Indian contributions to mathematics

Indian contributions to mathematics
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- Dr. Bhaskar Kamble

Generally accepted world view
Greece European Renaissance Modern Science 
“Most of the amazing science and technology knowledge
systems of the modern world are credited to have started
around the time of the Renaissance movement in Europe
around the 15th century. These knowledge systems are
generally traced back to roots in the civilization of Ancient
Greece, and occasionally, that of Ancient Egypt. Hence, most of
the heroes we are taught about in school and college are
European, or Greek.”*
*http://bharathgyanblog.wordpress.com/2013/09/21/calculus-was-discovered-in-india/

A brief history of science generally accepted today

Humans originate
from the apes

Humans originate
from the apes
ancient
civilizations
(Sumerian, Egyptian,
Babylonian etc.)

ancient
civilizations
Babylonian etc.)
Around 500 B.C.
Origin of mathematics
and philosophy in
ancient Greece
Precursor to the
European Renaissance
MAJOR
STEP I
Humans originate
from the apes

ancient
civilizations
Babylonian etc.)
Around 500 B.C.
and philosophy in
ancient Greece.
Precursor to the
Dark ages
in Europe
MAJOR
STEP I
Humans originate
from the apes

Renaissance in
Europe, based on
the Greek civilization
ancient
civilizations
Babylonian etc.)
Around 500 B.C.
and philosophy in
ancient Greece.
Precursor to the
MAJOR
STEP I
MAJOR
STEP II
Dark ages
in Europe
Humans originate
from the apes

Renaissance in
Europe, based on
ancient
civilizations
Babylonian etc.)
Around 500 B.C.
and philosophy in
ancient Greece.
Precursor to the
The birth of the
modern sciences
and mathematics-
Kepler, Galileo,
Newton etc. are
the heroes.
MAJOR
STEP I
MAJOR
STEP II
Dark ages
in Europe
Humans originate
from the apes

MAJOR
STEP I
MAJOR
STEP II
Retaining the skeletal structure…
The birth of the
modern sciences
and mathematics-
Kepler, Galileo,
Newton etc. are
the heroes.
Renaissance in
Europe, based on
Dark ages
in Europe
Around 500 B.C.
and philosophy in
ancient Greece.
Precursor to the
ancient
civilizations
Babylonian etc.)
Humans originate
from the apes

Renaissance in
Europe, based on
The birth of the
modern sciences
and mathematics-
Kepler, Galileo,
Newton etc. are
the heroes.
500 B.C.
Mathematics and
philosophy in Greece.

Renaissance in
Europe, based on
The birth of the
modern sciences
and mathematics-
Kepler, Galileo,
Newton etc. are
the heroes.
500 B.C.
Mathematics and
WHAT IS BEING GENERALLY MISSED IN THE ABOVE …

Renaissance in
Europe, based on
The birth of the
modern sciences
and mathematics-
Kepler, Galileo,
Newton etc. are
the heroes.
500 B.C.
Mathematics and

Renaissance in
Europe, based on
Surya Siddhanta,
(2000 B.C. or
earlier)
Sulbasutras,
800 B.C. or
earlier
Astronomical text
with
trigonometric,
algebraic and
geometric
principles
Geometry and algebra for making Vedic
altars, statement and demonstration of
the Pythagoras theorem 300 years
before Pythagoras.
Aryabhatta
500 B.C.
Mathematics and
Brahmagupta,
Bhaskara I
Bhaskara II
Kerala school of mathematics, (Madhava,
Nilakantha, Jyeshthadeva etc.) – used
differential and integral calculus 200 years
before Newton and Leibniz, infinite series,
spherical trigonometry, astronomy etc.
The birth of the
modern sciences
and mathematics-
Kepler, Galileo,
Newton etc. are
the heroes.

Renaissance in
Europe, based on
Surya Siddhanta,
(2000 B.C. or
earlier)
Sulbasutras,
800 B.C. or
earlier
Astronomical text
with
trigonometric,
algebraic and
geometric
principles
before Pythagoras.
Aryabhatta
500 B.C.
Mathematics and
Brahmagupta,
Bhaskara I
Bhaskara II
The birth of the
modern sciences
and mathematics-
Kepler, Galileo,
Newton etc. are
the heroes.
Also most of the high school mathematics attributed to the Greeks. Algebra, geometry,
trigonometry, the Pythagoras theorem, Trigonometry, decimal system, concept of zero,
rational, irrational and negative numbers, algebraic identities comes from India!

