We begin by commenting on the nature and limitations of the primary source material on ancient Indian astronomy.We then highlight the accomplishments of Indian astro-mathematical tradition and its place in world history.

1. National Workshop on Ancient Indian Scientific Heritage
Kurukshetra University, 11 November 2014
Ancient Indian astronomical tradition:
Characteristics and accomplishments
Rajesh Kochhar
President IAU Commission 41: History of Astronomy
Hon. Prof., Panjab University, Mathematics Department, Chandigarh
Indian Institute of Science Education and Research, Mohali
rkochhar2000@yahoo.com

2. Ever since human beings learnt to walk upright,
they have looked at the sky and wondered. The sky
has remained the same, but its meaning and
significance have been changing.
Human beings have tried to comprehend
their cosmic environment in their own
cultural framework, and derive
material benefit from knowledge so
gained. At the same time this
knowledge has been used to construct a
theological and philosophical
worldview.

3. We are of course part of the
Universe. But today, we tend to
look at it as if from the
outside. In ancient past, cosmic
environment was seen as an
integral part of human life and
affairs.

4. The beginnings of astronomy (and mathematics) are
related to the requirements of the ritual in early
cultures. Ritual was a means of securing divine
approval and support for terrestrial actions. To be
effective, the ritual had to be elaborate and well-timed.
In India sacrificial altars were specified to be
built of proper size and shape. Out of this
requirement, came the development of geometry as
well as arithmetic and algebra.

5. In addition, the ceremonies had to be performed at
the proper time. Since planetary motions provided a
natural means of time keeping, their refined study
became important. Early astronomical knowledge
went into the making of sacred literature, mythology
and cosmogonical models.
Even when astronomy developed as a
scientific discipline in its own right
in India, it very consciously sought
to retain links with the sacred texts.

6. The rather static Vedic astronomy, whose routes go
the Rigveda itself, prevailed in India for more than
a millennium. Its termination can be assigned a
precise date, namely, 499 CE. This is the year
when the terse Aryabhatiya, authored by
Aryabhata (b. 476 CE), appeared on the scene. The
work has remained influential ever since.
Since the basic astro-mathematical texts were
called Siddhantas (proven in the end), this phase
can be called Siddhantic.

7. The transition from Vedic astronomy to Siddhantic
astronomy is rather poorly understood. Before
discussing these phases and the intervening
transitional period in some detail, it will be useful
to review the nature and limitations of the
available source material .

8. India had taken to the composition and preservation
of texts long before writing began in India. Scripts
like Kharoshthi and Brahmi were introduced into
India in about third century BCE for writing Prakrit
languages rather than Sanskrit. The oldest
documents in Sanskrit are inscriptions from about
first century BCE. Writing of sacred texts began
later; the writing material came from plants or trees
and had a short life.

9. Ancient texts that can serve as source of
information for us are of five types: (i) Vedic
corpus, (ii) Puranas and the epics of Ramayana and
Mahabharata, (iii) Buddhist and Jain texts, (iv)
astronomical texts, and (v) texts from other
countries.
Each text in the vast Vedic corpus, from the
archaic Rigveda to the relatively late
Dharmashastras, has been preserved in its original
form without any addition, deletion or alteration
being made. This feature makes them an extremely
valuable source of information.

10. The early Vedic texts constitute Hinduism’s
heritage. Hinduism in action is represented by the
Puranas and the Epics which were often recast to
meet the contemporaneous requirements of their
custodians and their audience. There are 18 major
Puranas and various recensions of the Ramayana
and historically the more important Mahabharata.
At the level of individual texts and recensions
there are additions as well as deletions.

11. But, if we take this corpus as a whole, we notice
that whatever was composed at any time has
survived in one text or the other so that over all
there have been additions but no deletions.
In contrast to the sacred literature, astronomical
texts underwent deletion as well as addition. New
or revised texts appeared and many old ones
either totally vanished or survived partially. In
the following, I discuss some of the salient points
of ancient astronomical literature.

12. In imitation of the Rigveda, astronomical texts were
composed in metrical verse so that an astronomer had
to be a poet also. Poetry is not the ideal vehicle for
dissemination of scientific knowledge. Requirements
of meter compelled the poet-astronomer to use
synonyms or half-words and resort to allusions. This
introduced vagueness and imprecision.

13. These texts were not composed for the purpose
we are using them now. They were not designed
as library books in the sense of self-contained
self-study material. They required familiarity
with the context and personal intervention of a
teacher. In their time, there must have been
background knowledge to go with these texts
which was not preserved and is now lost. All
ancient texts are valuable for what they expressly
contain. Absence of mention does not necessarily
mean that the thing did not exist.

