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Books > History > Medieval > A Passage To Infinity (Medieval Indian Mathematics from Kerala and its Impact)

A Passage To Infinity (Medieval Indian Mathematics from Kerala and its Impact)

Description

This book traces the first faltering steps taken in the mathematical theorization of infinity which marks the emergence of modern mathematics. It analyses the part played by Indian mathematician s through the Kerala conduit, which is an important but neglected part of the history of mathematics.

A Passage to Infinity: Medieval Indian Mathematics from Kerala and Its Impact begins with an examination of the social origins of the Kerala School and proceeds to discuss it mathematical genesis as well as its achievements. It presents the techniques employed by the school to derive the series expansions for sine, cosine, arctan, and so on. By using modern notation but remaining close other methods in the original sources; it enables the reader with some knowledge of trigonometry and elementary algebra to follow the derivations. While delving into the nature of the socio-economic processes that led to the development of scientific knowledge in pre-modern India, the book also probes the validity or otherwise of the conjecture of the transmission of Kerala mathematics to Europe through the Jesuit channel.

The book straddles two domains: science and social science. It will appeal to those interested in mathematics, statics, medieval history of science and technology, links between mathematics and culture and the nature of movement of ideas across cultures.

George Gheverghese Joseph holds joint appointments at the University of Manchester, the United Kingdom and at the University of Toronto, Canada.

George Gheverghese Joseph was born in Kerala, India. His family moved to Mombasa in Kenya where he did his schooling. He studied at the University of Leicester, the United Kingdom, and then worked for six years in Kenya before pursuing his postgraduate studies at Manchester, the United Kingdom. He has travelled widely, holding university appointments and giving lectures at various universities around the world. He has appeared on radio and television programmes in India, the United States, Australia, South Africa and New Zealand, as well as the United Kingdom. In January 2000, he helped to organise an International Seminar and Colloquium to commemorate the 1500th year of Aryabhata's famous text, Aryabhateeyam; the seminar was held in Thiruvanthapuram, Kerala. In December 2005, he organised an International Workshop at Kovalam which was the culmination of a AHRB-fijnded Research Project on 'Medieval Kerala Mathematics: The Possibility of Its Transmission to Europe'. In 2008, he gave talks at Loyola University, Chicago, USA and the Mathematical Association of America (MAA), reporting on the findings of the AHRB project. He holds joint appointments at the University of Manchester and at the University of Toronto, Canada. He authored the bestseller The Crest of the Peacock: Non-European Roots of Mathematics (1991). His other works include George Joseph: The Life and Times of a Kerala Christian Nationalist (2003), Multicultural Mathematics (1993) and Women at Work: The British Experience (1983).

**Objectives and Original Features of This Book**

The genesis of this book may be traced to an earlier book, The Crest of the Peacock: Non-European Roots of Mathematics, first published in 1991. On page 20 of that book I wrote: One of the conjectures posed in Chapter 9 (p. 249) is the possibility that mathematics from medieval India, particularly from the southern state of Kerala may have had an impact on European mathematics of the sixteenth and seventeenth centuries. (Joseph 2000) This single sentence aroused more interest and controversy than other issues raised in the book. At a meeting soon after the book came out, the author was asked whether he was in the business of dethroning Newton and Leibniz! It was also interpreted by some as suggesting that the Indians invented calculus and transmitted the knowledge to Europe. In the course of talks and presentations in conferences presentations in 1990s the author had conjectured that the conduit lor such a transmission were the Jesuits whose presence in Kerala from the middle of the sixteenth century was well attested in historical records of the period. To understand the passions unleashed, we need to stand back and look at the beginnings of modern mathematics. Two powerful tools contributed to its creation in the seventeenth century: the discovery of the general algorithms of calculus and the development and application of infinite series techniques. These two streams of discovejy seemed in the early stages to reinforce one another in their simultaneous development by extending the range and application of the other. The origin of the analysis and derivations of certain infinite series, notably those relating to the arctangent, sine and cosine, was not in Europe, but in an area in South India that now falls within the state of Kerala. From a region covering less than a thousand square kilometres north of Cochin, during the period between the fourteenth and sixteenth centuries there emerged discoveries that anticipate similar works of European mathematicians such as Wallis, James Gregory, Taylor, Newton and Leibniz by about 200 years.