“As for India, it would appear that it has played a minimal role
in this magical story. Hence, many western accounts of the
“Ascent of Man” do not devote even a single line to India’s
contributions.”*
*http://bharathgyanblog.wordpress.com/2013/09/21/calculus-was-discovered-in-india/

Renaissance in
Europe, based on
Surya Siddhanta,
(2000 B.C. or
earlier)
Sulbasutras,
800 B.C. or
earlier
Astronomical
text with
trigonometric,
algebraic and
geometric
principles
before Pythagoras.
Aryabhatta
500 B.C.
Mathematics and
Brahmagupta,
Bhaskara I
Bhaskara II
The birth of the
modern sciences
and mathematics-
Kepler, Galileo,
Newton etc. are
the heroes.
Also most of the high school mathematics attributed to the Greeks. Algebra, geometry,
trigonometry, the Pythagoras theorem, Trigonometry, decimal system, concept of zero,
rational, irrational and negative numbers, algebraic identities comes from India!

Some of the references used
(Referred to as “Encyclopaedia” in the presenation)

(referred to as
“TAGS” in the
presentation)

India’s contributions to mathematics
Part 1

THE CASE OF THE
“PYTHAGORAS THEOREM”

From an article published in
The Transactions of the Royal Society of
Edinburgh, 1790
by John Playfair
“Remarks on the Astronomy of the Brahmins”
THE CASE OF THE “PYTHAGORAS THEOREM”

This article discusses:
Three sets of astronomical tables obtained from India dated to the
end of the Kali Yuga (3102 BC.)
(1) Brought from Siam in 1687 to Europe
(2) Sent from Krishnapuram in about 1750 to Europe
(3) Brought from Tiruvallur in 1772 from Europe

Indian contributions to mathematics

From ‘Hints concerning the observatory at Benares’ c. 1783 by Reuben Burrow
“we know that he (Pythagoras) went to India to be instructed;
but the capacity of the learner determines his degree of
proficiency, and if Pythagoras on his return had so little
knowledge in geometry as to consider the forty-seventh of
Euclid as a great discovery, he certainly was entirely
incapable of acquiring the Indian method of calculation,
through his deficiency of preparatory knowledge …
…each teacher, or head of sect that drew his knowledge
from Indian sources, might conceal his instructors to be
reckoned an inventor.”

Sulba Sutras
Baudhayana
(from the
Krishna Yajurveda)
Apastamba
(from the
Krishna Yajurveda)
Katyayana
(from the
Shukla Yajurveda)
Composed around 800 BC.
(although the authors emphasize they are merely
stating facts known since the early Vedic age.)
“Pythagoras theorem” in the Vedas

- Dr. T. A. Sarasvati Amma, “Geometry in ancient and medieval India” (Motilal
Banarsidass, Delhi (2007) )

Around 2,500 years ago Pythagoras went from Samos to
the Ganga to learn Geometry (evidence and reference
will be shown later).
He would certainly not have undertaken such a strange
journey had the reputation of the Brahmin’s science not
been long established in Europe.
Pythagoras and his followers were vegetarians and
believed in transmigration of souls, both of which are
Indian concepts (many more similarities of philosophical
type will be shown later).

Pythagoras theorem should be called
Baudhayana theorem!
(at least in India)

Trigonometry in the Surya Siddhanta
An astronomical text dated to 2000 BC:
some extracts from “On the trigonometric tables of the Brahmins”
by John Playfair
published in the Transactions of the Royal Society of Edinburgh, Vol. IV,
1798

This article discusses a table of sines calculated
for different angles in the Surya Siddhanta and
the possible working principle behind it

Working principle for constructing the table of sines in the
Surya Siddhanta, as deduced by Playfair
 sin)sin()cos2()2sin( 
1. Suppose sinθ, sin(θ+α), and cos α are known. Then from
the above equation we can find sin(θ+2α).
2. Next in the above equation replace θ by θ+α. So we get
3. sin(θ+2α) is known from the previous equation. So from the
above we get sin(θ+3α).
4. Again following the same procedure, we get sin(θ+4α), and
so on….
)sin()2sin()cos2()3sin(  

BABABA cossin2)sin()sin( 
Proof


2

BA
BA




B
A
 cos)sin(2)2sin(sin 
 sin)sin()cos2()2sin( 


 sin)sin()cos2()2sin( 

Regarding the antiquity of the
trigonometry contained in the Surya
Siddhanta

Pingala, Meru Prastara and the
Binomial theorem
“Math for poets and drummers”, R. W. Hall, Dept. of Mathematics and Computer Science, St. Joseph’s
University, Philadelphia
“Binomial theorem in ancient India” A. K. Bag, Indian Journal of History of Science 1966
 Pinagala, a scholar studying the mathematics of music
and rhythm, described the Meru Prastara in his treatise
Chhandah-Sutra in 200 B.C. (chhandah= meter of a poem,
e.g. Bhagavad Gita chapter 10.35).
 In the process he described the binomial theorem for integer index in
200 B.C. several centuries earlier than anywhere else in the world.
 It is also described by Halayudha’s commentary on the Chandah
sutras dating to the 10th century AD.
 Today it is known as Pascal’s triangle after the posthumous
publication of Traité du triangle arithmétique in 1665.