14. Although decimal system was invented in India,
astronomical texts express numbers in terms of real or
artificial words or word parts, opening the door for
deliberate or inadvertent mis-representation. While
elaborate schemes were devised to prevent corruption of
Vedic texts, no such mechanism was available in the
case of scientific works. Since texts were written on
plant material which had a very short life, old
manuscripts had to be regularly copied. During the
process, inadvertent errors could be introduced. At
times, a word was deliberately changed here and there to
suppress or modify the original meaning. Also, new
material was added to old texts without recording that
this was being done. To add to the confusion, entirely
different texts have identical names.

15. Sacred literature can be considered timeless, but
science advances, rendering older texts outdated.
Because of the oral tradition, once a text fell into
disuse because of arrival of a better or newer
text, the old one was forgotten except for the
excerpts that may have been incorporated in other
texts. From the above, it is clear that it is not
possible to construct a connected account of
history of ancient astronomy in India.
Chronology to an extent remains a problem and
there are significant gaps in our understanding
that cannot be filled.

16. The Vedic period
There are stray astronomical references in the
Rigveda (Rv) including to a solar eclipse. Rv
(5.40.5-9) describes how an asura, Svarbhanu by
name, pierced the Sun ‘through and through with
darkness’. The Sun appealed to rishi Atri who
through his prayers ‘caused Svarbhanu’s magic
arts to vanish’ and thus ‘found the Sun again’.
This passage occurs in the Rigveda’s fifth
mandala whose authorship is credited to the Atris.
This ‘episode’ is mentioned and embellished at a
number of places in the Vedic literature :

17. Kaushitaki (or Sankhayana Brahmana) of the
Rigveda (24.3); Panchavimsha (or Tandya)
Brahmana of the Samaveda (4.5.2; 4.6.13; 6.6.8;
14.11. 14-15; 23.16.2); Shatapatha Brahmana of
the Shukla Yajurveda (5.3.2.2); and Gopatha
Brahmana of the Atharvaveda (8.19) ( Dikshit
1896, Vol. 1, p. 58; Kane 1975, pp. 241-242).
What Atri probably did was to chant mantras
while the eclipse lasted. The Rigvedic description
is significant. An eclipse was seen as the demon’s
work in disrupting the cosmic order. Propitiation
was needed to restore that order.

18. Later Vedic texts rename the eclipse-causing demon
as Rahu. Note that the word Rahu does not occur in
Rv. As we shall see, to maintain semblance of
continuity with sacred texts, post-Aryabhata
astronomy/astrology texts came to employ Rahu as a
scientific term. There is thus a dichotomy in the use
of the term Rahu before and after 499 CE.

19. The Yajurveda proper as well as its associated
literature is a valuable source for a study of early
history of mathematics and astronomy. As already
noted, these intellectual disciplines arose from the
postulated requirements of ritual for which the
Yajurveda is the manual.
It introduces the concept of nakshatra, 27 (earlier
28) bright stars or star groups in the sky which are
used as markers for the Moon and the Sun’s orbits.

20. Yajurveda also refers to the four colures, the two
equinoxes and two solstices. There is an important
phenomenon associated with the colures, known as
precession of the equinoxes (ayanamsha). The
position of the equinox (or solstice) is not fixed in
sky but move in a retrograde manner completing
the cycle in 26000 years.

21. The precession of equinox serves as a clock, with
equinox or solstice as the hand of the clock and
the background nakshatras as the digits on the
dial. The phenomenon is thus a useful, though
approximate, chronological tool.
The oldest exclusively astronomical text is the
very concise Vedanga Jyotisha, which comes in
two overlapping recensions, one attached to the
Rigveda (attributed to one Lagadha) , the other to
the Yajurveda.

22. Taken together, the two versions contain some 49
independent verses, some of which have not yet
been interpreted. Of all the Vedic texts, it is the
most obscure. This is not surprising. As a
scientific text, it was made redundant, but being a
Vedic text it was memorized and passed on from
generation to generation as a relic. It contains an
interesting observation which can be
approximately dated. Both the versions say that
the winter solstice took place at the nakshatra
Shravishtha ( later called Dhanishtha ) . Making
some reasonable assumptions, one can assign the
date c. 1400 BCE to this observation.

