INDIAN MATHEMATICIAN VARAHAMIHIRA - sitamma

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INDIAN MATHEMATICIAN VARAHAMIHIRA
P.S. Avhale
S.B. Kohle
R.V. Waghmare
Dept. of Mathematics ,
Shivaji College ,
Kannad, Dist Aurangabad
Shivaji College ,
Kannad, Dist Aurangabad
Dept. of Mathematics,
Shivaji College ,
Kannad, Dist Aurangabad
16
RESEARCH PAPER - MATHEMATICS
ABSTRACT
Varahamihira was an Indian Astronomer, Mathematician and Astrologer
in Gupta era. His famous treatises are Pancha-siddhantika and Brihat-Samita.
He wrote the Mathematics and Astrology in poetic and metrical styles, no one
try after and before such style. Varahamihira’s writing give a comprehensive
picture of 6th century of India, his real interest lay in astronomy and astrology.
He repeatedly emphasized the importance of astrology
Introduction
Varahamihira is also called Varaha, or Mihira. Varahamihira was born in Kapitthaka
in India in the year 505 AD. According to his own statement in the penultimate verse of
his Brhajjataka “he was a native of Avanti (Western Malawa) the son of Adityadasa
(servant of sun) and in strutted by him having obtained the blessing of the sun-god at
Kapitthaka “. According to Utpala, Kapittha was a village where there was a sun-temple.
It is usually identified with modern Kayatha, a small village about 20kms from Ujjan on
the Ujjain-Maski Road. There is no doubt that Varaha belonged to a family of sunworshippers. Not only does he pay homage to the sun in almost all his works, but he
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himself was regarded as an incarnation of the Sun-god. He was devotee of demons
especially Mareecha, Subahu & astrologer who lived in Ujjain. He also describes himself
as Avantyaka and him commentator Bhajjataka Utpala styles him Svantikacarya. His son
Prthuyasas, also an astronomer, invokes the Sun-god in the opening verse of his
Satpancasika. He received his early education at Kapitthaka. He lived and worked at
Ujjan for most of his life. Ujjain was an important place for learing mathematics before
the Varahamihira around 400AD.
Works
Varahamihira is remembered for his famous work the panchasiddantika (PS). He
wrote Panchasiddantika in 575 AD, gives us information about Indian texts which is now
lost. This work is a compilation of the summary of major achievements of the Hindu
astronomers before the time of Aryabhatta and a compendium of Greek, Egyptian, Roman
and Babylonian origin. The work is a treatise on mathematical astronomy. Varahamihira
wrote on all the three branches of Jyotisa which are following
1)
Tantra or Siddhanta or Mathematical astronomy.
2)
Hora or horoscopy of wedding (vivaha) and nuptials (jataka) and prognostics
(Sakuna), for Journeys (yatra).
3)
Samhita or mudane astrology
He wrote Vatakanika is a work on omens exclusively, but it exists only in a
fragmentary form as quoted in other works on travels or what may be called
military astrology he wrote Brhadyatra, Svalpayatra and Yogayatra.
Panchasiddantika
He was the first one to mention in his work Panchasiddhantika that the ayanama
or the shifting of the equinox is 50.32 second. It is a compendium of Vedanga Jyotisa as
well as Hellenistic astronomy (including,Greek,Egyption and Roman element)
The Pancasiddhantika also contains many examples of the use of a place value number
system. There is however quite debate about interpreting data from Varahamihira’s
astronomical texts and from other similar works.Some believe that the astronomical
theories are Babylonian in origin, while others argue that the Indians refined the Babylonian
models making observations of their own.
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Suraya siddhanta:
Suraya siddhanta was written in Sanskrit in the form of poetry. It is divided
into fourteen chapters, which are following
1.
Places of the Planets
2.
The Motions The of the Planets
3.
Direction, Place and Time
4.
The Moon and Eclipses
5.
The Sun and Eclipses
6.
The Projection of Eclipses
7.
Planetary Conjunctions
8.
Of the Stars
9.
Risings and Settings
10.
The Moon’s Risings and Settings
11.
Certain Malignant Aspects of the Sun and Moon
12.
Cosmogony, Geography, and Dimensions of the Creation
13.
The Gnomon
14.
