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***Aryabhata the Elder
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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 of
power 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
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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
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Born: 1114 in Biddur, India
Died: 1185 in Ujjain, India
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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>b) = <Picture:
sqrt>((a+<Picture: sqrt>(a<Picture: ^2>-b))/2) <Picture: + or - >
<Picture: sqrt>((a-<Picture: sqrt>(a<Picture: ^2>-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
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1.Dictionary of Scientific Biography 2.Biography in Encyclopaedia
Britannica 3.B Datta, The two Bhaskaras, Indian Historical Quarterly 6
(1930), 727-736.
***Brahmagupta
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Born: 598 in (possibly) Ujjain, India
Died: 670 in India
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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 Brahma-
sphuta- 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
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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
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Born: about 520 BC in India
Died: about 460 BC in India
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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
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1.P Z Ingerman, 'Panini-Backus form' suggested, Communications of the ACM
10 (3)(1967), 137.
***Sripati
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Born: 1019 in (probably) Rohinikhanda, Maharashtra, India
Died: 1066
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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 lunar
eclipses, 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
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Born: 1903 in Triplicane, Madras, India
Died: 25 April 1978 in Madras, India
<Picture>
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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
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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.
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