Critical Podium Dewanand India
History of mathematics in India / source=hinduism.co.za
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SOUTH ASIAN HISTORY
Pages from the history of the Indian sub-continent: Science and Mathematics
History of Mathematics in India
In all early civilizations, the first expression of mathematical understanding
appears in the form of counting systems. Numbers in very early societies
were typically represented by groups of lines, though later different
numbers came to be assigned specific numeral names and symbols (as in
India) or were designated by alphabetic letters (such as in Rome). Although
today, we take our decimal system for granted, not all ancient civilizations
based their numbers on a ten-base system. In ancient Babylon, a sexagesimal
(base 60) system was in use.
The Decimal System in Harappa
In India a decimal system was already in place during the Harappan period,
as indicated by an analysis of Harappan weights and measures. Weights
corresponding to ratios of 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100,
200, and 500 have been identified, as have scales with decimal divisions.
A particularly notable characteristic of Harappan weights and measures
is their remarkable accuracy. A bronze rod marked in units of 0.367 inches
points to the degree of precision demanded in those times. Such scales
were particularly important in ensuring proper implementation of town
planning rules that required roads of fixed widths to run at right angles
to each other, for drains to be constructed of precise measurements, and
for homes to be constructed according to specified guidelines. The existence
of a gradated system of accurately marked weights points to the development
of trade and commerce in Harappan society.
Mathematical Activity in the Vedic Period
In the Vedic period, records of mathematical activity are mostly to be
found in Vedic texts associated with ritual activities. However, as in
many other early agricultural civilizations, the study of arithmetic and
geometry was also impelled by secular considerations. Thus, to some extent
early mathematical developments in India mirrored the developments in
Egypt, Babylon and China . The system of land grants and agricultural
tax assessments required accurate measurement of cultivated areas. As
land was redistributed or consolidated, problems of mensuration came up
that required solutions. In order to ensure that all cultivators had equivalent
amounts of irrigated and non-irrigated lands and tracts of equivalent
fertility - individual farmers in a village often had their holdings broken
up in several parcels to ensure fairness. Since plots could not all be
of the same shape - local administrators were required to convert rectangular
plots or triangular plots to squares of equivalent sizes and so on. Tax
assessments were based on fixed proportions of annual or seasonal crop
incomes, but could be adjusted upwards or downwards based on a variety
of factors. This meant that an understanding of geometry and arithmetic
was virtually essential for revenue administrators. Mathematics was thus
brought into the service of both the secular and the ritual domains.
Arithmetic operations (Ganit) such as addition, subtraction, multiplication,
fractions, squares, cubes and roots are enumerated in the Narad Vishnu
Purana attributed to Ved Vyas (pre-1000 BC). Examples of geometric knowledge
(rekha-ganit) are to be found in the Sulva-Sutras of Baudhayana (800 BC)
and Apasthmaba (600 BC) which describe techniques for the construction
of ritual altars in use during the Vedic era. It is likely that these
texts tapped geometric knowledge that may have been acquired much earlier,
possibly in the Harappan period.
Baudhayana's Sutra displays an understanding of basic geometric shapes
and techniques of converting one geometric shape (such as a rectangle)
to another of equivalent (or multiple, or fractional) area (such as a
square). While some of the formulations are approximations, others are
accurate and reveal a certain degree of practical ingenuity as well as
some theoretical understanding of basic geometric principles. Modern methods
of multiplication and addition probably emerged from the techniques described
in the Sulva-Sutras.
Pythagoras - the Greek mathematician and philosopher who lived in the
6th C B.C was familiar with the Upanishads and learnt his basic geometry
from the Sulva Sutras. An early statement of what is commonly known as
the Pythagoras theorem is to be found in Baudhayana's Sutra: The chord
which is stretched across the diagonal of a square produces an area of
double the size. A similar observation pertaining to oblongs is also noted.
His Sutra also contains geometric solutions of a linear equation in a
single unknown. Examples of quadratic equations also appear. Apasthamba's
sutra (an expansion of Baudhayana's with several original contributions)
provides a value for the square root of 2 that is accurate to the fifth
decimal place. Apasthamba also looked at the problems of squaring a circle,
dividing a segment into seven equal parts, and a solution to the general
linear equation. Jain texts from the 6th C BC such as the Surya Pragyapti
Modern-day commentators are divided on how some of the results were generated.
