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CHRIST COLLEGE, IRINJALAKUDA
A STUDY ON THE MEDIEVAL KERALA
SCHOOL OF MATHEMATICS
A Paper Submitted in Partial Fulfillment of the Requirements
for the Bachelor’s Degree in Mathematics
Department of Mathematics
By
Sumon Jose
Moderator
Ms. Seena V
Irinjalakuda
February 2013

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INTRODUCTION
It is said that Mathematics is the gate and key of the Science. According to the
famous Philosopher Immanuel Kant, "A Science is exact only in so far as it employs
Mathematics". So, all scientific education which does not commence with Mathematics is
said to be defective at its foundation. Neglect of mathematics works injury to all
knowledge.
However in the present day scenario this subject is not given its rightful place. So
much so declaring 2012 as the 'National Mathematical year' as a tribute to Mathematics
wizard Srinivasa Ramanuja, the Prime Minister of our nation, Dr. Manmohan Singh
voiced his concern over the "badly inadequate" number of competent mathematicians in
the country." It is in this context that the year 2012 was announced as the year of
Mathematics in honour of the Mathematics wizard Ramanuja. Being the year of
Mathematics we are exhorted to earnestly pursue the path marked out by the famous
mathematicians of our country. We are Heirs of a great patrimony of Aryabhatta,
Bhaskara, Brahmaguptha,Mahavira, Varahamihira, Madhava, Ramanuja etc.
One of the clear cut reasons why there are not many takers for this subject is that
we often are not aware of this great line of praiseworthy heritage that we have inherited
in this regard. We learn about Gregory series, Newton series and so on but forget those
geniuses who anticipated these western mathematicians by several centuries. So I feel
that the task I have at hand is to give a rightful place in the history to these great
predecessors of ours.
Being a student of Christ College, strongly believe that I have an added
responsibility to be a worthy follower of this great system since it is this very soil
that gave birth to Sangama Grama Madhava, the great leader of Kerala School of
Mathematics who pioneered the invention of the Power series expansions of sine, cosine
and tangent and the early forms of calculus, several centuries before Newton and Leibniz.

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Though in my own little way, I have tried to be faithful to the History by
enunciating the life and works of the great mathematicians of the Kerala School of
Mathematics of the Medieval times.
Countless thanks to our God almighty for his boundless grace and immense love
and guidance in bringing out this report successfully.
In this humble effort of mine, I place on record the help and guidance I received
from my erudite guide Ms. Seena V whose corrections and support have played a major
role in this work. I also would like to thank Prof .M K Chandran whose seminar on the
‘Literture and Mathematics’ acted as a starting point for this work. As I present this work,
I present it as a homage to those great men who opened new paths in this subject.
Sumon Jose

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CHAPTER 1
A CONTEXTUAL STUDY OF THE ORIGIN AND DEVELOPMENT
OF THE KERALA SCHOOL OF MATHEMATICS
1.1 Introduction
Studies that scientifically evaluate the factors that account for the distinctive
features of the development of an advanced school of Mathematics in Kerala would
testify to the undeniable and well noticeable effect of the cultural, historical and ethical
context that prevailed in Kerala during those times. The subject Mathematics can take its
root of development in two ways: Maths for Maths sake and Maths for the sake of other
subjects. If we could analyze the historical development of the Kerala School of
Mathematics we can notice it very well that it had its development mainly for the sake of
other branches of sciences and also to satisfy the needs of the human beings. To begin
with a study of the Kerala school of Mathematics, this chapter is trying to have a
contextual study of the origin and development of the Kerala School of Mathematics. For
that sake this chapter is further subdivided into two parts such as the Social Origins of the
Kerala School of Mathematics and the Mathematical origins of the Kerala School of
Mathematics.
1.2 The Social Origins of the Kerala School of Mathematics
The medieval period of the Kerala History is marked by the various historical
developments such as the spread of agricultural and village communities, the
development of overseas trade between various continents etc. There was also rising
competitions among the various small countries to become the ultimate and supreme
power not only by way of wars but also by being the intellectual champions of the times.