The Movement of the Heavens and Human Activity
Time cycles
The astronomical time cycles contained in the text were remarkably accurate at
the time. The Hindu Time Cycles, copied from an earlier work, are described in verses
11–23 of Chapter 1:
When computed, this astronomical time cycle would give the following results:
The average length of the tropical year is 365.2421756 days, which is only 1.4 seconds
shorter than the modern value of 365.2421904 days (J2000). The average length of the
sidereal year, the actual length of the Earth’s revolution around the Sun, as 365.2563627
days, which is virtually the same as the modern value of 365.25636305 days (J2000).
This remained the most accurate estimate for the length of the sidereal year anywhere in
the world for over a thousand years.
Planetary Diameters
The Surya Siddhanta also estimates the diameters of the planets. The estimate
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for the diamete r of Mercury is 3,008 miles, an error of less than 1% from the
currently accepted diameter of 3,032 miles. It also estimates the diameter of Saturn
as 73,882 miles, which again has an error of less than 1% from the currently accepted
diameter of 74,580. Its estimate for the diameter of Mars is 3,772 miles, which has
an error within 11% of the currently accepted diameter of 4,218 miles. It also estimated
the diameter of Venus as 4,011 miles and Jupiter as 41,624 miles, which are roughly
half the currently accepted values, 7,523 miles and 88,748 miles, respectively.
2.1.3 Trigonometry
The Surya Siddhanta contains the roots of modern trigonometry. It uses sine
(jya), cosine (kojya or “perpendicular sine”) and inverse sine (otkram jya) for the first
time, and also contains the earliest use of the tangent and secant when discussing the
shadow cast by a gnomon in verses 21–22 of Chapter 3: Of [the sun’s meridian zenith
distance] find the jya (“base sine”) and kojya (cosine or “perpendicular sine”). Consider
a circle of radius R, with centre O. let A’OA and B’OB be two diameters intersecting at
right angles, one being horizontal and the other vertical. Let AC be an arc such that
AOC= .
Then the arc AC is measured by
Draw CD perpendicular to OA. Then CD is called the and OD is called Then CD=R
sin.and OD=Rcos.One can easily verify that and consequently We shall prove the
following result found in Pancasiddhantika:
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Let B be the centre of a circle and AB, DB radii and ABD=2. The bisector of ABD
meets AD at E and the circle at F.Draw AC perpendicular to DW and AG perpendicular
to diameter through B perpendicular to BD.
We have DE=R sin
Now, AC=Rsin2
Therefore,
and therefore,
and
Therefore,
Varahamihira took R=120’ whereas Aryabhata took it as 3438’
From these we can infer that Varahamihira was aware of the following:
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Important contribution to trigonometry was his sine tables where he improved
those of Aryabhata I giving more accurate values.
Calendrical uses
The Indian solar and lunisolar calendars are widely used, with their local variations,
in different parts of India. They are important in predicting the dates for the celebration of
various festivals, performance of various rites as well as on all astronomical matters.
Vasishtha siddhaanta
Vasishtha Siddhanta is one of the earliest astronomical systems in use in India,
which is summarized in Varahamihira’s Pancha-Siddhantika (6th century). It is attributed
to sage Vasishtha and claims a date of composition of 1,299,101 BCE.
Pitamaha Siddhanta or Brahma-siddhanta
Three wings of astrology are the three parts of astrology, among which Siddhnatha
astrology is prominent. Above mentioned 18 ancient saints have contributed towards
Siddhantha astrology. Shastras of these saints are named after them. Pitamaha Siddhanta
is one of these shastras composed by Rishi Pitamaha.
Pitamaha Siddhantha was composed in the historical period of 8300 BC to 3000 BC.
Many great saints contributed to the field of astrology during this time. Following are the
names of those saints:
Siddhanta Jyotish : Name of 18 Rishi
Surya Pitamaha Vyaso Vashishthaoatri Parashara
Kashyapo Narad Garg Maarichimnu Angira
Lomash Polishashcahiva Chayawano Yavano Mrigu
Shoneko ashthadashadhaite jyoti Shastra Pravartaka.