There are several questions relating to the activities of this group (henceforth referred to as the Kerala School2), apart from technical ones relating to the mathematical motivation and content of their work. This book proposes to examine both the technical aspects of these mathematical discoveries as well as temporal and socio-economic context of the rise and decline of the Kerala School of Mathematics and Astronomy.

Such a study is timely given the growing interest in recent years in issues such as:

1. The comparative epistemology of Indian Mathematics as distinguished from Western (including Greek) mathematics and

2. The nature of the socio-economic processes that led to the development of scientific knowledge in pre-modern India.

The broad overlapping objectives of the earlier part of this book are: 1. to examine the mathematical genesis of the Kerala School from the earlier history of Indian mathematics in general and the Aryabhatan School in particular;

2. To explore the practices of mathematics/astronomy identifiable as those pursued by the Kerala School;

3.To outline the mathematical content of work done by individual members of the school;

4. To offer possible reasons for the neglect of the work of the Kerala School in standard accounts of the history of modern mathematics;

5. To examine the social origins of the Kerala School through a study of the structure and composition of contemporary society;

6. To explore traditional modes of generating knowledge prevalent in Kerala at the time and in particular how such knowledge was disseminated across space and time and

7. To re-examine the widely-held though mistaken view that Indian mathematics was mainly utilitarian, neglectful of proof and stagnant after the twelfth century in the light of what is now known about Kerala and other medieval Indian mathematics.

The overall objective of the later part of this book is to examine the conjecture of the transmission of Kerala mathematics to Europe, with a view to informing the wider history of mathematics. More specifically, it will examine the following questions:

1. Given the current evidence, what is the extent of the transmission of knowledge through Jesuits from Kerala to Europe during the sixteenth and seventeenth centuries? Was Kerala mathematics and astronomy part of this transmission?

2. What was the mode of diffusion of this transmitted knowledge within Europe?

As far as the second part of the book is concerned, there has been little substantive literature on the issue of transmission of mathematics from India to Europe after the twelfth century. Hence, this study will be of deep historical and cultural significance.

**Issues Raised and Some Resulting Hypotheses**

The mathematicians/astronomers of the Kerala School were predominantly Nambuthiri Brahmins with a few who came from other castes, such as the Variyar and the Pisarati, both traditionally associated with the performance of specific functions in the temple. The personnel of the temple of that time consisted broadly of three groups: scholars (bhatta), priests {santi) and functionaries (panimakkal). The scholars and priests were almost always Brahmins. Of the non-Brahmin functionaries, the Variyars looked after routine tasks of the temple, including keeping the accounts of the temple, and the Pisaratis were a subcaste of priests officiating-during the performance of rituals in their own temples. The latter sometimes acted as Sanskrit instructors as well. The two groups seems to have evolved and acquired a high social status over a long period—a period that saw the establishment of the hegemony of Brahmins over the Kerala society through their substantial landholding sanctified by custom {devasvam). The Brahmins had the resources and leisure to pursue higher learning, including the study and reinterpretation of certain mathematical and astronomical works of the earlier period from the North. This gives rise to the hypothesis that the genesis of the Kerala School is found in the older traditions prevailing in Kerala and elsewhere in India.

The corporate bodies of Brahmin landholders, organised as temple-centred oligarchies, continued to wield wider control through the creation of new institutional structures such asyogams (caste unions) and sanketams ('sacred territories' controlled by Brahmins). During the fourteenth century, which marked the start of the Kerala School, there were a number ofyogams controlled by powerful Brahmin landlords (known as Nambuthiri-Uralars). The Nambuthiris were organised along patriarchal lines following a strict primogeniture system of inheritance. In addition to the structures of political importance enhancing their social and economic powers, there was a customary practice called sambandham, a form of sexual alliance with the non-Brahmin castes, particularly women from the Nair aristocracy. The eldest son of a Nambuthiri family alone entered into a normal marriage alliance (veli) with a Nambuthiri female while his younger brothers, if there were any, could only form a sambandham relationship. This arrangement was in a way an interlocking institution of the patriachal Nambuthiri males and matrilineal Nair females. It had the effect of removing any family responsibility from the younger sons among the Nambuthiris while at the same time stabilising the system of matrilineal inheritance among the Nairs.