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Thus the kings of the times also tried to be hosts of intellectual giants in their courts. The
development of mathematical studies is very much influenced by all such factors.
1.2.1 Kerala: An Agrarian Society
The people of Kerala belonged to an agrarian society which depended much on
the monsoon season for its agricultural planning. In that context the prediction of rain, the
arrival of summer and also the tides were of atmost importance. In those regions of
Kerala where paddy was the main cultivation it was very much necessary to have a good
prediction of the climatic status of the state. Thus there arose a need to accurately study
the solar and lunar movements. This led to the development of attempts to accurately
prepare solar and lunar calendars. So much so it is the calendar based agriculture that
gave ascendancy to the Brahmins. Mathematics and Astrology were tools in pushing
forward such a development.
1.2.2 The Namboodithiri Culture of Kerala
The medieval namboodiri families of Kerala followed a patrimonial system that
gave the rites of the family property only to the eledest son of the family. This resulted in
the prominence and domination of the Karanavar of the family over the other members.
Often they were relieved from the responsibilities of the household life and were leading
as ‘free birds’ yielding them a lot of leisure time. This prompted the other siblings to
prove their talents in other areas such as science, music, art forms etc. And certainly one
of the main attractions of the times was the researches that were carried out in the field of
Astronomy. This slowly led to the entry of many to the field of mathematical speculation,
theoretical research and so on. Thus there was a growing number of people who
followed that path of mathematical research and promulgating that knowledge to others.
1.2.3 Temples as Cultural Centres
The temples of the medieval times were not only houses of prayer and worship;
instead temples acted at platforms for intellectuals to hold discussions, exchange
knowledge and impart the findings to others. It is certain that temples and the caste

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system of the times played a major role in maintaining a scientific temper among the
people. It acted as a media for exchange of scientific knowledge. Just like the European
monasteries that acted as houses of intellectual eruditeness, the temples of Kerala played
a major role in promoting and imparting knowledge.
1.2.4 The Medieval Educational System of Kerala
It is indeed pretty too exciting to note how knowledge was imparted at a time
when there was hardly any possibility of printing and modern means of communication.
The system of education prevailed in Kerala in those days was known as the “Gurukula”
which made the students to stay with their own teachers and to assimilate knowledge on
various subjects and life matters on a daily basis. The prominent gurus of the times were
also noted mathematicians. To further elaborate: it is interesting to note that the major
medieval mathematicians of Kerala were part of a GURUSRENI which had Madhava of
Sangamagrama as its first prominent Guru. Madhava was a prominent mathematician and
astrologer lived between 1340 and 1425. Parameswara (1360-1460) who later became the
main proponent of the findings of Madhava was his pupil. He educated Damodara (1410-
1510) yet another noted figure in the history of the Kerala School of Mathematics.
Damodara was the teacher of Jyeshtadeva (1500-1610) and Neelakanda (1443-1560).
Achyutha Pisharadi, Chithrabhaanu and Sankara Varier are also members of this
GURUSRENI.
1.2.5 Final Remarks on the Social Context of the Development of the Kerala School
of Mathematics
Between the 14th
and 17th
centuries, at a time when Mathematical and astrological
researches of the Kerala School was at its zenith, the Brahmins who did not have hectic
responsibilities of the family ties, engaged in study of the puranas, in writing poems and
slokas and a minority of them engaged in scientific- astrological-mathematical research.
It is this scientific seeking that sprouted before almost five centuries that paved a strong
foundation for the so called Kerala School of Mathematics that anticipated many of the
western parallels in the field of Mathematics by many centuries.

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1.3 Mathematical Sources of the Kerala School of Mathematics
Each science, whether it be pure sciences like Mathematics or it be applied
sciences, none of them grow independently. It is the findings of the past that stand as
stepping stones in the development of any science. The origin of the Kerala School of
Mathematics is not an exception to this principle. It too had its intellectual roots in the
ancient Indian wisdom that was spread across the country and beyond through the ancient
universities of Nalanda and Thakshasila. The findings of the early giants like Aryabhatta,
Brahmagupta, Bhaskara and so on have acted as launching pads for the medieval
mathematicians of the Kerala School.
1.3.1 The Ganita of the Aryabhatiya
Of late there have been much controversy among the historians and researchers with
regard to the native place of Aryabhatta. However there are conclusive proofs that he
spent most of his post university life (He studied at the Nalanda university) in
Kusumapura which was indeed capable of standing out with Ujjaini the most noted place
for mathematical researches in India during the early times. Thus the Medieval
mathematical enquiry in kerala have been much influenced by the ideas of Aryabhatta.
Some of the main ideas of Aryabhatta which were later on developed by the medieval
Keralese mathematicians are the following:
1. In his Aryabhatiya, (Exactly speaking in GITIKAPADA which is a pada of the
Aryabhatiaya) he speaks of a table of sine series which was ineed a launching pad
for Madhava and other mathematicians of the Kerala School to further obtain
more precise values.
2. The geometric progressions which were enunciated by Aryabhata in his works
induced especially in the medieval keralite mathematicians a taste to study about
progressions, series and sequences.
3. Aryabhata approximated the value of pi by five decimal points and building on
that Madhava of Sangamagrama approximated the value of pi correct to 13
decimals.