Other contributions by scholars of
music, language, and rhythm
• The sequence of numbers 0,1,1,2,3,5,… were first given by Virahanka (ca.
600-800 A.D.), Gopala (earlier than 1135 AD) and Acharya Hemachandra
(1150 AD). about 50 years before Fibonacci. (Today they are called Fibonacci
numbers). (“The so-called Fibonacci numbers in ancient and medieval India”,
P. Singh, Historia Mathematica 12, 229-244 (1985).)
• In computer science, the notation technique known as Backus-Naur form
was first described by Panini, (a linguist and Sanskrit grammarian from 4th
century BC born in Pushkalavati, Gandhara, (now in Pakistan)). The works of
modern day linguists and information theorists such as Leonard Bloomfield,
Zellig Harris, Axel Thue, Emil Post, Alan Turing, Noam Chomsky, and John
Backus, are based extensively on Panini’s works.
• Panini also anticipated the binary number system.
“On some rules of Panini”, Leonard Bloomfield, Journal of the American
Oriental Society, 47, 61 (1927).
“ ‘Panini-Backus’ form suggested ”, P. Z. Ingerman, Comm. of the ACM,
1967.

General Observations
 Techniques of the fundamental arithmetic operations: addition, subtraction,
multiplication, division; Extracting square and cube roots; the rules of operations with
fractions and surds;
 the Indian methods of performing long multiplications and divisions were introduced
in Europe as late as the 14th century AD…
 The rule of three, brought to Europe via the Arabs (from India) came to be known as
the Golden rule for its great popularity and utility in commercial computations…
 Modern methods of extracting square and cube roots, described by Aryabhata in the
5th century AD, were introduced in Europe only in the 16th century AD.
 The introduction of negative numbers and systematic use of symbols to denote
unknown quantities and arithmetic operations … the development of the algebraic
formalism.

And of course…
• (wrongly called Arabic numerals – should be called the Indian number
system)
Whole numbers, rational numbers, irrational numbers to any degree of
accuracy, addition, subtraction, multiplication, division, square roots, cube
roots, can express incredibly small or incredibly large numbers. (For e.g.
888 in Roman system is DCCCLXXXVIII ).
• “It is no coincidence that the mathematical and scientific renaissance
began in Europe only after the Indian notation was adopted. Indeed the
decimal notation is the very pillar of all modern civilization.”
(Amartya Kumar Datta, Resonance, April 2002.)
0,1,2,3,4,5,6,7,8,9, and ‘ . ‘
(Zero and the decimal place value system)

Some well known Indian mathematicians

Aryabhata (499 A.D.)
 His main work: Aryabhatia, written when he was 23.
 The earth is round and rotates on its axis, and the earth revolves
around the sun. Rotation of earth: 23h, 56m, 4.1s (Encyclopedia)
 Put forth the true scientific cause of eclipses (Encyclopedia)
 The moon reflects light from the sun.
 Astronomical findings were based on accurate astronomical
observations. (Encyclopaedia)
Main astronomical findings:

 “Add 4 to 100, multiply by eight, and then add 62,000. By this rule the
circumference of a circle with a diameter of 20,000 can be approached.”
 The Aryabhata algorithm, further developed later by Bhaskara in 621, is the
standard method for solving first order Diaphontine equations (ax+by=c). Also
known as Kuttaka (pulverizer) algorithm.
1416.3
20000
62832


 Modern methods for finding square roots and cube roots, these methods
were introduced in Europe only in the 16th century AD. (Amartya Kumar Datta,
Resonance, April 2002.)
irrationality!

http://en.wikipedia.org/wiki/Aryabhata
How the terms sine and cosine originated

Brahmagupta (598 - 670 A.D.)
 Important works: BrAhmasphuTasiddhAnta (revised and criticized the works of earlier
astronomers such as Aryabhatta) and KhanDa-KhAdyaka (astronomy and mathematics).
 Discovered what is known today as Brahmagupta’s lemma for solving the so-called Pell’s
equation Dx2 1 = y2, (Brahmagupta’s lemma) and solved it for D = 83 and 92. (A complete
solution for any D was later provided by Jayadeva and Bhaskaracharya II through the
chakravala method.)
 “Symmetric formula for area of a cyclic quadrilateral, appearing for the first time in the
history of mathematics” (see for e.g. ‘A modern introduction to ancient Indian mathematics’
by T. S. Bhanu Murthy, ), and expressions for the diagonals of a cyclic quadrilateral.
 “Gave a simple rule for forming a ‘Brahmagupta quadrilateral’, a cyclic quadrilateral
whose sides and diagonals are integral and whose diagonals intersect orthogonally.”
Reference: Encyclopedia


http://en.wikipedia.org/wiki/Cyclic_quadrilateral#Brahmagupta_quadrilaterals

Brahmagupta’s results are mentioned in, for e.g., “Geometry
revisited” by Coxeter and Greitzer, published by the
Mathematical Association of America (MAA):

some extracts from the book (pp. 56-59)

