Ancient Indian astro-mathematical tradition: Evolution and linkages

AIP Conf. Proc. 1283, pp. 156-160; doi: (5 pages)

MATHEMATICS AND ASTRONOMY: A JOINT LONG JOURNEY: Proceedings of the International Conference

Editor(s): Manuel de León, D. M. de Diego, R. M. Rosa

Date: 23–27 November 2009

Location: Madrid, (Spain)

Rajesh Kochhar

Indian Institute of Science Education and Research Mohali

Sector 26, Chandigarh 160019, India

[email protected]


Indian astronomical tradition is characterized by antiquity, continuity and interaction with the outside world. From 6th century CE till the time of Kepler’s laws, Indian astronomers were probably the only ones in the world who could calculate eclipses with any degree of accuracy. In the 12th century, an astronomer in Central India, Padmanabha by name, predicted the lunar eclipse of 8 November 1128 and was rewarded by the king with a land grant ( Mirashi 1933-34). The tradition was alive well into the 19th century. By means of shells arranged on the ground and using mathematical  tables memorized “by means of certain artificial words and syllables”, a “Kalendar maker residing in Pondicherry” calculated the lunar eclipse of 31 May -1 June 1825, with an error of no more than +4 minutes for the beginning (Neugebauer 1983, p. 436). Even now, traditional astronomical almanacs in India, known as panchangas, used in India for ritual and religious purposes base their calculations on ancient texts. It is only in the case of eclipse that they borrow data from modern sources.


The beginnings of astronomy are related to the requirements of the ritual in early cultures. Ritual was seen as a means of securing divine approval and support for terrestrial actions. To be effective, it had to be elaborate and well-timed, so that a careful distinction could be made between auspicious and inauspicious times. Since planetary motions provided a natural means of time keeping and were seen as embodiment of divine signals, astronomy developed as an intellectual discipline( see Yano 2003). Similarly mathematics grew as an aid to designing sacrificial altars. The oldest geometry texts in India are the Sulvasutras which dealt with questions like the square root of two. Different scholars place the earlier of these texts anywhere between 800 BCE and 400 BCE. Astronomy texts are decidedly older. Subsequent developments in mathematics came about as an astronomical aid.


Source material and its inherent limitations


Any enquiry into ancient Indian astronomy or for that matter into any aspect of ancient India must take note of the inherent limitations of the source material available. Scripts (Kharoshthi, Brahmi) were introduced into India about 3rd century BCE or somewhat earlier for writing Prakrit languages derived from Sanskrit.  Script for Sanskrit itself, the language of Hindu scriptures, was adopted much later.  Writing material came from plants or trees and had a short life. Paper was not introduced into India till about 8th century CE. The Vedic texts were in any case forbidden to be written.  (In the following Sanskrit words have been written by omitting diacritical marks.)


Ancient Indian intellectual tradition was oral. Texts were in the custody of specialist caste groups who memorized them and transmitted them to the next generation by word of mouth. What was not considered worth preserving at any point in time was lost for ever. Also, it is not possible to assign firm dates to any early event or development. (Western scholarship especially during the colonial period tended to deny antiquity or originality to ancient India. As a backlash, many researchers have tended to unduly stretch the chronology backwards.)


Ancient India has bequeathed us four types of texts. (i) The Vedic corpus, considered sacred, was faithfully preserved without any addition or subtraction. Methods were devised to prevent deliberate or unintended corruption. Strictly speaking, early Vedic texts are not texts of Hinduism. Rather they constitute the heritage of Hinduism.(ii) Hinduism in practice is represented by the Puranas and the epics of Mahabharata and Ramayana. Additions were made to these texts but no deletions.(iii) Buddhist , and to an extent Jain, texts. Buddhist texts in Pali and later in Sanskrit are particularly valuable because they can be dated. Valuable additional information   comes from texts from outside India. (iv) Astronomy texts proper experienced both addition and deletion.


The Vedic corpus, comprises priestly books composed by a large number of authors over a long period of time, which could be as much as two thousand years, say from 1700 BCE to zero CE and later (Kochhar 2000).The importance of the Vedic texts lies in the fact that scrupulous care was taken to preserve them in their original form. They are thus truly representative of the time of their composition even if that time is largely indeterminable. The pride of place in the Vedic corpus belongs to the oldest and the stand-alone text, Rgveda, containing about ten thousand stanzas. According to Kochhar (2000) it was composed over a period extending say from 1700 BCE to 900 BCE, although its earliest portions probably contain memories of still earlier time. (Some other texts, though closed later, may contain older matter.)The youngest texts, like Manavadharmasastra, or Manusmrti, which could be as recent as about zero CE, give or take a few centuries, represent transition to Hinduism proper.


