INDIAN KNOWLEDGE SYSTEM

PROJECT
(HS – 203)

TOPIC :-
Contributions made by Several Indian Astronomers,
Celestial Coordinate System, and the Indian Calendar

Submitted to: Submitted by:

 Dr. Preeti Patanjali GROUP – 01:
SUKHMANI KAUR (004)
LOVEPREET SINGH(054)
SIMARDEEP KAUR(023)
UJWAL SINGH(040)
TEJSVI BHAT(038)

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 ACKNOWLEDGEMENT

We would like to express our sincere gratitude to Dr. Preeti Patanjali, for her invaluable
guidance and support throughout this project. I would also like to extend my heartfelt thanks
to my parents for their unwavering encouragement and motivation. Lastly, I would like to
acknowledge the contributions of my classmates who helped me in various ways. Without
their help, this project would not have been possible. I am also grateful to the institution for
providing me with the resources and facilities necessary for the completion of this project.

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 CONTENTS

 WHAT IS ASTRONOMY?

 UNIQUE ASPECTS OF INDIAN ASTRONOMY

 HISTORICAL DEVELOPMENT OF ASTRONOMY IN INDIA

 TABLE OF INDIAN ASTRONOMERS & THEIR SEMINAL CONTRIBUTIONS

 THE CELESTIAL COORDINATE SYSTEM

 ELEMENTS OF THE INDIAN CALENDAR:

 ĀRYABHAȚIYA AND THE SIDDHANTIC TRADITION

 WHAT IS PAÑCĀNGA?

 BIBLIOGRAPHY

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 WHAT IS ASTRONOMY?

ASTRONOMY is a branch of science that studies

celestial objects, space, and the physical universe as
a whole using concepts mainly from Mathematics.
The sky and the outer space have been a great
source of curiosity for mankind forever, especially
Ancient Indians. Therefore, Astronomy has been a
branch of study right from the pre-historic times.
Ancient Indians have thus contributed o the study by Unknown Author is licensed under

of astronomy in very significant ways and have laid some foundation for its growth in
days to come. We shall see some of the salient aspects of Indian Astronomy in the
project.

 UNIQUE ASPECTS OF INDIAN ASTRONOMY :

Ancient Indians considered astronomy a important part of their lives . However, in
modern science (that has been influenced by the western approach), the study of
astronomy, celestial bodies and space have become just a matter of curiosity to
understand what all elements they consist and how it can be studied using
mathematical models . These do not have any relationship with the society or the
culture, except that they influence climatic conditions. For an instance, Sun, which
nowadays is imagined as nothing but an inert matter consisting of material, radiating a
large amount of energy, is considered as am living entity and a deity in many cultures
for it provides life to all the living beings on the Earth.

Knowledge of astronomy was widely used by different sections of Indian society not
just by the subject matter experts. Villagers, farmers , craftsmen and others would
know the meaning and significance of Rasi (Zodiac signs) , Naksatra(stars) and
months of the calender and certain details about every day(Pañcāṅga). They
developed some understanding of how seasons are formed, as seasonal changes affect
economic activities such related to celestial bodies such as farming, and the general
health of individuals, This is common knowledge, even if the citizens are not educated
in modern methods. There are cultural-religious aspects too.
For instance, in India, it is a common tradition for the newly married couple to be
shown the pair of stars known as
Vasistha and Arundhati
(corresponding to the visual
binary system in the
constellation of ‘Ursa Major’)
as part of marriage ceremony by
the priest or family elders. The
celestial binary is held up as a
model for the married life of the couple.

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There are several differences between the current (predominantly Western) approach and that
of the Indian and other ancient civilizations to astronomy. These are summarized as below:
 The Celestial entities are an integral part of all living beings on Earth. There is a
strong sense of mutual dependence between the earthly and the celestial entities.

 Astronomy, a study of the celestial entities and phenomena is interconnected with

cultural practices and daily life. Many events and planning of activities are done with
reference to the celestial entities. ‘Kalanirnaya’(Determination of time) was
considered the purpose of astronomy. Therefore, consulting the Pañcāṅga is a daily
necessity and often the day begins with this activity.

 Astronomy is not a study of some alien entities but an integral and important aspect of
one’s life. Several concepts and models were developed by ancient Indians to address
this requirement. This partly explains the growth and maturity of astronomical
thinking in Indian society right from the early times.

 Indians developed a systematic procedure of study, observation, data collection,

codification, pattern recognition, and analysis. This led to the development of
advanced mathematics including arithmetic, geometry, algebra, trigonometry, and the
basics of calculus.