Some believe that these results came about through hit and trial - as
rules of thumb, or as generalizations of observed examples. Others believe
that once the scientific method came to be formalized in the Nyaya-Sutras
- proofs for such results must have been provided, but these have either
been lost or destroyed, or else were transmitted orally through the Gurukul
system, and only the final results were tabulated in the texts. In any
case, the study of Ganit i.e mathematics was given considerable importance
in the Vedic period.
The Vedang Jyotish (1000 BC) includes the statement: "Just as the
feathers of a peacock and the jewel-stone of a snake are placed at the
highest point of the body (at the forehead), similarly, the position of
Ganit is the highest amongst all branches of the Vedas and the Shastras."
(Many centuries later, Jain mathematician from Mysore, Mahaviracharya
further emphasized the importance of mathematics: "Whatever object
exists in this moving and non-moving world, cannot be understood without
the base of Ganit (i.e. mathematics)".)
Panini and Formal Scientific Notation
A particularly important development in the history of Indian science
that was to have a profound impact on all mathematical treatises that
followed was the pioneering work by Panini (6th C BC) in the field of
Sanskrit grammar and linguistics. Besides expounding a comprehensive and
scientific theory of phonetics, phonology and morphology, Panini provided
formal production rules and definitions describing Sanskrit grammar in
his treatise called Asthadhyayi. Basic elements such as vowels and consonants,
parts of speech such as nouns and verbs were placed in classes. The construction
of compound words and sentences was elaborated through ordered rules operating
on underlying structures in a manner similar to formal language theory.
Today, Panini's constructions can also be seen as comparable to modern
definitions of a mathematical function. G G Joseph, in The crest of the
peacock argues that the algebraic nature of Indian mathematics arises
as a consequence of the structure of the Sanskrit language. Ingerman in
his paper titled Panini-Backus form finds Panini's notation to be equivalent
in its power to that of Backus - inventor of the Backus Normal Form used
to describe the syntax of modern computer languages. Thus Panini's work
provided an example of a scientific notational model that could have propelled
later mathematicians to use abstract notations in characterizing algebraic
equations and presenting algebraic theorems and results in a scientific
Philosophy and Mathematics
Philosophical doctrines also had a profound influence on the development
of mathematical concepts and formulations. Like the Upanishadic world
view, space and time were considered limitless in Jain cosmology. This
led to a deep interest in very large numbers and definitions of infinite
numbers. Infinite numbers were created through recursive formulae, as
in the Anuyoga Dwara Sutra. Jain mathematicians recognized five different
types of infinities: infinite in one direction, in two directions, in
area, infinite everywhere and perpetually infinite. Permutations and combinations
are listed in the Bhagvati Sutras (3rd C BC) and Sathananga Sutra (2nd
Jain set theory probably arose in parallel with the Syadvada system of
Jain epistemology in which reality was described in terms of pairs of
truth conditions and state changes. The Anuyoga Dwara Sutra demonstrates
an understanding of the law of indeces and uses it to develop the notion
of logarithms. Terms like Ardh Aached , Trik Aached, and Chatur Aached
are used to denote log base 2, log base 3 and log base 4 respectively.
In Satkhandagama various sets are operated upon by logarithmic functions
to base two, by squaring and extracting square roots, and by raising to
finite or infinite powers. The operations are repeated to produce new
sets. In other works the relation of the number of combinations to the
coefficients occurring in the binomial expansion is noted.
Since Jain epistemology allowed for a degree of indeterminacy in describing
reality, it probably helped in grappling with indeterminate equations
and finding numerical approximations to irrational numbers.