The system of primogeniture kept the eldest son of the Nambuthiri family busy looking after the property and community affairs while his younger brothers lived unencumbered lives with plenty of leisure time. Lacking in social status and power on a par with the eldest brother, the younger ones needed to attain social respectability through other means. Scholarship, both secular and religious, was one way available to them to make a mark. We therefore formulate the hypothesis that many of the well-known mathematicians/astronomers of the Kerala School may have emerged from among this unencumbered section of the Nambuthiris.

There exist records calledgranthavaris recounting the day-to-day accounts, from the late fifteenth century onwards, of prominent families (swarupams), Nambuthiri caste corporations (yogams and sanketams) and prominent Nair houses (taravads). These are in the form of palm leaf manuscripts now being studied by historians for reconstructing the socio-economic history of pre-colonial Kerala. They contain details about economic transactions, social relationships and cultural practices. A noticeable feature is the importance given to the practice of meticulously documenting the events and accounts of the economic transactions of the day.

The system of ownership during the period was not absolute. Even the ruler did not enjoy absolute ownership of land. The ownership of land was a conglomerate of multiple rights and privileges enjoyed hereditarily by different occupational caste groups and organised into a hierarchy. Among these various rights and privileges, the superior kind were vested with the landlords, predominantly the Nambuthiris. The landlords were not free to sell, gift or mortgage their lands to anyone they liked. They could transfer their rights only to equals and the act normally did not affect other rights held by other social groups. Each group could indulge in transactions possible with(in) their entitlements. Commonly, such transactions seem to have been confined to mortgages {panayam). The granthavaris recorded mortgage of various rights held by the concerned social groups and the deduction of interest out of their entitlements. Since the entitlements varied from a given number/measure of coconuts, areca nuts, paddy and other cereals and pulses, the interest against the cash drawn on mortgage had to be reckoned in terms of inter-commodity exchange ratios. The rights were mortgaged invariably to the landlords and the interest on the cash borrowed on mortgage was deducted out of the entitlement. The landlords thus had to maintain a systematic account of what was given to the subordinates/dependents and the deductions from their entitlements. As evidenced by the granthavaris, this accounting was done meticulously. It is obvious that the numerate were skilled in formulaic and tabular devices for deriving multiples and fractions beyond normal comprehension. This should suggest a link between the needs of the numerate with those of the pure mathematicians working at the frontiers of mathematical knowledge and hence a hypothesis about the social origins of higher mathematics in Kerala.

In the history of Kerala's agrarian expansion, the dissemination of calendrical knowledge played a vital role. The knowledge of solar calendar and the skills associated with agrarian management of seasons constituted a crucial source of Brahmin economic domination. Astronomy and mathematics were the two instruments of contemporary seasonal forecasts. The calendar was central to all socio-cultural practices of the period. This again points to the socio-economic relevance of astronomical/higher mathematical learning during the period providing a plausible hypothesis worthy of further examination.

A relevant question in the context of this book that is rarely addressed in studying the historical development of traditional sciences, is how knowledge got generated and transmitted in societies before the introduction of printing and modern schooling system. The traditional societies of Kerala were made up of specialised groups engaged in hereditary trades. It was the responsibility of the elders among each occupation group to train their younger ones in the respective trade. The acquisition of higher knowledge following the teacher-disciple (guru—sisya) mode was normally possible only for the elites. For promoting Vedic learning (i.e., learning from the scriptures) there were centres (salas), attached to the temples, almost exclusively for the Brahmins. Non-Vedic sciences known as shastras were taught by individual scholars at their residence. These sciences, codified in Sanskrit texts, were recorded in palm leaf bundles {granthas) that functioned as books. The palm leaf texts were preserved in the houses (Mams) of scholar teachers who were generally Nambuthiris. Many of the prominent Mams from the past have yielded Sanskrit manuscripts dealing with epics, grammar, philosophy, astrology and literary compositions. Few texts of higher mathematics and astronomy come down to us. This would lead to the hypothesis that the specialised texts were consulted only by a few. The larger society probably needed only the texts of calendrical calculations and astrological approximations which survived as they were periodically revised and rewritten onto fresh palm leaves by the succeeding scholars.