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4. Aryabhata discussed the notions of sine and cosine whereas Madhava and his
disciples worked out for the values of sine and cosine.
5. Aryabhata provided elegant results for the summation of series of square and
cubes:
and . And of course for
sure these equations were basic to the development of the sine series and the arc
tan series by Madhava.
1.3.2 Influence of the Works of Bhaskaracharya on the Kerala mathematicians
Bhaskaracharya was an Indian Mathematician of the Medieval Period who had been a
pioneer in many respects in the field of Mathematics. Some of his influences are the
following:
1. He was the first one to name the numbers such as eka(1), dasha(10), shata(100),
sahastra(1000), ayuta(10,000), laksha(100,000), prayuta (1,000,000=million), etc
which was later followed by all Indian Mathematicians.
2. He gave a proof for the Pythagorean theorem. Thus the method of mathematical
proofs came to Indian context which were then followed by the later
mathematicians. Thus we can find a mathematical approach being developed in
the post Bhaskaraic times in India.
3. He was a pioneer in introducing the preliminary concepts of infinitesimal calculus
and gave very notable contributions to the field of integral calculus. This was
further taken up especially by the mathematicians of Kerala who in effect
anticipated many of their western parallels by centuries following the path set out
by Bhaskara.
4. Several commentaries and also the original text of Lilavati written by
Bhaskaracharya have been escavated from various parts of Kerala and most of
those documents are dating back to the medieval times, which gives ample proofs

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to conclude that the works of Bhaskara have influenced the Medieval
mathematicians of the Kerala school of Mathematics founded by Madhava.
1.3.3 Elements from Narayana Pandit
He is a very prominent mathematician of the Indian tradition who was acclaimed
by many as one of the major mathematicians of the Indian sub continent. World famour
Mathematical Historian and researcher Plofker writes that his texts were the most
significant Sanskrit Mathematics treatises after those of Bhaskara II. Narayana Pandit had
written two works, an arithmetical treatise called Ganita Kaumudi and an algebraic
treatise called Bijganita Vatamsa. Narayana is also thought to be the author of an
elaborate commentary of Bhaskara II's Lilavati, titled Karmapradipika (or Karma-
Paddhati). So much so many of the translations of Lilavati found in various parts of
Kerala were written by Narayana Pandit. Thus we can trace a clear link between the
works of Narayana Pandit and that of the Medieval Kerala Mathematicians.
1.4 Conclusion
Clearly, no scientific advancement is an isolated one. It is the end product of the
social backgrounds, scientific inventions of the predecessors and the needs of the times.
The medieval Kerala School of Mathematics too is derived from the needs of the times
such as agricultural needs, intellectual curiosity, leisure time pursuits etc. More than all
that the medieval mathematicians of the kerala school were pioneers in many modern
fields of Mathematics centuries before the western claimers of those discoveries. Yet,
few modern compendiums on the history of Mathematics have paid adequate attention to
the often pioneering and revolutionary contributions of Indian mathematicians. However
it is crystal clear that a significant body of mathematical works were produced in the
Kerala by Sangama Grama Madhava and his disciples. The science of Mathematics
played a pivotal role in the life of the people from then onwards. No other branch of
science is complete without Mathematics. So much so we can find the influence of
Mathematics not only in scientific research but also in music, poetry, architecture etc.
The complicated and beautiful architectural works testify to a clear and well founded
knowledge of Mathematics from the part of our pioneers. If we closely look at the poetry