Buddhist literature also demonstrates an awareness of indeterminate and
infinite numbers. Buddhist mathematics was classified either as Garna
(Simple Mathematics) or Sankhyan (Higher Mathematics). Numbers were deemed
to be of three types: Sankheya (countable), Asankheya (uncountable) and
Philosophical formulations concerning Shunya - i.e. emptiness or the void
may have facilitated in the introduction of the concept of zero. While
the zero (bindu) as an empty place holder in the place-value numeral system
appears much earlier, algebraic definitions of the zero and it's relationship
to mathematical functions appear in the mathematical treatises of Brahmagupta
in the 7th C AD. Although scholars are divided about how early the symbol
for zero came to be used in numeric notation in India, (Ifrah arguing
that the use of zero is already implied in Aryabhatta) tangible evidence
for the use of the zero begins to proliferate towards the end of the Gupta
period. Between the 7th C and the 11th C, Indian numerals developed into
their modern form, and along with the symbols denoting various mathematical
functions (such as plus, minus, square root etc) eventually became the
foundation stones of modern mathematical notation.
The Indian Numeral System
Although the Chinese were also using a decimal based counting system,
the Chinese lacked a formal notational system that had the abstraction
and elegance of the Indian notational system, and it was the Indian notational
system that reached the Western world through the Arabs and has now been
accepted as universal. Several factors contributed to this development
whose significance is perhaps best stated by French mathematician, Laplace:
"The ingenious method of expressing every possible number using
a set of ten symbols (each symbol having a place value and an absolute
value) emerged in India. The idea seems so simple nowadays that its significance
and profound importance is no longer appreciated. It's simplicity lies
in the way it facilitated calculation and placed arithmetic foremost amongst
Brilliant as it was, this invention was no accident. In the Western world,
the cumbersome roman numeral system posed as a major obstacle, and in
China the pictorial script posed as a hindrance. But in India, almost
everything was in place to favor such a development. There was already
a long and established history in the use of decimal numbers, and philosophical
and cosmological constructs encouraged a creative and expansive approach
to number theory. Panini's studies in linguistic theory and formal language
and the powerful role of symbolism and representational abstraction in
art and architecture may have also provided an impetus, as might have
the rationalist doctrines and the exacting epistemology of the Nyaya Sutras,
and the innovative abstractions of the Syadavada and Buddhist schools
Influence of Trade and Commerce, Importance of Astronomy
The growth of trade and commerce, particularly lending and borrowing demanded
an understanding of both simple and compound interest which probably stimulated
the interest in arithmetic and geometric series. Brahmagupta's description
of negative numbers as debts and positive numbers as fortunes points to
a link between trade and mathematical study. Knowledge of astronomy -
particularly knowledge of the tides and the stars was of great import
to trading communities who crossed oceans or deserts at night. This is
borne out by numerous references in the Jataka tales and several other
folk-tales. The young person who wished to embark on a commercial venture
was inevitably required to first gain some grounding in astronomy. This
led to a proliferation of teachers of astronomy, who in turn received
training at universities such as at Kusumpura (Bihar) or Ujjain (Central
India) or at smaller local colleges or Gurukuls. This also led to the
exchange of texts on astronomy and mathematics amongst scholars and the
transmission of knowledge from one part of India to another. Virtually
every Indian state produced great mathematicians who wrote commentaries
on the works of other mathematicians (who may have lived and worked in
a different part of India many centuries earlier). Sanskrit served as
the common medium of scientific communication.
The science of astronomy was also spurred by the need to have accurate
calendars and a better understanding of climate and rainfall patterns
for timely sowing and choice of crops. At the same time, religion and
astrology also played a role in creating an interest in astronomy and
a negative fallout of this irrational influence was the rejection of scientific
theories that were far ahead of their time. One of the greatest scientists
of the Gupta period - Aryabhatta (born in 476 AD, Kusumpura, Bihar) provided
a systematic treatment of the position of the planets in space. He correctly
posited the axial rotation of the earth, and inferred correctly that the
orbits of the planets were ellipses. He also correctly deduced that the
moon and the planets shined by reflected sunlight and provided a valid
explanation for the solar and lunar eclipses rejecting the superstitions
and mythical belief systems surrounding the phenomenon. Although Bhaskar
I (born Saurashtra, 6th C, and follower of the Asmaka school of science,
Nizamabad, Andhra ) recognized his genius and the tremendous value of
his scientific contributions, some later astronomers continued to believe
in a static earth and rejected his rational explanations of the eclipses.