Approaches to History of Mathematics: Some Reflections

A history of mathematics may be approached in a number of different ways: as a chronological survey; by tracing the development of a particular theme or subject; through exploration of the life and work of individual mathematicians; or by focussing on specific mathematical communities at a particular time and place. The title of this book, A Passage to Infinity: Medieval Indian Mathematics from Kerala and Its Impact, could encompass all or some of these different approaches. But the emphasis here is on two main themes: a survey of Kerala mathematics (which is but a continuation of the history of Indian mathematics) and an examination of the conjectured transmission of certain fundamental elements of the calculus from Kerala to Europe through the Jesuits and other conduits. Now, mathematics is about ideas and their development. It is also about people and societies. Mathematics thus has a history worth knowing and telling. There are also those who feel that the history of mathematics, because it traces the genesis and development of ideas, has a role in education. Both historians and mathematicians have sets of expectations and preconceptions about the history of mathematics that may discourage them from even picking up such a book in the first place. Some may feel that a history of mathematics is little more than a chronology of names of great mathematicians, each associated with one or more theorems or axioms, and perhaps enlivened by an occasional colourful story retold from teacher to pupil down the years with about as much truth content as found in a number of fairy tales. Neither group is knowledgeable about the basic facts and methodology of each other's discipline. The problem then becomes one of persuading these two diverse audiences to pick up a book on the history of mathematics in the first place, and satisfy their expectations enough to keep them reading it. It would be even a greater bonus if one could alter those expectations sufficiently that they find themselves engaged in learning about and enjoying both mathematics and history as they read, without skipping bits concerning 'boring' background or the incomprehensible technical mathematics. For any serious author, there is one dominant standard to satisfy: getting the balance right between the facile and the impenetrable, between the historical context and the mathematical content, without patronising either the readership or compromising one's own intellectual integrity. This is no small task, for if you are reporting on the mathematics of a society from an older era recorded in an obscure language, your chance of success is even more remote. Yet, if you translate the mathematics of that society into the modern symbolic form, you are in the danger of misrepresenting or distorting the original sources and contexts and may well face the wrath of the purist and the pedant. In any case, ancient historians, however engaged they are with the intellectual histories of the cultures they study, have tended to shy away from the more numerate sources of their discipline. Mathematicians, on the other hand, may take one of three or more stances. Some feel that any pre-modern mathematics is trivial and irrelevant to modern mathematics. Indeed, to a number of research mathematicians the history they are concerned with is often three research papers deep. At the other extreme, there are those who already feel ownership of some version of that past through often-repeated legends in the standard textbooks. They may either strongly resent changes to the received wisdom, or simply assume that because it all happened so long ago that there cannot possibly be anything substantively new to tell. The difficult test remains: is it possible to convey the richness, complexity, difficulty and sheer 'otherness' to an audience that one counts not in dozens or even hundreds but perhaps into the thousands and beyond? Whether this book satisfies the test is a matter for the readers and the critics to judge.

List of Tables and Figures | viii | |

Acknowledgements | xi | |

1 | Introduction | 1 |

2 | The Social Origins of the Kerala School | 10 |

3 | The Mathematical Origins of the Kerala school | 41 |

4 | The Highlights of Kerala Mathematics and Astronomy | 67 |

5 | Indian Trigonometry: From Ancient Beginnings to Nilakantha | 82 |

6 | Squaring the Circle: The Kerala Answer | 108 |

7 | Reaching for the Stars: The Powers Series for Sines and Cosines | 142 |

8 | Changing Perspectives on Indian Mathematics | 156 |

9 | Exploring Transmissions: A Case Study of Kerla a Mathematics | 178 |

10 | A Finally Assessment | 198 |

Bibliography | 205 | |

Index | 215 | |

About the Author | 220 |

Item Code:

NAF658

Cover:

Hardcover

Edition:

2009

Publisher:

ISBN:

9788132101680

Language:

English

Size:

9.0 inch X 6.0 inch

Pages:

219

Other Details:

Weight of the book: 385 gms

This book traces the first faltering steps taken in the mathematical theorization of infinity which marks the emergence of modern mathematics. It analyses the part played by Indian mathematician s through the Kerala conduit, which is an important but neglected part of the history of mathematics.

