The mathematics of India : concepts, methods, connections

-- Chapter I Beginnings 1. Background: Culture and Language 2. Vedic Geometry 3. Antecedents? Mathematics in the Indus Valley 4. Decimal Numbers 5. Numbers in the Vedic Literature II The Aryabhatan Revolution 6. From 500 BCE to 500 CE 7. The Mathematics of the Ganitapada 8. From Brahmagupta to Bhaskara II to Narayana III Madhava and the Invention of Calculus 9. The Nila Phenomenon 10. Nila Mathematics - General Survey 11. The pi - series 12. The Sine and Cosine Series 13. The pi-Series Revisited: Algebra in Analysis IV Connections 14. What is Indian about the Mathematics of India? 15. What is Indian ...? The Question of Proofs 16. Upasamhara

This book identifies three of the exceptionally fruitful periods of the millennia-long history of the mathematical tradition of India: the very beginning of that tradition in the construction of the now-universal system of decimal numeration and of a framework for planar geometry; a classical period inaugurated by Aryabhata's invention of trigonometry and his enunciation of the principles of discrete calculus as applied to trigonometric functions; and a final phase that produced, in the work of Madhava, a rigorous infinitesimal calculus of such functions. The main highlight of this book is a detailed examination of these critical phases and their interconnectedness, primarily in mathematical terms but also in relation to their intellectual, cultural and historical contexts. Recent decades have seen a renewal of interest in this history, as manifested in the publication of an increasing number of critical editions and translations of texts, as well as in an informed analytic interpretation of their content by the scholarly community. The result has been the emergence of a more accurate and balanced view of the subject, and the book has attempted to take an account of these nascent insights. As part of an endeavour to promote the new awareness, a special attention has been given to the presentation of proofs of all significant propositions in modern terminology and notation, either directly transcribed from the original texts or by collecting together material from several texts.