 3000 would be leap years, but 4000 would not. This would make
an average calendar year of 365.24225 days, and the difference between
that and the current tropical year would be less than 5 seconds! It
would mean that further fixes to the calendar wouldn't be required for
over 7 million years. However, there's no real point in doing that. The
length of a tropical year is changing, constantly becoming shorter due
to the fact that the Earth is very gradually slowing down. No matter
how accurate any calendar is now, adjustments will have to be made.
Scientists now add a "leap second" to the length of a year every few
years!
Click
here: Download Perpetual Calendar for an example of a perpetual
calendar. It is written in EXCEL 4.0 but you may be able to open it
from many other spreadsheet application programs. When the program is
locked, you have access to only one cell. Enter any year from 1600 on,
and the correct Gregorian calendar for that year is displayed in a
format that will print on one page. It should work at least until 4000
CE.
More Historical Facts about Calendars
- The Maya and Aztec calendars had a 52 year cycle of 18,980 days,
mixing a ritual calendar of 260 days, with a seasonal 365 day calendar.
- The Inca calendar had months of 3 weeks, each with 9 days.
- The Hopi of Arizona celebrated New Year's Day in autumn, the Creek in summer, and other tribes at the vernal equinox.
- There were 7 planeta (Greek for wandering) bodies known in the
sky–the Sun, Venus, Mercury, Moon, Saturn, Jupiter, and Mars. At one
time the hours of the day were named in order for these seven bodies,
starting with the Sun's hour on the first day. After 21 hours, three
cycles, the last three hours of the first day ended with Sun, Venus,
and Mercury. This meant that the first hour of the second day would be
the Moon's and so this day was called "Monday."
- Many ancient civilizations paid more attention to the moon than
the Gregorian and Julian calendars do. It was common to have 12 months
alternating 29 and 30 days, with various fixes for the missing days,
such as the occasional addition of a leap month.
- Another simple plan uses 12 months of 30 days each, with five
extra days at the end belonging to no month. This very ancient concept
was used most recently in the French Revolutionary calendar (c. 1790).
It also had three weeks of 10 days each.
- The Russians switched to the Gregorian calendar in 1918, but
later changed to a five-day week which they kept until 1931. As
recently as 1940, they used a six-day week.
- Other than the Y2K computer problem, there is nothing significant
about changing to the year 2000. Most people will consider that date to
be the start of the new millennium and there is nothing wrong with
this! Astronomical time has a year 0 (zero; see the earlier reference
to year -45 which is also 46 BCE). Hence the first century can be
considered as running from 0 through 99 CE.
Student Research Questions
- Sosigenes was possibly Egyptian. Can you learn more about him?
- What does bissextile refer to?
- What occurred in 325 CE so that it was used for reckoning the date of the equinox?
- What was the length of a year computed by Hipparchus in 125 BCE?
- When the British adopted the Gregorian calendar, 11 days were skipped. Why?
- Does a calendar have to keep in step with the seasons? What are the advantages and disadvantages?
- How many days are there in the Muslim calendar year?
- How many days were in each month of the Mayan calendar? How many months per year?
- Can you find additional information about the Inca's calendar?
- Why are there seven days in a week?
- What else can you learn about the Russian Revolutionary or Eternal calendar? How many dates did Lenin have to drop?
- Why did the Romans place their Leap Day between February 24 and 25, instead of just adding it on at the end of the month?
I hope I have given you some ideas for fascinating research
projects. You can reach me with any questions or comments at
cmc-math@sbcglobal.net.
Read the Conclusion of A Perpetual Calendar.
References
An Act for Regulating the Commencement of
the Year; and for Correcting the Calendar Now in Use. Act of the
British Government–24 George II, Cap XXIII, 1751. Edited by Mark Brader, http://www.urbanlegends.com/legal/calendar_act.html
Compton's NewMedia CD ROM Encyclopedia, 1995
Meyer, Peter. The Julian & the Gregorian Calendars,
http://www.hermetic.ch/cal_stud/cal_art.html
Van Helden, Albert. The Gregorian Calendar, http://es.rice.edu/ES/humsoc/Galileo/Things/gregorian_calendar.html
World Book Encyclopedia. Chicago, 1986.
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This Page was last updated: Friday, April 8, 2005 at 5:08:51 AM
This page was originally posted: 5/14/2001; 3:49:26 PM.
Copyright 2008 cmcmath
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