There are a few stray astronomical references in the Rgveda, but for our purposes the more useful is the Yajurveda which is a manual for actual performance of ritual. (Sulvasutras, referred to above. are also affiliated to Yajurveda.) Moon’s position every night was marked with respect to bright star or star group that was seen near it. These 28 or 27 stars, known as naksatra, were enumerated beginning with Krttika (Pleiades) which then described the spring equinox. Later lists, prepared about 500 CE, begin with Asvini, that is Beta Arietis, in honour of the then spring equinox.


There is a solitary Vedic text, Vedanga Jyotisa, devoted exclusively to astronomy. Its oldest portions could be as old as 1400 BCE. It deals only with the movements of the sun and the moon.  Zodiacal signs and week days are conspicuous by their absence. They would be introduced into India about 100 BCE (see below). Being a scientific text it was superseded   by later texts and went out of use. But being part of the sacred Vedic corpus it  was still sought to be preserved. Consequently it is the most obscure and least understood of all Vedic texts.


The Puranas and the two epics, Ramayana and Mahabharata, were narrated to the public at large and often recast to suit the prevailing requirements of the narrators as well as the listeners.  Of these the Mahabharata has a certain historical value. It was open for a very long time. It was probably closed by 100 CE. The date is arrived on the basis of the fact that this vast anthology adheres to Vedic astronomy and is not familiar with zodiacal signs and week days (see below).


Greco-Babylonian astronomical inputs were received from about 100 BCE, after the Mauryan empire collapsed and the political vacuum was filled by new people who came into India from the northwest.  The early inputs comprise zodiacal signs (rasis) and  week days both introduced about 100BCE  followed by  an accurate luni-solar calendar ( known as the Saka calendar) introduced 78 CE. Greek astronomical elements made their documented appearance in India in 149 CE when a Greek astro-text was translated into Sanskrit by Yavanesvara. (The term Yavana , derived from Ionian, was used to denote Greeks.) It was versified in 269 CE by Sphujidhvaja under the title Yavanajataka ( Pingree 1959). The versification was a significant development, because it signifies assimilation of Greco-Babylonian elements into Indian tradition. And yet, Vedic astronomical tradition remained extant even after the introduction of Yavana texts. A Buddhist Sanskrit text known as Sardulakaranavadana and dated about fourth century CE still adheres to Vedic astronomy.


Revitalization of India astronomy was effected by Aryabhata (born 476 CE) with his influential  text Aryabhatiyam that appeared in 499 CE. Mathematical astronomy texts were called Siddhanta, meaning proven in the end. In the sixth century CE, false Sanskrit etymology was sought to be constructed for a manifestly Greek term like hora ( Varahamihira’s Brihjjataka 1.3;see Rao 1986, p.33)) This was proof that Indian astronomy had established its own identity and generated its own momentum that would last more than a millennium ( Kochhar & Narlikar 1995). The rather long period from 100 BCE to 499CE constitutes the transitional period and  is very poorly understood and needs further research by tapping sources from outside India if they exist.


Apart from eclipse calculations Siddhantic astronomy addressed other concerns as well: Calculation of mean and true position of the (geo-centric) planets; time of rise and setting of planets; conjunction of planets; conjunction of a planet and a star; heliacal rising and setting of stars; instrumentation; etc. Siddhantic astronomy received a new lease of life with  Paramesvara ( 1360-1455) who in a career spanning more than half a century  timed many eclipses and planetary conjunctions .He then set out to devise mathematical means to bring calculated times closer to observations. His singular contribution is the construction of Drgganita ( Drk system of computations). Paramesvara not only composed original texts but also wrote valuable commentaries on previous famous works. He had a number of eminent successors who constitute what is now referred to as the Kerala school. Mathematics was developed  as a tool for planetary calculations. There was very little work on mathematics for its own sake. A notable full-time mathematician is Mahavira (9th century CE). He for example worked out how a number can be cubed using an arithmetical progression. Many mathematical problems solved as part of astronomical exercises engaged the attention of famous European mathematicians centuries later( for example,  Diophantine equations). A world history of mathematics must take into account astronomy-related mathematical work carried out in India.


