11-10-2003, 08:56 PM
I found this contribution by Prasenjit medhi on the net.
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.
***Aryabhata the Elder
------------------------------------------------------------------------
Born: 476 in Kusumapura (now Patna), India
Died: 550 in India
Previous (Chronologically) Next Biographies Index
Previous (Alphabetically) Next Welcome page
------------------------------------------------------------------------
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).
***Bhaskara
------------------------------------------------------------------------
Born: 1114 in Biddur, India
Died: 1185 in Ujjain, India
Previous (Chronologically) Next Biographies Index
Previous (Alphabetically) Next Welcome page
------------------------------------------------------------------------
Bhaskara represents the peak of mathematical knowledge in the 12th Century and reached an understanding of the number systems and solving equations which was not to be reached in Europe for several
centuries. Baskara was head of the astronomical observatory at Ujjain, the leading mathematical centre in India at that time.He understood about 0 and negative numbers. He knew that x^2 = 9 had two
solutions. He gives the formula <Picture: sqrt>(a<Picture: + or - ><Picture: sqrt><img src='http://www.india-forum.com/forums/public/style_emoticons/<#EMO_DIR#>/cool.gif' class='bbc_emoticon' alt='B)' /> = <Picture: sqrt>((a+<Picture: sqrt>(a<Picture: ^2>-<img src='http://www.india-forum.com/forums/public/style_emoticons/<#EMO_DIR#>/cool.gif' class='bbc_emoticon' alt='B)' />)/2) <Picture: + or - > Picture: sqrt>((a-<Picture:
sqrt>(a<Picture: ^2>-<img src='http://www.india-forum.com/forums/public/style_emoticons/<#EMO_DIR#>/cool.gif' class='bbc_emoticon' alt='B)' />)/2).Baskara also studied Pell's equation x^2=1+py^2 for p=8, 11, 32, 61 and 67.When p = 61 he found the solutions x =1776319049, y = 22615390. He studied many Diophantine
problems.
Bhaskara's mathematical works include Lilavati (The Beautiful) and Bijaganita (Seed Counting) while he also wrote on astronomy, for example Karanakutuhala (Calculation of Astronomical Wonders).
References (3 books/articles) References for Bhaskara
------------------------------------------------------------------------
1.Dictionary of Scientific Biography 2.Biography in Encyclopaedia Britannica 3.B Datta, The two Bhaskaras, Indian Historical Quarterly 6 (1930), 727-736.
***Brahmagupta
------------------------------------------------------------------------
Born: 598 in (possibly) Ujjain, India
Died: 670 in India
Previous (Chronologically) Next Biographies Index
Previous (Alphabetically) Next Welcome page
------------------------------------------------------------------------
Brahmagupta was head of the astronomical observatory at Ujjain which was the foremost mathematical centre of ancient India.
He wrote important works on mathematics and astronomy. He wrote Brahmasphuta- siddhanta (The Opening of the Universe), in 21 chapters, at Bhillamala in 628. His second work on mathematics and
astronomy is Khandakhadyaka written in 665.Brahmagupta's understanding of the number systems was far beyond others of the period. He developed some algebraic notation. He gave remarkable formulas
for the area of a cyclic quadrilateral and for the lengths of the diagonals in terms of the sides.
Brahmagupta also studied arithmetic progressions, quadratic equations,theorems on right-angled triangles, surfaces and volumes.
The remaining chapters deal with solar and lunar eclipses, planetary conjunctions and positions of the planets. Brahmagupta believed in a static Earth and he gave the length of the year as 365 days
6 hours 5 minutes 19 seconds in the first work, changing the value to 365 days 6 hours 12 minutes 36 seconds in the second book. This second values os not, of course, an improvement on the first
since the true length of the years if less than 365 days 6 hours.
One has to wonder whether Brahmagupta's second value for the length of the year is taken from Aryabhata since the two agree to within 6 seconds, yet are about 24 minutes out.
References (4 books/articles) References for Brahmagupta
------------------------------------------------------------------------
1.Dictionary of Scientific Biography 2.Biography in Encyclopaedia Britannica 3.B Datta, Brahmagupta, Bull. Calcutta Math. Soc. 22 (1930),39-51. 4.H T Colebrooke, Algebra, with Arithmetic and
Mensuration from the Sanscrit of Brahmagupta and Bhaskara (1817).
