Kali ahargaṇa (Kali ahargaṇa number or Kalidina) is an integer associated with a civil day. The integer represents the number of civil days in a collection of consecutive days beginning with a special day called the kali epoch and ending with a specified day. Kali ahargaṇa is one of the basic parameters of Indian astronomy and it is extensively used in all sorts of astronomical computations.
Commencement of kali epoch The way how the date of the beginning of the Kali epoch was calculated can be summarized thus. The whole basis for the computation is the following cryptic statement by Āryabhaṭa in Āryabhaṭīya (śloka (stanza) 10 in Chapter 3 Kālakriyā):
"When sixty times sixty years and three quarter yuga-s (of the current yuga) had elapsed, twenty-three years had then passed since my birth." According to commentators, this stanza refers to the fact that sixty times sixty years, that is 3600 years, have elapsed since the beginning of the kali era. So, using this statement as the basis, to determine the date of commencement of the Kali epoch, one need to determine exactly on which day Āryabhaṭa made this statement. There is a fair degree of agreement among historians regarding the year in which the statement was made. Historians believe that Āryabhaṭa made this statement in 499 CE. But the exact day of the year on which the statement was made is still a matter of conjecture as it has not been mentioned in Āryabhaṭīya or anywhere else. However, according to one view, the statement was made on March 21, 499 CE perhaps because the day was calculated to be the vernal equinox day of that year or the day following it. According to Āryabhaṭa, the duration of a year is 365 days 6 hours 12 minutes 30 seconds, that is, 365.25868 days approximately. Hence, as per Āryabhaṭa, the number days in a period of 3600 years is 1,314,931.25 days. Since a Julian year is 365.25, the number of Julian years in a period of 1,314,931.25 days is 3600 years 31.25 days. Assuming that the statement was made at sunrise on March 21, the sunrise of 31 days before that would fall on February 18. The balance of 0.25 days is a quarter of a day and so, 3600 Āryabhaṭan years exactly before the sunrise of March 21 would fall at the midnight of February 17–18. Now, regarding the year, it may be noted that historians have never included a year zero and so 3600 years before 499 CE would be 3102 BCE. Thus, the beginning of the Kali epoch may be fixed as the midnight of February 17–18, 3102 BCE. There are two different conventions regarding the exact moment at which the kali epoch. According to one convention, called the ardharātrika convention, the epoch is the midnight of February 17–18, 3102 BCE. According to the other convention, called the audāyika convention, the epoch is the moment of sunrise on February 18, 3102 BCE.
Verification Many Indian Almanac makers routinely include the kali ahargana numbers of every day of the relevant year in the almanacs. From these almanacs we can see that the kali ahargana of 1 January 2024 is 1,871,845. It can be verified that the number of days during the period from 18 February 3102 BCE in the proleptic Julian calendar to 31 December 2023 CE in the Gregorian calendar (both days inclusive) is exactly 1,871,845. While making the computations, the following points should be noted:
There is no year 0. Year n CE is a leap year if n is divisible by 4. Year n BCE is a leap year if (n-1) is divisible by 4. Thus the year 1 BCE is a leap year. After 14 September 1752 CE, leap year rule: “Every year that is exactly divisible by four is a leap year, except for years that are exactly divisible by 100, but these centurial years are leap years if they are exactly divisible by 400. For example, the years 1700, 1800, and 1900 are not leap years, but the year 2000 is.” A summary of the computations is depicted in the following diagram. The diagram shows the numbers of days during certain subperiod of the period from 18 February 3102 BCE to 31 December 2023 CE.
Ahargana In Indian astronomical traditions, the term kali ahargana (also called kalidina) is an integer associated with a civil day. The integer represents the number of civil days in a collection of consecutive days beginning with a special day called the kali epoch and ending with a specified day. The Kali ahargana of a day is the number of days in the duration from the Kali epoch and the sunrise on the day under consideration or the previous midnight depending on which convention is followed regarding the kali epoch, audāyika or ardharātrika.
Computation of kali ahargana Given a date in the Common Era calendar, it is trivial and straightforward to compute the kali ahargana of that day. The Common Era calendar is the product of the evolution over centuries with intervening events like the Gregorian reform and the different dates of adoption of the reform in different countries. However, if the date is given in some other calendar, say the pre-modern Saka calendar, then the compuatation of the corresponding kali ahargana is indeed very complicated. The texts of classical Indian astronomy spend a lot of energy in explaining elaborately the procedure for the computation of kali ahargana.
The computational procedure Bhāskara I has given the following procedure for the computation of the kali ahargana. Brahmagupta, Lalla, Śrīpati and Bhaākara II all have given the same procedure the computation of the kali ahargana. Data for a yuga consisting of 4,320,000 years (constants)
MS = number of saura months in a yuga = 4,320,000 X 12 = 51,840,000 DS = number of saura days in a yuga = 51,840,000 X 30 = 1,555,200,000 ML = number of lunar months in a yuga = (number of lunar revolutions) - (number of solar revolutions) = 57,753,336 - 4,320,000 (data on lunar revolutions from Aryabhatiya) = 53,433,336 DL = number of lunar days in a yuga = 1,603,000,080 MI = number of intercalary months in a yuga = (number of lunar months) - (number of saura months) = 53,433,336 - 51,840,000 = 1,593,336 DO = number of omitted tithi-s in a yuga = (number of lunar days) - (number of civil days) = 1,603,000,080 - 1,577,917,500 (data on civil days from Aryabhatiya) = 25,082,580 Data for the relevant day
m = number of months elapsed from 1st Caitra d = number of days elapsed since the end of the last Amāvāsya y = saura years elapsed in śāka Computations
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