This is an extraordinary book. It is full of interesting and important information on Indian mathematics and its development. The interconnections of mathematics, science, culture, language, and religion are described with an expert’s touch.

Most standard history of mathematics textbooks provide incomplete pictures of the mathematics developed in India. Normally, they include a collection of achievements of a few Indian mathematicians that are then compared or contrasted with the mathematics of other cultures. Typically, they do not delve into the broader cultural history to link the various episodes. In this book readers will find a well researched account of Indian mathematics that is represented as a coherent and largely continuous intellectual tradition.

The possible cross-fertilization of mathematical concepts between the Indian and other civilization is presented in some detail. When the chief arguments for a particular viewpoint are presented, supporting evidence is included. In addition, the author carefully notes when such evidence is lacking and when it contradicts other hypotheses. The book is essential reading for anyone teaching a history of mathematics course or interested in the subject.

The problems facing anyone writing such a book are formidable: the astronomical and mathematical manuscripts are in verse, chronicles of historical events are rare in Sanskrit literature, and there is a lack of biographical detail of authors. One must deal with the existence of myriads of undocumented Sanskrit manuscripts and with the religious factor. Many texts were ascribed to the revelation of gods or to legendary sages, which attributions expunged the historical context of the work to stress the divine importance of their content.

The earliest extant texts in an Indian language are ancient religious collections of religious hymns known the Vedas. In the service of sacred rite, they leave out a lot of the technical, methodological, and historical details. Substantial mathematics was developed in ancient astronomical treatises. However, there is very little reliable observational data to enable researchers to check for accuracy of the cosmological models used and which would have allowed astrochronological attempts to date the manuscripts. Sanskrit learning, where the spoken word was emphasized, did not regard mathematical knowledge as providing a unique standard of certainty. We learn that in the majority of extant treatises, unlike with the Greeks, there was no conventional structure of proof consistently invoked as essential to the validation of mathematical statements.

The mathematics discussed in Sanskrit manuscripts is significant. It includes a decimal system of number words to express quantity, the use of positive and negative numbers and zero, the basic rules of algebra, the rule of three, geometric mensuration formulas, series expansions for basic trigonometry functions predating Leibniz and Newton, the use of the pulverizer for solving for finding integral solutions of simultaneous linear equations. Besides the cultural perspective, the mathematical highlights of the book include: Aryabhata’s sixth century treatise on astronomy, the undated Bakhshali manuscript, Bhaskara’s *Lilavati*, an eleventh-century account of science and mathematics in India described by the Arabic astronomer, al-Biruni, and the remarkable accomplishments of the Kerala School of Madhava that existed from the late fourteenth to early seventeenth century.

The discussion of the contents of early astronomical treatises includes a helpful glossary of astronomical terms. The book includes an explanation of the existing caste system, an appendix of the basic features of Sanskrit language and literature, a glossary of terms, biographical data on Indian mathematicians, and an invaluable bibliography.

Despite the difficulties inherent in the project, the author does a remarkable job presenting the mathematics of India. Anyone delving into this book, general reader or historian, will find straightforward explanations of the mathematics involved, learn of the culture that surrounded the subject, and come away with a clearer understanding of the Indian civilization and its mathematics.

Jim Tattersall is Professor of Mathematics at Providence College, in Providence, RI.