This can introduce a divergence of one or two days from the actual start of the month, although the notation of the day of the week makes it possible to identify the day referred to. Moreover, the Muslim month begins with the sighting of the first sliver of crescent Moon after the new Moon. For example, "Saturday night" refers not to the night between Saturday and Sunday-as in our calendar-but to the night between Friday and Saturday. In the Islamic calendar, each day begins at sunset. For agricultural purposes, and to determine the fiscal year for collecting taxes on crops, many Muslim countries have adopted a solar calendar. Because of this progressive seasonal shift, the Islamic calendar remains in use for religious purposes only. After a cycle of about 33 years, the first day of Muharram falls a full year earlier, so that 33 Muslim years correspond to about 32 in the Gregorian calendar. In other words, the first day of Muharram, the first month of the Muslim calendar, gradually "travels" backward through the seasons-as do all the months of the calendar. However, as in all purely lunar calendars, even the Islamic year is about 11 days shorter than the tropical year, i.e., the time it takes for the Sun to return to the equinox. This yields a nearly perfect concordance between the mean length of the calendar month-(354 days x 19 + 355 days x 11)/30/12 = 29 days, 12 hours, 44 minutes, and 2.8 seconds-and that of the synodic month. In every 30-year cycle, the "abundant years" are the 2nd, 5th, 7th, 10th, 13th, 16th, 18th, 21st, 24th, 26th, and 29th, that is, those whose number divided by 30 leaves a remainder equal to one of the numbers above. However, to harmonize the duration of the month with the lunation period (29 days, 12 hours, 44 minutes, and 2.8 seconds), each 30-year cycle is composed of 19 years of 354 days ("regular" years) and 11 years of 355 days, known as "abundant years"-i.e., in which the last month (Dhu al-Hijjah) has 30 days rather than 29. The months, alternately 30 and 29 days long, add up to a 354-day year. In its final form, probably defined shortly after his death, it comprised a cycle of 30 years of 12 months each. Having banned the calendar used in pre-Islamic Arabia, which he saw as tainted with paganism, Muhammad introduced a strictly lunar calendar. Despite these efforts, at the time of the later Julian reform, the civil year was about 90 days ahead of the solar year. To prevent the enormous confusion resulting from this practice and to curb the power of the Pontifices, attempts were made to define precise rules for the adjustment. From time to time, it neglected to intercalate the Mercedonius altogether. It did so in an empirical and arbitrary manner, often dictated by political motives. The College of Pontifices (priests) was in charge of adjusting the calendar. In the Numan calendar, the mean duration of the year was thus, on average, 366.25 days (= (355 + 377 + 355 + 378)/4), i.e., about a day longer than the actual solar year. In the years when it was added, February was shortened to 23 or 24 days, and the 5 or 4 days removed were added to the 22 or 23 of the Mercedonius, which was thus always 27 days long. The Mercedonius was alternately 22 or 23 days long. To make the two coincide, an additional month called Mercedonius was intercalated in alternate years after the Terminalia day (February 23). But the solar year lasts about 365.25 days, i.e., 11.25 days longer than the lunation cycle or lunar year. The two months were "intercalated" between December-the tenth and last month of the Romulean calendar-and March. The 57 missing days needed to reach the total 355 (31 x 4 + 29 x 6 = 298 days) were distributed between two new months: January, with 29 days, and February, with 28 days (still its length today). The remaining months were shortened by a day each, so that all the months now comprised an odd number of days. By contrast, the 31-day months in the old Romulean calendar-March, May, Quintilis, and October-were kept unchanged in the new calendar. Moreover, the Greek calendar alternated full months of 30 days with hollow months of 29 days. This may have been due to the Roman belief that odd numbers were lucky and even numbers were unlucky. However, the Greek version comprised 354 days (= 12 x 29.5 days, i.e., the mean length of lunation), while the Numan year had 355. Borrowed from the Greeks, the Numan calendar was a 12-month lunisolar calendar. Romulus's successor Numa Pompilius is credited with the reform of the Roman calendar.
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