 An interpolation formula equivalent to the modern Newton-Stirling
interpolation formula of second order.
 Applied this method to find the sines of intermediate angles from a given
table of sines.
 The need to make complex calculations and the ability due to a superior
number system, thus brought advances in numerical techniques as well
References: Encyclopedia, TAGS

Bhaskara I (600-680 AD)
 Important works – MahAbhAskarIya and LaghubhAskarIya (provided
explanations and interpretations of Aryabhata’s reasonings).
 AryabhatIyabhAshya – a commentary on
Aryabhatia (dated 628 AD.).
 Provided a compact classification of
mathematics into different specializations
(Encyclopedia).
 Responsible for evolving trigonometry in its present form (ardhajya etc.
see encyclopedia), and created the modern trigonometric circle.
 Gave an approximation for the sine.
 Elaborated on the kuttaka method of Aryabhata.

Bhaskara II (Bhaskaracharya)
(1114 AD)

Bhaskara II (AD. 1114)
• Several treatises – Lilavati, a standard work of
Hindu mathematics, covering arithmetic,
algebra, geometry and mensuration. Many
eminent Sanskrit mathematicians wrote
commentaries (bhashyas) on it.
• Bijaganita – standard treatise on Hindu
algebra. More advanced text than the above.
• Siddhanta shiromani – standard treatise on
Hindu astronomy.

 Developed a general algorithm (the chakravala algorithm, based on
Jayadeva’s earlier work of 11th cent.) to obtain integral solutions to the so-
called Pell’s equation:
22
1 yDx  where D is also an integer.
 Can be used to fond solutions for any D. D=61 and 109 are especially
difficult, but Bhaskara used the chakravala algorithm to find the solution
in a few lines!
Solution for D=61, x = 226, 153, 980 and y = 1, 766, 319, 049
Solution for D=109, x = 15140424455100 and y = 158070671986249
In 1657 Fermat (unaware of the chakravala method) proposed the
above equation with D = 61 to Frénicle as a challenge problem.
Encyclopedia

22
in a few lines!
Encyclopedi
“What would have been Fermat’s astonishment if some missionary,
just back from India, had told him that his problem had been
successfully tackled there by native mathematicians almost six
centuries earlier?”
-André Weil, in “Number Theory, an approach through history
from Hammurapi to Legendre” (pp. 81-82)

22
in a few lines!
“The chakravala method anticipated the european methods by
more than a thousand years. But no European performances in
the whole field of algebra at a time much later than Bhaskara’s,
nay nearly up to our times, equalled the marvellous complexity
and ingenuity of chakravala.” - Selenius
“What would have been Fermat’s astonishment if some missionary,
just back from India, had told him that his problem had been
successfully tackled there by native mathematicians almost six
centuries earlier?”
-André Weil, in “Number Theory, an approach through history
from Hammurapi to Legendre” (pp. 81-82)
Encyclopedi

 Laid the seeds of differential calculus as shown, for e.g., the following
formula as well as its geometrical demonstration, for calculating the
instantaneous velocity (tatkalika gati) of planets
 dd  cos)(sin
(which is a result of differential calculus arrived at by taking limits.)
 Knew that the derivative vanishes at the points of extrema.
 Discovered what is known today as Rolle’s theorem in mathematical
analysis/calculus.
Encyclopedia
The seeds of differential calculus:

1. All the arithmetic (except decimal representation) taught in school today is described
in Lilavati - addition, subtraction, and division, multiplication, finding square roots and
cube roots for integers.
2. All the rules taught in school for manipulating fractions: addition, subtraction,
multiplication, and division of fractions, as well as finding square roots and cube roots
of fractions are also clearly defined and described.
3. The basic algebra taught in school – that of representing an unknown quantity by a
symbol and setting up an algebraic equation and solving it to find the value of that
quantity, is described in Lilavati in full detail. In fact the chapter dealing with this
seems to be straight out of a modern school text book!
4. Explicit rules on handling zero and a clear notion of the limit which forms the base of
infinitesimal calculus.
5. A section on quadratic equations containing the standard method of solving such
equations by completing the square.
6. All the results on arithmetic progressions, geometric progressions, permutations and
combinations which are taught in high school are clearly described in Lilavati.
7. Other interesting results such as the sum of the first n whole numbers, the sum of the
squares of the first n whole numbers, and the sum of the cubes of the first n whole
numbers are also presented.
From Lilavati:

A few observations
• What is discussed is just a tip of the iceberg.
• It shows a thriving and vibrant scientific culture in India,
open to criticizing and evolving and building on the works
of earlier scientists. It was open to criticize and modify
earlier works and was not stuck in dogmas - much like
modern science is (supposed to be) practiced today.
• Presents a picture opposite to that depicted in the
mainstream historical accounts as India being a backward
and stagnant civilization.
• Seriously questions the belief that the Greeks were the only
mathematicians worth mentioning – this is a view for which
there is no proof, only the repeated claims of Eurocentric
scholars repeating the Greek – Renaissance sequence.