23. How old the general contents of the two recension
are is difficult to say. Vedanga Jyotisha describes a
rather inexact calendar and does not mention
zodiacal signs and weekdays which make their
appearance in India two millennia later. Vedanga
Jyotisha concepts remained in vogue for a very
long time.
Shulvasutras, attached to the Yajurveda, which
address the question of making of sacrificial altars,
are the world’s oldest texts on geometry. Among
other things, they make extensive use of what later
came to be known as Pythagoras theorem.

24. One of the problems taken up in the Shulvasutraa
is the construction of a circle or a square of twice
the size of a given one. This led the authors to
calculate a fairly correct value of . Similarly,
attempts to construct a circle of the same area as a
square or vice versa resulted in evaluating the
ratio of a circle’s circumference to its diameter,
Mahabharata simply sets pi =3. However by the
time we come to Aryabhata, we get a value
accurate to four decimal places.

25. Transitional period
The political vacuum in India caused by the
collapse of the post-Ashokan Maurayan empire
was filled by rulers who were based in Afghanistan
and Central Asia. From our point of view,
particularly important are the Greco-Bactrians and
the Shakas (Sakas, or Indo-Scythians).
Mahabharata (Vanaparva 188:34-36) and other
texts call them mlechchhas and dub them as sinful
and untruthful.

26. Unrighteous or not, they brought with them
elements of Greco-Babylonian astronomy, which
slowly got incorporated into the mainstream and
brought about modernization of Indian astronomy.
Most notably, India obtained an accurate luni-solar
calendar. It is surmised that the old Shaka calendar
was established by the Shakas in 123 CE to
commemorate their victory over the Parthians in
Bactria. It was used by the Shaka emperors and
Satraps in their Indian territories.

27. It is surmised that in 78 CE, in Ujjain, the
accumulated 200 years were dropped and the
suitably Indianized new Shaka era was ushered in.
There is direct archaeological evidence of the
depiction of zodiacal signs at Baudha Gaya,
dated c. 100 BCE. Weekdays were slow in
making an entry. It has been suggested that they
appeared in 4th or 5th century CE.

28. A number of old texts adhere to the Vedic astronomy.
Kautilya’s Arthashastra; the Ashokan edicts (3rd cent.
BCE); the Buddhist Sanskrit text, Shardulakaranavadana (
4th cent. CE); and the Jain works, Surya Pannati and
Chand Pannati.
It is remarkable that the zodiac and the weekdays do
not figure in the Mahabharata text. It is well known
that additions were made to Mahabharata over an
extended period of time, till it came to its present
size of a hundred thousand shlokas.

29. It is reasonable to suppose that if zodiacal signs and
weekdays had been in general vogue when the
Mahabharata text was still open, they would have found a
way in. Experts believe that the Mahabharata took its
present form in about 400 CE. One can therefore say that
the zodiac and weekdays, which later became an integral
part of Siddhantic astronomy, were introduced into Indian
mainstream in the fifth century CE.

30. In a significant scholarly exercise, Varahamihira
(d. 587 CE), a junior contemporary of Aryabhata,
made a comparative study of the five extant
Siddhantas. The compendium, which came to be
known as Panchasiddhantika, is actually a
Karanagrantha; it omits all theory and provides
concise rules for quick calculations.

31. Varahamihira grades the texts according to their
accuracy. Surya Siddhanta is the most accurate;
Romaka and Paulisa, which are obviously of
foreign origin, slightly less so. The two older ones,
Vasishtha Siddhanta and the Paitamaha Siddhanta,
were the least accurate, the latter more so than the
former.
Paitamaha Siddhanta is based on Vedanga
Jyotisha, and like it deals only with the Sun and
Moon. While in the other cases, the epoch is 505
CE, in this case it is 80 CE. It was obviously
included for its archival value.

32. It is not surprising that of the five, Surya Siddhanta
was the most accurate; it was an old text only in
name; it was recast in the light of Aryabhata’s work,
not the Aryabhatiya, but another one since lost.
Around 1000 CE, Surya Siddhanta was again recast;
it is this version which is still in use for making
panchangas, or traditional almanacs which depend
on it except for timings of eclipses which they take
from modern sources.

33. We know of three Surya Siddhantas: Pre-
Varahamihira ( known only by name), Surya
Siddhanta as redacted by Varahamihira after
Aryabhata; the present Surya Siddhanta.
Interestingly, astronomical works as text books were
known by their author. But when their elements were
incorporated into astrology-oriented texts, they were
given divine names to enhance their market value.

34. Siddhantic astronomy
Siddhantic astronomy focused on the 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. A
notable achievement of it was the calculation of
lunar and solar eclipses.

