Calendar and Time Review
A compendium for the new millennium
by Al R. Vilcius

Assuming that programmable computers and other electronic devices will continue to handle automatic date functions in the future, it must be concluded (from information presented below) that Y2K problems are not over on January 1, and that this is not likely the last calendar bug mankind will encounter..

Calendars are concerned with counting and classifying days; the measurement of fractions of a day is classified as timekeeping. U.S. Naval Observatory. Calendars also provide the illusion of understanding and controlling time itself, giving mankind a link with the cosmos.

The (tropical) year is defined as the time the earth takes in making one complete orbit around the sun, measured between two successive Vernal Equinoxes. Vernal Equinox Technically, "the year" is defined observationally as being the interval of time between two successive passages of the Sun through the vernal equinox. The vernal equinox defines the beginning of spring, and is the instant when the Sun is above the Earth's equator while observationally going from south to north.

Astronomers have observed that one (tropical) year consists of approximately 365.24219878 rotation of the earth, each rotation being defined as 1 day. Explanatory Supplement to the Astronomical Almanac.

There is no ISO, ANSI, NIST, or any other official standard, in the modern sense, for the current Gregorian calendar: it was instituted in 1582 by the Roman Catholic Church which was the only organization in a position to establish anything resembling an international standard at that point in history.

International Organization for Standardization number ISO 8601 specifies a date notation of YYYY-MM-DD , where YYYY is the calendar year, MM is the Gregorian month between 01 and 12, and DD is the day of the month from 01 to 31.

The Jesuit astronomer Christopher Clavius helped Pope Gregory XIII to introduce what is now called the Gregorian calendar.

Clavius proposed that Wednesday, Oct. 4, 1582 (Julian) should be followed by Thursday, Oct. 15, 1582 (Gregorian). He proposed that leap years occur in years exactly divisible by four, except that years ending in 00 must be divisible by 400 to be leap years.

Here is a diagram of the Gregorian calendar leap year algorithm:

leap year algorithm

Rule: A year is a leap year if it is divisible by four, unless it is divisible by 100 in which case it should be divisible by 400.

This rule makes the long run average length of a calendar year:

365 + 1/4 – 1/100 + 1/4000 = 365.2425 calendar days

which is approximately 365.2425 – 365.24219878 = 0.00030122 days longer than defined by actual rotations

hence it would take 1/ 0.00030122 or about 3,320 years for the Gregorian calendar to be out of phase with the seasons (rotations) by 1 day.

This rule was decreed by the Papal bull Inter Gravissimas. (Among the most serious) issued by Pope Gregory XIII in 1582, and defines what we now know as the Gregorian calendar. Hence the bull modified the older Julian Calendar which had been in use for the previous 1629 years - since 44 B.C. The bull also

The Gregorian calendar was implemented in Continental Europe in October 1582 by eliminating 10 days from the month - from Oct. 5th to Oct. 15th. In response, the people of Frankfurt rioted against mathematicians and the Pope, who they believed had conspired together to rob them of 10 days.

The Gregorian calendar was implemented in England some 170 years later, in September in 1752 by eliminating 12 days, from Sept. 2nd to Sept.14th, following an Act of Parliament in 1751, and approval by King George II. Also New Year's Day was changed from March 25 to January 1. This perceived loss of 12 days also caused riots in the streets. Russia didn't accept the Gregorian calendar until 1918.

Clavius' rules imply that The Year 2000 is a Leap Year for all those following the Gregorian Calendar (Western World); February 29, 2000 exists and is a Tuesday.

In addition to date notation, part of the well known Y2K bug is the leap year issue since it has never before in the history of computing been necessary to use the "400 rule". Consequently, automatic date controllers that have not implemented this rule will not treat the Year 2000 as a leap year. This gives rise to two problems:

Risks are that these problems could affect:

Conclusion: Y2K problems are not over on January 1.

Because the Gregorian calendar will get one day further out of phase with the seasons every 3,320 years or so, further modifications to shorten the calendar elapsed time are being suggested, such as "if the year is also divisible by 4000, it's not a leap year."

Such an additional rule would make the long run average length of a calendar year:

365 + 1/4 –1/100 + 1/400 – 1/4000 = 365.24225 calendar days

which is approximately 365.24225 – 365.24219878 = 0.00005122 days longer than defined by actual rotations

hence it would take 1/ 0.00005122 or about 19,523 years for the modified calendar to be out of phase with the seasons (rotations) by 1 day.

This means that:   Another calendar bug is in the pipeline.

Now it becomes a matter of determining which is the suitable person/institution to make such a decree, and how she would be empowered by the society, internationally. In 730 A.D., the Venerable Bede, a mathematically skilled Anglo-Saxon monk, pointed out that the 365 1/4-day Julian year was 11 minutes, 14 seconds too long, but no one did anything about it for another 850 years!

 Ancient History:

Prior to the Gregorian calendar, the Julian calendar was in common use, established by Julius Caesar around 46 B.C. The Julian calendar, developed by Caesar's astronomer, Sosigenes, had an average of 365.25 days a year. This produced an error of

approximately 365.25 – 365.24219878 = 0.00780122 days longer than actual rotations

hence it took 1/ 0.00780122 or only about 128 years for the Julian calendar to be out of phase with the seasons (rotations) by 1 day.

over 1,629 years of using the Julian calendar, it would have been out by 1629 × 0.00780122 or about 12.7 days, were it not for the mishandling of some 3 leap years early on. This is where Clavius got his 10 days to make the seasonal correction.

Prior to the Julian calendar, the Egyptian calendar, based on a 365 day year, was in use. With an error of approximately -0.2422 it would have been short by almost 1 day every 4 years, making it appear as if the seasons are migrating over the calendar.

Personal End-Notes:

Christopher Clavius did more than any other German scholar of the 16th century to promote a knowledge of mathematics. He was the first to use the decimal point, and was a gifted teacher and writer of textbooks: his arithmetic books were used by many mathematicians including Leibniz and Descartes.

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FrançoisViète (or Vieta) (1540-1603) French mathematician, revised concepts and notation of algebra, making algebra a separate discipline from arithmetic; he published "Canon Mathematicus" in 1597, created the "analytic art" of algebra, and gave new solutions to the cubic and quartic equations. Viète did not like Clavius's calendar.

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