I have included a few important details about just a few of the
most famous ancient Indian mathematicians from past years. To my
mind, the most important and most influential of these figures
were
Aryabhatta and Panini. Aryabhatta had an excellent understanding
of the Keplerian Universe more than a thousand years before
Kepler, while Panini made a remarkable analysis of language,
namely Sanskrit, which was not matched for 2,500 years, until
the modern Bacchus form, in the 20th century.

Born: 476 in Kusumapura (now Patna), India
Died: 550 in India

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Aryabhata wrote Aryabhatiya , finished in 499, which is a
summary of Hindu mathematics up to that time, written in verse.
It coveres astronomy,spherical trigonometry, arithmetic, algebra
and plane trigonometry.Aryabhata gives formulas for the areas of
a triangle and a circle which are correct, but the formulas for
the volumes of a sphere and a pyramid are wrong.

Aryabhatiya also contains continued fractions, quadratic
equations, sums ofpower series and a table of sines. Aryabhata
gave an accurate approximation for pi (equivalent to 3.1416) and
was one of the first known to use algebra. He also introduced
the versine ( versin = 1 - cos) into trigonometry.

Aryabhata also wrote the astronomy text Siddhanta which taught
that the apparent rotation of the heavens was due to the axial
rotation of the Earth. The work is written in 121 stanzas. It
gives a quite remarkable view of the nature of the solar system.

Aryabhata gives the radius of the planetary orbits in terms of
the radius of the Earth/Sun orbit as essentially their periods
of rotation around the Sun. He believes that the Moon and
planets shine by reflected sunlight, incredibly he believes that
the orbits of the planets are ellipses. He correctly explains
the causes of eclipses of the Sun and the Moon.

His value for the length of the year at 365 days 6 hours 12
minutes 30 seconds is an overestimate since the true value is
less than 365 days 6 hours.

References (4 books/articles) References for Aryabhata the Elder
------------------------------------------------------------------------

1.Dictionary of Scientific Biography

2.Biography in Encyclopaedia
Britannica

3.B Datta, Two Aryabhatas of
al-Biruni, Bull. Calcutta Math. Soc. 17 (1926), 59-74.

4.H-J Ilgauds, Aryabhata I,
in H Wussing and W Arnold, Biographien bedeutender Mathematiker
(Berlin, 1983).

This is what I know of Aryabhatta I (with two 't')

The man was born somewhere in south-central India in the year 476 AD
and as a
boy, went to study astonomy at the Univrsity of
Nalanda (near the eastern
city of Patna). He made significant contributions to
the field of astronomy
and probably propounded or came close to the
Heliocentric theory of
gravitation, almost a thausand years before
Copernicus.

Aryabhatta's Magnum Opus, the "Aryabhattiya"
was translated into Latin in the
13th century (about 700 years after his death). The
book explained methods for
calculating the areas of triangles, volumes and
surface areas of spheres as
well as square and cube root. Aryabhatta's ideas
about eclipses and the sun
being the source of moonlight did not have much of
an impression on European
astronomers as by then they were coming to know of
these facts through
independent observations of scientists such as
Copernicus and Galileo.
However, one can appreciate his contribution when
considering that the works
were done 700 years before the "translation", and
much before Copernicus or
Galileo came into being.

In that book, Aryabhatta is said to have written
"Just as a person travelling
in a boat feels that the trees on the bank are
moving, people on the Earth
feel that the Sun is moving".

Aryabhatta's biggest problem was he could not
make a good telescope with the
technology at hand in the 5th and 6th century AD.
This hindered further
advancement of ancient Indian astronomy. Though it
should be admitted that
with his unaided observations with crude
instruments, he was able to arrive at
sufficiently accurate measurement of astronomical
movements and movement of
the earth and moon around the sun to predict
eclipses.

Aryabhatta's methods of astronomical calculations
expounded in his
"Aryabhatta-siddhanta" were reliable enough for
practical purposes of fixing
the Panchanga (Hindu calendar). Thus, eclipses were
also forecast and their
true nature was perceived at least by other
astronomers that prepared the
Indian calendars.

I am not sure if concept of zero and the
"decimal" system was his. I thought
that happened in India centuries before Aryabhatta -
I could be wrong though
(needs looking into, this one).

I believe Indians in the first century BC had
words for each number to the
power of ten (10, 100, 1000 etc) going up to 10 to
the power of 53 as
"Tallakshanam". First archaeological proof of
inscription explaining the
meaning of the number "zero" was found on a copper
plate in Gujarat, India,
from 585 CE.