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of the olden times, we can duly say with sure footing that most of the poets were more of
mathematicians than poets. The concept of Vrittam, which was of vital role in the
Malayalam and Sanskrit works are very much mathematical. The Karnatik and
Hindustani music developed in our sub continent demonstrates many mathematical
elements. Thus we can conclude that Mathematics was very much close to the daily life
situations of the people of earlier Kerala history. The poetical trick of Kadapayadi found
in many manuscripts testify to the inquisitive mind and the intellectual advancements of
the people of that time. Thus we can remark that for the people of Kerala of the Medieval
times, Mathematics flowed from their everyday life.

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CHAPTER 2
PROMINENT MATHEMATICIANS OF THE KERALA SCHOOL OF
MATHEMATICS
2.1 Introduction
Free from the political, social and economic upheavals that engulfed the rest of
the Indian subcontinent, Kerala had a generally peaceful existence. Thus there flourished
a pursuit for knowledge especially during the medieval times. The period of the history
between the fourteenth century to the 16th
could be acclaimed as the Golden Era in the
history of Mathematics in Kerala. The Medieval school of Mathematics in Kerala was
founded by Sangama Grama Madhava and extends roughly upto the time of Sankara
Varier and Chithrabhaanu who lived in the 16th
century. This school of thought made
noteworthy contributions to the various fields and pioneered several braches of
Mathematics. A thorough knowledge of the life, works and contributions of these
prominent figures in this field is mandatory in order to rightly understand the Kerala
School of Mathematics of the medieval times.
2.2 Sangama Grama Madhava
Sangama Grama Madhava (1340-1425), renowned as the founder of the Kerala
School of Mathematics and Astronomy is believed to be from the town of
Sangamagrama, of present day Irinjalakuda. He opened the path to the infinite series
approximations of trigonometric series. His discoveries were very decisive in the
formation of the branch of Calculus. It is his works as well as that of his followers that
gave a firm foundation to the program of Mathematics in Kerala. Therefore it is indeed
of vital importance that we study the works and contributions of Madhava in detail in
order to have a glimpse of the Kerala School of Mathematics.

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2.2.1 Infinite Series Approximations by Madhava
He discovered infinite series for the trigonometric functions of sine, cosine, tangent and
arctangent. The famous work of Jyeshtadeva called Yuktibhasa sheds light on the
derivation and proof of the infinite series approximation for inverse tangent found out by
Madhava. Jyeshtadeva describes it as follows.
“The first term is the product of the given sine and radius of the desired arc
divided by the cosine of the arc. The succeeding terms are obtained by a process
of iteration when the first term is repeatedly multiplied by the square of the sine
and divided by the square of the cosine. All the terms are then divided by the odd
numbers 1, 3, 5, ....
Thus we can derive the following equation:
Or its equivalent expression:
2.2.2 Madhava’s Works in Trignometry
It is believed that the most accurate forms of sine table and cosine table of that time were
the results of the works of Madhava. He approximated those values using the following
formulas:

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Whereas until recently both these series were believed to be the sole works of Isaac
Newton (1670) and Wilhelm Leibniz (1676).
2.2.3 Madhava’s Attempts to Calculate the Value of
Madhava's work on the value of π is cited in the Mahajyānayana prakāra.
Though it cannot be taken for a reliable source because there is controversy among the
scholars regarding whether this is work was written by Madhava himself, it gives ample
proofs for us to conclude that Madhava anticipated the Gottfried Leibniz series by
centuries. Despite the fact that no surviving works of Madhava contains conclusive
proofs that he found out this series, we can find unambiguous proofs for the same from
the works of his followers life Nilakanda Somayaji, Jeyshtadeva etc who attribute the
series to Madhava in their works. So much so later on, this series was renamed as the
Madhava Gregory Leibniz Series. The series is given below.
He also gave a more rapidly converging series by transforming the original infinite series
of , obtaining the infinite series
2.2.4 Algebra
Madhava carried out researches in other braches of Mathematics also. He found
methods of polynomial expansion and also discovered the solutions of transcendental
equations by the method of iteration.
2.2.5 Madhava and Calculus
Calculus is the study of ‘Rate of Change’. It is branch of Mathematics that has
applications in many other sciences and until recently it was believed that Calculus was
invented by Sir Issac Newton and Wilhem Gottfried Leibniz independently in two
different parts of the world. However researches of the recent times into the mysteries of

