This book is a geometrical survey of the Sanskrit and Prakrt scientific and quasi-scientific literature of India, beginning with the Vedic literature and ending with the early part of the 17th century. It deals in detail with the Sulbasutras in the Vedic literature, with the mathematical parts of Jaina Canonical works and of the Hindu Siddhantas and with the contributions to geometry made by the astronomer mathematicians Aryabhata I & II, Sripati, Bhaskara I & II, Sangamagrama Madhava, Paramesvara, Nilakantha, his disciples and a host of others. The works of the mathematicians Mahavira, Sridhara and Narayana Pandita and the Bakshali Manuscript have also been studied. The work seeks to explode the theory that the Indian mathematical genius was predominantly algebraic and computational and that it eschewed proofs and rationales. There was a school in India which delighted to demonstrate even algebraical results geometrically. In their search for a sufficiently good approximation for the value of pie Indian mathematicians had discovered the tool of integration. Which they used equally effectively for finding the surface area and volume of a sphere and in other fields. This discovery of integration was the sequel of the inextricable blending of geometry and series mathematics.
Indian Mathematics gives a unique insight into the history of mathematics within a historical global context. It builds on research into the connection between mathematics and the world-wide advancement of economics and technology. Joseph draws out parallel developments in other cultures and carefully examines the transmission of mathematical ideas across geographical and cultural borders. Accessible to those who have an interest in the global history of mathematical ideas, for the historians, philosophers and sociologists of mathematics, it is a book not to be missed.
This book traces the first faltering steps taken in the mathematical theorisation of infinity which marks the emergence of modern mathematics. It analyses the part played by Indian mathematicians through the Kerala conduit, which is an important but neglected part of the history of mathematics. 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 its 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 to the 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 sciences. It will appeal to those interested in mathematics, statistics, medieval history, history of science and technology, links between mathematics and culture and the nature of movements of ideas across cultures.
This volume is the outcome of a seminar on the history of mathematics held at the Chennai Mathematical Institute during January-February 2008 and contains articles based on the talks of distinguished scholars both from the West and from India. The topics covered include: (1) geometry in the oulvasatras; (2) the origins of zero (which can be traced to ideas of lopa in Paoini's grammar); (3) combinatorial methods in Indian music (which were developed in the context of prosody and subsequently applied to the study of tonal and rhythmic patterns in music); (4) a cross-cultural view of the development of negative numbers (from Brahmagupta (c. 628 CE) to John Wallis (1685 CE); (5) Kunnaka, Bhavana and Cakravala (the techniques developed by Indian mathematicians for the solution of indeterminate equations); (6) the development of calculus in India (covering the millennium-long history of discoveries culminating in the work of the Kerala school giving a complete analysis of the basic calculus of polynomial and trigonometrical functions); (7) recursive methods in Indian mathematics (going back to Paoini's grammar and culminating in the recursive proofs found in the Malayalam text Yuktibhaua (1530 CE)); and (8) planetary and lunar models developed by the Kerala School of Astronomy. The articles in this volume cover a substantial portion of the history of Indian mathematics and astronomy. This book will serve the dual purpose of bringing to the international community a better perspective of the mathematical heritage of India and conveying the message that much work remains to be done, namely the study of many unexplored manuscripts still available in libraries in India and abroad.
Originally, my intention was to write a "History of Algebra", in two or three volumes. In preparing the first volume I saw that in ancient civiliza tions geometry and algebra cannot well be separated: more and more sec tions on ancient geometry were added. Hence the new title of the book: "Geometry and Algebra in Ancient Civilizations". A subsequent volume on the history of modem algebra is in preparation. It will deal mainly with field theory, Galois theory and theory of groups. I want to express my deeply felt gratitude to all those who helped me in shaping this volume. In particular, I want to thank Donald Blackmore Wagner (Berkeley) who put at my disposal his English translation of the most interesting parts of the Chinese "Nine Chapters of the Art of Arith metic" and of Liu Hui's commentary to this classic, and also Jacques Se siano (Geneva), who kindly allowed me to use his translation of the re cently discovered Arabic text of four books of Diophantos not extant in Greek. Warm thanks are also due to Wyllis Bandler (Colchester, England) who read my English text very carefully and suggested several improve ments, and to Annemarie Fellmann (Frankfurt) and Erwin Neuenschwan der (Zurich) who helped me in correcting the proof sheets. Miss Fellmann also typed the manuscript and drew the figures. I also want to thank the editorial staff and production department of Springer-Verlag for their nice cooperation.
Contributed articles presented at the AMS-India Mathematics Conference, held at Bangalore in December 2003.