 HISTORICAL DEVELOPMENT OF ASTRONOMY IN

INDIA:
The development of astronomy as a systematic area of study was driven by multiple
considerations by civilizations world over. They looked up to the sky for a few important
reasons. First, it helped them develop a calendar to predict seasons. The stars in the sky
served as a reliable aid for directional and navigational purposes and for astrological
predictions. Moreover, the sky was always considered as an abode of gods. Above all,
observing the sky has been a fascinating experience for many. The Indian civilization was
driven by similar considerations:
 The Vedic living required methods for fixing an auspicious time for performing
various rituals and activities. Knowledge of the cardinal directions is also required to
set the Vedic altars.
 In ancient times there were two major wealth generating activities: Agriculture and
Trade. Agriculture was the main driver of the economy and the wealth creation.
Ability to predict the weather conditions and understand the mechanism of season
formation was a key requirement to improve agricultural productivity.
 The other economic activities related to international trade. This required a good
understanding of the latitude, longitude and loxodrome. Moreover, there was a need
for calendar as well.
 Astronomy plays a crucial role in navigation. The basic idea navigation is to follow
rising or setting location of a particular constellation for a certain time and then turn
towards another constellation. Therefore, knowledge of astronomy becomes crucial.

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 Indians have been regularly using astronomy for maritime purposes and some studies
suggest that details of this have been well documented.
All these called for a calculation of time and timekeeping, which resulted in the formation
and continuous evaluation of Indian calendar system Pañcāṅga which is accurate system of
calendar even today. Indian astronomy developed continuously in time right from The Vedic
period. For example, in the Atharvaveda-samhita we are able to see references to stars,
planets and comets. Parasara-tantra consists of planet and comet observations made during
the 2nd millennium BCE. The text provides details of the side real periods of Jupiter and
Saturn.
Furthermore, the movement of Mars has been described quantitatively. The text also contains
a list of 26 comets. In the Vedanga Jyotisa the concept of Yuga consisting of 5 solar years of
62 lunar months has been defined. The concept of Yuga was introduced to synchronise the
Solar and lunar calendars. Two intercalary months Amhaspati and Samsarpa were added to
complete a Yuga. Recent studies use astronomy software for simulating Vedic sky. Using this
they analyse the statements in Satapatha-brahmana about the Krttika and other such
astronomical events mentioned in the Rgveda and conjecture that these could have been
observed around 3000 BCE. Another area of study in astronomy leads to cosmology (mainly
the origin of Universe). The Nasadiya-sukta of Rgveda (RV 10.129) speculates on the origin
of the Universe. The Yajurveda and the Atharvaveda gives a full list of 27 Stars commencing
from Kritika. A study of Itihasas (Ramayana & Mahabharata) reveals numerous references to
stars, planets, and their position in the sky. This indicates a good understanding by the
society of celestial entities and their movement in the sky.
The basic canonical texts of the Jains are around 45, besides a large number of subsidiary
texts. Of these, Sthananga and Bhagavati-sutra contain information on mathematics and
astronomy. Two other individual texts, the Nandi-sutra and Anuyogadvara deals with
numerous topics, including topics on astronomy and mathematics, which a Jaina monk was
supposed to know. Tattvarthadhigama-sutra of Umasvati (185-219 CE) discusses cosmology
and astronomy. Among later Jaina works on astronomy, a noteworthy addition is Jyotissara
by Thakker Pheru (14th century CE) in 238 verses, divided into 4 chapters.

No. Vedic Reference Statement Date

1 Śatapatha-brāhmaņa Krttikā (Pleiades) never swerve from the 2950 BCE

(2.1.2.3) east.

2 Winter solstice at the mid-point of the 1660 BCE

Maitrāyanīya-brāhmaņa Sravistha segment and the summer
Upanisad (6.14) solstice at the beginning of Mägha.
3 Vedāṅga-jyotișa Winter solstice at the beginning of 1300 BCE
Śravisthā and the summer solstice at the
mid-point of Åśleșā.
4 Taittiriya-āraņyaka Abhaya Dhruva (currently identified as 2800 BCE*
(II.19.1) alpha-Draconis) in the Simšumāra
Constellation is the
pole star.