However…
• The conventional belief is that the Greeks
invented their philosophy and mathematics by
themselves, without external influence.
• Even though India was at the time well known for
her scientific, mathematical, artistic and
philosophical knowledge,
• And even though several Greeks visited India to
acquire Indian knowledge, and there are, as a
result, several similarities between Greek and
Indian philosophies.

some points to note
• Greek knowledge did not originate in a
vacuum.
• There were several visits by Greeks to India
and similarities in philosophy.
• We now mention several instances where the
Greeks travelled to India and were influenced
by the Indian knowledge system and
incorporated it into their own.

Greek visits to India
• Darius I sent Skylax to explore the Indus in 519
BC. Later called frequent meetings between
Greeks and Indians for counsel and discussion.
(TAGS)
• Aristoxenes (350-300 BC) mentions a dialogue
between Socrates and an Indian philosopher.
(TAGS)

• Key Greek philosophers such as Plato,
Democritus, Pherecydes, and Pythagoras, are
known to have travelled to India.

“We know that he (Pythagoras) went to India to be instructed”
-Reuben Burrow, Hints concerning the observatory at Benares, 1783

-William Hamilton,
The history of medicine, surgery, and
anatomy, Vol. I (1831)

-William Hamilton,
The history of medicine, surgery, and
anatomy, Vol. I (1831)
“Journeys to India and indebtedness to
Brahminical wisdom are now ascribed to
numerous founders and leaders in Greek
thought, such as Plato, Democritus,
Pherecydes of Syrus and, quite often,
Pythagoras.”
-Wilhelm Halbfass, India and Europe: an
essay in understanding, Albany, State
University of New York Press (1988), p.
16.

• Several Greek philosophers travelled with Alexander
when he invaded India and interacted with Indian
sages. These include Onesicritius, Cynic, Democritean,
Anaxagoras, and Pyrhho.
• Pali Buddhist literature records religious and
philosophical dialogues between the Buddhist monk
Nagasena and the Indo-Greek ruler Menander.
• Gnostic philsopher Bardesanes of Edessa (ca AD 200)
travelled to India.
• The founder of the Neoplatonic school, Plotinus, went
to India in AD 242 expressly to study its philosophy.

• Several Greek philosophers travelled with Alexander
when he invaded India and interacted with Indian
sages. These include Onesicritius, Cynic, Democritean,
Anaxagoras, and Pyrhho.
• Pali Buddhist literature records religious and
philosophical dialogues between the Buddhist monk
Nagasena and the Indo-Greek ruler Menander.
• Gnostic philsopher Bardesanes of Edessa (ca AD 200)
travelled to India.
• The founder of the Neoplatonic school, Plotinus, went
to India in AD 242 expressly to study its philosophy.
O. P. Jaggi, Indian System of Medicine, Vol. 4 of History of Science and
Technology of India,
Delhi, Atma Ram and sons, 1973
Amiya Kumar Roy Chowdhury, Man, Malady, and Medicine – History of Indian
Medicine,
Calcutta, Das Gupta and Co. Ltd, 1988
TAGS

Similarities in philosophies
• Pythagoras and his followers believed in the
transmigration of the soul (reincarnation) – a
typically Indian (Hindu/Buddhist) concept. He
himself claimed having fought in the Trojan
war in a previous incarnation.
• The Pythagoreans also were strict vegetarians,
again a trait typical to Hindus/Buddhists/Jains.

• Pythagoras and his followers believed in the
transmigration of the soul (reincarnation) – a
typically Indian (Hindu/Buddhist) concept. He
himself claimed having fought in the Trojan
war in a previous incarnation.
• The Pythagoreans also were strict vegetarians,
again a trait typical to Hindus/Buddhists/Jains.
H. G. Rawlinson, “Early contacts between India and Europe”, in A Cultural
History of India, A. L. Basham (ed.) (Oxford University Press, 1975) (p. 427-428)

• The concept of karma is essential to Plato’s
philosophy.

