Abstract
Arithmetical calendar (or tabular calendar) is sometimes referred to as the Fātimid calendar but this is in fact one of several almost identical tabular Islamic calendars. This calendar introduced by Muslim astronomers in the 9th century CE to predict the approximate begin of the months in the Islamic lunar calendar. Chronologists adopted 11 leap years in a 30 year cycle. In the case of leap Hijri year they add one day to the last month of the Hijri year. The cycle of this calendar agree with the Smaller cycles (2–5.333 years) discovered by Galal and Rashed (2011) and coincide with the lag criterion given by Galal (1988).
We suggested the Islamic tabular calendar. The Leap years of this suggested Islamic tabular calendar may be 2, 5, 7, 10, 13, 15, 18, 21, 23, 26 and 29. Our suggested Arithmetical calendar satisfies the mathematical patterns, while the old Arithmetical calendar (or tabular calendar) does not satisfy a known fixed rule.
We conclude empirical formula for our suggested Islamic tabular calendar. From this empirical formula, we can calculate if the Hijric year after immigration is a leap or a non-leap year.
1 Introduction
Many societies have used the lunar phase's cycle as the basis for their calendars. Even today a variety of cultures, including the 22% of the world population that holds the Islamic faith still begin their months with the first sighting of the crescent Moon in the evening after a New Moon. Therefore the Islamic calendar is a strict lunar calendar. Because the beginning of the Hijric month is determined by observation, it cannot be accurately predicted. However, for secular use a tabular calendar is available to determine the beginning of the Hijric month by fixed rules.
Islamic calendar converter is based on the arithmetical calendar (or tabular calendar), introduced by Muslim astronomers in the 9th century CE to predict the approximate begin of the months in the Islamic lunar calendar, reasonably accurate arithmetic calendars were in practice, which is still used in most of the Islamic countries.
The Islamic tabular calendar consists of 12 lunar months. A lunar month on average has 29.530589 days. Assuming it is 29.5 days, and the lunar civil months are set to have 30 and 29 days alternately, we get a common year 29.5 × 12 = 354 days. This is 0.367068 days less than 29.530589 × 12 = 354.367068…. Since this is very close to 11/30 = 0.3666666..., Chronologists suggest 11 leap months every 30 to adjust the Islamic calendar. The error is 4 × 10−4 day per 30 years. This error accumulates to a day in 2492 years (CitationThe Hijric calendar, Umesh Nair, April 6, 2009).
Chronologists have made a rule that each of the odd-numbered months comprises of 30 days while each of the even-numbered months comprises of 29 days, starting at sunset. That is, there are 11 leap years in a 30 year cycle. In the case of leap Hijri year, Chronologists added one day to the last month of the Hijri year, CitationMcPartlan (1996).
Some scholars suggested alternate leap year schemes. There are four different alternate leap year schemes. The suggested schemes described in as follows.
Table 1 Leap years in each 30-year cycle and its usage.
2 Results and discussion
We note that the cycles of the leap and non-leap years of the Islamic tabular calendar agree with the smaller cycles (2–5.333 years) discovered by CitationGalal and Rashed (2011) and coincide with the lag criterion of the Moon after Sunset given by CitationGalal (1988).
In the present study we suggest that the leap years of the Islamic tabular calendar after immigration may be 2, 5, 7, 10, 13, 15, 18, 21, 23, 26 and 29. Therefore the Arithmetical calendar pattern as in
Fig. 1 The suggested Arithmetical calendar after immigration.
According to and , we can conclude the following consequences:
| (1) | The old Arithmetical calendar (or tabular calendar) does not satisfy forms according to the particular system or particular rule. | ||||
| (2) | Kuwaiti algorithm and popular calendar have a small variation from our suggested Arithmetical calendar. While Fatimid (Misri/Bohra) and al-Bīrūnī calendar have many variation from our suggested Arithmetical calendar. | ||||
| (3) | Our suggested Arithmetical calendar satisfies the mathematical Pattern. In this mathematical pattern:
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To know if a year after immigration leap year or non-leap year through our suggested Arithmetical calendar, we can conclude the following empirical formula:
| (1) | If H-└H/30┘ × 30 belongs to (2, 5, 7, 10, 13, 15, 18, 21, 23, 26 and 29), then the year is a leap year. | ||||
where
H = the year after immigration i.e. the Hijric year.
└H/30┘= the floor of H/30.
Examples:
| (1) | To know if the year 1435 after immigration is a leap or a non-leap year, As follows: | ||||
1435 – └1435/30┘ × 30 = 1435 – (47 × 30) = 25,
Therefore the year 1435 is a non-leap year.
3 Conclusions
We can conclude that our suggested Arithmetical calendar satisfies the mathematical pattern while the old Arithmetical calendar (or tabular calendar) does not satisfy any fixed rule. We conclude empirical formula for the suggested Islamic tabular calendar. From this empirical formula, we can calculate if the Hijric year is a leap or a non-leap year. Also the cycle of our suggested Arithmetical calendar agree with the Smaller cycles (2–5.333 years) discovered by CitationGalal and Rashed (2011) and coincide with the lag criterion given by CitationGalal (1988).
Notes
Peer review under responsibility of National Research Institute of Astronomy and Geophysics.
References
- McPartlan,Q. J. R. Astr. Soc.371996837842
- Galal.‘Helwan Observatory’Bull. Ser. A581988
- GalalRashedNRIAG J. Astron. Astrophys.2011305322 (special issue)
- The Hijri calendar, Umesh Nair, April 6, 2009. http//www.usvishakh.net/Umesh.