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 In this discipline of Indian astronomy a class of text called 'siddhāntas.' was developed
beginning 5th century CE. The siddhāntas. astronomy adopted more sophisticated
mathematics, incorporated the planets in the system, devised a of planetary revolution system
of coordinates for the determination the periods of planetary revolutions and the relative sizes
of the Earth, the Sun and the Moon. Aryabhata developed a mathematical approach to
astronomy in his work Āryabhațiya m(499 CE). This is considered the first full ledged
treatise on mathematical astronomy in India. Varahamihira in his Panca-siddhantika(530 CE)
refers to 5 different astronomical texts which were prevalent during his time. This include
Paitamaha, Vasistha, Romaka, Paulisa and Saura Siddhāntas.s. They deal with the true
motion of the moon, sun, diurnal problems, lunar and solar eclipses, movements of planets
such as Mercury, Venus, Jupiter and Saturn.
Jyotirmimamsa of Nīlakaņțha Somayājī written in 1504 CE, is another important work in
Indian astronomy. Nilkantha stressed the importance of astronomical observation and argued
for the necessity of correcting parameters periodically based on observation of eclipses, the
asun, moon and the planets. In 1500 CE Nilkantha revised the prevalent method for
calculating planetary positions and proposed a planetary model in which the planets orbit
around the Sun which itself moves around the Earth. This is essentially the modern picture in
a geocentric framework.

 TABLE OF THE INDIAN ASTRONOMERS AND THEIR

SEMINAL CONTRIBUTIONS:
S.no Details of the Period, Location Salient Contributions
. WORK/MATHEMATICIAN
1. Author not known - Sürya- Prior to 6th century CE It appears that there are several versions
siddhānta of it available. An ancient version is
summarised in Varāhamihira's
Pañcasiddhāntikā. A modern version is
very popular even now among the
traditional scholars and calendar
makers.
2. Varāhamihira - Pañca- 6th century CE Presents an updated summary of five
siddhāntikā ancient siddhāntas.
3. Āryabhața - Āryabhațiya Born 476 CE A section on mathematics, foundations
Kusumapura,near of trigonometry; Calculation of the sine
Pataliputra,Bihar function;
Rotation of the Earth; Accurate
algorithms for the
positions of the Sun, the Moon and
planets; Earth
in the cosmos; Eclipses.
4. Bhāskara I - Āryabhațiya- 7th century CE Commentary on Āryabhațiya explaining
bhāșya, its
Mahā-bhāskarīya mathematics and astronomy in detail;
Develops the Āryabhatan system in his
own texts.

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5. Brahmagupta - 7th century CE A detailed system of calculations
Brāhmasphuța- pertaining
siddhánta, Khandakhādyaka to the Sun, the Moon, and planets;
Many new
algorithms and explanations in
astronomy. path-
breaking results in mathematics like
vargaprakrti
(quadratic indeterminate equations), and
cyclic
quadrilaterals; Khaņdakhādyaka, a
practical
manual of Indian astronomy.
6. Lalla - Śişyadhīvrddhida-tantra 8th-9th century CE A textbook which expounds on the
Aryabhatan
system, with new algorithms.
7. Manjulācārya - Laghumānasa 10th century CE An explicit expression for the 'second
correction' to the longitude of the
Moon; Derivative of sine
function and instantaneous velocity of
the Sun and the Moon.
8. Śripati - Siddhānta-śekhara 11th century CE An important text quoted by the later
astronomers.
9. Bhāskarācārya II - Siddhānta- Born 1114 CE Most of the standard calculations and
širomaņi, Vāsanābhāsya, algorithms in Indian astronomy
Karaņakutūhala included, mistakes rectified,
generalizations made, a calculation-
manual using ready-made tables, and
arithmetical simplifications.
10. Kerala School 14th-19th Century This school made important
Mådhava of Sańgamagrāma- contributions to mathematical analysis-
Veņvāroha, Sphuțacandrāpti 1340-1425 CE
Derivation of infinite
series for x, sine and cosine functions,
Parameśvara of Vațasseri- much before the subject developed in
Drggaņita, Bhațadīpika, 1360-1455 CE Europe. A major revision of the
Siddhānta-dīpikā traditional planetary theory in 1500 CE.
Innovations in astronomical
Nīlakaņțha Somayājī or computations; Systematisation of the
1444-1550 CE
Somasutvan of Trikkantiyur- applications of spherical
Tantra-sańgraha, Āryabhațiya - trigonometry to astronomy; Improved
Bhāsya theory of eclipses.
Jyeşthadeva - Gaņita-yuktibhāșā
Acuyta Pisaroti- 1500-1610 CE
Sphuțanirnayatantra
Sankaravarman- Şadratnamala 16th century CE

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11. Gaņeśa Daivajña - Grahalāghava Born 1507 CE Simplified procedures for calculation of
planetary
positions, used for preparing almanacs
or Pañcāngas even now.
12. Kamalākara-Siddhānta-tattva- Born 1616 CE Elaborate work mostly based on Indian
viveka concepts and parameters but
incorporates elements of the Greek
astronomer Ptolemy's system.
13. Candraśekhara Sāmanta - Born 1835 CE Important modifications in planetary
Siddhānta-darpaņa parameters revised the lunar theory,
designed simple instruments, reformed
the traditional calendar of
Odisha.
14. Raja Sawai Jai Singh - Yantrarāja- 1688-1743 CE Built famous observatories in several
racanā, Zij Muhammad Shahi parts of North India.