Tantrasangraha, composed by the renowned Kerala astronomer Nīlakantha Somayājī (c.1444-1545 AD) ranks along with Āryabhatīya of Āryabhata and Siddhāntaśiromani of Bhāskarācārya as one of the major works which significantly influenced further work on astronomy in India. One of the distinguishing features is the introduction of a major revision of the traditional Indian planetary model. Nīlakantha arrived at a unified theory of planetary latitudes and a better formulation of the equation of centre for the interior planets (Mercury and Venus) than was previously available. In preparing the translation and explanatory notes, K. Ramasubramanian and M. S. Sriram have used authentic Sanskrit editions of Tantrasangraha by Surand Kunjan Pillai and K V Sarma. All verses have been translated into English, which have been supplemented with detailed explanations including all necessary mathematical relations, illustrative examples, figures and tables using modern mathematical notation.
What is algebra? For some, it is an abstract language of x's and y’s. For mathematics majors and professional mathematicians, it is a world of axiomatically defined constructs like groups, rings, and fields. Taming the Unknown considers how these two seemingly different types of algebra evolved and how they relate. Victor Katz and Karen Parshall explore the history of algebra, from its roots in the ancient civilizations of Egypt, Mesopotamia, Greece, China, and India, through its development in the medieval Islamic world and medieval and early modern Europe, to its modern form in the early twentieth century. Defining algebra originally as a collection of techniques for determining unknowns, the authors trace the development of these techniques from geometric beginnings in ancient Egypt and Mesopotamia and classical Greece. They show how similar problems were tackled in Alexandrian Greece, in China, and in India, then look at how medieval Islamic scholars shifted to an algorithmic stage, which was further developed by medieval and early modern European mathematicians. With the introduction of a flexible and operative symbolism in the sixteenth and seventeenth centuries, algebra entered into a dynamic period characterized by the analytic geometry that could evaluate curves represented by equations in two variables, thereby solving problems in the physics of motion. This new symbolism freed mathematicians to study equations of degrees higher than two and three, ultimately leading to the present abstract era. Taming the Unknown follows algebra’s remarkable growth through different epochs around the globe.
The present volume provides a fascinating overview of geometrical ideas and perceptions from the earliest cultures to the mathematical and artistic concepts of the 20th century. It is the English translation of the 3rd edition of the well-received German book “5000 Jahre Geometrie,” in which geometry is presented as a chain of developments in cultural history and their interaction with architecture, the visual arts, philosophy, science and engineering. Geometry originated in the ancient cultures along the Indus and Nile Rivers and in Mesopotamia, experiencing its first “Golden Age” in Ancient Greece. Inspired by the Greek mathematics, a new germ of geometry blossomed in the Islamic civilizations. Through the Oriental influence on Spain, this knowledge later spread to Western Europe. Here, as part of the medieval Quadrivium, the understanding of geometry was deepened, leading to a revival during the Renaissance. Together with parallel achievements in India, China, Japan and the ancient American cultures, the European approaches formed the ideas and branches of geometry we know in the modern age: coordinate methods, analytical geometry, descriptive and projective geometry in the 17th an 18th centuries, axiom systems, geometry as a theory with multiple structures and geometry in computer sciences in the 19th and 20th centuries. Each chapter of the book starts with a table of key historical and cultural dates and ends with a summary of essential contents of geometr y in the respective era. Compelling examples invite the reader to further explore the problems of geometry in ancient and modern times. The book will appeal to mathematicians interested in Geometry and to all readers with an interest in cultural history. From letters to the authors for the German language edition I hope it gets a translation, as there is no comparable work. Prof. J. Grattan-Guinness (Middlesex University London) "Five Thousand Years of Geometry" - I think it is the most handsome book I have ever seen from Springer and the inclusion of so many color plates really improves its appearance dramatically! Prof. J.W. Dauben (City University of New York) An excellent book in every respect. The authors have successfully combined the history of geometry with the general development of culture and history. ... The graphic design is also excellent. Prof. Z. Nádenik (Czech Technical University in Prague)
This book presents a complete and accessible description of the history of early India. It starts by discussing the origins and growth of civilizations, empires, and religions. It also deals with the geographical, ecological, and linguistic backgrounds, and looks at specific cultures of the Neolithic, Chalcolithic, and Vedic periods, as well as at the Harappan civilization. In addition, the rise of Jainism and Buddhism, Magadha and the beginning of territorial states, and the period of Mauryas, Central Asian countries, Satvahanas, Guptas, and Harshavardhana are also analysed. Next, it stresses varna system, urbanization, commerce and trade, developments in science and philosophy, and cultural legacy. Finally, the process of transition from ancient to medieval India and the origin of the Aryan culture has also been examined.
In this third installment of his classic 'Foundations' trilogy, Michel Serres takes on the history of geometry and mathematics. Even more broadly, Geometry is the beginnings of things and also how these beginnings have shaped how we continue to think philosophically and critically. Serres rejects a traditional history of mathematics which unfolds in a linear manner, and argues for the need to delve into the past of maths and identify a series of ruptures which can help shed light on how this discipline has developed and how, in turn, the way we think has been shaped and formed. This meticulous and lyrical translation marks the first ever English translation of this key text in the history of ideas.