But in spite of such setbacks, Aryabhatta had a profound influence on
the astronomers and mathematicians who followed him, particularly on those
from the Asmaka school.
Mathematics played a vital role in Aryabhatta's revolutionary understanding
of the solar system. His calculations on pi, the circumferance of the
earth (62832 miles) and the length of the solar year (within about 13
minutes of the modern calculation) were remarkably close approximations.
In making such calculations, Aryabhatta had to solve several mathematical
problems that had not been addressed before including problems in algebra
(beej-ganit) and trigonometry (trikonmiti).
Bhaskar I continued where Aryabhatta left off, and discussed in further
detail topics such as the longitudes of the planets; conjunctions of the
planets with each other and with bright stars; risings and settings of
the planets; and the lunar crescent. Again, these studies required still
more advanced mathematics and Bhaskar I expanded on the trigonometric
equations provided by Aryabhatta, and like Aryabhatta correctly assessed
pi to be an irrational number. Amongst his most important contributions
was his formula for calculating the sine function which was 99% accurate.
He also did pioneering work on indeterminate equations and considered
for the first time quadrilaterals with all the four sides unequal and
none of the opposite sides parallel.
Another important astronomer/mathematician was Varahamira (6th C, Ujjain)
who compiled previously written texts on astronomy and made important
additions to Aryabhatta's trigonometric formulas. His works on permutations
and combinations complemented what had been previously achieved by Jain
mathematicians and provided a method of calculation of nCr that closely
resembles the much more recent Pascal's Triangle. In the 7th century,
Brahmagupta did important work in enumerating the basic principles of
algebra. In addition to listing the algebraic properties of zero, he also
listed the algebraic properties of negative numbers. His work on solutions
to quadratic indeterminate equations anticipated the work of Euler and
Emergence of Calculus
In the course of developing a precise mapping of the lunar eclipse, Aryabhatta
was obliged to introduce the concept of infinitesimals - i.e. tatkalika
gati to designate the infinitesimal, or near instantaneous motion of the
moon, and express it in the form of a basic differential equation. Aryabhatta's
equations were elaborated on by Manjula (10th C) and Bhaskaracharya (12th
C) who derived the differential of the sine function. Later mathematicians
used their intuitive understanding of integration in deriving the areas
of curved surfaces and the volumes enclosed by them.
Applied Mathematics, Solutions to Practical Problems
Developments also took place in applied mathematics such as in creation
of trigonometric tables and measurement units. Yativrsabha's work Tiloyapannatti
(6th C) gives various units for measuring distances and time and also
describes the system of infinite time measures.
In the 9th C, Mahaviracharya ( Mysore) wrote Ganit Saar Sangraha where
he described the currently used method of calculating the Least Common
Multiple (LCM) of given numbers. He also derived formulae to calculate
the area of an ellipse and a quadrilateral inscribed within a circle (something
that had also been looked at by Brahmagupta) The solution of indeterminate
equations also drew considerable interest in the 9th century, and several
mathematicians contributed approximations and solutions to different types
of indeterminate equations.
In the late 9th C, Sridhara (probably Bengal) provided mathematical formulae
for a variety of practical problems involving ratios, barter, simple interest,
mixtures, purchase and sale, rates of travel, wages, and filling of cisterns.
Some of these examples involved fairly complicated solutions and his Patiganita
is considered an advanced mathematical work. Sections of the book were
also devoted to arithmetic and geometric progressions, including progressions
with fractional numbers or terms, and formulas for the sum of certain
finite series are provided.
Mathematical investigation continued into the 10th C. Vijayanandi (of
Benares, whose Karanatilaka was translated by Al-Beruni into Arabic) and
Sripati of Maharashtra are amongst the prominent mathematicians of the
The leading light of 12th C Indian mathematics was Bhaskaracharya who
came from a long-line of mathematicians and was head of the astronomical
observatory at Ujjain. He left several important mathematical texts including
the Lilavati and Bijaganita and the Siddhanta Shiromani, an astronomical
text. He was the first to recognize that certain types of quadratic equations
could have two solutions. His Chakrawaat method of solving indeterminate
solutions preceded European solutions by several centuries, and in his
Siddhanta Shiromani he postulated that the earth had a gravitational force,
and broached the fields of infinitesimal calculation and integration.