• The concept of karma is essential in Plato’s
philosophy.
“Metempsychosis, with the complementary doctrine of karma, is the key-
stone of the philosophy of Plato. The soul is for ever travelling through a ‘cycle
of necessity’: the evil it does in one semicircle of its pilgrimage is expiated in
the other. ‘Each soul’, we are told in the Phaedrus, returning to the election of
a second life, shall receive one agreeable to his desire.’ “
“…most striking of all is the famous apologue of Er the Pamphylian, with which
Plato appropriately ends the Republic. … ‘In like manner, some of the animals
passed into men, and into one another, the unjust passing into the wild, and
the just into the tame.”
-H. G. Rawlinson, “Early contacts between India and Europe”, in A Cultural
History of India, A. L. Basham (ed.) (Oxford University Press, 1975) (p. 427-428)

• The theory that matter consists of four
elements (earth, water, air and fire) was
taught by Empedocles (490-430 BC), disciple
of Pythagoras. The later Aristotelian
description of the physical world included the
ether (space) element as well.
• Indian philosophy also describes the physical
world in terms of these five elements.

• The theory that matter consists of four
elements (earth, water, air and fire) was
taught by Empedocles (490-430 BC), disciple
of Pythagoras. The later Aristotelian
description of the physical world included the
ether (space) element as well.
• Indian philosophy also describes the physical
world in terms of these five elements.
(Bhagavad Gita 7.4 and 7.5.) Rough translation - Earth, water,
fire, air, and ether, (describes space and matter) as well as
mind, intellect, and ego (describes consciousness) , are My
apara (‘lower’) nature (which keep on changing with time),
while the para (‘higher’) nature is the unchangeable Self
(Atman), which being beyond Time, is beyond change as well.

• Many similarities between Greek medicine
and Indian medicine.
• Pythagoras is known to have travelled to India
and upon his return to have influenced the
Hippocratic system of medicine.
• The Hippocratic collection mentions an Indian
regime for cleaning the teeth, as well as listing
drugs of Indian origin, some with corrupted
Sanskrit names.

Some observations…
• In Indian civilization, science (apara vidya), unlike the
materialistic world view of today, is not opposed to the
spiritual quest (para vidya).
• And the spiritual quest (what is mistranslated as religion in
today’s context) is not opposed to the pursuit of material
science, and is not a set of dogmas to be blindly adhered
to.
• In fact both spirituality (para vidya) and science (apara
vidya) are the two sides of the same coin, (‘the coin of
wisdom’, ‘the coin of knowledge’). None of the sides are
ignored at the others’ expense.

Some observations…
• In Indian civilization, science (apara vidya), unlike the
materialistic world view of today, is not opposed to the
spiritual quest (para vidya).
• And the spiritual quest (what is mistranslated as religion in
today’s context) is not opposed to the pursuit of material
science, and is not a set of dogmas to be blindly adhered
to.
• In fact both spirituality (para vidya) and science (apara
vidya) are the two sides of the same coin, (‘the coin of
wisdom’, ‘the coin of knowledge’). None of the sides are
ignored at the others’ expense.
“…the Vedic Hindu, in his great quest of the para vidya (absolute
truth), made progress in the apara (relative truth), including the
various arts and sciences, to a considerable extent, and with a
completeness which is unparalleled in antiquity.” –Bibhutibhushan
Datta, Ancient Hindu Geometry, 1993.
“…the culture of the science of mathematics or of any other branch
of secular knowledge, was not considered to be a hindrance to
spiritual knowledge. In fact, apara vidya was then considered to be a
helpful adjunct to para vidya.”
-B. Datta and A. N. Singh, History of Hindu Mathematics, 1962.

Genius down south:
The Kerala school (1300-1600 AD)

Kerala school (1300-1600)
• Pioneered by Madhava of Sangamagrama (1340-1425)
(Today Irinjalakuda in Thrissur district).
• Continues and develops upon the findings of the
Aryabhata school.
• The mathematicians and astronomers of this school
formed a continuous line till the 17th century and made
several important contributions to calculus,
trigonometry, spherical trigonometry and astronomy.
• Most of Madhava’s original writings are lost, but his
work survives in the bhashyas (commentaries) by later
scholars of the school.

• Vatasseri Parameshvara Nambudiri (1380-
1460)
• Damodara
• Nilakantha Somayaji (1444-1544)
• Jyeshthadeva (1500-1610)
• Achyuta Pisharati (1550-1621)
• Melpathur Narayana Bhattathiri (1559-1645)
• Sankara Varma (1800-1838)

• Tantra Sangraha in 1501 and Aryabhatia-Bhashya by
Nilakantha Somayaji (Sanskrit).
• Yuktibhasha by Jyeshthadeva in 1530 (Malayalam).
Elaborates further on the Tantra Sangraha.
• Kriyakramakari (a commentary on Bhskaracharya’s
Lilavati) andYukti-Dipika (commentary on
Tantrsangraha) by Shankara Variyar (1500-1550).
• Sadratnamala by Shankaravarman in 1819 (Sanskrit).
• Karana Paddhati
• These books are commentaries on the results of
Madhava and contain several new results developing
on his work.