***Panini
------------------------------------------------------------------------
Born: about 520 BC in India
Died: about 460 BC in India
Previous (Chronologically) Next Biographies Index
Previous (Alphabetically) Next Welcome page
------------------------------------------------------------------------
The dates given for Panini are pure guesses. Experts give dates in the 4th,5th and 6th century BC.
Panini was a Sanskrit grammarian who gave a comprehensive and scientific theory of phonetics, phonology, and morphology. Sanskrit was the classical literary language of the Indian Hindus.
In a treatise called Astadhyayi Panini distinguishes between the language of sacred texts and the usual language of communication. Panini gives formal production rules and definitions to describe
Sanskrit grammar. The construction of sentences, compound nouns etc. is explained as ordered rules operating on underlying structures in a manner similar to modern theory.
Panini should be thought of as the forerunner of the modern formal language theory used to specify computer languages. The Backus Normal Form was discovered independently by John Backus in 1959,
but Panini's notation is equivalent in its power to that of Backus and has many similar properties.
Reference (One book/article) References for Panini
------------------------------------------------------------------------
1.P Z Ingerman, 'Panini-Backus form' suggested, Communications of the ACM 10 (3)(1967), 137.
***Sripati
------------------------------------------------------------------------
Born: 1019 in (probably) Rohinikhanda, Maharashtra, India
Died: 1066
Previous (Chronologically) Next Biographies Index
Previous (Alphabetically) Next Welcome page
------------------------------------------------------------------------
Sripati wrote on astronomy and mathematics. His mathematical work is undertaken with applications to astronomy in mind, for example a study of spheres.His works include Dhikotidakarana (1039), a
work on solar and lunareclipses, Dhruvamanasa (1056), a work on calculating planetary longitudes,eclipses and planetary transits, Siddhantasekhara a major work on astronomy in 19 chapters. The
titles of Chapters 13, 14, and 15 are Arithmetic, Algebra and On the Sphere. Sripati obtained more fame in astrology than in other areas.
Reference (One book/article)
***Cadambathur Tiruvenkatacharlu Rajagopal
------------------------------------------------------------------------
Born: 1903 in Triplicane, Madras, IndiaDied: 25 April 1978 in Madras, India
<Picture>
Previous (Chronologically) Next Biographies Index
Previous (Alphabetically) Next Welcome page
------------------------------------------------------------------------
Rajagopal was educated in Madras, India. He Graduated in 1925 from the Madras Presidency College with Honours in mathematics.
He spent a short while in the clerical service, another short while teaching in Annamalai University then, from 1931 to 1951, he taught in the Madras Christian College. Here he gained an
outstanding reputation as a teacher of classical analysis.
In 1951 Rajagopal was persuaded to join the Ramanujan Institute of Mathematics then, four years later, he became head of the Institute. Under his leadership the Institute became the major Indian
mathematics research centre.
Rajagopal studied sequences, series, summability. He published 89 papers in this area generalising and unifying Tauberian theorems.
He also studied functions of a complex variable giving an analogue of a theorem of Landau on partial sums of Fourier series. In several papers he studied the relation between the growth of the mean
values of an entire function and that of its Dirichlet series.
A final topic to interest him was the history of medieval Indian mathematics. He showed that the series for tan^-1 (x) discovered by Gregory and those for sin x and cos x discovered by Newton were
known to the Hindus 150 years earlier. He identified the Hindu mathematician Madhava as the first discoverer of these series.
Rajagopal is described in [1] as follows:-
Rajagopal was a teacher par excellence and a reliable and inspiring research guide. No words can adequately describe his modesty. Rational thinking and interest in psychic studies were two
attributes which he imbibed with pride from his teacher Ananda Rau.
References (4 books/articles)
References for Cadambathur Tiruvenkatacharlu Rajagopal
------------------------------------------------------------------------
1.C T Rajagopal, Bull. London Math. Soc. 13 (5) (1981), 451-458. 2.C T Rajagopal: September 8, 1903, to April 25, 1978, J. Anal. 1 (1993), vii. 3.Y Sitaraman, Professor C T Rajagopal (1903-1978),
J. Math. Phys. Sci. 12(5) (1978), i-xvi. 4.M S Rangachari, Prof. C T Rajagopal, Indian J. Math.22 (1) (1980), i-xxix.
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.
