The beginnings of astronomy 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, it had to be elaborate and well-timed, so that a careful distinction could be made between auspicious and inauspicious times. (Note that mathematical problems such as obtaining the square root of two and approximate value of pi ( circumference of a circle divided by its diameter) were taken up in the context of preparation of fire altars and are discussed in the Shrautasutras.) Since planetary motions provided a natural means of time keeping and were seen as couriers of divine signals. Skies were therefore regularly monitored. This was the beginning of astronomy as an intellectual discipline.

1. Ancient Indian astronomy and
mathematics
Rajesh Kochhar
Honorary professor, Mathematics Department, Panjab
University, Chandigarh 160014
Indian Institute of Science Education and Research, Mohali,
Punjab
rkochhar2000@yahoo.com
Lecture delivered at SJB Institute of Technology, Bangalore, 5 May 2015

2. • Modern astronomy today is at the cutting
edge of scientific enquiry. Although we are by
definition part of the universe, we look at it as
if from the outside. In earlier times, however,
cosmic environment was seen as inseparable
from the terrestrial environment and human
affairs. The world was anthropo-centric. Earth
stood still and the whole world revolved
around it. As masters of the Earth, human
beings considered themselves to be special in
the eyes of the Almighty and worthy of his
attention.

3. • The beginnings of astronomy 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, it had to be
elaborate and well-timed, so that a
careful distinction could be made
between auspicious and inauspicious
times.

4. • (Note that mathematical problems such
as obtaining the square root of two and
approximate value of pi ( circumference
of a circle divided by its diameter) were
taken up in the context of preparation of
fire altars and are discussed in the
Shrautasutras.)

5. • Since planetary motions provided a natural
means of time keeping and were seen as
couriers of divine signals. Skies were therefore
regularly monitored. This was the beginning of
astronomy as an intellectual discipline.
• The above considerations are general; they
would apply to any geographical area or
cultural group.
• We now come to ancient India. Before
proceeding further, let us examine the nature
and limitations of the source material
available.

6. Source material: nature and limitations
• 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, the first use being for stone
inscriptions. Writing material came from plants
or trees and had a short life. Paper was not
introduced into India till about 8th century CE.
Paper in any case was not used for Brahminical
texts.

7. • Texts were in the custody of specialist caste groups
who memorized them and transmitted them to the
next generation by word of mouth. The extant texts
would have been supplemented with explanatory
“notes” to serve an immediate purpose. What was
not considered worth preserving at any point in
time was lost for ever.
• Astronomical results were seen as a revelation
rather than deduction. They were therefore
recorded in the Rigvedic format, that is, in metrical
poetry. An astronomer had to be a poet first and
then an astronomer. Because of constraints of
metre, synonyms or half-words had to be used or
allusions made. This introduced vagueness and
imprecision.

8. • These texts are not library texts. They were meant
for a select group which knew the context. Their
interpretation, out of context, is a difficult task.
• These texts are not complete. Knowledge which was
considered to be the requisite background would
not be mentioned. Absence of mention therefore
does not constitute proof of absence. More
specifically, we do not know how observational
parameters were obtained ( borrowing? actual
observations?)
• Although decimal system was invented in India, in
astronomical texts, numbers are expressed in terms
of real or artificial words or word parts, opening the
door for deliberate or inadvertent mis-
representation.

9. • Because of the sanctity of Vedic texts, elaborate
schemes were devised to preserve them as a
gramophone record. No such mechanism was
employed for astronomy, with the result that
astronomical ( and Ayurvedic) texts are like an
audio cassette.
• The Vedic corpus once created was preserved as
such. In the case of the Epics and Puranas
additions were made but no subtractions. But in
the case of scientific texts, both deletions and
additions were made. Once a new edition came,
parts of old edition would be deleted for ever.
• Also, it is not possible to assign firm dates to any
early event or development. It is therefore not
possible to construct a connected account of any
aspect of early India.

10. Vedic astronomy (1400-BCE-5th century CE)
• There are stray astronomical references in the
Rigveda. For our purposes, Yajurveda is more
useful because it was a manual for ritual . It
thus lists 27-28 nakshatras ( bright star or star
groups) which are seen near the Moon every
night. Later, when an accurate luni-solar
calendar was introduced, Vikrami month names
were so chosen as to contain an important
piece of information. For example, the month
Vaisakha is so called because in this month, the
Moon is full near the nakshatra, Vishakha.

11. • The oldest Indian text exclusively
devoted to astronomy is the
Vedangajyotisha (VJ). Because of the
sanctity attached to the Vedic corpus, it
has been faithfully preserved down the
ages. But as a scientific text it was
overtaken by new developments and
became obsolete.No wonder then, VJ is
the least understood of all Vedic texts.
Remarkably, it remained in vogue for a
very long period, till the 6th century CE.

12. • The oldest portions of VJ could be as old as
1400 BCE.
• Week days and zodiacal signs which are now
integral part of Indian astronomy/ astrology
were not known in the Vedic astronomy period.
• 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,
all adhere to the Vedic astronomy.

13. • The Mahabharata text is a vast collection,
of about 100,000 shlokas, composed
over an extended period of time. And
yet, the Mahabharata does not know of
week days or zodiacal signs.

14. Valuable inputs were received, in stages, from
the Greco-Babylonian world to the northwest
of India.
It is surmised that the (old) Shaka [Saka]
calendar was established by the Shakas in 123
CE to commemorate their victory over the
Parthians in Bactria [Balkh in Cental Asia]. It
was used by the Shaka emperors and Satraps in
their Indian territories. In 78 CE, in Ujjain, the
accumulated 200 years were dropped and the
suitably Indianized new Shaka era was ushered
in.

15. • 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.

16. Aryabhata
• The modernization of Indian astronomy,
by incorporating these and other inputs,
took place at the hands of Aryabhata
whose influential work, Aryabhatiyam,
appeared in 499CE. This was the
beginning of the Siddhantic astronomy,
so-called because the primary astro-
mathematical texts were called Siddhanta
[ proven in the end].

17. What was the situation before Aryabhata?
• 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.
Varahamihira grades the texts according to
their accuracy.

18. • Surya Siddhanta is the most accurate; Romaka
and Paulisha, 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. Note that we
learn about pre-Aryabhata works not from
their own time but from later redactions.

19. • 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.

20. • 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. 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.

21. Siddhantic astronomy
• Siddhantic astronomy focused on the
calculation of mean and true position of the
(geo-centric) planets, (arranged in order of
increasing orbital period, they are the Moon,
Mercury, Venus, Sun, Mars, Jupiter , and
Saturn. ; 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.

22. • 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);
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).

23. • 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. We know
of them from the reports of others such as
Nilkantha 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.

24. • His singular contribution is the construction of
Drgganita ( Drk system of computations). 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.

25. Development of mathematics
• 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. 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.

26. • 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, Bhaskara I, Bhaskara II
and Parameshvara also considered
special types of system of two linear
Diophantine equations.

27. • 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.

28. • 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 Gottfried Wilhelm
Leibniz.

29. • 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.

30. • 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. Also,
receipt of inputs from others is an
important ingredient of intellectual
development. There should not be any
self-consciousness while discussing this.

31. •Thank you