Kali ahargaṇa (Kali ahargaṇa number orKalidina) is an integer associated with a civil day. The integer represents the number ofcivil days in a collection of consecutive days beginning with a special day called thekali epoch and ending with a specified day.[1]Kali ahargaṇa is one of the basic parameters of Indian astronomy and it is extensively used in all sorts of astronomical computations.
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 inĀryabhaṭīya (śloka (stanza) 10 in Chapter 3Kālakriyā):[2][3]
According to commentators, this stanza refers to the fact that sixty times sixty years, that is 3600 years, have elapsed since the beginning of thekali 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 inĀryabhaṭīya 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.[3]
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 ayear 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 thekali epoch. According to one convention, called theardharātrika convention, the epoch is the midnight of February 17–18, 3102 BCE. According to the other convention, called theaudāyika convention, the epoch is the moment of sunrise on February 18, 3102 BCE.
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.[4][5] It can be verified that the number of days during the period from 18 February 3102 BCE in theproleptic Julian calendar to 31 December 2023 CE in theGregorian calendar (both days inclusive) is exactly 1,871,845. While making the computations, the following points should be noted:
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.
InIndian astronomical traditions, the termkali ahargana (also calledkalidina) is an integer associated with a civil day. The integer represents the number ofcivil days in a collection of consecutive days beginning with a special day called thekali epoch and ending with a specified day.[1] TheKali 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 thekali epoch,audāyika orardharātrika.
Given a date in theCommon Era calendar, it is trivial and straightforward to compute thekali ahargana of that day. The Common Era calendar is the product of the evolution over centuries with intervening events like theGregorian 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-modernSaka calendar, then the compuatation of the correspondingkali ahargana is indeed very complicated. The texts of classical Indian astronomy spend a lot of energy in explaining elaborately the procedure for the computation ofkali ahargana.
Bhāskara I has given the following procedure for the computation of thekali ahargana.Brahmagupta,Lalla,Śrīpati andBhaākara II all have given the same procedure the computation of thekali ahargana.[1][6]
Data for ayuga consisting of 4,320,000 years (constants)
Data for the relevant day
Computations
Theahargaṅa number obtained by applying the above procedure may sometimes in error by one day. The correctness or otherwise of the computed value can be tested by finding the day of the date given by theahargaṅa number and the day of the date of which theahargaṅa number was being calculated. Letr be the remainder when theahargaṅa number is divided by 7. Ifr is 0, then the day given by theahargaṅa number would be Friday, ifr is 1, the day would be Saturday, and so on. The defect or excess in theahargaṅa number may be corrected accordingly by increasing or decreasing the number by one.
The procedure is illustrated by computing thekali ahargaṅa of Tuesday 10 July 2001. The data relevant to this date are as follows:
The computations proceed as follows.
To check the correctness of the value, note that the remainder when 1,863,634 is divided by 7 is 3 which corresponds to Monday and the day of the date is Tuesday. So the calculated value of theahargaṅa number is in defect by one day. Increasing the value by one theahargaṅa number of Tuesday 10 July 2001 is obtained as 1,863,635.
If thekali ahargana of some recent date is known, then thekali ahargana of any desired date can be computed using the following formula: LetKD be thekali ahargana of a dateD and letX be the date on which thekali aharganaKX is to be computed. Then:
For example, thekali ahargana of 10 July 2001 can now be used to compute thekali ahargana of 15 August 1947.
The reverse problem of determining the Common Era date, that is, the date as per the commonly accepted Julian/Greogorian calendar, corresponding to a givenkali ahargaṅa is also important. Archaeologists have come across several inscriptions in which dates are recorded askali ahargaṅa and there are several Sanskrit texts which containkali ahargaṅa indicating the day of the completion of the work. Decoding these data into dates in modern calendar is important in fixing the dates of the monuments or texts. Tables have been constructed to solve this problem which help ease the difficulty of performing cumbersome computations.[6][7]
Historical records, literary and inscriptional, of the North India and Deccan are silent about thekali ahargaṇa. But in the legends of Kerala there are many dates expressed askali ahargaṇa (kalidina) in thekatapayãdi notation. A few of them are given below.[6][8]
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