***Aryabhata the Elder
------------------------------------------------------------------------
Born: 476 in Kusumapura (now Patna), India
Died: 550 in India
Previous (Chronologically) Next Biographies Index
Previous (Alphabetically) Next Welcome page
------------------------------------------------------------------------
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).
***Bhaskara
------------------------------------------------------------------------
Born: 1114 in Biddur, India
Died: 1185 in Ujjain, India
Previous (Chronologically) Next Biographies Index
Previous (Alphabetically) Next Welcome page
------------------------------------------------------------------------
Bhaskara represents the peak of mathematical knowledge in the 12th Century and reached an understanding of the number systems and solving equations which was not to be reached in Europe for several
centuries. Baskara was head of the astronomical observatory at Ujjain, the leading mathematical centre in India at that time.He understood about 0 and negative numbers. He knew that x^2 = 9 had two
solutions. He gives the formula <Picture: sqrt>(a<Picture: + or - ><Picture: sqrt><img src='http://www.india-forum.com/forums/public/style_emoticons/<#EMO_DIR#>/cool.gif' class='bbc_emoticon' alt='B)' /> = <Picture: sqrt>((a+<Picture: sqrt>(a<Picture: ^2>-<img src='http://www.india-forum.com/forums/public/style_emoticons/<#EMO_DIR#>/cool.gif' class='bbc_emoticon' alt='B)' />)/2) <Picture: + or - > Picture: sqrt>((a-<Picture:
sqrt>(a<Picture: ^2>-<img src='http://www.india-forum.com/forums/public/style_emoticons/<#EMO_DIR#>/cool.gif' class='bbc_emoticon' alt='B)' />)/2).Baskara also studied Pell's equation x^2=1+py^2 for p=8, 11, 32, 61 and 67.When p = 61 he found the solutions x =1776319049, y = 22615390. He studied many Diophantine
problems.
Bhaskara's mathematical works include Lilavati (The Beautiful) and Bijaganita (Seed Counting) while he also wrote on astronomy, for example Karanakutuhala (Calculation of Astronomical Wonders).
References (3 books/articles) References for Bhaskara
------------------------------------------------------------------------
1.Dictionary of Scientific Biography 2.Biography in Encyclopaedia Britannica 3.B Datta, The two Bhaskaras, Indian Historical Quarterly 6 (1930), 727-736.
***Brahmagupta
------------------------------------------------------------------------
Born: 598 in (possibly) Ujjain, India
Died: 670 in India
Previous (Chronologically) Next Biographies Index
Previous (Alphabetically) Next Welcome page
------------------------------------------------------------------------
Brahmagupta was head of the astronomical observatory at Ujjain which was the foremost mathematical centre of ancient India.
He wrote important works on mathematics and astronomy. He wrote Brahmasphuta- siddhanta (The Opening of the Universe), in 21 chapters, at Bhillamala in 628. His second work on mathematics and
astronomy is Khandakhadyaka written in 665.Brahmagupta's understanding of the number systems was far beyond others of the period. He developed some algebraic notation. He gave remarkable formulas
for the area of a cyclic quadrilateral and for the lengths of the diagonals in terms of the sides.
Brahmagupta also studied arithmetic progressions, quadratic equations,theorems on right-angled triangles, surfaces and volumes.
The remaining chapters deal with solar and lunar eclipses, planetary conjunctions and positions of the planets. Brahmagupta believed in a static Earth and he gave the length of the year as 365 days
6 hours 5 minutes 19 seconds in the first work, changing the value to 365 days 6 hours 12 minutes 36 seconds in the second book. This second values os not, of course, an improvement on the first
since the true length of the years if less than 365 days 6 hours.
One has to wonder whether Brahmagupta's second value for the length of the year is taken from Aryabhata since the two agree to within 6 seconds, yet are about 24 minutes out.
References (4 books/articles) References for Brahmagupta
------------------------------------------------------------------------
1.Dictionary of Scientific Biography 2.Biography in Encyclopaedia Britannica 3.B Datta, Brahmagupta, Bull. Calcutta Math. Soc. 22 (1930),39-51. 4.H T Colebrooke, Algebra, with Arithmetic and
Mensuration from the Sanscrit of Brahmagupta and Bhaskara (1817).
