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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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