 THE CELESTIAL COORDINATE SYSTEM:

A celestial coordinate system is a mechanism for specifying positions of planets, stars, and
other celestial objects related to physical reference points available to an observer situated on
the earth. It helps to locate a celestial luminary using a three dimensional or a spherical plane.
The figure below provides the rudimentary elements of the celestial coordinator system. The
celestial pole is an imaginary large sphere concentric to the earth. All the objects in the sky
can we projected on the celestial sphere. The celestial north pole and celestial south pole are
analogous to the north pole in south pole of the earth respectively. Similarly the celestial
equator is concentric to the earth's equator. Likewise, all the elements of the celestial sphere
such as the celestial hemisphere are analogous to the earth's hemisphere. The only difference
is that it is projected on to a great circle so that all celestial luminaries can be mapped on to
the sphere using some celestial coordinates.
Another useful coordinate system is ecliptic. Ecliptic is an imaginary line depicting the Sun's
movement around Earth. When an observer see the Sun's path from Earth's reference frame it
appears to move around the earth in a path that is tilted with respect to the spin axis at 23.5
degree. The figure alongside graphically illustrates the ecliptic. The ecliptic plain sets the
reference plane and is the basis for a ecliptic co-ordinate system which is used for all further
astronomical calculations. A third aspect of the
coordinate system is the horizontal coordinate
system from the perspective and the position of an
observer. The horizontal plane with the Zenith and
the Nadir are the two poles of the system. The
zenith is an imaginary point directly 'above' a
particular location (highest point), on the celestial
sphere and nadir is 180 degree from
zenith (lowest point). It helps to position the stars
and other luminaries in the sky related to the
observer's horizon. As we

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 know by experience, the position of the celestial
luminaries varies with time and is useful
for the observer to locate and track these celestial
entities. The figure below is a simple illustration
of this. The vector from the observer to a point of
interest (such as a star or any celestial body) is
projected perpendicularly on to a reference plane.
The angle between the projected vector and
a reference vector on the reference plane is
called the azimuth.

Every celestial object appears to rise in the eastern part of the sky, travel up along a 'diurnal
path' and set in the western part. The relative positions of the star are fixed, and they do
not appear to move with respect to each other. However, the Sun, the Moon, and the planets
seem to appear to move eastwards in the background of stars (apart from the daily motion
from east to west). Let us consider the movement of Sun as viewed from the Earth as shown
in th e figure below.

At the time of Sunrise on a particular day, the Sun is at A which is in the same direction as a
star S1. After one day, at the next Sunrise, it would be at B, which is in the same direction as
a star S2, east of S1. The change in the position of the celestial objects forms the backbone
of computation of various time units such as year, month and day. The calendaring and other
astronomical calculations are done using these basic principles. Further, some of the elements
of the seasons are also inferred from analyzing the trajectory of the Sun in the ecliptic.
A solstice occurs when the sun's path crosses the extreme north or south points on the earth's
equator. In the figure below, when the sun is at S4, it is at the southernmost point with respect
to the equator. This is known as the winter solstice. Similarly, when the Sn is at S2, it is at the
northernmost point whether respect to the equator. That point is known as the summer
solstice. At S1 and S3, the ecliptic intersects the equator, and these points are known as the
equinoxes. When the Sun is at
equinox it is an equinoctial day.
Between S4 and S2, the Sun
moves northwards (known in the
Indian system as Uttarayana), and
between s2 and S4 the Sun moves
southwards (known as
Daksinayana). We find Vedic
references to these important
astronomical concepts. Taittriya-

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samhita(6.5.3) observes, 'Thus the Sun moves southwards for 6 months and northwards for
6 months'. The equinoctial day (visuvat) is mentioned in Aitareya-brahmana (18.4). Methods
to compute the n that equinox is available in vedanga-jyotisa.

We know that ecliptic is the path of the Sun in the background of stars. The Moon and almost
all the planets are found within a belt of wind 8 degree on either side of the ecliptic, this belt
is known as 'zodiac' or Rasicakra. The zodiac has been divided into 12 equal parts from a
fixed initial point in the ecliptic in order to trace the trajectory of the moon and other planets
in the context of the stars, known as rasis. Each segment spans 30 degree. The figure below
shows the zodiac signs on the ecliptic. The concept of 12-fold division of the ecliptic is
praised to the early periods of Vedanga-jyotisa. The 12 Rasis are Mesa(Aries),
Vrsabha(Taurus), Mithuna(Gemini), Karka(Cancer), Simha(Leo), Kanya(Virgo),
Tula(Libra), Vrscika(Scorpio), Dhanus(Sagittarius), Makara(Capricorn),
Kumbha(Aquarius), and Mina(Pisces).