35. Siddhantic astronomy broke new ground. And yet, it tried
to maintain continuity with sacred literature by borrowing
terminology and concepts from the Vedic corpus.
Aryabhata himself made astronomical use of the Vedic
Yuga scheme, while the Vedic terms Rahu and Ketu were
incorporated, presumably by Varahamihira, into
astronomical/astrological literature pertaining to eclipses.
Yuga scheme
Manusmrti describes a Yuga scheme which postulates a
universe without beginning or end that continually
undergoes spells of creation and destruction. The scheme
is further elaborated on in the Puranas. Complete
description becomes available from Surya Siddhanta. The
main points of the scheme are summarized below.

36. In the Vedic times, a year comprised 12 months and
360 days. A human year was said to be a day of the
gods so that a divine year (Dyr) would consist of
360 human years (yr).
Four Yugas, Kali, Dvapara, Treta, Krta (or Satya),
were defined with their duration in the ratio 1:2:3:4.
Kaliyuga was the current one and the shortest.
Numerically, it was set equal to 1200 Dyr. The four
added together constitute a Chaturyuga [four-age]
or a Mahayuga [great-age]. A Mahayuga thus
consists of 12000 Dyr.
A still bigger unit called Brahma’s day or Kalpa was
defined as equal to 1000 Mahayugas.

37. To combine the celestial with the terrestrial, a
mythical ruler, Manu, was postu;lated who
presides of a Manvantara ( Manu’s interval)
comprising 71 Mahayuga. Since 1000 is not
divisible by 71, there is no simple way by which
Manvantara and Kalpa can be reconciled .The
equation is set up as follows.
It will be convenient to use mathematical notation
to properly understand the structure within a
Brahma’s day. Let us denote the duration of a
Kaliyuga by the symbol k; Dvapara, Treta and
Krta are then 2k, 3k and 4k respectively.

38. Let a Mahayuga be denoted by m, so that
m = k + 2k + 3k +4k = 10k.
Let us denote a Krtayuga (=4k) by s. Then
1 Brahma’s day
=1000m
= 994m + 6m
=14 x 71m +15s
=14 x 71m + 14 s+ s
=s + 14(71m + s).

39. Recall that 71m is a Manvantara. We can
now describe a Brahma’s day in words. A
Brahma’s day begins with a dawn equal to
a Krtayuga. This dawn is followed in
succession by 14 Manvantaras, at the end
of each of which there occurs a deluge
(pralaya) lasting a Krtayuga. This complex
scheme has perplexed many modern-day
commentators.

40. Thus, Rev. Ebenezer Burgess in his famous
1860 annotated translation of the Surya
Siddhanta wondered: ‘Why the factors
fourteen and seventy - one were thus used
in making up the Aeon [Kalpa] is not
obvious’ (Burgess 1860:11). I think this
scheme was constructed working
backwards from the neat round figure of
1000.

41. To sum up so far, the three basic building blocks,
expressed in human years, are as follows.
1 Kaliyuga=1200 Divine years=432,000 years.
1 Mahayuga=10 Kaliyuga=4.32 million years.
1 Brahma’s day or Kalpa = 1000 Mahayuga = 4.32 billion years .
For the sake of continuity with the scriptures, the Yuga
scheme along with the nomenclature was borrowed by
the astronomers. Instead of simply expressing
revolutions in a million or a billion years, an
astronomer would say that there were 146,568
revolutions of Saturn in a Mahayuga, implying an
orbital period of 29.4743 years.

42. Interestingly Aryabhata boldly modifies the Vedic Yuga
scheme to suit his purpose. He makes the four
components of a Yuga equal in length. He next defines
his Manvantara to comprise 72 Mahayugas and sets a
Kalpa equal to 14 Manvantaras, so that his Kalpa
consists of 1008 Mahayugas, rather than 1000.
Rahu and Ketu
In the Indian context, Aryabhata was the first person to
enunciate the mathematical theory of eclipses. According
to this theory, solar and lunar eclipses occur when the
moon is at either of its orbital nodes. These theoretical
points move in a direction opposite to that of the planets
and complete an orbit in the rather short period of 18.6
years.

43. This development was immediately taken note of
in astrological literature, which classified the two
nodes as planets, implying that they were now
amenable to mathematics. Since they were
hypothetical they were dubbed shadow planets.
The 6th century CE text Brihajjataka (2.2-3) by
Varahamihira includes Rahu and Ketu in the list of
planets, and even gives their synonyms The two
nodes are 180 degrees apart so that specifying one
fixes the other. It would thus have sufficed to
include just one of them. Both were listed no
doubt to bring the planetary number up to nine
which was considered sacred.

