A study of Parameśvara's record of eclipses
Aditya Kolachana and K. Ramasubramanian
Cell for Indian Science and Technology in Sanskrit
Dept. of Humanities and Social Sciences
Indian Institute of Technology Bombay, India

1. Introduction
India has a hallowed history of mathematics and astronomy. Starting with Āryabhaṭa (5th CE), there
were several illustrious names like Varāhamihira, Brahmagupta, Bhāskara – I , Lallācārya,Mahāvīrācārya, Śrīdhara, Śrīpati, Bhāskarācārya (12th CE), and so on, who made significant
contributions in the areas of arithmetic, geometry, algebra, mensuration and the like.
During the 14th to the 16th centuries, generally referred to as the golden age of mathematics in India,
the study of astronomy and mathematics really flourished in the Malabar region of Kerala, on thebanks of the Nilā river. A succession of scholars, collectively referred to as the Kerala School ofMathematics, made remarkable contributions to the advancement of the field that is referred to as
infinitesimal calculus today. This school was founded by a gifted mathematician and astronomer
named Mādhava who is best known for inventing the infinite series for pi, and other trigonometricfunctions for the first time in the history of mathematics. Some of the noted astronomers andmathematicians who followed Mādhava include Parameśvara, Dāmodara, Nīlakaṇṭha, Jyeṣṭhadeva,Śaṅkara Vāriyar and so on.
The primary aim of this paper is to make a systematic study of the eclipses recorded byParameśvara, who was a direct disciple of the illustrious Mādhava. Since the details of theoccurrence of eclipses are presented by Parameśvara using the Kali-ahargaṇa calendar (day countfrom the beginning of Kali epoch starting 17th / 18th February, 3102 BCE), we had to develop
necessary tools for the conversion of dates between Indian and Gregorian calendars in order tostudy these historical eclipses. This study, as may be seen from sections 5 and 6 of this paper, in
addition to contributing to the pages of the history of science, could also facilitate further researchin arriving at better estimates of fluctuations in the length of day (∆T) – which is a very interesting
and challenging problem pursued by astronomers in different nations across the world.

12. Parameśvara and his contributions
Besides being a disciple of the great savant Mādhava, Parameśvara (1360 – 1455) himself was aprolific writer and is credited with several works in astronomy and astrology. He hailed from thevillage Aśvatthagrāma situated at the confluence of the Nilā river and the Arabian sea 1.
Understandably, Parameśvara time and again refers to the banks of the Nilā river (िनिलातट) while
presenting the details of the eclipses observed by him (see section 5).
The works of Parameśvara include original writings (such as Dṛggaṇita and Goladīpikā), as well as
detailed commentaries on other major works. Some of his well known commentaries includeBhaṭadīpikā – a commentary on the Āryabhaṭīya of Āryabhaṭa, and Siddhāntadīpikā – a supercommentary on Govindasvāmin's Mahābhāskarīya-bhāṣya. He has also authored commentaries on
Laghumānasa, Sūryasiddhānta, Līlāvatī and the like, some of which are yet to be published.
In addition to being an expert theoretician, Parameśvara was also an avid practical astronomer.
Being situated on the vast sandy expanse of the banks of the Nilā river as it converges with the
Arabian sea, he seems to have carried out astronomical observations for over 55 years, and also
meticulously noted down the details of the occurrence of a variety of astronomical phenomena. In
order to elucidate the principles behind the computation of eclipses, Parameśvara composed three
short treatises exclusively on eclipses: Grahaṇāṣṭakam, Grahaṇamaṇḍana, and Grahaṇa-nyāyadīpikā. In his Siddhāntadīpikā, he records 13 eclipses (8 solar and 5 lunar) which he personallyobserved, along with a brief description of their characteristics. Nīlakaṇṭha, another famous
mathematician and astronomer from the Kerala school, and grand-disciple of Parameśvara, also
cites these observations of Parameśvara in his monumental treatise Āryabhaṭīya-bhāṣya (acommentary on the Āryabhaṭīya). In the same work, Nīlakaṇṭha extolls Parameśvara as Golavid(one who has excelled in spherics). He observes2:"परमेश्वराचायेर्येण पुनिः ग्रहणग्रहयेोगािदिकं येन्त्रैः पञ्चपञ्चाशद्वर्षकर्षकालं सम्येक् परीक्षितमक्षितम्। … तस्ये
गोलिवित्तमत्विं च गोलदिीक्षिपकािदिितमभिः तत्कृतैः प्रबन्धविविशेषकैः ज्ञायेते। अत अन्येषकां करणानिां स्विस्विकाल
येावित्सूक्ष्मत्विं
12ततोऽप्येतत्कृतस्ये