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the Kerala School of Mathematics has shown that Sangamagrama Madhava laid the
foundations for the development of the calculus, that he conceived the ideas that are basic
to the field of Calculus. In his books he speaks vividly of differentiation, term by term
integration, iterative methods for solutions of non linear equations and the theory that the
area under the curve is its integral.
2.2.6 Sangamagrama Madhava’s Works
K V Sarma (1919-2005), renowned Indian historian of science who was responsible
for bringing to light several of the achievements of the Kerala School of Mathematics has
identified the following as the works authored by Sangamagrama Madhava.
1. Golavada
2. Madhyamanayanaprakara
3. Mahajyanayanaprakara
4. Lagnaprakarana
5. Venvaroha
6. Sphutacandrapti
7. Aganita-grahacara
8. Candravakyani
2.3 Vatasseri Parameswara
Vatasseri Parameswara who is believed to have lived between c.1380CE and
c.1460CE was a disciple of Madhava of Sangamagrama. He was a prominent figure in
the field of observational Astronomy. As he was an astrologer he realized the need for
better mathematical tools to correct the astronomical parameters which were followed
traditionally. It is this need that brought out the mathematician in him. As he was a
prolific writer, there are almost 25 manuscripts identified as the works of Parameswara.

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2.3.1 Contribtutions of Parameswara
The most noted contribution of Parameswara is his Mean value type formula for
the inverse interpolation of the sine and he is believed to have been the first
mathematician to give the radius of circle with inscribed cyclic quadrilateral, an
expression that is normally attributed to Lhuilier. Parameswara concluded that a cyclic
quadrilateral with successive sides a, b, c and semi perimeter s has the circum radius
given by the equation
2.3.1 Works of Vatasseri Parameswara
 Bhatadipika - Commentary on Aryabhatiya of Aryabhata I
 Karmadipika - Commentary on Mahabhaskariya of Bhaskara I
 Paramesvari - Commentary on Laghubhaskariya of Bhaskara I
 Sidhantadipika - Commentary on Mahabhaskariyabhashya of Govindasvami
 Vivarana - Commentary on Suryasidhanta and Lilavati
 Drgganita - Description of the Drk system (composed in 1431 CE)
 Goladipika - Spherical geometry and Astronomy (composed in 1443 CE)
 Grahanamandana - Computation of eclipses (Its epoch is 15 July 1411 CE.)
 Grahanavyakhyadipika - On the rationale of the theory of eclipses
 Vakyakarana - Methods for the derivation of several astronomical tables
2.4 Vatasserri Damodara
Vatasserri Damodara Nambudiri, a famous astronomer and mathematician of the
Kerala School Of Mathematics was the son of the Vatasserri Prameswara Nambudiri.
Damodara was the teacher of Nilakanda Somayaji who initiated him into the science of
Astronomy and taught him the basic principles of mathematical computations. His name
is kept alive in the series of teachers of Mathematics from the kerala school not because
of any of his noted works or contributions but because of the fact that he had been

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instrumental in handing down the works of Madhava to later generations which led to
further developments and enquiries in the field of Mathematics.
2.5 Neelakanda Somayaji (1443-1560)
Noted for his comprehensive astronomical treatise named
TANTRASAMGRAHA, Neelakanda was one of the main proponents of the Kerala
School of Mathematics. As he had cared to record and preserve details about his own life
and times, we now have a few accurate particulars about him known to us. Referances in
his own writings propose that he was a member of the Kelallur family residing at
Trikkandiyur in modern Tirur. As he was a master of several branches of Indian
Philosophy and culture it is believed that the father of Malayalam Thunchathu
Ramanujan Ezhuthachan was a pupil of Somayaji.
2.5.1 Tantrasamgraha
It is a leading astronomical treaties written by Neelakanda Somayaji which was
completed in the year 1551 CE. It consists of 432 versus in Sanskrit divided into eight
chapters. It has inspired two commentaries namely Tantrasamgraha vakya of an unknown
author and Yuktibhasa authored by Jyeshtadeva. This book along with its commentaries
bring forth the depth and developments of the Kerala School of Mathematics as it
established several pioneering attempts in the field of Mathematics which came about in
an attempt to compute astronomical data accurately.
2.5.2 Somayajis Contributions to Mathematics
In his works he has discussed infinite series expansions of trigonometric functions
and problems of algebra. Several of his works testify to the fact that he had a clear idea of
spherical geometry. He has mentioned several trigonometric and spherical trigonometric
formulae in his writings. Several of writings especially his treatise named
Tantrasamgraha substantiates the fact that he knew elements of calculus especially
differentiation. Some of his writings mention in detail about operations of the sine
function which he used in his calculation of astronomical data.