A Passage to Infinity: Medieval Indian Mathematics from Kerala and Its Impact begins with an examination of the social origins of the Kerala School and proceeds to discuss it mathematical genesis as well as its achievements. It presents the techniques employed by the school to derive the series expansions for sine, cosine, arctan, and so on. By using modern notation but remaining close other methods in the original sources; it enables the reader with some knowledge of trigonometry and elementary algebra to follow the derivations. While delving into the nature of the socio-economic processes that led to the development of scientific knowledge in pre-modern India, the book also probes the validity or otherwise of the conjecture of the transmission of Kerala mathematics to Europe through the Jesuit channel.

The book straddles two domains: science and social science. It will appeal to those interested in mathematics, statics, medieval history of science and technology, links between mathematics and culture and the nature of movement of ideas across cultures.

George Gheverghese Joseph holds joint appointments at the University of Manchester, the United Kingdom and at the University of Toronto, Canada.

George Gheverghese Joseph was born in Kerala, India. His family moved to Mombasa in Kenya where he did his schooling. He studied at the University of Leicester, the United Kingdom, and then worked for six years in Kenya before pursuing his postgraduate studies at Manchester, the United Kingdom. He has travelled widely, holding university appointments and giving lectures at various universities around the world. He has appeared on radio and television programmes in India, the United States, Australia, South Africa and New Zealand, as well as the United Kingdom. In January 2000, he helped to organise an International Seminar and Colloquium to commemorate the 1500th year of Aryabhata's famous text, Aryabhateeyam; the seminar was held in Thiruvanthapuram, Kerala. In December 2005, he organised an International Workshop at Kovalam which was the culmination of a AHRB-fijnded Research Project on 'Medieval Kerala Mathematics: The Possibility of Its Transmission to Europe'. In 2008, he gave talks at Loyola University, Chicago, USA and the Mathematical Association of America (MAA), reporting on the findings of the AHRB project. He holds joint appointments at the University of Manchester and at the University of Toronto, Canada. He authored the bestseller The Crest of the Peacock: Non-European Roots of Mathematics (1991). His other works include George Joseph: The Life and Times of a Kerala Christian Nationalist (2003), Multicultural Mathematics (1993) and Women at Work: The British Experience (1983).

**Objectives and Original Features of This Book**

The genesis of this book may be traced to an earlier book, The Crest of the Peacock: Non-European Roots of Mathematics, first published in 1991. On page 20 of that book I wrote: One of the conjectures posed in Chapter 9 (p. 249) is the possibility that mathematics from medieval India, particularly from the southern state of Kerala may have had an impact on European mathematics of the sixteenth and seventeenth centuries. (Joseph 2000) This single sentence aroused more interest and controversy than other issues raised in the book. At a meeting soon after the book came out, the author was asked whether he was in the business of dethroning Newton and Leibniz! It was also interpreted by some as suggesting that the Indians invented calculus and transmitted the knowledge to Europe. In the course of talks and presentations in conferences presentations in 1990s the author had conjectured that the conduit lor such a transmission were the Jesuits whose presence in Kerala from the middle of the sixteenth century was well attested in historical records of the period. To understand the passions unleashed, we need to stand back and look at the beginnings of modern mathematics. Two powerful tools contributed to its creation in the seventeenth century: the discovery of the general algorithms of calculus and the development and application of infinite series techniques. These two streams of discovejy seemed in the early stages to reinforce one another in their simultaneous development by extending the range and application of the other. The origin of the analysis and derivations of certain infinite series, notably those relating to the arctangent, sine and cosine, was not in Europe, but in an area in South India that now falls within the state of Kerala. From a region covering less than a thousand square kilometres north of Cochin, during the period between the fourteenth and sixteenth centuries there emerged discoveries that anticipate similar works of European mathematicians such as Wallis, James Gregory, Taylor, Newton and Leibniz by about 200 years.