दिृग्गितमणताख्येस्ये

सूक्ष्मत्विमुपपद्यते।

अिपचThe renowned indologist KV Sarma (in his D.Litt Thesis Volume I, p.200.) has identified the modern names of thevillage and river as Alattur and Bharathappuzha respectively.
Pillai 1957, p.154.

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परमेश्वरोक्तमन्दिोच्चपिरध्येादिीक्षनिाम् अायेर्षभिटोक्तेभ्येः सूक्ष्मत्विितममदिानिीक्षमिप ग्रहयेोगािदिष्विस्माितमभिज्ञायेते।"“Eclipses, conjunctions of planets etc. were examined with the help of instruments by the great preceptor
Parameśvara for 55 years … His profundity in spherical astronomy is evident from some of the treatises
like Goladīpikā composed by him. His work Dṛggaṇita turns out to be more accurate than other karaṇa
(handbook) works of his time. Moreover, even today (ca. 16 th century), parameters of the circumferenceof epicycle etc. given by Parameśvara are more accurate than those stated by Āryabhaṭa as evident fromthe calculation of the conjunction of planets etc.”

In yet another work Siddhānta-darpaṇa-vyākhyā, in connection with specifying the magnitude of
the precision of the equinox (ayanāṃśa), Nīlakaṇṭha again adores Parameśvara as his paramaguru(guru's guru) with great reverence3:"...येतो

भिागर्षविपरमेश्वराचायेर्येण

अस्मत्परमगुरुणा

चलांशास्स्विं (४५३६) इतित कल्येब्दे परीक्षक्ष्ये

पञ्चदिशांशपूितिनिणीता।"“...since our paramaguru Bhārgava Parameśvara in the Kali year 4536 has specified the magnitude of theprecession of equinoxes as 15 degrees after experimentation.”

The above citations are provided not only to make it amply clear that Parameśvara was held in very
high regard, but also to show that he was considered an authority in the field of astronomy. But for
Parameśvara being an astronomer par-excellence, the great savant Nīlakaṇṭha would not havequoted him while discussing the problem of precession of equinoxes, and deciding on the value ofayanāṃśa. In fact, the works of Parameśvara have strongly influenced the works of subsequent
mathematicians and astronomers – particularly his Dṛggaṇita, which came as a revision of the older
Parahita system of computation.
3. The methodology adopted for our study
In order to study Parameśvara's record of eclipses, as mentioned earlier we had to first convert therecorded Indian dates into Gregorian dates, and then find out whether eclipses actually occurred on
those days. To achieve this, we first developed a date conversion tool to convert Indian dates
specified in Kali-ahargaṇa or Śaka systems to dates based on the Gregorian calendar. We also wrotea software application using modern astronomical algorithms to predict the occurrence of historical3

Sarma 1977, p.17.

3solar and lunar eclipses. Finally, we compared the dates given by Parameśvara against the resultsthat were obtained using our software application as well as the database of historical eclipses
maintained by the National Aeronautics and Space Administration (NASA), USA. The software
applications referred to above were built using the Python programming language.