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2.5.3 Works of Neelakanda
The following are some of his works that shed light on the astronomical and
mathematical advancements of the medieval times in Kerala.
1. Tantrasamgraha
2. Golasara : Description of basic astronomical elements and procedures
3. Sidhhantadarpana : A short work in 32 slokas enunciating the astronomical
constants with reference to the Kalpa and specifying his views on astronomical
concepts and topics.
4. Candrachayaganita : A work in 32 verses on the methods for the calculation of
time from the measurement of the shadow of the gnomon cast by the moon and
vice versa.
5. Aryabhatiya-bhashya : Elaborate commentary on Aryabhatiya.
6. Sidhhantadarpana-vyakhya : Commentary on his own Siddhantadarapana.
7. Chandrachhayaganita-vyakhya : Commentary on his own Chandrachhayaganita.
8. Sundaraja-prasnottara : Nilakantha's answers to questions posed by Sundaraja, a
Tamil Nadu based astronomer.
9. Grahanadi-grantha : Rationale of the necessity of correcting old astronomical
constants by observations.
10. Grahapariksakrama : Description of the principles and methods for verifying
astronomical computations by regular observations.
11. Jyotirmimamsa : Analysis of Astronomy
2.6 Jyeshtadeva
Jyeshtadeva was a mathematician, astronomer of the Kerala school of
Mathematics who is best known for his master piece work called Yuktibhasa.
2.6.1 Yuktibhasa
It is a commentary on the famous work by Neelakanda Somayaji. It is more than a
commentary as it gives the proof and complete rationale for the statements laid out in

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Tantrasamgraha. This was an out of the way effort for the traditional Indian Mathematics
until then. Some scholars call it as the first book on Calculus. There are various aspects
that make this book very special. First of all, unlike the earlier scholarly works that were
published in Sanskrit, this book was written in Malayalam. It is also special to note that
this book was written in prose contrary to the tradition until then. This books brought to
the Indian tradition, the idea of mathematical proof as it introduced proofs for the
thermos stated by Neelakanda.
2.6.2 Works of Jyeshtadeva
The works of Jyeshtadeva are the following:
Yuktibhāṣā
Ganita-yukti-bhasa
Drk-karana
2.7 Achyutha Pisharadi
He was a Sanskrit grammarian, scholar, astronomer and mathematician of the
Kerala School of Mathematics. He was educated by Jyeshtadeva. Though most of his
works are in the field of Astronomy, they contain several details regarding Mathematics.
He is also noted for his commentary on Venvoroha , the famous work of Sangamagrama
Madhava. Some of his famous works are the following: Praveśaka, Chāyāṣṭaka,
Uparāgaviṃśati, Rāśigolasphuṭānūti, Veṇvārohavyākhyā , and Horāsāroccaya
2.8 Sankara Variar
An astronomer, mathematician of the 16th
century, he was a pupil of Neelakanda
Somayaji. He pursued the goal of astronomical researches aided by the tools of
Mathematics. He has authored books that reveal to us the greatness of the medieval
mathematicians of the Kerala school of Mathematics. The known works of Sankara
Variar are the following:

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 Yukti-dipika - an extensive commentary in verse on Tantrasamgraha based on
Yuktibhāṣā.
 Laghu-vivrti - a short commentary in prose on Tantrasangraha.
 Kriya-kramakari - a lengthy prose commentary on Lilavati of Bhaskara II.
 An astronomical commentary dated 1529 CE.
 An astronomical handbook completed around 1554 CE.
2.9 Conclusion
Until recently there was a misconception that the branch of Mathematics made no
progress in India after Bahaskaracharya and that the later mathematicians were just
content by repeating the works of their predecessors. But of late, this misunderstanding
has been cleared. The period between 14th
and seventeenth century marks a golden era in
the history of Mathematics in Kerala. According to several scholars of the recent past,
some of the works of Madhava and his successors have been transmitted to Europe via
Jesuit missionaries and through traders who were active around the ancient port of
Muziris at that time. However, we ought to realize that we live in a land of giants whose
works are yet to be unveiled completely. There should be further pursuits in order to
better understand them and to reveal the marvelous works of the Keralite mathematicians
to the rest of the world.