Early healthcare knowledge resided in common people; it was Sanskritized later. In contrast familiarity gained about the skies was considered to be revealed knowledge and seen as an extension of scriptures. It was disseminated in the pattern of Rgveda, in terse metrical poetry. An astronomer had to be a poet first. Precision and exactitude that we expect from a scientific text today was sacrificed at the altar of poetical niceties. Although India originated numeral system, astronomical texts convey numbers through artificial words which could be mis-spelt inadvertently or deliberately, causing confusion. More often, certain important parts of mathematical formulas had to be omitted, and exact terms jettisoned in favour of terms, which though uncommon or vague, met the metrical requirement. Metrical composition however had one advantage as far as an author was concerned. Metre was his signature so that when later astronomers cited an earlier work, the credit could be given to the original author.


Siddhantic texts were not planned as text books. To facilitate teaching, commentaries were composed on famous texts. These commentaries did not aim at explaining the derivation of the rules. They often concerned themselves with expressing the rules in clearer language and in illustrating them with examples. These commentaries serve a very important historical purpose. They tell us about the state of the subject at the time of their composition. Indeed we know of many authors and works from there mention in preserved commentaries.


Siddhantic texts are not library books. These texts are not complete. They were meant for a small group of audience which knew the context. In their time they would have been accompanied by complementary notes which are no longer extant. What was common knowledge or expected to be known beforehand will not find place in a terse oral metrical text. In other words, conclusions can be drawn on the basis of what is explicitly mentioned in the texts. But no inference should be drawn from what is left unsaid. If the dog barked we know the dog was there. But if the dog did not bark we can never know whether the dog was absent or was present but chose to remain silent.


Chronology of ancient is a serious unsolvable problem. Situation in astronomy is not so hopeless as in other fields because epoch is inbuilt into astronomical tables.  But we know of the time period for only  a handful of  astronomers. Some astronomers have mentioned the year of birth in their work. But we do not know when they died. There are others who cannot be assigned any reliable date. For example an important astronomer Lalla can only be assigned a wide time bracket between 8th and 10th centuries CE (Ohashi 2009, p.25)


In short it is not possible to construct ancient India the way the exercise can be carried out for Egypt, China, or Iraq (Mesopotamia). Western scholarship, especially during the colonial period, tended  to deny antiquity, originality or practical knowledge to ancient India. This produced a backlash. Many researchers have tended to  unduly stretch the hronology backwards or in a spirit of misplaced patriotism claim for ancient astronomers more than is warranted.


Instead of saying directly that the orbital period of Saturn was 29.4743 years, Surya Siddhanta would say that there were 146,568 revolutions of Saturn in a mahayuga of 4.32 million years. It would have been more convenient to express such numbers in units of a million years, but obviously the driving force was the desire to show solidarity with the Vedic literature. Although the Rgvedic format was copied, there was no mechanism for preserving the original text. Given the fact that results were presented in a terse imprecise contextual way, even important results were left unanchored .It was easy to replace a word here and there   in an older text for expediency. Thus it became impossible to build on previous scholarship or to resolve controversies.


The following features of Indian astronomical tradition are noteworthy.

Major astronomical texts are known by their authors. But when their contents were incorporated into texts that were used for astrological purposes, the latter texts were given divine names like Surya Siddhanta. This has been and still is the most commonly used astronomical text. The name has remained the same but the contents have been revised.


Essential inputs into planetary calculations include orbital periods of the seven plants and the distance to noon in terms of earth’s diameter. We do not know how these values were arrived at, whether they were result of direct measurements or were borrowed.


Generally speaking, parameters were not seen as physically determined and therefore sacrosanct, but as mathematically adjustable. There was debate on whether the latitude of Ujjain, the ancient zero meridian, should be taken as 22 and a half  degrees or 24 degrees.  Commenting on this long-standing dispute, a commentator, Suryadeva ( b. 1191), wrote  : “ We do not see any use” in the former value. To enhance the usefulness of the latter value, a word was sought to be changed in an older, well-respected text  (Shukla 1976, pp124-125).


Precession of equinoxes


It was part of tradition to observe the winter solstice with respect  to a nearby bright star. Thus, Vedanga Jyotisa places winter solstice in Beta Delphini. A later text Mahabharata (100 BCE or earlier) records that winter solstice had moved backwards into Alpha Aquilae. Varahamihira (6th century CE) notes that in his time it has moved even further backwards into Alpha Lyrae. We do not know the accuracy of these observations nor, in the two earlier cases, the epoch of observation. In other words, though the  phenomenon was noticed qualitatively, no attempt  was made to ascertain the rate.