In the second part of this treatise, there are several chapters relating
to the study of the sphere and it's properties and applications to geography,
planetary mean motion, eccentric epicyclical model of the planets, first
visibilities of the planets, the seasons, the lunar crescent etc. He also
discussed astronomical instruments and spherical trigonometry. Of particular
interest are his trigonometric equations: sin(a + b) = sin a cos b + cos
a sin b; sin(a - b) = sin a cos b - cos a sin b;
The Spread of Indian Mathematics
The study of mathematics appears to slow down after the onslaught of
the Islamic invasions and the conversion of colleges and universities
to madrasahs. But this was also the time when Indian mathematical texts
were increasingly being translated into Arabic and Persian. Although Arab
scholars relied on a variety of sources including Babylonian, Syriac,
Greek and some Chinese texts, Indian mathematical texts played a particularly
important role. Scholars such as Ibn Tariq and Al-Fazari (8th C, Baghdad),
Al-Kindi (9th C, Basra), Al-Khwarizmi (9th C. Khiva), Al-Qayarawani (9th
C, Maghreb, author of Kitab fi al-hisab al-hindi), Al-Uqlidisi (10th C,
Damascus, author of The book of Chapters in Indian Arithmetic), Ibn-Sina
(Avicenna), Ibn al-Samh (Granada, 11th C, Spain), Al-Nasawi (Khurasan,
11th C, Persia), Al-Beruni (11th C, born Khiva, died Afghanistan), Al-Razi
(Teheran), and Ibn-Al-Saffar (11th C, Cordoba) were amongst the many who
based their own scientific texts on translations of Indian treatises.
Records of the Indian origin of many proofs, concepts and formulations
were obscured in the later centuries, but the enormous contributions of
Indian mathematics was generously acknowledged by several important Arabic
and Persian scholars, especially in Spain. Abbasid scholar Al-Gaheth wrote:
" India is the source of knowledge, thought and insight". Al-Maoudi
(956 AD) who travelled in Western India also wrote about the greatness
of Indian science. Said Al-Andalusi, an 11th C Spanish scholar and court
historian was amongst the most enthusiastic in his praise of Indian civilization,
and specially remarked on Indian achievements in the sciences and in mathematics.
Of course, eventually, Indian algebra and trigonometry reached Europe
through a cycle of translations, traveling from the Arab world to Spain
and Sicily, and eventually penetrating all of Europe. At the same time,
Arabic and Persian translations of Greek and Egyptian scientific texts
become more readily available in India.
The Kerala School
Although it appears that original work in mathematics ceased in much
of Northern India after the Islamic conquests, Benaras survived as a center
for mathematical study, and an important school of mathematics blossomed
in Kerala. Madhava (14th C, Kochi) made important mathematical discoveries
that would not be identified by European mathematicians till at least
two centuries later. His series expansion of the cos and sine functions
anticipated Newton by almost three centuries. Historians of mathematics,
Rajagopal, Rangachari and Joseph considered his contributions instrumental
in taking mathematics to the next stage, that of modern classical analysis.
Nilkantha (15th C, Tirur, Kerala) extended and elaborated upon the results
of Madhava while Jyesthadeva (16th C, Kerala) provided detailed proofs
of the theorems and derivations of the rules contained in the works of
Madhava and Nilkantha. It is also notable that Jyesthadeva's Yuktibhasa
which contained commentaries on Nilkantha's Tantrasamgraha included elaborations
on planetary theory later adopted by Tycho Brahe, and mathematics that
anticipated work by later Europeans. Chitrabhanu (16th C, Kerala) gave
integer solutions to twenty-one types of systems of two algebraic equations,
using both algebraic and geometric methods in developing his results.
Important discoveries by the Kerala mathematicians included the Newton-Gauss
interpolation formula, the formula for the sum of an infinite series,
and a series notation for pi. Charles Whish (1835, published in the Transactions
of the Royal Asiatic Society of Great Britain and Ireland) was one of
the first Westerners to recognize that the Kerala school had anticipated
by almost 300 years many European developments in the field.