***Panini
------------------------------------------------------------------------
Born: about 520 BC in India
Died: about 460 BC in India
Previous (Chronologically) Next Biographies Index
Previous (Alphabetically) Next Welcome page
------------------------------------------------------------------------
The dates given for Panini are pure guesses. Experts give dates in the 4th,5th and 6th century BC.
Panini was a Sanskrit grammarian who gave a comprehensive and scientific theory of phonetics, phonology, and morphology. Sanskrit was the classical literary language of the Indian Hindus.
In a treatise called Astadhyayi Panini distinguishes between the language of sacred texts and the usual language of communication. Panini gives formal production rules and definitions to describe
Sanskrit grammar. The construction of sentences, compound nouns etc. is explained as ordered rules operating on underlying structures in a manner similar to modern theory.
Panini should be thought of as the forerunner of the modern formal language theory used to specify computer languages. The Backus Normal Form was discovered independently by John Backus in 1959,
but Panini's notation is equivalent in its power to that of Backus and has many similar properties.
Reference (One book/article) References for Panini
------------------------------------------------------------------------
1.P Z Ingerman, 'Panini-Backus form' suggested, Communications of the ACM 10 (3)(1967), 137.
***Sripati
------------------------------------------------------------------------
Born: 1019 in (probably) Rohinikhanda, Maharashtra, India
Died: 1066
Previous (Chronologically) Next Biographies Index
Previous (Alphabetically) Next Welcome page
------------------------------------------------------------------------
Sripati wrote on astronomy and mathematics. His mathematical work is undertaken with applications to astronomy in mind, for example a study of spheres.His works include Dhikotidakarana (1039), a
work on solar and lunareclipses, Dhruvamanasa (1056), a work on calculating planetary longitudes,eclipses and planetary transits, Siddhantasekhara a major work on astronomy in 19 chapters. The
titles of Chapters 13, 14, and 15 are Arithmetic, Algebra and On the Sphere. Sripati obtained more fame in astrology than in other areas.
Reference (One book/article)
***Cadambathur Tiruvenkatacharlu Rajagopal
------------------------------------------------------------------------
Born: 1903 in Triplicane, Madras, IndiaDied: 25 April 1978 in Madras, India
<Picture>
Previous (Chronologically) Next Biographies Index
Previous (Alphabetically) Next Welcome page
------------------------------------------------------------------------
Rajagopal was educated in Madras, India. He Graduated in 1925 from the Madras Presidency College with Honours in mathematics.
He spent a short while in the clerical service, another short while teaching in Annamalai University then, from 1931 to 1951, he taught in the Madras Christian College. Here he gained an
outstanding reputation as a teacher of classical analysis.
In 1951 Rajagopal was persuaded to join the Ramanujan Institute of Mathematics then, four years later, he became head of the Institute. Under his leadership the Institute became the major Indian
mathematics research centre.
Rajagopal studied sequences, series, summability. He published 89 papers in this area generalising and unifying Tauberian theorems.
He also studied functions of a complex variable giving an analogue of a theorem of Landau on partial sums of Fourier series. In several papers he studied the relation between the growth of the mean
values of an entire function and that of its Dirichlet series.
A final topic to interest him was the history of medieval Indian mathematics. He showed that the series for tan^-1 (x) discovered by Gregory and those for sin x and cos x discovered by Newton were
known to the Hindus 150 years earlier. He identified the Hindu mathematician Madhava as the first discoverer of these series.
Rajagopal is described in [1] as follows:-
Rajagopal was a teacher par excellence and a reliable and inspiring research guide. No words can adequately describe his modesty. Rational thinking and interest in psychic studies were two
attributes which he imbibed with pride from his teacher Ananda Rau.
References (4 books/articles)
References for Cadambathur Tiruvenkatacharlu Rajagopal
------------------------------------------------------------------------
1.C T Rajagopal, Bull. London Math. Soc. 13 (5) (1981), 451-458. 2.C T Rajagopal: September 8, 1903, to April 25, 1978, J. Anal. 1 (1993), vii. 3.Y Sitaraman, Professor C T Rajagopal (1903-1978),
J. Math. Phys. Sci. 12(5) (1978), i-xvi. 4.M S Rangachari, Prof. C T Rajagopal, Indian J. Math.22 (1) (1980), i-xxix.