The 'sidereal period' of an object is the time taken by it to complete one revolution in the
background of stars. The Moon moves in an orbit around the Earth, which is slightly
inclined to the ecliptic. The Moon's sidereal period is closed to 27.32 days, which
constitutes one lunar cycle. In other words, the moon covers nearly 1/ 27th part of the
ecliptic per day. In the Indian system, the zodiac has been divided into 27 equal parts from
the fixed initial point in the ecliptic in order to trace the trajectory of the Moon in the context
of the stars. Each such division is known as naksatra, measuring 13 degree 20 minutes or 800
minutes of the arc of the ecliptic(360 / 27). Each division is named after a selected star that is
generally prominent already traditionally well-known and is broadly equally spaced in the
zodiac. Each day would be associated with an naksatra .They are Asvini, Bharani, Krittika,
Rhini, Mrgasiras, Ardra, Punarvasu, Pusya, Aslesa, Magha, Purva Phalguni, Uttara Phalguni,
Hasta,Citra,Svati, Visakha, Anuradha, Jyestha, Mula, Purvasadha, Uttarasadha, Sravana,
Dhanistha, Satabhisaj, Purva Bhadrapada, Uttara Bhadrapada, and Revati. The full list
beginning with Krittika is found in Taittiriya-Samhita, (4.4.10.1-3), and in Atharvaveda and
other parts of the Vedic repository . The 27 naksatras are mapped on to the 12 rasis.
Therefore, each rasi is associated with two or a quarter (27 / 12) naksatra.

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 ELEMENTS OF THE INDIAN CALENDAR:
A calendar is a method for counting systematically and continuously the successive days by
cycle periods such as year and month. Calendaring is done based on two luminaries in the
sky, namely the Sun and the Moon. One of the critical uses of astronomy is to calendar the
time and identify seasons, which are related to the relative of the Sun with respect to the
Earth. The Indian system of astronomy has dealt with this elaborately as evidenced in the
Vedic text. While in the Vedic texts we merely find the description of these elements, the
computation details of these are found in Vedanga Jyotisa, though approximate and
simple. Later day mathematicians introduced formal method for mathematical computations
of the calendar. There are three clear time-markers as even casual observations of the sky
reveal, viz., day, month, and year. All major civilizations grappled with the problem of how
these time units are related to each other.

 The Notion of a Year:

Solar Year Savana Year Solar Year
Year 12 Solar Months (365 12 Savana Months (360 12 Lunar Months (354
days) days) days)

Month 30.5 days 30 days 29.5 days

Yuga 60 Solar Months 61 Savana Months 62 Lunar Months

Remarks Known as Samvatsara Known as Idavatsara Known as Anuvatsara

The elapsed time for the Sun to return to the same star, which is nothing but the period of
the revolution of the Earth and the Sun, is the solar year. Solar calendars are based on
this. On the other hand, the period of the return of the Moon in a position (Full moon) or
conjunction to the sun (New Moon) in relation to the Earth, is a lunar month. 12 such
successive months form the lunar year, and this is the basis of all lunar calendars. Both
the calendars are in use today. The Solar-year calendar is followed for indicating the dates
of the years in the states of Tripura, Assam, Bengal ,Odisha , Tamil Nadu ,Kerala and
partly Punjab and Haryana while the lunar calendar is followed in other states for the
same purpose. But in all states for fixing the dates of religious festivals and for selecting
the auspicious time for undertaking many socio-religious activities, the lunar calendar is
used. The lunar calendar, however, is pegged to the solar calendar, are more precisely,
the lunar months are linked with the solar months to prevent these from getting delinked
with the seasons. Therefore, it is more apt to say that in the Indian tradition a luni-solar
calendar is followed. Knowledge of the components of the Solar and lunar calendar is
critical.

What institutes year has been variously defined in Indian system. The Vedic year
consists of 12 months each of 30 days (known as Savana). This results in a year
consisting of 360 days. This was synchronised to the seasons by adding 5 days to the
calendar. The Rgveda (1.164.11) depicts this as "The wheel (of time) formed with 12

12
 spokes, revolves around the heavens, without wearing out. O Agni, on it, are 720 sons
(viz., days and nights)". According to this verse a year has 12 months and 360 days. 5 or
even 5.25 days were added later . In Yajurveda, there are references 12 lunar months
(known as Vatsara) amounting to 354 days. The synchronisation was done using the
Ekasdasartra ceremony to account for 365 days in a year leaving an error of 0.25 days.
Subbarayappa and Sarma have compiled the list of astronomical references in various in
ancient Indian literature pertaining to these aspects. It appears that the authors of Rgveda
were aware of the discrepancies between the duration of the lunar year and the solar year
and the need to add an intercalary month for synchronising the two. The table below
summarises the differences between alternate a cycles in the Indian tradition.