These

applications are briefly described below.
Application I: Date conversion tool
The conversion of dates between the Indian luni-solar calendar (generally specified in terms ofŚaka year) and the Gregorian calendar is not a trivial problem. This is because of the introduction ofwhat are known as adhimāsa (intercalary months) and kṣayamāsa (omitted months) in the Indian
calendrical system. Twelve lunar months constitute only ~354 days (~29.54 * 12), while a solar yearis ~365 days. Therefore, a lunar year lags a solar year by approximately 11 days. In order to see thatthe lunar calendar does not lag behind the solar calendar, an adhimāsa is added to the lunar calendar
roughly once every 2.75 years on an average. However, this introduction is not done arbitrarily. It isbased on the occurrence of a certain physical phenomenon, namely a saura-saṅkramaṇa (transit of
the sun from one zodiac to another). An adhimāsa is said to occur when a saura-saṅkramaṇa does
not occur within a lunar month. Similarly, a lunar month is omitted in the Indian calendar when twosaura-saṅkramaṇas occur within a lunar month. Finding the precise occurrence of these two typesof months is not straightforward because they do not occur periodically, and also crucially depends
upon calculation of the instantaneous rate of motion of the sun and the moon as seen from the Earth.
However, we have written appropriate code to compute their occurrence as accurately as possible.
Taking a contemporary date as reference, and finding the correspondence between the Indian and
Gregorian calendars for this day, our tool starts back computing, and adjusts for the intercalary as
well as omitted months. Accordingly, the tool accurately maps the dates from the Indian calendar to
the Gregorian calendar and vice-versa.
Application II: Program to compute eclipses
This application was developed in order to predict the occurrence as well as the duration of solar
and lunar eclipses and is based on the algorithm given by Peter Duffet-Smith 4 for this purpose. The
classic text on spherical astronomy by W.M. Smart 5 was also consulted.

45Duffet-Smith 1981.
Smart 1949.

4

Any astronomical computation commences with the specification of a set of parameters that include
the initial positions of the planets, as well as other orbital elements for a particular day. This ischosen as the reference day or epoch for all further calculations. In our case, we have chosen theorbital elements of the Sun and Moon on January 1, 1980 as our standard reference, and thenproceeded to calculate their mean positions on any desired day using their mean motions. We thenapply corrections to the mean positions to arrive at the true longitudes of the sun and the moon. In
the case of the sun, we applied only the equation of center correction (known as mandasaṃskāra in
Indian astronomy) in order to arrive at the true position of the sun. In the case of the moon, besides
the equation of center, corrections for evection, annual equation, third and fourth corrections, andvariation are applied to arrive at its true longitude.
By testing for the occurrence of conjunction and opposition of the sun and the moon, we predict theoccurrence of new moon and full moon days respectively. If at the time of their conjunction or
opposition, the distance of the moon from its node is within a prescribed range (known as ecliptic
limits), then a solar or lunar eclipse is said to occur. We have developed the necessary code to
predict the occurrence of these events, and have also extensively tested its accuracy.
4. Verifying Parameśvara's record of eclipses in Siddhāntadīpikā
It was noted earlier that Parameśvara has recorded the details of 13 eclipses that were observed byhim during his lifetime in his Siddhāntadīpikā. These eclipses span over a period of more than three
decades (from 1398 to 1432) as can be seen in Table 1. The description provided by Parameśvara
invariably includes the date of occurrence of the eclipse in Kali-ahargaṇa, as well as the type of theeclipse (solar or lunar). The other details provided by him vary from eclipse to eclipse. These may
include the nature of the eclipse (full, partial), the location of its appearance, the time and durationof occurrence, and so on. Interestingly, all these descriptions are given in the form of verses
composed in the anuṣṭubh metre. The early part of his record describes eight solar eclipses – notnecessarily in chronological sequence – along with a very brief description of their characteristics.
The last four verses describe five lunar eclipses.
Having converted the dates of the occurrence of eclipses specified by Parameśvara in Kaliahargaṇa into the Gregorian calendrical dates, we checked for the occurrence of eclipses on those
dates using both our eclipse tool as well as the NASA database. The details of the comparison arelisted in Table 1. It may be noted that there is a complete concurrence between the dates specified in5the Siddhāntadīpikā and the ones obtained with our eclipse tool, as well as the NASA database. This
clearly indicates that the record of eclipses presented by Parameśvara forms a reliable historical
record which is sought after by the astronomers who are working on arriving at a better expressionfor ∆T.
Table 1: Comparison of Parameśvara's record with our eclipse tool and NASA database