A spiritual history of India draws on more than a decade of work by a Harvard University scholar and provides coverage of its sacred places, its core tenets and the historical events of specific regions while sharing a basic introduction to Hindu religious ideas and how they have influenced modern India. Reprint.
In recent decades it has become obvious that mathematics has always been a worldwide activity. But this is the first book to provide a substantial collection of English translations of key mathematical texts from the five most important ancient and medieval non-Western mathematical cultures, and to put them into full historical and mathematical context. The Mathematics of Egypt, Mesopotamia, China, India, and Islam gives English readers a firsthand understanding and appreciation of these cultures' important contributions to world mathematics. The five section authors--Annette Imhausen (Egypt), Eleanor Robson (Mesopotamia), Joseph Dauben (China), Kim Plofker (India), and J. Lennart Berggren (Islam)--are experts in their fields. Each author has selected key texts and in many cases provided new translations. The authors have also written substantial section introductions that give an overview of each mathematical culture and explanatory notes that put each selection into context. This authoritative commentary allows readers to understand the sometimes unfamiliar mathematics of these civilizations and the purpose and significance of each text. Addressing a critical gap in the mathematics literature in English, this book is an essential resource for anyone with at least an undergraduate degree in mathematics who wants to learn about non-Western mathematical developments and how they helped shape and enrich world mathematics. The book is also an indispensable guide for mathematics teachers who want to use non-Western mathematical ideas in the classroom.
This boxed kit contains a 33-card deck and an illustrated guidebook to tarot divination.
This book presents contributions of mathematicians covering topics from ancient India, placing them in the broader context of the history of mathematics. Although the translations of some Sanskrit mathematical texts are available in the literature, Indian contributions are rarely presented in major Western historical works. Yet some of the well-known and universally-accepted discoveries from India, including the concept of zero and the decimal representation of numbers, have made lasting contributions to the foundation of modern mathematics. Through a systematic approach, this book examines these ancient mathematical ideas that were spread throughout India, China, the Islamic world, and Western Europe.
Basic Approach Developed as a comprehensive introductory work for scholars and students of ancient and early medieval Indian history, this books provides the most exhaustive overview of the subject. Dividing the vast historical expanse from the stone age to the 12th century into broad chronological units, it constructs profiles of various geographical regions of the subcontinent, weaving together and analysing an unparalleled range of literary and archaeological evidence. Dealing with prehistory and protohistory of the subcontinent in considerable detail, the narrative of the historical period breaks away from conventional text-based history writing. Providing a window into the world primary sources, it incorporates a large volume of archaeological data, along with literary, epigraphic, and numismatic evidence. Revealing the ways in which our past is constructed, it explains fundamental concepts, and illuminates contemporary debates, discoveries, and research. Situating prevailing historical debates in their contexts, Ancient and Early Medieval India presents balanced assessments, encouraging readers to independently evaluate theories, evidence, and arguments. Beautifully illustrated with over four hundred photographs, maps, and figures, Ancient and Early Medieval India helps visualize and understand the extraordinarily rich and varied remains of the ancient past of Indian subcontinent. It offers a scholarly and nuanced¿yet lucid¿account of India¿s early past, and will surely transform the discovery of this past into an exciting experience. Tabel of Contents List of photographs List of maps List of figures About the author Preface Acknowledgements A readers guide 1. Understanding Literary and Archaeological Sources 2. Hunter-Gatherers of the Palaeolithic and Mesolithic Ages 3. The Transition to Food Production: Neolithic,Neolithic¿Chalcolithic, and Chalcolithic Villages, c. 7000¿2000 bce 4. The Harappan Civilization, c. 2600¿1900 bce 5. Cultural Transitions: Images from Texts and Archaeology, c. 2000¿600 bce 6. Cities, Kings, and Renunciants: North India, c. 600¿300 bce 7. Power and Piety: The Maurya Empire, c. 324¿187 bce 8. Interaction and Innovation, c. 200 BCE¿300 ce 9. Aesthetics and Empire, c. 300¿600 ce 10. Emerging Regional Configurations, c. 600¿ 1200 ce Note on diacritics Glossary Further readings References Index Author Bio Upinder Singhis Professor in the Department of History at the University of Delhi. She taught history at St. Stephen¿s College, Delhi, from 1981 until 2004, after which she joined the faculty of the Department of History at the University of Delhi. Professor Singh¿s wide range of research interests and expertise include the analysis of ancient and early medieval inscriptions; social and economic history; religious institutions and patronage; history of archaeology; and modern history of ancient monuments. Her research papers have been published in various national and international journals. Her published books include: Kings, Brahmanas, and Temples in Orissa: An Epigraphic Study (AD 300¿1147) (1994); Ancient Delhi (1999; 2nd edn., 2006); a book for children, Mysteries of the Past: Archaeological Sites in India (2002); The Discovery of Ancient India: Early Archaeologists and the Beginnings of Archaeology (2004); and Delhi: Ancient History (edited, 2006).