A solar year is approximately 365.2564 days(modern value) and 12 lunar months is

approximately 354.3671 days. There is a gap of nearly 10.89 days is between the two.
Therefore, to align the lunar year with the solar year which is connected with the seasons,
an additional lunar month has to be introduced in same years. In Yajurveda(4.4.11) there
is a mention of the need to add two intercalary months in 5 years. It had been noticed that
the Sun and the Moon return together to nearly the same position in the framework of
stars, after 5 years. The concept of Yuga was introduced to synchronise the Solar and
Lunar calendars. Two intercalary months Amhaspati and Samsarpa to complete a Yuga.
A 5 year cycle is mentiones in Taittriya and Vajasaneyi samhitas.

 Solar and Lunar Months

The Surya-siddhāntas. defines the solar month as the time taken by the Sun to traverse a rasi,
which is the difference between the time of ingress of the Sun to one rasi(Sankranti) and to
the time of Ingress to the next rasi. As Mesa is the first rasi , the time taken by the sun to
travel this rasi forms the 1st solar month, and it is called Mesa in Kerala, and Vaisakha in
Tripura, Bengal, Odisha, Haryana and Punjab,Bahag in Assam, and Citra in Tamilnadu. The
second solar month is similarly formed by the time taken by the Sun to traverse the second
rasi, what time which is Vrasabha, and so on for other months. The names of the 12 months
and 6 seasons are given in Taittriya-samhita(1.4.14; 4.4.11) as "madhu, madhava(vasanta
rtu), sukra, suci(grisma rtu), nabha,
nabhaya(varsa), isa, urja(sarad rtu), sahas,
sahasya(hemanta rtu), tapas, tapasya(sisira rtu)."
A lunar month(candra-masa) is the time interval
between two moons (Amavasya), or 2 full moons
(Purnima). It consists of two paksas: Sukla-
paksa(bright fortnight) from Amavasya to
Purnima, and the Krsna-paksa(dark fortnight)
from Purnima to Amavasya. The picture alongside
illustrates the idea pictorially. A normal lunar year
has 12 lunar months. The names of the 12 lunar
months are Caitra, Vaisakha, Jyestha, Asadha,
Sravana, Bhadrapada, Asvayuja, Kartika,
Margasira, Pusya, Magha, and Phalguna. During
most Caitra months the Moon will be close to the star Citra(Spica), on the full moon day of
the month. Similarly, during most the Vaisakha months, the moon will be near the star

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Vaisakha on the full moon day in that month. This is the logic behind the nomenclature of the
months.

 The Notion of Tithi:

The Indian calendar is based on the true longitudes of the Sun and the Moon and is related to
the actual happenings in the sky. The relative position of the Moon and the Sun on a date
determines the tithi . In the stellar background the Sun moves at the rate of 1 Degree per day
eastwards and the moon moves at a rate of 13 degree per day. Tithi captures the angular
separation between the sun and the moon. It is the time-unit during which the angle between
the Sun and the moon (as viewed from the Earth) increases by 12 degrees or its integral
multiples. At
amavasya, the sun and
the moon are perfectly
aligned (that is why we
are not able to see the
moon) and the angle is
zero degree and the
first tithi begins. At the
end of the first Tithi
the angle separation is
12 degrees and the
second tithi begins.
The second tithi ends
and the third begins
when the angle is 24
degree and so on. The
figure given illustrate the same. The 12 degrees merely an average. In reality, the actual
duration of the tithi is will vary on account of the perturbing forces affecting the true motions
of the Sun and the Moon. The actual duration of the tithi may vary from 26 hours 47 minutes
to 19 hours 59 minutes.

As we saw, there are rudiments of a calendar with adhikamasas (intercalary months), and 27
nakshatras as markers of the Moon's movements. However the description in the samhitas
are qualitative. A definite quantitative calendarical system is described in Vedanga Jyotisa. It
is available in Rgvedic and Yajurvedic versions. Vedhanga Jyotisa is the first text in India to
give mathematical algorithms in Astronomy. There is nothing on planetary motion in this
work. There are short algorithms for finding tithi, Nakshatra, Sun's position in the sky, etc.
The Vedhanga Jyotisa gives rules for calculating some astronomical quantities related to the
Sun and the Moon. The motions of the Sun and the Moon ( in the background of stars) are
assumed to be uniform.