There are several questions relating to the activities of this group (henceforth referred to as the Kerala School2), apart from technical ones relating to the mathematical motivation and content of their work. This book proposes to examine both the technical aspects of these mathematical discoveries as well as temporal and socio-economic context of the rise and decline of the Kerala School of Mathematics and Astronomy.

Such a study is timely given the growing interest in recent years in issues such as:

1. The comparative epistemology of Indian Mathematics as distinguished from Western (including Greek) mathematics and

2. The nature of the socio-economic processes that led to the development of scientific knowledge in pre-modern India.

The broad overlapping objectives of the earlier part of this book are: 1. to examine the mathematical genesis of the Kerala School from the earlier history of Indian mathematics in general and the Aryabhatan School in particular;

2. To explore the practices of mathematics/astronomy identifiable as those pursued by the Kerala School;

3.To outline the mathematical content of work done by individual members of the school;

4. To offer possible reasons for the neglect of the work of the Kerala School in standard accounts of the history of modern mathematics;

5. To examine the social origins of the Kerala School through a study of the structure and composition of contemporary society;

6. To explore traditional modes of generating knowledge prevalent in Kerala at the time and in particular how such knowledge was disseminated across space and time and

7. To re-examine the widely-held though mistaken view that Indian mathematics was mainly utilitarian, neglectful of proof and stagnant after the twelfth century in the light of what is now known about Kerala and other medieval Indian mathematics.

The overall objective of the later part of this book is to examine the conjecture of the transmission of Kerala mathematics to Europe, with a view to informing the wider history of mathematics. More specifically, it will examine the following questions:

1. Given the current evidence, what is the extent of the transmission of knowledge through Jesuits from Kerala to Europe during the sixteenth and seventeenth centuries? Was Kerala mathematics and astronomy part of this transmission?

2. What was the mode of diffusion of this transmitted knowledge within Europe?

As far as the second part of the book is concerned, there has been little substantive literature on the issue of transmission of mathematics from India to Europe after the twelfth century. Hence, this study will be of deep historical and cultural significance.

**Issues Raised and Some Resulting Hypotheses**

The mathematicians/astronomers of the Kerala School were predominantly Nambuthiri Brahmins with a few who came from other castes, such as the Variyar and the Pisarati, both traditionally associated with the performance of specific functions in the temple. The personnel of the temple of that time consisted broadly of three groups: scholars (bhatta), priests {santi) and functionaries (panimakkal). The scholars and priests were almost always Brahmins. Of the non-Brahmin functionaries, the Variyars looked after routine tasks of the temple, including keeping the accounts of the temple, and the Pisaratis were a subcaste of priests officiating-during the performance of rituals in their own temples. The latter sometimes acted as Sanskrit instructors as well. The two groups seems to have evolved and acquired a high social status over a long period—a period that saw the establishment of the hegemony of Brahmins over the Kerala society through their substantial landholding sanctified by custom {devasvam). The Brahmins had the resources and leisure to pursue higher learning, including the study and reinterpretation of certain mathematical and astronomical works of the earlier period from the North. This gives rise to the hypothesis that the genesis of the Kerala School is found in the older traditions prevailing in Kerala and elsewhere in India.

The corporate bodies of Brahmin landholders, organised as temple-centred oligarchies, continued to wield wider control through the creation of new institutional structures such asyogams (caste unions) and sanketams ('sacred territories' controlled by Brahmins). During the fourteenth century, which marked the start of the Kerala School, there were a number ofyogams controlled by powerful Brahmin landlords (known as Nambuthiri-Uralars). The Nambuthiris were organised along patriarchal lines following a strict primogeniture system of inheritance. In addition to the structures of political importance enhancing their social and economic powers, there was a customary practice called sambandham, a form of sexual alliance with the non-Brahmin castes, particularly women from the Nair aristocracy. The eldest son of a Nambuthiri family alone entered into a normal marriage alliance (veli) with a Nambuthiri female while his younger brothers, if there were any, could only form a sambandham relationship. This arrangement was in a way an interlocking institution of the patriachal Nambuthiri males and matrilineal Nair females. It had the effect of removing any family responsibility from the younger sons among the Nambuthiris while at the same time stabilising the system of matrilineal inheritance among the Nairs.