Information on precession became essential for later astronomers. Siddhantic astronomy employs a fixed zodiac beginning with the first point of Aries. To calculate the tropical longitude one must fix the zero precession year and define a rate for the precession. Since the first point of Aries coincided with spring equinox about 500 CE, early astronomers like Aryabhata (499CE) or Brahmagupta (628 CE) took no notice of it. But the phenomenon becomes perceptible as time passed, with the result that later astronomers had to quantify it.   A real problem was what value to take. There was in addition debate on whether the first point oscillated or revolved. Different astronomers took different epoch for the zero precession year, and used different year length as well as different rates of precession.


I have already referred to the handicaps of oral tradition. Precession illustrates it. The extant Surya Siddhanta apparently believes in trepidation theory; takes 27 degrees and the maximum amplitude; and 54 arcsec as the annual rate. Burgess (1860, p.114) remarks on the passages giving the above information: “nothing could well be more awkward than this mode of stating the  important fact of the precession of equinoxes…”


In this context Surya Siddhanta  quotes a figure of 600 revolutions in a yuga (We need not go into the technical details.) Bhaskara (12th century) while citing Surya Siddhanta quotes the number 30 in place of 600.The suspicion is that the version of Surya Siddhanta in the hands of Bhaskara used the Sanskrit word kritvah , while the extant text uses krityah The two words are similar sounding  , but  imply a difference of  a factor of 20!  ( “thirty twenties” in Surya Siddhanta as against  “thirty times” in Bhaskara’s version). Matter did not end here. Various astronomers made different and rather laboured attempts “to stretch the words to suit a desired interpretation” (Dikshit 1896:  Vol. 2, p.207). The controversy is typical of the versified astronomical literature. There is no unique interpretation, and there are unintended “misprints” and deliberate mis-readings. The discussion Instead of focusing on the nature of the natural phenomenon at times degenerated into “your prescription versus mine”.


Siddhantic astronomers’ goal was calculation of planetary positions that matched the observed sky. The input parameters were not assigned any intrinsic sanctity. They were numerically adjusted to have common factors in numerator and denominator for ease of calculation. In other words we will never know what exact value was obtained for parameters (say obliquity) and how it was arrived at. The main achievement of Siddhantic astronomers was in solving mathematical equations that arose while calculating planetary orbits.


Indian astronomy (and cosmology) traveled to China, Korea,  and thence to Japan. It even appeared in  Burma. The world-wide intellectual impact of the Indian astro-mathematical tradition can be gauged from the currently popular  English word algorithm. It is  derived from the last name  of Baghdad-based mathematician Abu Jafar Muhammad ibn Musa al Khwarizmi (c.780-c.850 CE), who subsequently became the instrument for transmission of Indic numerals into Europe.



To help place an author’s work in proper temporal context the date of the original publication is given . For bibliographical convenience, date of reprint, mostly facsimile, is also given.


Burgess, Ebenezer (1860) The Surya Siddhanta ( reprint , Delhi: Motilal Banarasidass,2005)


Dikshit, Sankar Balakrishna (1896) History of Indian Astronomy (English Translation in two volumes of Bhartiya Jyotish Sastra   originally published. in Marathi) ( New Delhi : India Meteorological Department,1968,1981) ( Pagination refers to English translation.)


Kochhar, Rajesh (2000) The Vedic People ( Hyderabad: Orient Longman)


Kochhar, Rajesh & Narlikar, Jayant (1995) Astronomy in India: A Perspective (New Delhi: Indian National Science Academy).


Mirashi, V. V. (1933-34) “The Sarkho plates of RatnadevaIl of the Chedi year 880”. Epigraphia Indica, Volume 22, pp 159-165)


Neugebauer, Otto (1983) Astronomy and History: Selected Essays (New York: Springer-Verlag)


Ohashi, Yukio (2009)  “The mathematical and observational astronomy”. In: Science in India, Vol. 13, Part 8. ( ed. : Jayant Narlikar) ( New Delhi: Viva Books),pp.1-88.


Rao, B.S.(1986) Varaha Mihira’a Brihat Jataka ( Delhi” Motllal Banarasidass)


Shukla, K.S. (1976) Aryabhatiya of Aryabhata ( Delhi: Indian National Science Academy).


Yano, Michio (2003) “Calendar, astronomy and astrology ”. In: The Blackwell Companion to Hinduism (ed.: Cavin Flood) ( Oxford : Blackwell)




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