The later astronomical text of the siddhāntas.. Are far more sophisticated and elaborate, with
procedures to calculate planetary motions, eclipses etc. The calculations are more accurate,
and the calendar is well defined and more advanced. All the texts adhere to the sutra format
and there is a continuity in the tradition. Precise trigonometric measurements were developed
and used to achieve accurate time measurement. This development and refinement continued
over thousand years. Trigonometric functions accurate to the first minute world developed
and used in Surya-siddhāntas., and Āryabhațiya . This was improvised to seconds(vikala) by
Vatesvara(904 CE), and thirds(tatpara) by Madhava(14th century CE).

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A Calendar System was developed using the Prime Meridian of Ujjayini(current Ujjain, also
known as Avantipura in Indian literature). Methods were developed for recalibration of the
calendar for any local place.

 ĀRYABHAȚIYA AND THE SIDDHANTIC TRADITION:

Compared to Vedanga Jyotish, the later astronomical texts of the siddhantic period are far
more sophisticated and elaborate, with procedures to calculate planetary motion, eclipses, the
time for sunrise and sunset, duration of the day, relation between shadow of the Sun and the
time, and many other quantities of astronomical interest. Aryabhata was the pioneering
figure in the tradition of full fledged mathematical astronomy in India called the siddhantic
tradition, though there were siddhāntas.s even before Āryabhațiya . There were many
innovations in techniques and the evolution of ideas in the tradition. Detained explanations,
derivations and demonstrations (geometric and otherwise) are found in the commentaries.
This development and refinement continued for over 1200 years.

Āryabhațiya is the first extant (discovered and available) text on mathematical astronomy in
India. It was a pioneering work that established the framework for mathematical astronomy
in India. It contains a systematic treatment of all the traditional astronomical problems. The
text was composed in 499 CE. Āryabhațiya describes the procedures for calculating the
positions of the Sun, the Moon, and the planets, astronomical variables associated with the
daily paths of these objects in the sky, especially the sun, and the important phenomena of
eclipses. In all these calculations, trigonometry plays a crucial role. The 'jyardha' or the half
chord introduced in Āryabhațiya is far more convenient than the Greek chord of
astronomical computations.This jyardha is nothing but the modern sine function (apart from a
constant factor). A very important recursion relation for the sines and an explicit table of
sines are given in Aryabhatia. Aryabhatia has only 121 verses and is organised into four
parts namely: Gitikapada, Ganitapada, Kalakriyapada and Golapada.

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According to Aryabhata, the Earth is a sphere at the centre of the framework in which stars
and planets move. According to a verse in Āryabhațiya :
" The globe of the earth stands (supportless) at the centre of the circular frame of asterisms
surrounded by the orbits (of the planets); it is made up of water, earth, fire, and air and is
spherical."
According to the earlier astronomical works in India and elsewhere, the Earth was stationery
and all the celestial objects rotated in the sky, completing one rotation in a day. The celestial
objects rise in the Eastern part of the sky and set in the western part and are visible when they
are above the Horizon. However, Aryabhatia presented a different view. The diurnal motion
of the celestial objects was me
ntioned in one of the verses as follows:
" Just as a man in a boat moving forward see the stationery objects as moving backward, just
so are the stationery stars seen by the people at Lanka (on the equator), as moving exactly
towards the west."
What is implied in this verse is that though it appears that objects in the sky are moving from
east to west ,they are indeed stationery and it is the Earth which is moving (rotating) along
with the entities situated on it.
As we had seen earlier, a yuga of 5 years was vogue at the time of Vedanga Jyotsa. On the
other hand, in smrtis and the old Surya-siddhāntas.(before Āryabhațiya ), we have the
concept of a maha-yuga of 43,20,000 years. A similar approach is taken in Āryabhațiya .
Moreover a Maha-Yuga is taken to be composed of four sub yugas namely : Krta, Treta,
Dvapara and Kali. In other texts, that duration of these sub-yuga are in the ratio 4:3:2:1
respectively. In Āryabhațiya they are all of the equal duration, namely, 10,80,000 years. It
can be insert that the beginning of the current Kali-yuga is on February 18, 3102 BCE which
was a Friday.
In the mahayuga all the planets and the auxiliary quantities associated with them make
integral numbers of revolutions. The number of revolutions in the Stellar background made
by the planets in the maha-yuga as per Āryabhațiya is given in the following table:
PLANET NO. OF SIDEREAL MODERN VALUE
REVOLUTIONS PERIOD
Sun 43,20,000 365.25868 365.25636
Moon 5,77,53,336 27.32167 27.32166
Moon’s Apogee 4,88,219 3231.98708 3232.37543
Moon’s Nodes 2,32,226 6974.7491 6793.39108
Mercury* 1,79,37,020 87.96988 87.96930
Venus* 70,22,388 224.69814 224.70080
Mars 22,96,824 686.99974 686.97970
Jupiter 3,64,224 4332.27217 4332.58870
Saturn 1,46,564 10766.06465 10759.20100