Parameśvara's record

Eclipse Tool

NASA database

Kali-ahargaṇa Type of Eclipse Converted Date (CE)

Date (CE)

Date (CE)16,43,524

Solar

9th Nov, 1398

9th Nov, 1398

9th Nov, 1398

16,47,156

Solar

19th Oct, 1408

19th Oct, 1408

19th Oct, 1408

16,48,722

Solar

1st Feb, 1413

1st Feb, 1413

1st Feb, 1413

16,52,000

Solar

23rd Jan, 1422

23rd Jan, 1422

23rd Jan, 1422

16,53,387

Solar

10th Nov, 1425

10th Nov, 1425

10th Nov, 1425

16,55,130

Solar

19th Aug, 1430

19th Aug, 1430

19th Aug, 1430

16,55,484

Solar

8th Aug, 1431

8th Aug, 1431

8th Aug, 1431

16,55,662

Solar

2nd Feb, 1432

2nd Feb, 1432

2nd Feb, 1432

16,52,694

Lunar

17th Dec, 1423

17th Dec, 1423

17th Dec, 1423

16,53,403

Lunar

25th Nov, 1425

25th Nov, 1425

25th Nov, 1425

16,54,614

Lunar

20th Mar, 1429

20th Mar, 1429

20th Mar, 1429

16,55,293

Lunar

28th Jan, 1431

28th Jan, 1431

28th Jan, 1431

16,55,647

Lunar

17th Jan, 1432

17th Jan, 1432

17th Jan, 1432

5. A sample of eclipses from the record of Parameśvara
As indicated above, since Parameśvara's record of eclipses is reliable, it can be used for various
historic and scientific studies including the calculation of a better expression for ∆T, which requiresa reliable historic record. Of the 13 eclipses recorded by Parameśvara, the following three solar
eclipses may be of interest for calculating ∆T, since they furnish details such as the location ofvisibility of the eclipse, the time of occurrence of the eclipse in that locality, and so on.
Parameśvara's original verses from the Siddhāntadīpikā6 recording these eclipses are presented
below, along with our translation.

6

Kuppanna Sastri 1957, pp. 329 – 331.

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1. Solar eclipse of 19th October, 1408
द्युगणे रसितथ्येिद्रिवेदिषकड्भिूितममसितमम्मते।
मुक्तेऽकर्केऽस्तमयेो दिृष्टः निाडीक्षपादिोऽितमस्त विान्तरे।।
O n Kali-ahargaṇa corresponding to 16,47,156 the sun was observed to be setting when it
was released (last contact) from the eclipse, or there was a difference of a quarter of a nāḍī(~6 minutes) between them (last contact and sunset).

2. Solar eclipse of 10th November, 1425
द्युगणे सप्तनिागाितमग्निगुणेषकुरसभिूितममते।
गोकणर्ये ग्रहणं भिानिोः दिृष्टं निात्र िनिलातट।।
On Kali-ahargaṇa corresponding to 16,53,387 a solar eclipse was observed in Gokarṇa, but
not here on the banks of the Nilā river.

3. Solar eclipse of 19th August, 1430
शून्येाितमग्निभिूशरेष्विङ्गभिूतुल्ये द्युगणे रवेः।
गोकणर्ये ग्रहणं दिृष्टं िनिलाब्ध्येोः संगमेऽत्र नि।।प्रोक्ते िदिनेऽिप िबबस्ये पाश्र्विर्ये विणर्षस्ये भेदिनिम्।
कैश्चितमश्चित् कुमारैरत्रािप कितमल्पतं विा िनिलातट।।
On Kali-ahargaṇa corresponding to 16,55,130 a solar eclipse was observed in Gokarṇa, but
not at the confluence of the Nilā river with the ocean. On the said day also, some boys
claimed to have observed a change in shade on one side of the solar orb here too, on thebanks of Nilā river.