The system of primogeniture kept the eldest son of the Nambuthiri family busy looking after the property and community affairs while his younger brothers lived unencumbered lives with plenty of leisure time. Lacking in social status and power on a par with the eldest brother, the younger ones needed to attain social respectability through other means. Scholarship, both secular and religious, was one way available to them to make a mark. We therefore formulate the hypothesis that many of the well-known mathematicians/astronomers of the Kerala School may have emerged from among this unencumbered section of the Nambuthiris.

There exist records calledgranthavaris recounting the day-to-day accounts, from the late fifteenth century onwards, of prominent families (swarupams), Nambuthiri caste corporations (yogams and sanketams) and prominent Nair houses (taravads). These are in the form of palm leaf manuscripts now being studied by historians for reconstructing the socio-economic history of pre-colonial Kerala. They contain details about economic transactions, social relationships and cultural practices. A noticeable feature is the importance given to the practice of meticulously documenting the events and accounts of the economic transactions of the day.

The system of ownership during the period was not absolute. Even the ruler did not enjoy absolute ownership of land. The ownership of land was a conglomerate of multiple rights and privileges enjoyed hereditarily by different occupational caste groups and organised into a hierarchy. Among these various rights and privileges, the superior kind were vested with the landlords, predominantly the Nambuthiris. The landlords were not free to sell, gift or mortgage their lands to anyone they liked. They could transfer their rights only to equals and the act normally did not affect other rights held by other social groups. Each group could indulge in transactions possible with(in) their entitlements. Commonly, such transactions seem to have been confined to mortgages {panayam). The granthavaris recorded mortgage of various rights held by the concerned social groups and the deduction of interest out of their entitlements. Since the entitlements varied from a given number/measure of coconuts, areca nuts, paddy and other cereals and pulses, the interest against the cash drawn on mortgage had to be reckoned in terms of inter-commodity exchange ratios. The rights were mortgaged invariably to the landlords and the interest on the cash borrowed on mortgage was deducted out of the entitlement. The landlords thus had to maintain a systematic account of what was given to the subordinates/dependents and the deductions from their entitlements. As evidenced by the granthavaris, this accounting was done meticulously. It is obvious that the numerate were skilled in formulaic and tabular devices for deriving multiples and fractions beyond normal comprehension. This should suggest a link between the needs of the numerate with those of the pure mathematicians working at the frontiers of mathematical knowledge and hence a hypothesis about the social origins of higher mathematics in Kerala.

In the history of Kerala's agrarian expansion, the dissemination of calendrical knowledge played a vital role. The knowledge of solar calendar and the skills associated with agrarian management of seasons constituted a crucial source of Brahmin economic domination. Astronomy and mathematics were the two instruments of contemporary seasonal forecasts. The calendar was central to all socio-cultural practices of the period. This again points to the socio-economic relevance of astronomical/higher mathematical learning during the period providing a plausible hypothesis worthy of further examination.

A relevant question in the context of this book that is rarely addressed in studying the historical development of traditional sciences, is how knowledge got generated and transmitted in societies before the introduction of printing and modern schooling system. The traditional societies of Kerala were made up of specialised groups engaged in hereditary trades. It was the responsibility of the elders among each occupation group to train their younger ones in the respective trade. The acquisition of higher knowledge following the teacher-disciple (guru—sisya) mode was normally possible only for the elites. For promoting Vedic learning (i.e., learning from the scriptures) there were centres (salas), attached to the temples, almost exclusively for the Brahmins. Non-Vedic sciences known as shastras were taught by individual scholars at their residence. These sciences, codified in Sanskrit texts, were recorded in palm leaf bundles {granthas) that functioned as books. The palm leaf texts were preserved in the houses (Mams) of scholar teachers who were generally Nambuthiris. Many of the prominent Mams from the past have yielded Sanskrit manuscripts dealing with epics, grammar, philosophy, astrology and literary compositions. Few texts of higher mathematics and astronomy come down to us. This would lead to the hypothesis that the specialised texts were consulted only by a few. The larger society probably needed only the texts of calendrical calculations and astrological approximations which survived as they were periodically revised and rewritten onto fresh palm leaves by the succeeding scholars.

