Contents: Preface. Introduction. 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 π Series. 12. The Sine and Cosine Series. 13. The π 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. Bibliography. Index. In the millennia-long story of the mathematical tradition of India, certain exceptionally fruitful periods stand out. This book identifies three of them: 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. Its main concern 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 story, 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 account of these nascent insights. As part of an endeavour to promote the new awareness, 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.