Will Zeller's Rule Work Indefinitely?

Date: 04/07/2004 at 00:56:29
From: hunter
Subject: Will Zeller's Rule work indefinitely?

I showed an equation involving Zeller's Rule to a college teacher and
he told me the equation may not work for very distant years.  He said
that it may be impossible to write an equation relating the day of the 
week to a given year; month; and day of month because the exact value 
relating the two may have irrational numbers involved.

His statement came as a shock to me; I thought that since the use of
the rounding down function is used throughout "Zeller's Rule," the 
equation would work indefinitely.  Who's right?  Is the equation sure 
to work in 4561?  Indefinitely?

Date: 04/07/2004 at 09:39:44
From: Doctor Peterson
Subject: Re: Will Zeller's Rule work indefinitely?

Hi, Hunter.

Zeller's formula exactly corresponds to the Gregorian calendar, and 
will work as long as that calendar is used.  It is true that, over a 
very long time, that calendar would need further adjustment, just as 
the Julian calendar did; but until a new calendar is defined, you 
can't say the formula is wrong!  The point is that a calendar is not a 
measurement of reality, but a legal concept established by law, and 
therefore it remains valid as long as, but ONLY as long as, the law 
is in effect.  So the formula you have does not refer to the 
astronomically measured length of a year, and does not depend on 
physical reality for its accuracy; on the other hand, it depends on 
the whim of governments, so no one can really say how long it will 
remain valid!

A calendar is only a way to fit a whole number of days into each 
year, while staying as close as possible to what is, as you were 
told, a theoretically irrational number of days per year 
astronomically (and also somewhat variable, and subject to errors in 
measurement).  Therefore no completely regular calendar can be exactly 
correct forever; but then, in a sense, no calendar is really exact 
anyway, since the whole point is to approximate the year with whole 
numbers.  It's just that an extra day will have to be added or dropped 
eventually.  What is surprising is that such a simple set of rules 
(and therefore a simple calculation) happens to be able to do such a 
good job of approximating the physical length of a year.

Our Calendar FAQ has links to several sites about calendars; here is 
one that explains the Gregorian calendar with links to other details, 
historical and astronomical:

  Gregorian Calendar
    http://scienceworld.wolfram.com/astronomy/GregorianCalendar.html 

This site summarizes how the rules were changed from the Julian to 
the Gregorian calendar, and mentions a proposed additional rule that 
would keep it accurate for 20,000 years:

  Calendars
    http://csep10.phys.utk.edu/astr161/lect/time/calendars.html 

  However, the Julian year still differs from the true year of
  365.242199 days by 11 minutes and 14 seconds each year, and over
  a period of 128 years even the Julian Calendar was in error by
  one day with respect to the seasons.  By 1582 this error had
  accumulated to 10 days and Pope Gregory XIII ordered another
  reform: 10 days were dropped from the year 1582, so that October
  4, 1582, was followed by October 15, 1582.  In addition, to guard
  against further accumulation of error, in the new Gregorian
  Calendar it was decreed that century years not divisible by 400
  were not to be considered leap years.  Thus, 1600 was a leap year
  but 1700 was not. This made the average length of the year
  sufficiently close to the actual year that it would take 3322
  years for the error to accumulate to 1 day.
 
  A further modification to the Gregorian Calendar has been
  suggested: years evenly divisible by 4000 are not leap years.
  This would reduce the error between the Gregorian Calendar Year
  and the true year to 1 day in 20,000 years.  However, this last
  proposed change has not been officially adopted; there is plenty
  of time to consider it, since it would not have an effect until
  the year 4000.

That is, the length of a year in the Julian calendar was

  365 + 1/4 - 1/100 = 365.24
  (off by 0.002199 days, or 1 day in 454 years)

and in the Gregorian calendar is

  365 + 1/4 - 1/100 + 1/400 = 365.2425
  (off by 0.000301, or 1 day in 3322 years)

while with the 4000 year rule it will be

  365 + 1/4 - 1/100 + 1/400 - 1/4000 = 365.24225
  (off by 0.000051, or 1 day in 19,607 years)

Given that such a simple addition (which has not been made only 
because it is not needed yet) would fix the Gregorian calendar so 
effectively, we can safely say that you can use Zeller's formula up 
to the year 4000.  After that--if the change is actually made in law-- 
you can just add a term to the formula and keep it correct.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/