124 5. COUNTINGThe Gregorian calendar. A modification of the Julian calendar was introduced in
1582 on the recommendation of Pope Gregory XIII. The Gregorian correction re-
moved the extra day from years divisible by 100, but restored it for years divisible
by 400. A 400-year period on the Gregorian calendar thus contains 303 years of
365 days and 97 leap years of 366 days, for a total of 146,097 days. It would be
perfectly accurate if the year were of the following length:365 + Ú " ú(5ï + 4ûä = 3652425 days'
Since this figure is slightly too large, there are still too many leap years in the
Gregorian calendar, and a discrepancy of one day accumulates in about 3300 years.
The Muslim calendar. The prophet Muhammed decreed that his followers should
regulate their lives by a purely lunar calendar. In a lunar calendar the months
are in close synchrony with the phases of the Moon, while the years need not have
any close relationship to the seasons or the position of the Sun among the stars.
The Muslim calendar, taking as its epoch the date of the Hijra (July 15, 622 CE),
consists of 12-month years in which the odd-numbered months have 30 days and
even-numbered months 29 days, except that the final month has 30 days in a leap
year. Thus, the year is 354 or 355 days long, and as a result, the years wander
through the tropical year.
The Hebrew calendar. More common than purely lunar calendars arc lunisolar cal-
endars, in which the months are kept in synchrony with the phases of the Moon and
extra months are inserted from time to time to keep the years in synchrony with
the Sun. These calendars lead to a need for calculation and therefore take us right
up to the development of arithmetic. Several such calendars have been used since
ancient times and continue to be used today in Israel, China, and elsewhere. It
must have required many centuries of record keeping for the approximate equation
"19 solar years = 235 lunar montlis" to be recognized. Since 235 = 12 • 12 + 7 · 13,
the addition of the extra month 7 times in 19 years will keep both years and months
in balance, with an error of only about 2 hours in each 19-year cycle, or one day in
220 years.The Julian day calendar. An example of what we have called a linear calendar is
the Julian day calendar, which is to be distinguished from the Julian calendar.
The Julian day calendar was invented by Joseph Justus Scaliger (1540 1609) and
apparently named in honor of his father Julius Caesar Scaliger (1484-1558). It was
advocated by the British astronomer John Frederick William Herschel (1792-1871).
In this calendar each day is counted starting from what would be the date January
1, 4713 BCE on the Julian calendar. Thus the day on which the first draft of
this paragraph was written (August 9, 2002, which is July 27, 2002 on the Julian
calendar) was Julian day 2,452,496.
The Maya calendar. The most unusual calendar of all was kept by the Maya. The
three Maya calendars account for a number of phenomena of astronomical and
agricultural importance. As discussed above, numbers were written in a place-
value system in which each unit is 20 times the next smaller unit, except that when
days were being counted, the third-place unit, instead of being 20 · 20 = 400, was
20-18 = 360. This apparent inconsistency was probably because there are 360 days
in the "regular" part of the 365-day Maya calendar known as the Haab, and the
other 5 days were apparently regarded as unlucky (and so, best not included in the