Yet, few modern compendiums on the history of mathematics have paid adequate
attention to the often pioneering and revolutionary contributions of Indian
mathematicians. But as this essay amply demonstrates, a significant body
of mathematical works were produced in the Indian subcontinent. The science
of mathematics played a pivotal role not only in the industrial revolution
but in the scientific developments that have occurred since. No other
branch of science is complete without mathematics. Not only did India
provide the financial capital for the industrial revolution (see the essay
on colonization) India also provided vital elements of the scientific
foundation without which humanity could not have entered this modern age
of science and high technology.
Mathematics and Music: Pingala (3rd C AD), author of Chandasutra explored
the relationship between combinatorics and musical theory anticipating
Mersenne (1588-1648) author of a classic on musical theory.
Mathematics and Architecture: Interest in arithmetic and geometric series
may have also been stimulated by (and influenced) Indian architectural
designs - (as in temple shikaras, gopurams and corbelled temple ceilings).
Of course, the relationship between geometry and architectural decoration
was developed to it's greatest heights by Central Asian, Persian, Turkish,
Arab and Indian architects in a variety of monuments commissioned by the
Transmission of the Indian Numeral System: Evidence for the transmission
of the Indian Numeral System to the West is provided by Joseph (Crest
of the Peacock):-
Quotes Severus Sebokht (662) in a Syriac text describing the "subtle
discoveries" of Indian astronomers as being "more ingenious
than those of the Greeks and the Babylonians" and "their valuable
methods of computation which surpass description" and then goes on
to mention the use of nine numerals.
Quotes from Liber abaci (Book of the Abacus) by Fibonacci (1170-1250):
The nine Indian numerals are ...with these nine and with the sign 0 which
in Arabic is sifr, any desired number can be written. (Fibonaci learnt
about Indian numerals from his Arab teachers in North Africa)
Influence of the Kerala School: Joseph (Crest of the Peacock) suggests
that Indian mathematical manuscripts may have been brought to Europe by
Jesuit priests such as Matteo Ricci who spent two years in Kochi (Cochin)
after being ordained in Goa in 1580. Kochi is only 70km from Thrissur
(Trichur) which was then the largest repository of astronomical documents.
Whish and Hyne - two European mathematicians obtained their copies of
works by the Kerala mathematicians from Thrissur, and it is not inconceivable
that Jesuit monks may have also taken copies to Pisa (where Galileo, Cavalieri
and Wallis spent time), or Padau (where James Gregory studied) or Paris
(where Mersenne who was in touch with Fermat and Pascal, acted as an agent
for the transmission of mathematical ideas).
1.Studies in the History of Science in India (Anthology edited by Debiprasad
2.A P Juskevic, S S Demidov, F A Medvedev and E I Slavutin: Studies in
the history of mathematics, "Nauka" (Moscow, 1974), 220-222;
3. B Datta: The science of the Sulba (Calcutta, 1932).
4.G G Joseph: The crest of the peacock (Princeton University Press, 2000).
5. R P Kulkarni: The value of pi known to Sulbasutrakaras, Indian Journal
Hist. Sci. 13 (1) (1978), 32-41.
6. G Kumari: Some significant results of algebra of pre-Aryabhata era,
Math. Ed. (Siwan) 14 (1) (1980), B5-B13.
7. G Ifrah: A universal history of numbers: From prehistory to the invention
of the computer (London, 1998).
8. P Z Ingerman: 'Panini-Backus form', Communications of the ACM 10 (3)(1967),
9.P Jha, Contributions of the Jainas to astronomy and mathematics, Math.
Ed. (Siwan) 18 (3) (1984), 98-107.
9b. R C Gupta: The first unenumerable number in Jaina mathematics, Ganita
Bharati 14 (1-4) (1992), 11-24.
10. L C Jain: System theory in Jaina school of mathematics, Indian J.
Hist. Sci. 14 (1) (1979), 31-65.
