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A Perpetual Calendar: Some Lessons in History and Mathematics (Part 2)
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Download Perpetual Calendar
by Mike Contino
CMC & CalifSU East Bay
In part 1 (A Perpetual Calendar: Some Lessons in History and Mathematics) we spoke of some connections between our current calendar
and Julius and Augustus Caesar, Pope Gregory XIII, Martin Luther, Henry
VIII, George Washington, George II, and Vladimir Lenin.
There
is a mistake in the first part of this article, both web and printed versions. The
length of a tropical year is getting shorter but the reasons are more
complicated than a simple "the Earth is slowing down." As the rotation
of the Earth slows, the year would get shorter since the length of a
day gets longer. But the year also gets longer since the Earth is
slowing as it revolves around the sun. The competing effects of these
slowing motions will make the tropical year about 45 seconds shorter in
the year 5000 than it was in -5000.
Another editing error, in
the printed version, involving the start of the New Millennium, has
already been corrected in the on-line version.
Some answers to the questions raised in part 1:
2. Bissextile is the term for leap years because in Roman times the
Leap Day was added six days before the Kalends of March, i.e., six days
before March 1. So in leap years, after 7 Kalends there was a 6 Kalends
plus a second day called 6 Kalends, or bis-sexto. Note that the Romans
used a "count down" method for dates: The First of the Month minus 5,
minus 4, … (see question 12).
3.
Dropping 10 days from the calendar was not essential to the Gregorian
reform in order to fix the length of the year. The days were dropped to
bring the equinox back to where it was in 325 CE when the Council of
Nicea fixed the computation of the date for Easter.
4.
Hipparchus computed the solar year to be approximately 365.2466 days.
Compare that to the Gregorian Year of 365.2425 days or 31,556,952
seconds, making it 26 seconds longer than a tropical year.
5.
After the 1582 adoption of the Gregorian calendar which dropped 10
days, 1600 was a leap year in both calendars but 1700 was one only by
Julian reckoning. This made for an additonal day discrepancy so the
British Act of Parliament called for the omission of 11 days to align
with other countries.
6. - 9. The Muslim year has 354 or 355
days. It does not have the months keep in step with the seasons. Both
the Mayans and the Aztecs used a 365 day year of 18 months, each with
20 days. There were 5 intercalated "days of evil omen" added at the end
of the year. Though their calendars did not match, they both also used
a ritual cycle of 260 days. Together these formed a repeating cycle of
18,980 days or 52 years. The Incas had years of 12 or 13 months, each
with three weeks of nine days.
10. One source says there is no
astronomical reason for having seven days in a week. But perhaps it was
because there are seven "planetae," wandering bodies visible in the
sky—Sun, Venus, Mercury, Moon, Saturn, Jupiter, and Mars. At one time
the hours of the day were named, in this order, for these 7 bodies,
starting with the Sun's hour on the first day. After 21 hours, or three
cycles, the Sun's day ended with the hours of Sun, Venus and Mercury.
This meant that the first hour of the sceond day would be the Moon's.
Continuing to choose every third one in a cyclic pattern gives: Sun,
Moon, Mars, Mercury, Jupiter (Jove), Venus, and Saturn. This yields the
names for the days of the week. Some correspond better in English
(Sunday, Monday, Saturday); others in Spanish/French (Lunes/ Lundi,
Martes/ Mardi, Miércole/ Mercredi, Jueves/ Jeudi, Viernes/ Vendredi).
English uses the Teutonic gods Tui (equivalent to the Roman god Mars),
Wotan (Mercury), Thor (god of Thunder, Jupiter, Jove), and Frigg or
Frigga (goddess of love, Venus).
11. Since 1700, 1800, and 1900
were not Gregorian leap years, Lenin had to drop 13 days when Russia
adopted the calendar (temporarily) in 1918.
There are many patterns that can be found in our
calendar; have your students look at any ordinary year and any leap
year to see what they can discover. For example, if we ignore whether a
month ends on the 30th or 31st, then the months of September and
December are always identical since the first day of each month will
start on the same day of the week. This is also true for April and
July. May and June are always unique. For example, since June 1, 2000
falls on a Thursday, no other month of the year will start on a
Thursday. This trait is shared by October in leap years, but by August
in other years.