It is evident from the table that Aryabhata's computations were amazingly close to the
modern value considering that it was estimated in the 5th century CE. The number of civil

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days in maha-yuga and known as Yuga-savana-dina(D) is also specified. As per Āryabhațiya
D = 15,77,917,500.

The angular position of the planet at any time with respect to a reference line is called
longitude(theta). In all Indian text, the reference line is the direction of the beginning point of
the Mesa rasi in the stellar background. The moon and the planets move in the plains that are
slightly inclined to the ecliptic. We ignore this inclination for the time being. Assume for a
moment that the planets move uniformly (with constant speed) in circles with the earth as the
centre in the plane of the ecliptic only. Then this angle is termed as the 'mean longitude' and
denoted by theta 0. This varies at a uniform rate. According to Aryabhatia the mean
longitudes of all planets are zero at the beginning of Kali-yuga.
The apparent motions of the Sun, Moon and the planets in the background of stars are not
uniform. Two corrections have to be applied to the mean longitude(theta 0) to obtain the
'true'(geocentric) longitude. These corrections are:
 Manda-samskara :This is due to the non-uniformity of the motion due to the
eccentricity of the planet’s orbit .It is called the ‘Equation of centre’ in modern
astronomy. This is the only correction th the Sun and the Moon (for Moon, there
are some other minor corrections specified in later texts).In the case of the actual
planets called taragrahas in India (traditionally only Mercury, Venus, Mars,
Jupiter, and Saturn), we obtain the true heliocentric longitude, that is, the
longitudes with respect to the Sun after the manda-samskara.

 Sighra-samskara- This correction is done only for the planets. This converts their
heliocentric longitudes (as observed from the Sun) to geocentric longitudes (as
observed from the Earth).
Aryabhatia discussed the above two corrections for the first time in Indian tradition. In the
early 17th century CE, the European astronomer Kepler had given the correct model for the
motion of the planets. According to him, planets move in elliptical orbits around the Sun. The
planetary model which is implicit in Āryabhațiya and also in the later Indian texts is broadly
equivalent to the Kepler model and gives approximately the same results for longitudes.
There were some problems with the interior planets, Mercury and Venus which were resolved
by Nilakantha Somayaji in 1500 CE in his Tantra-sangraha.
There were some problems associated with the calculational procedure for Mercury and
Venus in the traditional Indian model. Around 1500 CE, Nilakantha Somayaji modified the
procedure based on detailed scientific analysis and proposed a revised planetary model in his
Āryabhațiya -bhasya and other works. According to the model the planets move in eccentric
orbits around the Sun (that is, the centres of the orbits will be slightly displaced from the Sun)
whichitself orbits around the Earth this was before the well known heliocentric model of
Copernicus in 1542 CE, which has some shortcomings. These are absent in the revised
model of Nilakantha, which is in a geocentric framework. It is essentially the same as Tycho
Brahe's model for the planetary motion propose around 1580 CE, where the planets move
around the Sun ,with the Sun itself orbiting the Earth.

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 WHAT IS PAÑCĀNGA?
Pañcānga is the backbone of Indian astrology and calendar. All festivals and local events are
based on Pañcānga and it place a vital role in daily lives of the vast majority of people in the
country. The development of calendric astronomy in India in a systematic manner can be
traced back to Vedanga-Jyotisa where in the calendar was formulated based on on cycle
periods of 5 years called yuga. However, it was the Siddhāntas.-Jyotisa , which employed
scientific computations that ushered in the refined calendar and Pañcānga system that is in
vogue today.

As the name suggest, Panchanga has five components: tithi, naksatra, vara, karana, and yoga.
Based on astronomy concepts developed, Pañcānga computes the 5 components at any given
instant. In the Pañcānga used by the Indian society, such calculations are done, and the results
are made available in a ready reckoner format for direct reading and interpretation. The
calculations are based on the true longitudes of the Sun and the Moon.

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 o BIBLIOGRAPHY:

1.) BOOK:
Images and Content from “Introduction to Indian Knowledge Systems: Concepts and
Applications”, co-authored by Professor B Mahadevan

2.) INTERNET:
Images from different websites.

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