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CHAPTER 3
MAJOR QUESTIONS OF THE KERALA SCHOOL OF
MATHEMATICS
3.1 Introduction
Wikipedia defines a school of thought as “a collection or group of people who
share common characteristics of opinion or outlook of a philosophy, discipline, belief,
social movement, cultural movement, or art movement.” In the same way the Kerala
school of Mathematics and Astronomy has got its own major questions. Those are
questions that arose as a process and they depend mainly on the thrust of that school. In
Kerala the mathematical enquiry flourished as a result of man’s curiosity to know about
the planets, other heavenly bodies and the influence of the planetary movements on
man’s life. To sum up we can classify the major questions of the Kerala School of
Mathematics into the following kinds.
3.2 Astronomical Research
The planets and the movement of the planets were always a matter of curiosity
and enquiry for the people of Kerala. They tried to research on the measurements that
governed this process and were interested in collecting materials and details that would
supplement their research. And it is in attempting to solve astronomical problems that the
Kerala School independently created a number of important mathematical concepts.
Some of the major works of the Kerala School of Astronomy and Mathematics are
treaties regarding astronomical research and gradually they lead us to mathematical
enquiry. Some of the major texts that are noteworthy in this regard are the following:
1. Grahapareeksakrama is a manual on making observations in Astronomy based on
instruments of time.

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2. In Tantrasangraha revised model of Aryabhata’s model for the planets mercury
and venus are given. The equation of the centre for these planets remained most
accurate until the time of Johannes Kepler in the 17th
century.
3. Golasara: It is a brief description of basic astronomical elements and procedures.
4. Siddantadarpana: It is a short work in 32 slokas enunciating the astronomical
constants with reference to the kalpa and specify his views on astronomical
concepts and topics.
5. Chandrachayaganita: It describes the methods for the calculation of time from the
measurements of the shadow of the gnomon cast by the moon and vice versa.
6. Aryabhatiya bhasya: It is an elaborate commentary on the Aryabhatiya.
7. Sidhantadarpanavakya: It is a commentary on the siddantadarpana.
8. Chandrachayaganita Vyakhya: It is a commentary on Chandrachayaganita.
9. Sundaraja Prasnottara: It is the collection of answers from Nilakanta to Sundaraja,
an astronomer from Tamilnadu.
10. Grahanadi Granta: It involves the rational of the necessity of correcting old
astronomical constants by observations.
These are only a few examples of the major astronomical works from the Kerala
school of Mathematics. In fact there are many such works. A brief reading of these works
would testify to the fact that it is the need for supplementing astronomical data with
mathematical parameters that resulted in the progress of mathematical research in India.
One of the major barrier was to accurately calculate the circumference of the earth and
other planets. That is how the question of calculating the accurate value of pi came to the
scene of the Kerala School of Mathematics.
3.3 The Value of Pi (π)
To calculate the value of π had been one of the major quests of the Kerala School
of Mathematics. The works on the value of pi π cited in the Mahajyanayanaprakara
(Method for the great sines) is an ample testimony to the early efforts of the Kerala
Mathematicians to approximate the values of pi. The infinite series expansion of π

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presently known as the Madhava Leibniz series is a tangible example of the giant leap
that was taken by the Kerala mathematicians.
Using the notation for summation we can express the same as follows:
But this equation has an inbuilt error within it. However what is most exciting is the fact
that he also gave correction terms (Rn) for this approximation, that too in three forms. The
following are the correction terms obtained by him:
where the third correction leads to a highly accurate computations of the value of pi. The
most important fact is that they come as the first three convergents of a continued fraction
which can itself be derived from the standard Indian approximation to pi namely
62832/20000. As it is already mentioned while discussing Madhava the school gave yet
another infinite series for pi such as:

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By using the first 21 terms of this sequence they approximated the value of correct to
11 decimal points. It was calculated as 3.14159265359. The other method they used was
to add a remainder term to the original series of pi. They used the remainder term
in the infinite series expansion of to improve the approximation of pi to 13
decimal places of accuracy when n=75.
3.4 Infinite Series Expansions
The Kerala School has made a no of contributions to the field of infinite series. These
include the following:
This formula was already known from the works of the 10th
century mathematicians. The
mathematicians of the Kerala School of Mathematics used this result to obtain a proof of
the result;
for large values of n.
They applied ideas from differential and integral calculus to obtain infinite series (Taylor
and Maclaurin) for sine, cosine and arctangent functions.
The Tantrasamgraha Vakya gives the mathematical notation for the same:
or equivalently:
They made use of the series expansion of the arctangent function to obtain the infinite
series expansion of π. Furthermore Neelakanta’s demonstration of particular cases of the
series

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The following are the main series that are attributed to the Kerala school of
Mathematics and Astronomy.
1. A particular case of the Euler series was developed by the Kerala School of
Mathematics.
2. The following is an expression for the value of pi which was proposed by the
Kerala School.
3. The following series is equivalent to the Gregory series which was later on named as the
Madhava Gregory series.
4.
5.
6.
7.
The above series are also collectively known as the Madhava Taylor Series.
3.5 Conclusion
In short there flourished in the Kerala School of Mathematics and Astronomy, a
serious approach towards theoretical Mathematics during the medieval period. So much
so several pioneering contributions made by the mathematicians of the Kerala school
were brought to light at a later point of time only. One of the areas where pioneering
works had been initiated by the Kerala School of Mathematics and Astronomy was the
branch of Calculus. According to the standard story, Calculus was introduced by Leibniz
and Newton independently in two different parts of the world. However while there are
disputes with regard to who could be originators of the same, the branch of Calculus was

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anticipated centuries before them by the mathematicians of the Kerala School of
Mathematics. Several infinite series expansion that were named after European
mathematicians are nowadays being renamed with the Kerala mathematicians in the near
past. As already suggested by many of the science historians, there is a possibility that
this piece of knowledge was transmitted from Kerala to Europe through the
instrumentality of the traders who frequented Kerala during the time of Madhava and
other mathematicians of the Kerala School and the Jesuit missionaries. Though there are
no conclusive proofs for this, the ultimate message remains the same: we belong to the
land of erudite mathematicians.

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CONCLUSION
Doing this research work was an eye opener for me personally as it invited my
attention to this so far neglected area of the history of Mathematics. It was a new
experience in my whole learning of Mathematics.
To give in a nutshell, we can sum up this whole work in the following words.
Kerala has had a continuous tradition of astronomy and mathematics from much earlier times.
The school flourished between the 14th and 16th centuries and the original discoveries of the
school seems to have ended with Narayana Bhattathiri (1559-1632). In attempting to solve
astronomical problems, the Kerala school independently created a number of important
mathematics concepts. During the medieval times from the time of Madhava of Sangama
Grama there flourished in Kerala a pursuit of knowledge that gave rise to deep scientific
thinking, astronomical research and Mathematical speculation. Madhava being the
pioneer of this movement is known as the founder of this school of thought. His student
and follower Vatasseri Parameswara paved a new path in the development of the Kerala School
of Mathematics and Astronomy as brought about some corrections in the traditional methods of
Mathematics. The next great name is of Nilakanta Somayaji whose works are of great importance
in astronomical and mathematical enquiry. By his lasting contributions, Jyeshtadeva also deserves
a mention. All the others mentioned in this essay were proponents and were instrumental in
handing down the knowledge to the later generations.
However the undeniable fact is that there flourished once in this very land,
advanced thinking and theoretical knowledge only a few of which has been handed down. Many
of such writings were lost in time or some of them even not recorded. Therefore more effort and
research should go into this area. The areas of Indian music, architecture, art etc exhibit a very
clear knowledge of mathematical principles though not in explicit terms.
Finally I hope this area would invite more researchers and students who in the
future would write newer chapters to this work.

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