Likewise, the concept of "pi" as the ratio of the
circumference to the
diameter of a circle might be older than Aryabhatta,
although he calculated
that value upto the forth decimal. The old
Sanskrit text Baudhayana Shulba
Sutra of the 6th Century BCE mentions this ratio as
approximately equal to 3.
Aryabhatta in 499 BCE worked out the value to Pi to
the 4th decimal place, as
3(177/1250) = 3.1416. A few centuries later, in 825
CE, the Arab
mathematician, Mohammed Ibna Musa, said that "this
value 62832/20000 has been
given by the Hindus (Indians)".

Aparrt from that, there is some opinion that the
earlier mentioned
Sanskrit
book of 6th century BC, called "Budhayana Sulba
Sutra" (literally - knowledge
about the rule of the rope) explained a variant of
the Pythagoras theorem. The
book explained how to construct geometrically
perfect structures (such as
complicated fire alters with concentric circles and
squeares expanding out
from the centre, where each outer square will
tangetially touch the inner
circle) and has text which translates roughly as the
square of the area
produced by the diagonal of a rectangle is equal to
the sum of the squares of
areas produced by its two sides (dealing with
diagonals of a rectangle rather
than a right angle triangle - but essentially
meaning the same). Measurements
in construction of gemetric shapes were apparently
done with lengths of rope
those days, hence the name of the book. If this is
true, the theorem predates
Pythagorus by nearly two millenia !!

A few other Indian astronomers might merit
mention, such as Bhaskaracharya, of
almost the same time as Aryabhatta. Bhaskaracharya
calculated the length of a
solar year as 365.2587 days, against the current
accepted figure of 365.2596
days. Bhaskaracharya, in his ‘Surya Siddhanta’ (i.e.
study of the Sun) notes
"objects fall on the Earth due to a force of
attraction by the Earth.
Therefore, the Earth, planets, constellations, moons
and sun are all
influenced in their respective motion by this mutual
attraction".

Using various search engines on the net, once
should be able to find most of
the topics above, and perhaps even more, as more and
more info is made
available almost by the day.

Price: £17.00 (Paperback)
ISBN-13: 978-0-8218-0989-1
Publication date: 9 July 1998
American Mathematical Society
0 pages, mm
Series: Mathematical World number 12
Search for titles in the same series

Reviews

'Varadarjan spins a captivating
tale, and the mathematics is first-rate.
The book belongs on the shelf of any
teacher of algebra... The great treasure
of this book is the discussion of the
work of the great Hindu mathematicians
Aryabhata (c.476-550), Brahmagupta
(c.598-665), and Bhaskara (c.1114-1185).
The book contains many exercises that
enhance and supplement the text and that
also include historical information.
Many of the exercises ask readers to
apply the historical techniques. Some of
the exercises are quite difficult and
will challenge any student.' - The
Mathematics Teacher

'"Varadarajan gives us nice
treatment of the work of Indian
mathematics on the so-called Pell
equation as well as a very detailed yet
teachable discussion of the standard
story of the solution of cubic and
quartic equations by del Ferro,
Tartaglia, Cardano, and Ferrari in
sixteenth-century Italy." Mathematical
Reviews' -

Description

This text offers a special account of
Indian work in diophantine equations during
the 6th through 12th centuries and Italian
work on solutions of cubic and biquadratic
equations from the 11th through 16th
centuries. The volume traces the historical
development of algebra and the theory of
equations from ancient times to the
beginning of modern algebra, outlining some
modern themes, such as the fundamental
theorem of algebra, Clifford algebras and
quarternions. It is geared toward
undergraduates who have no background in
calculus.
This book is co-published with the Hindustan
Book Agency (New Delhi) and is distributed
world-wide, except in India, Sri Lanka,
Bangladesh, Pakistan, and Nepal, by the
American Mathematical Society.

Readership: Undergraduate mathematicians,
graduate students, and research
mathematicians and historians interested in
the history of mathematics.

Contents

Some history of early
mathematics :
Euclid-Diophantus-Archimedes

Pythagoras and the Pythagorean
triplets

Aryabhata-Brahmagupta-Bhaskarsa

Irrational numbers: construction
and approximation

Arabic mathematics

Beginnigs of algebra in Europe

The cubic and biquadratic
equations

Solutions for the cubic and
biquadratic equations :
Solution of the cubic equation

Solution of the biquadratic
equation

Some themes from modern algebra
: Numbers, algebra, and the
physical world

Complex number systems and the
axiomatic treatment of algebra

References

Chronology

Index

Authors, editors, and
contributors

Edited by V.S.
Varadarajan, University of
California, Los Angeles

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