Charles Whish (1794-1833), (civil servant for the East India Company)
Transactions of the Royal Asiatic Society of Great Britain and Ireland, Vol. 3, No. 3, (1834)

Kerala school (1300-1600): Key
discoveries
• Irrationality of π
From the Aryabhatia of Aryabhatia
“Add 4 to 100, multiply by eight, and then add 62,000. By this
rule the circumference of a circle with a diameter of 20,000
can be approached.” (i.e. it is an approximation.)

discoveries
• Irrationality of π
From the Aryabhatia of Aryabhatia
“Add 4 to 100, multiply by eight, and then add 62,000. By this
rule the circumference of a circle with a diameter of 20,000
can be approached.” (i.e. it is an approximation.)
Nilakantha in Aryabhatia-bhashya:
“Why has an approximate value been mentioned here (in Aryabhatia) instead of
the actual value?”
And goes on to give the answer:
“Given a certain unit of measurement in terms of which the diameter specified
has no fractional part, the same measure when employed to specify the
circumference will certainly have a fractional part…even if you go on a long way
(i.e. keep on reducing the measure of the unit employed), the fractional part
will only become very small. A situation in which there will be no fractional part
is impossible, and this is what is the import of the expression Asanna (can be
aproached).”
-Development of calculus in India: contribution of kerala school (1350-1550 CE),
K. Ramasubramanian, IIT Bombay

discoveries
• Infinite geometric series
Jyeshthadeva’s proof in Yuktibhasha:




























 x
xx
x
xx
x
x
x 1
1
1
1
1
11
1
1
1
1
1 2









x
x
xxx
n
n
1
1
1
2









x
xxx
1
1
1 32
 for |x|<1
nlimitthetakeNow

discoveries
• Series expansions for sin, cos and arctan functions
along with the error after n terms
• An approximate proof for the arctan series in
modern notation follows

!7!5!3
sin
753
xxx
xx

!6!4!2
1cos
642
xxx
x

753
tan
753
1 xxx
xx

discoveries


 642
2
1
1
1
1
tan xxx
x
x
dx
d
Integrating both sides from 0 to x, we get

753
tan
753
1 xxx
xx
Derivation of series for arctan:

discoveries


 642
2
1
1
1
1
tan xxx
x
x
dx
d

753
tan
753
1 xxx
xx
Putting x = 1:

discoveries


 642
2
1
1
1
1
tan xxx
x
x
dx
d

753
tan
753
1 xxx
xx
Putting x = 1:
Putting x = 1:
7
1
5
1
3
1
1
4


discoveries


 642
2
1
1
1
1
tan xxx
x
x
dx
d

753
tan
753
1 xxx
xx
Putting x = 1:
Putting x = 1:
7
1
5
1
3
1
1
4

The so-called Gregory series, published by Gregory in
1668, but discovered by Madhava 300 years earlier!

discoveries
• Not just the series, the error after retaining n
terms was also obtained.
• The Madhava (Gregory) series converges very
slowly: the first 200 terms add up to one-
fourth of 3.1466.
• Nilakantha in Tantra Sangraha used the
knowledge of the error to obtain series with
much faster convergence.

discoveries
fourth of 3.1466.







77
1
55
1
33
1
4
1
1
4 333


discoveries
fourth of 3.1466.







77
1
55
1
33
1
4
1
1
4 333








545
4
343
4
141
4
4 555


discoveries
fourth of 3.1466.







77
1
55
1
33
1
4
1
1
4 333








545
4
343
4
141
4
4 555

and several others…

Verses from Sadratnamala

753
tan
753
1 xxx
xx

Verses from Sadratnamala
value of π is calculated to 17 decimal places and expressed by the famous kattapayadi system

discoveries
• Several discoveries in spherical trigonometry
and astronomy: an early planetary model
which was identical to the one proposed by
Tycho Brahe.
• Several of their works are still subjects of
research by modern mathematicians (much
like Srinivasan Ramanujan’s works)!

A case for the possible transmission
of the mathematics and astronomy
of the Kerala school to Europe

Historical background
• In the middle ages Europe is centuries behind India in mathematical
knowledge.
• In 1202 the Indian number system is popularized in Europe by Fibonacci.
• At the same time, Europe is engulfed in the Dark ages and abject poverty.
Hence trade and conquest with wealthy nations such as India assumes
importance.
• But trade implies navigating the seas, which needs knowledge of
astronomy, the ability to calculate latitude and longitude, which in turn
requires knowledge of trigonometry, tables of sines etc. Also, a reliable
calendar is a must.
• Neither did Europe have the knowledge, nor was its calendar reliable
enough for navigation. Thus several ships were lost accompanied by
severe economic and human losses.
• Thus navigation and calendar reform become priority programs by the
church. Lucrative prizes are offered for anyone who could provide accurate
techniques.