6. The relevance of this record for ∆T calculations
Of late, Morrison and Stephenson in one of their recent publications 7, have explained theimportance of historical eclipse records in arriving at a better estimate of ∆T. The value of ∆T,
which takes into account the effect of variation in the Earth's rate of rotation, is very crucial in
7

Morrison et. al. 2004.

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calculating eclipses in the remote past. According to them, as of now, there seems to be “... a degreeof confusion and misapprehension among the historians of astronomy over the choice of values of∆T...”. In this connection, it was felt that the 'reliable record' of eclipses given by Parameśvara could
be of some relevance.
In the first of the three examples presented above, it is stated that the eclipse occurred close tosunset. Since it is also stated that the eclipse was over just 6 minutes before sunset, this could beparticularly useful for ∆T calculations. The path traced by this eclipse is shown in Figure 1 8.
In the second example, Parameśvara states that the eclipse was observed in Gokar ṇa (14.55o N,
74.3167o E) but not on the banks of the Nilā river (10.77o N, 75.9o E). This gives us latitude andlongitude limits for the path of the eclipse which also may prove to be useful in the calculation of∆T. Figure 2 shows the path of this eclipse.
Similarly, in the third example it is stated that this eclipse too was observed in Gokarṇa but not at
the confluence of the Nilā river with the ocean. Therefore, we may use the same longitude andlatitude limits as above in trying to arrive at a better estimate of ∆T. Figure 3 shows the path tracedby this eclipse.
A more detailed investigation into Parameśvara's record, as well as other records of historicaleclipses found in certain inscriptions in India could further assist in arriving at better estimates of∆T.
Acknowledgement
The authors would like to place on record their sincere gratitude to MHRD for the generous support
extended to them to carry out research activities on Indian science and technology by way of
initiating the Science and Heritage Initiative (SandHI) at IIT Bombay.

8

Source of all figures: NASA. Weblink given in the list of references.

8

Figure 1: Path of solar eclipse on 19th October, 1408

9Figure 2: Path of solar eclipse on 10th November 1425

10Figure 3: Path of solar eclipse on 19th August, 1430

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References

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Govindasvāmin and the Super-commentary Siddhāntadīpikā of Parameśvara. Madras:
Government Oriental Manuscripts Library.
2. Pillai, S.K. (1957). Āryabhaṭiya of Āryabhaṭācārya with the Mahābhāṣya of Nīlakaṇṭha
Somasutvan, Part III Golapāda. Trivandrum: Trivandrum Sanskrit Series 185.
3. Sarma, K.V. (1977). Siddhānta-darpaṇa of Nīlakaṇṭha Somayājī with Auto-commentary.
Punjab: Punjab University.
4. Subbarayappa, B.V., and Sarma, K.V. (1985). Indian Astronomy: A Source Book. Bombay:
Nehru Centre.
5. Duffet-Smith, P. (1981). Practical Astronomy with your Calculator. 2nd ed. Cambridge:
Cambridge University Press.
6. Meeus, J. (1991). Astronomical Algorithms. 1st ed. Richmond: William – Bell Inc.
7. Ramasubramanian, K., and Sriram, M.S. (2011). Tantrasaṅgraha of Nīlakaṇṭha Somayājī.
New Delhi: Hindustan Book Agency.
8. Smart, W.M. (1949). Spherical Astronomy. 4th ed. Cambridge: Cambridge University Press.
9. Five Millennium Catalog of Solar and Lunar Eclipses. National Aeronautics and Space
Administration website, 2014 [viewed 7th November, 2014]. Available from http://eclipse.gsfc.nasa.gov/eclipse.html
10. Morrison, L.V., and Stephenson, F.R. (2004). Historical values of the Earth's clock error ∆T
and the calculation of eclipses. Journal for the History of Astronomy, 35(3 – 120), 327 –
336.

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