44. For naming these nodes, Varahamihira
turned to Vedic literature. The eclipse-causing
Vedic demon Rahu now became the
ascending node. The term Ketu was merely a
common noun employed variously to
describe comets, meteors, etc. It was now
made into a proper noun to denote the
descending node. The Rahu-Ketu theory
travelled to China in course of time, where it
was integrated into the mainstream.

45. Siddhantic astronomers
Illustrious names in Indian astronomy following
Aryabhata include Latadeva (505 CE) who was
Aryabhata’s direct pupil; Varahamihira (already
mentioned ) a compiler and integrator rather than an
original scholar, and an expert on omens; Bhaskara I (c.
574); Aryabhata’s bête noire Brahmagupta (b. 598)
whose works were very influential and were later
translated into Arabic; Lalla (c. 638 or c. 768); Manjula
or Munjala (932); Shripati (1039); and the celebrated
Bhaskara II (b. 1114).

46. It has often been stated Siddhantic astronomy, on
the basis of old scholarship that Indian
mathematics went into decline after Bhaskara II.
This is not true.
Indian astronomy and mathematics received a new
lease of life with Madhava (c. 1340-1425), who
founded what has come to be known as the Kerala
School of Astronomy. His own mathematical works
have been lost.

47. We know of Madhava’s work from the reports of
others such as Nilakantha who lived 100 years later.
Madhava’s pupil Parameshvara (1360-1455), 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).

48. The unbroken tradition of eclipse calculation was
alive till as recently as early 19th century. A Tamil
astronomer computed for John Warren , a French
astronomer in the service of British East India
Company, the lunar eclipse of 1825 May 31-June 1
with an error of +4 minutes for the beginning,-23
minutes for the middle, and -52 minutes for the end (
Neugebauer 1983:435).
Critique
The most remarkable feature of ancient Indian
astronomical tradition from Aryabhata to the Kerala
school has been the development of mathematical
tools for astronomical calculations.

49. The 19th and early 20th century Western
historiography viewed mathematics as a triumph of
pure thought and accepted ancient Greek as
standard for judging the rest of the world. In such a
framework, Indian contribution came to be belittled.
There is now greater appreciation of cultural
plurality and the realization that historical
developments should be examined in their own
framework.
The earliest known systematic treatment of linear
Diophantine equations in two variables was given by
Aryabhata who proposed a continued-fraction like
solution of ax+by=c. Subsequently, Brahmagupta ,

50. Bhaskara I, Bhaskara II and Parameshvara also
considered special types of system of two linear
Diophantine equations.
Brahmagupta found integer solution of many Pell
equations x2-Ny2=1, but was not able to apply it
uniformly to all values of N. The general solution
was obtained by Bhaskara II.
Madhava discovered infinite series for sine, cosine
and arctangent functions and for as early as 14th
century. The European names associated with these
‘discoveries’, made more than 200 years later, are
Colin Maclaurin, Isaac Newton, James Gregory and
GottfriedWilhelm Leibniz.

51. 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.
As I pointed out earlier, Western appreciation of Indian
mathematical achievements is a recent phenomenon. This
calls for rewriting of the world history of mathematics.
How Indian achievements influenced developments in
Europe in their time can be seen from the etymology of
terms. The numbers 0 to 9 came to be known as Arabic
numerals, because Europe learnt them from the Arabs. In
Arabic they are called Hind-se, from India.

52. Three terms in English, (trigonometrical) sine,
algebra and algorithm come from the 12th
century Latin translation of works of a noted 8th
- 9th century Baghdad mathematician known by
his short name al-Khwarizmi who became the
conduit for transfer of Indian mathematical
knowledge to Europe. He came from a small
historical place called Khiva to the south of Aral
Sea, which is now part of Uzbekistan and whose
ancient name is Khwarizm. In the translation of
his book on arithmetic, his name was Latinized
to Algoritmi which in turn gave rise to
algorithm.

53. The Latin/English term sine comes from his
algebra. Indian astronomy introduced the term
jya , which literally meant a bowstring and was
given the technical meaning of half-chord. It
was also called jiva, was rendered in Arabic as
jaib. Now, jaib was an existing word in Arabic
meaning fold of a dress; this was literally
translated as sinus in Latin.

54. We have seen that requirements of ritual,
astronomy and astrology gave a great fillip to the
development of approximate methods in
mathematics as a versatile tool for solving
practical problems. Now that the historians are
sensitive to the fact that different cultures had
different characteristics and the developments in a
particular cultural setting must be examined in
context, there is greater appreciation the world
over for Indian astronomical-mathematical
tradition.

55. Thank you