• At the same time, Indian mathematics had all this
information. Indian navigators used to do trade with several
countries. Thus it became important to acquire this
knowledge.
• But Hindus were ‘pagans’, ‘heathens’, and ‘idol
worshippers’ who had to be ‘civilized’ (christianized). So
though it privately sought ‘pagan’ learning, publicly it
continued to deny that there was any learning among the
‘pagans’.
• Anyone who acknowledged ‘pagan’ sources of knowledge
would be burnt at the stake for being a heretic. Thus,
although ‘pagan’ knowledge was appropriated (as shall be
seen), the sources were never acknowledged.

• At the same time, Indian mathematics had all this
information. Indian navigators used to do trade with
several countries. Thus it became important to acquire
this knowledge.
• But Hindus were ‘pagans’, ‘heathens’, and ‘idol
worshippers’ who had to be ‘civilized’ (christianized).
So though it privately sought ‘pagan’ learning, publicly
it continued to deny that there was any learning among
the ‘pagans’.
• Anyone who acknowledged ‘pagan’ sources of
knowledge would be burnt at the stake for being a
heretic.

Opportunity and means
• 1499- Vasco da Gama arrives at the Malabar coast in Kerala and
establishes a direct link to Europe via Lisbon.
• 1540 – Francis Xavier arrives in Goa and makes Kerala a hub of missionary
activities (missionary activity is still vigorous in Kerala ).
• Jesuit mathematician and astronomer Christoph Clavius includes
mathematics in the curriculum of Jesuit priests at Collegio Romano.
(Clavius later headed the calendar reform committee.)
• The first batch of Jesuit priests mathematically trained by Clavius reach
Malabar (including the city of Cochin, the epicenter of the Kerala
mathematicians) 1578 onwards. These include: Matteo Ricci, Johann
Schreck, and Antonio Rubino.
• It is clear that the express purpose is to acquire Indian knowledge on
navigation, astronomy and the calendar (panchang). They learn the local
language and are in close touch with local scholars and royal personages.
• Also, Rubino and Ricci have been recorded in correspondence as
answering requests for astronomical information from Kerala sources.

Circumstantial Evidence
• 1597 - Tycho Brahe becomes the Royal astronomer of the Holy Roman
empire upon the invitation of emperor Rudolph II to Prague.
• In this capacity, he is a natural recipient of Indian astronomy texts
obtained by Jesuit priests from Kerala.
• Is it just a coincidence that his model of planetary motion, the ‘Tychonic
model’, is identical to the one proposed by Nilakantha in his Tantra
Sangraha in 1501?
• Jyeshthadeva’s Yuktibhasha gives a formula involving a passage to infinity
to calculate the area under a parabola. The same formula was used by
Fermat, Pascal, and John Wallis.
• The chronology of the events, and the circumstantial evidence is too
strong to be a mere coincidence.

Circumstantial Evidence
• 1597 - Tycho Brahe becomes the Royal astronomer of the Holy Roman
empire upon the invitation of emperor Rudolph II to Prague.
• In this capacity, he is a natural recipient of Indian astronomy texts
obtained by Jesuit priests from Kerala.
• Is it just a coincidence that his model of planetary motion, the ‘Tychonic
model’, is identical to the one proposed by Nilakantha in his Tantra
Sangraha in 1501?
• Jyeshthadeva’s Yuktibhasha gives a formula involving a passage to infinity
to calculate the area under a parabola. The same formula was used by
Fermat, Pascal, and John Wallis.
• The chronology of the events, and the circumstantial evidence is too
strong to be a mere coincidence.
“This very strange current-day belief that only Christians, or their theologically correct
predecessors in Greece have developed almost all serious knowledge in the world
demonstrates the strength of the continuing cultural feeling against ‘pagan’ learning. There is
nothing ‘natural’ or universal in hiding what one has learnt from others: the Arabs, for
instance, did not mind learning from others, and they openly acknowledged it. This is
another feature unique to the church: the idea that learning from others is something so
shameful that, if it had to be done, the fact ought to be hidden. Therefore, though the church
sought knowledge about the calendar, specifically from India, and profusely imported
astronomical texts,…this import of knowledge remained hidden.”
-D. P. Agrawal
References:
(1) D. F. Almeida and G. G. Joseph, Eurocentrism in the history of mathematics: the case
of the Kerala school, Race and Class (2004).
(2) Cultural foundations of mathematics, C. K. Raju

Indian contributions to mathematics

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