Current astronomical data tells us that it takes the Earth roughly 365 days, 5 hours, 48 minutes, and 42.2 seconds to complete one orbit about the Sun. If we fix the position of the Earth relative to the Sun at noon on January 1, then after exactly 365 days (at noon on January 1 the next year) the Sun has not quite reached its original position. Indeed, since the orbital period is approximately 365 1/4 days, then four years later at noon on January 1, the Sun's position has fallen one day behind with respect to a 365-day calendar. With a calendar comprising years solely of 365 days, the synchronization of the Sun with respect to the calendar will fall further and further behind. After 4 · 180 = 720 years, the position of the Sun will have fallen 180 days behind, and so midsummer will occur in January (at least in the Northern Hemisphere)!

Julius Caesar was aware of this problem with the calendar, and is responsible for the introduction of leap years. Since the Sun falls behind approximately one day every four years, one solution is to add one day to the calendar every four years. The resulting calendar, with 366 days every fourth year, is known as the Julian Calendar.

The Julian Calendar is only approximately correct. A decimal approximation to the orbital period of the Earth, based on the time quoted above, is 365.24219 days. Thus, the Julian Calendar overcompensates by an average of 0.00781 days each year. The calendar will be off by a whole day after 1/0.00781 years. Let's compute that:

In the sixteenth century, it was determined that the Sun's position had advanced ten days with respect to the Julian Calendar, and steps were taken to modify the calendar once again. In 1582, Pope Gregory devised a new calendar, which is still in use today. Different countries adopted the new Gregorian Calendar at different times. In the United States, this occured in 1752, by which time the error had grown to 11 days. The dates September 3-13, 1752 were skipped entirely as a one-time correction! September 2 was followed immediately by September 14.

To further improve accuracy, the rule for leap years in the Gregorian Calendar was changed slightly from the Julian Calendar:

- every year numerically divisible by 4 would have an extra day (February 29) . . .
- except for century years (1800, 1900, 2000, . . . ) which would not have the extra day . . .
- except for years numerically divisible by 400 (2000, 2400, . . . ) which would have an extra day after all!
With the Gregorian calendar, an extra day is added 97 out of every 400 years. So, the calendar year averages 365 and 97/400 days. Since 365 + 97/400 = 365.2425, this is still not quite right. In practice, this error is managed by using leap seconds which are "skipped" from time to time at midnight on New Year's eve.

We can use continued fractions to look for a simpler scheme for fixing the calendar. We start with the basic continued fraction information for the length of Earth's orbit around the Sun.

## Exercise 3

The difference between the orbital period of the Earth and 365 days is

365.24219 – 365 = 0.24219. If we used continued fraction convergents to approximate this number, we could make up new calendars which compensate for the difference between the orbital period of the Earth and 365 days. What would those new calendars be like? Give as many answers as seem reasonable for the precision we have for the orbital period of the Earth. For each answer, describe where you would put the leap days.

Section 13.1 | Section 13.2 | Section 13.3 | Section 13.4

Copyright © 2001 by W. H. Freeman and Company