There are just 14 different calendars, seven for
regular years and 7 for leap years—one for a year which starts on a
Sunday, another for when it starts on Monday, and so on. You can build
a set of 14 dodecahedron-calendars like those on the back of the
Special Conference Edition of the ComMuniCator each year, and you will
have built your own perpetual calendar. Many perpetual calendars number
these calendars from 1 (regular year starting on a Sunday) through 14
(leap year starting on a Saturday). Since computing the calendar to use
involves looking at the remainder upon dividing by 7, I prefer instead
to number these from 0 to 6 regular and 0 to 6 leap; the clock (or
modulo 7) arithmetic fits this scheme better.
Have your students
number the calendars in one of these methods and then write out the
progression of calendars. To do this they will only have to know that
365/7 gives a remainder of 1, and thus calendars generally progress by
1 except after a leap year. For example, January 1, 1989, the year
after a leap year, was a Sunday; it used calendar 0 and it is a good
place to start looking for a repeating pattern. The next years used
calendars 1 and 2. Then came the #3 leap year calendar. Skip #4 and 1993
thus was a year 5—it started on a Friday. Have your students continue
the pattern to see when the process repeats. This will occur when
calendar 0 is again used on the year after a leap year. The progression
should start 0, 1, 2, 3L (in 1992); 5, 6, 0, 1L (in 1996); 3, 4, 5, 6L
(in 2000); 1, 2, 3, 4L…. Note that in this cyclic pattern, day 6 is
followed by day 0 and also that the 6L falls in the year 2000 which
does, indeed, start on day 6, Saturday.
What patterns will your
students discover? They should be able to continue this to see that
calendars repeat every 28 years—at least from 1901 until 2099. So the
year 2028 will be a leap year that starts on a Saturday, just as 2000
does, and just as 1972 did. This fact might help someone compute the
day upon which they were born. During the 28 years, we use each leap
year calendar exactly once and we use each regular calendar 3 times.
Although this 28 year cycle works at present, the progression is
interrupted by the non-leap centuries in our Gregorian calendar. If
they carry it out far enough or do some arithmetic with the number of
days in a century, students can discover that the actual progression of
calendars follows a 400 year cycle. So the 1599 calendar was identical
to the calendars of 1999 and 2399.
We can check this 400 year
cycle. Ask your students how many leap years in a century. There are
usually 24 with 25 in a leap century. So in four centuries there are 97
leap years and 303 regular years. Recall that a regular year advances
the calendar by 1 day, and a leap year by two. In four centuries the
calendar is thus advanced by 303 + 2*97 days = 497 days. This gives a
remainder of 0 when divided by 7. In other words, dates 400 years apart
have the calendar number in use advanced by 0 days; they are the same.
Some notes on the Perpetual Calendar download:
Most
of the calculations involved in figuring out which calendar to use are
done in the single cell A6 in my perpetual calendar. It uses 1600 as a
base year, adds a day for every year since then, an extra day for every
4th year, looses a day every century (the non-leap years), and gains
another every 400 years (leap centuries). It gets the remainder after
dividing by 7 and adds this to 6 (since 1600 used calendar 6; January 1
was a Saturday). This answer is copied to A12, giving the starting day
for January 1. Subsequent months are even easier. Months of 30 days (4
weeks plus 2 days) are followed by a month which begins 2 days later in
the week. Likewise there is a 3 day advance after months of 31 days,
and March starts on the same day as February—plus 1 in a leap year.
These starting numbers are computed in A12 through A25 and carried over
to the calendar proper to get each month to start on the correct day of
the week. The rest of the month simply uses "today is one more than
yesterday" and, at the end of each month, blanks out any impossible
dates such as March 33, April 31, and February 30.
I hope you
find this of use, both in itself and as a source of research and
mathematical activities for your students. I have skipped many details
of the calendar itself but please write cmc-math@sbcglobal.net if I can
answer any questions.
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This Page was last updated: Monday, December 18, 2006 at 5:35:01 AM
This page was originally posted: 5/14/2001; 4:48:42 PM.
Copyright 2008 cmcmath
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