Approaches to History of Mathematics: Some Reflections

A history of mathematics may be approached in a number of different ways: as a chronological survey; by tracing the development of a particular theme or subject; through exploration of the life and work of individual mathematicians; or by focussing on specific mathematical communities at a particular time and place. The title of this book, A Passage to Infinity: Medieval Indian Mathematics from Kerala and Its Impact, could encompass all or some of these different approaches. But the emphasis here is on two main themes: a survey of Kerala mathematics (which is but a continuation of the history of Indian mathematics) and an examination of the conjectured transmission of certain fundamental elements of the calculus from Kerala to Europe through the Jesuits and other conduits. Now, mathematics is about ideas and their development. It is also about people and societies. Mathematics thus has a history worth knowing and telling. There are also those who feel that the history of mathematics, because it traces the genesis and development of ideas, has a role in education. Both historians and mathematicians have sets of expectations and preconceptions about the history of mathematics that may discourage them from even picking up such a book in the first place. Some may feel that a history of mathematics is little more than a chronology of names of great mathematicians, each associated with one or more theorems or axioms, and perhaps enlivened by an occasional colourful story retold from teacher to pupil down the years with about as much truth content as found in a number of fairy tales. Neither group is knowledgeable about the basic facts and methodology of each other's discipline. The problem then becomes one of persuading these two diverse audiences to pick up a book on the history of mathematics in the first place, and satisfy their expectations enough to keep them reading it. It would be even a greater bonus if one could alter those expectations sufficiently that they find themselves engaged in learning about and enjoying both mathematics and history as they read, without skipping bits concerning 'boring' background or the incomprehensible technical mathematics. For any serious author, there is one dominant standard to satisfy: getting the balance right between the facile and the impenetrable, between the historical context and the mathematical content, without patronising either the readership or compromising one's own intellectual integrity. This is no small task, for if you are reporting on the mathematics of a society from an older era recorded in an obscure language, your chance of success is even more remote. Yet, if you translate the mathematics of that society into the modern symbolic form, you are in the danger of misrepresenting or distorting the original sources and contexts and may well face the wrath of the purist and the pedant. In any case, ancient historians, however engaged they are with the intellectual histories of the cultures they study, have tended to shy away from the more numerate sources of their discipline. Mathematicians, on the other hand, may take one of three or more stances. Some feel that any pre-modern mathematics is trivial and irrelevant to modern mathematics. Indeed, to a number of research mathematicians the history they are concerned with is often three research papers deep. At the other extreme, there are those who already feel ownership of some version of that past through often-repeated legends in the standard textbooks. They may either strongly resent changes to the received wisdom, or simply assume that because it all happened so long ago that there cannot possibly be anything substantively new to tell. The difficult test remains: is it possible to convey the richness, complexity, difficulty and sheer 'otherness' to an audience that one counts not in dozens or even hundreds but perhaps into the thousands and beyond? Whether this book satisfies the test is a matter for the readers and the critics to judge.

List of Tables and Figures | viii | |

Acknowledgements | xi | |

1 | Introduction | 1 |

2 | The Social Origins of the Kerala School | 10 |

3 | The Mathematical Origins of the Kerala school | 41 |

4 | The Highlights of Kerala Mathematics and Astronomy | 67 |

5 | Indian Trigonometry: From Ancient Beginnings to Nilakantha | 82 |

6 | Squaring the Circle: The Kerala Answer | 108 |

7 | Reaching for the Stars: The Powers Series for Sines and Cosines | 142 |

8 | Changing Perspectives on Indian Mathematics | 156 |

9 | Exploring Transmissions: A Case Study of Kerla a Mathematics | 178 |

10 | A Finally Assessment | 198 |

Bibliography | 205 | |

Index | 215 | |

About the Author | 220 |

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