11. L C Jain and Km Meena Jain: System theory in Jaina school of mathematics.
II, Indian J. Hist. Sci. 24 (3) (1989), 163-180
12. K Shankar Shukla: Bhaskara I, Bhaskara I and his works II. Maha-Bhaskariya
(Sanskrit) (Lucknow, 1960).
13. K Shankar Shukla: Bhaskara I, Bhaskara I and his works III. Laghu-Bhaskariya
(Sanskrit) (Lucknow, 1963).
14. K S Shukla: Hindu mathematics in the seventh century as found in
Bhaskara I's commentary on the Aryabhatiya, Ganita 22 (1) (1971), 115-130.
15. R C Gupta: Varahamihira's calculation of nCr and the discovery of
Pascal's triangle, Ganita Bharati 14 (1-4) (1992), 45-49.
16. B Datta: On Mahavira's solution of rational triangles and quadrilaterals,
Bull. Calcutta Math. Soc. 20 (1932), 267-294.
17. B S Jain: On the Ganita-Sara-Samgraha of Mahavira (c. 850 A.D.),
Indian J. Hist. Sci. 12 (1) (1977), 17-32.
18. K Shankar Shukla: The Patiganita of Sridharacarya (Lucknow, 1959).
19. H. Suter: Mathematiker
20. Suter: Die Mathematiker und Astronomen der Araber
21. Die philosophischen Abhandlungen des al-Kindi, Munster, 1897
22. K V Sarma: A History of the Kerala School of Hindu Astronomy (Hoshiarpur,
23. R C Gupta: The Madhava-Gregory series, Math. Education 7 (1973),
24. S Parameswaran: Madhavan, the father of analysis, Ganita-Bharati
18 (1-4) (1996), 67-70.
25. K V Sarma, and S Hariharan: Yuktibhasa of Jyesthadeva : a book of
rationales in Indian mathematics and astronomy - an analytical appraisal,
Indian J. Hist. Sci. 26 (2) (1991), 185-207
26. C T Rajagopal and M S Rangachari: On an untapped source of medieval
Keralese mathematics, Arch. History Exact Sci. 18 (1978), 89-102.
27. C T Rajagopal and M S Rangachari: On medieval Keralese mathematics,
Arch. History Exact Sci. 35 (1986), 91-99.
28. A.K. Bag: Mathematics in Ancient and Medieval India (1979, Varanasi)
29. Bose, Sen, Subarayappa: Concise History of Science in India, (Indian
National Science Academy)
30. T.A. Saraswati: Geometry in Ancient and Medieval India (1979, Delhi)
31.N. Singh: Foundations of Logic in Ancient India, Linguistics and Mathematics
( Science and technology in Indian Culture, ed. A Rahman, 1984, New Delhi,
National Instt. of Science, Technology and Development Studies, NISTAD)
32. P. Singh: "The so-called Fibonacci numbers in ancient and medieval
India, (Historia Mathematica, 12, 229-44, 1985)
33. Chin Keh-Mu: India and China: Scientific Exchange (History of Science
in India Vol 2.)
Another view on Indian Mathematics:
Indic Mathematics: India and the Scientific Revolution
Dr. David Gray writes:
"The study of mathematics in the West has long been characterized
by a certain ethnocentric bias, a bias which most often manifests not
in explicit racism, but in a tendency toward undermining or eliding the
real contributions made by non-Western civilizations. The debt owed by
the West to other civilizations, and to India in particular, go back to
the earliest epoch of the "Western" scientific tradition, the
age of the classical Greeks, and continued up until the dawn of the modern
era, the renaissance, when Europe was awakening from its dark ages."
Dr Gray goes on to list some of the most important developments in the
history of mathematics that took place in India, summarizing the contributions
of luminaries such as Aryabhatta, Brahmagupta, Mahavira, Bhaskara and
Maadhava. He concludes by asserting that "the role played by India
in the development (of the scientific revolution in Europe) is no mere
footnote, easily and inconsequentially swept under the rug of Eurocentric
bias. To do so is to distort history, and to deny India one of its greatest
contributions to world civilization."
Development of Philosophical Thought and Scientific Method in Ancient
Philosophical Development from Upanishadic Theism to Scientific Realism
History of the Physical Sciences in India
Technological discoveries and applications in India
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