Hence, following are the names of the saints that took astrology to its height:
RishiSurya, Rishi Pitamaha, Rishi Vyaso, Rishi Vashishthaoatri, Rishi ParasharaKashyapo,
Rishi Narad, Rishi Garg ,Rishi Marichi, Rishi Manu, Rishi Lomash, Rishi Polish, Rishi
Chawana. Rishi Yavan, Rishi Mrigu
5 verses about the Brahma Siddhanta have survived till today in Varahamihira’s
compilation, explanation and treatise PanchaSiddhanthika.).
1.
Pitamaha Brahma computed that 5 years would cause a yuga of the Sun,
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Moon and Dhanishta Nakshatra. (See Also: How many kinds of Yugas
are there?)
2.
After 30 months an adhikamasa (extra month) and after 62 days a loss of a
day (avama – kshaya tithi).
(It is necessary to add and drop months and days periodically just as leap
days are added in leap years as a correction to the calendar.)
3.
Varahamihira tells us that if we subtract 2 from the Saka Year of his reference,
we come to the beginning of a Paitamaha Yuga. Then we divide it by 5 and
the remainder gives us the number of years since the beginning of the
paitamaha yuga. Now we can compute the Ahargana or the count of days. ,
starting from the Sukla Paksha of the Magha Masa.
(See : Varahamihira – Really 427 of Saka Era? :
Pancha Siddhantika, Kalahana’s Rajatarangini : Date of Mahabharata War
and How many kinds of Sakas (Eras) are there?)
4.
Since the Paitamaha yuga contains 1830 savana (solar) days and 1860 tithis,
(see Date of Sri Rama as per Balakanda for explanation on tithis), you can
get the tithis by 1860/1830 times ahargana = 62/61 times ahargana.
5.
The sun passes through each of the 27 nakshatras, 5 times in a yuga of 1830
savana days. So in the ahargana, the sun passes through (ahargana/1830) *
27 *5 nakshatras = (9/122)*ahargana
6.
One paitamaha yuga contains 67 sidereal (star-based) revolutions of the
moon. So the moon passes through 27*67 nakshatras in a yuga. Therefore
(ahargana/1830)*27*67 = 603/610*ahargana = ahargana – (ahargana*7/610).
Romaka Siddha and Paulisa Siddhanta
The “The Romaka Siddhanta”(“Doctrrine of the romas”)and the Paulisa
Siddhanta (“Doctrine of paul”) were the works of Western origin which influenced
Vara hamihira’s thought. Though this view is controversial as there is much evidence
to suggest that it was actually vedic thought indigenous to India which actually first
influenced Western astrologers and subsequently came back to India reformulated.
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Hera
In Hora, he wrote many treatises on shakuna(augury) as well as the Brihajjataka (“great Birth”) and Laghu jataka(“short Birth”). He was also an astrologer
and write on all the three main branches of jyotisha astrology.
Samhita
Samhita or Brihat-Samhita is the second most famous treatise of varahamira. It
includes the topics of human interest such as astrology, eclipses, transits of planets, comets,
gems, pearls, rituals architecture, iconography, omens, cosmetics, water-driving, rain fall,
clouds aphrodisiacs, horticulture, growth of plants, manufacture of perfume, matrimony,
domestic relations, species of men and woman, weather forecast, details of Indian
geography etc.
Conclusion
Varahamihira also made important contributions to the mathematics. He was
young contemporary of the senior Aryabhata (born in 476 AD) and also well known
exponent of Indian astronomy. Though not an originator in astronomy or mathematics, he
was a prolific writer and produced several works, big and small, which had a tremendous
impact on later astronomers particularly astrologers.
References
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Ancientindians.wordpress.com/tag/pitamaha
Astrobix.com/.../surya-siddhanta-history-of-astrology-ancient-indian.
En.wikipedia.orgwikiVarâhamihira .
Pendit Bapu Deva Sastri Translation of the surya siddahanta.
Prof.S.Madhavan Triuvanthapuram;Varahamihira:A Versatile Genius
Shashi .S. Sharma Mathematics & Astronomers of Ancient India
www.astrojyoti.comvarahamihirainfo.htm.
www.britannica.comEBcheckedtopic623232Varahamihira.
www.enotes.com/topic/vasishta-siddhanta.
www.gap-system.org~historyBiographiesVarahamihira.html .
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www.jatland.comhomeVarahamihira.